Interested in mentoring a finitist?
TLDR: This is not a typical discussion post, I'm a finitist looking for a mentor (who is preferably not a finitist).
I don't believe in actual infinities, only potential infinities. I have no formal training in mathematics, only an engineering degree. Nevertheless, I've got some original, informal ideas on how a math free of actual inifinites might look and I'm hoping to talk to someone about them. I don't think I'd get much traction in starting a discussion on my ideas here, in part because it would require a time commitment to hear me out and in other part because they're not formalized ideas so they would just annoy some members. I think it'd be much better if I paid someone to hear me out and provide constructive criticism.
I tried getting tutoring from a PhD in mathematics but the philosophy of infinity wasn't his thing so that was just a waste of time. Comparatively, I think there are a lot of members of this community who are experienced with infinity and interested in exploring the more philosophical aspects of mathematics.
If you are experienced and trained in this area and would be up for helping me out through paid mentoring, please let me know.
Thanks and sorry for posting a topic which is not a typical discussion.
I don't believe in actual infinities, only potential infinities. I have no formal training in mathematics, only an engineering degree. Nevertheless, I've got some original, informal ideas on how a math free of actual inifinites might look and I'm hoping to talk to someone about them. I don't think I'd get much traction in starting a discussion on my ideas here, in part because it would require a time commitment to hear me out and in other part because they're not formalized ideas so they would just annoy some members. I think it'd be much better if I paid someone to hear me out and provide constructive criticism.
I tried getting tutoring from a PhD in mathematics but the philosophy of infinity wasn't his thing so that was just a waste of time. Comparatively, I think there are a lot of members of this community who are experienced with infinity and interested in exploring the more philosophical aspects of mathematics.
If you are experienced and trained in this area and would be up for helping me out through paid mentoring, please let me know.
Thanks and sorry for posting a topic which is not a typical discussion.
Comments (313)
And refreshing it is for these times. Not a mention of God or Jesus or climate change.
I was a professional mathematician for many years, focusing on complex analysis, teaching and writing a few papers, and still do modest research. And although I accept ideas like the set of real numbers and associated cardinalities, I have never used infinity as anything more than unboundedness. To all intents and purposes my mathematics has been infinity free.
The forum has had a number of discussions about this topic, but that's no reason for you to avoid bringing it up in a new thread or resurrecting an old thread. There are some sharp people here.
:rofl: Don't be so sure.
[quote=Wikipedia]To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications he identified the Absolute Infinite with God, and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world.[/quote]
God = [math]\infty[/math].
The OP is an atheist, mathematically speaking that is.
The tone of the OP does not suggest Cantor's theological nonsense.
That's true. Apologies if my comment offended you in any way.
The OP mentions Aristotle's distinction of actual vs. potential infinities. The Wikipedia page on the subject doesn't explain the difference between the two all that well. My own take is that Aristotle assumes that for something to be actual, that thing has to have an "end"; since [math]\infty[/math] has no end (it's synonym is endless) it can't be actual. The only alternative then is to "exist" only as pure potential.
Finitism is kinda sorta echoed by Agent Smith in The Matrix [math]\downarrow[/math]
That can't be true. Calculus is all about infinity.
Quoting jgill
If you believe that mathematics represents some sort of idealistic ultimate reality, perhaps Cantor's infinities could be seen as "theological." But then, all of math is. Math is all fun and games, that just turns out to work. People used to think that "0" is absurd. And negative numbers. And "i."
The set of real numbers is at the center of my discontent. It puts points as the foundational building block of mathematics and I find it hard to imagine that something (an n-dimensional continuum) can be constructed from nothing (0-dimensional points).
Quoting jgill
This is why my concerns are more philosophical than mathematical. I think changing our philosophy (removing points from the foundation) will have little impact on the actual mathematics that we do.
Quoting jgill
If I can't find a mentor, I might do just that.
Quoting jgill
I do feel that there is a little bit of Cantor's nonsense implied in any view that supports actual infinities. I've heard people say that the paradoxes entwined with actual infinities are beautifully mysterious...I just think they demonstrate the flaws of the concept of actual infinity.
Quoting T Clark
I would argue that calculus done right (with limits) is all about potential infinities.
All mathematics is about "potential" entities. So what we gonna do? Round pi off to 3.14? 3.14159? How many decimal places do I need to get to the real pi?
Quoting keystone
History shows that is a bad standard by which to judge a concept.
I'll stop now. I'm not a mathematician, just an engineer like you.
Why can't we just say that pi is not a number? Instead, it is an algorithm (e.g. pick your favorite infinite series for pi) used to generate a number. This algorithm is potentially infinite in that we can never complete it, but we can certainly interrupt it to generate a rational number. If you interrupt it, maybe you'll get 3.14. Actual infinity only comes into play if you claim that the algorithm can be completed, in which case it would generate a real number - a number with actually infinite digits. This is what I would like to challenge.
Quoting T Clark
Perhaps I should have written that I believe it is impossible to imagine assembling points to form a continuum. A bit of magic is needed to make the leap from a finite collection of points forming nothing to an infinite collection of points forming a continuum.
That makes the real numbers a challenging and intriguing subject.
[math]x\to \infty [/math] means x gets larger without bounds. Limits
Maybe not as challenging as you think. I think the application of real numbers would remain largely unchanged, it just requires a reinterpretation. If convention says that in between the rational points lies irrational points, I would say that in between the rational points lies continua (e.g. little lines). With this reinterpretation,
There's a simplicity and intuitive appeal to this reinterpretation. What challenge do you see?
I was speaking of currently accepted set theory, not challenges of it.
Oh sorry, I see what you're saying. I think I have a grasp of how real numbers play into accepted set theory but it is challenging for me to envision the existence of a set of all natural numbers. Without assuming its existence, accepted set theory doesn't get far off the ground.
A limitation of that conceptualisation, is that it asserts what might be considered an unnecessarily rigid ontological distinction between functions (intension) and data (extension), which is surely a matter of perspective, i.e the language one uses. Also, recall incommensurability; the length of diagonal lines in relation to a square grid have a length proportional to sqrt(2). The decimal places of sqrt(2) are only "infinite" relative to the grid coordinates.
Any computable total function N --> N can be regarded as a number, whose value is equal to the potentially infinite sequence of outputs it encodes. e.g '3' can be identified with the constant function f(n) = 3, whilst pi can be identified with the computable function whose values if executed are the potentially infinite sequence g(0) = 3, g(1) = 3.1, g(2) = 3.14 ... These numbers can be compared positionwise, with arithmetical operations defined accordingly. However, there are only a subcountable number of such functions, meaning that any set that contains some of these functions either doesn't contain all of them,or contains errors i.e partial functions that fail to halt on certain inputs, to recall the halting problem.
That said, it could be argued that the concept of exact and correct computation, whereby a computer program or function specification is translated by man or machine to a precise and correct result of execution, is an ideal platonistic notion that is incompatible with the austere epistemic and metaphysical conservatism of finitism. In which case one wants a purely extensional treatment of mathematics that doesn't appeal to any notion of computation, in which case see Brouwers intuitionism for a calculus built around choice sequences that appeal only to the existence of resources for memorising data generated by a creating subject.
Like many who are philosophically inclined, I am happy to accept actual infinities as a useful mathematical simplification an epistemic trick but not something that makes proper ontological sense.
But note also that simply switching your ontological support from parts to wholes from 0D points to 1D lines doesn't fix the deeper issues. You just set yourself up for the same puzzle at the next geometric level where we have to glue together the infinity of 1D lines that construct the 2D plane, the infinity of 2D planes that make up the 3D volume, the 4D hyperspace, and so on, presumably all the way up until we hit ?D space.
So the idea of 0D points some kind of absolute notion of discreteness is offensive to the ontic intuition. But the same should apply to its dichotomous "other", the idea of an absolute continuity as the alternative.
We need a more subtle metaphysics. We need an intuition that itself sees parts and wholes, the discrete and the continuous, as the two emergent parts of the one common rational operation.
This is simple enough. It is just the unity of opposites of ancient times, the dialectics of Hegel, the semiotics of Peirce (in particular).
This would see the discrete and the continuous as being each others limiting case. Each stands as the measure of its other. The two form an inverse or reciprocal relation. A dichotomy is that which is both mutually exclusive and jointly exhaustive. You boil everything down to two precisely opposed limitations. And they are the measure of each other in the form that the discrete = 1/continuity, and continuity is similarly = 1/discreteness.
What does this mean for number lines? It says that while we must think of the 1D whole being constructed of 0D points, that claim must be logically yoked to its "other" of each 0D point existing to the degree the 1D continuity of the line has in fact been constrained.
Constraint is the "other" of construction. Global constraint is the downward causality of holism that matches the notion of construction, which is the upwards causality of reductionism.
So the number line is composed of its 0D parts to the degree that it is also able to limit the continuity that would deny the presence of such 0D parts. And we see this in the emergence of the matching limits of the infinite and the infinitesimal.
Infinite = 1/infinitesimal, and infinitesimal = 1/infinite. Each value relies on its "other" as the source of its own crisp identity. The number line as our little dichotomised universe of possibility has to be able to express both polar extremes to express either polar extreme. The points of the line are only as unboundedly tiny as the unbounded extent of the line can ensure this to be the case.
Step back and this fits the kind of metaphysics where what is real emerges as a limitation on an Apeiron or Vagueness an unbounded potential or limitless everythingness.
The number line would start off as any and every kind of possibility. The principle of noncontradiction would fail to apply to any statement about it, and so it would be formally "vague".
But then a reciprocal operation gets going. You have a separation in which a kind of locality starts to appear within a kind of globality. You get a move towards a collection of parts that simultaneously - in equal reciprocal measure reveals the larger context that is the continuous whole.
I mean it doesn't even make sense to talk about 0D points except in the context, or in contrast, with the presence of the 1D line, right?
And the more you dig down and keep finding ever more definite points the infinitesimals the stronger and better defined becomes the global continuum that has the kind of structure in which such a set of points could clearly exist. You get the co-arising of the continuum. And indeed, the whole Cantorian kit and kaboodle if you keep adding richer structure to your story.
Why not? The more you develop the definition of "the discrete", the more that is based on the refinement of the definition of "the continuous". The two directions of determination are reciprocally yoked. And once you see that, the whole deal becomes less ontologically contentious.
Although it also becomes more ontologically contentious in that you have to shift folk from thinking that the discrete vs continuous issue is an either/or question, to realising that it is a dialectical one. Both co-arise from the common ground of a logical vagueness. Hence without both, you ain't got either.
Can you explain this to me from a computer programming perspective? In your comparison, is the data the output of the function? A function can return a function, but it can also return another object type, like a string. In the latter case, there is a type distinction between between the function and its output, but I don't see how this is unnecessarily rigid. I suspect I'm missing your point.
Quoting sime
Can you conceive of a computer that can display a line of exact length sqrt(2)? In reality something will give, whether it's the gridlines or the diagonal, some imperfection (or uncertainty) will be inserted into the system to allow everything drawn to be 'rationally'. Might the abstract world face the same constraints? If we are certain that the grid lines are perfect, why can't we just claim that we are uncertain of the actual length of the diagonal and instead label it with a potentially infinite algorithm for calculating the length (corresponding to sqrt(2))? In the end, the math is the same, but the philosophy is different since I'm not assuming that sqrt(2) is a number and I'm not assuming that unending processes can be completed.
Quoting sime
Yes, I agree to some extent. As mentioned above, if we move away from actual infinity I think we need to allow some uncertainty to creep into our systems. But what's so bad about that? Why do we need our computers to complete an endless computation of sqrt(2)? When we work with sqrt(2) we never actually work with the decimal expansion.
I'm not saying that 1D lines are the fundamental object and all other objects are constructed from them. As you say, we run into the same issue when constructing 2D surfaces from 1D curves. I am proposing that instead of constructing the whole from the parts, that we construct the parts from the whole. We start with the highest dimensional continuum of interest. Think about how we draw a graph on paper. We don't make use of pointillism. We draw a square on our paper and then cut up that square by drawing some lines, for example the line x=0 and y=0. Only when these lines are drawn does the point (0,0) emerge. The more lines we draw, the more points emerge. Might the same apply to objects in the abstract world? Might continua be fundamental instead of points?
Quoting apokrisis
But conversely, we can talk about a 1D line in the absence of points. In the example above, when I draw the first line and assign to it the function x=0 I haven't drawn any points yet. All I've done is draw a line and described how that line will interact with other lines IF they are added to my drawing. And I can keep adding more lines to my drawing to no end, adding more and more points, but never will I have infinite points. Or in other words, no matter how many times I cut up a piece of paper, never will it vanish to nothingness. Never could a continuum be decomposed into points. For this reason, I have to disagree with you when you say that continua as foundational objects are offensive to the ontic intuition.
Quoting apokrisis
The thing is that we can't go the limit. We can't complete the computation of the infinite series 0+0+0+0+... but as far as I can tell it looks like it should add to 0. I find it hard to believe that in some cases it adds to 1 and in other cases it adds to 2. But this is what we're doing when we assemble 0-length points to create lines of length 1 and 2.
Instead, to me it makes more sense to start with a continuum. Start with a string of length 10. Cut it in half, then you get strings of length 5. Cut one string in half, then you get two strings of length 2.5. Keep going and as you go strings of potentially infinite different lengths emerge. There's substance there from the start. You don't have to go the limit to have useful objects when starting with a continuum. With this parts-from-whole construction, objects are finite and processes are potentially infinite...and there are no paradoxes. This is in direct contrast to set theory where the whole is constructed from the parts and the objects of study (sets) are actually infinite.
I don't understand why you want to challenge this. I use approximations to pi all the time. When I want a quick and dirty approximation of the area of a circle inscribed in a square with sides x, I use 3/4 * x^2. I can round pi off anywhere I like depending on the precision I need. To say that irrational numbers are not really numbers doesn't make any sense to me. Of course they are.
Quoting keystone
I really don't get this. I have no problem imagining continuity arising from discreteness. I learned, saw it, got it, in 6th grade algebra. As @apokrisis wrote in a later post:
Quoting apokrisis
Holding two apparently contradictory ideas in your mind at the same time is a required skill, e.g. waves and particles. It's no big deal. I learned that, saw that, got that in 12th grade physics.
What advantage is there in seeing things your way. Expecting abstract concepts such as mathematical entities to have some sort of ontological reality doesn't make sense. Mathematicians love math for math's sake. Engineers such as me just want something that works - no ontological interpretation necessary. I assume the same is true for most scientists. How does your way work better than the way it is handled normally?
It makes ontological sense to me. I do agree that is a useful, abstract simplification. Really, all math is. All reality is.
Quoting apokrisis
I can imagine these apparently contradictory ways of seeing things could be difficult to grasp, but it's something you need to do if you want to use math. As I said, from the beginning, I could see that resolving this dichotomy is inherent in understanding all of math, at least the more practical math that engineers and scientists use.
Quoting apokrisis
Agreed.
Quoting apokrisis
Right.
Yep. So construction gets replaced by constraint. And then my point is you go the next step of seeing construction and constraint as the two halves of the one system. Which is where you arrive at a triadic metaphysics the one where a vague potential gets organised by the reciprocal deal of construction~constraint.
Quoting keystone
Which would be the "infinite dimensional" continuum ... unless you can find some larger argument that tells you what actually regulates the emergence of algebraic structure.
So when algebra is placed under the constraint of having to preserve normed division, you get the reals, complex, complex, quartonions, octonions and even the 16D sedonions. But you also get a petering out of the mathematical properties you are trying to preserve.
To me, that is a good structuralist argument. You might think any number of dimensions might still contain algebra. Or you might think that only one dimension contains "real algebra". Well actual algebra acts as a constraint that gives you something intrinsically more complex when it is run over the whole space of what seems possible.
And quantum theory even suggests complex numbers as the true centre for algebraic structure as it makes commutativity count out in the real world of particles and symmetry breaking.
So I am saying I wouldn't deal with the metaphysics of the number line in isolation. It is illustrative of the far bigger conversation we need to have about how holism in mathematical conception plays out. The same principles have to cover mathematical structure in general as category theory argues, having absorbed the metaphysics of Hegel and Peirce.
Quoting keystone
This is a rather basic level of discussion. Again, how could it even be a continua unless it could be cut? How could it even be a 0D point except as the positive absence of any dimensioned extension?
I draw attention to the fact that you want to make one thing "the fundamental". This is monism. This is reductionism. This is not holism.
Holism says everything is a system of relations. And so the first number you need to get to is two. Some pair of "fundamental things" in relation. This in turn only makes sense when you get to three "fundamental things" in relation.
You have to get to the point where the relating of things is itself dualised in some limit case fashion. You have to arrive at a metaphysical dichotomy or reciprocal relation where one of the things is your local scale of being your world-constructing collection of individual parts and the other is then your global scale of being, or the constraints, habits, laws, emergent macro-properties, downward causality, etc, that is the generalised holism of the system in question.
So if you want to apply the strength metaphysics to questions about mathematical structure, you have to count to three in terms of "fundamental things". But fortunately three is then enough. Its a theorem in network theory that all networks of relations reduce to "threeness". :grin:
Quoting keystone
But each piece also gets more pointlike. So eventually the matter becomes obscured by arriving at that even more fundamental thing of being a vagueness.
The cut has to be sandwiched between the two ends of two lines. Each end of the line is a point. At what point does the point marking the cut that is, the absence of a point at that point get marked off from the other two points marking the starts of a pair of now separated continua?
You will be familiar with these kinds of arguments. And they make no real sense because they talk about dimensionless points and dimensioned lines without any clear definition of the relation between the two. There is no operation connecting them.
But if you have an argument based on a reciprocal relation, then each only exists to the extent it is a limitation its other. And that is how you recover the intuitionist ontology of Brouwer.
An infinite amount of cutting will result in an infinite number of number line pieces. And an infinite amount of gluing will put the number line back together.
But that then means you have to be able to both cut and glue. The two actions go hand in hand. If either action runs out of steam, so does its "other".
So it is easy to picture just forever cutting a line. Or instead, just forever gluing points. Yet both are equally one-eyed perspectives in a mathematical reality where this has to be in fact a reversible operation. The two sides of the equation need to be resepcted by our ontic interpretation.
And what do we get when we roll back the reciprocal operation of cutting~gluing back to its primal origin where it is first detectable as a structure-forming relation?
Points and lines are what we see by the end in a rather black and white fashion. But what were they as some kind of distinction of this type first started to swim into view?
Quoting keystone
But the fact that we can approach the limit both limits with arbitrary closeness is how we know they are there. The limit is precisely that which isn't reachable in the end. But it certainly defines the direction we need to keep going from the start.
And this only makes sense if we are starting from the symmetry breaking of a dichotomy. The initial split is simply a reaction against the logical "other".
So each limit is defined as heading in the opposite direction of what is being left behind. Continuity is the impulse to put as much distance from discreteness as possible. Discreteness is likewise the intent of becoming as discontinuous as can be imagined.
So at the start of things the foundational conditions which is a logical vagueness the need is for a direction that leaves something behind. So there is the need for two things in fact. And both of them are trying to leave each other behind. If there are more than two things trying to leave each other thing behind then nothing really gets left behind in a maximal or extremitising way. Again, that is the definition of a dichotomy mutually exclusive and jointly exhaustive.
Thus logic can define the start of a distinction in terms of a reciprocal desire to simply separate. From there, both sides will go as far as they can go towards matching limits. But because this dividing is only real for as long as distance is being created between the two, then neither can actually become separated "in the limit" as they remain yoked together by their need for this opposition.
The point and the line, or the infinitesimal and the infinite, are actualised to the degree they are actively divided.
Quoting keystone
Again, this suffers all the usual problems of an object-oriented ontology. Reality is better understood in terms of relations processes and structures.
This may be true, but I don't think everybody qualified to have an opinion agrees with you. There are physicists who believe the universe is infinite. That doesn't really make sense to me, but a lot of things that don't make sense to me turn out to be true, so I'll remain agnostic.
[quote=Ms. Marple]Most interesting[/quote]
[math]\infty[/math] isn't and object like for example an elephant or the number 10[sup]100[/sup] or the word "elephant", it's simply a shorthand for the procedure 1. n = 0; 2. print n; 3. n = n + 1; 4. go to 2]. :chin:
I was thinking the same thing.
Once again, calculus is about LIMITS, as my mathematical genealogical ancestor, Karl Weierstrass would have explained.
That damn lemniscate and the problems it produces . . . :roll:
With circular reasoning. Perhaps a label for endless but not quite infinite in a physical sense ?
Quoting keystone
For physics, isn't that the driving force behind quanta, to put a stopper into space leaking out ?
Okay. Try this:
An object is at rest. It is not moving.
Now the object is moving at a velocity V.
How many different velocities did the object move at, to get from zero velocity to V velocity?
If your answer is not "infinite" then you don't deserve the name "mathematician". Because calculus presupposes that there are infinite velocities there.
Get off your high horse.
True, but in many a calculus problem and theorem the limit IS infinity.
You say there exists a number called pi with infinite digits and you use a truncated approximation of it when you calculate the approximate area of a circle.
I say that what exists is a (finitely defined) algorithm called pi that doesn't halt but you can prematurely terminate it to produce a rational number to calculate the approximate area of a circle.
The difference is that you are asserting the existence of an infinite object, something beyond our comprehension. My approach seems more in line with what us engineers actually do, so why bother asserting the existence of something impossible to imagine if you don't even need to?
Quoting T Clark
Do you believe that 0+0+0+0+... can equal anything other than 0? If not, then how can you claim that 0-length points could be combined to form a line having length?
Quoting T Clark
Sounds like double-think from 1984. There are no contradictions in wave-particle duality.
Quoting T Clark
With my view many paradoxes (Zeno, Dartboard, Liar's, etc) are easily resolved. While this should be enough, it also aligns far better with what us finite beings actually do. Also, while I haven't gotten into it here, it makes quantum physics less weird. Ultimately, I'm talking about the philosophy of mathematics, not the application of it. The day to day mathematics of engineers don't change with this new foundation.
Quoting T Clark
Yes, all reality is void of actual infinities. So why do we need to believe that reality is just an approximation of some ideal infinity-laden object that we can't comprehend or observe? Why can't we stop at reality?
Quoting T Clark
They think it's possible only because modern math welcomes actual infinity. If mathematicians rejected actual infinity then I'm sure physicists would be less inclined to accept it.
I'm basically warning against logicism, the ideology that there is a single correct logical definition of a mathematical object. Thinking in this way leads to unnecessary rejection of infinite mathematical objects, for such objects aren't necessarily infinite in a different basis of description. e.g the length of a diagonal line doesn't have infinite decimals relative to a basis aligned with the diagonal.
Also, the algorithm for approximating sqrt(2) to any desired level of accuracy can itself be used to denote sqrt(2) without being executed.
I take it you must mean dis-solved?
You might be interested in Norman Wildberger on YouTube. He seems to hold positions on infinity similar to yours.
I wasn't endorsing the infinite universe argument, just pointing out that it has been seriously considered by qualified scientists.
I still don't get it. I don't see any advantage in your way of seeing things. For me, pi is clearly a number. I guess numbers and mathematics began with counting. Even that simple step was an simplified abstraction. Since then, the mathematical universe has been expanded to include non-counting elements - 0, rational numbers, negative numbers, real numbers, imaginary numbers. All of those are also simplified abstractions and are also numbers, even if it's hard to find a real world analogue.
Seems like you're asking for an abstract, human invention to match up with your understanding of reality. It doesn't work that way. As they say, the map is not the terrain.
Quoting keystone
A number is not an object. It doesn't have a physical existence. Also, it's not beyond my comprehension. That way of seeing things has always made sense to me.
Quoting keystone
Abstract entities, i.e. all human concepts, are always simplified reflections of the world. I can't think of any that aren't. That's why math is so wonderful. It's a game of pretend that just happens to work really, really well.
Quoting keystone
Of course there are. Particles and waves are different kinds of physical entities. One is extended, spread out, in space and the other is found in a specific location. That's contradictory, and, just like numbers, both are simplified, abstract ideas. The fact that they seem contradictory, at least to most people, is a failure of human imagination.
Quoting keystone
Agreed. I think, like many mathematicians, you are expecting math to have a precise correspondence with reality. That never works.
Quoting keystone
That's kind of a circular argument:
You - Mathematics shouldn't include elements with infinite properties because that doesn't match reality. Nothing infinite actually exists.
Me - There are qualified people who believe that infinite phenomena exist.
You - They've been fooled by their reliance on mathematics which include infinite elements.
Normally [math]x\to \infty [/math] arises in the following context:
[math]\underset{x\to \infty }{\mathop{\lim }}\,f(x)=L\text{ }\Leftrightarrow \text{ }For\text{ }\varepsilon >0\text{ }\exists M>0\text{ }such\text{ }that\text{ }x> M \Rightarrow \left| f(x)-L \right|<\varepsilon [/math]
or [math]\underset{x\to \infty }{\mathop{\lim }}\,\,f(x)=\infty \text{ }\Leftrightarrow \text{ }For\text{ }N>0\text{ }\exists M>0\text{ }such\text{ }that\text{ }x> M\Rightarrow \left| f(x) \right|> N[/math]
Infinity as a mathematical object is not used in calculus with one exception that I can think of, in complex analysis (calculus of complex variables) where the "point at [math]\infty [/math]" corresponds in a projective sense with the north pole of the Riemann sphere.
See? You started an entertaining discussion that drew in some pretty good thinkers. Probably better than paying a PhD student. :cool:
The need to complicate things by forming your triadic metaphysics is unclear to me. You say that the idea of an absolute continuity as the alternative is offensive to the ontic intuition. Please explain why. [I do agree that a triadic metaphysics is required, but not the one you propose.]
Quoting apokrisis
What is the dimension of purely empty abstract space? One might say that it is infinite dimensional, another might say that it is 0-dimensional. What matters is what you do in that space. If you're in elementary school and learning to draw X-Y graphs, then the highest dimensional continuum of interest is 2D. I see no need for one to assert the existence of continua of which they're not interested in.
Quoting apokrisis
One step at a time :)
Quoting apokrisis
The continua that I'm describing can be cut. Consider cutting a string (continua). When doing so, end-points emerge. I see no need to say that infinite points reside within the string.
Quoting apokrisis
In reality I think that the three fundamental things are space, strings, and observers.
In math I think that the three analogous fundamental things are continua, cuts, and computers/minds.
Quoting apokrisis
No, no matter how many times you cut a continua it never becomes more point-like. In a similar way, no matter how many times you cut a string it never becomes 'nothing-like'...since it always remains 'something-like'.
Quoting apokrisis
I would draw it like this: ----o o---- (note that the o is like an open interval)
Of this diagram, the cut is this: o o (note there's nothing actually there, the point is not an actual object)
Quoting apokrisis
I can picture cutting a line and gluing lines back together. I can't picture gluing points together...that's just gluing nothing. I don't see the need for points being actual objects.
Quoting apokrisis
0+0+0+0+0+0+.... approaches 0. Nothing comes from nothing, no matter how much of it you have. Continua are not the limiting case of points.
Now consider the following summations:
5
2.5 + 2.5
1.25 + 1.25 + 1.25 + 1.25
...
I can keep going down this route whereby each term gets smaller and smaller, but the overall sum of each line remains 5. Something evolves to something, no matter how many times you cut it up. Points are not the limiting case of continua.
There is not a duality between points and continua where they define each other. Continua are fundamental whereas points are not.
Quoting apokrisis
I don't understand your objection. What is the usual problem of an object-oriented ontology that I'm facing?
Pi is a ratio. Diameter~circumference. So it is actually an algorithm. And it can vary between 1 and infinity as it is measured in a background space that ranges from a sphere to a hyperbolic metric.
How all that actual physics translates to claims one might want to make about numberlines and irrational values is another issue.
There are so many ways to undermine the metaphysics implicit in the continuum that perhaps this ought to be taken as confirmation that it is a useful concept that doesnt demand further justification in terms of realist explanations?
Maths just defines it and gets on with it. And that is fine. It is what maths does.
I like to highlight the many unreal aspects of the conception from a realist metaphysics point of view. Another big one is the assumption the ground of counting aint divergent as you zero in on some arbitrary point.
Chaos theory and fractals illustrate how this might be the case. Scale matters. As you zoom in, you can no longer be sure that any point belongs to the line or a gap in the line if you are dealing with something fractal but space filling like a Cantor dust or Peano curve. If everything diverges on the finest scale, how do you plonk down your finger on some defined spot with any true certainty?
But the fact that the real world undermines the simplicities of the metaphysics that maths finds useful is part of the epistemic game here. The more holes there are in the story, the more we can take it as all just a story about reality - that works with unreasonable effectiveness.
We can take comfort in the transparency of it being a model. We can get on with using it within the limits in which it looks to be useful.
I agree. However, I would go one step further and say that infinity-laden objects such as real numbers are also not objects like the number 10100. Instead, they are simply a shorthand for a procedure.
Quoting Agent Smith
Or maybe a certain type of unstable mind searches to understand infinity as it grapples for an absolute to hold on to.
The following is philosophy, not mathematics. Please treat it as such. I only have an undergraduate degree in math and at the university where I got the degree calculus was just a wee bit tougher than high school calculus. On the whole, the whole thing had been 40 years ago, I haven't used any of it since then, and my memory is not perfect; I don't remember much math.
What about approaching something infinitely small? dividing a given integer by a larger and larger integer in succession. Until the ratio almost becomes zero.
f(x)=sin(x)/x. It can't be evaluated at zero. Yet calculus succeeds in doing so. It does by using smaller and smaller numbers for x, and not actually using zero, but yet it gets an evaluation at zero, which means that it accepts that there is such a thing as an infinitely small number. It does have an end point, which can be only achieved by having an end point in the denominator of the fraction in the previous example. The end point in the denominator approaches infinity for x to approach 0; yet it's not a point. Yet, the actual value of sin(x)/x given by calculus is a real number, and that can only happen if Calculus does use infinity as a mathematical point. If infinity were not a mathematical point in calculus, sin(x)/x could not be evaluated at zero. yet it is evaluated at zero. So infinity, despite itself not being a point, does act as a mathematical point in calculus.
Somewhat related, my understanding is that the planck length applies to measurement, not space itself. My understanding is that the continuity of space is still to be understood. Until then, it is possible that there is no limit to how small one can divide space.
Quoting magritte
Yes.
In a quantum reality we can only talk about it's velocity when measurements were made. Since we can only ever make a finite number of measurements in any given time interval, I would answer 'a finite number'. I believe that the answer in pure mathematics should be the same.
I see. I'm not really heading down the logicism route at the moment.
Quoting sime
Agreed. That is why I what to give existence to the algorithm and not the completed output of the algorithm (which would be for example the complete decimal expansion of pi).
I really like Norman Wildberger. I think his issues with the foundations of mathematics are valid. However, I do not like his resolution to the issues. He still believes that points are fundamental. At the bottom he explains natural numbers using tic marks, which is no different than spaced points on a number line.
Sin(x)/x as x approaches zero is an entity itself, a ratio that converges to one. Look at the simpler ratio (x^2)/x as x approaches zero. It's an indeterminate form that reduces to x, so goes to zero.
What I'm saying is that pi isn't the string of digits that begin with 3.14. Pi is the algorithm(s) such as the infinite series beginning with 4/1 - 4/3 + 4/5 - 4/7 + .... Since we can't actually complete the computation of an infinite series, we never produce a number. So let's just say that pi is the algorithm. The beauty of the algorithm is that it's definition is entirely finite (I just wrote it in finite characters) and it's execution is potentially infinite (i.e. it would compute to no end). There's no need to say that pi has any association with actual infinity. If we say that it's a decimal number then we must say that it has actually infinite digits.
Quoting T Clark
I don't need mathematics to align with reality. I just think reality has a clever way of avoiding actual infinity and making sense. Reality is a good sign post. If a computer can't do the math (in principle), maybe there's something wrong with the math.
Quoting T Clark
A number is an object of computation. Computers do stuff with numbers. I don't think you can fill your head with all the digits of the decimal expansion of pi. The best you could do is fill your head with an algorithm for calculating pi. That's what I'm saying exists - the algorithm.
Quoting T Clark
Interesting view. I doubt that many hold this view. I think the traditional view is the inverse, that abstract objects are the ideals and reality is just an approximation of the ideal. Nevertheless, I don't hold either view. The concept of a unicorn is not a simplified reflection of any real world object.
Quoting T Clark
Imagine me flipping a coin. While it's in the air is it heads or tails? I'd say it's neither. Instead it has the potential to be heads or tails. Only when it lands does it hold an actual value. In between quantum measurements objects are waves of potential. When they are measured they hold an actual state. I see no reason why the potential should behave the same as the actual so I see no contradiction. In fact, I think if they behaved the same then change would be impossible.
Quoting T Clark
I'm not expecting that, I just believe that truth rhymes.
Quoting T Clark
The universe has a wonderful way of avoiding actual infinities. Maybe we could to the same in math. If we do, maybe people would be less open to the unsupported idea that the universe is infinite.
This is true, and I sincerely appreciate everyone's responses.
Think of how the speed of light is an absolute limit on the velocity of a mass. The mass can be accelerated to some arbitrary speed approaching light speed, but it cant actually arrive at light speed. The limit bounds the velocity of mass as an absolute. But the velocity of the mass is always some shade within that limit.
The reciprocal argument makes that explicit. The approach to a limit is asymptotic as it is always yoked to its divergence from its other pole. For a mass, it likewise can never be absolutely at rest, although it can approach that minimum velocity with arbitrary closeness.
So as I argued, continuity is measurable as the absence of discreteness. The fact you can choose to truncate your decimal expansion in search of some specific numberline value only shows you didnt exhaust its capacity for discreteness and thus also failed to demonstrate it is as securely continuous as you might want to assume.
Quoting keystone
I think the maths of manifolds and topology would want to give a more sophisticated answer than that.
And physics likewise would give you something more complex as all actual spaces come with time and energy too. Vacuums are quantum.
As I mentioned, if we are talking spaces with algebraic dimension, then there is a whole structuralist story about that as well.
So arguing against the infinite numberline in terms of a Euclidean geometry conception might already be heading off in the wrong direction - even if it is a hardy perennial of philosophical debate.
Quoting keystone
But a string has a width. And so you can eventually chop it so much that the width exceeds the length. At which point, your analogy is in trouble.
The width of your string would have to shrink every time the length of your continua is cut. That would preserve your claimed geometric relation.
But now we are into the log/log realm of the fractal. We are into the reciprocal relation of two processes yoke together that my triadic metaphysics describes, and not the monistic notion of a single process - a continuity of cutting - that you want to claim.
Quoting keystone
There is a huge literature on how to handle terminal points of continua. Im not arguing against the maths that maths finds useful.
But I am pointing to the deeper metaphysical issue that this kind of discussion reveals. We wind up with a threeness because you are demanding that two continua separated by a cut is still also the one continuum.
So rather than finding a point on a line, we create two lines with a cut that leaves them with a point sealing their bleeding ends, and some kind of gap inbetween that is not a point, just the absence of even points now? An anti-point perhaps? Or what?
It all falls apart to the degree we try to apply everyday folk metaphysics - a Euclidean form of realism.
A more sophisticated metaphysics would let you analyse the situation in terms of a unit of opposites. A dialectical process. The numberline doesnt need to get cut, but neither is it ever whole. The numberline instead always exhibits its twin reciprocal properties of being both limitlessly integrated and limitlessly differentiable.
To have this kind of character - to have as emergent properties the opposite conceptions of being cleanly cut and being smoothly unbroken - then requires the third thing of a logic of vagueness.
The line that can be both cut and connected is describing a state that is maximally binary in its ontology. That claim of absolute bivalent crispness in turn must find its grounding contrast in its own opposite of being the minimally vague. The numberline could be other. It could be just a swamp of vagueness. It could be a fractal Cantor dust for instance, where you could never know whether you land on a cut or a line.
So again, maths can take a simple view and dispose of its metaphysical issues as cheaply as it wants. The whole numberline debate is then of metaphysical interest perhaps largely as it reveals how quickly we indeed do stop short in our metaphysics generally.
We find some kind of monistic formula that identifies a fundamental thing. And thats it. Job done.
Im just arguing that the numberline debate is another example revealing that any holist ontology has to be triadic.
Quoting keystone
But with a Dedekind cut approach, arent you just stabbing your finger down on the point marking one or other side of the cut continua and never knowing which of the two terminating points you have touched. There is always the third ground thing of a vagueness?
Each cut leaves a left and right point. But you dont know which side of the divide you are pointing to. And so left vs right is a radically indeterminate claim. The PNC fails to apply.
Quoting keystone
It binds you to a monistic and reductionist conception of nature.
As I say, thinking that way is fine if all you want to do in life is construct machines. That can be your reality model.
But for natural philosophy and metaphysics, not so much.
I have no trouble with this way of seeing it.
Quoting apokrisis
Agreed, but I think many people don't see it that way. Some think that math somehow produces reality. That if math doesn't track common sense, everyday reality exactly, there needs to be an explanation. That something is wrong.
Quoting apokrisis
Agreed. That's consistent with my understanding of metaphysics in general - it is not true or false, it works or it doesn't.
The Planck length emerges out of the triad of dimensional constants, c,G and h. Which happen to be reciprocally yoked.
So zoom in on the Planck scale and you find the same metaphysics I have described. The smallest length is also the hottest temperature. Spacetime becomes so buckled that it dissolves into the vagueness of a quantum foam. It has neither length nor points, flatness nor curvature, in any proper contextual fashion.
Event and context become the same size. And so neither can be distinguished from its other. The Planck scale speaks to a fundamental cut-off for all such metrical relations.
Sure. But reality scales. It runs its couplings in physics-speak. The maths that best describes reality has to do the same.
So the everyday folk conception or reality - and the maths that might describe that - is based on the current experienced state of the Cosmos, when it is vast, cold, and a couple of degrees from the limit of its heat death.
That is a world in which an object#oriented ontology of medium sized dry goods seems to make fundamental sense.
But physics tells us that this is not fundamental, just a passing stage. The Big Bang had quite a different kind of ontology. And physics has worked up a decent account of the maths required to track how each stage evolved into its next.
And again, it is the kind of triadic/holistic/dialectic systems view Im talking about. Peircean semiosis.
I don't see any advantage to the fact that your way of conceptualizing pi is "entirely finite."
Quoting keystone
There's always something wrong with the math. That's why people keep having to add on new concepts to keep up with our understanding of reality.
Quoting keystone
It is my understanding that computers do not generally store the algorithm for generating pi, they store the actual number rounded to a specified number of decimal laces. If computers think pi is a number, why shouldn't I?
Quoting keystone
Perhaps Plato and some mathematicians and logicians think this way. Not most people.
Quoting keystone
When I measure light one way, it's always a wave. When I measure it another way, it's always a particle. It's not a wave that becomes a particle. It's always both at the same time.
Quoting keystone
Again, sez you.
I think you and I have taken this as far as we're going to get. I don't see the need for or value of the way of seeing things you propose. You obviously disagree. Neither of us is going to convince the other.
I think I understand what you are saying about scaling, but I am not familiar with this phrase. What does it mean?
Quoting apokrisis
Seems like physics is always trying to compress all this multitude of stories about reality at many scales and stages into a single narrative that covers everything at once.
I don't think you and I are disagreeing much.
Thanks.
I'm just rereading through your older messages and I have a few additional comments:
Let's imagine a line where cuts have been made to mark all rational points (I don't believe this is possible, but let's go with it for now). I believe you cannot mark any more points on this line. If you throw a dart in between the rational points then you will hit an indivisible line segment. That is as discrete as it gets, and even then the line is securely continuous.
Quoting apokrisis
Agreed. I don't think my statement there was critical to my argument though.
Quoting apokrisis
True, the element proportions change with further divisions but still you can't cut a string out of existence. I think your argument doesn't attack the essence of my argument.
Quoting apokrisis
If we cut y=0 with the 'knife' x=0, then there is a void between the two newly produced line. We then have line x<0, void x=0, and line x>0. For all practical purposes this void is a point, the only difference is philosophical: that the void is not an object. It is the absence of an object (continua).
Quoting apokrisis
Both of which are performed on continua, not points.
Quoting apokrisis
The cuts are 0-dimensional so they are illusions of convenience. If you throw a dart at the line you will always hit the line, never the cut. The cuts have measure 0 after all. Instead of a number line, let's consider the curves x=0, y=0, and y=x^2 in 2D Euclidean space. These curves cut each other at (0,0) so we have one 'point' in this system. Is there vagueness? Yes. Without constraining the curves at all points the system is more topological than geometrical. But what's wrong with that? When anyone or any computer draws this system it's always imperfectly drawn, but by labelling the curves with their equations, we know precisely what will happen if we make additional cuts.
Quoting apokrisis
In part you're preaching to the choir since my view is triadic (computers/cut/continua). What is the triad in your view? Is it continuous/discrete/vagueness?
Quoting apokrisis
I suppose the triad that I'm proposing isn't solely object-oriented. It has a subject (computer), a verb (cut), and an object (continua).
Quoting apokrisis
I don't want to dive into what happens at the Planck scale, in part because I'm not informed enough on the topic and in part because my understanding is that we can't observe the universe at this scale.
we were talking in terms of Calculus, and that is a very integral and important circumstance to my question. Perhaps I should have pointed that out.
Damn.
In Calculus 1 classes, there is not a concern that the subject be axiomatized. But if we are concerned with having the subject axiomatized, then the ordinary mathematical context is one in which there are infinite sets. Just take an infinite sequence of real numbers. A sequence is a kind of function, and every function has a domain, and the domain of an infinite sequence is the infinite set of natural numbers.
You mean the continuum is everything. That is the opposite of nothing. Then what you call continua are the line segments that are fall inbetween these two complementary extremes.
Quoting keystone
These are your words. But if the line is cut, then you are also talking about a lack of line with some infinitesimal length, not a 0D point.
This just helps show that the idea of a 0D point is ontically problematic and in need of much better motivation than you are providing. You assume too much without providing the workings-out.
Quoting keystone
Nothing and everything are really the same. A void and a plenum are either too empty to admit change, or too full to admit change. White noise is both every song ever written, or that even could be written, played all a once, and no song being played at all.
Continua are certainly then something. And since something cannot come from nothing, continua must exist as a constraint on a state of everything. This is the better route to getting towards the intuitionist view of the continuum as having numbers with as many decimal places as you care to produce them.
The constraint on mining the number line for some particular value at a point is time and energy. If you develop a tight enough context, you can produce a matching degree of certainty.
Again, it is all about the reciprocal relation. The one physics cashes out in terms of entropy and information these days.
Quoting keystone
Sure. Behind it all is symmetry and symmetry breaking. Numbers are based on the maximum symmetry that is their identity operation - 0 for addition, 1 for multiplication. This first step suffices to produce the integers. Then more complex algebra gives you further levels of symmetry to populate the number line more densely with other symmetry breakings.
There are generators of the patterns. You start with the differences that dont make a difference. Then this yields a definition of the differences that do.
Again the logic of the dialectic and the basis of semiotics. Stasis and flux are a dichotomy. Mutually dependent and jointly exhaustive. Each is the measure or the other.
Quoting keystone
The operation is crisp or determinate to the degree it is robustly dichotomised. It is vague to the degree there could be some doubt.
To use the usual example, when you say x=0, are you talking about 0.00 . to some countable number of decimal places. Have you excluded x=0.0000 .a gazillion places later 0001?
There could be a vagueness about this nought of which you speak so freely. There could be some uncertainty you have failed to eliminate.
The adjective 'is infinite' is mathematically defined in the formal theory, set theory. I have seen no formal theory in which the adjective 'is potentially infinite' is mathematically defined.
A mathematical definition of an adjective 'P' is of the form:
Px <-> Q,
where 'Q' is a formula that has no free variables other than possibly 'x' and no symbols other than the primitives or previously defined symbols.
It's something to keep in mind that people who use 'is infinite' are at least backed up by a mathematical definition, while those who rely on a notion of 'potentially infinite' are not.
First, there are two different notions of 'the continuum'. One is that the continuum is the set of real numbers R. The other is more specifically that the continuum is R along with the standard ordering on R, or formally the ordered pair
Second, where can one read of a notion of the real continuum as an "n-dimensional continuum"? What does it mean?
Third, the set of real numbers is not constructed from nothing. The set of real numbers is constructed as the set of Dedekind cuts of rational numbers (alternatively, as equivalence classes of Cauchy sequences of rational numbers), and the rational numbers are constructed as equivalence classes of integers, and the integers are constructed as equivalence classes of natural numbers, and the set of natural numbers is derived axiomatically from the set theory axioms. That is not nothing.
/
Suggestion: Since you are interested in formulating an alternative to infinitistic mathematics, then you would do yourself a favor by first reading how infinitistic mathematics is actually formulated, as opposed to how you only think it's formulated, and also you could read about non-infinitistic alternative formulations that have already been given by mathematicians.
/
Quoting keystone
That's what you feel. But you've not supported it. If by "Cantor's nonsense" you mean his religious beliefs, then it is plain, flat out false that axiomatic infinitistic mathematics implies Cantor's religious beliefs.
This is important to recognize:
(1) Cantor's work is not axiomatic. His work was from before the modern axiomatic method reached a satisfactory stage. There are mathematical difficulties with Cantor's work due to the fact that it's not axiomatic.
(2) No matter what one thinks of Cantor's religious beliefs and how he related them to mathematics, we can separate the wheat from the chaff by recognizing the conceptual advantages of Cantor's set theoretic work without also including his religious views about it.
(3) The inconsistencies that he tried to explain by religious notions do not (as far as we know) occur in actual axiomatizations by later mathematicians.
(4) Given (2), other than for historical appreciation and for informal motivation, it is not needed for mathematicians to refer to Cantor at all. Once we had Zermelo's work, and then as it itself is rendered by the methods of symbolic logic, for purposes of formal mathematics, we could forget that Cantor even existed.
What specific paradoxes do you refer to?
Keep in mind that no contradiction has been found in ZFC.
Fine. But it's not easy to axiomatize real analysis that way.
One can philosophize all day about how one thinks mathematics should be. But other folks will ask "What are your axioms?" They ask because they expect that an alternative mathematics should have the objectivity of set theory, which is utter objectivity in the sense that, by purely algorithmic means, we can definitively determine whether a purported proof is actually a proof.
As I mentioned, that is not how it is done. You would do yourself a favor by reading a good textbook on the subject so that you would have a basis to critique the actual mathematics rather than what you only imagine is the actual mathematics.
You don't.
The existence of the set of natural numbers is derived axiomatically. Granted, the key axiom is that there exists a successor inductive set, which is an infinitistic assumption.
On the other hand, the notion of "potential infinity" demands alternative axioms.
Take just the non-infinitistic axioms of set theory. What axioms does the "potential infinity" proponent add to get real analysis?
That's right.
In previous threads I've pointed out that in set theory there is not a noun 'infinity'.* Rather there is the adjective 'is infinite'.
This is a crucial point for understanding how the subject of infinite sets is approached in set theory. To continue to ignore this point is to commit to continual confusion about the very foundational notions.
* Notations such as 'as n goes to infinity' are abbreviations for formal writing that dispenses with 'infinity' as a noun. And points of infinity and negative infinity are something different too.
No. You're just being glib and not even thinking about what I wrote; just pouncing in an ill considered way about it.
First, 'finitist' has many different senses in the philosophy of mathematics.
Second, recognizing that set theory doesn't have 'infinity' as a noun but rather 'is infinite' as an adjective, does not at all entail that one shouldn't also assert that there exist sets that are infinite.
(1) By what source do you assert that Cantor "lost his mind"?
Cantor had collapses and severe depression. I don't know of any source that says he "lost his mind" in the sense of insanity such as schizophrenia, delusions or hallucinations. One does not ordinarily say of people who are depressed that they "lost their mind".
(2) By what source do you assert that Cantor was in a "lunatic asylum" (thus suggesting that he was himself a lunatic)?
Cantor went to sanitoriums for his collapses and depression. I don't know of any source that says he was instutionalized as a lunatic.
(3) It is not a safe inference that Cantor's mathematical work itself caused his collapses. Famously there were other stressors at work.
/
Now, let's deal with the heart of this. Whatever mental problems Cantor had, they do not refute the insights of his work. Otherwise, would be an ad hominem argument. And I mentioned also that the core of his work is easily detachable from his religious beliefs.
It's fine to be interested in Cantor's biography, but it's beneath the dignity of even as common a forum as this to argue or even insinuate that his mental difficulties enter into a fair evaluation of his work.
Yeah, you often ditch an exchange with that arch "Good day" sign off, while not ingesting a single bit of the information and explanation given to you.
I comprehend the notion of an infinite set.
That is not what logicism is.
The word 'more' there is excess.
Quoting TonesInDeepFreeze
Well, truth is you're too technical for my taste. Not your fault though!
:smile: You're the expert, you would know!
[s]Cantor doesn't do that. In fact, Cantor proved that that CAN'T be done. It's his MOST famous result.
You have it completely backwards.
What articles have you read about Cantor that have led you to your terrible misunderstandings?[/s]
EDIT: I struck out my message here. I misunderstood the poster. He did not say that Cantor said the continuum can be listed; rather he said that Cantor said that that cannot be done, which is correct. My message is fully retracted.
Actually, you could start by just refraining from making claims for which you have no basis.
But if you do want to know more about Cantor's life then there is the Dauben biography.
Quoting Agent Smith
That's such a cop out. When a person such as you posts a bunch of wildly intellectually irresponsible, incorrect and confused bull, it's not being "too technical" for me to flag it and sometimes even, gratis, provide explanation regarding it.
Anyway, I am truly curious why you thought of extrapolating from the fact that Cantor had breakdowns and depressions and was in sanitoriums to claiming that he "lost his mind" and was in a "lunatic asylum". Or you just like to bolster your point of view about mathematics by making stuff up and post it as if it's fact?
I'm not sure how your post relates to my quote. To clarify, I've taught both undergraduate and graduate courses in (mathematical) analysis and published papers in the subject, and I have only very rarely had to resort to a transfinite argument or even read such an argument. In fact, the only time I can recall is the Hahn-Banach theorem in the functional analysis grad course I took many years ago. The proof involves the Hausdorff Maximal Principle (i.e. Axiom of Choice or Zorn's Lemma) and even there if one strengthens the hypotheses just a tad HMP or AOC or ZL can be avoided.
All math majors learn a little naive set theory and some ZFC these days, so mathematicians and students work in an environment underlaid by the fundamentals of the real line. It's just that conversations involving cardinalities beyond [math]{{\aleph }_{1}}[/math] don't usually occur in classical or even much of modern analysis.
Wiki says this:
If you want to argue for potential infinities over actual infinities, then the real world is surely the better place to test your case.
Arguing against maths using physicalist intuition becomes Quixotic if maths simply doesnt care about such things. Physics at least cares.
What I have said is that - as the history of metaphysics shows - there are two camps of thought about the physical world. Broadly it divides into the reductionism of atomism and the holism of a relational or systems approach.
The systems approach is triadic and says reality is a self organising hierarchical structure. So it deals with the potential and the actual by saying, yes, well both exist. It aint either/or. You have being and becoming. This makes sense as you also have necessity to complete the triad.
This triadic metaphysics reveals itself in many guises. Aristotles hylomorphism, Hegels dialectics, Peirces semiotics. So it gets confusing. But it offers a structure that can be used to see that modern physics is recapitulating the same holistic moves. The Planck triad of constants is a key sign of that.
So I am arguing from a particular metaphysical point of view. And I am saying reductionism is a monistic, concrete and object-oriented metaphysics in which the concept of potential doesnt even make sense. Reductionism is comfortable with a world in which things either exist or fail to exist. Atoms and void. The nearest reductionism gets to a state of potential is allowing for some concrete ensemble of statistical possibilities.
This entified approach to existence carries over to talk about dimensionality. Dimensionality doesnt develop. It simply exists, or fails to exist, in a concrete countable way. A point is a 0D object. A line is a 1D object. End of discussion. There is no why as to how it might be the case, given the limited ontological options that reductionism employs.
It you are fed up with hitting that brick wall, then that is where a systems metaphysics is worth a spin. It takes the bigger story seriously. It offers models that speak to the notion of pure potentiality - as that which can then birth self-organising reciprocal limits - in things like Anaximanders Apeiron, Aristotles prime matter, Peirces logic of vagueness, the quantum foam of quantum gravity physics.
So I have urged stepping back and considering how one could even talk about continua except within the limits of a dichotomy - the dichotomy of the discrete~continuous.
You are talking about the fundamentality of the interval - a concrete or actual object that is a finite length that is thus both discrete and continuous at the same time. Well how does the finite interval get to be a combination of apparently contradictory properties. How could that state of affairs develop from something more fundamental? What is the deeper story on how finitude itself can arise?
So the choice is not between the zero-D point that makes no ontological sense and the truncated 1D interval that suddenly makes sense. You can claim to have no problem with an infinity of cuts and yet have a problem with an infinity of points.
I would say the 0D point and truncated interval are in the same class of question-begging objects. Both are atomised entities lacking a properly motivated existence.
If you want to move the argument forward, well at least talking about truncated intervals highlights the contradiction that a reductionist metaphysics embodies, but a systems metaphysics seeks to explain.
How do the discrete and the continuous combine in the one actualised object? How does such a state of affairs develop? And out of what?
You said, "I have never used infinity as anything more than unboundedness."
I don't opine as what 'used' means there. I only pointed out that the calculus uses certain infinite sets, even if not explicitly. Just the real line alone is based on having the infinite set of real numbers.
Quoting jgill
(1) I only said that infinite sets are used. I didn't say infinite sets with cardinality greater than aleph_1 are used.
(2) Just to be clear, we don't know what aleph is the cardinality of the set of real numbers.
(3) What is the difference you have in mind between classical and modern? Ordinary contemporary analysis is classical analysis.
/
As to the Wikipedia quote, of course I agree, and I mentioned the distinction between the notion of infinite size and the notion of points of infinity.
Clarification:
When you said "I have never used infinity as anything more than unboundedness", perhaps I misunderstood you. I thought you meant 'infinity' in the sense of infinite sets. That is, I thought you meant that you recognize that certain sets are infinite, but you don't use them.
But maybe you didn't mean that you don't use those sets. But that you do use them, but you don't use the extended real line with its points of infinity? As instead you simply deploy the fact that the reals are unbounded?
The basic set theoretic structure of the reals underlies almost everything I have done, but I haven't used infinity as a "point" (nor the axiom of choice). Infinity is a limit in the language of calculus.
Quoting TonesInDeepFreeze
Classical means the tools of analysis like limits, differentiation and integration and all those entail. Nitty gritty. Actual specific results vs broad generalities. The more modern you get the more abstract the subject becomes with broad generalities and topological arguments. It's vague to an extent.
Hard & Soft Analysis
Quoting keystone
The problem here is that the real number line is the mathematical object that was in question, surely? So as a construction, it hosts both the rational and the irrational numbers as the points of its line.
Now, like Peirce and Brouwer, you might want to make more ontic sense of this by employing the notion of intervals.
And so the claim becomes that reality has a fundamental length the unit one interval. There is a primal atom of 1D-ness or continuity. There is an object or entity that begins everything by already being both extended and truncated. And in being both these things as a primal symmetry state possessing a canonical oneness both as a contradiction and an equality it can then become the fundamental length that then gets either endlessly truncated, or endlessly extended.
The equality of 1 can be broken by a move in either direction. And each move creates a ratio - a rational number - that speaks to the number of steps taken away from home base. If you can count upwards to a googol/1, then you can divide downwards to 1/googol. The reciprocal relation between extension and truncation is right there explicitly in the symmetry breaking of the unit 1 interval.
Now as reciprocal directions of arithmetic operations, each would seem to extend infinitely. Or at least, they are unbounded operations. The higher you can count with this system, the smaller you can divide its parts. But could you reach actual zero, or actual infinity? Well the obvious problem is that this would mean squeezing your origin point - the unit 1 - out of actual existence. The whole way of thinking would lose its anchoring conception of the truncated interval that set the whole game of approaching its dichotomous limits going.
So yes. If you just think of numbers as rationals ratios that embody reciprocal actions on a fundamental length then actual completed infinities and actual 0D points or cuts become anathema to the metaphysical intuition.
What then happens when you add the irrationals? Does that change anything?
The usual way of picturing it is that the number line becomes so infinitely crowded by markable points that it is effectively a continuum. Infinite extension and infinite truncation become the same thing. A new state of symmetry. A new unit 1 state. The continuum is that which is neither truncated nor extended. The concept of a finite length anchoring things is dissolved and becomes vague.
In hierarchy theory terms Stan Salthe's "basic triadic process" this is a familiar state of affairs. The small has become so small that it just fuses into a steady blur. The large has by the same token become so large that it has completely filled the field of view.
You wind up in a world where there is a global bound that arises because continuity has been extended so much that its truncated ends have crossed the event horizon (as cosmology would call it). And likewise, the local bound has become so shrunk in scale that it is a fused blur of parts (a wavefunction as quantum physics would call it). And then that leaves us, as the observing subject, surrounded by a bunch of medium sized dry goods objects that embody both truncation and extension in terms of looking like composite wholes made of divided parts.
So we can make sense of the rationals as intervals on a continuum the symmetry of the unit 1 being equally broken in both its complementary directions. And then when this asymmetry is maximised, you wind up in this hierarchical situation where you live in a world of truncated lengths, but then are semiotically closed in by a global bound of holographic continuity (synechism or constraint, in Peirce speak) and a local bound of quantum discreteness or fluctuation (tychism or spontaneity, in Peirce speak).
But that is connecting with the physics. A richer notion of mathematics that includes time and energy along with space. I'll get back to the issue of the irrationals.
Now for me, it seems clear enough that the familiar irrational constants pi, phi, e, delta are again unit 1 ratios, but ratios generated by growth processes. So e for example scales the compounding growth of a unit 1 square, not a unit 1 interval. That is why it is incommensurate. It is a unit 1 dropping in on the numberline from a higher dimensionality.
Surds have the same story. That then leaves the unbounded cloud of numbers with random decimal expansions. Some like pi, phi, e, delta have their generators in a higher dimension as I argue. But that isn't even a drop in the ocean of all the infinitesimals that seem to exist and so make the numberline infinitely dense with dimensionless points or cuts.
One view that appeals is that all these meaningless numbers with random decimal expansions mark nothing in particular. They are in fact the tychic spontaneity of Peircean semiotics the inability of nature to suppress or constrain all its surprises. Down at the ground level of truncation, it is just fluctuation the seething instability of the quantum vacuum, filled with its virtual particles and zero point energy (to use some much abused terms).
But maybe also, all these random fluctuations are actually ratios of some kind visitors from another dimension like the growth constants. Maybe they all have a generator that makes each of them a unit 1 story in a bigger picture, one with infinite dimensionality. Or unbounded dimension.
So the infinitesimals becomes a blur of unit 1-ness that results from a numberline living in infinite relational dimensionality, just as Cantorian infinity establishes its own unit 1-ness by becoming the point where counting goes "supra-luminal" crosses the event horizon in physical terms.
Something is certainly going on here with the irrationals. Some definitely have their higher dimensional generators like the unit square, the unit rectangle, the unit circle, the unit Pythagorean triangle. Crisp and necessary mathematical structure arises out of the swamp of algebraic symmetry breaking.
Do the rest of the irrationals have a similar story of geometric necessity behind them? Or are they just a blur of surprises and accidents. A blur of differences that don't make a differences and hence that which serves as the blank and continuous backdrop to the constants of nature which in turn have the most supreme importance.
I think Peirce and Brouwer argue towards a world where the number line starts off from the conception of truncated intervals the infinity of truncation~extension operations that can act on the unit 1 length. That gives us the rationals that are completely at home in the world so defined.
Then you get the intrusions from higher mathematical dimensions ratios or symmetry breakings from a larger universe of rational shapes. These pop on the line in ways that don't fit so exactly.
This then leads to a view of the number line that is a continuum of fluctuations. But now we are counting all digits with random decimal expansions as a something rather than a nothing.
It is information theory all over again where both meaning and nonsense are assigned the same bit-hood status. Signal and noise are in the eye of the beholder. What matters in the new counting system is there is the truncated interval the ensemble of microstates. Information theory becomes about counting all actualised differences, not the differences that make a difference (even if that dichotomy can then be recovered with other relational measures like the notion of mutual information).
Anyway, the picture I have in mind is a continuum which is composed of rational intervals truncated lengths that appear over all possible length scales. Eventually these lengths become either too large or too small and so exceed our pragmatic grasp. We live in a world where energy and time matter, along with (3D) space. And so it is simply a fact of living in that world which imposes a cosmic event horizon and a Planck scale cut-off. As an intuitionist would say, we haven't got the time or energy to pursue either the promise of unbounded interval expansion or unbounded interval truncation.
But into this rational continuum creeps some irrational numbers that really matter the rare visitors from higher-D ratios.
And also then, we have the noisy background crackle that is the randomness of all the other unbounded decimal expansions we can imagine as truncation operations. If they have generators, then in the spirit of Kolmogorov complexity, there is no actually simpler algorithm available but to print out every entropic digit. They represent the limit of the generatable. Another way of saying they are merely the meaningless noise that can't be squeezed out of any system. The quantum uncertainty that one knows one finds at the Planck cut-off of any system made of actual material stuff.
Makes sense.
Quoting jgill
That differs from how I find 'classical' is used. I find that 'classical' mathematics means all and only those results that can be formalized as theorems of ZFC with classical logic. And classical logic means the first order predicate calculus including the law of excluded middle.
Well, that's interesting. I learned something. Thanks. Classical analysis of course means more or less what I said, going back to Weierstrass and Cauchy - and I forgot, the study of special functions - but in foundations classical has another meaning.
Wiki:
:cool:
I'll give details if requested. :cool:
I think it boils down to the real line being composed of points and each point having a unique number. Instead of focusing on pi, let's return to the real line having length but its constituient points all having no length. Does that bother you? Does it bother you that a probability of 0 does not mean impossible? Or that there are the 'same number' of even integers as there are integers? These are age old paradoxes that don't bother many educated people so perhaps you see things more clearly than I do...
Quoting T Clark
Yes, they store and work with rational numbers...not real numbers.
Quoting T Clark
Are you talking about the double-slit experiment or some other experiment?
Quoting T Clark
I think I was unclear in what I meant. When physics equations result in singularities/infinities we take that as a sign that there's something wrong with our equations. Time and again we have made progress at understanding the universe by eliminating those singularities/infinities.
Quoting T Clark
I don't think you're seeing my point, but fair enough.
I agree that in terms of the orthodox point-based interpretation of calculus that there would be infinite points along a line. However, a continuum-based interpretation which involves the exact same computations (and which I would argue is more consistent with the limit definitions) follows the same measurement restrictions as our quantum reality.
No, I don't believe the continuum is everything. I think that the computer/mind lies outside the continuum. For example, when you imagine a sphere your mind exists outside that sphere.
Quoting apokrisis
In our mind, we are neither thinking of everything nor nothing. We can only think of something. I don't believe in the existence of either extreme.
Quoting apokrisis
When a line is cut, none of the line is lost. It is just divided. "Nothing" exists between the cuts, and nothing has no size. We can call this 'nothing' a point.
Quoting apokrisis
I'm only conveying my ideas piecemeal and even then my ideas are not formalized. I don't come to this thread with any notion that I have it all figured out. I'm greatly appreciative of your feedback here. You're the first to ever entertain my idea on cutting a continuum. (or perhaps you have the same idea)
Quoting apokrisis
To some extent I agree with this. The part I have trouble with is your use of 'everything'. I think your 'everything' is every 'potential' thing. My 'everything' is every 'actual' thing (which doesn't include objects/events that don't exist/happen).
Quoting apokrisis
What exactly do you mean by this? I don't think 'a state of everything' needs to exist for 'something' to exist.
Quoting apokrisis
I understand how we start with natural numbers > integers > rational numbers > real numbers, etc. I'm not sure what to make of this comment though. You return to the point that 'each is the measure of the other' so I think that's key to your argument, I'm just not comprehending it yet...
Quoting apokrisis
When I say 0, I don't mean 0.1, 0.01, or 0.001. I mean exactly 0 the rational number. Is that still vague?
That's true and I don't think you've seen mine and I don't think either of us is going to change.
Huh? Werent we about talking about how we picture the continuum just as much any other mathematical object like a sphere? Non sequitur here.
Quoting keystone
Can you picture a hypersphere as easily as a sphere? Does that make you doubt that it is a constructable object? Is your whole argument going to be based on what you personally find concretely visible in your minds eye? Thats a weak epistemology that wont get you far.
Quoting keystone
And when the line is joined, is nothing gained? If there is no gap due to the cut then is there no connection if there is a join?
You just seem to be saying stuff. I cant picture a cut which doesnt result in a gap. Are you now claiming you can see that just as vividly as a sphere? Is this an argument where we just accept your word on everything? The rules for constructing mathematical objects is becoming unclear.
Quoting keystone
Its a standard kind of idea. For instance - https://en.wikipedia.org/wiki/Dedekind_cut
Quoting keystone
Yep. I mean an everythingness that is a universal potential and not some set of actual things.
Actualisation indeed eliminates possibilities. Which is how one would argue for infinity as an unbounded process rather than an actualised value.
Quoting keystone
Can you picture getting something from nothing? Can you picture being left with something having carved away most of everything?
One of these two is more picturable, no?
It is also the central principle of physics in being the principle of least action. The sum over possibilities or path integral.
Its the logic of a reciprocal or inverse operation. How do we recognise the discrete except to the degree in lacks continuity. How do we recognise continuity except to the degree it lacks the discrete. One is present to us to the degree the other is absent.
Grey is the vague. It can then become white to the degree it sheds its blackness, and black to the degree it leaves behind its white. We have two ultimate limits where black = 1/white and white = 1/black.
If it's not a rhetorical question (and apologies to the OP if this is off topic)...
Quoting Goodman, Languages of Art
So not necessarily a matter of degree. Arguably a matter of discrimination. Which can be all or nothing. Witness digital reproduction. Where black and white are kept safely apart by grey, and there is no need for any collapse (or refinement) into 50 or more shades.
(Easy with those abstract nouns please, Apo...)
I am talking about how the spectrum that allows your 50 shades of grey arises. This is confusing for sure. But after the separation of the potential, you get the new thing of the possibility of a mixing.
So we start with a logical vagueness - an everythingness that is a nothingness. We have a greyness in that sense. Something that is neither the one nor the other. Not bright, not dark. Not anymore blackish than it is whitish. You define what It is by the failure of the PNC to apply. You are in a state of radical uncertainty about what to call it, other than a vague and uncertain potential to be a contextless anything. It is not even a mid-tone grey as there are no other greys to allow that discriminating claim.
But then you discover a crack in this symmetry. You notice that maybe it fluctuates in some minimal way. It is at times a little brighter or darker, a little whiter or blacker. Now you can start to separate. You can extrapolate this slight initial difference towards two contrasting extremes. You can drag the two sides apart towards their two limiting poles that would be the purest white - as the least degree of contaminating black - and vice versa.
Once reality is dichotomised in this fashion, then you can go back in an mix. You can create actual shades of grey by Goodmans approach.
I once had a holiday job mixing industrial paint in monster vats. It was amusing how my recipe for the morning for some companys shade of white, used to paint their fridges, requires drums and drums of bright white, and then a few teaspoons of black, and indeed a touch of red, to make it theirs.
So once you separate, then you can create. With a vat of white and a fat of black, you can mix every shade of grey inbetween. But if you only have a vat of light grey and a vat of dark grey, your range is way more limited. And if you have two middle of the range greys, then it might be hard to know whether the two have already been mixed as your efforts at mixing look to be making no further difference.
This is the ontology. Potential gets divided, the divisions produce the further thing of a space of free possibilities - a spectrum of mixed states.
Everythingness Is turned into a set into of local particulars within some global bounding constraints, At which point, our everyday notions of t reality as a collection of medium sized dry goods takes over. We take the atomic particulars for granted and get on with constructing the mixtures that have become concretely possible. We dont seem to need a theory of how this generalised somethingness itself came to be.
I know, but as usual you don't see where I'm coming from.
Quoting apokrisis
Yours not mine.
Quoting apokrisis
With that attitude...
Quoting apokrisis
...and with those abstract nouns.
Quoting apokrisis
Do you mean, an indiscriminate application of colour words to the domain of things (or patches)?
Quoting apokrisis
Do you mean, you are able to apply the words in a manner that begins to distinguish two different though still overlapping colours?
Quoting apokrisis
But you're anticipating the later refinement (the bipolar continuum) and assuming it's intrinsic to the earlier distinction. I was pointing out that it isn't.
Quoting apokrisis
Goodman's approach is concrete and clear. Yours is abstract and poetic.
A discrete classification in no way has to imply a continuous one.
Quoting TonesInDeepFreeze
Is it possible for a continuum to exist and be defined mathematically without relying on numbers?
Quoting TonesInDeepFreeze
I'm referring to a curve (1D continuum), surface (2D continuum), etc.
Quoting TonesInDeepFreeze
I'm not referring to the construction of the set of real numbers but construction of a line. Can you comment on whether points can be assembled to construct a line without making use of real numbers?
Quoting TonesInDeepFreeze
I acknowledge that I could really benefit in reading more textbooks. But just practically speaking, I could waste a lot of time sinking my head in textbooks without connecting with the community/a mentor to make sure I'm headed in the right direction. There's certainly value to me in having these discussions on this thread at this intermediate point along my journey.
Quoting TonesInDeepFreeze
What are your thought's on Hilbert's Hotel Paradox? In this paradox, he describes a hotel having infinite rooms. In this story we can't describe the hotel using inductive sets. The hotel simply has actually infinite rooms. Do you think it's a gross misrepresentation of infinite sets?
Most notably Hilbert's paradox of the Grand Hotel, but also the following:
I hope this is a prelude to you making an attempt to explain then. :meh:
Quoting bongo fury
So abstractions are banned from a metaphysical discussion. :up:
Quoting bongo fury
Nope. As usual you don't see where I'm coming from.
Quoting bongo fury
Back to the abstract nouns I guess. Semiotics as a maximally general theory of meaning tells us signs point out the differences that make a difference, not just merely the differences. But then there must be a spectrum of differences of some kind such that there could indeed be the differences that make a difference that are then different from the differences that don't. ie: we want to be able to separate signal from noise in a crisp and dichotomous fashion.
So before acts of signification can be a thing, there must be a spectrum of differences to be thus divided into its opposing classes. And where does this spectrum arise in a fashion that can leave it also the generalised indifference which is merely the noise against which the difference that makes a difference stands out?
Confused yet? :lol:
Primal difference must be resolved back to an indifferent sameness that allows meaningful difference to exist be actualised as a second order contrast.
Its thermodynamics. Let loose a bunch of particles in a box. They all have different velocities and directions. In time, they will still all be moving differently, but they will have collectively arrived at some constant macrostate equilibrium. You have a baseline of indifferent difference that can now support a measure of significant difference. A negentropic fluctuation.
Whatever linguistic distinction that Goodman might want to make in terms of how the world is can now be made in a measurable fashion. We can have the particularity of actual events as we have the generalised anonymity of an indifferent ground a state that is concrete and actual too.
So maybe you don't even realise that to make a mark on reality, you need a stable surface on which such a mark can endure. There are steps to take to reach the place in which you want to set up your metaphysical camp.
Quoting bongo fury
Goodman can kiss my arse then.
I agree. IF there is any merit to my view, then the hard work hasn't even begun.
I don't know whether a geometric theory can pick out one particular line and definite it as 'the continuum'? I'm too rusty on the subject.
Quoting keystone
Thanks.
Quoting keystone
Hilbert's Hotel is an imaginary analogy that seems fine to me.
Why can't we talk naively about points combining to form a line? It seems a little disturbing that to discuss what is seemingly a very simple concept requires significant training.
Yes, those are paradoxes. But my point is that they are not contradictions in ZFC* (and I'm not claiming that you claimed that they are contradictions in ZFC).
Zeno's paradox is actually resolved thanks to ZFC (I mean thanks to ZFC for providing a rigorous axiomatization for late 19th century analysis).
Cantor's paradox was met by ZFC by not adopting unrestricted comprehension.
Galileo's paradox strikes me a "nothing burger". I am not disquieted that there is a 1-1 between the squares and the naturals.
Gabriel's horn. I don't know enough about it.
Ross-Littlewood. Another limit problem. Doesn't bother me. Indeed, set theory provides a framework for rigorously distinguishing between terms of a sequence and the limit of the sequence .
Thompson's lamp. A non-converging sequence, if I recall. Again, rather than this being a problem for set theory, it's a problem that set theory (as an axiomatization of analysis) avoids.
Dartboard paradox. I don't know enough about it.
/
Tarski-Banach. I'm not expert on it. But my impression is that it strikes as paradoxical only when we overlook that points are not physical things. Points are abstract, used in a conceptual armature to "model" physical things but we don't contend that those physical things are actually made of abstract points. At least that's my naive layman's take on it.
/
But Lowenheim-Skolem. The problem for me is that I'm not sure that my write-up to myself about it is correct in all details. But even with the technical explanation, it does raise for me some puzzlement.
/
* A contradiction in ZFC would be a theorem of the form:
P & ~P
No such theorem has been shown in ZFC.
Wikipedia: The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in ZermeloFraenkel set theory (ZFC), but only by showing that it follows from the existence of the natural numbers.
I feel like you could give me a little more slack here on my phrasing.
Quoting TonesInDeepFreeze
I cannot answer this question.
My point is that your description is not an accurate or even reasonable simplification of how set theory proves that there is a complete ordered field and a total ordering of its carrier set. (The carrier set is the set of real numbers and the total ordering is the standard less-than relation on the set of real numbers.)
Your phrasing struck me as polemical and misleading by saying "magic" and "leap", which does not do justice to the fact that set theory is axiomatic, and while the set of naturals is given by axiomatic "fiat", the development of the integers, rationals and reals is done from the set of naturals in a rigorous construction.
I should have explained explicitly what I meant when I wrote "(Cantor)" as you interpreted my intention backwards.
"Most notably Hilbert's paradox of the Grand Hotel, but also the following:
Gabriel's horn
Galileo's paradox
RossLittlewood paradox
Thomson's lamp
Zeno's paradoxes
Cantor's paradox
Dartboard paradox"
The Diagonal Paradox can be extended in principle to any curve in 2D. For example, a circle of radius 1 has a circumference of 2pi, but if I apply my system of sine curves to the circumference I find that as they converge uniformly to the existing circumference, their lengths tend to infinity. Hence I am staring at what appears to be the simple circle I began with, but I now have one with infinite circumference, and hence infinite area.
Thus infinity is everywhere in plane geometry where it shouldn't be.
I consider the Paradox an aberration that results from collapsing one dimension to a lower dimension in certain circumstances and insignificant although bizarre. But Wolfram claims that this crops up in Feynman diagrams. It goes to the very nature of lines and points.
Of course, non-infinitistic systematizations for mathematics are interesting and of real mathematical and philosophical import. And there are many systems that have been developed. Personally though, I am also interested in comparisons not just on the basis of having achieved the thing, but also in how complicated the systems are to work with, the aesthetics, and whether fulfilling the philosophical motivations are worth the costs in complication and aesthetics.
I think I see now. You didn't mean that Cantor claims that we can list the points in the line, but rather Cantor showed that we can't do that?
If you let me know that the above is correct, then I should retract what I said earlier.
I agree about the importance of the real world, and perhaps investigating the sub Planck scale is important for the deepest insights. I just don't think it's required here. Maybe I'm wrong.
Quoting apokrisis
Thanks for this detailed explanation. These concepts are brand new to me. I would think that my views fall within holism.
Quoting apokrisis
I have a problem with an infinity of anything, including cuts. I believe that the only thing that is infinite is potential.
Quoting apokrisis
Okay, I accept that substance (continua) and void (0D points) and are both fundamental!
I'm thinking of something more irreducibly complex. A dimensionality that is "completely" void can't help but have some residual degree of local fluctuation. And likewise, a dimensionality that is "completely" full, can't help but have some residual degree of fluctuation but of the opposite kind. Particles can appear in the coldest vacuum state. But holes or local voids can appear in the hottest vacuum states.
So you have here a system like the Universe with its reciprocal Planck cut-off conditions. Finitude that seals both its ends. The hot Big Bang is where there is so much of everything that there is no room at all for local somethingness except as the smallest void-like fluctuations. Some fleeting patch of coolness.
This becomes the eternal spawning multiverse of inflation, for example, where the inflation field rages, but here and there, by a quantum fluctuation, some spot cools just slightly and that results in a phase transition. Another bubble universe such as our Big Bang starts to form at that place.
And then that nipped off bud of dimensionality keeps growing, keeps cooling and expanding, until it eventually flips over into its de Sitter state of a vast void a space as cold and empty as it can quantumly get. And now it is the expression of the opposite thing of a nothingness with its residual minimum entropy particles. The void is now the hole that hosts the faintest possible sizzle of its own blackbody radiation photons so cold that their wavelength is about 38 billion lightyears.
The point is that mathematics can conjure up all kinds of models based on simple premises. It can just take concrete starting points for granted, and take the resulting paradoxes as something to either ignore or even be a little proud of.
But physics is now pushing maths rather harder. It is time to be a bit more serious about eliminating those confusions. It is time to stop being so content with a reductionist metaphysics and to get serious about the modelling of holistic reality.
Physics and cosmology are highly concerned about how the Universe could exist how finitude could be extracted from potential infinity.
Space, time and energy all look to have had a definite start at the Planckscale that defines the dimensionality of the Big Bang. The continuum was born of one cut-off that could separate the nothingness that could be found in the everythingness. The fluctuations that were the holes.
And then this dimensionality by expressing the reciprocal actions of spreading and cooling is on the long path to the other end of space, time and energy. The story gets inverted. The cosmos has become "all hole" the largest nothingness with only the faintest possible remaining sizzle of particle fluctuation.
So maths can have its petty wrangles over how to model infinity. It's inconsequential. But to the degree that the interpretation is holistic, it is going to be on the right side of history. And so the intuitionists and finitists feel more correct for that reason.
However what really matters if we are interested in models of reality as it actually is is the fact that finitude can be extracted from pure unboundedness. Closure can be extracted from openness ... if that openness is also being transformed from a vagueness to generality (in Peirce-speak).
The universe can exist as it is making the heat sink that it is falling into.
Although there are still big questionmarks. We still seem to need eternal inflation at the front end as a kind of somethingness to get the Big Bang ball rolling, and dark energy at the back end as also a kind of somethingness to deliver the de Sitter state that ensures an eternalised Heat Death cut-off at the other end.
The metaphysical riddle isn't yet solved. However the physics of the residual "somethings" has become highly constrained. And overall, they point to a holistic or pansemiotic view of existence the triadic systems story where the container and its contents co-emerge from unbounded potential.
The small grows large. The hot grows cold. Symmetries are broken in ways that are themselves symmetric. By heading to infinity in either of these directions, you encounter the infinitesimal as a consequence.
The universe has an irreducibly complex generator in that it is a triadic and recursively self-referencing knot of relations. And any proper notion of a continuum would have to pull off that trick too.
I would argue that the 'real number line' should instead be called the 'real line' since it's composed of more than just numbers. Consider the proof that sqrt(2) is an irrational number. I would argue that the proof only demonstrates that sqrt(2) is not a rational number and that something beyond rational numbers must exist on the real line. It does not prove that sqrt(2) IS a number. I believe that irrationals are algorithms which describe this mysterious other object - continua. For example, if we conventionally said that two curves intersect at a point with irrational coordinates I would say that they intersect but that we cannot precisely determine the coordinates of that point. All we could do is use those irrationals to identify a point with rational coordinates arbitrarily close to the intersection point. To me, this is what we do in practice. We can go down the wrong path (philosophically at least) in assuming the existence of an object that is, in principle, beyond our reach. In QM we have come to accept a certain level of uncertainty. Why can't we do the same in math?
Quoting apokrisis
I don't believe there is a fundamental length since any length can be divided. If there is any fundamental unit related to rational numbers it would be the unit step along a branch in the Stern-Brocot Tree. With each step down the tree we add an L or an R to the string representation of the number above it.
I appreciate that you are using a lot of physics analogies here but I feel like you've gone to far. Your explanation (involving higher-dimensional ratios, virtual particles, etc.) seems to complicate things far more than it simplifies.
We do prove "sqrt(2) is a [real] number".
More exactly:
We prove that there is a unique positive real number r such r^2 = 2, and then we prove that r is not the ratio of two integers.
I wouldn't argue that we can't. I suppose people already have made logic systems with values such as 'uncertain' that can be be applied to a different mathematics. And I can imagine that certain scientific enquires might be better served by such systems.
But that doesn't erase the rewards meanwhile of classical mathematics.
Adding to my response about the particular paradoxes. Even if we granted that they indicate flaws in the concept of infinitude, then that is a concept of infinitude extended beyond set theory into imaginary states of affairs for which set theory should not take blame. Those paradoxes don't impugn set theory itself.
It's not about me. It's about computers in general. I can imagine a computer picturing a 4D hypershere as easily as a sphere. And if you don't accept that computers can picture things, then I can imagine a mind that lives in a 4D universe that can picture a 4D hypersphere as easily as a sphere. I can imagine a computer of arbitrarily large capacity and processing power, but I can't imagine an computer with infinite capacity and processing power.
Quoting apokrisis
Here's an analogy that closely relates to how I see it. Consider a magic-eye (stereogram) puzzle. In this analogy, the printing on the page is the continuum and I, the observer, am the computer. When I look at the page I never physically do anything to it. However, if I look at it just right, pieces of it appear to float above the page and form an image. The interaction between the observer and the page (the computer and the continuum) result in a beautiful outcome. In this analogy, the interaction is the act of cutting. Mathematics occurs when 'computers cut continua'. But then a moment later I get disctracted and the image vanishes. All that's left is the unobserved page and the observer.
But for the analogy to more closely align with my view of math, each time I look at the page a totally different image could pop up. The page contains the potential of infinite images, but only one image is actualized at any moment.
Quoting apokrisis
Returning to my magic eye analogy, what actually exist are the page and the observer. For a brief moment a single image pops out and contingently exists as well. If all potential images popped out simultaneously then the whole page would pop out resulting in no image at all. So while a cat could potentially pop out, if it's a dog that does actually pop out, then the cat doesn't actually exist...not now. And so, there can be no set of all images.
Quoting apokrisis
That is true.
Returning to my magic eye analogy, the image pops up only because the background does not. Can the image and background be the 'measure of the other' that you're referring to? If so, then that makes sense to me.
I know one can easily imagine imagining the hotel (i.e. that above each floor there is another floor) but can you imagine the actual endless hotel as a whole? I'm trying to get at whether you can imagine a set of all natural numbers.
I will have to trust you on your claims related to ZFC since I don't have a good understanding of it.
Quoting TonesInDeepFreeze
What was the 19th century analysis resolution to Zeno's paradox?
Quoting TonesInDeepFreeze
Hopefully you don't mind returning to Hilbert's hotel since I'd rather work on actual objects then relationship between sets. Are you not disquieted that a subset of rooms is equinumerous to the full set of rooms?
Quoting TonesInDeepFreeze
Wikipedia: If a dart is guaranteed to hit a dartboard and the probability of hitting a specific point is positive, adding the infinitely many positive chances yields infinity, but the chance of hitting the dartboard is one. If the probability of hitting each point is zero, the probability of hitting anywhere on the dartboard is zero.
Are you not disquieted that a probability of 0 does not mean impossible?
Quoting TonesInDeepFreeze
Achilles travels half the distance from A to B in 1 second, half the remaining distance in 0.5 seconds, half the remaining distance in 0.25 seconds, etc. Does he reach B? The answer is yes, in 2 seconds. This implies he completes an infinite set of actions. However, what happens when he holds Thompson's Lamp and each step switch the state of the light on/off? What's the final state of the lamp? If he completes one infinite set of actions (Zeno) he must be able to complete the other set of actions that are paired with it (Thompson). Without resorting to axioms, does this bother you?
Quoting TonesInDeepFreeze
Right. And to be honest I don't even know if I have any beef with ZFC (since I don't fully understand it). I don't think calculus needs actual infinity to work. All I'm proposing is that we reinterpret it and keep the math unchanged. Similarly, I suspect (with no evidence to provide) that ZFC doesn't need actual infinity to work either. Perhaps it just needs a reinterpretation. For example, I have no problem forming a 1-1 relationship between n and the n^2. I just don't think there's an actual set that contains all n (and similarly all n^2). In other words, my qualms are not with the math, they are with the philosophy.
Not as a physical object. On the other hand, I know so little of cosmology that I don't know how to dispel my bafflement that the universe could be finite or my bafflement that the universe could be infinite.
Quoting keystone
Yes, I do conceive it clearly. I conceive the abstract property of being a natural number. Then I conceive the set of all and only the natural numbers to be the set that pertains to all and only those things having said property. I know what the property is; so it takes only a shift to conceive of a set that corresponds to the property. (Of course, that can't be applied always, lest we get contradiction from unrestricted comprehension. In that case, my naive intuition just needs to adjust to accept restriction.)
That is not at all an argument that there exists such a set. It's only a description of my own intuition.
/
Or another view: If there is any mathematical reasoning that can be considered safe, then it's manipulation of finite strings of objects or symbols (whether concrete sticks on the ground, or abstract tokens). In that regard, I can see the formal derivation in Z set theory of the theorem that we read off as "there exists a set whose members are all and only the natural numbers", though the actual formal theorem doesn't have English words like that. Then, in some a worst case scenario, some crisis where my ability to conceive abstractly is terribly diminished, if I really had to, I could fall back to extreme formalism by taking the theorem to be utterly uninterpreted, but a formula nevertheless to be used in mathematical reckoning.
I'm not sure what you're referring to.
In Hilbert's full Hotel, move guest 1 to room 2, guest 2 to room 3...yada yada yada...it can accept an additional guest. Nobody explained the paradox with this phrasing, but I think this captures my frustration. I feel like the 'yada yada yada' skips over the most important part. It is in this sense that I feel like some magic is happening. I understand the standard explanation that since there is no last room each guest moves to a new room, but consider an alternate interpretation: in the first step 1 there is a dislodged guest, in the second step there is a dislodged guest, in the third step there is a dislodged guest, how does the yada yada yada result in no dislodged guests.
Infinite summation: convergence of an infinite sequence to a limit.
Quoting keystone
I don't conceive of an infinite set of physical rooms.
As to sets, I already mentioned that I am not bothered that the squares (a proper subset of the naturals) is 1-1 with the naturals.
/
Dartboard paradox. I'm rusty in probability theory. I'd have to go back the books to refresh myself.
Thompson's Lamp and Zeno's paradox. I already addressed those. I don't have more to say about them.
Quoting keystone
It does in its common form.
But there are non-infinitistic systems too. I know a little about them, but not enough to say how well they perform.
Quoting keystone
It wouldn't be ZFC then.
We could delete the axiom of infinity, but then we don't get analysis.
Or we could negate the axiom of infinity, but then we don't get analysis but instead a theory inter-interpretable with first order Peano arithmetic.
Quoting keystone
The axiom of infinity and the results from it are mathematics. If you want a mathematics without the theorems that we read as "there exist infinite sets" and "there exists a set whose members are all and only the natural numbers", etc., then that is not just philosophical but also mathematical
The imaginary hotelier can do that also.
I'm referring to the fact that set theory proves there exists a complete ordered field and a total ordering of its carrier set. And then I highlighted that the carrier set is the set of real number and the total ordering is the standard less-than relation on the set of real numbers. I don't know how to put it more simply. Other than perhaps to add that that set together with the ordering is the continuum. I don't recall whether I mentioned it earlier in this thread, though it is utterly basic to discussion of the continuum.
Hah! In the spirit of the infinite fractal coastline paradox. Nice paper.
Quoting jgill
No, I think it is significant and general. To relate it to my own interests the Cosmos there is this same virtual vs actual wrangle over renormalisation in quantum theory. Either renormalisation is some horrible kluge to be eliminated by better maths ... or in fact finitude and its dichotomous cut-offs have to be brought into the maths of the infinite somehow.
What I would point out again is how complex numbers may have their Penrosian "magic" as they speak simultaneously to the symmetries or conservation principles of rotation and translation. The incommensurability arise at that point the foundational distinction between spinning at a spot and moving away in a straight line.
A 0D point could be spinning, it could be moving. It has no context, so we can't yet tell. It is technically vague. No symmetries have been broken and so no symmetries have been revealed.
But as soon as we imagine anything happens, the foundational symmetry breaking is the Noether symmetries that close the world for Newtonian mechanics.
Newton starts the world already in motion. The first derivative. The zeroeth doesn't really exist even at a Gallielean level of relativity. And this Newtonian world is defined by its twin inertial freedoms to rotate and translate. One keeps things anchored at locales. The other sends them moving and so carving out the global largeness of space.
This would be why you find that converging to a limit has this fractal coastline property. Rotation and translation want to be different. They exist by being incommensurate.
It is the old squaring the circle story. Pi is irrational as curves and straight lines are at root incommensurate. They have to be that to co-exist in the same world and so be the symmetry-breaking making that world.
Complex numbers then speak to how the two directions of free action one closed and cyclic, the other open and expansive can be united under a unit 1 symmetry description. They explain how the symmetry is selectively broken in a way that is special to a 3D continuum. You get the chirality and commutative order arising as something that materially matters unique in that it can produce those local knots, or Newtonian particles, that can't unknot, in the style modelled by twistor theory or fibre bundles..
So it is not a mistake resulting from the simple collapse of a higher dimensionality into a lower one. It is about a complex collapse a reduction of dimensionality to a 3D continuum that then sets you up in a world where the number of spin directions finally match the number of momentum directions.
The collapse produces the internal gauge symmetry where rotation and translation become the new thing a breaking of the symmetry achieved in 3D. You can get a cosmos founded on Newtonian mechanics with the Noether dichotomy to close it as a world, make it safe for local knots of energy that can't come untied, just endlessly shuffled about like bumps in a rug.
In short, you can collapse dimensionality to SO(3) and discover it then spits out SU(2). The magic happens. Commutative order becomes a thing. Time is born. Space is anchored in a way that can be described symmetrically by spherical or Cartesian coordinates. Etc, etc. :grin:
And what of the cut off issue? I'm thinking this gives an argument in terms of the Planck scale representing a dimensional ratio.
The problem with infinities and infinitesimals is this urge to give them a concrete value. Or if not that, then they are processes without bounds. And neither answer is truly very satisfying in the light of physical reality.
The point about the Universe is that it is a story of fractal dimensionality or scalefree growth. The universe exists as a log/log story of cooling and spreading. This is both open and closed in some sense. And I've explained that in terms of it being an inversion at a deeper level.
It is going from very hot to very cold by going from very small to very large. Something is increasing as something is reducing. And if we look at a light ray, we see it is the same thing from opposite perspectives. Light waves as your simplest sine wave or helix (the helix making the rotation~translation deal explicit) are both stretched to their maximum possible extent and also redshifted to their maximum possible extent. They become as big as the cosmic event horizon and as cold as absolute zero.
So while we do like to measure all this using yardsticks like rulers and thermometers, it is essentially a dimensionless ratio. The Planck triad does not stand for some actual independently measurable number. It stands for a Platonic ratio - just like e scales unit 1 growth or pi scales unit 1 curvature. The cut off becomes simply the point where local spin and global translation first measurably come apart. And this is a qualitative distinction, not a quantifiable one.
The same goes for the Heat Death where everything arrives at its other end where the Planck triad are inverted - 1/planck - to give the cut off marking the other end of the effective 3D continuum. (Again, inflation and dark energy are unsolved aspects of this view.)
And this is why I argue for the numberline in these same terms - as the infinitesmal and the infinite as each other's reciprocal measure. A way to have the small and the large being both unboundedly open and yet also finitely closed.
The missing bit is that the numberline is based on naked spatial intuition. Peirce and Brouwer were trying to bring in time and energy (the two being complementary under quantum uncertainty) as the way to achieve the trick of an "open closure". Sure you can count forever. But that then takes time an energy.
You have two actions in opposition. This means that you can only expand until you run out of energy. And you can only contract until you run out of space.
Adding this to the numberline conception builds in the self-limiting finitude that the conventional spatial version lacks.
As I've said, Turing computation has run into a similar story of physical constraint. Infinity only reaches so far in a world where the gravity of being a hunk of circuits curling up into a black hole at a certain critical energy density a cut off point.
So rather than either/or when it comes to taking sides on virtual infinity vs actual infinity, I've been pushing "both" in the dichotomistic sense for this reason. Openness and closure are what must emerge as themselves something that co-arise from the firstness of Peircean vagueness and become a combined continuity within the thirdness of Peircean generality.
A tricky business.
One way of perceiving the sum of all paths in Feynman's path integral for paths that consistently move towards a target is to have the particle pass through a series of finite parallel plates, each with "portholes" where the particle pops through, from one plate to the next in a system where both the number of holes and the number of plates increases without bound, with plates being squeezed closer and closer together.
A path then is zig-zag, roughly like the original diagonal paradox except with random configuration. If all possible paths of this type arise, then there will be some with lengths growing towards infinity that are nevertheless contained in a closed and finite environment and go from left to right, let's say. Like the sequence of contours in my note. One meter or infinite meters? The "same" path. :chin:
I've already raised this point, asking if sqrt(2) is the exception or the rule. Higher dimensional generators could produce generators of some number that looks to be an irrational point of the line. But then these numbers - growth constants like e and phi are ratios and so are dimensionless unit 1 values more than they are some weird real number.
The status of any regular irrational seems different. They would lack generators apart from decimal expansion. Something else is going on.
Quoting keystone
Can the Planck length be divided? Not without curling up into a black hole.
What you believe and what the Universe would like to tell you seem two different things. Who wins?
Quoting keystone
Reductionism in either maths or physics is showing its age. I am just exploring the holism that would give the larger view. And in being triadic, that is irreducibly complex. Tough to deal with perhaps. But it is what it is.
Quoting keystone
Yea, nah. I'm not buying these feats of your imagination.
Quoting keystone
That's taking a point of view. Performing a figure~ground gestalt discrimination. So it would indeed be a good analogy of how an observer reads information into the world. It is how the brain sees reality in terms of collections of bounded objects.
We are getting towards Peirce who completed Kant's project of creating a metaphysics that begins in the world-constructing mind. But we can only go far taking this semiotic path before wondering how to take the step to a pansemiotic view the one where the world is "thinking itself" into definite being in ontic structural fashion.
Quoting keystone
Well the trick is that there is enough contour information to allow only the one possible stereoscopic reading once the brain has filtered out all the pixillated noise of the random colours. The hidden rabbit or seagull is merely hidden while the brain finds a way to suppress the shapelessness of the coloured pixels from the intelligibility of a depth perception-based contour.
So as a stimulus it is fixed - designed to play off two different boundary constructing visual subsystems. We are being informed of an incoherent surface by the random field of colour pixels. We have to stop looking at it as a flat incoherent surface to find the quite different depth-based reading.
Quoting keystone
The problem is that it doesn't. It plays on a dichotomous rivalry of brain subsystems. You have to switch off the one and employ the other. The search is for the single hidden interpretation. Only one of the two points of view can spot it.
Quoting keystone
That is the standard logic of Gestalt psychology. Figure and ground arise as a holistic calculation. The whole brain is organised by this contrast-creating logic. Neuroanatomy is a collection of useful dichotomies or symmetry breakings we can impose on the world to make in intelligible.
This is why the brain is not a computer. It works holistically. It imposes contrast so as to separate the world into signal and noise. It doesn't crunch data. It constructs meaning by suppressing randomness. To be attentively focused is to have defocused on everything else.
So some things about your magic eye analogy hold. It certainly explains why reductionism has such a grip on our psychology. The brain is so skilled at ignoring backdrops and seeing only the foreground events that we might indeed believe that suppressing meaningless noise is an effortless and costless mental affair.
But the brain has vastly more inhibitory connections than excitatory ones. It has to work on ignoring as much of the world as it can so as to then see it in an object-oriented fashion.
Putting a finger on an irrational value has this sort of extreme cost. Unless we have some generating algorithm to shortcut the whole decimal expansion, we just have to plough on using infinite resources. Every next digit is just as much a surprise at the last. Even if we might be thinking after even five decimal places, well how many more do I pragmatically need here.
So your general approach to root the question in actual psychology is right. And that is Brouwer/Peirce for you. There is a cost to decimal expansion where there is no shortcut algorithm. This is the Kolmogorov complexity approach I mentioned. What could be more pointless(!) than a numerical value without the constraint of a generator?
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IF f(n) [math]\geq[/math] MAX THEN f(n) = MAX
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A line of code in our (simulated) universe where f(n) is any function on n (any number) and MAX is the largest number permissible within the simulation.
All impossible propositions have probability 0, but not all probability 0 propositions are modally impossible. For instance, the probability of number being picked by randomly selecting a random
real number between 0-5 is zero, but a number will be selected. The fact that a number will be selected is not impossible, in fact, it will actually occur in this situation.
Yes, this will seem very counterintuitive. The simplest way I can explain it in a non-technical fashion is that selecting any non-zero probability for each number will force us to add up way over 100%, because there are infinitely many other "participants" (numbers), which means the only probability we can assign to each participant is zero.
There's actually a way out of this being nonstandard analysis giving us infinitesimals and particularly nilpotent infinitesimals in several hyperreal systems (*R). These can be very elegant in several applications including this one, where we can include numbers so small that even their squares (n^2) are zero but the numbers themselves, n, may not be.
If the infinitistic systemization for mathematics are more powerful, beautiful, and simple then there's not much appeal to a non-infinitistic systemization. As mentioned earlier though, I do wonder whether our infinitistic systemization can simply be reinterpreted from being based on actual infinity to being based on potential infinity. However, I'm not far enough in my learning journey to answer this.
The above is correct.
That message is retracted by me.
I don't know definitively that that is the case, but it seems to me to be so.
Quoting keystone
There are systems - such as intuitionistic* ones and others - that are said to embody use of potential infinity. But, personally, I don't know of one that can define within the system 'is potentially infinite', while set theory does define 'is infinite'. So, at least as far as I can tell, saying 'potentially infinite' is not yet, at least, a formalized notion but rather a manner of speaking.
* Though intuitionistic set theory does have an axiom of infinity. The problem though is that we can't directly compare statements in intuitionistic theories with those in classical theories, because the semantics are so radically different.
I had never heard that expression in a mathematics discussion before joining TPF.
Whether by that name or not, the idea goes at least as far back as Aristotle.
In terms of using the truths of physics as a guide in the hunt for truth in mathematics, I'm totally with you. However, I feel like you're mentioning a lot of physical phenomenon but not explaining clearly how they relate to mathematics.
Quoting apokrisis
I'm not puzzled by unboundedness. I can draw a line with open ends on a piece of paper and label the ends negative and positive infinity. This unbounded object is entirely finite.
Quoting apokrisis
It appears that you are looking at the universe from a point-based perspective in that there's a first instant which is followed by the next instant, and so on. Just as points cannot form a line, instants cannot form a continuum of time (i.e. a timeline). We must start with the entire timeline of the universe and only then can we make cuts in the timeline (i.e. observations/measurements) at different points in time to actualize reality (i.e. collapse the wave function of the universe). And if the timeline is like the one I just drew on the paper above, it is meaningless to talk about what existed at t=0 because nothing actually existed at that time.
My impression is that we do not prove sqrt(2) is a number, but instead we assume it is a number by means of the completeness axiom.
One would be crazy to say that classical math is wrong. I'm in no position to say this with certainty, but I believe that our finite descriptions of any given real number is valid and useful. Similarly, our finite descriptions of any given infinite set is valid and useful. I just don't think the real number or the infinite set itself actually exists, nor does it need to. We never work with the decimal expansion of pi, nor do we work with the infinite set itself (e.g. by explicitly listing all its elements). We always work with the finite descriptions. We work with algorithms. And if we want to work with an actual number, we halt the potentially infinite algorithm to generate a rational number and use that in our computation. I believe this is what classical mathematics does. At least that's what engineers do.
One could always be clearer. But then there is also the issue of how much you are equipped to understand. I can't be held responsible for all the work you might need to do.
However I'm not complaining. At least I'm make progress on making my ideas clearer to myself at times. :grin:
Quoting keystone
You can draw a sign that you then interpret in a certain habitual fashion. The issue then is how does this sign relate you to the reality beyond. Does is create a secure bridge? Or is it wildly misleading?
Quoting keystone
Nope. I've said I'm starting the world from its Planckscale cut off. To start the world from a point would be to start it from the singularity where all physics has been scrambled to nonsense.
And the logic is the same whether we are talking in terms of fundamental intervals or fundamental durations.
My triadic systems approach says intervals and durations are irreducibly complex entities. They are born not as some monistic given but from an emergent, self-referencing, dichotomy.
And this is Planck scale physics. It is what the maths says.
How do we measure time? In the usual approach we spatialise it. We call upon the fundamental contrast between a rotation and a translation. We imagine a length that is constrained to a self-repeating local cycle a clock hand that goes round in a local circle. And then we use that locally fixed symmetry as the standard unit to then create some unbounded sequence of length intervals. Every time the big hand completes one sweep, the day adds another hour of temporal distance travelled.
And there is nothing in this mental picture to say why it can't continue forever. Physical change has been anchored at one of its ends by being made to spin forever in the one spot. And that then frees up the possibility of change being turned into an infinity of time steps in its other direction. Globally, time becomes as infinitely large as it likes.
So you can have a proper definition of an instant or an interval only when you have nailed the rather ambiguous or vague notion of "a change" to a dichotomous coordinate system a symmetry breaking. One end has to be fixed. You do this by creating a cycle that just goes round and round the same 0D point. The other then is allowed to flap completely free. It can follow a straight line forever as a straight line is simply an endless repetition of steps that never repeat rather than the other thing of an endless repetition of steps that only repeat.
A moment in time a durée - thus is an irreducibly complex object. It combines rotation and translation to create the emergent thing of a "fundamental time step". And length intervals are also the same trick. They just get stripped of the philosophical nod to energy and change (the things time must measure) and become a story of the 0D point and its 1D line.
At least with clocks you can see how repetition and difference are the partners in crime. The idea of the spatial interval becomes shrouded in mathematical mystery.
But if we add back the physics the irreducible quantum indeterminacy found in nature then the clock of the length interval becomes visible. Any point must vibrate. It will have a resonant motion ... because QFT says so. And we can see the reciprocal relation between location and momentum as plain as your face in Heisenberg's uncertainty principle.
Nature has a fundamental frequency. Physics says it is so. A systems metaphysics says it is only to be expected.
If maths has been left behind in this grand and still unfolding adventure, tough shit.
(1) In set theory, there is no completeness axiom. Rather, we prove as a theorem that the system of reals is a complete ordered field.
(2) We assume axioms. (Or, in another view, we don't even assume them but rather merely investigate what their consequences are.) The theorems we derive from the axioms are not "assuming by means of". Okay, in a broad loose way of speaking, someone might say that the theorems are essentially just "assumptions" unpacked from the axioms. But that really muddies the matter terribly. Granted, everything we prove is, in a sense, "already in the axioms", but that obscures:
Yes, often we adopt axioms to prove the theorems we already know we want to have. But so what? That is, as they say, a feature not a bug of the axiomatic method.
And any alternative mathematics that is axiomatized is itself going to have that feature. So there's no credit in faulting set theory in particular for that.
And yes one might want for the axioms to be intuitively correct ("true") even if the theorems might be surprising. And with set theory, people's mileages vary. I find the axioms of set theory to be exemplary in sticking to only principles that are in concordance with the intuitive notion of 'sets'.
(3) So getting back to my earlier point: We prove that there is a unique positive real number r such r^2 = 2, and then we prove that r is not the ratio of two integers. Not the other way around as, if I recall, you suggested.
?? Bergson's time intervals? What? Rotation & translation = the two spools? :chin:
The cosmos is structured in terms of the scales at which interactions have had time to complete whatever change was going to happen. And beyond that event horizon ate all the possibilities yet to be actualised.
So time has a fundamental grain determined by c. A moment or duree is the completion of a change. And the Planck scale is the size of the smallest such moment.
It is indeed measured by a rotation and translation as it is the the time and space large enough to contain the first Planck scale energy fluctuation - a wave with the Planck frequency. So a single beat of a helical spiral. A sine wave. A rotation and translation that begin at the same size. But then a beat later, the space has doubled and so the energy halved. The fluctuation has already been redshifted. The difference between local spin and global translation is already established.
The fact we measure time with clocks is just showing how general the logic is. We have to have something that changes and yet that makes no difference to be able to measures the changes that do make the difference.
But then really accurate clocks wind up actually using atomic vibrations.
Food for thought.
Wiki:
To correct myself, e and phi are different in that e is transcendental and phi is algebraic. That's something I need to dig into further.
My argument is that the whole potential vs actual infinity thing comes from the fact that our ideas about numbers are based on systems of constraints. We want to close some space of values in a way that creates a set of numerical objects with shared properties. And so we end up with a set of "all the integers" that is a class of objects with tightly specified properties. And then relax some of those constraints and we find the integers nested in the larger set that is "all the rationals".
This is a subsumption hierarchy in hierarchy theory. Each successive set becomes larger as you relax the constraints that form its boundary. So from the point of view of the integers, they are potentially infinite when seen from inside their closed world - a closed world that is unbounded in its openness as an integer generator and then become viewable as actually infinite in being a crisply restricted domain within the larger and more relaxed space that is the set of rationals.
So virtual vs actual infinity is a point of view. Each level of number can contain within it a more constrained notion of number. And also itself be contained within a less constrained one.
Each such set is a machine for cranking out sequences according to some set of rules. It is a symbol generator, and so is open in that it possesses the finite means for spitting out infinite steps. But then from the point of view of some less restricted generating algorithm, the set looks to be closed in terms of all the values that can be produced in that way. There is now some larger set of values that becomes possible by a generator with less constraints producing values with less intrinsic structure.
This nested view is illustrated in diagrams like this...
I would say that those paradoxes don't necessarily impugn set theory itself. Just as actual infinities (rather infinitesimals) were banished from (mainstream) calculus and replaced with potential infinities (through limits) leaving the applied math essentially uneffected, I wonder if something similar could be done to the actual infinities of Set Theory.
I think it's quite easy to imagine a closed finite universe, for example a sphere of finite radius. It's a lot harder to fit in one's mind a sphere of infinite radius.
Quoting TonesInDeepFreeze
I completely agree. I can't imagine a scenario where the work that mathematicians have been doing for decades using set theory could suddenly be worthless. Maybe some ideas would need to be re-evaulated (e.g. Axiom of Choice and the BanachTarski paradox) but by and large I am completely convinced that the vast majority of modern math would retain it's value even IF actual infinities were banished.
No, limits use infinite sets. The standard axiomatization of analysis is ZFC. Ordinary modern analysis is decidedly infinitisitic. Maybe you're thinking of the banishment of infinitesimals?
A sphere has infinitely many points in it.
And is there such a thing as a sphere with an infinite radius? If I'm not mistaken the radius of a sphere is a real number, right?
Quoting keystone
We've come around full circle. Now circling for another orbit:
To axiomatize mathematics sufficient for the sciences, without infinite sets requires not just deleting the axiom of infinity from (Z\R)+CC (ie. Z without Regularity but with Countable Choice), but a very different system. There are systems without infinite sets, but I don't know how well they do or how easy they are to understand and use.
Quoting TonesInDeepFreeze
I believe that, contrary to the mainstream interpretation, 19th century analysis is the mathematics of continua (not the mathematics of points). I do believe that the mathematics of continua (i.e. 19th century analysis) does indeed resolve Zeno's paradox. So in this partial sense I agree with you.
However, I don't believe that a point-based interpretation of 19th century analysis allows for Zeno's paradox to be resolved. If it does, then answer this: If I travel along a line from x=0 to x=1, what is the next point that I travel to? As you know, there is no 'next point' so Zeno would claim that it's impossible to take a first step. As long as we hold on to points Zeno's paradox stands.
Quoting TonesInDeepFreeze
It sounds like you don't think Hilbert's Hotel is a good analogy then. Is that true? I like discussing in terms of the hotel because it's clear that we're talking about an actually infinite object not a potentially infinite process. For example, I have no problem with programming a computer algorithm to print n:n^2 for all n. Such a program could be written in a few lines of code. My issue is with the notion that such a program could be executed in completion. The complete output of the program would be an actually infinite object which I don't believe in.
Quoting TonesInDeepFreeze
Please provide the simplest example you can think of where it is needed? I'd like to see whether I can challenge it.
Quoting TonesInDeepFreeze
Just as I can write a program that outputs all natural numbers, I think it is fair game to talk about an infinite set. Might it be possible that ZFC is useful mathematics that talks about things that don't actually exist? Might the useful thing not be the objects but the 'talking'. Might we be able to reinterpret the axiom of infinity as 'we can talk about infinite sets'?
Please allow me to indulge in Hilbert's Hotel a little more just so that we can stick with actually infinite objects instead of potentially infinite processes. As such, I want to avoid resorting to algorithms or properties to ensure that we are clear on our differences. In the lobby there is a lamp that is currently off. Each room has a switch to invert the state of the lamp. When each guest is moved to the next room, the guest flips the switch. After infinite guests move rooms, is the lamp on or off? There's no way around it...Hilbert's imaginary universe (where infinite processes can be completed) doesn't make sense.
How is that substantively different from Thompson's lamp?
I already responded regarding Thompson's lamp.
/
I don't know a theorem of set theory that is rendered as "infinite processes can be completed".
Set theory doesn't axiomatize thought experiments.
The distinction makes sense if we approach numbers as entities generated by a process, an algorithm. An instruction set that generates the natural numbers (vide infra) will go on forever, it's a task that's endless ([math]\infty[/math]).
1. n = 1
2. print n
3. n = n + 1
4. goto 2
An actual infinity would be impossible, algorithmically speaking because for that, the algorithm must terminate, but as you can see it's a bloody loop.
It's a non finito and perhaps that's the whole point.
But that doesn't prove that there does not exist a set whose members are all and only the natural numbers or that there does not exist an infinite set.
And it doesn't prove even the weaker claim that it is not provable that there exists a set whose members are all and only the natural numbers or that it is not provable that there exists an infinite set.
The non sequitur is:
R does not provide us with W, therefore there is no T that provides us with W.
One might as well say, "A hammer won't lift a beam, therefore nothing will lift a beam".
If we want for it to be provable that there does not exist an infinite set, then we need axioms to do that.
If we delete the axiom of infinity from ZFC and add the negation of the axiom infinity, then we prove that there does not exist an infinite set. But that theory is inter-interpretable with first order PA, which does not provide a calculus for the sciences.
And if we merely want for it not to be provable that there does not exist an infinite set, then we merely need to delete the axiom of infinity from ZFC, and then we can't prove that there exists an infinite set and we can't prove that there does not exist an infinite set. And that theory does not provide a calculus for the sciences.
This is to say that it is fine to mention the well known ostensive illustration of the notion of 'potential infinity', but that's all that is - an ostensive illustration of a notion; it doesn't even hint at how we would make a theory from it.
Youve got it back to front. It has been shown there is a generator that can produce an arbitrarily large number. Youve accepted that. Now what needs proving is that there is also this set containing an infinity of numbers.
Why do you keep shifting the burden in your posts?
Does finitism mean some domains of math vanish into thin air?
(2) Perhaps it is not necessary to have infinite sets for an axiomatization of mathematics for the sciences. It's just that in order to evaluate a non-infinitistc axiomatization, we need to have it specified.
Undefinable real numbers have no place in my view. Focusing solely on definable irrationals, I believe an irrational is irrational, no matter how efficient/beautifully it can be expressed. My point is the rule not the exception.
Quoting apokrisis
It has not been determined whether space is discrete or continuous (LINK). I'm inclined to believe that the planck length is a limitation that is applied to measurement, not the divisibility of space itself.
Your trust in physics has made you believe something obviously counter to your mathematical intuitions - that any positive number can be divided.
Quoting apokrisis
Seriously? My mind can only hold so many bits therefore there is a largest number that I can retain in my memory. However I can easily imagine a mind whose memory could retain a larger number. It's the same thing with dimensions. No leap of faith is required here. The real feat of imagination is imagining a mind whose memory could hold an infinitely large number. That I cannot do.
Quoting apokrisis
In my stereogram analogy I do not exist within the page, and neither does the computer exist within the continuum. Or perhaps more digestible, neither does the computer exist within the simulation.
Quoting apokrisis
It sounds like you're saying that the rabbit exists even when not observed. I would argue that it only has the potential to exist and it actually exists when observed correctly. In any case, I think you're nitpicking my analogy. If I handed you a blank white piece of paper you could argue that it is a picture of a polar bear playing in the snow but I would argue that it only contains the potential to be such a picture, and it would actually be that picture only once you cut out the bear figure with scissors.
Quoting apokrisis
Perhaps I'm missing your point. Do you agree that given a continuum there's infinite potential to how you cut it up?
Quoting apokrisis
I would prefer to say 'this is why the brain operates different from a modern computer'. I agree with Max Tegmark when he said 'we should reject carbon-chauvinism'. However, this is already a broad discussion, I don't think we should extend it to debate this.
I understand how we got here, it's just that to me this screams that what got us here is a mistake.
Quoting Kuro
I have a hard time accepting real numbers, I have an even harder time accepting hyperreal numbers.
But yes there is a very simple way out of this, and that is that points don't exist. When you throw a dart at a dartboard, you don't hit a point, you hit an area. Any discretization of a dartboard into areas produces a finite number of areas each with a finite probability, all summing to a probability of 100%. What's wrong with this view?
:up:
Quoting TonesInDeepFreeze
As I've argued, science is founded on the pragmatism of the semiotic modelling relation. That changes the view of the situation. Axiomatic deduction becomes just the middle part of a three stage process. The formality of the deductive step is matched by the informality of the initial abductive step (the inductive leap to a possible answer) and then the informality (or pragmatism) of the inductive confirmation the act of measurement, the act of actually applying the idea of counting to the world which needn't necessarily be crisply countable.
You say the idea of the stick as an atomistic object gives safe ground for counting in terms of abstract objects. But are any two sticks ever alike outside your willingness to grant them the status of being "sticks" of near enough similarity in terms of size, shape, weight, etc, to ignore the differences and declare them "the same"?
So the idea of counting is different from the physical act of counting. One deals in abstract objects, the other in real objects. And all real objects full of contingency. They have any number of defects and blemishes. Indeed, the ontology of real objects for the Peircean pragmatist is that they are simply a state of contingency that has been constrained to some suitable degree where the differences statistically cease to matter. The set of physical objects is constructed as the range under the generator of some agreed distribution curve. Near enough becomes exact enough in a way that is outside of deductive reason, but within grasp of the feedback loop of a hypothesis-testing pragmatism.
It is great that maths tries to make itself as robust as possible by an axiomatic approach. The methods of deductive proof. This part of the modelling relation needs to be as watertight as it can be.
But then science is equally concerned with the business of measurement. It may be informal in the logical sense, but it has to be strict in the methodological sense.
Then we must have a robust approach to abduction as well. And here things become murky. How does one bottle true creative insight of the kind that is the leap to the model that is going to work and achieve its pragmatic goals?
There are clearly ways to cultivate or train folk in abductive reasoning. But most education systems are more concerned to bash you over the head with a textbook.
Given your concern for the proliferation of cranks obsessed with offering original solutions to the biggest problems, the secrets of abductive thought should give clues to what goes wrong there.
Largely of course it is a failure to be iterative. To guess small and test. Then guess larger and larger in conjuction with broader and broader reality testing of the formal theory being constructed.
ELI5.
Quoting apokrisis
You believe that math itself has some fundamental limits, perhaps a frequency, a duration, or a length. You may be right but I think you're wrong in applying it to continua. Instead, you should include the computer as a fundamental participant in mathematics and apply these limitations to the computer. Apply them to measurement. It makes sense, after all. Every computer has its limits. It has a finite memory so there is a limit to the size of the numbers that it can store. It also operates at some frequency. This is all common sense and it requires no philosophizing. Mathematical objects don't exist eternally in the Platonic realm. They exist when we (computers) compute!
Must an ordered field necessarily be a field of numbers? Could it instead be a field of equations? For example, I would imagine that adding a 1 to each number in the field does not negate the field.
Quoting TonesInDeepFreeze
I largely agree with you about axioms. My only issue is that the axioms of set theory are not in concordance with the intuitive notions of 'finite sets'. And since the only sets we ever work with directly are finite, I think we should be cautious accepting axioms that oppose them. But as I mentioned before, a small tweak to the language of the axioms will be consistent with my intuitions. Instead of saying 'there exists an infinite set' I would be comfortable saying 'there exists an algorithm that describes an infinite set'.
I am triggered by this. If you start to think anything and than stop, where does it go? If we think of an infinite line, it seems to have to vanish as one's mind caters to finite things. Not just this fact, but the fact that we do not have infinite life-spans also makes it impossible to somehow span one's consciousness in infinite directions. Birth and death create this finite life, and probably all forces of life thereof.
You reject vagueness then. That is certainly the usual thing to do. :wink:
Quoting keystone
And how do you know there is a rule unless you have ever seen some exception? How can you define a definite limit unless you actually have indeterminate challenge of the borderline cases?
A rule is meant to range as a constraint over all possible exceptions. In practice, we live by rules that simply limit exceptionality to that which doesn't matter. We reign in the differences that matter and ignore the differences that don't.
I prefer the practical approach to these things rather than the Platonic. The advantage of indeterminacy is that it then gives you something to determine. There is a reason to have been thinking deeply about what matters and thus a choice that has been made which constructs a world with some actual meaning to it.
Quoting keystone
But the Plank scale measures the fundamental grain of indeterminacy quantum uncertainty or vagueness. Determination can start from that point where the symmetry is dialectically broken. We can start to say that some particle has both a limited location and a limited momentum in some strong sense.
The limit of measurement is indeed vagueness or indeterminacy the state where the PNC finally fails to divide the world distinctly.
Quoting keystone
But the image is constructed by having the contour information split into a pair of representations which you then have to fuse bak into one by changing your depth of view. So you could only ever see the suggestive contour of whatever object has been plugged in.
Quoting keystone
That is a better analogy. I prefer my own still the static on the TV screen which is both every show you could ever see, but all at once ... or else just meaningless noise.
Quoting keystone
The mathematical picture of the continuum is of something that can be unlimitedly cut. But physics tells us that the real world works with cut-offs.
That doesn't mean there is some smallest interval a point where the line and its gaps are the same size and so you can land cleanly either, and with equal probability, on the line or the gap. It means that there is some scale where the very distinction between lines and gaps becomes vague. It ceases to be a meaningful distinction.
If you agree with a cut based approach to the continuum, then this is what lies at the end of the trail. The cuts must have some size to actually disconnect the line at a point. The more cuts you make, the more the continuous line becomes a tissue of cuts. When you get down to the scale where the line segments and the cut widths are the same, it is 50/50 whether you have one or the other. And the next cut - in a single step - then removes the line completely.
So there are some issues if you want to claim bivalence all the way down to the limit.
Quoting keystone
What does Tegmark know about carbon then?
Biologists learn that carbon chemistry is unique because of things such as that it forms 29,000 hydrogen compounds to nitrogens 65, oxygens 21, or silicon's 55. And carbon makes metastable monomer chains where silicon only makes crystals.
In other words, if the physical world wanted to do "computing" in the style of genetic codes providing the constraints to organise chemical potentials to do work, carbon was the only thing that made it a true possibility.
Rather than reject carbon-chauvinism, we ought to reject basic biological ignorance.
No. But all complete ordered fields are isomorphic with one another. So all complete ordered fields are isomorphic with the system of reals.
Quoting keystone
I know there are field equations (but I know really nothing about them), but I don't know what a field of equations is.
Quoting keystone
All the axioms are in that concordance, except one.
Quoting keystone
I think there's something to that. Indeed, it is, roughly speaking, right in line with Hilbertian finitism. Hilbert's idea was that we can work in infinitistic mathematics if we have a finitistic proof of the consistency of infinitistic mathematics. Famously, we found out that there is no finitistic proof of the kinds of systems we'd like to use, not only not of set theory but even of PRA, the system itself that we may take as exemplifying finitistic reasoning at its "safest". Yet, if I understand correctly, Hilbert's condition was a sufficient condition not a necessary one.
Quoting keystone
That's interesting. But, if that is to be a statement in the system, we'd need to see "described" couched mathematically. I have a hunch that your notion is pretty much the same as 'there exist potentially infinite sets', and as I've said, I don't know a system that says it.
I disagree. I think that potential infinity comes naturally to us, whereas we reluctantly hold on to actual infinity. We do so because we are not used to the notion of the observer (mathematician/computer) playing an active role in mathematics. As with classical physics, we want the observer to play a passive role by making passive observations upon the objects of math that exist in completion for eternity in the Platonic realm.
But you're a fan of physics so you should appreciate that in QM the observer plays an active role converting the potential to the actual (by means of making measurements to collapse the wave function). Consider the possibility that mathematics is contingent upon the interaction between the computer and the continuum (by means of cuts), in the same way that reality is contingent upon the interaction between the observer and the wave function (by means of measurements).
How many numbers are there?
A mathematical platonist would have to say infinite. I would go in a different direction. I would ask 'where?' How many numbers are 'where'? In other words, in what computer/mind are you talking about. You have to be specific about where is because there is no Platonic realm. And right now, as I'm getting ready to go to bed, if you ask me 'how many number are there in my mind?' I would answer zero.
We need another Cantor to make the next leap in our understanding of [math]\infty[/math]. I hope he's out there somewhere. Fingers crossed.
My interest in finitism hearkens back to the Greeks I suppose. [math]\infty[/math] defied common sense intuitions e.g. an infinite set and its proper subset have the same cardinality which in colloquial terms means a part is equal to the whole. Some might disagree on this point, but there are peeps who say this is exactly what the mathematics says. [math]\infty[/math] maybe the internal combustion engine of math - creates more problems than solutions.
I can't speak to the standard axiomatization of analysis, but the informal definitions that us engineers were taught didn't use sets. As written on Wikepedia: a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. I take "approaches" to be a potentially infinite process.
Just to be clear, an infinite set has the same cardinality as some of its proper subsets, but not all of them.
Quoting Agent Smith
No it does not. It means a proper part is in 1-1 correspondence with the whole, not equal to the whole. A proper subset S of T is never equal to T. I've mentioned this before, but you repeat your disinformation.
Quoting Agent Smith
Who specifically?
Quoting Agent Smith
I can only guess at what vague thing you mean every time you write the leminscate.
I don't know what problem, especially a practical one, you think is caused by set theory.
That isn't my argument. Maths can be treated as its own abstract game with its own generating axioms. That game generates a bunch of abstract nonsense, but also occasionally hits on something "unreasonably effective" so far as modelling realty is concerned.
The issue is with the abductive end of the story the intuitions that might guide the further progress of our reality modelling as a community of philosophers, scientists and mathematicians.
If your intuitions are wedded to the concrete machinations of deductive algorithms, then you might be locking yourself into a world picture where you can't see it as instead a story of self-organising emergence with natural cut-offs.
Finitism indeed might not be the best way to extend the reach of the formalisable. But it may instead be the best way to reconnect the formalist project to the reality it left behind.
That is the debate I am focused on.
Quoting keystone
But I've already said this about computers. The abstract formalism the Turing machine has an infinite length of tape and infinite time to move it back and forwards through the single digital gate. However real computers live with real physical restrictions and their maximum information capacity is constrained by the Bekenstein bound. This is the point at which the computer turns into a black hole under its own gravitational pull.
So yes, we have theories about how to distinguish between the mathematical fictions and the real world realities. Time and energy become constraints on counting or cutting as the intuitionists argued.
Quoting keystone
And yet Universal Turing Computation is a mathematical object conceived in Platonia. This is the kind of "paradox" we are meant to be figuring out here, not simply saying one is the other as if the differences were moot.
Let's drop a dimension for simplicity. I can think of x^2+y^2=1 without having to think of any points. I can also draw that function without drawing any points (I can show you if you're curious). The points come from the circle, not the other way around.
No, there is no such thing as a sphere with infinite radius...that's explains why I can't imagine it.
Mathematics, in many branches, is brimming with sets. Analysis, topology, abstract algebra, probability, game theory... Can't even talk about them, can't get past page 10 in a textbook, without sets.
But of course, one can use the theorems of mathematics for engineering without tracing the proof of those theorems back to axioms, in particular the set theoretic axioms. That's not at issue.
A sequence is a set. And it has a domain, which is a set, and a range, which is a set. An infinite sequence is an infinite set with an infinite domain.
Of course, one can leave that unconsidered, not in mind, when working in certain parts of calculus. That is not at issue. But when we trace the proofs of the theorems of analysis back to axioms, then, in ordinary treatments, those are the axioms of set theory.
Of course. No one says that you have to.
Quoting keystone
A circle is a certain kind of set of points. I don't know what you mean.
Quoting keystone
Yes, and since infinitistic mathematics doesn't have spheres with infinite radi, infinitistic mathematics doesn't call on you to imagine it.
:up:
That means you understand why it's not such a good idea to post disinformation that set theory claims that a (proper) part is equal to the whole?
And maybe, you'll say who you thinks says that set theory does make that claim?
I'm a fan of semiotics the modelling relation. So that is a formal theory of how observers and their realities relate by acts of measurement.
I'm working one step up. I'm speaking for those like Peirce, Rosen, Pattee, etc, who are modelling the modelling relation in a rigourous fashion. And yes that does then lead to a better metaphysical understanding of both classical and quantum physics.
Quoting keystone
I am saying much the same thing. But the question is not where the numbers need to be represented or stored. It is how many decimal places do you really need for the task in hand?
What this means is that the whole business of counting becomes self-limiting. The aim of "good mathematics" is instead to represent the fewest digits you can reasonably get away with. An infinity might be available, but I would rather save time and energy for other things by only having to remember just one.
This is where we get down to binary bits of logic. The only numbers needed are 0 or 1. Or indeed, the tape head of a Turing machine that can either make or erase a mark.
But it takes the semiotic view to create this reciprocally self-limiting Nirvana where infinite information is available, yet you can boil it all down to a simple yes/no response.
Is it a 0 or a 1? That's all I need to know to be completely certain rather than maximally uncertain. And a world that is boiled down to yes/no certainty demands hardly any time or energy to live in it.
This is what maths looks like when it does involve an intelligent observer with some actual purpose. A big enough range of possibilities to cover all eventualities. But then the complementary operation that narrows the field to a single actuality of complete certainty or counterfactual definiteness.
That just takes the conversation back to where we were before. One can have whatever concept of limits one wants to have, including conceiving in terms of potential infinity. Indeed, there are systems that do (and I little doubt succeed) in axiomatizing large amount of analysis without infinite sets.
And lots of people aren't concerned with axiomatization, so they don't even care whether or how analysis is axiomatized, as they at the same time prefer to conceive in terms of potential infinity rather than there being infinite sets. Not at issue.
But, to the extent that one is interested in axoimatization, one would want to go the next step, which is to ask, okay, what are the axioms, or at least, what initial ideas are there for what the axioms might be?
If you say, that's not your concern, then fine. Then we just have different roads we want to travel. But one can't fairly criticize the road of set theory if one is not addressing it as an axiomatization. And even if not criticizing set theory but instead just saying mathematics can be done with unformalized "potential infinity" instead, then it's not a fair comparison since one is an axiomatization and the other is not.
Okie dokie cranky spanky.
[quote=Ms. Marple]Most interesting.[/quote]
While I do agree that mathematicians have achieved some kinda broad consensus on the role of [math]\infty[/math] in mathematics, probably this is your area of expertise, the existence of finitism suggests to me that there's trouble in (Cantor's) paradise!
There are critics of X, therefore there is something very wrong with X.
That is a risibly stupid argument.
Criticisms by mathematicians (finitism is a mathematical movement), to my knowledge, aren't usually baseless.
Then you write a post having nothing to do with your quote of me. It still stands that it is disinformation to say that set theory claims that a (proper) subset of the whole is equal to the whole.
The argument for the claim I made is a simple one and perhaps it doesn't meet the standards of rigor required in mathematics.
There's nothing in the world that doesn't have detractors. So, by your logic, everything is wrong.*
Finitism has detractors, so by your logic, there's something very wrong with finitism.
* Or, to put it not so sweepingly, let's unpack what you've said.
(1) As I've pointed out to you probably half a dozen times already, finitism has many forms. Some finitists work in infinitistic set theory.
(2) Many finitists are critics of infinitistic set theory. But your own criticisms are ignorant, self-malinformed, and disinformation. You don't like the concept of infinite sets and want to see a mathematics without them. That is fine. But you grab any chance to indulge your confirmation bias about the subject.
In this case, you don't even appeal to specifics but instead ludicrously reason that since there are critics then we can pretty much bet that they're right. That is so remarkably irrational. It's in the same league of irrationality as people who say, "The conclusion that climate change is anthropogenic is wrong, which I know because there are scientists who say it's wrong."
No, to claim that the conclusion is wrong requires actually comparing the work of the dissenting scientists with the work of the preponderance of other scientists.
To claim that set theory is wrong requires comparing the arguments of the detractors of set theory with the arguments of the mathematicians and philosophers in favor of and in defense of set theory.
Just grabbing random out of context quotes against set theory, one-liner tidbits, and polemical irrelevancies to argue against set theory is like viewers of Fox News who base their claims about politics on whatever chyrons happen to cross the screen, whatever infantile propaganda memes are splashed and whatever disassociated falseoids happen to spill from the mouths of the on-screen anti-pundits.
It's not so much that you don't meet a standard of rigor, it's that you lie about the subject.
Quoting TonesInDeepFreeze
:rofl:
If it's any consolation I was a math major at two big state universities in the 1950s and I can't recall of ever studying any aspect of foundations beyond skimpy material on rationals and irrationals and the continuity of the real line. As a freshman at Georgia Tech I was placed in an experimental course in introductory calculus that began with epsilons and deltas - very confusing at first. The engineers had it much easier, as I learned when I moved back into the standard curriculum. But when I started grad school at another university in 1962 one of the first required courses was an introduction to foundations using Halmos' Naive Set Theory and the Peano Axioms. It was quite illuminating.
What? You're the one spreading disinformation, notwithstanding the little touch you give with Latin phrases.
You say that set theory claims that a (proper) part can be equal to the whole. I explained to you twice exactly the way you are incorrect. My going to Google or not doesn't affect that you're spreading disinformation.
You'd be right if that emoticon meant 'QED'.
Gaudeo te relinquere, domine.
There, see, I took you up on your suggestion to visit Google.
Quoting TonesInDeepFreeze
I think saying "there exists a set of all natural numbers" is equivalent to writing a program to print all natural numbers and running it through to completion. However, I think set theory can be reframed to correspond to potentially infinite algorithms instead of actually infinite sets. After all, we never directly work with the infinite sets themselves, but instead the finite strings of characters that describe them.
Thought experiments are beneficial because they make it clear what we're talking about - with set theory (as it is framed today) we're talking about actually infinite objects, not potentially infinite processes. And I'm not convinced by your response to Thompson's lamp because your answer lies outside of the thought experiment where it's unclear to me whether your resolution requires the completion of an infinite process. In Hilbert's hotel universe, infinite processes can be completed so there must be a definite final state of the lamp. If you can't provide the state of the lamp then it's worth questioning whether infinite processes can truly be completed in that universe. And if they can't even be completed in an imaginary universe, why would we think they can be completed in reality?
A finite being cannot exhibit or work with an infinite set directly. To do so requires an infinite being. Since one cannot disprove the existence of an infinite being (e.g. God) one cannot disprove the existence of infinite sets. But does the burden of proof lie on the atheist? Paradoxes are a good option because they demonstrate that actually infinite universes (in which infinite sets can exist) harbor contradictions, such as in Hilbert's Hotel Universe where infinite processes can and cannot be completed.
That "A Hilbert Hotel has a countably infinite subject" refers to a sentence of ZFC, and not an actual hotel.
The hotel is not finite. It has infinitely many rooms.
Quoting sime
ZFC doesn't distinguish among hotels, real or mythical.
And the point is not that there is a countably infinite subset. Every countably infinite set has a countably infinite subset.
Quoting sime
I don't see what the axiom of choice has to do with it. The rooms are enumerated by room numbers. Choice is not invoked.
Quoting sime
Hilbert's Hotel is an imaginary analogy to ('S' for the set of positive natural numbers):
For any natural number k>0, S is 1-1 with S\{1 2 ... k}.
For k = 1, as a new guest arrives, we move each already staying guest to the room above.
For k>1, as k number of new guests arrive all at once, we move the already staying guests up more than one room, as we move them up k number of floors.
/
For me, the problem is not so much that there is anything counter-intuitive about this, but rather that it's rude and bad business practice to keep waking guests up in the middle of the night and make them pack and move to another room, especially an infinite number of times. Not only that, but the poster keystone has added lamps that keep turning off and on, which is extremely annoying when people are trying to get a good night's rest for the next day when everybody is going out to see Zeno's 10K Charity Run where Achilles will have to run through an infinite number of distances and suffer the ignominy of getting beat by a turtle.
Nothing is wrong with this view except that it misses the point of the paradox, which isn't related to a literal physical dart (we have no idea ifphysical space is discrete or not; this is a debated metaphysical topic I won't get into) rather the very fact that anytime you have probability with infinitely many "contestants", whether it's dense space or whatever, you will necessarily either give the "contestants" a probability of 0 or be faced with adding up over 100% (since reiteratively summing any non-zero quantity indefinitely will approach over 100% at some point).
Your "solution" isn't a solution in that it doesn't talk about what the problem talks about. The "problem" is referring to continuity in dense contexts: it's not at all a "problem" in nondense contexts, this is equivalent to solving the Liar paradox by just saying "what if the guy doesn't lie?"
In any case, there are two mathematically respectable solutions to this 'paradox'. One is the philosophical thesis of saying that zero-probability is not the same as modal impossibility, which is so far the most widely accepted solution. The other solution is to introduce hyperreal numbers, particularly nilpotent infinitesimals, such that each contestant has probability ? but reiterative summation does not eventually yield anything over 100%, primarily because while ? isn't 0, ?+? can still equal 0. This approach is not as widespread.
A perpetually growing hotel that always has a finite number of rooms is still an infinite set, because there isn't a bijection between any finite set and the number of rooms in the hotel. But such a hotel isn't describable in ZF if the axiom of choice is assumed, because it forces Dedekind-infiniteness upon every infinite set.
As I recall, it's not a perpetually growing hotel. Rather, it's a hotel with denumerably many rooms and denumerably many guests, one to each room.
The "paradox" is not about potential infinity, but rather about the set of natural numbers (or any denumerable set along with a given enumeration of it).
Quoting sime
You have it backwards. That the set of rooms is Dedekind infinite is what makes the hotel "paradoxical".
Moreover, we don't need any choice axiom to prove that the set of natural numbers is Dedekind infinite.
Let there be a hotel with denumerably many rooms with room numbers 1, 2, 3 ...
Suppose there are denumerably many guests in rooms and that each room has a guest.
Suppose a new guest arrives. Then the hotel manager moves the already staying guests this way: If a guest is in room j, then that guest is moved to room j+1. And the new guest is put in room 1.
Or suppose k (k a natural number such that k>1) number of new guests arrive. Then, for all guests, if the guest has been in room j, then the guest moves to room j+k. And the k new guests are put in rooms 1 through k.
/
Indeed it is the failure of the pigenonhole principle for infinite sets (i.e. that infinite sets are Dedekind infinite) that allows the "paradox".
That's right. But we have to distinguish between the extensional concept of a number of hotel rooms that can be built, visited, observed, realized etc, versus the intensional concept of a countably infinite set of rooms. The latter refers not to a hotel, but to a piece of syntax representing an inductive definition of the natural numbers.
The paradox is due to conflating intension with extension. Keystone is right to raise objection.
Your analogy betwen mathematics and theology is not apt.
One can disprove 'there exists an infinite set' by stating axioms that disprove 'there exists and infinite set'. The obvious choice for such an axiom is 'there does not exist an infinite set'.
Anyway, I never asked you to disprove anything at all.
Quoting keystone
A contradiction is a certain kind of sentence. But, of course, there is no world in which a contradiction is true.*
* The context here is ordinary logic.
And there is no contradiction in set theory.*
* As far as we know.
We don't intend or claim that a domain of discourse for set theory is a world such as a physical world of physical particles and physical objects. At the beginning of this discussion, if asked, I would concede that immediately.
Now, if you wish to have a mathematical theory, adequate for science, that does have a domain of discourse of physical particles and physical objects, then no one is stopping you from saying what that mathematical theory is, or might be, or some idea of it. Saying, [paraphrase] "We'll keep set theory except infinite sets and use potential infinity instead" is suggestive of an idea, but not much more.
Quoting keystone
I know of no theorem of set theory that there are infinite processes that both can and not be completed.
You've described your notion of potential infinity a few times (in another thread especially). And I've replied about it each time. Now, you're coming back to restate it, but still not addressing the substance of my previous replies. As in another thread, this just brings us around full circle.
Quoting keystone
Same as above.
Quoting keystone
The thought experiment is suggestive of an analogy with set theory, but suggestiveness is not an argument about set theory itself. One sets up a thought experiment, then suggests an anology with set theory, as that analogy however is only informal and imaginary. Then I immediately concede that set theory doesn't explain informal, imaginary analogies to it. The terms of the thought experiment are not set theory.
Set theory doesn't have lamps that turn off and on infinitely but such that there's a final state in which the lamp is either off or on.
But set theory does have infinite sums if there is convergence. So, set theory does not say there is such a "final state" for a non-converging sequence of 0s and 1s. Set theory doesn't have a contradiction that there is a final state for such a sequence. That's something good about set theory.
But one can say, what about the fact that set theory has the finite ordinals, but then the least infinite ordinal that comes after all the finite ordinals? Yes, but no one says that there is "process" by which we go trough all the finite ordinals and then arrive at the least infinite ordinal.
Quoting keystone
We don't! Set theory doesn't say there's a "completion in reality". Set theory doesn't have that vocabulary.
There's no consideration of intensionality in the illustration.
Quoting sime
I have no idea what you think you're saying, and, after a number such exchanges with you, my bet is that you have no idea what you think you're saying.
Edit: On second thought, I do see your point: The uncountable co-idemponality topos decountably isomporphizes with the muti-hyperprimes to induce a Booleantype semi-constructive syntax field.
Yes, how could I have missed that?
You do recognise that Dedekind-infinite sets aren't extensionally meaningful, right?
So if one writes down an inductive definition of the natural numbers
1 + N <--> N
where <--> is defined to be an isomorphism, then to say N is "Dedekind-Infinite" means nothing more than to restate that definition.
Inappropriate extensional analogies for understanding dedekind-infinite sets , such as unimaginable and unobservable completed infinite sets of hotel rooms are going to appear paradoxical .
Definition of 'extensionally meaningful'?
And, if you respond with word salad, as you are wont to do, then I can't help you.
Quoting sime
That's not a definition of anything, let alone the set of natural numbers.
Quoting sime
'N is Dedekind infinite' means that there is a 1-1 correspondence between N and a proper subset of N. There's no need to drag isomorphism into it. The function {
And intensionality has nothing to do with this.
Your messages each time are as good as scribbled postcards from the Rabbit Hole.
The extensional meaning of a set are the items it refers to, in contrast to the definition of the set in terms of a formula, that is to say it's intentional meaning. Countably infinite sets cannot be given an extensional definition for obvious reasons, which is why finitists object to the reality of such objects, even if conceding that such 'sets' have instrumental use for generating numbers.
Quoting TonesInDeepFreeze
Its just short-hand for the inductive definition of the Naturals in terms of an F-algebra with respect to an arbitrary category in which 1 --> N represents '0' and N --> N represents the successor function that corresponds to the Dedekind infiniteness property.
Quoting TonesInDeepFreeze
That's a fair enough remark, given that only the right arrow is involved.
Quoting TonesInDeepFreeze
I never said that Choice was involved in the definition of dedekind infinity, i said that the presence of Choice causes all infinite sets in ZF to become Dedekind infinite by default, which is a major failing of ZFC in ruling out the only sort of "infinite" sets that have any pretence of physical realisability in the sense of extension.
If we don't agree on how we speak about the subject or our context for the way we speak about the subject, then we can't communicate.
My context is ordinary mathematical logic, which is the context of Hilbert himself. Though his Hotel predates the fuller context of mathematical logic, at least we're in the ballpark.
If you have a different context in mind, then one needs to know where to can reference it.
So, from my context, here goes:
Quoting sime
I don't look for sets to have a meaning. Word, symbols, sentences have meanings. Where do authors speak of sets having meaning?
Quoting sime
Sets don't refer to items. Sets have members. Where do authors speak of sets referring to items?
Quoting sime
Granted, authors often speak of 'defining sets'. But to be accurate, there are two senses of defintion:
(1) Syntactical. Defining a symbol.
Example
'N' is the symbol.
The definition of 'N' is
N = x <-> (x is an inductive successor set & x is a subset of every inductive successor set)
Sometimes, we have definitions such as this:
B = {0 5 8}
I take that is an example of what you mean by an 'extensional definition'.
(2) Semantical. For a given interpretation of a language, showing a formula using n free variables that is satisfied by certain n-tuples of members of the domain of the interpretation. (That's putting it roughly, though.)
Example (with first order PA)
With the standard model
x = x
defines the set of natural numbers.
Quoting sime
I didn't know you're using category theory.
Quoting sime
Of course, though I wouldn't state that way.
Quoting sime
It rules out that there are infinite sets that are not Dedekind infinite. But I don't know your definition of 'physical realizability'. Sets are not physical anyway (though, outside of set theory, we could have sets whose members are physical objects). Also, even without choice, we can't prove that there do exist infinite sets that are not Dedekind infinite.
There's an interesting question: In ZF we cannot prove 'There does not exist an infinite set that is not Dedekind infinite'. But do we ever have this situation?:
A definition of 'Q' and a proof of 'Q is infinite' but such that there is not also a proof of 'Q is Dedekind infinite'.
In other words, it's conceivable that even though we can't prove that every infinite set is Dedekind infinite, for any infinite set we define, we can prove that it is Dedekind infinite. Hmm, not sure, but I think that might be the case, since a definable set will be constructible (?) and the axiom of choice holds for constructible sets. (?)
(1) Your bit about choice didn't make sense to me earlier, but I might have been incorrect to mention it in connection with proving that N is Dedekind infinite.
(2) I said your remarks about N were not a definition. Perhaps they are in a context of category theory.
(3) Ordinarily I take 'intensionality' to be about intensional contexts such a modal, epistemic, etc. operators. Now that you've clarified your sense of 'intensionality' and 'extensionality', I don't dispute that, in that sense, the ordinary definitions regarding infinite sets are not extensional. But I don't like that terminology because it jams against the ordinary sense of 'extensionality' we use in set theory.
(4) Perhaps you're not down the Rabbit Hole. But I'm cautious not to fall into it.
/
On the other hand, only by extraordinary shifting of context would you not be incorrect in your description of the hotel as finite but always expanding. Hilbert meant an infinite hotel, not a finite one always expanding. If you want to set up a different story from Hilbert's then of course that is fine. But we shouldn't conflate them.
EDIT NOTE: I meant to include 'not' before 'down' in the comment about the Rabbit Hole. It's corrected now.
I believe that cuts made to a continuum are perfectly precise since I can draw it with no vagueness. For example, consider this drawing of y=0 and y=x^2-2:
There's no blurriness to my drawing. However, when I start to measure it (usually through calculus), my measurements may be imprecise. For example, if I try to measure the decimal expansion of the coordinates where they intersect I will generate a rational approximation which does not correspond to the exact point (however I can get arbitrarily close as long as my calculator has got the time and capacity).
Hopefully this explanation makes more clear my view of physics where the planck constant applies to a limitation of measurement, not of space/time itself. This is how I think space/time can be continuous but measurement discrete. And I like to think of our reality as something like a simulation by a computer which has finite time and capacity. The planck length and other limits of resolution are direct consequences of limitations of that finite computer.
Quoting apokrisis
I'm confused with where we've landed with this and I think I we need to step back. You said "I've already raised this point, asking if sqrt(2) is the exception or the rule." I don't see sqrt(2) as a rule. I see it as a description, perhaps of an infinite set. As I've discussed with TonesInDeepFreeze, I see no problem in there being a description of an infinite set, even if the infinite set doesn't actually exist.
Quoting apokrisis
I like my stereogram analogy more because it highlights that the observer plays an active role in actualizing the rabbit all the while doing no actual manipulation of the page (not actual cutting of the continuum). But I agree that it's not a perfect analogy because the creater of the stereogram encoded the rabbit in the page, whereas in mathematics it's a true blank slate.
There's an Australian mathematician, Norman Wildberger, on YouTube who doesn't accept infinities.
Here's a link to one of his videos.
Difficulties with real numbers as infinite decimals
https://www.youtube.com/watch?v=tXhtYsljEvY
You might try contacting him.
Quoting TonesInDeepFreeze
Ok, if we've only proved that the reals are an ordered field, then is it possible that we haven't proved that sqrt(2) is a number?
Quoting TonesInDeepFreeze
Not an axiom, but at the heart of set theory is the definition of equinumerosity of infinite sets which is not in concordance with MY notion based on finite sets. Let me try to explain using finite sets A and B and hopefully my use of set theory terminology is correct.
When I think of A being equinumerous to B, I think that there exists a bijection AND no injection between A and B.
When I think of A being more numerous than B, I think that there exists an injection from B to A AND none from A to B.
When I think of A being less numerous than B, I think that there exists an injections from A to B AND none from B to A.
If I apply this intuition to infinite sets A and B, then A is neither equi/more/less numerous to B. The only way that this is possible is if A and B are both the null set. In other words, my intuition based on finite sets leads me to believe that infinite sets are all empty.
I want to highlight again that I'm not saying that set theory is wrong, I'm just proposing that set theory might not be about actually infinite sets, but instead the potentially infinite algorithms that describe the infinite sets.
Quoting TonesInDeepFreeze
Interesting!
Quoting TonesInDeepFreeze
Yes, I use the word 'described' without explaining exactly what I mean but essentially I'm referring to the descriptions already used in set theory today to describe the contents of an infinite set. I'm not proposing that we reframe set theory to be based on potentially infinite sets as that would not be satisfactory. For example we can't describe a circle using an endlessly increasing list of points. That list will never suffice (proof Cantor). Instead, we describe a circle using a potentially infinite 'algorithm' for generating the actually infinite list. For example, the equation x^2+y^2=1 describes a circle in completion.
Maybe we both agree that our universe is like a simulation by a finite computer and as such our universe has some limitations in measurement, e.g. as it relates to Planck length. What I want to highlight is that our universe's computation is performed by one computer (I actually think it's more like a cluster of computers but let's leave that aside). Conversely with mathematics we're not constrained to a single computer. The math that I do on my pocket calculator might offer a different precision than the math that can be done on a supercomputer. We need our understanding of how computers participate in mathematics to be general and apply to all computers. As such, we cannot say that mathematics has limits of resolution, but instead that the computer does. And to keep things connected with reality, let's stick with finite computers having arbitrarily fine (but in all cases finite) precision.
Okay, I'll concede this point and agree that set theory is used to formalize many branches of mathematics...and I don't think it's going away. But as I said before, we never actually manipulate the infinite sets directly. We 'describe' infinite sets and work with the 'descriptions'.
I concede that there may be no better way to formalize calculus than set theory. I don't want to challenge the efficacy/validity of set theory. I only want to challenge the philosophy/interpretation of it. We don't actually work with the infinite sequence itself, we work with descriptions of the infinite sequence.
Yes, the conventional way to think of a circle is as an (actually) infinite set of points. What I'm proposing is that we think of the circle as the description (a finite string of characters) of those points. For example, the equation defining the circle. For example, consider this unorthodox drawing of a circle:
In this drawing there are no points. You certainly cannot identify a point and provide its coordinates. However, this drawing captures everything 'circular' about the green curve because it's associated with an equation. This equation is the description I've been referring to in my other posts. It describes, not the points the curve is made of but instead the potentially infinite points that would emerge IF I add more curves to the system. For example, let's add y=0:
In this case, two points emerge with this 'cut'. And because we know the equations for these two curves we can 'measure' the coordinates of these points.
With this approach we start with a continuum and progressively cut it. At any stage the objects are finite in number and the potential for additional cuts is never exhausted.
Conversely, if we start with points to build a continuum we're going to need infinity of them...and even then, I think such an approach is problematic and paradox filled.
I believe that calculus is more closely aligned with this parts-from-whole approach than it is with the conventional whole-from-parts approach.
If you're going to describe a continuum with numbers you will need infinite decimal places. But of course, it doesn't have to be decimal. Binary is more pure. But that doesn't reduce the number of digits required. You can't do much math with one bit.
How would you catalogue all continuous curves? That would be a starting "point". In order to have derivatives and integrals you would need some kind of function derivable from a catalogued example. Sorry, but the whole approach sounds absurd.
Quoting TonesInDeepFreeze
Nowadays it's an extremely popular rabbit hole, complete with patios, restaurants and theaters. Doesn't do anything for me. Both Category theory and K-theory arose in the mid 20th century. I've mentioned before that one of my instructors from a half century ago was at a talk about the latter when the lecturer started entertaining comments and questions from the audience, became so annoyed with the flack he was receiving, he practically yelled, "Just believe me and I can prove anything!!"
Well, not in mathematical analysis.
Ultimately I want a mathematics that is formally supported by axioms and my preference is to keep what we've got (ZFC) because I'm guessing that so much is based on ZFC because it works so well. Also, I don't have the skill to craft new axioms, ha!
But here's my beef.
With geometry (specifically drawings) we can do things that cannot be capture with algebra. For example, I can draw a perfect finite picture showing the intersection of y=x^2-2 and y=0 even though I couldn't do the algebra to tell you the infinite decimal coordinates of the intersection.
With 'algebra' (specifically the manipulation of equations through calculus) we can do things that cannot be captured with geometry. For example, I can perform a finite set of operations (derivative) to easily calculate the slope at a point but I could never actually draw the infinite steps as described by the limit definition.
However, we want there to be an equivalence between algebra and geometry and so infinity leaks in. If instead we see algebra and geometry to be complementary (instead of equivalent), then we don't need infinity. In geometry we would limit ourselves to what is finitely possible. In algebra we would do the same. And sometimes we would need to switch back and forth between the two, similar to how in physics we switch back and forth between complementary wave and particle descriptions of light.
But to accomplish the above, we need to rethink how geometry (specifically how we draw things) and set theory (specifically whether set theory is about points or algorithms). But as I've said before, I don't think this changes much math...I think we've been doing it right all along. I just think what we say doesn't correspond to what we do.
We say that there are infinite numbers of points on a graph when we actually only explicitly draw a finite number. We say that sets are about points when we actually only work with 'algorithms'/descriptions.
As this conversation progresses, I've warmed up to set theory (although I currently hold an unorthodox view that set theory might not actually be about sets).
LoL.
What I'm proposing is that there is no "contest" involving infinitely many "contestants". For example, I'm proposing that to do calculus we don't need to assume that a continuum is built from the assembly of infinitely many points. Can you provide the simplest possible example in calculus where we need to assume that there are infinitely many points?
This is Russell's argument. This photograph of Keystone's face might be blurred, but Keystone's face itself is not. Therefore vagueness is merely epistemic and not ontic. It exists in our representation of reality and not reality itself.
But the reverse argument also applies. The representation can be sharper than what it represents. The right facial recognition algorithm could separate a dim CCTV image of Keystone in a hoody from all the other faces stored in a police data bank. Signal processing can extract structural information that stands behind any amount of confusing surface detail.
So sure, one can eliminate vagueness by using well-defined algorithms like your equations. They are linear and not non-linear after all. Errors in measuring initial conditions don't matter as the uncertainty only grows in polynomial time and not exponentially.
But how wide are your lines even mathematically? How sure are you they are single lines and not a small bundle of lines sharing a neighbourhood with infinitesimal spacing? And when does this vagueness start to matter? Doesn't it matter if your rigorous mathematical edifice must also fit a physical world were nonlinearity is in fact the generic condition?
So having signal processing to sharpen up your view of an uncertain world is great. Really useful. We can see why maths is "unreasonably effective" in that regard.
But that doesn't engage with the foundational issue of whether reality itself is vague or crisp at base. And hence what kind of ontology we are correct to import into our "picturing" of math's epistemology.
I see your point. In reality there is one truth (e.g. God either exists or not). I have this romantic/naive notion that in Mathematics there is similarly one truth and one ultimate set of axioms that captures it. But as it is today that is not the case with mathematics. So we may both be right when I say infinite sets don't exist and you say that they do because maybe we're starting with different axioms. Point granted.
Quoting TonesInDeepFreeze
In Set Theory we say 'There exists a set...'. What do we mean by this? I take it to mean that the set must literally exist somewhere and my understanding is that the mainstream view is that it exists in the Platonic Realm. This is the world I want to explore. While Hilbert's Hotel cannot exist in our world, it should have no problems existing in the Platonic Realm which is infinite. Same goes for Thompson's Lamp. And what I'm trying to get at is that in the Platonic Realm, where infinite operations are completed (i.e. each guest flicks the light switch) something breaks because Thomson's Lamp can't decide whether it should be on or off after all switches are flicked. Or in other words, if an inhabitant of the Platonic realm is computing a geometric series and Grandi's series simultaneously term by term, then they both must either sum to a number or not. We are not allowed to say that one does and the other does not. That would be a contradiction that would cause the Platonic Realm to explode. I want to argue that the Platonic Realm necessarily explodes leaving no place for infinite sets to exist.
Quoting TonesInDeepFreeze
I apologize if I'm not addressing the substance of your previous replies. It's not intentional, I thought I was.
Quoting TonesInDeepFreeze
As I mentioned in my other post, it's not just about analogies - it's about the Platonic Realm. It's the only place that can harbor infinite sets. It's also the only place that can harbor the thought experiments so they're inseparably linked.
Quoting TonesInDeepFreeze
In the Platonic Realm, can infinite objects exist but never be constructed? My view of the Platonic Realm is that it's a world where infinite processes can be completed, where someone can add 1 + 0.5 + 0.25 + ... term by term and in a finite amount of time complete the calculation to yield exactly 2.
Thanks. I have admired him for many years. I loved his videos on real numbers. I've reached out to him a couple of times but didn't get much of a response aside from a dismissive response when I posted a comment on one of his videos. I think he's a canary in the coalmine but his proposed solutions don't target what I believe is the heart of the issue - that points cannot be used to construct a continuum. He's also way too much of a ultra-finitist for me. I believe potential infinite is essential in mathematics.
Can you explain what you mean by 'catalogue all continuous curves'?
I just now watched this video by Wildberger:
https://www.youtube.com/watch?v=U75S_ZvnWNk
In that video, he's an intellectually disorganized, sneaky, weaselly lying sophist and a fool. He's an insult to intelligence.
First, the title of the video is "Modern "Set Theory" - is it a religious belief system?" Yet he mentions nothing about that, let alone supporting it. Indeed, it is unsupportable.
Religions involve some combination of (1) a god, gods, deities, angels, demons and other non-physical beings - usually with personalities and utterances attributed to them, and acting causally, and usually miraculously, on the physical world, (2) explanations of the universe and creation, (3) human souls and accounts of what happens to those souls after death, (4) mythic narratives about human history, including religious founders and prophets, (4) divinely mandated moral codes, (5) prayers, chants and incantations that have power upon the physical world, (6) teleologies and (7) eschatologies. And I confined just to aspects of belief, since Wildberger didn't say that set theory is also a religious practice.
So, with that provocative title, coyly cast as an interrogative, what the sneaky sophist does is wet the water slide for thinking that set theory is religious while he does nothing toward engaging the very meme he's disseminating.
/
In the opening he says he's going to "dismantle" "what we currently are doing now".
Surprise: he doesn't.
He says he's going to "examine the details of what we're currently doing". But instead of "details", all we get in the video are atrocious oversimplifications, and not a single one of those articulates an actual problem with set theory.
Then he says the video presents a fair case that mathematics can have a solid foundation. But it turns out his "fair case" is just to wave his magic wand (actually he uses an old drum stick) and say that the natural numbers are the foundation for all the other branches of mathematics. But he says not a single word showing how that would be done except for a chart with 'the natural numbers' as the base of the pyramid of mathematical subjects. As if doing that carries even a mote of an argument. I'm going to make a chart with avocados at the base of a pyramid of the hierarchy of the U.S. federal government. And, presto, the avocado on my table is now the president, head of state and supreme commander of the U.S. armed forces.
Then he states his gravamen: "Set theory is a logically inadequate foundation for mathematics". But in the entire video he does not cite a single actual problem with the logic of set theory. His only criticism is that it's axiomatic. Yeah, set theory tells you what its axioms are - so you can take them or reject them - and then shows you exactly, step by algorithmically checkable step, the derivations of the theorems. So what would Wildberger's own foundation be if not to present axioms? He gives not a clue, except that pyramid chart.
* He is creative with that drum stick. He uses it not just as a pointer, but he holds it up to illustrate the real number line. And at other times he uses it like a magic he waves to declare that he's shown some supposed problem with set theory, analysis, and mathematics. Like Trump magically declassifying documents.
/
He says about the natural numbers, "as numbers become big, arithmetical problems arise". Okay, what is one them? He mentions no problems. This is from the guy who had a few minutes ago promised to get detailed in the video.
/
He says the problem with the continuum is how do we model what happens when we look closely at it. Well, that's merely a vague question. What do you mean by "model"? What do you mean by "happens"? What do you mean by "look closely at it"? He then asks, "What is the mathematical way of magniying and subdividing and looking more clearly and carefully at smaller and smaller subdivisions?" Um, how about doing that with proofs of theorems from axioms?
He says, "This is hugely problematic". Yes, Norman, your vagueness and lack of the details you promised is kind of a problem."
He mentions the irrationals, with an actual smirk on his face. Argument by facial expression. (The video is full of those kinds of smug faces, while his words are saying, at best, nothing really, if not, at worst, outright lying.) He says we are faced with how to set up an arithmetic for irrational numbers, and "This is hugely problematic. It's hugely problematic." (Like when Trump* says, "It's a terrible thing. A terrible thing", as we're supposed to be convinced by the mere fact that he says it, and twice.) Wildberger simply skips that mathematics does, from axioms, define the operations on the real numbers with proofs about them.
He says that when mathematicians did lay out foundations for the reals, "[This] is really where the difficulties lie. It's really where the difficluties lie." - Norman "Double Assertion" Wildberger.
* No political comparison is intended between Wildberger and Trump. Just that they're both crackpots and liars.
/
Here's the first lie:
He says that ZFC "was not really a framework based on precise defintions and clear theorems".
ZFC is nothing but utterly rigorous definitions and utterly rigorous proofs of theorems. One may reject the axioms, but it is beyond dispute that the defintions are rigorous and the proofs rigorously derived from the axioms.
So that's Wildberger lie #1.
He says the problem is that set theory is axiomatic. Of course it is. Because it's with the axiomatic method that one states rigorous definitions and show rigorous proofs.
Wildberger is not just lying; he's also showing himself to be completely confused.
Then, "It was as if we had abandonded the effort to try to set up this foundational issue very carefully and clearly."
ZFC was exactly an effort to more carefully use set theory than Cantor did. Even if one rejects ZFC, it is beyond dispute that ZFC is a full formalization where Cantor's work was not.
So that's Wildberger lie #2.
He says the set theorists said, "Let's just assume that it works".
The set theory mathematicians didn't just assume that set theory provides a foundation for analysis, they proved that it does, by constructing a complete ordered field and the foundational theorems about it, from which analysis can take over from that point.
So that's Wildberger lie #3. The set theory mathematicians didn't just assume that set theory provides a foundation for analysis, they proved that it does, by constructing a complete ordered field and the starting theorems about it, from which analysis can take over from that point.
Then, "That's really what happened. 'Let's assume that the kind of things we want to be true really are true'.
That's really what did not happen.
So, reiteration of Wildberger lie #3.
Then, "[They say] 'We'll dress this up as if it's an axiomatic framework'".
Argument by characterization rather than substance. They didn't "dress it up" as an axiomatic framework. It is an axiomatic framework.
Then, "The job of framing this all was outsourced to logicians, which is really philosophy, a branch of philosophy that overlaps with mathematics".
He's insinuating that the set theory was not really rigorous because it was done by philosophers not mathematicians.
(1) Actually they were mostly mathematicians. Many of them were not just set theorists but worked in other fields of mathematics too. Some of the greatest names in mathematics were key in the development of the logic, set theory and type theory: Hilbert, Godel, von Neumann, Tarski, Whitehead, et. al (2) Set theory and mathematical logic are mathematics. (3) Many were all three - mathematicians, logicians and philosophers. But mathematical logic is astoundingly rigorous; it's all about rigor
Then, "Its not too much of an exaggeration to say mathematicians outsourced the foundations of their subject to the philosophers".
First "the philosophers", as if there are two distinct sides - the mathematicians and the philosophers. Second, yes many of the logicians were both mathematicians and philosophers. But again, Zermelo, Fraenkel, Bernays, Godel, et. al, where mathematicians no matter what else they were or weren't.
So it's not too much of an exaggeration to say that Wildberger is lying on this point too, and to call it Wildberger lie #4.
Then, "There's kind of an agreement not to examine closely what this logical foundational of set theory is".
You have got to be kidding me! Set theory and its logic have been examined, discussed, critiqued, reworked and reworked until the cows home. In literally library stack upon library stack upon library stacks of books and journals, examining set theory from every angle it can be examined from. I guess Wildberger doesn't have access to even a single library, not even at the university where he teaches.
So that's Wildberger lie #5.
Then he wiggles his hand as he calls set theory, "crazy things".
Argument not just by hand waving, but hand wiggling too!
/
That's just some of it, in just one nineteen minute video.
You seem to be well read in math, philosophy, and science. Out of curiosity, what are you trained in?
Quoting apokrisis
I disagree. While an algorithm can indeed tune an image to make it better suited for the intended audience (e.g. colorize by infrared for those who are unable to see in the infrared spectrum), it cannot add these colors if the infrared is not actually there in the first place. No computer processing can make the image higher resolution than reality. Or more mathematically, if I tell you that I'm thinking of a number that begins with 3.14159 there is no way that a computer can increase the precision of this number to tell me the next digit.
Quoting apokrisis
For a fully accurate depiction, the line width should be exactly 0. I can't accomplish that with a drawing. Perhaps I should have cut a paper instead, but that wouldn't work perfectly either as the paper would fold and bend and cast shadows and leave small gaps. Even better, I could have made a stereogram whereby each area pops out at a different height. But alas, I am lazy so you have to forgive that my lines have some width when in reality they should not.
Quoting apokrisis
About as sure as I am that there is no teapot orbiting between the Earth and Mars revolving about the sun in an elliptical orbit. Give me a reason to believe otherwise. I don't believe in infinitesimals either so you'll need to also give me a reason to believe in them too. The figure (Apokrisis' face) is exact and so is the measurement (the photo of your face), it's just that they're not equivalent. Do you consider this mismatch to be vagueness?
Quoting apokrisis
How does my view break down with nonlinearity?
Quoting apokrisis
Perhaps we need to figure out whether the objects of mathematics are vague or crisp at base. This might help us guide our investigations into reality. This is why I think it's so important to have a sound philosophy for mathematics.
I agree that this is the biggest weakness of his view. He complains about the foundations of math but his foundation is set of tick marks on a white board.
Quoting keystone
We proved they are a complete ordered field.
I answered that in exact detail in a previous post. Pretty much, you're asking for notes from the first week of Calculus 1.
Quoting keystone
All bijections are injections. So you're confused to begin with.
What you mean is that you take equinumerous to mean: A and B are equinumerous iff there is bijection between A and B and no injection from one into a proper subset of the other.
Of course, in ZFC that does hold for finite sets but not for infinite sets.
B is more numerous than A iff there is an injection from A into B and no bijection between them. Same in set theory.
B is less numerous than A iff A is more numerous than B. Same thing with set theory.
You're wasting our time. We already know that in set theory, infinite sets differ in this salient way from finite sets. Galileo's paradox, Hilbert's hotel, Dedekind infinitude, and this latest point about equinumerosity are all variations on the same point: infinite sets map 1-1 with proper subsets of themselves. You don't need to keep giving examples. We already agree that infinite sets map 1-1 with proper subsets of themselves. For you, it's counterintuitive and you won't accept it. Fine. I have no motivation to convince you otherwise.
Quoting keystone
'Infinite sets are empty' is a contradiction. And set theory does not have that contradiction. So if it's your intuition, then set theory is not for you.
Quoting keystone
I know. You've said that a dozen times.
Day one of high school Algebra 1:
"Students, we start with the set of real numbers and the real number line."
And you don't have to tell me yet again your finitistic interpretation of that or how we can work with only a finite number of those points, etc. I know that's your view.
When math starts with points that are then assembled into curves, there is a way of describing those points on the real line, identifying a point with .5 for example. If you are starting out with curves or geometric figures you need to be classify them, order them somehow, for you then wish to create points by intersections I suppose. You need rigorous definitions for curves, then an axiomatic structure. All of which seems far-fetched. But who knows?
The reason old-fashioned terminology is not so bad. One-to-one onto, etc. Bourbaki may be to blame? Just nit-picking, ignore me.
Depends on what you mean by 'manipulate'.
Set theory, most strictly, is a certain set of formulas. In mathematical logic, when we say 'ZFC' we are referring exactly to the set of formulas.
Of course, those formulas are "read off", or "rendered" in, say, English as if they are English sentences. But they are not English sentences. Also, most people do have in mind that the formulas pretty much "say" what the English renderings are. However, again strictly speaking, the formal meanings are given by the method of models.
In that context, its hard to say what 'manipulate' means. For myself, I do recognize that I manipulate symbol strings. But I don't at all think that somehow, like a puppeteer, I'm manipulating mathematical objects. Rather, I am writing formulas as lines in proofs. I do intuitively think of those formulas as saying something about whatever objects are in the domain of discourse, but I am not manipulating those objects themselves. Rather, I am writing formulas about them.
But what about having not just a pre-philosophical intuition about what set theory "says" but a truly articulated philosophical position about it? Realist, nominalist, structuralist, fictionalist, consequentialist, instrumentalist...? That's for each mathematician to decide for herself or himself. Or not decide. There's no law of the universe that one can't just chug along proving theorems without declaring a philosophical position. Of course, then one can't provide a philosophical justification for set theory. But then, of course, one can say, "I don't need to provide a philosophical justification. The question of a philosophical justification is a fine one indeed, and I like peeking in on the debates now and then. But no matter how the arguments turn out, I'll still enjoy set theory and I'll correctly point out that it is a recursively axiomatized theory that proves the theorems used for the mathematics of the sciences, and such that, at least in principle, all the proofs can be written in complete formality so that their correctness can be algorithmically verified".
Cool. The formulas can be about whatever you want them to be. Hilbert's tables, chairs and beer mugs.
I don't know whether platonism and/or variations on platonism are the majority view among those who have a view, but I wouldn't bet against it.
Quoting keystone
In that realm, there is no final state for the lamp. Poof. Done. Still no contradiction.
Your fallacy is in setting up an imaginary world, with states-of-affairs like set theory, but then adding a state that doesn't exist in the set theory and thus rightfully as an analogue not in the imaginary world, so you get a contradiction but outside the original terms you set: an imaginary world corresponding to set theory.
Basically, you have your concepts of how the statements of set theory should be understood, so you posit a realm to embody those concepts. But your concepts are not analogues of any actual statements in set theory. So it's a cheat.
You require that sets can be "built" only in finite "processes". Then, since it's all processes to you, when you see set theory posit objects not built with such processes, you incorrectly saddle set theory as claiming that certain states are achieved only after an infinite process. Your arguments about the realms then are based on conflating your concept with set theory.
By the way, are you capitalizing 'Platonic Realm' as a proper noun to shade discussion, even if just by hair, about mathematical realism?
Quoting keystone
If you reread the posts (and in the other thread) then you'll see exactly where.
Quoting keystone
Set theory has no theorems about mathematical agents constructing or not constructing things. So you can't plop them into the middle of one of these realms you're making and have it be about set theory. (For an approach to mathematics that does have something like agents constructing things, see intuitionism.)
Quoting keystone
Fine. It's not a realm of set theory. You don't know anything about set theory. But you keep burdening it with what you misunderstand it to be. You imagine a realm that has some similarities with set theory, but also has things not corresponding with set theory. Then you blame set theory.
Basically your realms are like models. But a model of a theory is one in which the vocabulary of the theory is interpreted in the model and every theorem of the theory is true in the model. But you're adding things that are not the interpretation of anything in set theory and even worse, posting states-of-affairs about them don't model the theorems of set theory. Sorry, no go.
Quoting keystone
Let your '...' be "such that P".
For a given model M,
ExPx
is true in M iff there is at least one member of the domain such that that member of the domain is also in the subset of the domain that is the denotation of 'P'.
That means, in terms of your "realms" (whatever their ontological or metaphysical character), there is at least one object in that realm that has the "property" denoted by 'P'.
It's a non-converging sequence.
Set theory doesn't have a "final state" with that.
But here's what set theory does have:
Let N = the set of natural numbers.
Let f be a function.
Let dom(f) = N
Let for all n in dom(f), f(n) = 1/(2^n)
So f(0) = 1, f(1) = 1/2, f(2) = 1/4 ...
0 is not in ran(f).
Let g be a function.
Let dom(g) = ran(f)
Let ran(g) = {"off", "on"}
Let for all r in dom(g), g(r) = "off" iff En(r = f(n) & n is even)
So g(1) = "off", g(1/2) = "on", g(1/4) = "off" ...
So I've mathematically "translated" Thompson's lamp.
What about the "final state"? There is no final state. The mathematics doesn't have a "final state" here, exactly because the mathematics doesn't have nonsense like "an infinite process with a final state". Of course you get a paradox from having an infinite process with a final state. It is the very point that mathematics is not capable of such nonsense.
But this:
Let (h) = g u {<0 "off">}
So the "final state" of h is "off". No contradiction.
Let j = g u {<0 "on">}
So the "final state" for j is "on". No contradiction.
Choose whichever "final state" you like - the "final state" with h or the "final state" with j. But neither is determined by g.
NOTE: I am not claiming that there aren't mathematical treatments of supertasks that, with advanced definitions and construction, formalize notions of "infinite number of steps in finite time" or "final states for infinite processes". But if the treatments are formalized with ZFC, then it can't be the case that they contradict ZFC. I'm not expert in that matter, but I would put my proverbial money on it.
Your proposal is finitism. It's a cool proposal and an interesting philosophical topic on its own right, but entirely unrelated to the paradox you were trying to solve (i.e. irrelevant).
That's a classic '50s movie, 'Blame It On Bourbaki'. Cary Grant and, if I recall, Anita Ekberg. Grant's character is trying to get Ekberg's character to take some experimental injections for her terminal illness, but she doesn't want the new experimental drug. Her famous line is, "No, sir, jections!"
My focus is systems science. Which means all of the above really. But neuroscience in particular.
Quoting keystone
The point was that the processing removes vagueness - the unlimited number of shades of grey - by imposing a binary, black and white, constraint on the image. It boils faces down to a grid of points representing exact distances between the most informative features.
But if you just want to resist the concept of vagueness, thats your lookout. I can only say it was about the single most paradigm shifting thing I ever learnt.
Quoting keystone
LOL. Just as the cuts in the line should be exactly 0 length. Your arguments here seem all over the place.
How do you glue actually 0D points together to make a continuous line? How do you glue 0D width lines together to make a plane?
The issue to be resolved is how divisibility can co-exist with the continuum that it divides. That is where my point about the discrete and the continuous being a dichotomy that emerges from a logical vagueness comes in. It justifies treating both the cutting and the gluing as complementary limits on actuality. The infinite and the infinitesimal are two ends of the one spectrum of possibility.
But I cant see what your answer is at all.
You've repeated this a lot so it's clearly important. Maybe my view is in disagreement with Set Theory then. In any case, I'm not in a position to challenge Set Theory directly. I'll continue this in a response to another one of your posts.
Quoting TonesInDeepFreeze
How is that a contradiction?
Quoting TonesInDeepFreeze
I know that we are taught that the real line is composed of infinite points. What I'm asking for is you to provide an example where we need to assume that the real line is composed of infinite points. Saying we do things a certain way is not evidence that we need to do it that way.
Why do we need to order the curves?
Quoting TonesInDeepFreeze
I don't think you're being reasonable here. You say that many of the paradoxes are resolved by Set Theory (essentially it is you who is making the link between the two) but then you say that criticisms of the paradox can't touch Set Theory. You can't have it both ways.
In Zeno's Paradox Achilles only travels the distance after completing an infinite process. For example, if Set Theory has nothing to say about completed infinite processes then it cannot be used address Zeno's Paradox.
Quoting TonesInDeepFreeze
Maybe just to make it clear that we're talking about an actual place.
In Achilles' journey he arrives at the destination. He takes a final step. Does Set Theory model Achilles' journey or not?
Quoting TonesInDeepFreeze
If mathematics does not allow for such nonsense then it cannot claim to resolve Zeno's paradox.
I think I've said it about two or three times. It is important because it is the very heart of the matter, which is that set theory axiomatizes the mathematics of the real numbers. But, for some reason, I don't see the passage in the post you linked to. If you look at where I said it, then you'll see how it was a response to a comment or question of yours.
Quoting keystone
I'm sorry, but are you serious?
Quoting keystone [bold original]
I've answered that before. (In this thread or another one.)
Here's another answer: Because if it had only finitely many points, we wouldn't be able talk about points greater than any of the positive points you chose or less than any of the negative points you chose. And we wouldn't be able to talk about those that are in between any two adjacent points.
If you're the teacher and you tell the students there are only these certain points, then the student comes to class the next day and says, "I wanted to use mathematics to figure out how to build a table, but my mom she needs an overhang between 1/8 inch and 1/16 inch, but there is no number between 1/8 and 1/16 among the numbers you'll allow me to use; it's not in your list of all the points."
Infinitely many points are needed so that we can speak in greatest generality, no matter what degree of tolerance you might think is the greatest degree needed.
The infinitude of the set of real numbers is a consequence of the definition of 'real number'. Set theory axiomatizes that courtesy of the axiom of infinity. If we took out the axiom of infinity, then we wouldn't have the set theoretical construction of the complete ordered field that is the system of the reals.
Think of it this way: I show you an airplane.
You say, "Why do you need that big plate at the bottom; couldn't you make an airplane without that kind of thing?"
I say, "I don't know whether you can, but it is needed for this airplane. And it's not that just that it has a specific purpose, but that if you took it away, then all the rest of the parts wouldn't work together; the whole thing is an interdependence. If you want to build an airplane without that kind of plate, then go ahead and do it."
If it doesn't make sense to think that something comes from nothing, maybe we need to revisit the belief that something comes from nothing.
If it doesn't make sense to think that continua come from points (i.e. dartboard paradox), maybe we need to revisit the belief that continua come from points.
You're not being reasonable. Do you want to inform and enlighten yourself about the subject, or you just want to raise objections premised in not knowing anything about it? Better to know the thing, then you could critique it fairly.
Quoting keystone
I said that, and it's okay. But in the last posts, I realized that the more pertinent point is the one I'm making now.
Quoting keystone
What are you talking about? You are mentioning the paradoxes to impugn set theory. Then I reply: (1) The paradoxes don't show a contradiction in set theory, (2) the set theoretical, mathematical methods don't end in contradiction, so, in that way they "solve" the paradoxes; not that the paradoxes dissolve on their own, but rather when we do actual mathematics instead, we don't incur contradiction, (3) the realms in the paradoxes are not models of set theory anyway.
Indeed, if no one mentioned a connection between the paradoxes and set theory, I'd have no objection at all!
Quoting keystone
You are completely missing the point. And now your lily pad jumping from Thompson's lamp to Zeno.
I gave you a rigorous answer regarding Thompson's lamp, but instead of comprehending that, you skip it as if it doesn't exist and frog hop to another pad.
You're not in good faith.
Quoting keystone
I answered about Zeno's paradox many posts ago (in this thread or another one).
How about having some attention span and look at my rigorous response to Thompson's lamp?
If you then just jump to another subject (and you're incorrect about it also) then I take it you are not in good faith.
Quoting TonesInDeepFreeze
No, because mathematics shows how to calculate that Achilles did finish the race.
You just keep showing that you don't care about understanding any of this but instead want to continue roving among already answered points as if they weren't already answered.
You have it backwards, and show that you didn't bother to read my previous post about modelling.
The realm of the paradox is not a model of the mathematics. Formally, the mathematics is not a model; it's a theory. Models are of theories, not the other way around.
With the mathematics, we have that certain functions are continuous. This allows that mathematics is modelled by such things as Achilles crossing the finish line. And we know that Achilles does cross the finish line. So the mathematics works. The paradox though is a model in which Achilles does not finish the race. So we're not so attracted to that model, and indeed thankful that it is not a model of the mathematics.
Can you see that? Can you be reasonable enough to see that that is reasonable.
If you want to start with curves (or continuous objects) in order to derive points you need a systematic way of talking about them. If you plan to assemble contours or curves by gluing together tiny straight lines, then you are doing nothing more than is done when one calculates the length of a contour in complex analysis.
But I guess what you are really interested in is potential infinity and its kin. So my comment is what difference does it make how you deal with that? We've had posters here who spend years working up what they consider astounding revelatory articles, only to be more or less ignored. They become so enraptured with their ideas they get caught in that spiral in which the more effort you exert the more you think your product is of value, losing their objectivity.
But if you enjoy your project as a personal challenge, have at it. I have certainly been in that boat!
Quoting TonesInDeepFreeze
If I write N = {1, 2, 3, ...} it seems that N has infinite elements. But appearances can be decieving. If someone proved that 1=2=3=... then N actually only contains one element.
Quoting TonesInDeepFreeze
I'm not proposing that the real line is composed of an unchanging, finite set of points. I'm proposing that the real line has infinite potential to 'give birth' to points as they are needed. You never did respond to my post where I drew a circle for you. That post depicts what I mean, but in any case, I think the following paragraph will as well.
If you're a teacher and you tell the students to measure a table, you wouldn't hand them a bunch of nothing (points) to make the measurements. You would hand them a ruler (continuum) which has perhaps tic marks every 1/8 inch. If the table lines up halfway in between the tic marks, then and only then do you add a new tic mark based on the average of the adjacent tic marks. And if matter weren't discrete, you would have the potential to endlessly add tic marks to the ruler. It's just that you never would complete the job. And in fact, if you somehow were able to completely populate the ruler with tic marks the ruler becomes useless. It's just one big black tic mark. In this example, The distinction between numbers is lost and the set of real numbers is no more useful than a set containing only one element.
In my view, the ruler comes before the tic marks. And because we can manipulate the ruler as we go, we can speak about it's potential in greatest generality no matter what degree of tolerance you might need. If we took out the axiom of infinity, then we would acknowledge that we could never produce a useless ruler. We could never cut a string to the point where it vanishes to nothingness.
Here's my airplane analogy. Smart people build a well engineered airplane but before they launch it they go through a non-scientific ritual of blessing the plane to ensure it will fly. I'm acknowledging that they're good engineers and they've built a good plane but is that ritual really necessary? Is an actually infinity of points really necessary?
Quoting TonesInDeepFreeze
I knew I'd make a mistake with those terms, but at least you knew what I was trying to say.
Quoting TonesInDeepFreeze
The way I would phrase it is that we already know that in set theory, infinite sets don't conform at all to the intuitions we've developed from all sets that we've actually worked with direction.
Quoting TonesInDeepFreeze
It reminds me of the 'shut up and calculate' phrase that sometimes is said when people try to interpret the meaning of Quantum Mechanics. I agree that we can shut up but I think philosophy is important. I believe that with a proper philosophy of mathematics we can refine our intuitions and apply them in our quest to understand our universe. I also believe that paradoxes are the most important guidepost in our quest to see truth.
Quoting TonesInDeepFreeze
I believe all worlds (physical and abstract) are simulated by computers. Not necessarily a computer made of silicon, just an entity that computes. One might say that 'God is a mathematician'. I'm not trying to squeeze set theory into a specific world, I'm just trying to imagine a computer with enough capacity to hold it. If you say that set theory is a finite set of rules, axioms, properties, etc. from which theorems could be derived then Set Theory could exist on a finite computer and all is good.
However, if you say that set theory describes the behavior and asserts the existence of another computer which directly works with the sets themselves then such a computer would necessarily be infinite in capacity (and infinite in speed as well to complete infinite operations in finite time).
I'm willing to begin by assuming that such a computer exists, and then explore whether it would explode. I do not have the skill to prove that the computer that holds set theory will explode. However, I think I have the skill to at least discuss whether the computer that holds the infinity paradoxes will explode.
Quoting TonesInDeepFreeze
I'm willing to explore computers that perform supertasks, like one that simulates Zeno's Paradox where he performs infinite steps in finite time. And IF that computer explodes and IF I can form a 1-to-1 correspondence between the processes on that computer and the processes on the computer that holds the infinite sets then it's reasonable to say that that computer also explodes. If you think I haven't proven this yet...well that is true!
Quoting TonesInDeepFreeze
When I say something exists I mean there is a computer where it is in memory. Today at work I simulated fluid flowing in a mixing tank so during the simulation that virtual mixing tank existed, even though there was no physical counterpart. I just can't envision any computer holding even just the natural numbers without exploding.
So when you write "For a given model M, ExPx is true in M iff there is at least one member of the domain such that that member of the domain is also in the subset of the domain that is the denotation of 'P'", are you assuming the existence of an infinite computer?
I mean what a computer does, either to the virtual objects or the bits themselves, when it computes.
Quoting TonesInDeepFreeze
Let's start with my main claim - Nothingness (i.e. points) cannot be assembled to form a something (i.e. a continuum), no matter how much of it you have. Even infinite nothingness is still nothingess.
Quoting TonesInDeepFreeze
I honestly thought you were covering both Thompson's Lamp and Zeno's Paradox simultaneously since you included the geometric series in your description. I took it that you were giving an algorithm on how to construct the following table:
Step #,incremental distance, current state of lamp
0, 1, on
1, 1/2, off
2, 1/4, on
3, 1/8, off
etc.
I would like to add an additional column to the table to represent the total distance travelled.
Step #, incremental distance, current state of lamp, dist. travelled
0, 1, on, 1
1, 1/2, off, 1+1/2
2, 1/4, on, 1+1/2+1/4
3, 1/8, off, 1+1/2+1/4+1/8
etc.
The pressing question is whether there is a last row to this table - one might call it row aleph_0. On one hand, we say that there is no last row because we want to say that there is no final state of the lamp. However, on the other hand, we imply (but don't explicitly state) that there is a last row because Achilles completes the journey and the total distance travelled equals exactly 2. There is no finite Step # for which the distance travelled is exactly 2.
Quoting TonesInDeepFreeze
I argue that any valid demonstration uses a parts-from-whole (points-from-continuum) construction.
Quoting TonesInDeepFreeze
Aha! Your answer rests upon continua, not points! The mathematics works because you're considering the journey as a whole. The complete journey exists first and only then can we choose to talk about what happens at t=0.5 or t=0.95 or any other instant. But we cannot talk about the journey starting from t=0 proceeding to the adjacent instant.....because there is no adjacent instant. Atalanta cannot even begin her journey.
Quoting jgill
Agreed. But you've got to start somewhere and fortunately it's easy to understand what I'm talking about, even though I'm talking informally.
Quoting jgill
No, I'm going the other way around. I'm not proposing a whole-from-part (curve from infinitesimal line segments) construction but a part-from-whole (smaller curves from larger curves) construction.
Quoting jgill
In one sense, physics would be unaffected by a new philosophy of mathematics because applied mathematics doesn't deal with actual infinities. However, in another sense a new philosophy of mathematics might yield deep insight into the philosophy of physics (in particular, Quantum Mechanics). Philosophy has a powerful way of shaping our worldview.
Quoting jgill
No matter how good I think some of my ideas are, they are informal and several years old and over the years I've shed all my great expectations. At this point, I'm just here to talk about it.
Quoting apokrisis
Very cool.
Quoting apokrisis
I'll keep it in my back pocket in case it will be useful to me in the future :D
Quoting apokrisis
In my view 0D points are not objects so you don't glue them together. You don't make a curve from points. You don't make a surface from curves. You don't build the whole from the parts. In my view you go the other way around. You start with the whole and cut it up. Consider how we draw a line. We don't use pointillism until a line emerges. We draw a line and then put some tic marks on it to identify points. I don't see what part of my explanation you're not getting.
Quoting apokrisis
What exactly do you mean by this?
My systems view says you have to go still further into true holism. You are simply replacing one kind of constructive action - gluing - with an opposite one, that is cutting.
The systems view instead opposes construction to the other thing of constraints. And even here, both construction and constraints arise out of a common unity in vagueness. So the ontology is fundamentally complex. And hence not widely understood by folk.
Anyway, what this means is that my view you do indeed start with the whole that you mean to divide into its constituent parts. So rather than constructing the line from a set of points, I would talk about constraining the line to arrive at the limit where it would become indistinguishable from a point. The interval would be made so small that the length of the line was the same size as the width of the line - both being now effectively zero.
You dont come at it top down by chopping up a whole line - an infinite number of constructing actions. You come at it top down by increasing the degree of the limitation. Length gets shrunk down to the point that it is no longer possible to be certain about describing it as indeed a length.
A line in turn would be arrived at as the constraint on the quality of plane-ness. Squish the 2D plane from either side and the limit of its compression becomes how a 1D line arises.
This sets up the basic dichotomy of global constraint vs local construction. And whereas a simple metaphysics, when faced with a dichotomy, demands that you now back one side or the other, a systems metaphysics says the two instead make for a unity of opposites. Each entails the other. Reality is what emerges in a triadic fashion because there is now a world in which top-down constraints enable the existence of local acts of construction - be they cuttings or glueings - and, reciprocally, these local acts of construction generally tend to reconstruct the system of constraints that were shaping them in the first place.
It is a synergistic metaphysics. Constraints define freedoms (as that which aint constrained). And freedoms construct constraints, on the whole, as that ensures the continued existence of a world that indeed has precisely those kinds of degrees of freedoms.
An electron is somehow the right kind of fundamental stuff for the laws of the universe to produce. They will keep being produced for as long as they prove a productive way to maintain a universe that has exactly those kinds of laws.
But good luck applying this kind of advanced systems logic to the simplicities of number theory. Peirce did try, but only got as far as a reasonable sketch. He never published the logic of vagueness he was working towards. We have only the notes.
You answered the question. You're not serious.
Quoting keystone
Quoting keystone
You asked me about finitely many points, not about potentially infinitely many points. Be clear.
Quoting keystone
I bet it was yet another way for you to say that you like the idea of potential infinity. No, I haven't responded to you at all on that. I mean, the dozens and dozens of my posts now on display in at least two threads don't exist.
Quoting keystone
You didn't have to waste your time typing that. I know your notion of potential infinity.
And you egregiously obfuscate the terminology. 1/8 increments is not a continuum. You could at least give the consideration of not appropriating terminology in a blatantly incorrect way.
Quoting keystone
Another of your arguments by analogy. Mathematics doesn't cover rulers with ink. The existence of the set of real numbers doesn't stop you from considering only a finite number of numbers for a given problem. The rest of the infinitely many numbers are not there waiting to spill themselves like ink all over your favorite finite set.
This is the problem: In a context such as this, it's fine to deploy analogies to illustrate intuitions and things like that. But the argumentative force is limited, at best. I shouldn't indulge you more.
Quoting keystone
Right. Since you can't be bothered to see the point of my analogy*, you skip it and just jump to your own analogy. Barely read the posts, not taking moments to reflect on what's been said, to ingest, so the posts are just jumping off points for you to say yet again how you think mathematics should be. You've got this down to an art, if not a science.
By the way, my analogy was not offered as an argument but merely lagniappe for you to (hopefully) grasp an idea that is not your own for a change.
Quoting keystone
What on earth are you talking about? Sounds like a pitch for an opening scene of a movie or something. Who do you have in mind to direct? Spielberg? Soderbergh maybe?
The mathematicians don't just assume the theory will work. Rather, they prove that it does, by deriving the existence of the real number system, then proving the theorems of mathematics used by the sciences.
Again, we go around in circles, because you keep re-insisting on points that had long ago been answered. You are a sinkhole.
Quoting keystone
For heaven's sake! I've answered that and answered it and answered it already.
The answer is that the ordinary axiomatization of the mathematics for the sciences has an axiom that implies that there exist infinite sets. If we remove that axiom from the rest of the axioms, then we don't get analysis. Period. Final answer, Regis. Got it?
But that does not preclude that one can devise a different system that yields the theorems for mathematics for the sciences, with different axioms, and if needed a different logic, in which we don't have the theorem that there exist infinite sets. Got it?
So if you really are interested in a mathematics without infinite sets, then go look at the other systems already!
Some of your quote links are not going to the posts in which the quotes occur.
Quoting keystone
Where 'some' includes the most important ones.
Quoting keystone
"We". There are mathematicians and philosophers for whom set theory is intuitive. You don't get to declare for all "we".
Quoting keystone
Of course. And I said that mathmematicians and philosophers may choose among many philosophies. You left that out. Probably you didn't even take in that I said it.
Quoting keystone
I don't know that they are the most important subject, but they have been at the very heart of philosophy of mathematics. It's instructive then how mathematical thoeries are presented to avoid contradiction.
Quoting keystone
I have been pointing out that your main arguments are to try to make set theory fit models that are not models of set theory. And your response to that? Another writeup in which you view set theory per a model that is not a model of set theory!
Quoting keystone
I'm glad it's not my job to reconstruct verbiage like that so that it makes sense. In the meantime, why don't you learn something about computability theory?
Quoting keystone
Wow. You just cannot help yourself from continuing to ask me to make set theory fit your Prorcustean beds.
And you took not a millisecond to think about my own answer.
Points are not "nothingness".
Quoting keystone
I exactly stated it as regarding Thompson's lamp.
There is no geometric series in my writeup.
Quoting keystone
I said nothing whatsoever about "distance" or "travel". And I said nothing whatsoever about sums.
You're adding things into what I wrote that are not there.
I recommend that you read what I wrote without imposing your preconceptions about it.
Quoting keystone
Meanwhile I showed you math.
Quoting keystone
Aha! You're making stuff up again!
There is no notion of continuity mentioned whatsoever in my writeup.
Quoting keystone
There is no "journey" mentioned in my writeup. The writeup has nothing to do with "journeys".
I suggest that you clear your mind for just one moment and read my writeup free of incorrect preconceptions about it.
You have a lot of patience. The "vagueness" principle apokrisis advances might work better. It would parallel the actual history of math.
Quoting apokrisis
Quoting apokrisis
The sequence of contours of the form [math]{{Z}_{n}}(t)=t+i\tfrac{1}{\sqrt{n}}Sin\left( 2\pi tn\sqrt{n} \right)[/math] demonstrate convergence to both an infinitesimal straight line (on the x-axis) and the origin, 2D down to 1D down to a point. [math]t\to \tfrac{1}{\sqrt{n}}[/math] and [math]n\to \infty [/math] with [math]{{Z}_{n}}(t)\to 0[/math].
Oh oh, I almost forgot [math]\text{Length }{{Z}_{n}}(t)\to \infty [/math]. You can ignore that I suppose :cool:
"Behind it all is surely an idea so simple, so beautiful, that when we grasp it - in a decade, a century, or a millennium - we will all say to each other, how could it have been otherwise? How could we have been so stupid?"
John Archibald Wheeler
I'm just not feeling your theory. I don't see the point.
Oooh...I've been quoting wrong. Moving forward I'll do it the right way.
Its not my theory. And likewise, your feelings are irrelevant to the argument it makes. So whatever. :cool:
Of course, I'm not saying that the natural numbers are actually equal. I'm saying that natural numbers as defined in an inconsistent system can be easily proven to be equal. IF ZFC were inconsistent and IF someone proved that to be the case then wouldn't this be the celebrated conclusion? But IF that were the case, that doesn't mean that math is wrong, it just means that ZFC is not a good foundation for mathematics.
Quoting TonesInDeepFreeze
It's funny how you criticize me if I don't respond to some of your points but then you criticize me if you don't respond to some of my points.
Quoting TonesInDeepFreeze
I'm saying that the wooden stick upon which tic marks are placed is the continuum.
Quoting TonesInDeepFreeze
I don't think distinction between numbers (e.g. 1 and 2) can be made without accounting for the continuum that lies between them. The 1/8 tics on the ruler have purpose because in between them lies a space.
Quoting TonesInDeepFreeze
I'm referring to the objects of set theory being beyond our grasp.
Quoting TonesInDeepFreeze
I get what you're saying, but I don't agree with it.
Quoting TonesInDeepFreeze
Sometimes there's not enough room for two conflicting ideas.Quoting TonesInDeepFreeze
It occupies zero space.
Quoting TonesInDeepFreeze
I'm confused why you embedded a geometric series into the definition.
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@TonesInDeepFreeze: Earlier today I was seeing that you responded to my posts and I was looking forward to our continued conversation on this thread but based on your responses it's clear that this conversation has run its course. As you've mentioned multiple times, we're going in circles. We both think the other is not listening or being reasonable. In any case, I do appreciate that you gave me a lot of your time at composing well written responses. I sincerely thank you.
I do appreciate the discussion we've had though. Thanks!!!
You don't know what you're saying.
ZFC uses a method of definitions such that no contradictions can be introduced through definitions. ZFC could be inconsistent, but not because of any definitions. And if ZFC were inconsistent then still so would be the sentence "infinite sets are empty".
Quoting keystone
What in the world? I don't criticize you if I don't respond to some of your points.
Anyway, my point stands: Your circle bit is yet another variation on your theme. I've responded over and over to such variations, even if not to the circle in particular. I've done enough.
Quoting keystone
So you dispute the continuum by posting a continuum. I take it that you consider that you need the stick to put the marks on.
Quoting keystone
Wrong. Look up the math sometime.
Quoting keystone
You are very kind to your unresponsive dogmatism to just now omit the key point in my reply:
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
You don't get to disagree with it. It's a plain fact, no matter anyone's philosophy.
I'll explain it yet again for you: Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity).
Quoting TonesInDeepFreeze
Don't worry about two. There's not enough room in your mind for even one coherent idea.
Anyway, my point stands that it is poor thinking to always try to jam the sense of one framework into another one incompatible with the first.
Quoting TonesInDeepFreeze
Distance is between points. That doesn't make points "nothingness". It doesn't make them nothing, let alone nothingness.
Quoting TonesInDeepFreeze
Another doozy by you. I said that you add things into what I wrote, and you reply by adding it again!
There is no geometric series in my writeup about Thompson's lamp. Period. No geometric series. Look at it again, hopefully this time not hallucinating, and you will see that there is no geometric series there.
We both think that, but you're wrong about it and I'm right about it. I have mulled over your remarks a pretty fair amount. I have turned them around in mind, including from what I understand to be your point of view, and I have responded on point to them as exhaustively as feasible for me. And I do have at least a little exposure to finitistic approaches and systems, and an interest in learning more about them.
You, on the other hand, just slide across what I say, not even taking a moment to understand, and instead getting it quite wrong, quite confused, and, at key points, essentially putting words in my mouth. And without a mote of intellectual curiosity to learn even the very first things about set theory.
And I don't insist on putting finitistic alternatives into the framework of set theory. But the core of your arguments, repeated over and over in same form or with variations, is to to insist on putting set theory into your finitistic frameworks.
I could easily switch roles with you, to play devil's advocate for, say, some given finitistic point of view critical of set theory. I could play that role. You couldn't do the same in reverse.
So nope, it's not a parity.
In ZFC, is the equation 1+1=2 a definition, a theorem, or something else? My understanding is that if ZFC were inconsistent then one could prove both that the natural numbers are distinct and that they are equal.
Quoting TonesInDeepFreeze
I dispute a continuum composed of points. I take it that you consider the points to equal the continuum.
Quoting TonesInDeepFreeze
I know the standard construction, starting with natural numbers then integers then rationals then reals, etc. And often we say that the naturals are defined as nested sets of sets. I am disturbed by this approach but I know in another thread you are already debating the definition of a set so let's leave it at that.
Quoting TonesInDeepFreeze
Then maybe ZFC is inadequate for analysis. Once this discussion has concluded I'm going to start a new post with my argument supporting this view. Perhaps we can discuss this further at that time.
Quoting TonesInDeepFreeze
I see points as emergent from distance...but we've been here before...
Quoting TonesInDeepFreeze
Care to try?
Okay, I see. I forgot the details of the Thompson's Lamp paradox. f(n) corresponds to the incremental time of light switching, not the incremental distance Achilles travelled. To me that is a moot point, but let's hold off on distance for the moment and focus on your complaint that I'm trying to fit set theory into my finitist perspective.
Do you agree that your formal definition describes the informal notion that there exists a complete table (having no last term) as described below?
Step #[n], incremental time [f], current state of lamp [g]
0, 1, on
1, 1/2, off
2, 1/4, on
3, 1/8, off
etc.
Also, if you look at the Wikipedia page (https://en.wikipedia.org/wiki/Thomson%27s_lamp) you will see a table which is more closely aligned with the paradox:
Step #, cumulative time, current state of lamp
0, 1, on
1, 1+1/2, off
2, 1+1/2+1/4, on
3, 1+1/2+1/4+1/8, off
etc.
Do you think that the incremental time table and the cumulative time table convey the same information, just in a different format?
You tendered the notion that infinite sets are empty. I said that's a contradiction (more exactly, it's inconsistent). Then you replied that if set theory were inconsistent then set theory has that infinite sets are empty. And above you quoted me yourself instructing you that if set theory is inconsistent then still "infinite sets are empty" is inconsistent. (!!!)
Quoting keystone
Non responsive. You say there is no continuum, but in the imaginary world you describe, you have a ruler that you say is the continuum. Have cake or eat it. Choose one.
Quoting keystone
No, I never said that. I posted explicitly (in this thread or another that I think you were in) what the continuum is. The continuum is:
In other context, it's okay to say that the continuum is:
where P = { | x in R> and M = { | t in P & s in P & Exy(t = & s = & Lxy)}
Quoting keystone
No, let's not leave it at that.
(1) I did not debate the definition of 'set'.
(2) That you are "disturbed" doesn't change the fact that in set theory, distinctness of natural numbers doesn't require consideration of a continuum. You are just plain flat out wrong.
Quoting keystone
That is an idiotic non sequitur.
Quoting keystone
Goody, your undefined 'nothingness' is justified by your undefined 'emergent from distance'.
Quoting keystone
At SAG-AFTRA rates. For that matter, I should already be charging you at least AFT rates for the instruction I'm giving you.
Quoting keystone
It's an important point, since it and the alternating states are what makes Thomson's lamp a different problem from Zeno's paradox.
Quoting keystone
No last entry. But keeping in mind that 'time' and 'current state' and 'lamp' are not in the mathematics itself.
Quoting keystone
We can add whatever math you want to my writeup. Define:
s(0) = 1
s(n+1) = s(n)+(s(n)/2)
j(n) =
And still my point about the writeup stands. We have an infinite sequence. There is no last entry in that sequence. (And we can also throw in the infinite series too, though it doesn't change the point). There is no contradiction there. Thomson's lamp is not a description of physical events. And it's not even model abstract set theory. Thomson's lamp does not show that set theory is inconsistent nor that set theory fails to provide mathematics for the sciences.
Ha! I hadn't read this until just now:
https://plato.stanford.edu/entries/spacetime-supertasks
"Benacerraf (1962) pointed out [that the] description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit. It may still be possible to complete the description of Thomsons lamp in a way that leads it to be either on after 2 minutes or off after 2 minutes. The price is that the final state will not be reached from the previous states by a convergent sequence. But this by itself does not amount to a logical inconsistency."
And that is what my own writeup says too.
Here's what we have:
A putative description of an imaginary world (which is not a physical world).
The description is not coherent, since it posits that there is a last state for a process that does not have a last state.
Since the description is not coherent it does not specify even an imaginary world. Perforce, not a world that is a model of set theory.
Set theory is consistent*.
Set theory does provide a mathematical version of infinitely many steps. But not with a last step that is the successor to the previous step.
It's not the fault of set theory that it doesn't have a version of an impossible world. Indeed, it is a virtue of set theory that it doesn't have a version of an impossible word.
It is a fail to claim that Thomson's lamp impugns set theory. Indeed, if Thomson's lamp imgugns anything, it's the supertask that is described. Just as set theory does not assert that there exists such a supertask.
* Presumably it is consistent, (1) since no inconsistency has been found, and (2) by arguments regarding its hierarchical nature (see Boolos).
Quoting keystone
The von Neumann characterization, which has been standard for a long time now is:
The set of natural numbers is the least successor inductive set.
df. Sn = n u {n}
thm. n is a natural number <-> (n = 0 or Ek(k is a natural number & n = Sk))
But now matter how we define the set of natural numbers, starting element, the successor operation and the starting element, as long as it is a Peano system*, then we get distinct natural numbers.
Distinctness does not depend on a particular characterization of the natural numbers (such as von Neumann's).
* And recall that all Peano systems are isomorphic with one another.
Of course, one may adopt a thesis that mathematics should only mention what can happen with a computer (call it 'thesis C'). Then, go ahead and tell us your preferred rigrorous systemization for mathematics for the sciences that still abides by thesis C.
And one can reject thesis C. And there is a rigorous systemization of mathematics for the science that does not abide by thesis C.
I got on an airplane that flied well, getting me from proverbial point A to point B. Show me your better airplane.
Any bird, bumblebee or dragonfly?
Perhaps maths, like logic and computation, is a view of nature that is accurate if nature were a machine. And perhaps nature is more than just that.
So no problem that you find maths useful. But nature seems larger in ways that still nag at the philosophical imagination.
Hence systems science.
Quoting TonesInDeepFreeze
I'm not sure if this is what you're referring to but I don't mention finitely many points. As you know, I don't think the real line is composed of points. However, I can see the confusion resulting from my loose use of terminology. When I say real line or continuum you interpret that as the real number line because that's how it used. So that's my bad. Anyway, I'm talking about a continuous object, for example a line, a string, a surface, etc. These continuous objects can be used to do all the math that we typically do (e.g. graphing curves) but they're not composed of points.
But one more tidbit.
Some remarks here made my wonder how (using only first order PA (and theorems I already know proven in first order PA)) to prove:
There are no natural numbers m and n such that m < n < m+1.
I got stuck, so I looked it up. It's a bit trickier than one might think.
Toward a contradiction, suppose m < n < m+1.
Since ~ n < 0, let n = m+p, for some p, with ~ p = 0
Since n < m+1, let m+1 = n+q for some q, with ~ q = 0
So n+q+p = m+1+p = m+p+1 = n+1
So p+q = 1
Since ~ p = 0, for some j, let p = j+1
Since ~ q = 0, for some k, let q = k+1
So j+k+2 = 1
So j+k+1 = 0, which is impossible
Ok, I get what you're saying now.
Quoting TonesInDeepFreeze
Okay, maybe I need to refine my statement. How about, if ZFC is inconsistent then you can prove that infinite sets are empty and you can prove that infinite sets are infinite?
Quoting TonesInDeepFreeze
If I stop using the word continuum and instead say that a ruler is a continuous object does that sit better with you?
Quoting TonesInDeepFreeze
I'm referring to your discussion with MetaphysicsUndercover where he says that a set by definition cannot be the empty set. This I do not want to debate here.
Quoting TonesInDeepFreeze
This conversation is going in so many directions I'm willing to set this point aside for now. Suffice it to say that I see nested sets of sets containing no objects in a similar way as I see geometry constructed from points - they're both creating something from nothing. I believe I'd be rehashing the same sort of arguments that I've already provided which you surely don't want to hear again, nor will you be convinced by it. After all, I feel much more comfortable talking about geometry than set theory so let's please drop this for now and I will acknowledge that as far as we know today, in set theory distinctness of natural numbers doesn't require consideration of a continuum.
Quoting TonesInDeepFreeze
For the table involving cumulative time, there is no row corresponding to a cumulative time of exactly 2 min. Does that mean that the table (and your set theoretic description) only describes the state of the lamp as time approaches 2 min?
Quoting TonesInDeepFreeze
I know it's not physically possible due to the physical limitations related to flicking a switch but I think we can set that detail aside. In this fictitious realm, can set theory can be used to describe the thought experiment of Thomson's Lamp all the way to 2 min? If Set Theory can't allow the cumulative time to reach 2 min, it seems that Set Theory fails to provide mathematics for the 'sciences' of this fictitious realm.
It sounds like by a similar reasoning we cannot say that 1+1/2+1/4+1/8+... = 2.
I agree, but as I just mentioned, by this logic we also cannot say that 1+1/2+1/4+1/8+... = 2 since this implies that there must be a last term to the summation. In other words, if we just look at the first two columns of the table...
Step #, cumulative time
0, 1
1, 1+1/2
2, 1+1/2+1/4
3, 1+1/2+1/4+1/8
etc.
...time never reaches 2 minutes.
Quoting TonesInDeepFreeze
Does the geometric series have the last step as a successor to the previous step?
Quoting TonesInDeepFreeze
It seems like the clock that counts to 2 minutes is performing a supertask. Can we say that set theory cannot describe clocks?
I'm not in a position to argue that Peano systems are inconsistent so I'd like to set this aside for now.
Of course.
Quoting keystone
There is only one set that has no members. That it is called a 'set' is extraneous to the formal theory. The formal theory doesn't even need to mention the word 'set'. We could just as well say "the object that has no members". And we don't even have to say "object". We could just say "There is unique x such that x has no members".
Quoting keystone
'continuous' is defined in mathematics. I don't know what you mean by it.
Quoting keystone
That's close enough to what I already said. But, again, keep in mind, the mathematics does not describe the imaginary scenario in every respect - indeed, because the imaginary scenario is not coherent while the mathematics is consistent.
Quoting keystone
That's not even the issue. The problem is as I mentioned in my remarks.
Quoting keystone
There is only an incoherent description of something that can't even be a fictitious or abstract model of anything, because it can't be the case that there is a final state that is a successor state where, for each state, there is a successor state.
Especially a finitist would see that immediately. For a finitist there is no such realm, and for an infinitist too.
In that context, I don't mean 'system' in the sense of axioms and a theory. I mean it in the sense of a tuple with a carrier set with a distinguished object and an operation, like an algebra. In that sense, 'consistency' or 'inconsistency' do not even apply.
Every formal theory begins with an intuition. I don't have a formal theory. I also don't think you want to discuss the intuition further since it's not formal...but I believe it is a better airplane. Based on all of the infinity paradoxes, I can't help but think that the current "rigorous systemization of mathematics for the science that does not abide by thesis C" is inconsistent. I cannot prove that formally, but I can discuss the infinity paradoxes.
WRONG. Yours is the typical claim of someone who knows not even the first week of Calculus 1.
An infinite summation is a LIMIT, not a final term in the sequence, as the sequence has no final term.
This is at the very heart of the paradox puzzles. The final state of the lamp is NOT a limit. There is no convergence between alternating "On" and "Off". So not only is there not a final state, but there's not even a limit of the sequence. The mathematics can't help, because the "realm" is impossible even on its own.
On the other hand, with Zeno's puzzle, the mathematics can offer that there is a limit to the sequence, thus infinite summation IS defiined.
Why don't you learn at least the first chapter in a Calculus 1 book?
It seems that this fundamental particle of set theory needs to be defined then.
Quoting TonesInDeepFreeze
How about instead of continuous object, I say topological manifold?
Quoting TonesInDeepFreeze
It's almost like you're saying that constructing the whole from the parts provides an incoherent description. I do see that immediately.
Mark that as one of the very rare times a light goes on in a crank's mind. Alas, though, even when it happens, the crank will later double back to commit the error again.
I can accept it as an algorithm for generating the set, but not as a completed set....but we've been here before....
From the axioms, we prove that there is a unique x such that x has no members:
E!xAy ~y e x
Then we define ['0' here for the oval with a slash through it]:
0 = x <-> Ay ~ y e x.
And, informally, we nickname that "the empty set".
So do you agree that the = in an infinite summation means something different than the = in a finite summation?
It's not an algorithm. It's not "generating" a set. It is a certain tuple.
It's amazing to me that cranks are FULL of criticisms to mathematics but they know nothing about it!
I'm going to let you have the last word on this point for now since I'm trying to keep the discussion focused on Thomson's lamp and continuous objects vs. points.
Look at the history, I've tried to end the discussion on this point but you keep looping it back it!
Absolutely I do not agree.
'=' stands for equality. Period.
So, now I have to give you a free lesson from the first chapter of Calculus 1.
The infinite sum here is:
Let f(0) = 1. Let f(k+1) = f(k)/2
df. SUM[k e N] f(k) = the lim of f(k) [k e N]
thm. the lim of f(k) [k e N] = 2
/
Did you not take Calculus 1 when you were in school?
Well, you asked me.
And, very likely you'll have the last word anyway, since I really am out of time for a while.
Wrong and contradicts what I've said about that, and contradicts the fact that I (unlike you) DO read in the philosophy of mathematics about different intuitions, including the range of finitism. And I( (unlike you) DO read about formalizations of those different intuitions.
I have nothing against expressing intuitions. But you do more than that. You compare your intuitions with set theory, including criticism of set theory, but you don't know anything about set theory so your criticisms are woefully ill-premised. That's where I come in and say, "Whoa whoa whoa there, pardner, it's fine to have criticisms of set theory, but you better know what you're talking about."
Quoting keystone
What in the world? Your comment about Peano systems was ludicrously ignorant and so wrong it's not even coherent. So I corrected you. There's no "looping back" by me.
Quoting keystone
I have developed a paradox which (I think) shows the problem of using limits to define equality. Let's shelve this discussion until then. If you're up for it, I'll hold off on submitting a new discussion on that paradox until you're back as I'm sure you'll have an interesting opinion. Of course, I'm not expecting for any commitment from you whatsoever. It's a free world...this is just a nod to you, noting that I'd benefit in you seeing my post....
Yet you went on to ignorantly argue about it!:
Quoting keystone
You don't get say that you don't want to argue about it, then argue about it anyway, then blame your interlocutor from replying. Not on this planet at least.
Hey, I'm all for you dropping whatever you want to drop. Saves me from cleaning up your messes. But "setting aside" doesn't Orwellian mean "I continue to remark on it, then say I want to set it aside, then if you reply you're "looping" back"."
Fair criticism
Here is my careful philosophical response: Bull.
All you did there is argument by mere repeated assertion, as you ignored the rebuttals I've already given.