Of Determinacy and Mathematical Infinities
Ontic determinacy, or the condition of being ontically determined, specifies that which is determined to be limited or bounded in duration, extension, or some other respect(s) - this by some determining factor(s), i.e. by some determinant(s).
A mathematical infinity (in contrast to metaphysical infinity) is not limited or bounded in only certain respects and is thereby countable (metaphysical infinity is not limited or bounded in all possible respects and is thereby uncountable - and I wont be addressing this concept). A geometric line, for example, is limitless in length but not in width (technically, it has 0 width, which is a set limit or boundary). An infinite set, as another example, is limitless in terms of how many items of a certain type it contains but is limited or bounded in being a conceptual container of these definite items. And, of course, one can have quantities of both infinite lines and infinite sets.
Determinacy specifies a process: namely a process via which states of being are obtained. These obtained states of being are defined by the limits or boundaries set by determinants or, in the case of indeterminacy, the absence of these.
Mathematical infinity specifies a state of being. This state of being is defined by the lack of limits or boundaries.
That then sets the stage for this question:
-- Can mathematical infinities (e.g., geometric lines, infinite sets, and so forth) be ontically determinate?
If yes, wouldnt that then imply that mathematical infinites are unchanging, unvarying, fully set, and fully fixed (as they at least to me so seem to be) but, then, further entail that they have all their possible limits or boundaries determined? But then this sounds like a blatant contradiction: a mathematical infinity has all possible limits or boundaries set and, at the same time and in the same respect, does not have all its possible limits or boundaries set (for at least some of its possible limits/boundaries will be unset in so being in some way infinite).
If no, wouldn't that then imply that mathematical infinities are indeterminate, hence ontically vague? Yet this directly contradicts the precise demarcations which define mathematical infinities such as that of a geometric lines length.
Or, else, are mathematical infinities to be taken as beyond the dichotomy of determinacy/indeterminacy - such that, for one example, in a system of causal determinacy all mathematical infinities are taken to be beyond the causally deterministic system? But, then, this wouldn't be causal determinism by definition.
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Anyone have any idea of where the aforementioned goes wrong? My best current guess is that mathematical infinites can be determinate but that this determinacy sets the limits of that which is not limited yet this seems so convoluted a notion that to me it currently borders on being nonsensical.
A mathematical infinity (in contrast to metaphysical infinity) is not limited or bounded in only certain respects and is thereby countable (metaphysical infinity is not limited or bounded in all possible respects and is thereby uncountable - and I wont be addressing this concept). A geometric line, for example, is limitless in length but not in width (technically, it has 0 width, which is a set limit or boundary). An infinite set, as another example, is limitless in terms of how many items of a certain type it contains but is limited or bounded in being a conceptual container of these definite items. And, of course, one can have quantities of both infinite lines and infinite sets.
Determinacy specifies a process: namely a process via which states of being are obtained. These obtained states of being are defined by the limits or boundaries set by determinants or, in the case of indeterminacy, the absence of these.
Mathematical infinity specifies a state of being. This state of being is defined by the lack of limits or boundaries.
That then sets the stage for this question:
-- Can mathematical infinities (e.g., geometric lines, infinite sets, and so forth) be ontically determinate?
If yes, wouldnt that then imply that mathematical infinites are unchanging, unvarying, fully set, and fully fixed (as they at least to me so seem to be) but, then, further entail that they have all their possible limits or boundaries determined? But then this sounds like a blatant contradiction: a mathematical infinity has all possible limits or boundaries set and, at the same time and in the same respect, does not have all its possible limits or boundaries set (for at least some of its possible limits/boundaries will be unset in so being in some way infinite).
If no, wouldn't that then imply that mathematical infinities are indeterminate, hence ontically vague? Yet this directly contradicts the precise demarcations which define mathematical infinities such as that of a geometric lines length.
Or, else, are mathematical infinities to be taken as beyond the dichotomy of determinacy/indeterminacy - such that, for one example, in a system of causal determinacy all mathematical infinities are taken to be beyond the causally deterministic system? But, then, this wouldn't be causal determinism by definition.
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Anyone have any idea of where the aforementioned goes wrong? My best current guess is that mathematical infinites can be determinate but that this determinacy sets the limits of that which is not limited yet this seems so convoluted a notion that to me it currently borders on being nonsensical.
Comments (123)
Maybe mathematical infinities only make sense in relation to the metaphysically infinite
How so? As background: I've read a little on Cantor's "Absolute Infinity" and find it to be a nice poetic conception. But I don't see any relevant connection between, for example, a geometric line and the attributes which Cantor said of God - aka of Absolute Infinity - e.g., that of being "the supreme perfection".
More to the point: how would this in any way clarify whether mathematical infinities are determinate, indeterminate, or neither?
To be more explicit on this matter, metaphysical infinity, in being perfectly devoid of all possible limits and boundaries (as defined in the OP), would then be perfectly undetermined in all respects. As such, it would then be perfect indeterminate (aka, nondeterminate). Yet the infinite length of a geometric line is definite, and so I take it in at least some meaningful way determinate; but, then, if it is determinate this brings back the issue addressed in the OP of apparent contradiction in regard to limits (contradictions which don't occur for metaphysical infinity on account of it being perfectly indeterminate).
[BTW, to my way of thinking, one can conceptualize metaphysical infinity as perfect being (i.e., God) just as readily as perfect nonbeing (i.e., nothingness ... as in, "why is there something rather than nothing") - this though the two are polar opposites. And neither have been either empirically or rationally evidenced to be to the satisfaction of most. Whereas at least some mathematical infinities - like the irrational number pi (whose decimal expansion is infinite despite the sequence of its decimals being, by all apparent accounts, determinate) - do occur in nature, or at least can influence our reality as though they do. But discussion of metaphysical infinity to me appears to enter a whole different ballpark than what the OP is asking.]
Can you elaborate? Do you mean that the line is measurable?
I know so little about math, but I'm always eager to learn.
Uh huh
Oh boy, here we go again . . . :roll:
I'm in no way a mathematician; not my personal forte. Wanted to be forthright about that. But sure on elaboration of my philosophical reasoning regarding the matter:
As I tried to point out in the OP: a geometric line is defined by an uncurved infinite length of zero width. Its length's expansion in both directions is not limited or bounded, yes. Its length is then of itself immeasurable. But its width and shape is subject to fully set limits or boundaries, thereby endowing the geometric line with a definite uncurved length. Devoid of this definite state of being brought about by fully set limits or boundaries - namely, of having zero width and a straight length - we wouldn't be able to discern it as a geometric line. Hence, as with all other mathematical infinities I currently know of, a geometric line is not perfectly infinite in all respects but only infinite in some respects while being finite in others. Due to its finite aspects, we hold a definite idea of what a geometric line is (it then becomes measurable in this sense; else expressed, it becomes countable).
OP's question is one of whether mathematical infinity - your field I take it - is determinate, indeterminate, or neither?
Just don't say that. It has a specific meaning in mathematics, and the length of a line is not countable in that sense.
Doesn't matter to whatever you're saying. Carry on.
I specifically said the length is immeasurable. If one can discern the quantity of lines specified, then lines as a whole are indeed countable. Or would you disagree with what I actually said?
Set theorists and foundations people might be interested in such distinctions, but for me infinity simply means unbounded. Going back millennia to study what the ancients thought is of historical interest, but there has been progress since then. If that's your goal, then don't even mention mathematics. Once you do you are out of your depth. Not criticising, just fact.
Is a geometric line - a maths concept - to you unbounded in all possible respects?
Besides, again, the issue is one of whether such unbounded things that we discern via definite demarcations are determinate, indeterminate, or neither.
Ah, I see, you meant countable as a unit, as a line. Sure.
Right, and its countable as a unit on account of having some limits or boundaries via which it can be so distinguished.
If we're in agreement on this, cool. :smile:
So for your question about the determinateness of mathematical infinities, you would say here that a line is I guess 'determinate enough' that we can pick it out as an object?
So, if I have a countable collection of lines, they are countable? I suppose that's a step in the right direction.
Quoting Srap Tasmaner
If it's in a countable collection that would seem to be the case. But if a line is the shortest distance between two points, it could depend upon the metric you are using. For example, in the taxicab metric the shortest distance between two points is greater than in the Euclidean metric.
So, a geometric line for example, once its placed on a geometric plane it is - in one sense - fully determinate. Its direction and figure are fully fixed in place. No variance; no vagueness. It is not as though the geometric line is semi-fixed or semi-vague. Yet in a different sense, it is only semi-limited or semi-bounded: being infinite only in length (but not in figure or, as a one-dimensional object, in width).
Yet this doesn't make it semi-determinate in the sense of being only partially fixed, or set.
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BTW, not naval gazing. I'm trying to address three metaphysical possibilities in regard to determinacy - namely, that of being a) completely determinate, b) completely nondeterminate, or the possibility of c) being only partly determined by determinants (i.e., of being semi-determinate). And my stumbling block is that by defining determinacy as I did in the OP (i.e., having limits or boundaries set by one or more determinants), I run into this stubborn paradox of having to differentate semi-determinacy from what I've so far termed "mathematical infinities" ... which are, again, only partly infinite in some respect while yet being finite in others.
I'm assuming this is somewhat dense, but there's the background to the OP and an answer to your question.
What makes them countable if they are completely devoid of any boundaries? So might staying on topic be another step in the right direction.
Is there any property in taxicab geometry analogous to curvature?
Maybe the average distance of the intersections at which you turn from the impossible direct route. Not sure what the point would be, but it's interesting to think of taxicab routes as approximations of the direct routes that are unavailable. (Or vice versa.)
Hmmm. I was hoping you'd say you were okay with this example so we could compare it to another that you feel differently about.
Does it help at all to look at how mathematics handles this? Vaguely similar questions do arise in mathematics.
So, for instance, we say any two points in a plane determine a unique line. But if we go up a dimension, thus allowing that third coordinate to vary without bound, two points are not enough to pick out a single plane, and there are infinitely many planes that contain the line they determine. You need one more point, not on the line, to uniquely determine a plane.
Just an example. Mathematics does sometimes directly address how determinate its objects are, at least in this sort of sense, whether there's a unique solution, finitely many, infinitely many, etc.
Is this sort of determinateness any use to you?
So width is length?
And what is "uncurved" length?
Still eager to learn.
I would like a better definition of determinacy. You seem to be implying that the line is determinate because the line exists in its entirety in the plane. Is this correct?
Well, the problem starts here:
Quoting javra
...with the supposition that any of this makes sense.
Oh, Banno. You're ruining our fun.
So far I don't find it being of use to alleviate the issue. Thanks for the input, though. What you say addresses determinacy in the sense of "that determined has its limits or boundaries set by one or more determinants". I'm so for robustly in favor of this definition.
Yet the same issue results: if a unique line is so determined by any two points on a plane (in the sense just provided above) how does one then commingle this same stipulation with the fact that that which is being so determined is - at the same time and in some respect - infinite and, hence, does not have some limits or boundaries in any way set by its determinants. Here, the two point on a plane do not set the limits or boundaries of the line's length - despite setting the limits or boundaries of the unique line's figure and orientation on the plane. Again, once the line is so determined by the two points on a plane, it is fully fixed or fully set; hence, fully determined in this sense. But going back to the offered definition above, this would imply that "the line, aka that determined, has it limits or boundaries fully set by one or more determinants" - which it does not on account of being of infinite length.
no
Quoting Real Gone Cat
a straight extension in space
Quoting Real Gone Cat
see my latest post for the definition also mentioned in the OP
Quoting Real Gone Cat
no
Quoting Real Gone Cat
Let him play! As an self proclaimed anti-philosophy philosopher enamored with Witt, he's into games. :wink:
OK Banno. What would you say "to be determined" is? This in the ontological sense rather than the psychological.
Quoting javra
...so it's anything that is bound, presumably meaning anything with a boundary, an edge. So ask what is excluded here - can you think of something that does not have a boundary? So ontic determinacy includes whatever you want. And off we go. Quoting javra Here we have games without frontiers. So we get things such as
Quoting javra
But of course infinities are bounded - the odd numbers are infinite yet do not include the even numbers, and so on.
So what we have here is the now ubiquitous stringing of words together out of context, the pretence of rational discourse, sitting on a hollow foundation.
Please do. Answer this question:
Quoting javra
If a line (not a line segment) is ontically determinate, I assume you can draw it in its entirety. No?
I can't. Can you?
OK, I get that. Tis why I've started the thread. But then, would you say that it is instead indeterminate? Neither determinate nor indeterminate?
It has a bunch of uses, which we might set out one by one, but which change and evolve over time - like all such words. Trying to capture it with something like
Quoting javra
...should fall flat; but unfortunately it just leads to long threads that say precious little.
In proposing and taking on the OP there's a wilful rejection of clear thinking in favour of confusion that is the antithesis of mathematics.
Quoting jgill
This is the right response.
Aright. What use do you take it to presently hold in the notion of causal determinism in particular? If you find that it holds different uses in this context, I'm more than happy to listen.
Taxicab Fun
Quoting javra
I don't know what you're talking about. Provide an illustration, please. Show me how you count them, 1,2,3,4,5,...
I don't have an opinion; that's rather my point. If you think there is such a use, then it is over to you to provide a coherent account of how and why.
Quoting jgill
Nor I.
and ...
Quoting Banno
I can only interpret this as implying that to you causal determinism is meaningless or nonsensical, as is its notion of determinacy.
But you're still butting in as the measure of all that can be understood.
OK, then.
If we admit that a line is not ontically determinate, then I suppose it's ontically indeterminate. I think your problem lies in equating "indeterminate" with "vague". One may draw part of the line in a specific location, just not the line in its entirety. Is this what you're looking for?
Thanks for the offer.
Unfortunately, not to my satisfaction as expressed, no.
Can not two points in a plane (with the plane itself determined by a multitude of points) determine a unique line, this as offered? In which case, the line here then has determinants and is thereby not indeterminate (i.e., undetermined). An indeterminate line so far makes little sense to me, as it would not be determined by determinants (here, namely, by points).
... as it is, been sitting on my own ass a little too long today. Going to take a break.
Well, see this. So that's not quite right, and certainly does not follow from what I said.
Quoting javra
It's what I do. You asked where you went wrong. The answer is that your OP needs substantial clarification.
Once you have a line, whether any other point in the plane is on it or not can be determined; it becomes an absolute yes/no question. Within a plane, every point is either on the line, above it or below, so the line perfectly bifurcates the plane. (Not for nothing, but given a line and a point, you figure out its relation to the line using a mathematical construction called a 'determinant'.)
There's also a sense in which a line, like any other function, gives a perfectly clear answer to how a segment of it can be extended: go on exactly like this.
It's altogether very well-behaved, and as sharply defined as, say, a triangle or some other sort of figure.
With regard to a segment, are you saying this segment is determinate in that it is finite and indeterminate in that it has infinite points in it?
By Jove, you're getting there! Two points do indeed determine a unique line segment joining those points, But there are lots of line segments including and extending beyond this initial segment, aren't there? The big Kahuna here is extending this segment infinitely in both directions.
this is what I'm questioning.
In attempts to simplify the reason for this thread:
We have two well-established concepts: that of determinacy (such as can be found in the notion of causal determinism), on one hand, and that of non-metaphysical" (aka, countable, mathematical) infinity (such as can be found in a geometric line of infinite length), on the other. On their own, both concepts are cogent (to most folks, at least). However, when attempting to define infinity (which describes a certain state of affairs) via determinacy (which describes how a certain state of affairs comes to be), inconsistencies emerge.
(Non-metaphysical) Infinity can thus either be:
a) determined, hence determinate
b) undetermined, hence indeterminate
c) neither (a) nor (b)
If determinate, then you run into problems such as given by
If indeterminate, then this directly contradicts the fact that, for example, a geometric line can be determined by geometric points as well as having properties specified by once so determined
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Seeing how Im having a hard time in even getting people to understand the problem, my only current conclusion regarding this problem is that its so dense that I neednt concern myself with it when specifying metaphysical possibilities of determinacy.
Thanks for the input.
Um, the points of a line may be put into one-to-one correspondence with the set of real numbers, which Cantor proved to be uncountably infinite in 1874. In fact, the points in a tiny line segment are uncountable.
I'm unsure why you're hung up on causal determinism. Do you think two points in the plane cause a line to be? I.e., the line was not there before? How else is countable infinity determinate? Because the act of counting gives us the set in its entirety? (OK, try it - count to infinity. We'll wait.)
Quoting javra
You seem genuinely interested in the topic. Depending on your math background, you could try to find a source that discusses the concept of infinity in math that you can use to begin to understand it. You could google texts on Set Theory for beginners, or find nice presentations on Youtube (this might be a good starting point).
You seem to be asking me to explain a commonly established attribute. If youd bother to check the link to infinity I posted youd find the following:
Quoting https://en.wiktionary.org/wiki/infinity
The definitions can of course be questioned, but they are commonly established, at the very least as best approximations of, as Banno would say, the terms usage.
Ill again try to explain. A metaphysical infinity has absolutely no limits or boundaries. Due to this, it cannot be discerned as a unit: it is immeasurable in all senses and respects and hence, when ontically addressed (rather than addressed in terms of being a concept) it is nonquantifiable. As a thought experiment, try to imagine two ontically occurring metaphysical infinities side by side; since neither holds any delimitations (be these spatial, temporal, or any other) how would you either empirically or rationally discern one from the other so as to establish that there are two metaphysical infinities? In wordplay games, we can of course state, two metaphysical infinities side by side but the statement is nonsensical. More concretely, ontic nothingness, i.e. indefinite nonoccurrence - were it to occur (but see the paradox in this very affirmation: the occurrence of nonoccurrence, else the being (is-ness) of nonbeing) - is one possible to conceive example of metaphysical infinity. Can one have 1, 2, 3, etc., ontic nothingnesses in any conceivable relation to each other? (My answer will be no for the reasons just provided regarding metaphysical infinity. However, if you believe this possible, please explain on what empirical or rational grounds.)
With that distinction hopefully out of the way, you can then have limitlessness or unboundedness that applies to a certain aspect of what nevertheless remains a unit. That which is limitless or unbounded about the unit cannot be measured of counted to completion - this as I've previously mentioned. The unit itself - which is a unit only because there are limits or boundaries which so delimit it - can however be counted. A geometric line does not have limiteless or unbounded width; its width holds a set limit or boundary, namely that of zero width. Because of this, one can quantify and thereby count geometric lines on a plane as individual units.
It bears note that Im not arguing for a novel concept here. As I pointed out to jgill, these are established notions: you have dictionary definitions such as those provided by Wiktionary and SEP entries on infinity in reference to this.
Quoting Real Gone Cat
It was given as one possible concrete example of ontic determinacy, primarily on account of all-knowing people such as Banno not getting the context of the usage of the term "determinacy". But no, causal determinism does not hold the only conceivable type of determincay: there can be already established notions of non-causal determinacies, this as @Srap Tasmaner illustrated in this post.
Quoting Real Gone Cat
Yea, I am. And as opposed to what? (a rhetorical question)
Every object is bounded in its identity, that is, it has a boundary that differentiates the object from what it is not. Does "ontically determinate" mean having such a boundary? Then it doesn't seem important whether the object is in some way infinite.
I warned you this would be trouble.
The usual way of using these words in mathematics is pretty straightforward. 'Countable' means there is a one-to-one correspondence between the set you have and a subset of the natural numbers, maybe all of them. So either finite, or 'countably infinite' like the natural numbers. We're talking about sets where you can write down the members in a list, even if that list goes on forever. 'Uncountable' is for bigger infinite sets. The real numbers, to start with, cannot be written down in a list that goes on forever, no matter how clever you are.
Obviously countable is nicer to deal with, because you can use algorithms that iterate (or recurse) their way through a list and you know that will get you not to the end but as far as you'd like to go.
(Also: Zeus could write out all the natural numbers in a finite amount of time just by doing the next one faster each step; not even Zeus could write out the real numbers in a finite amount of time. Lists are friendlier, even when they don't terminate.)
I'll say. Go deeper: Countable
Quoting javra
An infinite line is a line, therefore, I suppose, a "unit". But they can't be counted since the points in the Euclidean plane cannot be counted and so pairs of these points - defining lines - cannot be counted.
I too wonder how a continuum makes up something discrete
If I understand you right, yes, every individual cognition as identity is delimited from other cognitions and hence bounded. Yes, and this holds true for the concept of metaphysical infinity as well - in direct contrast with that supposedly ontic occurrence that the concept of metaphysical infinity specifies.
Ontically occurring metaphysical infinity is devoid of any ontic identity for it has no boundaries via which such an ontic identity can be established. Nothingness, for one conceivable example of such, can be identified by us on grounds of being different from somethingness, so to speak. There thereby is a conceptual boundary between nothingness and somethingness via which nothingness can be identified. But on its own, where this to be possible, nothingness would hold no ontic identity - for an identity would be something.
As to your conclusion, thanks for offering. Ill think about it some.
There is such a thing as equivocation between two or more meanings or usages of a term, right? I repeatedly described countability in its non-mathematical sense of able to be counted; having a quantity. As does the Wiktionary definitions previously posted.
Quoting Srap Tasmaner
Are the infinities of natural numbers and of real numbers two different infinities? Or are they the same nonquantifiable infinity?
In other words, countable can only hold the valid usage in its mathematical senses when addressing things such as lines. Therefore, the concept of there being 2 lines is invalid and nonsensical. This as all mathematicians know, in contrast to the stupidity of common folk.
And I must take my own head out of my own tunnel-visioned ass in order to realize this.
Got it.
By no means in agreement, but I got it.
Yea. That appears to roughly sum up the issue.
I don't wish to be mean, but this strikes me as complete word salad. The Wayans brothers (In Living Color) used to do a skit where two self-educated street preachers have a nonsensical conversation by stringing together unrelated words. Reminds me of that.
The bolded sentence ends in a question mark, so I assume its a question. And it seems to represent the crux of your argument. Can somebody translate?
You have reached an absurd conclusion. Of course there can be "two lines". Any finite collection of lines is clearly countable. And there are countable infinite collections of lines such as all lines parallel to the x-axis that pass through y= 1, 2, 3, ....
What are not countable are all lines in the plane.
Quoting Gregory
As the seconds tick by in the continuous flow of time we have minutes and hours which are "discrete".
I was just about to post this very same response.
Each line consists of an uncountable infinity of points, each plane consists of an uncountable infinity of lines, and 3D space consists of an uncountable infinity of planes. But two points are always discrete, two lines are always discrete, and two planes are always discrete (except where they may intersect).
So an infinite line has no ontic identity?
An infinite line is not metaphysical infinity. An infinite line is infinite only in length, not in width. Whereas metaphysical infinity would be infinite in length, in width, and in all other possible manners.
Except (a) you want specifically to talk about mathematical infinities, and there's prior art there you might as well become familiar with; and (b) the mathematical usage of 'countable' is actually something a lot like 'able to be counted', because listable.
I think what's throwing the discussion off is that we don't normally talk about the cardinality of a line except when we're considering it as a collection of points, the continuum, which is not countable. But that's not really measuring its length, different deal. If you have an infinite ruler marked off in centimeters, you'll be counting again.
Quoting javra
Yes. The cardinality of the set of natural numbers is aleph-0; the cardinality of the set of real numbers is aleph-1, aleph-0 raised to the aleph-0 power. It is not known whether there is a size in between, but I think most mathematicians think not. Could be wrong.
Quoting jgill
All emphasis mine. Um. Okay. I certainly don't understand what your stance is on whether or not infinite lines are countable. But I'm glad others like Real Gone Cat can make sense of your writing.
Quoting Srap Tasmaner
By being 2 different infinities, they are thereby quantifiable as infinities wherein each individual infinity is demarcated from the other by some limits or boundaries. These infinities are thereby countable (in a non-mathematical sense): 2 infinities.
This cannot be so of metaphysical infinity (for reasons I've become tired of repeating).
I mean, they're different in quantity, not quality. They're both cardinal numbers, just of different sizes.
Now there are transfinite ordinals, but you'd have to ask someone else about those.
Why can't you count metaphysical infinity? I assume you only recognize one metaphysical infinity, so haven't you counted it? One.
But such a metaphysical infinity would still have a boundary of its identity because it would be differentiated from what it is not, for example from finiteness or from infinite lines.
Not to be rude but, in this thread, you've made it a habit to assume things I haven't expressed and most often don't believe.
Quoting litewave
This will be true when it comes to it being a concept (a map of the territory). But it cannot be true of it as an ontic occurrence (as the territory itself). The boundaries you specify would nullify the possibility of its occurrence.
Mind, I've made no claim as to whether or not metaphysical infinity can ontically occur - and am intent on leaving this issue open ended. But - as with a) the infinity of nothingness or b) the infinity of at least certain understandings of God (each being a different qualitative version of what would yet be definable as metaphysical infinity) - it is possible for certain humans to conceptualize its occurrence.
Friend, as litewave pointed out, by your own argument, when you name a thing, you are placing a boundary on it. If you can, name two metaphysical identities. Now count them. Two.
Anyway, too many folks are trying to explain to you why you're wrong, but you keep doubling down. It's starting to feel like piling on. I'll keep reading comments, but I think I'm out.
Nothingness cannot have an ontic occurrence since it has nothing to occur, and if there were an infinite God he would be different from other objects, for example from us humans, so he would have a boundary of his identity too.
OK. So you uphold that the concept (which has been around in the history of mankind for some time) is vacuous. While neither agreeing nor disagreeing with you, I see nothing wrong with that.
Quoting Real Gone Cat
As has been typical, there's a reading comprehension problem. By my very own argument, the concept is quantifiable whereas that which the concept refers to is not.
Quoting Real Gone Cat
That makes the two of us.
Are you talking about a single infinite line being somehow countable? Like the points on the line?
Or are you talking about the set of all infinite lines being countable?
Neither are countable. Countable means this is #1, this next is #2, the next is #3, etc. It means some sort of algorithm for actually counting.
Maybe you are using the word differently. Like "I can be counting on you to do the best you can." Rather than counting 1, 2, 3, ...
Real Gone Cat is a fellow mathematician.
In presuming that - unlike at least the initial posts of Real Gone Cat - youre not posting this to have a good time at the expense of a poster you assume to be stupid (because laughing at retards is such an admirable trait in todays world):
Im sincerely bewildered at the chasm of understanding (your previous unwarranted rudeness aside).
Nowhere did I state either possibility you offer.
Countable in the sense of: one infinite line and another infinite line make up two infinite lines.
Or: the infinity of real numbers and the infinity of natural numbers and the infinity of transfinite numbers make up three numerically distinct infinities. More technically, make up three numerically distinct infinite sets.
As in, the infinity of real numbers is infinity #1, the infinity of natural numbers is infinity #2, and the infinity of transfinite numbers is infinity #3. Each of these three infinities is in turn other than the infinity of surreal numbers, for example, which on this list would be infinity #4, making a total sum of four infinities that have been so far addressed.
As it is I'm wanting to bail out. But on the possibility that your latest post was sincere in its questions, I've answered.
Certainly you have two infinite lines. And each line has the cardinality of the reals. Also, both lines, together, have that cardinality. And so on. But the "three numerically distinct infinite sets" are not really distinct, in that the set of integers is a subset of the set of reals, etc. So the cardinality of the integers is "less than" that of the reals.
The real numbers are quite complicated. And I didn't intend to belittle your efforts, but it sounded to me like you were thinking it is possible to count all lines in the plane through some sort of algorithm.
This will be true only when one assumes the occurrence of actual infinities, in contrast to potential infinities. As an easy to read reference: https://en.wikipedia.org/wiki/Actual_infinity From my readings the issue is not as of yet definitively settled - or at least is relative to the mathematical school of thought.
At any rate, the issue of whether infinities (in the plural) are determinate, indeterminate, or neither has dissipated from this thread some time ago. I'm looking to follow suit. Best.
Not familiar with that technical term, I googled "ontic determinacy" and found an article on "ontic vagueness"*1. Mathematical Infinity is vague only in the sense that it is off-the-map of real numbers. For example, in Fractal Graphics places where the computer encounters infinities, it stops calculating and renders the area as black, signifying merely "undefined" or "unbounded" or "indeterminate". However, physicists studying sub-atomic particles, also encounter off-the-map Math. Although that aspect of reality is beyond our ability to comprehend or to define, Heisenberg labeled it as an essential feature of the quantum level of reality : "Uncertainty" or or "Indeterminacy".
Unlike ancient mapmakers, who put warnings on the uncharted areas of their maps "here be dragons", Heisenberg merely warned that empirical quantum physics faded into theoretical metaphysics at the margins. So, he defined the ontically indeterminate zones in terms of "the old concept of 'potentia' in Aristotelian philosophy*2" (i.e. metaphysics). Then he bracketed that mysterious unknown zone as "something standing in the middle between the idea of an event and the actual event, a strange kind of physical reality just in the middle between possibility and reality". In Statistics, its called "Probability", and in Metaphysics, it's known as "Potential"
He didn't go so far as to label the state of Superposition as supernatural or unreal or nonbeing. Instead, he placed it in the statistical realm of infinite Possibilities, within which some things are more Probable than others. This is just a different way of looking at what we think of as Reality. So, he rejected the option of throwing-hands-up and declaring "absurd". "The other way of approach was Bohr's concept of complementarity . . . . as two complementary descriptions of the same reality". Two different states-of-being.
Unfortunately, to some of his colleagues that compromise smacked of equating empirical Physics with spooky Metaphysics. Nevertheless, in the 21st century, physicists have been forced to accept the unacceptable position of Overlapping Magisteria. Reality, on the quantum scale, is a vague gray area that is neither fully real & physical, but also not-yet-real (possible) & meta-physical (ideal or mathematical). :smile:
*1. Ontic Vagueness :
[i]A note about terminology: for this paper, I?m using "ontic vagueness? because that?s been perhaps the most common term in the literature on the subject. "Metaphysical vagueness? is probably the better
term (see Williams (2008)b for discussion). Perhaps even better would be to stop talking about
vagueness altogether and just talk about metaphysical indeterminacy.[/i]
https://elizabethbarnesphilosophy.weebly.com/uploads/3/8/1/0/38105685/ontic_vagueness_final.pdf
*2. Physics and Philosophy, by Werner Heisenberg, 1958
THE BLACK AREAS OF A FRACTAL ARE INFINITE, HENCE INDETERMINATE
No, I don't think this is correct.
To show that a set has the same cardinality as the set of natural numbers, you only have to show that the elements of the set can be placed in one-to-one correspondence with the set of natural numbers. You don't actually have to complete the set. To show that an infinite set has a greater cardinality, you only have to show that at least one element exists that cannot be placed in the one-to-one correspondence (ala Cantor's diagonal proof). No actual infinities need be assumed.
If the context is mathematics, then usually the notion of 'infinite' is referenced per set theory.
Of course, we are free to philosophize and use terminology in a non-mathematical way, but throughout your post, you mix mathematical terminology with your own personal meanings, whatever they may be. That is an invitation to confusion at the very onset.
To keep things straight, at least as to the mathematics itself, here is what mathematics provides:
In set theory, there is the defined adjective 'is infinite'.
Definitions:
A set is finite if and only if there is a one-to-one correspondence between the set and a natural number.
A set is infinite if and only if the set is not finite.
A set is countable if and only if (there is a one-to-one correspondence between the set and a natural number, or there is a one-to-one correspondence between the set and the set of natural numbers).
A set is uncountable if and if the set is not countable.
A set is denumerable [aka 'countably infinite'] if and only if (the set is countable, and the set is infinite).
Theorems:
There exist finite sets.
There exist infinite sets.
There exist denumerable sets.
There exist uncountable sets.
So, contrary to your assertion, it is not the case that every infinite set is countable.
Quoting javra
No, it doesn't have a 0 width. It just doesn't have a width at all.
Quoting javra
If that is to have any mathematical import, then it requires mathematical definitions of 'limitless', 'in terms of how many items of a certain kind it contains', 'limited', and 'conceptual container of these definite items'.
However 'bounded' does have a mathematical definition. Per certain orderings upon which we evaluate boundedness, some infinite sets are bounded and other infinite sets are not bounded.
Quoting javra
You say 'mathematical infinity', so I take it you're talking about mathematics. And about mathematics your are incorrect. 'is infinite' is not defined in terms of 'limits' or 'boundaries'.
Quoting javra
I take it that by 'mathematical infinites' you mean sets that are infinite.
"unchanging, unvarying, fully set, and fully fixed" as you use those words, are not ordinarily mathematical terminology (at least not in this introductory stage of discussion), but looking at mathematics from outside mathematics, and to indicate how mathematics is informally regarded, yes, we ordinarily think of the subjects of mathematics to be definite mathematical objects.
Quoting javra
A contradiction is a statement and its negation. There is no known theorem of set theory that is a contradiction. Also, your argument fails because you have a false premise, which is that 'is infinite' is defined in terms of 'limit' or 'boundary'.
Quoting javra
It goes wrong in these ways:
(1) Mixing formally defined terminology of mathematics with your own personal undefined informal terminology. (2) Adopting the premise that 'is infinite' is defined in terms of 'limit', 'boundary' or 'bounded'. (3) Thinking there is some kind of contradiction when only there is a pseudo-puzzle that results from the kind of strawman you set up by mixing terminologies and applying a false premise.
You're using the word 'countable' differently from the definition of the word in mathematics. So, of course, confusion will ensue.
1. is not the mathematical definition.
2. is not the mathematical definition, and it is in error by claiming to be so.
/
It's fine to make whatever arguments you want about a non-mathematical use of 'infinite' in metaphysics, but it is a disaster to mix that up with the mathematical definitions.
Mathematics is a special subject matter that defines terminology in a special way. It is not at all to be taken that mathematical usage is the same as either everyday usage or usage in non-mathematical areas such as metaphysics.
The mathematical context is not the same as your personal metaphysical context. Be clear what context you are in at any given point in a discussion. Otherwise, we get yet more incoherent discussions that devolve into even greater incoherence.
That is yet another variation on conflating the mathematics with personal undefined terminology.
If only you would carefully read the mathematical treatment of 'line' (either as an undefined primitive of axiomatic geometry, or a defined terminology of geometry developed set theoretically), 'length' and 'bounded'.
Then not mix up those definitions with your own personal undefined notions.
The cardinalities of the set of natural numbers and the set of real numbers are both infinite cardinalities, but not the same infinite cardinalities.
'nonquantifiable'. What is your mathematical definition?
It's fine to philosophize about mathematics. But it's silly to philosophize about it when you know virtually nothing about it.
It's fine to work out one's own metaphysical notions and try to make them not have conflicts. But then those are your notions, not notions of mathematics, so of course when you mix them together then you can get conflicts.
If your point boils down to the observation that mathematics handles notions in ways that conflict with the way you handle the notions, then, yes, of course, that is fully granted.
If it is the sum of the issue, then the sum of the issue is meaningless until 'discrete' is given a definition.
Meanwhile, the continuum is the pair
If there is a problem with that, then it awaits a clear statement of the problem.
Yes, a set of two lines is a countably (and finite) set.
The set of points in a line is uncountable. And the union of the sets of points in any number of lines is uncountable.
Quoting javra
'transfinite' is just another word for 'infinite'. There are many infinite sets, such as the set of natural numbers and the set of real numbers.
As to your notion of "three", there are only two sets you mentioned: the set of natural numbers and the set of real numbers. And there are two kinds of infinite sets, countable ones (such as the set of natural numbers) and uncountable ones (such as the set of real numbers), and every set is either countable or uncountable. But we are not limited to only that bifurcation. Among uncountable sets, we can specify even more adjectives, such as 'inaccessible', etc.
But one trifurcation (among many we could define):
finite
denumerable (countably infinite)
uncountable
Both finite and denumerable sets are countable.
No, that is plainly wrong.
Both the set of natural numbers and the set of real numbers are transfinite sets.
Why are you making pronouncements on a subject that you know virtually nothing about?
If there are no infinite sets, then there is no set of all the integers nor set of all the reals.
But the observation about them could still hold in the sense of recouching, "If there is a set of all the integers and a set of all the reals, then the cardinality of the former is less than the cardinality of the latter.'
Moreover, in any case, even without having those sets, we can show that there is an algorithm such that for every natural number, that natural number will be listed; but there is no such algorithm for real numbers.
/
Your readings about infinity are grossly inadequate. Among the vast vast number of mathematicians, it is settled that there exist the set of natural number and the set of real numbers; that infinite sets exist. Meanwhile, there are some, but relatively very few, mathematicians who insist on countenancing only "potentially infinite" sets (even though 'potentially infinite' has only a heuristic but not formal mathematical definition). But to say it is not "settled" is as good as a truism in the sense that any question will always have dissenters, but not a substantively correct claim since infinite sets are basic in mathematics, including ordinary calculus.
Hey, though I hope I'm wrong, your forgone conclusions regarding me and what I was addressing, your indignation, and your seeing of red is evident to me.
But so its said: There are at least two distinct senses of mathematical.
I have nowhere stated - nor to my mind insinuated - that I am addressing infinities as they are defined by schools of mathematical thought, i.e. as they are defined by mathematics.
I have instead used mathematical infinities in laymans terms from the get-go, as I initially thought (mistakenly) the OP made clear by the way terms were defined in relation to each other: in the sense of infinities that can be quantified and that furthermore pertain to quantities, and which are thereby, in this sense alone, mathematical. Mathematical in the sense that if a bird can count to ten, then this bird holds a respective measure of mathematical skill - despite this bird having no cognizance of any theoretical underpinnings devised by humans for the quantities it can count. In the sense that any appraisal of quantities, such as 1 + 1 = 2, is a mathematical ability that makes use of mathematical notions - irrespective of how these notions are established (hence, with or without the symbols that we humans use to express 1 + 1 = 2).
The thread was in part because of this placed in the category of General Philosophy rather than Logic and Philosophy of Mathematics.
Nor do I personally take established concepts in mathematics to the foundational cornerstone of what "infinity" at large can signify.
Notwithstanding, for my part, I have learned from this thread not to term quantifiable infinities that pertain to quantities mathematical - nor countable for that matter. Yes, my bad for not understanding beforehand how use of these terms would be strictly understood by those with a mindset such as your own.
I have no interest in addressing the many comments made in your many posts. And presently intend to let you have the last word, laugh, insult, or what have you.
You said you were distinguishing the mathematical notion from a metaphysical notion. You didn't say anything about the mathematical notion being a layman's notion. Anyway, what layman's notion would that be? Which laymen? There is not a distinct layman's notion about infinite sets as they occur in mathematics.
And your claim is belied by a passage such as this:
Quoting javra
The differences in cardinalities among infinite sets is not a notion a layman has even ever heard of. (Let alone surreal numbers.)
Quoting javra
Gotta admire the knack that goes into turning a retraction regarding terminology back around as a sarcastic dig such as "mindset such as your own". What mindset would that be? The mindset of someone who happens to know how the terminology is actually used in the subject under discussion.
Quoting javra
It's not much of an insult to point out that someone is throwing around mathematical terminology without knowing what it means and that they know virtually nothing about the subject while spouting opinions about it nonetheless.
Or as a great filmmaker said:
https://www.youtube.com/watch?v=9wWUc8BZgWE
I hate to forestall this thread's death, but I am curious about this.
I looked back at the OP yet again, the centerpiece of which is this question:
Quoting javra
You're talking specifically about the mathematical versions of concepts in wider (and vaguer) use and that wider use is what some of us assume lies at the foundation of mathematics, our intuitions about shapes, collections, counting, patterns, all that. I gather it's something like those intuitive, pre-theoretical ideas you really wanted to address, not their mathematical axiomatization.
Which is fine, and I can imagine doing a phenomenology of boundedness and unboundedness, that sort of thing. No doubt that would be interesting.
But there is still something odd about your decision though maybe I've misunderstood you to exclude mathematics. After all, we've had a few thousand years now of people thinking about just these things, and some of that thinking is what we call mathematics. The history of mathematics is far messier and various than your grade school textbook led you to believe, precisely because it's the history of people thinking about the sorts of things you've expressed interest in. Mathematics as it is now may strike you as somewhat rigid and narrow, and therefore of no use to you, but it is still a body of serious, rigorous thought about things like the infinite, so even if there's more to say than you can get out of established mathematics, it is surely the natural starting point, not the natural body of work to be excluded.
Maybe this thread would have gone differently if I had asked you directly to explain this:
Quoting javra
Looking back at our exchange, I realize I hoped that what you're talking about here would become clear as we worked through some examples, but it didn't.
So I still don't have the faintest idea how what you call "mathematical infinities" inserted themselves into whatever you were working on, and why their arrival was such a problem.
If actual mathematics is no use in solving your problem, then presumably these "mathematical infinities" obtruded for non-mathematical reasons; but I can't figure out what sort of non-mathematical problem would drag in a bunch of as a matter of fact, somewhat recondite, even for math mathematical concepts.
If you're of a mind, and not burned out on the topic, take another swing at it. It is, after all, a philosophy forum not a math forum. Maybe if you could explain a little more clearly how your problem relates to mathematics without being a mathematical problem, we could make some progress.
To properly address this, I believe there first ought to be a commonly understood or accepted, philosophical (rather than one pertaining to established schools of mathematics) differentiation between types of concepts regarding the notion of infinity (again, infinity not as its is mathematically defined but as a general, sometimes philosophical, notion: commonly defined by the absence of (non-mathematically defined) limits to that addressed).
I dont know. Once bitten twice shy. This threads issue can to my mind be easily overtaken by a broader philosophical issue and possible underlying stance. Namely, one of whether a) mathematics subsumes all reality (such as, for example, by grounding all of physics and, via further inference, thereby all of physicality this being one example of what could be termed a neo-Pythagorean view) or, else, b) reality holds aspects which can be in part and imperfectly modeled by what we humans have devised - over the long course of history you address - as various schools of mathematical thought.
So, does the mathematicians specialized definition of countability thereby take precedence over what layman understandings (such as the two Wiktionary definitions of infinity previously provided) of countability are? This on account of stance (a). Or else is the so here termed mathematical notion of countability a specialized understanding that is subsumed by commonly accepted every day notions of countability in general, namely "the capacity to count quantities"? As would naturally be the case in scenario (b).
Must I express non-mathematical for every term I use in attempts to define a certain species of concepts regarding infinity?
But then, what on earth would the non-mathematical countably of infinity signify to a general audience?! To be countable but not mathematical is a bit of a conceptual contradiction.
Yes, Im a little frustrated, maybe blowing off my own steam. But help me out a bit if you can.
Can we at least mutually understand a differentiation between quantifiable infinity which can thereby be addressed in the plural and nonquantifiable infinity which cannot thereby be properly addressed in the plural?
All infinities defined by mathematics would then be stereotypical examples of quantifiable infinity.
In contrast, the infinity of nothingness would then be one example of nonquantifiable infinity - such as when nothingness is conceptualized to have once existed/been/occurred devoid of anything. Those who claim the possibility that before the big bang was nothingness (e.g., https://en.wikipedia.org/wiki/A_Universe_from_Nothing) can be ascribed to implicitly make use of such concept of infinity.
If we cant conceptualize - and then properly term - this differentiation between species of infinity which humans at large can historically conceive of, then I dont see much point in further addressing the topic of the OP and, by extension, in answering your inquiry.
And this, in part, because nonquantifiable infinity (if this term doesnt get lambasted as well) can only be completely non-determined and thereby completely indeterminate ontically (though, again, it will be a determinate concept). The OPs inquiry, however, applies only to those infinities that are quantifiable and thereby in some way definite due to some demarcation or other occurring in that being addressed. Furthermore, to simplify the variety of such, the OP limits itself only those quantifiable infinities that are themselves made up of discrete quantities (like a geometric line being made up of discrete geometric points, or an infinite set made up of discrete numbers ).
To further complicate matters, then there needs to be a commonly held understanding of what to determine signifies (when the term isnt used to address psychological processes of mind or states that thereby result). Theres again been much criticism of how Ive attempted to define it (in short, as to set the limits or boundaries of - a standard dictionary definition, spelled out by me in greater explicit detail in the OP for an intended greater accuracy); none of this criticism being constructive in offering any alternative definition.
Im not hopeful this can work out, but Ill check back in some time. Thanks, however, for the offer.
No. It just needs to be clear what the context is.
Quoting javra
As I mentioned, one of those is claimed as a mathematical definition. But it is not.
Quoting javra
Whatever your questions about it, it would be best to start with knowing exactly what it is.
df. x is countable iff (x is one-to-one with a natural number of x is one-to-one with the set of natural numbers).
As far as I can tell, that is different from the everyday sense, since the everyday sense would be that one can, at least in principle, finish counting all the items, but in the mathematical sense there is no requirement that such a finished count is made.
Quoting javra
Best would be to state that you are using your own vocabulary as adapted from various everyday senses, then to state your definitions, and not just in an ostensive manner, or listing of cognates, or blurry impressionistic mentions using more terminology that is itself undefined.
Quoting javra
Indeed! You are the one who claims to represent a layman's non-mathematical notion. It's a safe bet that no one unfamiliar with set theory or upper division mathematics has any notion of all of a countably infinite set. There is no layman's notion of this. So it's silly trying to represent it.
Quoting javra
Define 'quantifiable infinity' and 'unquantifiable infinity'. The comments you then added are not definitions.
In sum: You seem to want to investigate notions of infinity in non-mathematical senses or contexts. Fine. But then you'd do well to leave mathematics out of it if you don't know anything about the mathematics.
:up:
Mathematics is mental content and cannot exist except in individual brains that physically exist in the physical present. It seems some hold a magical view of how mathematics is physically done. Yes, lots of good work over the millennia to build on.
I don't know what 'ungraded' means there, but there many many library shelves worth of articles and books in the philosophy of mathematics on the subject of abstractions and concepts vis-a-vis questions of existence and truth.
Quoting Mark Nyquist
Who?
You again blatantly misunderstand what I was saying.
Curious to see if you might comprehend what Ive been intending from the commencement of this thread if I were to use the rather pompous term numeration:
One can numerate geometric lines and infinite sets. Therefore, these and like infinities are capable of being numerated. As in 2 infinite lines or 2 infinite sets.
In contrast, the infinity of - for one example - a complete nothingness cannot be numerated for, if there were such a thing (linguistic problems in so saying aside), the infinity referenced would have no limits by which to be discerned nor, for that matter, would there occur any sentient being to psychologically delimit or define its presence.
In a similar vein, a non-mathematical numeration is a conceptual contradiction, this because to numerate is a mathematical faculty of mind.
Also, countability as it is defined in mathematics cannot occur in the complete absence of numeration - and can be viewed as a specialized format of numeration.
(Yes, though, to numerate is defined as to count.)
And although this thread is not intended to debate the properties of infinities as defined by mathematics,
Quoting TonesInDeepFreeze
No, not in my neck of the woods. The everyday sense would be that one could, in principle only, count an infinite series of elements/units/items for all of eternity yet to come and still never get to finish. This then addresses the notion of a "potential infinity" as first coined by Aristotle - in contrast to the Aristotelian notion of "actual infinity" which Cantor played a major role in making mainstream in part via use of the one-to-one correspondence you address.
In regard to this, from a previous post:
Quoting TonesInDeepFreeze
So, when the conceptual grouping (to not irk mathematicians by saying "set") of all natural numbers is taken to be a potential infinity it is still taken to be an infinity - else an infinite grouping - just not one that claims to be complete or else whole. Here, one can contrast the conceptual grouping of all natural numbers with - to keep thing as simple as possible - with the conceptual grouping all natural numbers that are even. There will be a one-to-two correspondence between them: for every one even natural number in the grouping of even natural numbers there will be two natural numbers in the grouping of all natural numbers. When both groupings are taken to be compete wholes, then the grouping of even natural numbers will contain a lesser cardinality than (more precisely, half the cardinality of) the grouping of all natural numbers contains - with both groupings yet being infinite. But when both groupings are taken to be never-complete, then for ever one item added to one grouping there will likewise be one item added to the other, and this without end. Such that one cannot compare the cardinality of infinities in each grouping, other than by affirming that they are both infinite in the same way.
Actual infinities can nowadays be very easily expressed and manipulated - and, so, have become of great mathematical use. But that it makes sense to conceive of any infinity composed of discrete items as "actual" rather than as "potential" (this in Aristotle's usage of these terms within this context - rather than what we understand by these term today) is not something that, for example, is amicable to mathematical proofs. Opinions can differ. This though, yes, the mathematics which Cantor introduced is nowadays mainstream.
Then again
Quoting TonesInDeepFreeze
I may not be a mathematician but I can take care of my own bank-account via numerations of various sorts just fine, and still have the occasional leisure to philosophically contemplate issues regarding quantities.
Wikipedia defines information as an abstract concept so that is a good indicator of common usage and mathematics is the same way.
The problem is abstract concepts exist as brain states in individuals and that part is left out of the definitions.
Pleroma?
Wasnt familiar with the Pleroma. Dont yet know how this is intended but, as Carl Jungs Gnostic understanding, sure, the Pleroma qualifies as nonquantifiable infinity.
Other possible candidates include certain understandings of God, Moksha, Nirvana, the Ein Sof, Brahman, and what some claim to be the ineffable (as in G-d) with all of these being traditionally understood as being the form which perfect Being takes. Then, again, theres the concept of nothingness as the absence of all being, which also qualifies as a possible candidate.
Whether or not any of these concepts are anything else but vacuous is irrelevant to the issue. The issue being that such type of infinity can and has been conceptualized by humans at large for a good sum of human history and that it differs from types of infinity that can be quantified and thereby numerated.
And, again, this thread was not supposed to be about such type of infinity, but about those infinities that can be numerated. As in two infinite lines on a plane can either intersect or be parallel.
Oh, you've been comparing math to woo all along. Seems like a category error to me. Carry on.
Sure.
So a line. On the one hand, there's a sort of procedure, which is repeatable, by which you can keep extending a line; there may be more than one way to do that physically different techniques, for example but they're all equivalent in the long run, because there's the usual asymmetry here: there is exactly one way to extend a line as a line and an infinite number of ways to extend it otherwise, with curves, angles, gaps, and so on. So we have a pretty strict formal constraint. On the other hand, we want to extend it forever, which requires the procedure to be repeated forever, without constraint.
If you think of the possible figures you could draw in a plane, you're constrained to the plane, but otherwise have complete freedom. If you compress and channel that freedom in a particular way, you can get a line: completely constrained in one dimension, but completely unconstrained in the other.
Is this the sort of thing you had in mind?
That stubborn reading comprehension problem again. Have I not termed the type of non-finitude you address as woo as metaphysical from the very commencement of this thread?
You dont strike me as the type of person who takes metaphysical enquiries and topics seriously, hence considering them to be woo. But correct me if I'm wrong.
At any rate, glad to see you find readings such as A Universe from Nothing to be woo - despite this notion being proposed by a well-established physicist.
Yes. Precisely.
Here understanding "constraints" as being determined or else determining factors (again, when it comes to maths, for one example, this not in causal ways - such as, per previous posts, how determinants can be addressed in maths; e.g., two geometric points can determine a geometric line ... this in non-causal manners).
Fyi, since the begining of this thread, I think I've figured the issue out. In the logical trichotomy of metaphysical possibilities regarding determinacy - namely: a) being completely determined, b) being completely nondetermined, and c) being semi-determined - quantifiable infinities will then be categorized by (c). Importantly though, when regarding quantifiable infinities as specified by maths, this semi-determinacy will always be devoid of causal determinacy.
Unless you find reason to disagree with this generalization regarding the determinacy of such infinities, I think I'm good to go.
There is one other little hitch though: a line, for example, not only can or may contain all the points in a plane colinear with it (that is, with any two of the points on the line), but it must and does.
Do we still call it freedom, absence of constraint, if you must actualize every open possibility?
I've been speaking of a line as embedded in a plane, because it's simpler to visualize that way, and you can contrast a line to the other possible figures in a plane, but a line is, by itself, simply a dimension. It is in one sense a result of constraining a plane, but in another sense a constituent of an infinite number of planes, whether seen as an infinite collection of zero-dimensional points, or more importantly here, I think seen as a formal constituent of the plane, as representing one of its dimensions. And here's the kicker: any line can itself be considered a constraint that partially determines a plane, as can any point.
By definition, of course. This will be the determinate aspect of it. But then
Quoting Srap Tasmaner
Even in the concept of "actual" or complete or whole infinity, can every open possibility be actualized?
I'm very open to learning otherwise, by what I currently understand by infinite length is that actualizing every open possibility would entail a limit/boundary/end of open possibilities ... thereby negating its affirmed infinitude. Am I misinterpreting something in the terminology?
Quoting Srap Tasmaner
Hm. Not disagreeing.
I've been intending to keep the topic as simple as possible, but I am personally recognizing at least four distinct types of determinacy: including two which could be here termed "top-down" determinacy or "constraint" (e.g., a line's occurrence can be deemed to of itself concurrently determine the placement of all points that constitute the line) and "bottom-up" determinacy or "constraint" (e.g., two points concurrently determine a line) - neither of which are causal. And via this somewhat simple understanding, things can get complex very quickly - especially when taking into account all four determinacy types I'm entertaining (the other two being causal determinacy and teleological determinacy). But maybe this is neither here nor there.
G-d? Brahman? Pleroma? This isn't woo? Merriam-Webster :
But whether you value such things is beside the point. I stand by my assertion : it's a category error.
The category implicitly addressed is that of "infinity". Do tell: how is the distinction between metaphysical infinity and quantifiable/mathematical infinity of itself a "category error" of the concept of infinity?
Your assertion is a bit nonsensical at it stands.
Well there's a formal out, if you want to take it, and then there are new questions.
The formal out is that in modern logic (Frege's logic, which he developed specifically for formalizing mathematics), "every" is of course no sort of number at all. "Every" indicates a conditional: "Every sperm is holy" says "If something is a sperm, then it is holy." This veers somewhat sharply away from the old treatment of universal generality (from Aristotle and medieval logicians, the square of opposition) in that universals are no longer taken to have 'existential import'; in this case, the existence of sperm is not entailed, and the claim is vacuously true if there are no sperm to be holy or otherwise. (Frank Ramsey was even of the opinion that universal generalities were exactly this, habits or rules of inference, nothing more, and not really quantification in the way people think.)
For our case, "Every colinear point is included" says "If a point is colinear with any two points already included, it's also included." Now that doesn't say, "If a point is colinear with any two points already included, add it"; it looks like a rule for adding points, but instead it claims directly that they are all already there. The rule is the line. You don't really construct the line at all, and then know what you have constructed, but by knowing the rule, know the line.
This is how mathematics makes the infinite comprehensible. No human being will ever have the opportunity to observe a one-dimensional line of any length, much less of infinite length; but any human being is capable of understanding the rule that defines such a line.
Of course, one can say, that's not really infinity; or one can say, that really is infinity and thus no one really understands such a rule, they only know how to work with it formally, as a bit of symbolism. (I think I've now alluded to all the principle schools of the philosophy of mathematics: realism, intuitionism, and formalism, for what that's worth.)
Not sure how this fits your thing, but there it is.
Honestly, @apokrisis is the only guy I know around here who's comfortable with this sort of metaphysics, and I learned the habit of looking for constraints from him. He'll mainly tell you that whatever system you're cooking up is a partial reconstruction of his own, but he'll understand what you're up to. You know the drill.
I do think it might be worth thinking a little more about how dimensions work, because they are so explicitly a matter of adding degrees of freedom, each of which is constrained by what was previously an added degree of freedom. That's a curious pattern. There are weirdnesses we're passing by, like fractals and space-filling curves, but gotta walk before you can run.
Hope some of this has been helpful.
It's a category error because you're judging mathematical notions of infinity by some dubious metaphysical standard. One that is vague at best.
You keep coming back to a line as being "constrained" in one dimension but not another. Are you aware that a plane consists of an uncountably infinite set of lines? And 3D space consists of an uncountably infinite set of planes? Now, by your understanding, is 3D space "constrained"?
Finally, it can be shown that the cardinality of the set of points in 3D space is equal to the cardinality of points in a line. I.e., the line can be mapped onto 3D space (and vice versa). So how is the line constrained again?
Before accusing another of nonsense, try picking up a math book.
Right. In general agreement. Thoughts go back to Cantor's popularization of actual infinities.
As I've previously mentioned, I've learned that this issue - that of how determinacy (or constraint) applies to infinities - is so esoteric (such as to most of the posters on this thread) that I need not concern myself with addressing it directly. For what its worth, I've at least gained an understanding - fallible though it is - regarding the issue which the OP addressed - in part, due to the interactions in this thread.
Quoting Srap Tasmaner
Actually, it in fact is a partial reconstruction of Aristotelian causes (predating apokrisis and his system by some time). Instead of addressing these causes as "explanations to why questions", I'm addressing them (in short) as distinct determinacy types.
Quoting Srap Tasmaner
It has, and thanks for it.
In your mind this sure seems to be the case. In reality as written in all of my posts, I have only differentiated between the two - without in any way judging one by the other.
Dont know why but not answering these questions bothers me. Might be your added in snide insult.
Yes: 3D space is by its very demarcation constrained to three dimensions rather than to two, one, zero (cf. geometric points), or else more than tree dimensions (cf. the ten dimensions of space in string theory).
I grant my non-mathematician mind doesnt comprehend how the first sentence entails the second, but yes: lines will still be constrained to individual units that can be numerated. Else we wouldnt be able to discern them as lines.
A point and any geometric extension are completely dissimilar from each other. It is strange that there is no thing in-between them. A point goes from itself into segments that have as many points as as any 3D object. There is something unintuitive about this and seems to resemble something from nothing
If you are thinking of a line segment between points A and B, then philosophically there really is nothing there - its merely a hypothetical path in Euclidean spaces of shortest length from A to B. We draw it with a pen, but that always gives us a two dimensional version having width. Modern math of course says otherwise.
Quoting Srap Tasmaner
I agree. As a biophysicist/philosopher this is the goto guy. Were we talking about foundations/set theory @Tones is the resident expert. Fellow mathematician, @Real Gone Cat for math in general. For modern or theoretical physics I'm not sure who that would be. @Kenosha Kid qualified, but he has left the room. Speak up, anyone.
The history of "infinity" is over two millennia old and progress over that period was done by philosophers/mathematicians. Were Aristotle to rise from the dust today he would tell philosophical devotees to pay attention to what has been achieved and not to refer to his ancient ideas. Or do you think he would eschew progress?
I'm not an expert.
No it doesn't. There are points and there are segments between points. A point doesn't "go from itself" to something else.
I don't have "a magical view of how mathematics is physically done", so you can leave me out.
But I thought you might have some particular mathematicians or philosophers in mind. Or do you have a magical view of people existing that don't actually exist?
That is such an inpenetrable mess that it would be a task to unsort it all. But a couple of points:
If we said that there are half as many even numbers as natural numbers, then we can also say there two-thirds as many even numbers as natural numbers, and three-fourths as many even numbers as natural numbers, ad infinitum. For example:
0 2 1 4 6 3 8 10 5 12 14 7 ...
Between every odd number, there are two even numbers.
Moreover, the notion that there are half as many even numbers as there are natural numbers induces that there are infinite subsets ("groupings" or whatever you call them) of the the natural numbers in smaller and smaller size ad infinitum. We would have "half as many even numbers as natural numbers", "one-third as many multiples of three as natural numbers", "one-fourth as many multiples of four as natural numbers", ad infinitum.
Detractors of set theory are put off by the fact that an infinite set has the same cardinality of certain of its subsets. Okay. But then we would ask, "So what is your axiomatic alternative?" But the notion of infinite subsets of the naturals in infinitely descending chains of smaller and smaller cardinality is itself utterly unintuitive. You see, at least set theory does preserve the most basic and most intuitive notion of even everyday mathematical thought: Sets are the same size if there is a one-to-one correspondence: sheep and counting stones.
/
As to your "quantified mathematical infinites", "metaphysical infinities", etc., I would suggest that instead of getting vocabulary all mixed up with mathematics, you could stipulate terminology such as:
q-infinite for "quantified mathematically infinite"
m-infinite for "javra's personal metaphysical notion of infinite"
etc.
And perhaps you'd be so gracious as to provide crisp definitions of each.
That would at least show some respect to the people reading your posts by allowing it to be clear in which of the different contexts you are claiming.
Because javra is never snide, you see.
It doesn't invite mathematical proof because 'actual' and 'potential' are not mathematically defined terms.
Meanwhile, formal set theory does not use the term 'is actually infinite' but instead plain 'is infinite', which is rigorously defined. And set theory is made of rigorous, formal, objectively verifiable, indeed machine-algorithmically checkable mathematical proofs.
On the other hand, the notion of 'is potentially infinite' has not, as far as I've ever found, been given a formal definition, let alone a system in which it used. Instead, it is an informal notion that is thought to be captured by (but not defined in) certain systems. Though, countenancing adoption of such systems raises questions about how much they can prove of mathematics, their own intuitive strengths and weaknesses, and their complexity in formulation and ease or difficulty in using.
No, I clearly see what you actually posted. In earlier posts, you mentioned that there is a layman's notion of countably infinite. Then later you asked what that could mean. And I replied to the effect that that question is your problem alone, since indeed there is no layman's notion of countably infinite.
Ask any person at a busy street corner what their notion of counting the infinite is. Here are the three possible answers you will get:
"Huh?"
"You can't count infinity. Everybody knows that."
"I'm just trying to catch a cab here. Do you know any good Thai restaurants uptown?"
/
There's a bunch more written by javra that I'd like to address, but I'm out of time now.
What I have said is "countable infinity" ... not "countably infinite".
Sorry, but I have better things to do that to spend more time in addressing such replies.
Oh come on! How captious can a person get?
'countably infinite' and 'countable infinity' are tantamount to each other.
'countable infinity' though is less apt, since there is no object that is infinity. Rather there are different things that are infinite.
And lately you wrote:
Quoting javra
'countably of infinity' is not even English.
To move past your ridiculously captious objection that I said 'countably infinite' (a coherent notion, used as a favor to you) rather than 'countable infinity' (an unclear notion since there is no object that has the name 'infinity'), instead I'll couch using only 'countable infinity':
You said there is a layman's (or whatever synonym of 'layman' you used) notion of countable infinity. Then you asked what "the non-mathematical countably of infinity would mean to the general audience (in other words, presumably, a layman audience).
And my point stands, there is no layman's notion of 'countably of infinity' (let alone that it's not English), nor layman's notion of 'countable infinity', nor (put better) layman's notion of 'countably infinite'.
/
Quoting javra
So your excuse for not dealing with the point that there is no layman's notion of a countable infinity (better put, of countably infinite) is that I used such a slight variation in phrasing.
Meanwhile, there's a trail of other falsehoods, blatant misconceptions and nonsense you've posted, and I explained your errors. As well as your posts in this thread are an impenetrable morass of your ersatz undefined terminology, with various ersatz undefined qualifiers popping in and out and out and in again. Any fault for you not being understood is yours, not your readers.
The notion that some infinite sets are countable is a special mathematical notion. It is technical and has a rigorous formal definition. There is no layman's version of it.
No one doubts that you are a grown up person who can do numerical reckoning just fine. The point is that you don't know anything about the mathematical notions that you have referred to in this thread. And I don't mean your philosophical notions, but rather the specific mathematics you have also referenced. Being able to balance a checkbook is nice, but it's not an informed understanding of the mathematics you've commented on.
Your neck of the woods is a fantasy place. People in everyday life don't take 'countable' to mean "one could, in principle only, count an infinite series of elements/units/items for all of eternity yet to come and still never get to finish."
I'd like to see just one person at your local supermarket in your neck of the woods who says anything like, "My understanding of what 'countable' means? Oh, it means that in principle you could count an infinite number of things for eternity but still not finish."!