We Are Math?
When we drill down to the deepest level of matter, we get the quantum wavefunction, a mathematical object that uses regular numbers (real numbers) and complex numbers which are based on i, the square root of negative one. We begin with matter, trace its source, and end up with a complex-valued mathematical wavefunction. Hm.
We have quantum entanglement, which says that signals can travel faster than light. Einsteins relativity theory says nothing can travel faster than light. We have an apparent contradiction. The contradiction can be resolved if we restate Einsteins relativity theory as nothing can travel through spacetime faster than light. If the signals somehow bypass spacetime, then the contradiction is resolved. Hm.
But what is outside of spacetime? Abstract objects like thoughts and numbers. For instance, the number 2 exists outside spacetime. A complex-valued mathematical wavefunction is an abstract object which exists outside spacetime. Hm.
Which suggests that realitythat me, you, Earth, universe, etc.is fundamentally some sort of abstract object existing outside spacetime. Hm.
Notes:
1. The status of abstract objects is an open philosophical question. The answer I accept is that they exist outside of spacetime. In particular, mathematical objects exists outside space time. Reference: https://plato.stanford.edu/entries/platonism-mathematics/
2. Some wavefunctions are functions of time. That doesnt mean the function itself exists in time. As a simpler example, the equation
v(t) = u + a*t (velocity v equals initial velocity u plus acceleration a times time t)
is taught in high-school physics. Velocity is a function of time. But the equation itself is unchanging. Its a thought outside spacetime.
We have quantum entanglement, which says that signals can travel faster than light. Einsteins relativity theory says nothing can travel faster than light. We have an apparent contradiction. The contradiction can be resolved if we restate Einsteins relativity theory as nothing can travel through spacetime faster than light. If the signals somehow bypass spacetime, then the contradiction is resolved. Hm.
But what is outside of spacetime? Abstract objects like thoughts and numbers. For instance, the number 2 exists outside spacetime. A complex-valued mathematical wavefunction is an abstract object which exists outside spacetime. Hm.
Which suggests that realitythat me, you, Earth, universe, etc.is fundamentally some sort of abstract object existing outside spacetime. Hm.
Notes:
1. The status of abstract objects is an open philosophical question. The answer I accept is that they exist outside of spacetime. In particular, mathematical objects exists outside space time. Reference: https://plato.stanford.edu/entries/platonism-mathematics/
2. Some wavefunctions are functions of time. That doesnt mean the function itself exists in time. As a simpler example, the equation
v(t) = u + a*t (velocity v equals initial velocity u plus acceleration a times time t)
is taught in high-school physics. Velocity is a function of time. But the equation itself is unchanging. Its a thought outside spacetime.
Comments (256)
Yes, we humans are essentially "abstract mathematical objects"*1 in space-time. I have arrived at a similar conclusion, except I typically use a more general term for reference to both the subjective objects of minds, and the objective things of physical senses : Information. From a scientific perspective, Mathematics*2 may be the fundamental aspect (essence) of reality. But, for Philosophical purposes Information*3 may be more broadly applicable. Math seems to be the most abstract form of Generic Information*4, yet it is the logical structure of the physical world.
Abstract objects*1 are not knowable by physical senses, but only by mental introspection or by communication with other minds. So, they are in the space-time world, but not of the physical world. Space-time is itself an abstract concept, that we measure indirectly by observing physical changes in the environment. Even the causes of change, Energy & Forces, are abstract concepts, not material things. We only know them indirectly by their effects on matter.
In my personal thesis, I refer to the universal power-to-enform (causation) as EnFormAction*5. And the logically necessary First Cause (the Enformer, the Programmer, the Great Mathematician, etc) is the only abstract thing that exists prior-to and outside the evolving (self-enforming) space-time world. I assume that you lean toward Platonism instead of Nominalism. Can you see the connection between Enformationism and your own proposal of a Mathematical universe? :smile:
*1. Abstract Object :
One doesnt go far in the study of what there is without encountering the view that every entity falls into one of two categories: concrete or abstract. . . . Though there is a pervasive appeal to abstract objects, philosophers have nevertheless wondered whether they exist. The alternatives are: platonism, which endorses their existence, and nominalism, which denies the existence of abstract objects across the board.
https://plato.stanford.edu/entries/abstract-objects/
Note : while Mathematical Objects are typically accepted as real, in some sense, by pragmatic physicists & mathematicians, their Ideal (abstract ; non-concrete) existence puts them in the same ontological category as Souls & Ghosts. Hence, philosophically controversial.
*2. Mathematical universe hypothesis :
The theory can be considered a form of Pythagoreanism or Platonism in that it proposes the existence of mathematical entities; a form of mathematicism in that it denies that anything exists except mathematical objects; and a formal expression of ontic structural realism.
https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis
*3. Information :
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed.
https://en.wikipedia.org/wiki/Information
*4. Information :
Knowledge and the ability to know. Technically, it's the ratio of order to disorder, of positive to negative, of knowledge to ignorance. It's measured in degrees of uncertainty. Those ratios are also called "differences". So Gregory Bateson* defined Information as "the difference that makes a difference". The latter distinction refers to "value" or "meaning". Babbage called his prototype computer a "difference engine". Difference is the cause or agent of Change. In Physics its called "Thermodynamics" or "Energy". In Sociology its called "Conflict".
http://blog-glossary.enformationism.info/page11.html
Note -- Information (EnFormAction) is generic in the sense of causing all forms of being in the universe.
*5. EnFormAction : The creative act of enforming; to give form to the formless.
Ententional Causation. A proposed metaphysical law or principle of the universe that causes random interactions between forces and particles to produce novel & stable arrangements of matter & energy. Its the creative force of the universe. AKA : The innovative power of Evolution; the power to enform; Logos; Change.
http://blog-glossary.enformationism.info/page8.html
This makes no sense to me. A category error at least. "Existing" north of the North Pole ... :roll:
If this is actually the case, however, I also don't see what non-trivial difference being "fundamentally some sort of abstract object" would make with respect to (our) human existence (pace M. Tegmark).
No it doesn't! From the physics stack exchange:
[b]Entanglement between two qubits means that if a measurement is made on one of them, the other one is decided instantaneously.
This is true, but this does not allow for faster than light communication. If you have one qubit with you and I have one qubit with me and you make a measurement on your qubit, that will mean my qubit is decided . But how does that send any signal ?
Later on, when I make a measurement on my qubit, I get a measurement, just as I would have got some measurement had you not measured first. There is no way for me to know that I got this measurement after you had measured yours or before you have measured yours. Hence, no signal can be sent faster than light using entanglement[/b]
Definitely, but one that is made repeatedly hereabouts. Folk want this sort of nonsense to be the case. What's that about, then?
If the distance between us was 1 light year. Then I could send you a message that says.
Measure your entangled particle the moment you receive this message.
I would then have to know very precisely (I think) when to measure my item just before you do, 1 light year away. I would then send a message to you 1 light year away, asking you what your measurement was. If I had measured a 1 then you will have read a 0.
But it will take 2 years after the actual event, for me to confirm this, so no signal is travelling faster than light in this thought experiment.
I don't know if this is correct but it's what I understand as quantum entanglement at the moment.
Defibrillation for dying theosophies imo.
Sorry, I didn't read this part carefully enough before responding. I speculate that there is no travelling needed in certain conditions, due to QFT. Every 'point' in space can manifest any particle/excitation at any moment of time. So the entangled state does not 'travel' as it is already there and only requires measurement.
Which is not matter, just a way to predict changes in matter, energy, etc.
I keep bringing up my late Corgi, Jake, and his inability to understand algebra, though I'm sure he tried. Humans may never "understand" matter and energy at some ultimate level, but, like Jake, they can enjoy chasing metaphysical sticks and learn how to manipulate physical entities. (He learned how to manipulate me)
You'll get some negative feedback for that assertion. Actually, at first experimenters were baffled by the "entanglement effect" which seemed to imply faster than light communication. Since then though, other explanations for the instantaneous correlation between particles have been proposed. I'm not a physicist, so I prefer a model that fits into my personal information-theoretic worldview. From that perspective, there is no movement of matter, energy, or information between entangled particles. Instead, the opposite spins are metaphorically two sides of one particle. And all particles in the universe are non-local & unreal (virtual) until triggered by an interaction to manifest with physical properties. In other words, the world is a single holistic (non-space-time) reservoir of infinite Potential, until transformed into Actual bits of matter/energy. The particle doesn't have to go anywhere, because it's already there.
Unfortunately, that holistic description will not make sense to those with a Reductive scientific paradigm of reality. But it fits neatly into the philosophical Enformationism worldview, in which abstract (non-concrete) Information is the fundamental substance (essence) of the world. Another way to look at it is to say that abstract Mathematics is the logical structure (interrelationships) of reality. Mathematics (numbers ; ratios ; equivalences), like Logic, exists only ideally, with no physical properties at all. The science of mathematics is a product of human inference & imagination, hence Idealistic instead of Realistic -- a theory instead of an observation.
However, the human brain is programmed, by pragmatic evolution, to interpret abstract relationships in concrete terms. Consequently, our worldviews are seen through a matter-based frame. So, Materialists and Nominalists are merely saying what they are seeing with their eyes. But, as cognitive scientist Donald Hoffman has proposed, those real things we think we see are merely "icons" or symbols (mental representations) of the underlying reality, which is mathematical or informational. So, Realists are seeing their own conceptual models of reality, not ultimate reality.
Hoffman uses the metaphor of a computer interface to describe how our brains are deceived by our own pre-conceptions. But, in keeping with the Enformationism thesis, I like to use The Matrix movie as a metaphor. In one scene, Cypher is showing Neo the green raining code, and remarks that "I don't even see the code anymore". Like computer screen icons, the code is an abstraction of an underlying reality -- or in this case a simulated reality. Perhaps the real world your senses perceive is a simulation of the true reality : the mathematical information (code) that constructs the world of the senses. :smile:
The Evolutionary Argument Against Reality :
The world presented to us by our perceptions is nothing like reality
https://www.quantamagazine.org/the-evolutionary-argument-against-reality-20160421/
Don Hoffman :
The Case Against Reality
https://en.wikipedia.org/wiki/Donald_D._Hoffman
SEMIOLOGY : REFERENCE vs REFERENT
Simulated Reality Code :
Matrix digital rain, Matrix code or sometimes green rain, is the computer code featured in the Matrix series. The falling green code is a way of representing the activity of the simulated reality environment of the Matrix on screen by kinetic typography.
https://en.wikipedia.org/wiki/Matrix_digital_rain
Don't see the code anymore :
https://kugelmass.wordpress.com/2012/03/29/i-dont-even-see-the-code-anymore/
SEE THE REFERENT (the object being described) NOT THE REFERENCE (the code, symbol, model)
Quoting 180 Proof
Does it make sense to you that our deepest description of matter is the wavefunction?
Does it make sense that the wavefunction is a mathematical function?
Does is make sense that mathematical functions exists outside spacetime?
"Deepest description" so far ... Why wouldn't that "make sense"?
This question, Art, doesn't make much sense. What else would / should "the wavefunction" be if not a mathematical function?
Another incoherent question. Abstract objects subsist in minds and minds exist are embodied spatiotemporally.
What book?
No, quantum entanglement says measurements will be correlated - a very different thing. As physicist Asher Peres noted, "relativistic quantum field theory is manifestly local." (longer quote here).
Quoting universeness
Yes though the term "instantaneously" can be misleading. The other qubit has to be measured and the results compared. As Peres also noted, "unperformed experiments have no results" (see counterfactual definiteness).
Quoting universeness
That's correct. But note that, per relativity of simultaneity, the order of the measurements can potentially differ in each particle's reference frame. The point is that each measurement is local. One measurement doesn't cause or influence the other measurement.
How do you know it's not interdimensional vibrating superstrings?
How do you know our universe is not a result of interacting branes?
:up: But if the two items 1 light year apart are entangled, then surely "However, if the events are causally connected, precedence order is preserved in all frames of reference." From the wikipedia article: https://en.wikipedia.org/wiki/Relativity_of_simultaneity applies?
Good point. Some philosophers say that material objects exist but abstract objects subsist.
So, let's change the third question.
Quoting Art48
Does it make sense to you now?
My understanding is that Einstein's famous "spooky action at a distance" concern was about something going faster than the speed of light. Here a quote from an article of Astronomy.com
- https://astronomy.com/news/2022/10/what-is-quantum-entanglement
"The strange part of quantum entanglement is that when you measure something about one particle in an entangled pair, you immediately know something about the other particle, even if they are millions of light years apart. This odd connection between the two particles is instantaneous, seemingly breaking a fundamental law of the universe. Albert Einstein famously called the phenomenon spooky action at a distance.
No. But I did look at another of his long essays, and he seems to be generally well-informed. In this thread, I'm only responding to the concepts expressed in this thread, not to Art's book. I'm aware that some of his ideas are fringey, but so are mine. That's why I try to encourage him to explore beyond the known into terra incognita, despite negative feedback.
For philosophical purposes, I'm not concerned about compatibility with "settled science", as long as the general idea makes sense to me (sounds logical). The notion of Mathematics as the foundation of physical reality, corresponds to my own understanding that General Information (which includes Math & Energy) may be the essential structure of Reality. That's not "settled science", but some prominent scientists are enthusiastic about such non-physical (abstract) aspects of Nature/Culture. :smile:
PS__I don't think that the mathematics of physics & minds is "outside of space-time". But, as non-physical abstractions, mathematical concepts only exist mentally & ideally, so not directly affected by the causal changes that we interpret as space-time. However, I do go so far as to postulate a timeless First Cause to explain the existence of our physical -- and meta-physical (mental) -- world. But I don't presume to speak for that hypothetical entity.
No. "Outside spacetime" is as incoherent as north of the North Pole. And to subsist is to be thought by minds which are, as I've pointed out already, embodied spatiotemporally; so the question remains doubly nonsensical to me.
OK, I suppose that's one view of abstract objects. Another view is that they [s]exist[/s] subsist outside spacetime. For instance, "two plus two equals four" subsisted and was true before the big bang.
In your view (if I understand it correctly) the thought "two plus two equals four" didn't exist immediately after the big bang because there were no minds to think it. Is that right?
People like to make up stories. We pass them from one person to another, keeping the bits we like and adding new stuff as we go. The talk "grows in the telling", as Tolkien put it.
We've found, over time, that there are ways in which we can make our stories of how things are more useful. So we find that a story that is liked by more folk will make it easier for folk to work together, for example. A story that explains a wider variety of situations or circumstances will have more uses than one that explains only a small part of the world. We've also found that we can check our stories, one agains the other, and modify them so that they are more consistent.
Where the story is about how the world is, this leads to taking a conservative approach to story telling, Small changes are preferred, because it is easier to see what their repercussions are, how they effect the other parts of the story as a whole.
On the other hand a desire for excitement or entertainment might lead one to posit large-scale changes to the story. Their ramifications will be greater, and their flaws more easily seen.
Either way, one evaluates changes in the story in terms of the other parts of the story. The process is holistic, consisting in a critique of the web of belief.
Put otherwise, anyone can make shit up. We need an evaluative eye that can spot the crap.
's suggestion that reality is an abstract object fumbles the clear and happy distinctions that we make between abstract or concrete, on the one hand, and real or not real, on the other. It muddles ideas from the every day and ideas from quantum theory. It is far more direct and reasonable to posit that abstractions such as property, marriage, and complex numbers are stuff we made up than to imagine them exiting in the way chairs and trees do, but in some parallel reality.
Along side a desire for stories of breadth and completeness, we need to foster a critical attitude. That seems to be missing here.
Sure. And philosophers tend to be good at "making sh*t up". Some of it turns out to be pragmatically useful, in which case science takes over to make use of the ideas. And this forum is an arena for presenting ideas to a wide variety of critical eyes. Some here find Hegel's ideas useful for their insights into the "teleology of history", while others find fault for the same "sh*t". Suum cuique."To each his own"
Yet all too often what we get is not constructive criticism, but censorious or condemnatory attacks on ideas that don't conform to a personal belief system. My comments in this thread are not intended to be scientific criticism, since I'm not a scientist with expertise in the mathematical concepts presented here. Instead, I'm trying to be supportive of philosophical exploration of ideas that are of interest to me personally --- especially the philosophical implications of Information Theory and Quantum Theory, which are ripe targets for both positive & negative criticism. :smile:
PS__FWIW, I did quibble about his notion of Mathematical objects as existing "outside spacetime". Does that count as critique, from your perspective? Math is indeed something that humans "make up", but based on observations of relationships that exist or persist within space & time. Then again, space-time is also an abstract concept (mental model), which is intended to describe observed changes in location and in relation. Yet the generalized or universalized concept of Mathematics seems to point beyond any particular brain/mind. So, where could it be located in space-time?
Critique vs. Criticism :
In general, criticism is judgmental and focused on finding fault, while critique is descriptive and balanced.
https://medium.com/storygarden/critique-vs-criticism-36ddf0d191ff
Was mathematics invented or discovered? :
Both discovered and invented. When humans perceive the world through consciousness, everything is an abstract entity without a pre-defined representation. When we discover something in the world (such as ability, physical object, event, causality, pattern, etc.), we start to use our minds to describe it. Our consciousness will create personal representations of everything for reasoning and thinking
https://www.quora.com/Was-mathematics-invented-or-discovered-1
Note -- We "discover" consistent patterns of inter-relationships between objects & actions, and then we "invent" formal symbols & language to allow us to discuss the invisible Logic that serves as the underlying inter-connection structure of the physical world.
As an ultimately abstract entity, I enjoy the company of so many numerous abstract entities. We often discuss if there really are concrete objects, but conclude they are just grammatical fictions.
More or less, although most math people give this question little thought. In my case, I was introduced to a notion years ago in my PhD studies. A little later on I decided to extend this idea to a more general realm - a sort of creative step. Once the basic ideas of the concept were set, then came the acts of discovery - finding what flows forth logically.
The varied origins of mathematical concepts involve both complicated regressions and sheer moments of genius. My advisor would say there are no original ideas in the subject - they all trace back to antiquity (I would disagree). Take the way we measure distances (Euclidean); some time back the notion of an abstract "distance" was conceived, in a space of "objects". Thus was born the study of metric spaces, which eventually led to or coincided with the development of topological spaces. And on and on . . .
I would think that reality is everything that exists including appearances of it sentient beings are aware of and everything they aren't aware of too. Even so these days I tend to regard the two dimension, massless particle-wave realm 'outside' or beyond the Higgs field as more fundamental than the three-dimension world of experience.
Tegmark speculated the Multiverse is made of math; that a logical structure exists in it that corresponds to reality and is based on oure math. THe idea of a mathematical universe brings one to think that there is a logical structure to the universe including the quantum world that seems necessary for temporality, motion, being and nothingness, change, evolution and so for th yet of course it could all be in the thought of God. An infinite Multiverse pre-existing from eternity in an Everett-Tegmark Level 4 Mutliverse construction perhaps. It's interesting to consider. String theoriy's Mutliverse paradigm with mathematical, logical structure would be interesting for mathematicians to consider and for others like myself to read about.
The particle measurement events aren't causally connected (i.e., correlation is not causation). So the precedence order need not be preserved in all frames of reference.
Quoting Causality (physics) - Wikipedia
Quoting Art48
Yes, that was his concern (which, more generally, was about wavefunction collapse). However QM neither specifies nor requires non-locality. Further, locality is one of the axioms of quantum field theory.
Quoting Principle of locality - Wikipedia
How is two plus two equals four subsisting outside space-time different to two plus two equals four existing outside space-time ?
I don't think its as clear cut as you suggest as the nature of the relationship between two 'entangled' items is not fully understood when we consider the following description:
Causation
[i]Causation is an action or occurrence that can cause another. The result of an action is always predictable, providing a clear relation between them which can be established with certainty.
Causation involves correlation which means that if an action causes another then they are correlated. The causation of these two correlated events or actions can be hard to establish but it is certain.
Establishing causality between two correlated things has perplexed those that are involved in the health and pharmaceutical industries. The fact that an event or action causes another must be obvious and should be done with a controlled study between two groups of people.
They must be from the same backgrounds and given two different experiences. The results are then compared and a conclusion can then be drawn from the outcome of the study. The process of observation plays a significant role in these studies as the subjects must be observed over a certain period of time.[/i]
Correlation
[i]Correlation is an action or occurrence that can be linked to another. The action does not always result to another action or occurrence but you can see that there is a relationship between them. Although the action does not make the other thing happen, the possibility of having something happen is great.
Correlation can be easily established through statistical tools. The correlated events or actions can be because of a common cause. Establishing correlation can be made certain if there are no explanations that will prove causality.
When you say that exposing kids to too much violence on television and films causes them to become violent adults can be untrue. Although violence on television and films can influence behavior, adults who are violent might have acquired the habit due to other factors such as poverty, mental illness, physical, mental, and emotional abuse as children.
It is therefore wrong to assume that violent behavior is due to television and films because there are several different aspects to consider. It is safer to say that there is a correlation between watching violent television shows and films and violent behavior than to say that violence in television and films causes violent behavior.[/i]
Summary
[i]1. Causation is an occurrence or action that can cause another while correlation is an action or occurrence that has a direct link to another.
2. In causation, the results are predictable and certain while in correlation, the results are not visible or certain but there is a possibility that something will happen.
3. Establishing causality is harder while there are many statistical tools available to establish correlation between events or actions.[/i]
I think the use of the term correlation for quantum entanglement is a wise use but really just indicates that the detailed nature of the relationship is not yet well understood.
I got an MA and did 2 years towards a math PhD (a PhD dropout, in other words). To me, math objects just seem to be there, much like a tree is there. I feel I can see numbers, fractions, etc., much like I see a tree. Mathematical Platonism seems to describe my experience.I read about Formalism but it doesn't "click" with me. P.S. there's a math prof on YouTube who questions if real number "really" exist. His name is N J Wildberger. https://www.youtube.com/watch?v=fXdFGbuAoF0
Quoting RussellA
I usually use "exist" for both cases. Another person brought up the exist/subsist distinction, so I used the word "subsist". Some philosophers (ex, Russell's "On Denoting" if I recall correctly) use the exist/subsist distinction where "exists" applies to things in spacetime, and "subsist" applies to abstract objects.
Quoting Art48
I will never look at my calculator in the same way again
In order for my calculator to have access to numbers, if numbers exist outside space-time, then my calculator during a calculation must also exist in part outside space-time.
The interesting question is that once the calculation is finished, how the calculator is always able to return to this world at the same time and location it left.
If we could discover how this happens, we could perhaps manipulate the return of the calculator to a slightly different time and space, thereby creating a Tardis-like machine.
The implications of numbers existing outside space-time are certainly truly staggering.
When we look up, at a highest level of matter, we see the Universe ...
Microcosm and macrocosm ... Small order and great order ...
We have formulated laws for both. And in some cases or from some aspects we can see a similarity.
OK, if the number 2 is in spacetime, where is it? And when?
How can we be what we have created them?
Can we be Ido or Esperanto?
Mathematics is a language. The language of Science. All languages are products of human thought. They are symbols that we have created in order to describe things in our environment and communicate with each other.
In its most basic form, Mathematics is numbers. They came into existence the moment man has started to count.
Mathematics are inherent to humans. Infants start to count before even they know what numbers are. And when they do that, they come into contact with the nature of Mathematics.
However, the various systems --or what we call fields-- of Mathematics are not inherent to humans. These include algebra, geometry, trigonometry, calculus, probabilities, etc. There also exist or have been existed different mathematical techniques, which are not even included in the list of the ones used in the Western civilization. For instance, Vedic Mathematics. All these are developed by humans, based on a multitude of conditions, with culture being the main one, as well as factors that have to do with the purpose of their usage and so on.
But the main point is this:
We cannot be the product of our thought. We cannot be our thoughts. We cannot be what we produce.
It exists in the mind as a concept, and it exists when I think about it, in the same way that government, love, apple, despair, mountain, etc exist in the mind as concepts.
It helps that there is a regularity in nature, and our numbers can model that regularity.
Even if numbers did exist outside our space-time, not only would we not know about them but also we wouldn't be able to access them. But as we do use numbers, accounting for the success of science and mathematics, the numbers we use must exist within our space-time.
Taking a simple example, an abacus can manipulate numbers, yet the abacus neither needs to nor would be able to access anything outside its own space-time.
Other folk say the same thing about angels and fairies.
Numbers and other mathematical entities are not a thing we talk about but a way of talking, a grammatical form. Like money, property and institutions they are a construct of our collective intent. They do not "exist" in someone's mind, nor in some unseen parallel reality.
How do you know that?
Institutions
1 does not refer to anything
Happy to consider alternatives. This is an approach that does not rely on confabulating mysterious entities.
Sounds sort of like trope theory.
You're offering a particular theory. It's about as well founded as any other, isn't it?
Nope.
Do you have a salient point?
Just noting that you state your theory as if it's a fact. May just be a custom.
Use your own bait.
So you're fishing? Have at it.
So you're fishing? Have at it.
Depends on the definition of "thing". My thing may not be your thing, but anything is better than nothing. :cool:
English is a language. Yet it can refer to objective reality, to things which exists independently of us. (There's a tree in my yard.). An image of the tree exists in your mind. But no actual tree is to be found between your ears. Similarly, math is a language that refers to objective reality, for instance, the number 2.
Quoting RussellA
Does your mind create the concept of 2? Does the concept of 2 cease to exists when you stop thinking about it? And if you create it, can you make it anything you wish? Can your 2 be an odd number? If it's your concept, why not? Why can't your 2 be greater than your 3? Because numbers have objective properties.
In contrast, Sherlock Holmes existed as a concept in the mind of Arthur Conan Doyle. Therefore, Doyle had the freedom to describe Holmes. He could have made Holmes short, tall, British, Scots, or even French.. But 2 is objectively real. That's why you cannot give your 2 any properties you wish. When we say "2+2=4" we are talking about objective reality, not any particular 2 in the mind of any individual person.
Oh oh. Metaphysician Undercover might challenge that! :gasp:
Math is usually associated with numerical Quanta, while Logic is associated with semantic Qualia. Ironically, both are expressed in "values" (numerical & meaning), and both are forms of Consciousness. That may help to explain why math overlaps both classes of experience. We become aware of individual objects, and infer their quantitative relationship to a collection of objects. Then we can deal with the group as-if it was a singular object (set ; whole system ; holism). So, maybe once we discover the "basic idea" of objective things & groups, we can discover (create) their subjective value (meaning) to the observer.
Sorry, I'm just riffing on your "creative step" notion. Our senses become aware of non-self things, that have only numerical value. But then, rational inference discovers a possible (logical) connection between thing and self. Hence, external objective Quanta (impersonal value) is transformed into personal Qualia via the "creative step" of inference (imagining thing & self together). The Measurement problem of Quantum Physics may be a case of crossing the line between Quanta/Qualia, numerical/personal, object/subject. I have to go now, but I may try to "extend this idea" at a later time. :smile:
Oh, of corse. But we might be able to avoid things such as...
Quoting Art48
, words can be used to talk about stuff, sure. Are you suggesting that what is being referred to is the image of the tree in your mind rather than the tree in your yard? That's not an uncommon confusion.
Quoting Art48
What if instead of "one mind creating the concept of 2", it is a construct of our communal capacity to use language - a way of talking about cases where we have two things? "There are two trees in my yard" as a way of talking about the trees, and so not a reference to some platonic form.
The concept "2" is just the capacity to do things like count to two, share two things, give someone two apples, and so on. The properties of 2 that seem objective are constructed, in the same way as spelling or sentences having a subject and a verb.
Quoting Art48
Notice that Sherlock is not restricted to the mind of Doyle - after all, Sherlock is still around whilst Doyle's mind is long gone.
The fixity of the properties of 2 might be much the same as the fixity of the spelling of "cat".
Consider the root of -1; it wasn't considered "real" until a way to talk about it in a consistent fashion was found.
All this by way of pointing out that there are other, better, options than platonic realism.
And that by way of showing that we are not maths.
It's a problem not unlike the difficulty dualism has in explaining how mind interacts with a body.
How does two interact with trees in order for there to be two trees?
See Benacerrafs Epistemological Argument. While that argument uses causal constraint, there are other variations that do not.
There is no need to assume any such thing as a number. We have numerals which are symbols, and the symbols having meaning which is dependent on the context of usage, like all symbols. The assumption of numbers is just a useful fiction employed by mathemagicians, which allows the ontology of Platonism to overrun the sciences.
In responding to "math is a language", I pointed out that language can refer to objective reality. The word "two" refers to the objectively real number 2, just as "tree" refers to an objectively real tree. I meant to say the tree image (or concept) in our mind corresponds to an objectively real tree, and the concept of "two" refers to something objectively real.
Quoting Banno
I just don't believe the concept of "2" is created. Yes, we come to apprehend it. But when we come to apprehend a tree, we don't believe we created it. I believe intelligent aliens would have the same concept of "2" as us.
Quoting Banno
If Sherlock is still around, where? Somewhere in spacetime? No, it seems to me concepts exists outside spacetime.
That would only be the case if a Platonist ontology is true, that there is such a thing as the number 2. Since "2" is used in numerous different ways, it's very doubtful that there actually is an object referred to by "2".
Quoting Art48
Things get complicated very quickly here. Look again at "The word "two" refers to the objectively real number 2, just as "tree" refers to an objectively real tree". So here I take it that we are talking about, say, the tree in your yard? So "tree" here is a reference to an individual. Is 'two" an individual in this way? So are you saying that when I talk about your two feet and then the two dollars in my pocket, I'm talking about the very same, individual, two? That "two" is a proper name for an individual? The same individual, two, can't be both in your feet and in your pocket, so it must be outside of space and time, is that the thinking?
But if it is outside of space and time, how is it we can talk about it being in your pocket at all? If the two dollars is not in your pocket, where is it? If your two feet are not in space and time, where are they?
Hence,
Quoting Banno
Sorry, good try, but an appreciation of creativity and discovery comes with involvement, not philosophical chatter.
Like the Four Horsemen, leaving a trail of despair . . . :cry:
Quoting Art48
With which I agree. But I think it is in conflict with your first statement, namely that numbers are objective (reality). It may be a question of interpretation ...
Anyway, this is my view on the subject:
Numbers are abstract objects. They do not actually exist.
The same applies to words.
They can be both spoken and written, but what we have then is only sounds and visual images, not the words themselves.
Numbers, like words, are not material. They only exist in our mind.
Note:
Stanford Encyclopedia of Philosophy says that they don't even exist in our mind:
"numbers are neither material beings nor ideas in the mind"
https://plato.stanford.edu/entries/abstract-objects/
***
Addendum
Re: "Numbers are abstract objects". The term "objects" might be confusing because it normally refers to something physical. I could use the word "things" --which is more general and can refer to anything-- but it's too commonplace and banal. So I prefer to use neither and say, "Numbers are abstract".
I may be misunderstanding what you mean by "outside spacetime". I think of "spacetime" as what we exist in, the three dimensions of space and one dimension of time. I think of a chair as being inside a room and the moon as being outside a room. How can my concepts, which I believe exist somehow within my brain, within the three dimensions of space and one dimension of time, be outside spacetime ?
The different glyphs are just different ways to combine images of a Greco/Roman architectural column.
So if numbers are just picture representations then suggesting they exist outside of spacetime, suggests that all human manifested symbols or imagined images exist outside of spacetime, which just brings us back to BS ways to try to find some significance in woo woo thoughts of 'outside this universe.'
I assume aliens will have an efficient way to describe how many planets are in the solar system they come from but that does not mean quantities have an objective reality.
I don't think quantities have an objective reality. I think they are subjective. Do you reference a football team or 11 football players or 22 arms or 110 fingers etc.
Do you see a pint of water or a combination of 2 half pints or so many water atoms?
How many branches make two trees?
Quantities are subjective. 1 star or 2 gases (hydrogen and helium)?
If locality is the case, then the common cause of the entangled particles' correlation is their initial preparation (see spontaneous parametric down-conversion).
Quoting universeness
:up:
In set theory, I think we can say that the number two is an individual in this way.
Quoting Alkis Piskas
It might appear like a very acceptable approach, to say that abstract objects are objects, only a different kind of object from physical objects, but then we need acceptable principles to set the two apart, or else we'll have equivocation between two types of "objects" in logical proceedings. As Banno indicated, this is problematic, because it presents the issue of interaction between the two types.
What Plato showed, is that if there is a distinct class of objects which are abstract (intelligible objects), then we must place all the ideas, including moral ideas such as "just" and "virtue", and aesthetic ideas like "beauty" into this class of abstract objects. Then the subjectivity of the supposed 'abstract objects' becomes apparent.
What Plato described is that the objectivity of abstract objects is provided for by "the good", because the good is the "object", or "objective", in the sense of the goal. Abstract objects are "objective" in the sense that they are useful toward goals.
This sets up the solution to Banno's problem of interaction, the good is the means by which the two types of objects interact. Furthermore, we have the principles here to properly distinguish the two types of objects in an effort to avoid the logical fallacy of equivocation. The abstract, or intelligible objects are associated with intention, (goals), as "the good" which is desired, and the other type of objects, sensible objects, are associated with the material world as particular things, which Aristotle assigned the law of identity to. By the principles demonstrated in Plato's cave allegory, the intelligible objects are more 'real' to us because we understand them directly with the mind, rather than through the unreliable medium of sensation. That this is truly the case is supported by the fact that there is a separation between the identity which we assign to a particular sensible object, and the identity which the object has in itself.
Phew! That link was to a physics level that is a bit high for me. I clicked on sub-links such as 'non-linear crystal,' 'etc to gain a better insight. But I found I had to click on further and further sub-links eg 'Schwinger limit' and then 'Birefringence,' to gain any clarity. I will go back to it, but you have moved past my current width and depth of physics understanding.
My mind creates my private concept of something in the world publicly named as "two".
Imagine at a particular place and time in the world there is something. The public name "two" is attached to this something by the authorities. From my observation of this something, in my mind I have the private concept two. Someone else observing the same thing will also have the private concept two. However, it may well be that my private concept two is different to their private concept two, but as we are both part of the same community, we will both name our private concepts as "two".
My mind has created my private concept two, someone else has created their private concept two. But as we are both part of the same community, the public concept "two" continues to exist even if I stop thinking about it.
Quoting Art48
Some aspects are objective, others subjective
The something that I have observed in the world about which I have the concept two is objective in the sense that it exists independently of me. Because the public name "two" has been attached to this something, the concept "two" is objective in the sense that it exists independently of me within the community. As regards my private concept two, it is objective in the sense that it somehow exists within my physical brain, but it is also subjective in the sense that no one else can ever know my private concepts.
Numbers don't refer to individuals, they describe the parts of the individual
In the world is something that has been given the public name "two" about which I have the private concept two. Starting with two things in the world each of which exists, the question is, when brought together, does a new existence come into being, where this new existence has each thing as a part. Does the number two exist as a new whole in addition to the existence of the two parts that make it up ? Not according to Hume, Kant, Frege or Russell.
As the problem of numbers involves language, Bertrand Russell's On Denoting may shed light. For example, in the sentence "the author of Waverly was Scott", the phrase "the author of Waverly" is not a referring term, in that it doesn't refer to an individual having an independent existence. It is a quantificational expression, a definite description of a set of properties that makes up "the author of Waverly". Frege and Russell believed that existence was not the first-order of an individual, but the second-order of a concept.
Similarly, the phrase "the number two" is not a referring term, in that it doesn't refer to an individual having an independent existence, but rather is a quantificational expression, a definite description of the separate parts that makes up what is known as "the number two".
In language, the phrase "the number two" doesn't refer to an individual having a unique existence, but is a description of the separate parts that make up what is known as "the number two".
Quoting Art48
It cannot
I observe something in the world that has the public name "one" and I have the private concept one. I observe something different in the world that has the public name "two" and I have the different private concept two. I observe "one" added to "two", and when observing this new something, I have the private concept three. As my concept of three has resulted from an addition to my concept of two, my concept of two cannot be "greater" than my concept of three.
Good question What do you think of the following explanation for explaining interaction?
Regard a human being as having four-parts: body, emotions, intellect, and consciousness. Consciousness receives input from seven sources: the five bodily senses of sight, sound, smell, taste, and touch. It also receives input in the form of emotions and thoughts. (If an eighth sense, like ESP, exists, that wont impact the argument.) So, when we see a tree, we actually see only light. (It could not be otherwise because we lack a specific tree-sensing sense). The actual input is patches of green and brown. Based on that input, the idea tree arises in our mind. If we touch the tree, we experience a rough surface, which gives us confidence that the tree is not an hallucination. If we see multiple patches of green and brown, the idea of number arises: we see two trees. So, it appears that tree and two are on equal footing: they are ideas which arise in our mind which help makes sense of the seven inputs.
This view, I think, is somewhat similar to Kants view that all we experience are phenomena. It differs in that it limits the phenomena we experience to sight, sound, smell, taste, touch, emotion and thought.
In this view, mathematical entities are not a distinct type of reality. They are ideas, just like tree.
You are right. The term "objects" I ised in saying "Numbers are abstract objects" might be confusing because it normally refers to something physical. I could use the word "things" --which is more general and can refer to anything-- but it's too commonplace and banal. So I prefer to use neither and say, "Numbers are abstract".
So, I believe we should use the word "object" only for material things, things we can perceive with our senses, things that actually exist. Thus, we can talk about objective things, which actally exist, indepedently of us, in contrast to subjective things, which are abstact and exist only for (each one of) us.
In this case, I think, there would be no equivocation, as you say, neither any kind of interaction of two types of objects.
Note: Of course, no confusion should be produced if the context, phrase or expression in which the word "object" appears, make it clear about its nature and meaning. E.g. my use of the expression "abstract objects" shows that the the word "object" is used as something non-physical, since something that is abstract can never be physical. Yet, as I said earlier, it is better to avoid the use of the word "object" altogether in these cases. Look what it has created! :grin:
Sorry, I was just jotting down some preliminary ideas related to the OP, and to your notion of "Creative Step" and "Discovery". When you "decided to extend this idea to a more general realm" (specific-to-general) you were doing Inductive Reasoning, which is one kind of creative act in Philosophy, and in Mathematics. But, another approach is to break-down a broad general concept into more particular applications (general-to-specific) Deductive Reasoning. I suppose both can be creative, depending on their practical or theoretical implementation (involvement??).
It seems that the OP is an attempt to generalize from spacetime observations to something beyond spacetime : specifically Mathematics & Mental Images. The abstract concepts of Mathematics exist "beyond" space-time in the sense that Math objects are not affected by the physical laws that govern the behavior of material objects. I suppose that is trying to extend that spaceless & timeless aspect of Math, to the abstractions that we humans create to represent the Self (or Soul, if you prefer). That is not exactly a new idea, except for the connection to immaterial Mathematics & Logic, which some thinkers imagine existing eternally out-there beyond the limits of space-time (Ideality instead of Reality).
So, he seems to be expressing an ancient concept (we are souls) in more modern language : "we are math". Whether that's a creative step may depend on how he develops the basic notion into a philosophical position. Unfortunately, even the Ontological status of Mathematics is subject to philosophical debate. So, the notion of a soul-man is not a slam-dunk. :smile:
PS__"Philosophical Chatter", as you put it, seems to be how philosophers get involved in discovering new ways to look at old ideas. Are the mathematician's chalk-marks on the blackboard more involved than text-marks on a philosophical forum?
"More or less, although most math people give this question little thought. In my case, I was introduced to a notion years ago in my PhD studies. A little later on I decided to extend this idea to a more general realm - a sort of creative step. Once the basic ideas of the concept were set, then came the acts of discovery - finding what flows forth logically." ___jgill
"Which suggests that realitythat me, you, Earth, universe, etc.is fundamentally some sort of abstract object existing outside spacetime. Hm." ___Art48
If the meaning of "two" is a private concept in my mind, and is different to a private concept in your mind, then you and I literally do not share the same concept of two. When I talk about two, I am talking about something utterly distinct from what you are talking about when you talk about two.
But suppose it so happens that your private concept and my private concept have some overlap, such that we agree publicly on certain aspects. Those public aspects are what make a difference, are what we use to count and do things. Any private, unshared aspects have no impact on what we do - if they did, they would thereby be public.
So those private aspects of the concept two make no difference, and it is only the public aspects that have a place in our affairs.
This is part of the Private Language Argument. Meaning is not found in private concepts inside minds or brains, but is public and shared.
The remainder of your argument hangs on this. It needs rethinking.
In my view, it is not a private concept. It's a pre-existent idea which we encounter in the "mindscape," just as we encounter a pre-existent tree in the landscape. In my view, ideas are pre-existent.
Excuse my being blunt, but it is wrong on multiple levels.
There are far more than five senses. Indeed it is problematic to treat the senses as discreet, as being seperate "inputs". Contemplate how difficult it is to seperate even smell and tase, for example.
Emotions and thoughts are more reactions rather than "inputs". I'll use scare quotes, because the analogy of an "input", derived from computer science, is misleading. It gives the impression of a one-way process. What we sense is very much dependent on our state of mind - on our emotions and thoughts. Sensing is better thought of as an interaction between our bodies and the world. Think of the way the eye changes it's physiology immediately that there is a change in the light, and of the various optical illusions you have seen. Similar things happen with each of the other sense.
We see the tree using the light reflected from it, and using our eyes and the light from the sun. We certainly do not see the light. If we did we would not have needed Young and Maxwell and others to set out the physics of light. What we see is the tree.
Your "actual input" is a misleading notion. One's neural network, starting at one's retina, constantly and actively re-works the signal it receives in order to construct the sense of green and brown. The "idea of tree" is constructed much later in the neural net, perhaps involving the areas of the brain that handle language. Our resident Neuroscientist, @Isaac, might be able to explain with greater clarity.
(Excuse my invoking you yet again, Isaac)
@Art48, you seem to be working with a homunculus-like view of the self, as if you were sitting inside your head looking out, receiving raw inputs of information that you interpret using a priori scripts. That is a view often attributed to Kant, although there are Kantians who deny it. The homunculus is, for several reasons, to be rejected.
In conclusion, trees are not ideas. They are things found in the world.
But you seem to be claiming that the numbers are found in the world in the way trees are, and so are not individuals from some parallel wold of forms. I'd therefore refer you back to this argument:
Quoting Banno
And I put it to you that there are alternatives that do not require such obtuse ontologies:
Quoting Banno
And not a reference to something else in the world, besides the two trees.
The homunculus, siting in a head looking out, using its Kantian a priori scripts to interpret what it sees.
Nuh. I don't think so.
Quoting Banno
Heres an enactivist perspective:
traditional neuroscience has tried to map brain organization onto a hierarchical, input-output processing model in which the sensory end is taken as the starting point. Perception is described as proceeding through a series of feedforward or bottom-up processing stages, and top-down influences are equated with back-projections or feedback from higher to lower areas. Freeman aptly describes this view as the "passivist-cognitivist view" of the brain.
From an enactive viewpoint, things look rather different. Brain processes are recursive, reentrant, and self-activating, and do not start or stop anywhere. Instead of treating perception as a later stage of sensation and taking the sensory receptors as the starting point for analysis, the enactive approach treats perception and emotion as dependent aspects of intentional action, and takes the brain's self-generated, endogenous activity as the starting point for neurobiological analysis. This
activity arises far from the sensorsin the frontal lobes, limbic system, or temporal and associative corticesand reflects the organism's overall protentional setits states of expectancy, preparation, affective tone, attention, and so on. These states are necessarily active at the same time as the sensory inflow (Engel, Fries, and Singer 2001; Varela et al. 2001).
Whereas a passivist-cognitivist view would describe such states as acting in a top-down manner on sensory processing, from an enactive perspective top down and bottom up are heuristic terms for what in reality is a large-scale network that integrates incoming and endogenous activities on the basis of its own internally established reference points. Hence, from an enactive viewpoint,
we need to look to this large-scale dynamic network in order to understand how emotion and intentional action emerge through self-organizing neural activity.(Evan Thompson, Mind in Life)
What's wrong with the homunculus? That seems to almost exactly describe my conscious experience. It seems like I'm inside my head looking out, only not sitting, and I don't know if any of the "scripts" (they seem more like memories to me) are apriori or not.
Quoting Alkis Piskas
Why assume no interaction? I see no need for this assumption, and the human use of mathematics and engineering in creating things in the physical world demonstrates that there definitely is interaction.
I don't think inductive reasoning in its usual interpretation is appropriate in this comment. I am not arguing that since it works here and there in a number of different venues it probably works in general. I am going the other direction and investigating, using deductive logic, whether ideas hold in various more or less distinct areas.
Quoting Gnomon
Good point. Nice and appropriate comparison. The chatter I was referring to was efforts to abstract "creativity" in math without using actual examples.
Being blunt (or frank) is virtue. We get a lot more accomplished that way. And it helps me clarify my own thinking.
You write There are far more than five senses. As I noted when I mentioned ESP, more senses arent a problem. Emotions and thoughts, like our physical sensations, are inputs to consciousness. If its not one-way, if our emotional or mental state impacts what we sense, that, too, is not a problem.
The basic picture is that we have transitory physical, emotional, and mental sensations. Sensations imply a sensor, an experiencer. A criticism Ive seen of the homunculus idea is infinite regress: who sees what the homunculus senses? I dont see the same problem with consciousness. In effect, the buck stops with consciousness.
I see consciousness fulfilling the role of experiencer. Of course, this assumes we have an enduring self. Hume just saw sensations, not the self. I think the answer was Humes selfin my view, his consciousnesswas the self he couldnt find, just as the eye can see everything but itself.
So, consciousness is one answer to the question of how personal identity persists through time. To me, its the best answer. Historical continuity, for example, gives a kind of personal identity, but it seems a somewhat superficial type of identity.
You write actual input is misleading and mention neural network processing. But I mean input to consciousness, after all processing has been done.
If trees are found in the world, who finds them? Can you explain the tree-sensing sense that allows us to sense trees? Is there such a sense? If so, what is the organ of our tree-sensing sense?
When I introspect, my consciousness experiences a stream on physical, emotional, and mental sensations. I feel my consciousness is me while the sensations, because they are transitory, are not me. So, my view is not merely mental. It describes more or less how I experience myself.
SPDC converts a photon into two entangled photons, each with half the energy of the original photon. For Type II SPDC, this prepares the singlet state (represented below by qubits):
[math]\ \ \ \ \ |\psi\rangle = \frac{|01\rangle - |10\rangle}{\sqrt{2}}[/math]
So that anti-correlation is the consequence of a local physical process (i.e., the SPDC process).
Subsequently, the two photons (qubits) can be separated by a large distance. QFT, by construction, says that spacelike-separated events, including measurements, cannot be causally connected. So the measurement of a photon is a single localized event in spacetime which has no effect on the measurement probabilities for the other photon until a signal arrives there (not exceeding the speed of light).
Note that the singlet state is perfectly anti-correlated in every basis (X, Y, Z, etc.). That is, if each qubit is measured in the same basis, then one will be measured as 0 while the other will be measured as 1.
Compare QFT to the case of Bertlmanns socks, which John Bell recounts:
Quoting Bertlmann's socks and the nature of reality - John Bell, 1981
Observing one sock to be pink doesn't cause the other sock to not be pink. Instead the common cause of the sock anti-correlation is Bertlmann's initial choice. That's analogous to QFT, where the SPDC process is the common cause of the photon anti-correlation. In both cases, locality is maintained.
The key difference is that the observation of Bertlmann's socks is explained by the pre-measurement colors of the socks. Whereas the observation of the entangled qubits can't be explained by pre-measurement values, as shown by Bell's Theorem. Which is why quantum interpretations become relevant (note the Local dynamics column).
It could well be, as in theory no one other than me knows what's in my mind. However, in practice, as we share more than 99.9% of our DNA, and we both have the same ancestor, "mitochondrial Eve", I would infer that our private concepts are very similar.
Quoting Banno
I agree. Elaborating, given something in the world that has been given the communal name "two", it may well be that my private concept of this something is actually three and your private concept of the same thing is four.
However, when I see something named "two" and have the private concept three, I will interact with the world in a particular way. Consistency is important, in that the next time I see something named "two", even though my private concept is still three, I will interact with the world is the same way as before. As you say, as regards my interactions with the world, " those private aspects of the concept two make no difference".
I'm just talking about the word "object" and it's use. The interaction I mentioned was between the two uses and meanings of the word. Mathematics and engineering have nothing to do here.
Anyway, nevermind. Too much has been already said about the subject. Let's move on.
Ok, It seems to me that your the sock example you site is similar to the glove example.
So if both gloves are individually boxed and are sent a great distance away from each other, then the moment I open my box, I know instantly, the contents of the other box, based on one left and one right handed glove, regardless of the distance between the two boxes. I understand that in this sense, the two gloves are set at their creation. Is SPDC simply asserting the same for these two entangled photons?
So, when you locally open one of the boxes, that act, does not affect the state of the gloves.
But a qubit can have more that two states due to superposition states. A qubit does not just resolve to 1 or 0, it can be in a superposition of 1 and 0.
The nature of the gloves as left or right handed is there from the beginning, just like in QFT, a coordinate in spacetime can manifest any of the known particle states/field excitations (almost like an interdimensional vibrating superstring). Entanglement may 'correlate' (as a field excitation 'travels') the states of two spacetime coordinates, regardless of the distance between them.
What's wrong with the imagery I am invoking, if it's incorrect.
There's a need to sort out two seperate discussions here. You were replying to my critique of the Platonic view of there being a distinction between the world of the senses and the world of forms. In your OP you propose that the world of the sense is the world of forms.
What you are proposing in the OP is not a dualism such as I was addressing, a physical world and a world of mathematical forms, but that the physical world is exactly the world of forms. Some form of idealism.
So you set consciousness as the pivot on which everything depends. Hence, you can talk without apparent irony of an Quoting Art48
I'd submit that consciousness is the very processing you dismiss. Again, you are not sitting inside your head looking at the results of the processing, but rather you are the processing.
The obvious difficulty, as for all forms of idealism, is how you can avoid solipsism. If all you have is your own consciousness, then, by that very fact, you are on your own.
Almost there.
Now we can go back to this:
Quoting RussellA
You go to the grocer and hand over a piece of paper with "Two apples" written on it. The grocer pulls out the bin with a picture of apples in it. She then looks at a chart showing the numbers with tally marks next to them, scans down the chart until she find "two", and pulls out one apple for each tally mark.
At no stage does she refer to a private conception, in her mind or in yours. The process is entirely public.
Can you now see that in using "two", say in asking for two apples, what counts is that you receive an agreed quantity of apples, which is a public activity? That this satisfies your private conception of "two" is of no consequence to the action involved.
Hence any private mental stuff is irrelevant to the meaning of "two".
Im not sure idealism applies. Id say our consciousness directly experiences its physical, emotional, and mental sensations, and so we can be certain the sensations exist. (Much like I think therefore I am although Id replace think with experience.) What causes the sensations? Are we a brain in a vat? Or are we experiencing the world more or less as it is? Or is what we experience Platonic forms? Or is there a monist entity responsible for what we experience? I can make some intelligent hypotheses, but I just dont know.
Quoting Banno
I think of consciousness as what is aware of the sensations. I think some philosophers view consciousness in the same way. Thus, the hard problem of consciousness. And, thus, the concept of philosophical zombies, which have all the sensations but no consciousness.
Neuroscience can explain (to a certain degree) our bodily, emotional and mental sensations but I dont think it can yet explain consciousness (except for the view that its what the brain does,, i.e., consciousness has a physical basis).
If locality is the case (per QFT) then the correlation is a consequence of the SPDC. However, unlike with the gloves, the final measurement values are not predefined (see counterfactual-definiteness).
Quoting universeness
Yes. Or in the case of the singlet state, the qubit pair is in a superposition of 01 and 10.
Quoting universeness
I'm not sure I follow the imagery. There isn't an image or picture that explains the correlation, unless one goes with an interpretation (such as Many Worlds or Superdeterminism).
Very interesting stuff! I will continue to try to understand all of the aspects of it, better than I do at the moment.
I'm sorry about the length of reply.
Buying two apples needs both private concepts and public names
Does meaning is use have implications for the status of numbers
I agree that the meaning of "two" is how the word "two" is used. But what is the implication for the status of numbers ?
Objects are publicly named in performative acts
Prior to the interaction between me and the shopkeeper, it is necessary that we both have the same chart. Alongside the picture of one apple the name "one", alongside a picture of two apples the name "two", alongside the picture of an apple the name "apple", etc.
However, it could well have been that alongside a picture of one thing was the name "red"
and alongside the picture of two things was the name "yellow", but we can assume that in some prior performative act by someone in authority, a picture of one thing had been named "one" and a picture of two things had been named "two", thereby establishing a public language.
I wake up hungry and have the image of two apples in my mind. I compare the image in my mind to the pictures on the chart, and see the name "two". I go into the shop, tell the shopkeeper "two apples", who looks at the chart, and by comparing the picture on the chart to the image of what is in the bin, is able to give me two apples.
The number "two" is redundant in this transaction
In fact, the number "two" is redundant in this transaction. I could just have shown the shopkeeper the picture of two apples. Numbers may be convenient, in that the number "two hundred" is more convenient than a picture of 200 things, but fundamentally, within this transaction, what can be done in numbers could equally well have been done in pictures.
Perhaps this is the point of Hartry Field's nominalism, an opponent of the Quine-Putnam Indispensability Argument for mathematical Platonism. Field rejects the claim that mathematical objects are indispensable to science, arguing that it is possible to reformulate scientific theories in such a way that mathematical objects are replaced by relationships.
The transaction couldn't happen without private concepts
What is fundamental in using numbers is our ability to compare two images, either a memory of an image with a picture on a chart, or a picture on a chart with an image of something in the bin. Yet it is inevitable that the image of two apples in my mind, the picture of two apples on the chart and the image of two apples in the bin will be different. A judgement will need to be made that two things having some differences and some similarities both fall under the same concept, in that we have the concept apple even though no two apples are the same. It is an inherent human ability to be able to look at several different things and discover a commonality within them, and discover that they fall under the same concept.
The transaction would not have been possible if either me or the shopkeeper had no concept of either an apple or the number two, in that without concepts we would be still sitting in the corner of the room motionless. It may well be that my private concept of "two" is actually three, and the shopkeeper's concept of "two" is actually four, but we will never know, and is in a sense irrelevant. What is essential is consistency of concept, in that yesterday when I saw "two" my concept was of three, today when I see "two" my concept is of three, and tomorrow when I see "two" my concept will still be of three.
It is true that for the transaction to proceed, no reference is ever made to our private concepts, in that the shopkeeper does not need to know my private concepts of either apple or two, but it is equally true that the transaction could never have happened if either of us had no private concept of either apple or two. The process of buying two apples needs both a private aspect and a public aspect. As regards the private aspect, each participant must have a private concept of both two and apples, and as regards the public aspect, there must have been a priori performative act by someone in authority linking a picture of an apple to the name "apple" and linking a picture of two objects to the name "two".
Concepts don't exist in a mind-independent world
As regards the public aspect, two objects are linked to the name "two". What exactly is this link? It is the same problem Achilles had with the tortoise. When the tortoise started to move his castle diagonally, Achilles said that that move wasn't in the rules. The tortoise replied "where is the rule that I have to follow the rules". Similarly, there is the public rule that what is pictured is given the name it is linked to, such that when an apple is linked to "apple", then "apple" means apple, and when two objects are linked to "two", then "two" means two objects. But as the tortoise would say "where is the rule that a name means what it is linked to"
These linkages are relations, and as relations don't ontologically exist in a mind-independent world, then neither do these linkages. But as we do perceive linkages in the world, and as these linkages don't exist in the world, they can only exist in our minds, meaning that things like apples and two can only exist in our minds.
How is the our concept of apple related to our word "apple", and our concept of two related to our word "two". In On Denoting, Russell argued that words such as "apple" and "two" are not referring terms, in that they are not referring to an individual having its own existence, but is in fact describing those properties or parts that it is composed of. As Russell wrote "Every proposition which we can understand must be composed wholly of constituents with which we are acquainted", where those things we can think about directly are sense data, universals, relations and oneself. Similarly concepts such as apple and two are not referring terms, referring to an individual having its own existence, but in fact describe the properties or parts that it is composed of and with which we are directly acquainted. Therefore, our concepts and words do the same job in describing the properties they are composed of and which we are directly acquainted.
The process of buying two apples needs both a private and public aspect
In summary, the process of buying two apples could not happen without both a private and public aspect. If either me or the shopkeeper had no private concept of either apple or two, we would remain motionless in the corner, unable to act. If either apple or two had not been publicly named in a performative act "apple" and "two", I wouldn't be able to communicate with the shopkeeper.
Outstanding.
https://www.jstor.org/stable/3609188#:~:text=Russell%20definition%20of%20number%20is%20the%20identification%20of,1%29%20related%20to%20some%20class%20of%20n%20members.
Re the OP :
Quoting Art48
Others have commented on this already, but here's my take : most materialists and idealists (recognizing that there are various flavors of each) must disagree with this premise, since both camps consider mind-stuff to be part of the world (i.e., "spacetime"). Only dualists would agree that thoughts are divorced from the "world out there".
With apologies, the better question is, "Do numbers exist independent of minds?" * And of course, debate over mathematical Platonism has raged for many decades, if not centuries, and will not be answered on TPF any time soon. Still, it's fun to plant your flag and defend it. My own view, following from the Frege-Russell definition, is that numbers do NOT exist independent of minds because the act of placing objects in a one-to-one relationship is a cognitive one.
*Assuming most idealists assert that ALL is mind, their answer to the question of mind-independence must be no, correct?
I tried to include a reference to Kant's philosophy of mathematics and a priori intuitions, but I know Banno isn't a fan.
Yeah, well, you know: horse, water. Smart horse drinks, stubborn horse wont unless its chilled Perrier.
I'm not. I agree with . Well done.
But I see some issues.
I might start with Russell's theory. Russell's logical atomism, as you so well describe it, involved working up from simple atomic statements setting out things of which we are directly acquainted, using Frege's logic. Philosophical analysis for Russell was a process of breaking down complex propositions into their constituent parts, and then reassembling them. By doing this, he argued, we could identify the truth of any proposition by looking at its separate components.
Wittgenstein, in PI, argued that what counts as a simple was dependent on the task at hand. The same thing can be simple in one context, but complex in another. The task of analysis is to understand the context and the language game at play. He argued that there was no fixed set of atomic propositions, and that the task of philosophy was not to break things down into their component parts, but rather to describe the complex interaction of language and context.
There are also the more involved criticisms of Russell's theories deriving from Kripke's Possible World Semantics. We might set those aside in this context, noting that it is not clear that Russell's work on denoting solves the syntactic and epistemic issues it was supposed to.
So we have here two differing approaches to the nature of the apples being purchased at our grocer. On the one hand we have Russell's view that the apple consists in a concatenation of "constituents with which we are acquainted", something like "Green or red and round and waxy and smooth and tart or sweet". On the other hand we might set out the nature of an apple by setting out the roles it might play as we go about our daily activities: The thing we pick, sell, bite, stew, bake in a pie and so on.
We have a similar divergence of approaches to the numbers. On the one hand we might have Russell's failed attempt to build arithmetic from logic by creating a set of axioms and definitions that could be used to construct mathematical proofs. He failed in this endeavour, as his work contained paradoxes that undermined the logical consistency of the system. These paradoxes showed that the foundations of mathematics could not be reduced to pure logic, and that mathematics contained elements of self-reference that could not be reconciled within a logical framework.
On the other hand we have accounts of how numbers are used in our everyday practices, which can include, for the mathematicians amongst us, quite complicated and sophisticated machinations. Numbers are to be understood not by setting up definitions from first principles, but by learning to make use of them.
Now comes an argument in favour of the second of these alternatives. Suppose Russell had met with more success, and been able to produce a definition of two that did not lead to inconsistency. How would we judge that what his definition sets out is in fact the number two? How would we verify tha this definition was accurate?
We would verify his definition by comparing it to our use of the number two, checking that what Russell defines is indeed adequate for the everyday tasks we set for that number. We would verify or falsify his definition by comparing it to our use of "two". after all, any stipulated definition is evaluated by comparison with the empirical facts of language use.
That is, the use of the number two has priority over any contrived stipulation.
Hence Wittgenstein's approach may claim some priority over Russell's.
So what are we to make of the supposed private concept of two? There is after all some appeal to your 'I compare the image in my mind to the pictures on the chart, and see the name "two"'. What do we make of the image in your mind?
The picture that holds us captive is of a concept of "apple" held before the mind, as an object of desire. Supposedly we make use of that concept in order to pick the apple from the pear, to construct our list of fruit for the grocer, or to argue on philosophy forums. But what is it?
The picture is of a piece of mental furniture. A page with "Green or red and round and waxy and smooth and tart or sweet" written next to the word "Apple" and a picture, near at hand for the homunculus to check in order to make sure he has his concept right.
But you might agree that such a notion of "concept" is far too passive. The concept of "apple" is what we use when we pick apples, when we sell and buy them, when we make a pie; it is interactive, showing itself in the vast range of activities in which it is involved.
Indeed, the argument used above applies here: How would we tell that the homunculus' page was correct? Only by comparison to what we do with apples.
And here the thought is that, if there is such a concept of "apple", it is a back-construction; it is something we began to build when mum cut up the thing she bought at the shop and served it for lunch, saying "Eat your apple", that we added to when we climbed an apple tree to reach the apples that were not blown, and now rely on for philosophical purposes.
Our interactions with the apple preceded and put in place any essentialist definition.
The concept of apple is not a list, but a family resemblance of our interactions with apples.
And those interactions are public.
So if there is a private concept of "apple", it is only there because of your public interaction with apples in the wide world. The private concept is public.
This approach also serves to sort the dilemma found in supposing that either concepts are in the mind or in the world. The dilemma relies on picturing concepts as mental furniture. If instead the concept "apple" is seen as what we do with apples in the wide world, then the question of whether they are in the world or in the mind dissipates. "Apples" are just our interactions with those things.
So again we can go back to
Quoting RussellA
and see that
Quoting RussellA
Thanks again for an excellent reply.
I need to communicate what I want to the seller.
I can simply point to an apple by referencing my stored image of one. I can then hold up two fingers or take the two apples myself and hand them to the server or I can say 'two' or I can even communicate some indication of 'repeat' or 'again' after I point to an apple and want to indicate 'two.'
It seems to me that its the public communication that matters between me and the seller.
Would I need the concept of 'two,' if I had no-one to communicate with?
Does it matter what my private concept of two is when I have to communicate it to another?
My goal is to get two apples so my goal is to communicate that to the server in a way that achieves my goal and broadly satisfies/ is acceptable to the servers notion of two apples.
Whole numbers are either 1 or multiple replicates of 1. That matches my private concept of 'two.'
You might conceive 'two' to be its own amount, rather than as multiples of 1.
'Can I have two apples please?'
'Can I have an apple and then can I have another apple?'
Both would have the same result but the second one is just a bit less efficient.
If there are 'private' notions of what 'two' is, I don't think they matter as much as your personal ability to communicate 'two,' publicly.
I had an Uncle who I used to go drinking with. He would often go to the bar and order a small packet of peanuts. After he finished them he would go get another bag and then once more.
He did this routine on many occasions. So of course people would ask him, with some incredulity and annoyance, why he did not just 'f******' ask for three bags of peanuts? He would just say 'that's just me!'
Maybe he had some strange private notion of 'three,' which clashed with the public notion of three.
This is really not the case, as the process of learning mathematics is more complex than what you represent it as. At the most basic, fundamental level, we simply learn usage, as you say. But then as we progress in our education, we must enter into a learning of abstract concepts. At this point there is a change, so that the student's mind evolves from learning simple operations of using numbers, to learning specific rules for use. That is actually a big difference, and you might see that it aligns roughly with the difference between arithmetic and mathematics.
The point here is that we cannot accurately make blanket statements like yours, " Numbers are to be understood not by...but by...", because "numbers" in the sense of arithmetic, and "numbers" in the sense of mathematics, is two different uses of the word.
Quoting Banno
In light of the difference described above, this statement is very problematic. In the higher levels of mathematics we are definitely taught to follow stipulations, axioms, while in the lower levels of arithmetic we are taught to follow demonstrated usage. The difference marks the development of the student's mind from practical application, to the understanding of theory. The understanding of theory is based in the learning of rules, rather than a simple learning of use.
The result is that within mathematics we have a sort of struggle, or disequilibrium between theory and practise. You say that practise has priority over theory (your statement "the use of the number two has priority over any contrived stipulation"), but this is not really the case. Mathematics, being a high level abstract form of logic, proceeds in the opposite way, theory is prior to practise. This is obvious in the history of modern math, theory precedes the application, and therefore shapes practise.
The difficult issue is that the theory must be derived from somewhere, and it is common practise in the development of mathematical axioms, to produce axioms which are derived as a description of common usage. This creates the appearance that usage has priority over theory.
That presents a problem which I've pointed to numerous times on this forum, of which many members are ignorant, and even actively deny. If the proposed axioms are meant to be a description, or representation of usage, and they are not an accurate representation, then falsity is allowed to enter into mathematics. Mathematics is such that the rules must be followed, and the rules are given priority over usage, so rules which are supposed to be representations of usage, which are faulty representations, must be followed, thereby allowing self-deception within mathematics. Usage appears to have priority over theoretical rules, because the theory appears to be a description of usage. But in reality, the theoretical rules are what shape usage, and that this is true is evident from the fact that faulty descriptions may be used as rules.
I agree that Russell's work on denoting is not without criticism, and Wittgenstein's meaning as use, the language game and family resemblances are important aspects. But perhaps both are needed to arrive at an understanding of the process of buying two apples.
The mind and the language it uses need both Russell's elementary concepts and Wittgenstein's compound concepts
The elementary concepts of "logical atomism" and the compound concepts of "meaning as use"
At the moment , it seems to me that apple as a thought in the mind and "apple" as a word in language may be understood as a combination of the elementary concepts of Russell's logical atomism and the compound concepts of Wittgenstein's meaning as use, in that neither is sufficient by itself, but each provides an essential part of the whole.
Elementary concepts
Following Russell, there are things with which we are directly acquainted: green or red, round or square, rough or smooth, tart or sweet, hot or cold, acrid or fragrant, loud or quiet, etc, and the mind can judge the difference between these binary opposites.
In Kant's terms, trying to add a chilled Perrier moment, the human ability to judge between such binary opposites is an a priori intuition, an epistemic condition, an innate ability we are born with. It is the product of 3.5 billion years of life evolving in synergy with the world within which it finds itself, an Enactivist understanding whereby a person's understanding of the reality they observe in the world has been determined by the evolution of life within the world before they were born. Sentient life is a physical expression of the world it finds itself within. IE, the function of schools is not to teach children how to distinguish between green or red, round or square, etc as these abilities are innate, but without these innate abilities, being taught more complex concepts would be impossible.
Compound concepts
Given these simple concepts we can then combine them in various ways into compound concepts. Any combination is possible, but some combinations are more useful than others. For example, I have discovered that the combination round, sweet and red/green is of particular use, in that I have discovered that the apple is beneficial to my existence in the world. For convenience, rather than keep saying "pass me the thing that is round, sweet and red/green", I could name it "apple" and say "pass me the apple". I could equally as well have named it "camel", and said "pass me the camel", with the intended meaning pass me the apple, but as it has turned out, in the English language, something round, sweet and red/green has been named "apple".
But any possible combination of elementary concepts can be named, regardless of whether the particular combination is useful or not. For example I could name the combination green, square and smooth as "grasquim", not something that I have ever discovered to be useful to me.
The "apple", as a compound concept, exists as a relationship between the elementary concepts round, sweet and red/green. "Grasquim", as a compound concept, exists as a relationship between the elementary concepts green, square and smooth. As Russell in On Denoting showed, neither "apple" nor "grasquim" refer to an individual having its own existence, but describe the parts, the properties, that make it up. As both "grasquims" and "apples" have the same existence as a set of properties, if we said that "grasquims don't exist", then we would have to say that "apples don't exist", and if we said that "apples exist", then we would have to say that "grasquims exist". But Russell's On Denoting overcomes this problem in that neither "grasquims" nor "apple" are subjects that are predicated as either existing or not existing, rather, they are descriptions of a set of properties, not individuals being referred to.
It may well be that the "apple" plays an important role in our daily activities, and the "grasquim" plays absolutely no role in our daily activities, but both "apple" and "grasquim" have a meaning, in that "apple" means round, sweet and red/green and "grasquim" means green, square and smooth.
When Wittgenstein says "meaning as use", " meaning" can be interpreted in more than one way. In one sense of meaning, the "grasquim" has meaning even though it has no use. In another sense of meaning, the "grasquim" has no meaning because it has no use, in the same way that someone could say " travelling to Mars doesn't mean anything to me", knowing that they will never travel to Mars. Perhaps Wittgenstein's "meaning as use" refers to the second interpretation.
Kripke criticised Russell's Descriptivist Theory using a modal and epistemic argument
As regards the epistemic argument, Kripke pointed out the flaws in Russell's treatment of compound concepts as being able to be known a priori, inferring that compound concepts such as "government" can be known a priori, which is certainly not the case. Kant is different, in that Kant treats elementary concepts as being a priori, not compound concepts, which is certainly the case, in that humans are born with the innate ability to distinguish green from yellow, for example.
As regards the modal argument, Kripke said names should be rigid designators, true in all possible world. This requires that the elementary concepts building up a compound concept must be necessary rather than contingent, in that "apple" is true in all possible worlds, providing the elementary concepts building it up are round, sweet and red/green and not round, sweet, red/green and on the table.
Both Russell's "logical atomism" and Wittgenstein's "meaning as use" are needed
In summary, humans for survival and development within the world need both compound concepts and the elementary concepts they are built from. Some compound concepts mean more to us than others because of the use we can make of them, in that the "apple" means more to us than "grasquim", ie, Wittgenstein's "meaning as use"
Yet, we wouldn't have compound concepts without the elementary concepts they are built from, the constituents with which we are acquainted, as it were those fundamental indivisible atoms on which the rest of matter is made, where such atoms have been discovered through logical reasoning rather than intuitive feeling, ie, Bertrand Russell's "logical atomism".
Ok, but I take exception to compound conceptions. I know what is meant by it, but I think it a misunderstanding. Some thing, with a set of properties in the form of conceptions subsumed under it, is still represented only by its own conception.
I can have the concept of a single thing such as the colour yellow, or I can have the concept of an apple, which is a set of things, round, sweet and red/green.
When just looking at something round, my concept will be of something round, when just tasting something sweet, my concept will be of something sweet, when just looking at something red/green, my concept will be of something red/green.
However, what happens when I experience all of these things at the same time, something round, sweet and red/green, ie, an apple ?
Either i) I experience a single concept made up from a set of concepts, a unity of apperception, or ii) I will experience a set of concepts, discrete and separate ?
By "compound concept" I mean compound in sense i) rather than sense ii).
However, there may be a more technical term than elementary concept and compound concept.
I) works just fine:
But the conjunction of representations into a conception is not to be found in objects themselves, nor can it be, as it were, borrowed from them and taken up into the understanding by perception, but it is on the contrary an operation of the understanding itself, which is nothing more than the faculty of conjoining à priori and of bringing the variety of given representations under the unity of apperception. This principle is the highest in all human cognition.
All that presupposes I think has some irreducible meaning. Whether we actually do think or not, is irrelevant, insofar as the very seeming of it requires an account.
My tentative explanation of how Ideas interact with Real things is similar, but based on a philosophical simulation of Quantum & Information Theory. The dual entities are distinct only in the sense that the same mind can distinguish between a Thing and the idea of the Thing. Real & Ideal things are conceptually distinctive, but not epistemologically exclusive -- they are not in parallel worlds, but in the same world. You don't have to go out of this world to create an imaginary replica of a physical object.
Plato's Ideals are often portrayed as existing in some aethereal heavenly realm. But they are differentiated from mundane Reality only in the sense that mental Meta-Physics is distinct from material Physics. The human brain is physical, and interacts (communicates) with its own material body via electro-chemical signals (material information). Meanwhile, the brain also interacts (communicates) with its own ideas via something like Quantum Signal Processing : conversion of physical processes to mathematical ratios & algorithms.
In other words, specific physical energy patterns are converted into coded information functions -- in this case, the function we call "imagination" or "conception". This transformation from physical Energy to Meta-physical Information happens within the holistic system of a Person, not in some parallel world.
Of course, the detailed "how" is far over my pointy little head. So, this is just a crude macro description of the micro mechanics of Thinking. Perhaps it could only be really/ideally understood by a Mathematical Physicist. :smile:
PS__The Brain deals with Material neuro-logical patterns while the Mind works with Logical/Mathematical patterns : inter-relationships.
Quantum Signal Processing is a Hamiltonian simulation algorithm
https://en.wikipedia.org/wiki/Quantum_signal_processing
The Hamiltonian of a system specifies its total energyi.e., the sum of its kinetic energy (that of motion) and its potential energy (that of position)
https://www.britannica.com/science/Hamiltonian-function
A half-way position. I think Wittgenstein's approach can wholly replace Russell's. Here are two follow-on points.
You say
Quoting RussellA
But I think this a bit too fast. While a baby has the potential to distinguish many colours, that potential is developed and reinforced over time by their interaction with the world. They learn to treat green and red differently because treating them differently has different results. The difference between red and green has public consequences. If the difference between harlequin green and neon green had similar consequences, she might learn to differentiate them before red and green. We know from studies of colour names across cultures that what counts as a significant difference in colour is at least partly dependent on where one grows up. While we are born with the potential to differentiate many colours, the ones we actually differentiate are dependent on our culture and environment. If it made no difference to us if the apple was red or green, we might well never make the distinction.
The capacity to differentiate colour is there, but it is trained by our interaction with others.
It follows that what is to count as an "elementary colour" is not entirely innate, but learned by interaction with the world. Similarly, what counts as an elementary concept, a simple, is dependent on one's interactions with the world, including other people, and language.
But we can go further and see that what counts as a simple in one case might not be a simple in another. Take a look at this example from Wittgenstein.
Is it the colours here that are the simples? Or are the colours irrelevant, and the fact that there are squares instead of circles what is important? Or that the grid is three by three, and not two by four? The point is that what is significant here is far from clear until one understands what is at stake.
One of the major differences between the Tractatus and the Investigations was Wittgenstein's realisation that what is to count as a simple is dependent on the task at hand. The meaning of "simple" varies with use.
For more on this, see The Dismantling of Logical Atomism, and
Quoting Kevin Klement
Just a tidbit of info: innateness usually includes capabilities that develop through some sort of engagement. For instance, walking upright is an innate feature of humans, but there are needed structures that won't develop until walking is attempted. The physical stress of trying to stand triggers their development.
This meaning of innateness goes back at least to Leibniz.
Maybe one of the most profound passages in Investigations that seems most do not appreciate. It dissolves away much of philosophys pretentious foundations.
There's a need to tie all this back to @Art48's OP. I lost track of the argument somewhere along the line. Seems he is treating quantum wave functions as the mathematical simples from which everything is derived. I suppose that's one view...
Quoting Art48
That seems to me to be a reification; I'd call consciousness the act (activity) of having sensations, thoughts, and so on; a more active notion than taking it as a thing that does the experiencing. A step further form the homunculus.
How does this step us away from the homunculus? If consciousness is an activity then there must be something which is doing that activity. It cannot be the human body which is performing this activity because there is no observable act of the body which could be called the act of being conscious. So the thing which is performing this act of consciousness must be something other than the body, but it sure appears to be within the body. Therefore we are lead from consciousness is an act, to the homunculus as the actor.
I hope you are never called upon to perform first aid.
Cant be the act of; it is only that to which the unity of all our representations belong, such that it is then possible for them all to be my representations, which, in turn, makes explicit a singular self, which gives I think. Consciousness does not act; it is merely indicates a relative quality of being acted upon.
The activity of the having of sensations is a function of physiology, the exchange of the affect of the sensation to the representation of it, is intuition, the object of which is phenomenon.
The act of having thoughts is understanding, the representations of which are conceptions, the objects of which are judgements.
. this principle of the anticipation of perception must somewhat startle an inquirer whom initiation into transcendental philosophy has rendered cautious .
The problem is that you refer to a number of very different acts "sensations, thoughts, and so on", and conclude that they comprise a single act called "consciousness". Don't you think that the unification of these vastly varying acts requires something like a "homunculus"? Or do you appeal to magic as the source of such a unification?
I know that I can think of an apple and I know the concept of an apple, therefore thoughts and concepts must exist.
Kant Critique of Pure Reason A108 - "Just this transcendental unity of apperception, however, makes out of all possible appearances that can ever come together in one experience a connection of all of these representations in accordance with laws. For this unity of consciousness would be impossible if in the cognition of the manifold the mind could not become conscious of the identity of the function by means of which this manifold is synthetically combined into one cognition."
Consciousness, the unity of apperception in the mind is mysterious.
It seems that when the mind perceives a whole, which may be a set of parts, the mind is able to concurrently perceive each possible combination of parts as a unity, where each unity is distinct and irreducible. For example, the mind when perceiving a set of parts such as circular, sweet and red/green is able to perceive these parts as a distinct unified whole, an apple, and having a unity, irreducible. It will also be the case that when the mind perceives each possible combination of parts making up the whole, such as circular and sweet, the mind will also treat that combination as a distinct unified whole, and having a unity, irreducible
Similarly, each thought, such as the thought of an apple, is a distinct unified whole and as a unified whole is not only irreducible but has meaning.
Aint that the truth. Especially since mind is itself merely a conceptual placeholder for whatevers going on upstairs. Gotta call it something, right? Calling it something isnt enough, in that it still needs be explained what the hell its for, what its doing, and how do we know all that.
I prefer reason over mind, myself.
Quoting RussellA
Agreed, but the ultra-moderns will insist each thought is reducible to its meaning, which is directly related to its communal, collective use. Without, of course, a strict methodology by which that actually happens.
Because they dont like metaphysics, they kill it.
So the homunculus is only a logical problem if we're using it to explain something about consciousness. Otherwise it's no more a problem to refer to consciousness as a thing than it is to refer to gravity that way.
Insisting that consciousness is a set of actions implies knowledge about the nature of consciousness that we just don't have at present. There's no good reason to adopt that pretense.
Russell and Wittgenstein fundamentally differ in that Russell's logical atomism requires both knowledge by acquaintance and description, whereas for Wittgenstein's meaning as use, knowledge by description is sufficient.
The question is, is it possible that Wittgenstein's approach includes knowledge by acquaintance.
I don't think it does. As he wrote in On Certainty, the proposition "here is a hand" is more about how the proposition is used rather than making an empirical claim about hands in the world. It may be objected that Wittgenstein's language games are circular, in that the meaning of the word comes from the game. As there is no external link, there is one problem of how to choose between different games, and another problem that there is no allowance for discourse between different games. For example, an atheist using one language game may not be able to criticise a religious believer using a different language game. A particular language game within a particular society may well be coherent, but such a language may not correspond with the world that the society lives within.
IE, Wittgenstein's language game of knowledge by description includes no link to knowledge by acquaintance.
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Quoting Banno
Russell distinguished between two ways of thinking about things. He made the contrast between knowledge by acquaintance and knowledge by description, those things we think about directly and those things we think about indirectly. Knowledge by acquaintance includes sense data, universals, relations and oneself. As regards universals, he wrote "Not only are we aware of a particular yellows, but if we have seen a sufficient number of yellows and have sufficient intelligence, we are aware of the universal yellow"
The question is, are Russell's universals in fact not innate but learned by interaction with the world. If so, then Russell's knowledge by acquaintance becomes part of Wittgenstein's knowledge by description
I don't think they are. Consider those elementary concepts such as green or red, round or square, rough or smooth, tart or sweet, hot or cold, acrid or fragrant, loud or quiet, etc. I may have learnt many things over the past few years, but my perception of green, for example, one of these elementary concepts, has remained constant throughout my life. I certainly may have learnt more about the occurrences of green within the world, grass is green, traffic lights become green etc, but my innate ability to see green has not changed since the day I was born.
I agree that even in the absence of green I have the potential to see green, but this potential hasn't been taught, it was something I was born with. It is true, however, that I had to be taught that the name of my elementary concept of green is "green". It is also true that even though I have the potential to see green, I have to interact with the world, otherwise there would be no green for me to see. .
My ability to see green is innate, though I can learn by interactions with the world its occurrences in the world and can learn by interactions with other people its name.
IE, Elementary concepts such the innate ability to see the colour green cannot be learnt by description within a language game.
===============================================================================
Quoting Banno
We can call our perception of the colour green a "simples". Is Russell correct in treating such a simple as independent of context and as knowledge by acquaintance or is Wittgenstein correct in treating such a simple as being dependent on context and as such knowledge by description.
In fact, Russell and Wittgenstein are talking about different things. Russell's simples are within the philosophy of the mind and epistemology, where such simples have neither meaning nor can be true or false, Wittgenstein's simples are within language, can have meaning and can be either true or false. As noted by the SEP article on Wittgenstein's Logical Atomism "The so-called colour-exclusion problem is a difficulty that arises for the Tractatuss view that it is metaphysically possible for each elementary proposition to be true or false regardless of the truth or falsity of the others (4.211)."
Wittgenstein's simples as being within language cannot be independent of the context they are within, as Wittgenstein explains, whilst for Russell, simples in existing independently of meaning, truth and falsity can be independent of any context they are in.
IE, Wittgenstein's approach of knowledge by description within language cannot include Russell's knowledge by acquaintance outside of language.
In summary, Wittgenstein's approach cannot wholly replace Russell's, as Wittgenstein's approach doesn't include knowledge by acquaintance, which Russell's does.
That strikes me as a much better starting place than a supposed passive absorption of impressions.
I'll not enter into a competition between Wittgenstein's logical atomism and Russel's. Both fall to the expediency of what counts as a simple. That ought be clear from the SEP articles. What is to count as a simple depends on the task in hand.
Back again to to the begining, and
Quoting RussellA
In place of this I offer a picture of "two" as part of a family of activities that we engage in together. What is significant in these activities is what is shared, since the activity is that sharing. What is not shared, is not part of that public activity, and so of no significance.
I really don't see why the homunculus is a logical problem, maybe you could explain this problem for me. I realize consciousness presents us with a problem, but I think it's more of a problem of premises rather than a problem of logic. If the homunculus is inconsistent with some other premise, maybe it's the other premise which is the problem.
The idea of a Cartesian theatre is subject to the development of an infinite regress if we imagine that the stream of data coming into the CNS is being witnessed by an internal person.
Quoting Metaphysician Undercover
You know, it's really that we're at the very beginning stages of even theorizing about the nature of consciousness. We're still grasping for conceptual tools while wondering if such a science is even possible.
Quoting Metaphysician Undercover
Yes, the faulty premise is that the psyche is a full fledged being that is somehow independent of the body and the body's environment. For a lot of reasons, we know that can't be what's happening. The homunculus fallacy is just part of that.
I agree that "two" is part of a family of activities that we engage in together, but where did "two" originate, allowing us to use it in our activities.
In answer to the question what are objects such as apples and what are numbers such as two, I can refer to the Standard Model, Kant's Critique of Pure Reason, Russell's On Denoting and Wittgenstein's Tractatus and Philosophical Investigations.
Within the Standard Model, in the world are fundamental particles, fundamental forces, time and space.
We are born with certain innate abilities, which have evolved over 3.5 billion years, elementary concepts such as the ability to distinguish between time and space, green or red, round or square, rough or smooth, tart or sweet, hot or cold, acrid or fragrant, loud or quiet, etc. In Kant's terms, from the Critique of Pure Reason, these are a priori pure and empirical intuitions. His term for the mind's ability to combine distinct parts into a unified whole is known as unity of apperception. Given innate elementary concepts, we can then discover correspondences between them and what we observe in the world.
From Russell's On Denoting, these innate elementary concepts may be combined by the mind into compound concepts. For example, the elementary concepts circular, sweet and red/green may be combined into the compound concept of apple.
From the Picture Theory of Wittgenstein's Tractatus, it may be discovered that these elementary and compound concepts in the mind correspond with what can be discovered in the world, and once a correspondence has been discovered, that concept can be named. For example, in discovered that our elementary concept of red corresponds with pictures of red in the world, we can name this concept "red".
From Wittgenstein's Philosophical Investigations, these named elementary and compound concepts can then become part of a coherent language. For example, in the statement "an apple has the properties circular, sweet and red/green in colour".
Using the above, an object, such as an apple, is a set of related properties, such as circular, sweet and red/green. But as relations don't ontologically exist in the world, apples can only exist in the mind. Similarly, a number, such as two, is a relation between two individuals. But as relations don't ontologically exist in the world, the number two can only exist in the mind. Therefore, objects such as apples and numbers such as two exist only the mind as compound concepts.
In answer to the question posed in the OP, We Are Math?, the answer is yes, we are math.
This proposed internal regress is bogus. The Cartesian dualism holds that the supposed "internal person" is non-material, therefore its method or mode of witnessing the data cannot be represented as a stream of data which needs to be witnessed again. Therefore the propose infinite regress is actually broken at the very first step, and only created by a strawman representation which does not properly represent substance dualism.
Quoting frank
How do you understand "independent" here? Does it simply mean not dependent on? Obviously it is completely unrealistic to present the two substances of substance dualism as completely independent of each other, and no form of dualism actually makes this claim. Platonic dualism for example, places the material as dependent on the immaterial, as the immaterial is understood as prior in time to the material.
So representing the homunculus as "independent of the body" is not necessarily a faulty premise. If we adhere to Platonic dualism, the homunculus could be properly independent of the body (not dependent on it), while the body is dependent on the homunculus. Further, since the homunculus is supposed to be of a completely different substance the infinite regress is avoided.
The question then can be apprehended as an issue of how the homunculus can observe, and interact with the material world. This interaction is understood through the concepts of intention, final cause, free will, and choice. When we understand that the homunculus is not dependent on the material body, we have the premise required to apprehend these concepts, because we establish the order of necessity in its proper one-way representation, as required by the one-way nature of time. Time makes necessity a one way direction.
The homunculus is prior in time to the body, such that the existence of the body is dependent on the homunculus, as the effect is dependent on its cause. But the reciprocal relationship is not one of necessity. The cause is not dependent on the effect. Nor is a cause necessarily an effect, because this would produce the dreaded infinite regress which we must avoid. The body has a relation with the homunculus which is a relationship of necessity, the homunculus is necessary for the existence of the body. But the homunculus has a relationship with the body which is not a relationship of necessity, the body is not necessary for the homunculus. The body therefore is contingent, and its posteriority in time, from the perspective of the homunculus, makes its existence better described or understood in terms of possibilities. This is what allows for the reality of the freedom of choice, and the reality that the internal homunculus is better described in accordance with the principles of dualism as the operator of the body, rather than the passive "sitting inside your head looking out" which Banno stated.
It's just clear that who you are is culturally and chemically mediated. Whether you are a lawyer or a gangster, that stuff depends on your environment. Was there lead in the water you drank as a child? Did you inherit schizophrenia? Were you sexually abused? Was your father a billionaire? Did you become a heroin addict?
You'll be a very different person in each of these cases, with very different emotions and cognitive functioning. This leads us to ask what the homunculus is supposed to be.
Ive had a dozen occupations, both professional and incidental, yet Im still just lil ol me.
Can the interest which makes one good at something, and conversely the lack of it that makes him not so good, be predicated on cultural or environmental influences?
There are certain kinds of childhood trauma that result in dissociative personality disorder. People who have that don't report what you do.
The fact that you do indicates that you didn't have that trauma, and your short term memory is being stored properly. A lot of this happens when you're asleep. That's just the tip of the iceberg of environmental, cultural, and biological elements that go hand in hand when your sense of self. So it's just hard to imagine how your self could be independent of your body.
Quoting Mww
Sometimes. If you teach a girl that females are bad at math, voila, she doesn't put any effort into it, and subsequently sucks at it.
.is just to be a bad teacher.
Well, your enamoured with that picture, despite the misgivings expressed. A theory that implies a conclusion as misguided as that just quoted doesn't have much merit.
Quoting RussellA
And fish are mortgages.
That'll be that we treat names as rigid designators. Your properties change, yet you remain Mww. That's as distinct from there being some group of properties that set out what it is to be Mww. Individuals need not have an essence.
According to Kripke, they always do.
Yes. If being made of wood is essential to what you mean by "that lectern" then that's an essential property, even though you learn about it a posteriori.
Mww and I were just talking about long-term memory, and we could say Mww's evolving memory is an essential feature of the guy in question, even though things could have been different.
It's tricky, but cool.
Yep.
One can suppose that Mww might have had different memories - or have lost them all together. Now if you ask "Who is it that lost their memory?" the answer has to be "Mww". Hence his memories are not essential to his being Mww.
So without his memories Mww would be a different Mww? Possible world semantics would say he has different properties, but still be Mww.
They are if they're essential to the Mww we're talking about. Look back at the aposteriori necessity.
I suppose you could maintain that Mww was only Mww up until the time he lost his memory, after which he was someone else. But in that case it is not Mww who has no memory, but someone else. The sentence "Mww has lost his memory" could not be about Mww.
When we say his memories are essential, we're saying that in all possible worlds where this particular Mww exists, he has these memories. It's basically part of our definition of him.
Just like Kripke's wood lectern. In every possible world where that lectern exists, it's wood. That's why wood becomes essential. It's a matter of the object of the statement.
If Mww subsequently loses his memory, what is now essential? It's all a matter of what we're trying to say.
I am this, my properties change, therefore the this I remain is not my properties.
I am an individual, but I dont need an essence.
I am not my properties, and my essence is not necessary.
Then I have absolutely nothing by which to judge myself, thus I cannot know anything about myself.
Yet, I do.
But then Mww would cease to be an individual, rigidly designated by "Mww".
"That wooden lectern" is a description, not a name. So yes, obviously, in every possible world in which that wooden lectern exists, it is a wooden lectern. In possible worlds in which the lecture is stone, it's a different lectern.
Again, who has lost their memory?
Quoting Mww
You still have properties. Judge yourself as you see fit. Those properties are just not essential.
That's incorrect. It appears that you ignored most of the essay.
"Mww" is a rigid designator. It picks out the same individual in every possible world. It picks out Mww in those possible worlds in which Mww lost his memory.
Hence we might say "Mww lost his memory", and not resort to "There was someone who was once Mww, but they lost their memory, and so are no longer Mww".
Are you, @frank, saying that this is incorrect?
I don't see how this is relevant. The homunculus is what allows oneself to adapt to such a wide range of environmental factors, like what you describe. Without such an "inner being" the living creature would not be able to change itself in the ways required to make the best of those various different circumstances. Simply put, it allows us to learn, which is to make changes to ourselves. So it is the homunculus itself which allows for what you describe, that every person is a very different person according to one's adaptations, yet still a person.
Remember that possible world semantics is about analyzing particular statements. This starts with understanding what a speaker intends.
It looks like you're trying to pin down "Mww" to the same meaning in every statement. It doesn't work that way.
Quoting Banno
Sure. This does not preclude the making of statements in which a particular memory, or a particular evolution of memory is essential to the subject of the statement.
It looks like you're pretty firmly wedded to the idea of a Cartesian theatre. I'm not, but it does occasionally jar me to know that I'm a product of chemicals and customs. :grimace:
An odd tangent.
We know that a rigid designator picks out the very same individual every possible world. We agree that "Mww" is a rigid designator. We agree that there are possible worlds in which Mww lost his memory. We agree that "Mww might have lost his memory" is a sentence about Mww.
In contrast, if having certain memories were essential to Mww's being Mww, and without those memories Mww is no longer Mww, then absurdly "Mww has lost his memory" is not a sentence about Mww.
Quoting frank
Well, that's the point of using rigid designators.
Quoting frank
So you assert. The argument presented here seems to suggest otherwise.
In every possible world where that individual occurs. When we say Nixon might have lost the election, the only possible worlds we're looking at are the ones in which he ran. That he ran for office is made essential to "Nixon" by the intentions of the speaker.
I'm not going to explain that again. Just read the essay.
Quoting Banno
Oh dear.
Sure. That's a plain English rendering of accessibility. Worlds in which Nixon lost are only accessible from worlds in which he ran.
But so what? Nixon remains Nixon in worlds in which he lost his memory...
Quoting frank
Oh dear indeed.
Again, who is the sentence "Mww lost his memory" about? I say it is about Mww. But if his memory is what determines that "Mww" refers to Mww, as you appear to be saying, then who is it about?
I didn't say that. The intentions of the speaker determines what "Mww" refers to.
Oh, dear.
:lol: Read the essay.
Then I am left to judge myself by that which changes, which makes explicit I cannot know myself as a singular self, insofar as my self must change as do the properties of me. And insofar as it is always and only myself that judges, and that which judges, and that which is judged by, always changes, then it is impossible for there to be a singular identifiable self which is judging.
What a mess am I.
Youve linked to trope theory and Cartesian theater. Which of these is the essay?
We were talking about Naming and Necessity, by Kripke. Banno got the naming part, the necessity part, not so much.
:chin:
I'm surprised you remember. I think there's something specific you wanted out of it and you ignored the rest.
Oh. Language philosophy. Hard pass.
I speak, you listen. You speak, I listen. Figure out whatever differences there might be.
End language philosophy 101.
No, actually I'm not, I think it's a very simplistic representation of dualist principles. I just hear people asserting that this representation has been disproven (when to my knowledge it hasn't), and I want to know if I've missed something, or if someone has come up with something new. But all you had was the false representation of infinite regress, which I've seen before.
Quoting frank
If it jars you then why believe it? When something is so highly counter-intuitive, then you ought not believe it unless it is well proven. That it's pop-culture obviously does not imply that it's been proven.
Quoting Banno
There is no such thing as the same individual in different possible worlds. That's a bad fiction which allows that the same individual has contradicting properties in different possible worlds, which implies that they cannot be the same individual. In other words, violation of the law of non-contradiction occurs if you call them "the same individual". Therefore "the individual" must be proper to one possible world only, and any other possible world would have a different individual. Never the twain shall meet. Frank's got the right idea, you are lost.
From your insistence that I "read the article", it seems you didn't until reminded.
https://thephilosophyforum.com/discussion/4545/naming-and-necessity-reading-group/p1
https://thephilosophyforum.com/discussion/4857/naming-and-necessity-lecture-three/p1
The second link has the discussion of Lecture Three, and so of kinds.
Yeah, you won't like it. He shows that there are a posteriori necessities.
Quoting Metaphysician Undercover
:roll:
That was pre-pandemic. I'm not the same person I was then.
Face it Banno, you're wrong. An individual cannot exist in numerous possible worlds. If the designator picks out something which is common to numerous possible worlds, it is clearly not an individual.
?
If there was a single feature of the designated individual which was common to all possible worlds, we would say that this is a necessary, or essential feature of the individual. We could not entertain the possibility of a representation of the individual without including that feature.
Kripke brought up possible worlds as an aid to understanding how modality works. There are ways of parsing modal expressions that turn them into nonsense, and I think MU would be inclined to do that. He'd say we can't assert that Nixon could have lost, because if he lost, that wouldn't be Nixon.
I think this confusion arises from trying to do something ontological with modal expressions, when that's not the intent behind them. We're generally just playing with logical or metaphysical possibility, and that's the way possible worlds should be taken: as logical hypotheses.
For instance? Quick and easy and to the point, please.
So we might be able to work out where exactly we differ in our readings of Kripke.
Quoting frank
According to Kripke, kinds have an essence, but not individuals.
Now an essence is here understood as a property that belongs to the item in every possible world. So there need be no property of Mww that belongs to him in every possible world, but the property of being made from H?O is true of water in every possible world. It's being made of H?O is essential to water.
Agree?
...the property of being made from H?O is true of water in every possible world, but is known a posteriori.
I think you're missing the point/meaning of possible-world semantics, MU. Aside from people like Lewis (who is a realist wrt possible worlds), "existing in a possible world" is (essentially) just a different way of saying that something isn't contradictory, that it does not entail a contradiction. That's it. So saying an individual exists in a possible world is only to say that some particular description, predicate, or state of affairs involving that individual is logically possible- it doesn't involve any contradiction or inconsistency.
So yes, an individual "exists" in numerous, maybe even uncountable, possible worlds, because there are numerous, maybe even uncountable, logically-possible propositions, predicates, etc that we can say of a given individual.
Yeah it occurred to me that engaging with MU on matters involving AP/philosophy of language/modal logic/etc might not be the brightest idea I've ever had, but I've been out of the mix for so long I think my tolerance for philosophical shenanigans is fairly high at the moment (something I expect will change/fade rather quickly).
This is a notion that still mystifies me.
According Kripke, his wooden lectern is made of wood in every possible world where that lectern exists. There are all sorts of properties we could change and still have the same lectern, but being wooden isn't one of them.
It's an essential property. Do you disagree with him about this?
Yes, that's exactly the point, "an individual" speaks of something in a completely different ontological category from what a "logical hypothesis" speaks of. So Banno's attempt to bring the individual into the logical hypothesis was a category mistake.
We see this same mistake quite often when people speak of "possible worlds". They will say for instance, that one of the possible worlds is the actual world. But the possible worlds are just representations, logical hypotheses, and although one of the possible worlds might be judged as the correct representation, or some feature might be common to a whole set of possible worlds, it is still not the actual world, as this is a distinct category from the representation.
Quoting busycuttingcrap
No, that's the category mistake described above. An individual does not exist in any possible worlds. As frank explained, possible worlds are logical hypotheses. Individuals are actual objects in the physical world. There might be individuals represented by such hypotheses, but no individuals exist in these hypotheses
Agreed. If he exists at all, in whichever of the possible worlds he exists in, that world must be necessary.
Carry on with your terribly misguided philosophy (if one can call it that) then.
Quoting frank
Trying to get through to Banno is like banging your head on a brick wall. Banno's been trying to argue that the essential property (what's common to every possible world), is the individual.
The example is found in the article Identity and necessity, not Naming and Necessity. Bottom of p.178. (the link is a dreadful PDF - anyone have a better copy?).
So we have two principal choices, either "X" refers to a different individual in possible world #1, from what it refers to in possible world #2, or else it is just like a placeholder within those possible worlds, as a representation of a single individual which is supposed to exist in a separate world all together. The latter is the conventional interpretation. The contradicting propositions are statements of possibility for an individual believed to exist in a separate actual world. But this means that "X" refers neither to an individual in possible world #1, nor to an individual in possible world #2, but to an individual in some separate world. In those 2 possible worlds, "X" just provides a means for us to talk about possibilities for that individual which exists in a completely separate world.
It appears like the existence of the Many Worlds interpretation of quantum mechanics, has produced an acceptance of the other principal interpretation. When the numerous different possible worlds are each assumed to have actual existence, as in MWI, then X must refer to a different individual in each possible world. Being a part of a different world in each case would render the individuals as different individuals. If X is taken to refer to the same individual existing in many different worlds at the same time, incoherency results from the contradiction of saying that numerous different things (different by way of each having a different description) are actually one and the same thing.
Oh. Thanks for the correction.
Where do lecterns exist ?
Kripke gives the example of "here is a lectern" as a description of something made of wood, something that can only be known a posteriori and is an essential property.
However, what happens when we move from the demonstrative pronoun to the definite article.
There is no single property that lecterns have. Some are made of wood, some of metal, some have a flat base, some a legged base, some are grey in colour, some brown, etc. But as Wittgenstein pointed out, objects such as lecterns do have a family resemblance, such that a human observer can judge the difference between a lectern and a non-lectern.
As lecterns have no essential property, then lectern is more like a rigid designator than a description, as Mary as a name is a rigid designator, having no properties.
If lecterns exist only as a family resemblance between "this lectern" and "that lectern", and family resemblances is a human judgement, how can lecterns exist in the world, unless family resemblance is also something that exists in the world ?
Known a posteriori in this world. Empirical knowledge obtained in a given world cannot translate to empirical knowledge in some possible world without contradicting the conditions for empirical knowledge. Ever been to a possible world, observed what is already cognized as water, analyzed it to find H2O in it, or not? Unless that happens, knowledge by experience is utterly irrelevant.
So it must be that is hardly an a posteriori necessity. Only if a myriad of presuppositions hold, the very epitome of contingent identity, would water on any possible world perfectly replicate water as it is known a posteriori on this one, the presuppositions we have logical .you know, one of those cursed a priori scripts but no empirical, justifications whatsoever, to hold.
So, yeah, true enough, water is made from H2O in any possible world, iff every single antecedent condition by which that criteria is met here, is met as well there. THAT is what we have no warrant to authorize, insofar as the plethora of antecedent conditions makes explicit there are some of which we have no knowledge, which means we could never claim the criteria there is met because we dont even know the totality of the criteria here. Therefore, a posteriori necessity is, while not absolutely false, is not necessarily true.
Philosophy was warned about this misuse of reason, but apparently, chose to disregard it.
It seems like you didn't hear me the first time. Every time you read "there is/exists a possible world such that X, Y, or Z", mentally substitute "it is not contradictory/inconsistent that X, Y, or Z". Possible-world semantics isn't actually making an ontological claim (at least, not if you're not David Lewis), its making a claim about logical consistency. And so obviously, the actual world is a possible world, since "being a possible worlds" means "not being contradictory/logically inconsistent". And the actual world is not self-contradictory or logically inconsistent, so, it is a possible world. I mean, obviously, how could the actual world be actual, if it wasn't possible?
Sure it does, because "existing in a possible world" isn't an existence claim about other worlds, despite appearances to the contrary. Replace "existing in a possible world" with "being logically possible/self-consistent", and your objection disappears.
You're just misunderstanding what possible world semantics means, for which you don't deserve blame: it can be highly misleading to say that "there exists a possible world" when all you mean is that something is logically possible or self-consistent. But it is a useful way to conceptualize possibility and modality, in certain contexts. Its far from the weirdest thing philosophers are prone to talk about.
Quoting Metaphysician Undercover
Understanding how certain technical terms are actually used in the relevant sub-field is actually just about the opposite of misguided philosophy: its trying to understand what philosophers mean, on their own terms. So its sort of necessary for a proper understanding of any philosopher. But I was admittedly tentative about engaging with you on this, because you are, frankly, sort of known for being stubborn about these things and not amenable to correction. But I assure you, as someone who spent most of their undergrad philosophy degree focusing on contemporary analytic philosophy and philosophy of language (so, for instance, folks like Saul Kripke), you're simply misunderstanding what these terms usually mean, in the context of contemporary philosophy/modal logic.
I'm sure you are both aware of this. It's a shame, since there are interesting problems in modal reasoning that we might discuss, but any discussion on these forums is plagued by such background noise from 'modal sceptics". I do not expect to achieve any depth.
This ties in with the recent discussion of forum quality, and I'd suggest following 's suggestion there of restricting the discussion to a particular essay. That way at the least it will be easier for mods to identify and deal with off-topic, dissimulating or obstructive comments.
I'm also aware that the link between this discussion of modality and the OP is not obvious.
Perhaps a thread on Identity and necessity?
Good question. In modal logic it is important to be clear about different ways in which individuals, kinds and descriptions are treated. So a lectern which is picked out by a description will be treated differently to say an individual picked out by a proper name.
I'd say the wooden lectern (definite article) is necessarily made of wood - if it were not, it would be some other lectern. But perhaps that lectern (demonstrative pronoun) might have been made of ice.
Subtle stuff, prone to induce long tedious threads.
Sounds fun!
OK, I understand what you are saying here. Now the problem is that when someone like Banno says that X,Y, or Z refers to "an individual" this is an ontological claim. So you can have your X, Y, and Z referring to nothing if you like, or even refer to a type, and claim logical consistency, but as soon as you say that one of these refers to an individual then you need to account for the existence of that individual because you have made an ontological claim.
Quoting busycuttingcrap
This does not solve the problem, because Banno's claim was that the designated individual exists in numerous possible worlds. And this produces logical inconsistency because the description of the designated individual is different in the different logical possibilities. Therefore these cannot be the same individual in distinct logical possibilities because of that inconsistency. Each logical possibility must necessarily represent a distinct individual.
So for instance, it is possible that MU is male, and it is possible that MU is female. In these two logical possibilities (these two possible worlds), "MU" does not refer to the same individual. In one possible world the individual is male and in the other possible world the individual is female. Therefore if the claim is that MU refers to an individual within each of those possibilities, these are necessarily two distinct individuals, on male, one female.
Quoting busycuttingcrap
Great, now are you ready to address the issue, which is the existence of the individual, in relation to logical possibilities, because you seem to have completely skirted the issue in this post.
The claims in question aren't ontological claims; that's the entire point. They sound or look like ontological claims, but they are not. So when I say that "there is a possible world such that X", for instance if I say "there is a possible world such that MU is president of the United States of America", I am not making an ontological claim, I am not asserting the existence of anything: the phrase "there is a possible world such that X" is synonymous and interchangeable with the phrase "X is logically possible/self-consistent/non-contradictory". So I'm not asserting that there exists any such world, I'm just saying that the proposition of MU being the president of the USA is logically possible/does not entail a contradiction.
Quoting Metaphysician Undercover
Sure it does: "existing in a possible world" means not entailing a contradiction. And there are numerous claims we can make about a given individual that do not entail contradictions (remember, "there exists a possible world" is synonymous with "does not entail a contradiction").
And so this suffices to address your concern about "the existence of the individual": as far as modality goes, the existence of an individual in different possible worlds is the same thing as having multiple logically possible/self-consistent propositions or predicates we can assert of that individual. MU "exists" in multiple possible words... because there are multiple propositions or predicates we can assert of MU that do not entail contradictions.
That's literally all that's going on here, "possible worlds" talk is just a different way of talking about contradiction and logical possibility. Which is admittedly confusing, since talking about the existence of possible worlds sounds like an ontological/existential claim... but its not.
I dont think whats now called modal reasoning is all that contrary to Kant, but more a unnecessary extrapolation of it. Or, to be gentle about it, a modernization. Kantian speculative metaphysics, after all, employs the very same modalities, just without the fancy symbols, and at a MUCH more fundamental reasoning level.
I don't get Kant, so I'm going by what it says in the text books. I'll try to reply to your previous post soon.
This notion of empirical knowledge of possible worlds is... confused.
Best way to think of the process is that the facts in a possible world are stipulated. They are certainly not observed.
Think of possible world semantics as a way of setting out or parsing a counterfactual English sentence. So the counterfactual "Banno might have put on the green shirt this morning" would be rendered as "In some possible world, Banno put on his green shirt this morning".
One does not peer into possible worlds; one constructs them.
Now that water is H?O is known a posteriori - discovered at some stage by burning hydrogen, it seems.
But on Kripke's account, it is a necessary fact; in any possible world you might specify, water is composed of H?O; or, if you prefer, you might stipulate a possible world in which the word "water" did not refer to H?O, but to XYZ, but even in that world water would still be H?O. It just would not be called water.
Aslo, note Quoting busycuttingcrap
None of what has been said here is making ontological claims; it is only setting up consistent ways of talking about counterfactuals.
Quoting Moliere
Try
Quoting Banno
Banno already covered this very well and I don't have much to add to what he said, except to emphasize that possible-world semantics- i.e. "there is a possible world (such that X, Y, or Z)" - are not actually making ontological claims (despite appearances to the contrary- "there is a possible world..." certainly sounds like an existential claim!), claims about the existence of some other world out there somewhere existing in... different dimensions or universes, I guess?
Instead, possible-world semantics is just a different way to think/talk about modality, a conceptual tool for thinking/talking about logical space: in other words, a way to think or talk about logical possibility (i.e.non-contradiction). We say that something is logically possible iff it does not entail a contradiction. And if something is logically possible, then we may also say that "there is a possible world" where that something is true or is the case. That's it. The bar for being a possible world is pretty low- as long as something doesn't entail a contradiction, then there is a possible world for that something. .
And possible worlds talk is admittedly quite confusing and misleading for people not familiar with this particular area of logic/philosophy- it sounds like asserting the existence of some unknowable world out there in the great beyond.. But its not. So ignore how it looks/sounds, and when you see the phrase "there exists a possible world such that X Y or Z" just mentally replace it with "X is logically possible" (i.e. X does not entail a contradiction) and a lot of these problems and questions should disappear..
Quoting Banno
Why would we do that?
-
Got it. Thanks.
Thoughts may actually be concrete objects (parts of a brain) and abstract objects in the sense of universals, like numbers, may actually be similarity relations among concrete objects. What is outside of spacetime then? Perhaps other concrete objects, including other spacetimes. According to theory of relativity, spacetime is actually just a special kind of space, a 4-dimensional space with one dimension (time) somewhat different from the other three. And according to topology, a space is just a special kind of collection. All mathematics seems to be reducible to concrete collections, from the empty collections (non-composite objects) to infinitely large collections (infinitely large composite objects). That's why set theory (the ultimate theory of collections) is regarded as a foundation of mathematics.
So according to set theory, all logically possible (consistent) collections exist, from the empty collections to infinite collections, and they, or relations between/among them, constitute all known mathematical objects, relations or structures. Note that all of this exists necessarily/automatically because nothingness constitutes the content of empty collections and empty collections constitute the content of larger collections, and larger collections constitute the content of even larger collections, and so on. And a spacetime is one of those collections and we are collections that are parts of a spacetime.
So is everything math? Well, there seems to be something about collections that is extra-mathematical. There is a composition relation (or set membership relation) between a collection and a larger collection that includes it. So collections are somethings (not nothing, as there can be no relations between nothing), but mathematics doesn't tell you more about these somethings than that one something includes another something. Mathematics is just about relations between these somethings and these relations are reducible to the composition relation. These somethings are not relations; they are what stands in composition relations. These non-relations might be called "things" or "qualities".
Quoting Mww
Parsing counterfactuals in terms of possible world semantics makes explicit the relation involved in the counterfactual.
So parsing Banno might have put on the green shirt this morning" as "In some possible world, Banno put on his green shirt this morning" sets it out with an explicit, consistent grammatical structure. Much the sam as "the shirt is green" can be set out as "there is something that is green and is a shirt"
(?(x)(g(x) & s(x))
And what is the difference between a logically possible world and a real world?
Ok, thanks. Im good with Banno might have put on a green shirt this morning. Im aware of the logical entailment that in some possible world he did, but my knowledge of either of those is exactly zero, so .
Sure, this is what you say now, but both you and Banno were making ontological claims. Banno said that in every logically possible world, mww is still the same individual, the same person. That is an ontological claim about the person named mww, which is completely independent of the logical possibilities you are talking about. And you yourself said the following:
Quoting busycuttingcrap
Notice, you assume the existence of an individual here. That is an ontological claim. Without the assumption of the existence of the named individual, logical possibilities take on a completely different role. Consider your example "MU is the president of the United States of America". If we assume the existence of a person named MU, then you are saying that it is possible that this person (with ontological status) is the president. But if we do not assume an ontological person named MU, then you are saying something completely different. You are saying that it is possible that the person who is the president is name MU. That is because we haven't given any necessary existence to an individual named MU.
These differences are dependent on the ontological assumptions made. So in this quote above, you are assigning ontological status to "an individual", then you are proposing to use modal logic to make statements of possibility concerning this assumed ontic individual. And you conclude that the individual "exists" in each of these numerous different logical possibilities. But that's where you are wrong. Each of the logical possibilities is a description, a predication, which could possibly be assigned to the individual. The 'possible predication' is not being assigned to the individual, it is proposed only as a possibility. Therefore the individual is really not there, in that logical possibility, because no actual predication is being made in that scenario of logical possibility (possible world). The individual must maintain an existence, separate from the possible predication, to maintain logical consistency, and ensure that the predication is a possible predication rather than an actual predication.
In this case, the 'possibility' was maintained to exist between the individual and the predication. We have the actually existing person, name MU, and the possible predication "is the president...". In the other case I described, there is no assumed person named MU, just the possibility "MU is the president...". The two cases have very different meaning, and the difference is due to one's ontological assumptions concerning the individual, MU.
Quoting busycuttingcrap
But the claim was that the individual exists in the possible world, not that what is said about the individual exists in the possible world. We know that the predication, the claim about the individual is a possibility, and therefore exists in the possible world. What is at question is whether the individual exists in the possible world.
So I'll tell you again, and maybe you'll make more sense of it this time. If the designator ("MU" for instance) is assumed to name a real individual, with existence in the world, this is an ontological assumption which denies the possibility that the named individual is a part of any logical possibilities proposed (therefore not a part of the possible worlds). So in this case, we cannot say that the named individual has any existence in any of the logical possibilities. This is already denied, because the real existence, the reality, or actuality of the named individual is already assumed by that ontological assumption, therefore no possibilities about the existence of that individual can be entertained. The reality is that the existence of the person is completely removed from, and irrelevant to the logical possibilities scenario.
Quoting busycuttingcrap
This is where your mistake lies. The problem is that with logical possibilities we can make contradicting predications, because they are only possible predications. So we can say for example it is possible that the person we know as MU is the president, and also that it is possible that the person we know as MU is not the president, if we do not have the actual predications for MU required to make that decision.
And this is why these cannot be considered as predications, they must be considered as possible predications. And, as I explained, this puts the division between possible and actual between the predicate and the individual, such that the individual is actual and completely separated from the predicate is a possible predicate, and therefore there is no proper predication.
If we claim as you state, that the same individual, the one we know by MU, exists in many possible worlds, then we have logical inconsistency because the law of identity and non-contradiction would be violated. We'd have to say that this same person, MU, is president in this possibility, and also not president in a different possibility, but in both scenarios is still the very same person. Well we cannot say that these are the very same person without contradiction, so the two scenarios would have to involve different individuals. Instead, we must say that just the predications are possibilities, and the individuals are separate from these possible predications (worlds), being actual and real. Therefore only the possible predications are within the possible worlds, while the individuals are not. In Aristotelian terms, the individuals are primary substance.
The concept of "counterfactuals" has ontological assumptions intrinsic to it. By designating something as counter to fact, you assume to know the fact, and that's an ontological claim.
That's the problem with your way of looking at logical possibilities. You make ontological assumptions, like the existence of the individual, such that the individual becomes a necessity within your possibilities (in all possible worlds). But this necessity is not a logical necessity at all, its just produced from your ontological assumption, the existence of the named individual. If you remove the necessity of the individual, to support your claim of making no ontological assumptions, then the logical possibilities (possible worlds) look completely different (explained above).
Yes please.
Where does Kripke's Identity and Necessity say that numbers exist
As regards Kripke's chapter on Identity and Necessity in his book Philosophical Troubles: Collected Papers, Volume 1, he writes:
"Independently of the empirical facts, we can give an arithmetical proof that the square root of 25 is in fact the number 5, and because we have proved this mathematically, what we have proved is necessary. If we think of numbers as entities at all, and let us suppose, at least for the purpose of this lecture, that we do, then the expression the square root of 25 necessarily designates a certain number, namely 5. Such an expression I call a rigid designator. Some philosophers think that anyone who even uses the notions of rigid or nonrigid designator has already shown that he has fallen into a certain confusion or has not paid attention to certain facts. What do I mean by rigid designator? I mean a term that designates the same object in all possible worlds."
On the one hand, he writes that numbers necessarily exist in all possible worlds, meaning that numbers ontologically exist in the world. However, he doesn't specify whether this world exists in the mind or is mind-independent. On the other hand, he writes that we are able to manipulate numbers independently of the empirical facts, meaning independently of any mind-independent world.
For Kripke's Identity and Necessity, as numbers ontologically exist in the world, and as we can manipulate numbers independently of any mind-independent world, the world he is referring to must be in the mind. IE, Kripke's Identity and Necessity infers that numbers exist in the world of the mind, not in a mind-independent world.
If numbers did exist outside our three dimensions of space and time, one wonders how a calculator physically existing in space-time when adding numbers is able to access numbers existing outside of space-time.
If numbers did ontologically exist mind-independently, as numbers exist as relations between individuals, one wonders how the ontological existence of relations in a mind-independent world can be justified.
It is not true that according to set theory all logically possible (consistent) collections exist. First, it's not even clear how that would be stated as a mathematical statement in set theory. Second, it's not even clear how that would be stated as a rigorous philosophical principle regarding set theory. Third, even if we did have in front of us a rigorous statement of such a philosophical principle, it's not a given that it is the consensus of set theorists and philosophers of mathematics that it is true.
Moreover, there are infinitely many statements formalizable in the language of set theory that state the existence of sets with given properties but such that it is consistent with set theory there exists such a set, but it is not a given that set theorists endorse that any given one of those sets exists. For example, it is consistent with set theory that there is a set that has cardinality strictly between the cardinality of the naturals and the cardinality of the reals, but it is not a given that it is the consensus of set theorists and philosophers that such a set exists.
Moreover, even stronger, set theory does preclude certain kinds of sets that otherwise it would be consistent to say they exist. In particular, the axiom of regularity precludes certain kinds of sets that otherwise would be consistent to say they exist.
Also, granted that some writers refer to "consistent collections", but that may cause misunderstanding, since it's not collections of sets that are consistent or not, but rather collections of statements about sets that are consistent or not.
Regardless of what you "actually hold", it is what you actually said. That's the problem, what you say is not consistent with what you actually believe. So you are diabolical in your attempts to describe things in ways which you do not yourself believe.
And the Kripke explanation quoted by RussellA above, makes the very same deceptive statement, stating something which nobody actually believes
Quoting RussellA
There simply isn't any objects in logical possibilities (possible worlds), and nobody actually believes that there is, despite the fact that many people like busycutter, and Banno, argue that there is.
Like what?
What I said was perfectly consistent with what I believed; you've merely been fooled by your willful ignorance RE how possible-world semantics works.
I mean, I'm sorry that you object to people using ontological-sounding language to talk about modality and possibility rather than existence, but you're not the language police, you don't get to tell people how they can or can't use technical technical terminology, or what their terms mean. If we stipulate that we're using phrases like "there is a possible world such that X" to mean "X is logically possible", then that's what we mean when we use those phrases- if you don't like it, too bad.
All you can do is yourself refrain from using possible-worlds talk, or you can stipulate that when you use possible-worlds talk you are using it as e.g. literal existential propositions. And the rest of us can and will continue to use them in the way that has been explained to you here, i.e. as a useful alternative way to talk/think about logical space that is not ontologically-committing.
The axiom of regularity precludes that there exist non-empty sets that don't have a minimal element. Most saliently, the axiom of regularity precludes that there is a set that has itself as a member.
I suppose we can elect @busycuttingcrap to moderate the thread. I'm looking forward to it.
Well, shoot. I was looking forward to that thread. I wish I could start it...
:cheer:
Why would there be less to do in a merely possible world? Some worlds may be simple and others more complex, whether they are merely possible or real.
I mean "set theory" in the most general sense, also known as naive set theory. It just says that a set is a collection of objects. This general concept of set is elaborated in uncountably many axiomatized set theories, for example the famous ZFC set theory. I refer to all these axiomatized concepts of set, as long as they are consistent.
Quoting TonesInDeepFreeze
For me, as long as such a set is consistently defined, it exists. In some axiomatized set theories it may exist while in others it doesn't. That's because every axiomatized set theory selects a limited collection of possible (consistently defined) sets. This is what Joel David Hamkins has called "set-theoretic multiverse".
...with respect to our world.
I just needed to finish that sentence to avoid confusion.
It seems you misunderstand the situation. I have no problem with you using the modal language mentioned here. Sure, "there is a possible world..." means ... is logically possible. That's obvious, and not an issue.
The problem is when people like you and Banno, also Kripke, use ontological language to talk about how an individual "exists" and the "existence" of an individual within your modal logic. This is what creates confusion for people. Aristotle set this separation years ago, to combat sophism. The individual exists as primary substance, and is therefore separated out from the logical structures. That is the difference between the subject which serves for predication, and the object which has separate, independent existence.
Quoting busycuttingcrap
The problem is that you use "existence" to talk about something other than existence. What's the sense in that? If you're talking about modality rather than existence, then obviously the appropriate thing to do is not to use "existence". Let me remind you again what you said.
Quoting busycuttingcrap
It appears that you stand corrected. The same individual does not exist in numerous different possible worlds, because if individuals did exist in these possible worlds they would be distinctly different individuals, according to the possibilities proposed. Therefore they would not be the same individual. If they were considered to be the same individual, the law of non-contradiction would be violated.
Sure, the first time they hear the phrase "there is a possible world such that blah-blah-blah". Then someone explains it to them, and they're all good. The only problem here is your stubborn insistence that people can't or shouldn't use terms in a way you don't like or agree with. But that's a problem on your end: possible-world semantics works, it is a useful tool, and so logicians and philosophers are going to continue to use it. If you don't like it, you're free to not participate.
In the expression "an individual exists in a possible world", the word "exist" is being used metaphorically, not literally, in the same way that it is being used metaphorically in the sentence "I existed on my desire for vengeance". The problem with a metaphorical language is that meaning depends on context and if the context is vague then the meaning is vague.
The problem is, that if we said "an individual exists in our actual world", are we still using "exists" metaphorically or literally ?
And then again, where does this "actual world" exist. I think it exists in the mind, though others would disagree. But even "the mind" is a metaphor.
IE, an individual exists in a possible world metaphorically, a possible world is a metaphor, exists in our actual world is being used either metaphorically or literally, and our actual world exists either metaphorically in our minds or literally as mind-independent.
(CPR, A61/B86)
Possible world semantics: amusing to play with, but dont think for a minute theres any knowledge to be gained by it. And if theres no knowledge to be gained, whatever amusement there is, is time poorly lost.
Again, you're failing to grasp the issue. The separation between the logical subject and the physical object provides the force for Aristotle's refutation of Pythagorean idealism, commonly known as Platonism. So the fact that logical subjects are not individuals, or particulars, is the premise whereby the illusions of Platonist fantasies can be dispelled.
Take the proposition of the op for example. "We are math". If the "we" of this statement refers to a multitude of physical individuals (conventional usage), then the answer to the question of the op is no, because there is a separation of category between these physical objects referred to with "we", and the logical subjects of mathematics. To say "we are math", when "we" is understood in this conventional way would be a category mistake. But if "we" is understood as a logical subject instead of a collection of particulars, then there is no such category mistake, and Platonism is allowed to flourish. Then there is nothing to prevent "we are math" from being a true proposition.
Of course, it ought to be obvious to you, that "we" is not a proper logical subject, it is vague, ambiguous, and not well defined. So the latter use of "we" ought not be allowed into any logical proceeding because of the ambiguity it brings with it. And this is exactly the problem with your and Banno's use of "individual". You insist on allowing an individual to be a logical subject (above mentioned category mistake), thereby introducing this ambiguous, ill-defined, form of logical subject into your logical proceedings, 'the individual'. I've argued against this practise in many threads on mathematics, where the ill-defined logical subject which is claimed to be a particular entity is called a mathematical object. But this is just a well-known category mistake, which was thoroughly exposed by Aristotle in his efforts to disclose the pervasiveness of sophistry in his time.
So I agree completely with you in your assessment of "your stubborn insistence that people can't or shouldn't use terms in a way you don't like or agree with". But I disagree with your characterization of this being a "problem", in this particular instance. My stubbornness and insistence is well justified and supported by the fact that what you and Banno propose constitutes a well-known category mistake. And this type of behavour, of insisting on allowing such ambiguity into your descriptions of logical possibilities, displayed by you and Banno, has been well documented as the basis for logical sophistry. So my insistence is warranted as well. Therefore my insistence that you use terms in a way consistent with good philosophical practise is not a "problem" at all, but has already been well demonstrated to be the solution to a problem, while your practise is the problem.
Quoting RussellA
RusselA, we know that metaphor has its uses. But creating ambiguity in terms which already are well-defined in philosophy, for the purpose of sophistry, is clearly not a good use. That is very poor epistemology.
Quoting RussellA
Metaphors do not provide good premises for logical proceedings. That is why we separate out ill-defined things like "particulars", "individuals", "objects", and speculate metaphysically about the existence of these things, rather than allowing them into our logical premises. When we allow ambiguity into the premises, soundness suffers.
True. In the sentence "Mary exists in a possible world", "exists" means "could exist", so the sentence is incorrect. It should be "Mary could exist in a possible world".
However, if it was a deliberate intention to use "exists" as meaning "could exist", then this would have been a valid metaphorical use of language. Confusing, but valid.
Sorry, but it's entirely legitimate to ascribe the predicate of existence of Mary in a possible world. Why is there so much confusion about counterpart theory or possible world semantics?
The point I was making, is that in this situation, the predicate "exists", is predicated as a possibility, therefore a possible predication, as is the case in "possible world" language use. "Mary", as the subject, on the other hand, must be given a place in relation to the possible predication. "Mary" does not signify a part of the possibility, the predication is the logical possibility. We can say that "Mary" represents an individual, but we still cannot assign "existence" to this individual without justification, as is the case with all such logical subjects. That it is logically possible that the individual represented by "Mary" does not exist demonstrates that we cannot assign existence to that proposed individual without justification. Only when "Mary" is shown to refer to a real physical individual (substance), can we say that Mary exists.
Otherwise "Mary" just signifies an individual in the general sense, in abstraction. And when "Mary" signifies an individual in the general sense, the logical possibility that Mary exists, takes on a completely different meaning. When "Mary" is not assigned to any specific individual, then the logical possibility of Mary's existence just means that it is possible that there is an existing person named Mary.
And you very well may be. My condolences.
However, you may exist as only a possibility in another philosophical realm where the word "you" can mean annihilation by fly-swatter. Or not. This is serious stuff.
The confusion is not about possible world semantics, the confusion is about the mixing up of metaphoric and literal meaning.
There is no confusion as to what "Mary exists in a possible world" means, as there is no confusion as to what "Mary has a heavy heart", "Mary is down in the dumps" or "Mary is as happy as Larry" mean.
There is nothing wrong with using poetic or metaphoric language, as such words are an integral part of language. The problem arises when poetic and metaphoric language becomes mixed up with language that is intended to be literal, after all, this is a philosophy forum where the meaning of words is important, not a poetry forum.
A possible world may or may not exist. If the possible world doesn't exist, Mary cannot exist in it, so "Mary does not exist in a world that does not exist" is true. If the possible world exists, then it is not a possible world, it is an actual world, so "Mary exists in a world that exists" is true. As a possible world is a modal world, if Mary exists within it, then Mary's existence should also be a modal existence. Therefore it would be better say "Mary may exist in a possible world", "Mary might exist in a possible world", "Mary can exist in a possible world" or "Mary could exist in a possible world".
Your claim was:
Quoting litewave
I refuted that claim:
https://thephilosophyforum.com/discussion/comment/766312
Your reply does not refute my refutation, as well as there are other errors in your reply:
First, here are the points in my refutation, most of which you skipped:
(1) It's not even clear how "all logically possible (consistent) collections exist" could be stated as a mathematical statement in set theory.
(2) It's not even clear how "all logically possible (consistent) collections exist" could be stated as a rigorous philosophical principle regarding set theory.
(3) Even if we did have a rigorous statement of such a philosophical principle, it's not a given that it is the consensus of set theorists and philosophers of mathematics that it is true.
(4) There are infinitely many statements formalizable in the language of set theory that state the existence of sets with given properties but such that it is consistent with set theory there exists such a set, but it is not a given that set theorists endorse that any given one of those sets exists.
You did reply to that point, but your reply fails, as I'll explain later in this post.
(5) Set theory does preclude certain kinds of sets that otherwise it would be consistent to say they exist. In particular, the axiom of regularity precludes certain kinds of sets that otherwise would be consistent to say they exist.
Since you did not reply to that, I'll add: I surmise that naively (informally, intuitively) most set theorists' notion of 'set' includes that sets are not members of themselves, and that, more generally, every set has a minimal member. That is especially witnessed as the axiom of regularity is a standard axiom, which is especially relevant since you say that naive set theory is "elaborated upon" by axiomatizations such as ZFC. This is a point blank refutation of your claim that "according to set theory, all logically possible (consistent) collections exist", as indeed both the naive notion of sets and the standard axiomatizations exactly preclude the existence of certain kinds of sets that would not be inconsistent to assert their existence otherwise. That point cannot be skipped and it alone decisively refutes your claim.
Added errors in your reply:
(6) When we say 'set theory' in the last 100 years, we mean one of the axiomatized set theories, not naive set theory. So saying 'according to set theory' would not be understood as 'according to naive set theory'.
(7) 'naive set theory' is ambiguous, as it means different things to different people and in different contexts. In any case, it is not usually understood as "it just says that a set is a collection of objects". And even if that were the meaning of 'naive set theory', it would not follow that according to naive set theory "all logically possible (consistent) collections exist" (whatever exact claim that might be).
The most salient sense of 'naive set theory' is the inclusion of the informal principle that to each property there is the set of all and only the objects that have that property. Or, formally, the axiom schema of comprehension: For every formula P in which 'y' is not free, we have E!yAx(xey <-> P). That schema is famously inconsistent. So taking 'naive set theory' in that sense is of no use to your claim.
And taking 'naive set theory', as you mention, as merely meaning an informal understanding that is nevertheless formalized in a theory such as ZFC also is of no use to your claim, since such theories exactly preclude the existence of certain kinds of sets that would not be inconsistent to assert their existence otherwise.
(8) It is not clear what you intend with "a set is a collection of objects". Is that intended as a definition of 'set'?
As far a quite informal notion, it's perhaps okay though it merely shifts from 'set' to 'collection'. But it is also widely viewed that 'set' is an informal notion that is not defined, especially as you say that naive set theory is explained by axiomatic set theory. Also, formally, in class theory (a variation of set theory), we make take 'is a set' as a primitive (undefined) predicate. But also, even in set theory, we may define 'is a set' as follows:
df. x is a set <-> ((x=0 or Ey yex) & Ez xez)
Or put informally: a set is not a urelement and not a proper class.
Also, the understanding of sets has been refined greatly since early definitions such as Cantor's. Especially we countenance the iterative concept. (For an excellent argument see Boolos's essay "The Iterative Conception of Set" in the great volume 'Logic, Logic, and Logic'.)
Edit: Also, ordinary set theory take sets to be hereditarily sets.
Returning to the main point: Such notions do not entail that all logically possible sets exist. We already saw that the axiom of regularity exactly disputes that all logically possible sets exist, but also it is just a non sequitur to jump from a definition of 'is a set' to asserting the existence of "all logically possible collections". From the definition of 'unicorn' we don't infer that unicorns exist (presumed counterfactual), let alone that all possible unicorns exist. From the definition of 'extraterrestrial creature' we don't infer that extraterrestrial creatures exist (unkown), let alone that all possible extraterrestrial creatures exist. From the definition of 'dog' (known fact but not inferred merely by definition), we don't thereby infer that dogs exist, let alone that all possible dogs exist.
(9) You mention Hamkins's multiverse view. But a multiverse view decidedly contradicts naive set theory (in the sense of the schema of comprehension). As to naive set theory in your sense of informal understanding anticipating formal axiomatization, the multiverse view and your remarks about it actually hurt your claim that "all logically possible collections exist". Indeed, the multiverse notion suits my argument: It depends on what particular theory is considered. For example, if we adopt CH as an axiom, then there does not exist a set whose cardinality is strictly between the cardinality of the set of naturals and the cardinality of the set of reals. But if we adopt the negation of CH as an axiom, then there do exist sets whose cardinality is strictly between the cardinality of the set of naturals and the set of reals. There is not in set theory itself a universal principle that "all logically possible collections exist" (even setting aside, as I've mentioned, that it's not clear how we would rigorously articulate such a principle).
And you say, "every axiomatized set theory selects a limited collection of possible (consistently defined) sets." But that contradicts your own claim that "according to set theory, all logically possible (consistent) collections exist":
A set theory (a) proves the existence of certain sets, and certain kinds of sets, having certain properties, and (b) disproves the existence of certain sets, and certain kinds of sets, having certain properties, and (c) for certain kinds of sets, leaves neither proven or disproven that they exist. So, even the most common set theories preclude the existence of certain sets and leave unanswered whether other certain kinds of sets exist. So, again, it is not the case that "according to set theory, all logically possible (consistent) collections exist" (let alone, as mentioned, it is not clear how "all logically possible (consistent) collections exist" could even be exactly stated in the language of set theory or even as a rigorous philosophical claim).
(10) You say, "For me, as long as such a set is consistently defined, it exists."
First, using the method of formal definition, there is no such thing as an inconsistent definition. (See many a book in mathematical logic for explanation of the method of formal definition, while I think Suppes's 'Introduction To Logic' is the best one on the subject.)
Second, and most telling, that for you something is the case about sets doesn't imply that "according to set theory" it is the case. You overstated. You jumped from your own glib view to a sweeping claim about set theory itself.
/
It is true that set theorists have different perspectives: Some favor a "wider" view of sets; that our theory should allow a more "liberal" acceptance of kinds of sets. And other set theorists favor a "narrower" view of sets. But, again, to understand those perspectives as rigorous requires a lot more work. And, again, since there are such disagreements, it is not the case that "according to set theory, all logically possible (consistent) collections exist".
Look what happened to this guy!
I think I can clarify a lot by addressing this part of your post:
Quoting TonesInDeepFreeze
As I said, by "set theory" I mean set theory in the most general sense. I supposed that this is what is commonly understood as naive set theory, but to clarify, I mean the concept of a set or collection of objects that is elaborated in all consistent axiomatized set theories together. In other words: if a set is included at least in one consistent axiomatized set theory, then such a set exists. As I said, there are uncountably many axiomatized set theories. ZFC set theory is just one of them. So I include also sets that are members of themselves and sets that don't have a minimal member, as long as these sets are consistently defined. Elsewhere you stated that there is no such thing as an inconsistent definition, so let me give you an example of an inconsistently defined set: an empty set that has one member.
Also, the example with CH may be clarifying. If there is a consistent axiomatized set theory that includes a set with cardinality between the cardinalities of the set of naturals and the set of reals, then such a set exists, simply because it is included in a consistent axiomatized set theory (that has an axiom that is the negation of CH). It's no problem that in a different axiomatized set theory, which has CH among its axioms, such a set doesn't exist. A set theory with CH as an axiom simply selects only certain sets among which a set with a cardinality between naturals and reals is not included. Every consistent axiomatized set theory selects certain sets, and by "all logically possible (consistent) collections" I mean sets or collections selected by all consistent axiomatized set theories together - that's the multiverse view in set theory. If a set is included at least in one consistent axiomatized set theory, then it exists in the set-theoretic multiverse.
I grieve for my people.
No one could predict that by "set theory" you meant your own personal concept (a concept that is not at all what people ordinarily mean by "set theory"). So my original point stands, per an ordinary understanding of "set theory", it is not the case that "according to set theory, all logically possible (consistent) collections exist", as well as we don't have a suggestion as to how that would be rigorous mathematical or even philosophical statement.
But now we have a revision: According to littwave's ersatz concept of set theory (not one ordinarily understood as "set theory"), call it 'L-theory', all logically possible (consistent) collections exist (setting aside how that would be a rigorous or mathematical or even philosophical statement).
Quoting litewave
And that is not what anyone means by 'naive set theory'. So your notion is not set theory and it's not naive set theory. I'll call it 'L-theory'. Moreover, you skipped my point that you enlist a multiverse view, yet a multiverse view clearly contradicts L-theory, as indeed a multiverse view is the very opposite of L-theory.
Quoting litewave
What are all the axiomatized set theories? There is no definitive list, and there is no conceptual limit. For that reason alone your notion is fatally vague. And what does "included" mean? Does it mean that a theory proves a theorem of the form E!xAy(yex <-> P) so that we can name that unique set? That would entail that there are only countably many sets (since there are only countably many names). Otherwise, in what sense does a set "exist" in your framework? Exist as a member of some model of some set theory (as, again, it is not known what theories are allowed to be considered for this purpose)? Or does it mean that there is a proof that there are certain classes of of sets having a certain properties? All and only properties expressible as a formula of set theory? Or sets that are subsets of model theoretic universes? Or what? 'exist' has at least two set theoretic meanings: To be named by a constant symbol in a definition or to be a member of a universe of a model of set theory.
Without providing something a lot more mathematically or philosophically substantive than you have, your notion is arm waving in a "dorm room".
Quoting litewave
And that is not set theory. And, for consistency then, you have to exclude the ordinary set theories, including ZFC, since they prove that there are no such sets. Your notion is not set theory; it is, at best, L-theory.
Also, how would you know that such a stew that provides that "all sets exist if they exist in at least one theory" is itself consistent?
Quoting litewave
That's not a definition of a set. That's a definition of a property, viz. the property of having no members and having a member.
A definition of a set is of the form (where 'c' here is a constant, so it's actually a definition of a constant):
x = c <-> P
where 'P' is a formula in which 'c' does not occur, and at most 'x' occurs free, and we have a previous proof of E!xP.
A definition of a property (actually a predicate symbol) is of the form:
Fx <-> P
where 'F' is a predicate symbol, and 'P' is formula in which 'F' does not occur, and at most 'x' occurs free.
Moreover, any theory that has the axiom of regularity makes inconsistent this formula.
Ex xex
since the axiom of regularity implies ~Ex xex.
And, since you place no limit on how we may extend any of the set theories, we can have many theories that preclude that there exist sets of certain kinds, including sets with cardinality strictly between the cardinality of the naturals and the cardinalities of the reals, or inaccessible cardinals, etc. Indeed, we can have consistent theories that, by dropping some axioms and adopting others, consistently preclude any kind of set we want.
Your thinking about all of this is thoroughly half-baked.
Quoting litewave
No, you skipped what I pointed out:
Quoting TonesInDeepFreeze
A theory does not "select only certain sets". There are properties such that a consistent set theory leaves unanswered whether or not there are sets having that property. So it is utterly vague to say what sets a given theory "selects" beyond those we know that the theory proves to exist. And even worse to hand wave that there is some universal criterion of existence based on a union of an undetermined set of theories. And again, since ZFC+CH is now ruled out by your requirement that there exists a set of cardinality strictly between the naturals and the reals, your vague L-theory cannot speak for set theory itself, since set theory itself is not settled as to CH.
Quoting litewave
I gather that you mean "in at least one" and not "selected by all" [emphasis original]. Using 'all' as you do confuses your own point.
But more importantly: Please cite where Hamkins says anything tantamount to "[all sets exist that are] selected by [at least one] consistent axiomatized set theory."
/
You've been corrected on a number of points: What set theory is. What naive set theory is. That L-theory is a sufficiently definite notion - mathematically or philosophically. That L-theory doesn't work out the way you think it does. What a definition of a set is as opposed to a definition of a property of sets. And I wonder from what exact specific passage you infer that Hamkins holds that all sets exist that are "selected" by at least one consistent axiomatic set theory.
/
Every consistent first order theory is extended by a consistent maximal first order theory, in the sense that every sentence in the language is a theorem of the maximal theory or its negation is a theorem of the maximal theory. But it's doubtful whether that would help L-theory: (1) The maximal theory is taken as an infinite union of theories with a sequence, as at each step the consistent alternative is included, but if both are consistent, then either (but not both, of course) may be included. This is nothing like the handwaving of L-theory. (2) From (1), there is not just one maximal theory. Not a definitive union of theories by which it is determined what sets exist or not. (3) The maximal theory is not necessarily recursively axiomatizable, so, to the extent that L-theory might be hoped to provide a formal theory, the theorem of maximal theories does not in and of itself provide a recursively axiomatizable theory.
The fly-people?
Well, Wikipedia article on naive set theory just mentions the general concept of a set as a collection of objects, and related general concepts like set membership relation, equality, subset, union, intersection etc. There is no requirement that a set cannot be a member of itself or that a set must have a minimal member, which you tried to impose on naive set theory.
Quoting TonesInDeepFreeze
Sure, the number of axiomatized set theories is uncountable. I see no reason to set any arbitrary limit to them.
Quoting TonesInDeepFreeze
Look at Hamkins' paper on multiverse in set theory. As an example, he talks about a world defined by axioms of ZFC + CH and a world defined by axioms of ZFC + negation of CH. He claims that both worlds exist in the set-theoretic multiverse. That means that a set with cardinality between naturals and reals doesn't exist in the (ZFC + CH) world but it exists in the (ZFC + negation of CH) world and thus exists in the multiverse. I cannot speak about set theory with the same rigor as Hamkins or you but this seems to be what I am trying to say.
Quoting TonesInDeepFreeze
ZFC+CH is not "ruled out", it just defines a part of the multiverse, a part in which there is no set with cardinality between naturals and reals.
Quoting litewave
Contrary to your mischaracterization of my remarks, I didn't opine as to naive set theory regarding the axiom of regularity, but rather as to set theory:
Quoting TonesInDeepFreeze
And that was before you ever mentioned 'naive set theory'.
In a following post, I said:
Quoting TonesInDeepFreeze
And that is true. I don't impose any particular axioms or principles onto naive set theory, since the rubric 'naive set theory' is not definite enough to say what its axioms (if any) are. Whether described as 'naively', 'informally' or 'intuitvely' (let alone formally), the vast number of set theorists regard sets as not being members of themselves.
I said what obtains in set theory and naively, informally, or intuitively with most mathematicians; I didn't say what does or does not obtain in an undefined 'naive set theory'. For that matter, I don't have any need to even mention 'naive set theory' other than that you brought it up.
So, please do not put words in my mouth again.
Quoting litewave
First, never take Wikipedia as authoritative on mathematics.
Second, please read the very article you cite. Wikipedia mentions three notions of 'naive set theory':
"Naive set theory may refer to several very distinct notions. It may refer to
Informal presentation of an axiomatic set theory, e.g. as in Naive Set Theory by Paul Halmos.
Early or later versions of Georg Cantor's theory and other informal systems.
Decidedly inconsistent theories (whether axiomatic or not), such as a theory of Gottlob Frege[6] that yielded Russell's paradox, and theories of Giuseppe Peano[7] and Richard Dedekind." [Wikipedia]
So, even according to Wikipedia, it is not a given that we take 'naive set theory' as you stipulate.
Quoting litewave
That is a dense article. (And I certainly don't trust that you have sufficient technical background to accurately represent anything it says.) So, please quote the specific passages you contend claim that all sets exist that are "selected" by at least one consistent axiomatic set theory. Tell me the exact formulations you have in mind that Hamkins mentions in his own words. And please cite where Hamkins says there is a set with cardinality between the naturals and the reals and that that is decided by the fact that ZFC+~CH is consistent.
Quoting litewave
You switched from my point:
Quoting TonesInDeepFreeze
The point there, as I mentioned previously, yet you still don't get it, is that you can't speak for "set theory" when set theory in and of itself does not determine CH.
Rather than address refutations head on, and then gracefully admit that there are the errors in your posts I've mentioned, you instead put words in my mouth, cite a complicated and highly technical article (of which I see no reason to think you have even the basics for a background to understand the article), and then also misconstrue my point about CH.
Keeping a running tab (otherwise, the discussion does not progress healthfully as instead the items get buried by your continuing to ignore them or evade them:
You've been corrected on a number of points: What set theory is. What naive set theory is (as you didn't even read the very article about it that you cited). That your own personal glib hand waving is a sufficiently definite notion - mathematically or philosophically. That you conflate your own personal glib hand waving with what set theory and naive set theory actually are. That your own personal glib hand waving doesn't work out the way you think it does. What a definition of a set is as opposed to a definition of a property of sets. And I wonder from what exact specific passage you infer that Hamkins holds that all sets exist that are "selected" by at least one consistent axiomatic set theory. And now, falsely putting words in my mouth about regularity, and evading my point about CH by misconstruing it. And you present no even remotely definite sense of what you mean by 'exist' - whether by syntactical definition, or by membership in a universe for a model, or otherwise. And you show no recognition of the distinction between defining a particular set versus a claim that there exist sets having a certain property - which is a distinction crucial to allowing this subject to be discussed intelligibly.
on page 2: "In this article, I shall argue for a contrary position, the multiverse view, which holds that there are diverse distinct concepts of set, each instantiated in a corresponding set-theoretic universe, which exhibit diverse set-theoretic truths."
Quoting TonesInDeepFreeze
The example with CH is in part 7: "Case study: multiverse view on the continuum hypothesis". See it for yourself. I tried to express the gist of it in my previous post.
And that quote is not at all tantamount to saying that we take as existing all the sets that are proven to exist according to different set theories. As it stands in and of itself, it could be mean the exact opposite - that there is no single universe that determines the totality of the sets. That what exists depends on each individual theory. He says there is are distinct concepts; yet your notion is that there is a unified concept that is arrived upon by collecting from all the distinct concepts, or from the union of what is proven among an uncountable number of theories. It remains to study Hamkins further to see exactly how his notion works, but at least from that passage, we cannot infer that it works as you claim it does.
But the more basic point is that, no matter your own views (or even Hamkins's, for that matter), it is not the case that "according to set theory, all logically possible (consistent) collections exist".
Set theory does not say that. There is no theorem of set theory that is anything like that. It's not even seen how it could be formulated in set theory.
Again, you conflate your own personal preference about how you wish to (mis)understand set theory with set theory itself. For that matter, I doubt you even know what set theory is.
And please don't dodge. Please say exactly what passages in part 7 you regard as saying that there is a set with cardinality between the naturals that is decided ('settled' in context) in general (not just per particular theories) by the consistency of ZFC+~CH.
And no retraction from you that you falsely put words in my mouth by claiming that I said naive set theory must obey the axiom of regularity. That you falsely put words in my mouth and then elected not to retract when it was pointed out witnesses intellectual dishonesty and poor faith.
And no admission by you that you omitted the key passages in the Wikipedia article that you cited yourself, as those passages directly support my point that naive set theory is commonly understood to be an informal framework that is informally inconsistent (and formally inconsistent if we formally spelled out the comprehension principle). Again, that's witness of your lack of sincerity to understand the subject matter on which you so freely opine and claim.
And no recognition by you that you missed my point, said more than once, about ZFC+~CH.
And by now you've been corrected on these points:
* What set theory is. (I doubt you even know what the term 'set theory' actually refers to.)
* What naive set theory is (as you didn't even read the very article about it that you cited).
* That your own personal glib hand waving is not a sufficiently definite notion - mathematically or philosophically.
* That you conflate your own personal glib hand waving with what set theory and naive set theory actually are.
* That your own personal glib hand waving doesn't work out the way you think it does.
* What a definition of a set is as opposed to a definition of a property of sets.
* That you have not shown a determining passage from Hamkins regarding your own notions vis-a-vis his.
* That you have not supported that Hamikins says what you claim he does about CH.
* That you falsely put words in my mouth about regularity, and without subsequent retraction, and evaded my point about CH by misconstruing it. And you present no even remotely definite sense of what you mean by 'exist' - whether by syntactical definition, or by membership in a universe for a model, or otherwise.
* You show no recognition of the distinction between defining a particular set versus a claim that there exist sets having a certain property - which is a distinction crucial to allowing this subject to be discussed intelligibly.
The union of all consistent axiomatized set theories does.
Quoting TonesInDeepFreeze
Hamkins regards the world defined by ZFC+~CH as equally real as the world defined by ZFC+CH and that both worlds exist in the multiverse. So that's how the continuum hypothesis is settled.
"Since we have an informed, deep understanding of how it could be that CH holds or fails, even in worlds very close to any given world, it will be difficult to regard these worlds as imaginary."
Quoting TonesInDeepFreeze
I didn't put words in your mouth. I thought that when you used the word "naively" you referred to naive set theory.
I mentioned ZFC just as an example of axiomatized set theory.
Yes, I deleted that post, as I realized it failed to track with what you did say.
That is an inconsistent theory. And even then it doesn't say what you say it does. You keep skipping the point that there is no apparent way to put your claim in the language of set theory.
And earlier you said ZFC is just an example. But now you premise on the union of all theories, of which ZFC is obviously one. But ZFC says no set is a member of itself, while you say that there are sets that are members of themselves, while you take existence from the union of all the theories. Now, granted, you might say that you are taking only from the individual theories the theorems that assert existences, not those that deny existences. It would take at least a bit of thinking to figure out how that would actually work, but at least it is overwhelmingly clear that it is nothing close to what set theory says.
Quoting litewave
That is the bare gist of it. And it doesn't say what you say it does. So I just repeat what you don't address:
Quoting TonesInDeepFreeze
Quoting litewave
Yet, I did not refer to naive set theory there. And I made clear previously that I was talking about set theory. And, if I now recall correctly, I had a least alluded to the fact that naive set theory is not definitely axiomatized (except usually it is understood to include the principle of comprehension). And in the second instance I made clear that I was talking about a naive, intuitive, informal way of thinking; and I did not in that regard say I was talking about naive set theory; moreover, since set theorists now do not work in naive set theory, it would make no sense to interpret me in the least charitable way possible - that I was talking about set theorists working in naive set theory.
So, please do not read my posts as carelessly as you read about set theory (I have no idea where you got your ersatz notions that you claim to represent "set theory", and you didn't even bother to read the Wikipedia article that you purported refuted one of my points, when actually it supported that point) and then purport that they say things that in fact they don't.
Back to the original point: You are incorrect that "according to set theory, all logically possible (consistent) collections exist". And you make it even worse with your handwaving about unions of theories and appropriating Hamkins (while you know not even the bare basics of the technical exposition). You are incapable of even conceding a single mistake, including the initial one.
I suppose that you also think that a union of ZFC+CH and ZFC+~CH theories is an inconsistent theory. Yet according to Hamkins the worlds defined by these two theories are parts of a consistent multiverse.
It is as if you took these two statements:
(1) "This ball is red."
and
(2) "This ball is not red."
and concluded that these statements are contradictory. But you didn't notice that these statements are not about the same ball but about two different balls and thus there is no contradiction between them. Same with axiomatized set theories: they define different worlds and thus are not contradictory.
Wow. You most clearly demonstrated your ignorance of the basics of this subject, and continue to carelessly misappropriate Hamkins.
Yes, ZFC+CH+~CH is inconsistent. That's clear on its face.
And worlds are models. Models are not consistent or inconsistent. Theories are what are consistent or inconsistent.
Yes, just as with Hamkins, different theories may have a different class of models from one another. Even certain kinds of theories by themselves have models that are not isomorphic with one another.
A (consistent) set theory has many models not isomorphic with one another. ZFC itself has models in which CH is true and models in which CH is false. But there does not exist any model of ZFC+CH+~CH, since inconsistent theories do not have models.
Hamkins points out that we are free to work separately in different models. He doesn't say that we combine a model of ZFC+CH with a model of ZFC+~CH.
You know virtually nothing about the subject of set theory and models of set theory.
As I mentioned, I doubt you even know what ZFC IS.
That's a variation of your abysmal ignorance of the basics of the subject of models of theories.
(1) and (2) are a contradiction. However, there is a model in which (1) is true and (2) is false, and a model in which (1) is false and (2) is true. And that may be the case on account of 'this ball' referring to different objects per different models (or, also, on account of whatever 'this ball' refers to being in the subset of the universe named by 'red', or not in the subset of the universe named by 'red' per given models).
Learn the basics of mathematical logic and model theory, toward the subject of models of set theory.
* You are utterly confused on even the most basic notions of a theory and models of a theory.
He combines the world of ZFC+CH with the world of ZFC+~CH into a multiverse. And since a set with cardinality between naturals and reals exists in the world of ZFC+~CH, it also exists in the multiverse.
No, he mentions that there are separate universes. That is the multiverse: The collection of separate individual universes. He doesn't combine universes all into one big clump. There is no such thing when there is not even what a model when the theories contradict one another. And there is no "the world of ZFC+CH" or "the world of ZFC+~CH". Rather, for each set theory, there are many non-isomorphic models.
Get it in your head: There is no model of an inconsistent theory. There is no model of ZFC+CH+~CH, let alone a model from combining two models.
And Hamkins doesn't say that he obtains a model made by combining models of ZFC+CH and of ZFC+~CH. That would be ludicrous.
And if Hamkins says that, aside from particular theories, in general and simpliciter, there is a set of cardinality between the naturals and the reals, then we would need to see the exact passage in which he says that.
What Hamkins mentions is that there are two approaches: (1) A universe view in which some particular model is taken as the one that determines all mathematical matters. (2) The multiverse view in which there is not one particular model that is taken as the one that determines all mathematical matters, but instead there is PLURALITY (his word, my emphasis) of models, and mathematical matters therefore are not settled simpliciter but rather with a plurality of answers, each depending on particular models. You completely mixed that up to think he's saying that the multiverse is a stew of all models thrown into a single pot to make another model itself. He says the OPPOSITE of that. And the question whether there exists a set of cardinality between the naturals and reals is not answered 'yes' according to some single multiverse, but rather answered 'yes' or 'no' according to different models that are in the collection of models that is the multiverse.
You don't know anything about what a theory or model is. Yet you make stubborn false claims about the subject, even misrepresenting Hamkins. That is intellectually shameful.
Rather than keep repeating your ignorance, you would do better to grab a book on the basics of the subject of theories and models and properly learn the concepts.
"Clump"? Is that supposed to be another technical term? A multiverse is a collection of universes or worlds, that's all.
Quoting TonesInDeepFreeze
Then there are "CH worlds" and "~CH worlds". You are missing the point, which is that a set with cardinality between naturals and reals exists in the multiverse.
Then the multiverse is a model of what? Or what is it?
Your snark is ludicrous in context.
You entirely skipped my specific and exact explanation, to instead try to gain a pathetic bit of snarky upper hand.
Actually 'clump' is not technical, because there IS NO technical notion of combining models of contradicting theories.
Quoting litewave
There are many models of CH and many models of ~CH.
Quoting litewave
No, you are missing the point that such a set exists in some models in the multiverse and not in other models in the multiverse.
You utterly got Hamkins backwards. Not surprising, since you are an ignorant, intellectually dishonest and (now seen to be) petty crank.
I already told you. It's a collection of models (or "worlds" informally).
Are you interested in understanding anything about theories, models, set theory and models of set theory? Or are you just going to continue insisting that you're right about everything even though you know nothing about the subject?
Really, where did you get that? I said that such a set exists in a ~CH world and does not exist in a CH world.
Quoting TonesInDeepFreeze
Well, whether you call the multiverse a model or a collection of models, the fact remains that a set with cardinality between naturals and reals exists in this collection, although it doesn't exist in all its subcollections (models), which is fine. And the same applies to the existence of any set that is defined by a consistent axiomatized set theory; that's why I said that all logically possible (consistent) collections exist.
From several textbooks and articles on mathematical logic and set theory.
Quoting litewave
As far as I can tell (based on the page you mentioned, and the surrounding pages) the multiverse is not a model. It is a collection of models.
Quoting litewave
Sets exist in universes (domains of discourses) for model. The collection is a collection of models. The only things that exist in that collection are models. The set exists in the universe of one of the members of the collection.
Quoting litewave
And now we're exactly where we started. Read the conversation again, if you wish to understand. Or, better yet, get a book on the subject so that you can understand its basics. Or continue to ignorantly demand that you're right, no matter how carefully it is explained exactly why you are not.
Still, it is incorrect that "according to set theory, all logically possible (consistent) collections exist." It's your own personal notion, not found as a theorem of set theory and not even as an agreed upon, let alone well articulated, informal meta-principle regarding set theory.
What do you think a theory is?
What do you think set theory is?
What do you think an inconsistent theory is? (You claim ZFC+CH+~CH is not an inconsistent theory, so it's clear you don't know what an inconsistent theory is.)
What do you think a model is? (You think there's a model (or "world") of ZFC+CH+~CH, so it's clear you don't know what a model is.)
What do you think a model of set theory is?
If a set X exists in a collection (model) which exists in a collection (multiverse), I see no problem in saying that the set X exists.
Quoting TonesInDeepFreeze
Set theory is a description or explanation of sets. An axiomatized set theory is a set of axioms about sets. A model of a set theory is a set, or a collection of sets, that is described by the theory. An inconsistent set theory would be one that affirms and denies the same property to the same set.
I am sure your people respect the advice of their beloved king/god/malevolent controller, when it is exclaimed/heralded :rofl:
But seriously you preferred big Vincent's 1958 original?