I hardly understand anything in this thread, as my knowledge of mathematics is rudimentary. — Wayfarer

I'm with you. And I was a professor of mathematics. I am still puzzled over what precisely "true" means beyond verification by formal proof.

If you point to a number, '7', what you're indicating is a symbol, whereas the number itself is an intellectual act. And furthermore, it is an intellectual act which is the same for all who can count. It's a very simple point, but I think it has profound implications — Wayfarer

I like your clarity.

Thanks! Coming from you, that is high praise.

My view is that numbers are real, but not physically existent. — Wayfarer

By itself, this could be framed within a conceptualist framework, where mathematics is reduced to psychology — not just platonism.

If you point to a number, '7', what you're indicating is a symbol, whereas the number itself is an intellectual act. — Wayfarer

The distinction between type and token.

And furthermore, it is an intellectual act which is the same for all who can count. — Wayfarer

Is it though? You may say this because you are a platonist, so you believe there is some unambiguous universal accessible to all.

So you don't accept that 7=7?

apropos of this general discussion, I've just downloaded a rather interesting textbook, What is Mathematics, Really? Reuben Hersh, 1999. Quite approachable.

So you don't accept that 7=7? — Wayfarer

Outstanding.

Hersh asks right upfront: 'in what sense do mathematical objects exist?' That was the question that hooked me on the subject, and I see it as a philosophical, not mathematical question.

It is (ZF\I)+~I that is bi-interpretable with PA. — TonesInDeepFreeze

And actually we don't need the 'F'.

(Z\I)+~I is bi-interpretable with PA.

[EDIT CORRECTION: I think it is incorrect that (Z\I)+~I is bi-interpretable with PA. This is correct: If every set is finite, then the axiom schema of replacement obtains and (Z\I)+~I = (ZF\I)+~I. But I don't think that works; I was thinking that the negation of the axiom infinity implies that every set is finite. But I think that itself requires the axiom schema of replacement.]

So you don't accept that 7=7? — Wayfarer

In that regard, Victoria Gitman writes the following alarming statement:

https://victoriagitman.github.io/talks/2015/04/22/an-introduction-to-nonstandard-model-of-arithmetic.html

In particular, a nonstandard model of arithmetic can have indiscernible numbers that share all the same properties.

Even though the law of identity is certainly applicable in the standard model of the natural numbers, it may fall apart in nonstandard models of arithmetic.

So, ω+7 ¬= ω+7 may be true in a nonstandard context, with ω the infinite ordinal representing the order type of the standard natural numbers. If it is false in any other nonstandard context, then this statement is even true but unprovable. I am not sure if this can be the case.

Victoria Gitman points to the following publication for a more elaborate explanation on what's going on:

R. Kossak and J. H. Schmerl, The structure of models of Peano arithmetic, vol. 50.

Unfortunately, the publication is not available online. It can be ordered in paper-based format for $180 from Oxford University Press:

https://global.oup.com/academic/product/the-structure-of-models-of-peano-arithmetic-9780198568278?cc=us&lang=en

So, we already had ineffable numbers. Now we also have indiscernible ones. What other monstrosities are they going to discover in the melted plutonium core of Chernobyl reactor number four?

Presumably nothing abstract.

My intuition says that Yanofsky is probably right, and that it is Cantor's theorem that is at the root of it all, but I am currently still struggling with the details of what he writes. — Tarskian

I agree, also with Yanofsky.

Cantor's proof is the simplest form of diagonalization that has all the "problematic" consequences, once we start to look at infinite sets (with finite sets Cantor's theorem is quite trivial). As Yanofsky say's:

If you try to express all the truth about the natural numbers, you are effectively trying to create an onto mapping between the natural numbers and its power set, the real numbers, in violation of Cantor's theorem.

And of course, with the proof of the theorem, using diagonalization, we showed that a surjection / onto mapping is not possible. This shows just how close making a bijection is to giving a proof. We understand that an infinite set is incommensurable to a finite set and that we cannot count finite numbers and get to infinity. However, this isn't the only thing we have problems once we encounter the infinite.

After all, if a formal system can express Peano Arithmetic, then Gödel's second incompleteness theorem holds that the system cannot prove it's own consistency.

What other monstrosities are they going to discover in the melted plutonium core of Chernobyl reactor number four? — Tarskian

Any observations on the arguments for or against mathematical platonism as outlined in this post?

Any observations on the arguments for or against mathematical platonism as outlined in this post? — Wayfarer

I subscribe to the following take on Platonism:

https://en.m.wikipedia.org/wiki/Philosophy_of_mathematics

Davis and Hersh have suggested in their 1999 book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism.

In my opinion, you cannot actively do mathematics if you do not believe that its objects are real while you are doing it.

Godel also thought that talent for Platonism is a prerequisite for being successful at mathematics:

Kurt Gödel's Platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly.

It is, however, mentally very easy to switch to formalism.

You can simply switch off the lights and declare that it is all just meaningless symbol manipulation and about nothing at all, which it actually is, if you take the time to think about it.

You can simply switch off the lights and declare that it is all just meaningless symbol manipulation and about nothing at all, which it actually is, if you take the time to think about it. — Tarskian

Except it doesn’t allow for the iunreasonable effectiveness of mathematics in the natural sciences.

As I said above, the reason the most people won’t defend platonism is because they don’t understand or can’t live with the metaphysical commitment it entails. Myself, I have no such difficulty.

To reify is to 'make into a thing'. Numbers don't exist as objects, except for in the metaphorical sense of 'objects of thought'. — Wayfarer

The problem which I have encountered in this forum, is that there is an attempt by many, to represent numbers, and other mathematical objects like sets, as things which are subject to the law of identity. The law of identity states what it means to have an identity as a thing, and it is known to be applicable to material objects. By representing mathematical objects as subject to the law of identity, which applies to things, mathematical objects and material objects are implied to be of the same type, each having the identity of "a thing".

The result of this is that there are significant conceptual structures, set theory, and mathematical logic in general, which are based on the assumption that there is no difference between 'objects' of thought' and material objects. This leads to absurd ontologies like model-dependent realism.

It is my opinion that this conflating of the two is the reason why quantum observations are so difficult to understand, and quantum theory interpretations are many and varied. Within quantum theory there are no principles which would allow for a distinction between the material object and the 'object of thought' so that the two are combined in a confused model of wave/particle dualism.

Except it doesn’t allow for the iunreasonable effectiveness of mathematics in the natural sciences. — Wayfarer

Viewing mathematics as just string manipulation highlights a different aspect of the same thing. The same holds true for structuralism. You can see mathematics as mostly templates with template variables. There are circumstances in which an alternative ontological view is actually the most inspiring one.

As I said above, the reason the most people won’t defend platonism is because they don’t understand or can’t live with the metaphysical commitment it entails. Myself, I have no such difficulty. — Wayfarer

I intuitively believe that arithmetical truth and physical truth are structurally similar. This explains why it is unreasonably effective in a physical context. For exactly the same reason, it should also be unreasonably effective in a metaphysical context.

I fully endorse Pythagoras' view on the matter:

https://en.wikipedia.org/wiki/Pythagoreanism

Pythagoras, in his teachings focused on the significance of numerology, he believed that numbers themselves explained the true nature of the Universe. Numbers were in the Greek world of Pythagoras' days natural numbers – that is positive integers (there was no zero).

In modern lingo, arithmetical theory, i.e. the theory of the natural numbers (PA), and the unknown theory of the physical universe exhibit important model-theoretical similarities.

For example, the arithmetical universe is part of a multiverse. I am convinced that the physical universe is also part of a multiverse.

The metaphysics of the physical universe is in my opinion nothing else than its model theory.

Model theory pushes you into a very Platonic mode of looking at things. In my opinion, it is not even possible to understand model theory without Platonically interpreting what it says.

thank you for the clarification.

A well-known mathematician takes a look at "truth" in mathematics:

Desparately Seeking Mathematical Truth

The "intuition" to perceive "mathematical objects" can be, and has to be, imagination of instatiations of those mathematical facts, even if it is just symbol manipulation. Even if math is just symbol manipulation, I can apply the relationship between those symbols to calculate the size of a building from its shadow. That is the intuition. When I think of a triangle I don't necessarily have the same picture as someone else. Anyone who thinks they are perceiving platonic objects directly is not paying attention.

In modern lingo, arithmetical theory, i.e. the theory of the natural numbers (PA), and the unknown theory of the physical universe exhibit important model-theoretical similarities.

For example, the arithmetical universe is part of a multiverse. I am convinced that the physical universe is also part of a multiverse.

The metaphysics of the physical universe is in my opinion nothing else than its model theory.

Model theory pushes you into a very Platonic mode of looking at things. In my opinion, it is not even possible to understand model theory without Platonically interpreting what it says. — Tarskian

If we don't differentiate between objects sensed and ideas grasped by the intellect. then there is nothing to prevent us from believing that the universe is composed of numbers. This is known as Pythagorean idealism, and often called Platonism. But Plato, along with Socrates, was very skeptical of this type of idealism, revealing its weaknesses. Aristotle, following Plato is often claimed to have decisively refuted Pythagorean idealism. He developed the concept of matter as a principle of separation between human ideas and the independent universe.

If we don't differentiate between objects sensed and ideas grasped by the intellect. then there is nothing to prevent us from believing that the universe is composed of numbers. This is known as Pythagorean idealism, and often called Platonism. — Metaphysician Undercover

I do not believe that the universe is composed of numbers.

What I believe is limited to the idea that the arithmetical multiverse is structurally similar to the physical multiverse.

For example, if there are five people in a group, this situation is structurally similar to a set with five numbers. It does not mean that a person would be a number.

You could conceivably make a digital simulation of the entire universe and run it on a computer. This simulation of the universe would consist of just numbers. What you would see on the screen will be an exact replica of what you would see in the physical world. It would still not mean that this collection of numbers would be the universe itself.

Pythagorean idealism is actually a widespread fallacy:

https://en.m.wikipedia.org/wiki/Map%E2%80%93territory_relation

The map–territory relation is the relationship between an object and a representation of that object, as in the relation between a geographical territory and a map of it. Mistaking the map for the territory is a logical fallacy that occurs when someone confuses the semantics of a term with what it represents. Polish-American scientist and philosopher Alfred Korzybski remarked that "the map is not the territory" and that "the word is not the thing", encapsulating his view that an abstraction derived from something, or a reaction to it, is not the thing itself. Korzybski held that many people do confuse maps with territories, that is, confuse conceptual models of reality with reality itself.

A map of the world can help us understand the world. The map will, however, never be the world itself.

Now, if it is about an abstract world, then the perfect map of such abstract world is indeed the abstract world itself. There is no difference between a perfect simulation of an abstract world and the abstract world itself.

That is why an abstraction cannot be truly unique. An abstraction can only unique up to isomorphism.

Physical objects, on the other hand, can be truly unique in this physical universe (but almost never in the physical multiverse).

Not everything that Pythagoras said was necessarily correct. The same for Plato. The same for Aristotle. It is just that they have managed to also say things that are amazingly insightful.

For example, if there are five people in a group, this situation is structurally similar to a set with five numbers. It does not mean that a person would be a number. — Tarskian

It's structurally similar because what constitutes "a group" is artificial, just like what constitutes "a set" is artificial. So you are just comparing two human compositions, the conception of a group and the conception of a set..

You could conceivably make a digital simulation of the entire universe and run it on a computer. This simulation of the universe would consist of just numbers. What you would see on the screen will be an exact replica of what you would see in the physical world. It would still not mean that this collection of numbers would be the universe itself. — Tarskian

If it's not the same as the universe, but a replica, then there is no limit to the difference which there may be between the two. I could show you a piece of paper and say that it's a replica of the universe. How would your proposed computer simulation provide a "better" replica of the universe? That's the thing about maps, they only show what the map maker decides ought to be shown.

A map of the world can help us understand the world. The map will, however, never be the world itself. — Tarskian

Then there's something more to reality than maps and the world which is mapped. There must also be something which makes one map "better" than another. This cannot be shown by the map nor is it a part of the world which is mapped.

Now, if it is about an abstract world, then the perfect map of such abstract world is indeed the abstract world itself. There is no difference between a perfect simulation of an abstract world and the abstract world itself. — Tarskian

This makes no sense. what would make an abstract world the perfect abstract world? Do you see what I mean? If there is no difference between the perfect simulation and the abstract world which is simulated, then they are one and the same thing. So now we have an abstract world which you claim is |a perfect simulation". What makes it perfect? It's just an abstract world like any other.

Physical objects, on the other hand, can be truly unique in this physical universe (but almost never in the physical multiverse). — Tarskian

This place is not real.

It's structurally similar because what constitutes "a group" is artificial, just like what constitutes "a set" is artificial. So you are just comparing two human compositions, the conception of a group and the conception of a set.. — Metaphysician Undercover

The notion of group may indeed be an abstraction, a way of perceiving things, but there are still five people, which are physically there.

How would your proposed computer simulation provide a "better" replica of the universe? — Metaphysician Undercover

Fewer differences.

You could measure a random sample of these differences, add up their squares, take the root, and rank replicas according to their sampled "deviation" from the original. This is actually done very routinely. It is even standard practice.

what would make an abstract world the perfect abstract world? Do you see what I mean? — Metaphysician Undercover

A perfect map of an abstract world is the abstract world itself. Perfect means "isomorphic" in this case.

According to the structuralist ontology, an abstraction consists of only structure. An abstraction is structure, turtles all the way down.

Hence, an isomorphic mapping of a structure is equivalent to the structure itself:

https://en.m.wikipedia.org/wiki/Isomorphism

The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism.

Since two isomorphic abstractions have the same properties, they are essentially identical:

https://en.m.wikipedia.org/wiki/Law_of_identity

Wilhelm Wundt credits Gottfried Leibniz with the symbolic formulation, "A is A".[4] Leibniz's Law is a similar principle, that if two objects have all the same properties, they are in fact one and the same.

Two abstraction are not truly identical. They are identical up to isomorphism.

For example, the symbols "5" and "five" are identical up to simple translation (which is in this case an isomorphism). Two maps can also be isomorphic. In that case, they are "essentially" identical.

The number "5" is an abstraction and is therefore not truly unique. It has numerous isomorphisms, such as "2+3" or "10/2" that represent essentially the same abstraction.

Abstraction are never truly unique.

This place is not real. — Lionino

The existence of the multiverse is a Pythagorean belief.

It is not possible to prove that the physical universe is part of a multiverse, simply because it is not possible to prove anything at all about the physical universe.

Leibniz's Law is a similar principle, that if two objects have all the same properties, they are in fact one and the same. — Tarskian

Hence Wheeler’s conjecture of the One Electron Universe

Nor do I lean back and say, Wow, that's true! I simply don't use the words "true" or "truth" when doing math. I don't even think the words. But that's me, not other math people. — jgill

Did you think your work was "about" anything? Or pure symbol-pushing?

I'm pressing you on this point because I don't believe you did not believe in the things you were studying!

I don't think of myself doing anything. I only do. Or did. I'm pretty old and not in such great shape to do much of anything. — jgill

I'm sorry to hear that.

Doesn't surprise me. I am (was) a humble classical analysis drone, far from more modern and more abstract topics. Maybe young math profs these days use the word "truth" frequently. — jgill

I believe you are making too much of what someone on the forum might have said about truth. You are the only professional mathematician on this site and you are more authoritative on what mathematicians do than anyone.

(On the other hand I did point out what I considered the truth of a form of rock climbing many years ago by demonstrating and encouraging a more athletic, gymnastic perception of the sport. Even then I didn't use the word "truth".) — jgill

You know, I would think there is much truth to rock climbing. A famous theoretical physicist, Lisa Randall, famously had a rock climbing accident. She is a specialist in the most advanced theories of gravity. I always thought that was ironic, a world-class expert on gravitation being injured by that very force.

Gravity is true, wouldn't you say? You probably have a visceral sense of gravity, more so than most physicists.

Did you think your work was "about" anything? Or pure symbol-pushing?
I'm pressing you on this point because I don't believe you did not believe in the things you were studying! — fishfry

I never spent any time thinking about what I was doing. I did it, and still do it because it is a fascinating realm of exploration. As was rock climbing when I was a lot younger. I never puzzled over the fundamental nature of mathematics. And I doubt my colleagues did either.

Gravity is true, wouldn't you say? — fishfry

No. Gravity simply is. Some aspects could be said to be true. Word babble IMO.

I never spent any time thinking about what I was doing. I did it, and still do it because it is a fascinating realm of exploration. As was rock climbing when I was a lot younger. I never puzzled over the fundamental nature of mathematics. And I doubt my colleagues did either. — jgill

Interesting.

No. Gravity simply is. Some aspects could be said to be true. Word babble IMO. — jgill

You don't believe in the word truth, or that anything in the world is true, even outside of math?

Hence Wheeler’s conjecture of the One Electron Universe — Wayfarer

That was an interesting phone call:

proposed by theoretical physicist John Wheeler in a telephone call to Richard Feynman in the spring of 1940.

It still generates a flurry of new articles in 2024!

But these articles don't particularly say anything new. They do not seem to make much progress in investigating the matter.

I'm always intrigued why a conversation about math morphs to conversation about physics.

Why wouldn't a discussion of mathematics morph into a conversation about the US elections? In elections mathematics plays a pivotal part too: who gets the largest number of votes. And when you have these different kinds of electoral systems, then it can happen that the candidate that gets the most vost votes isn't actually elected. Yet elections are math, aren't they? :wink: