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

When you were never taught how to go around the territory you are exploring, you tend to wander outside of that territory as the walk goes on. Same thing. It doesn't help that one of the chatters here is using wiki links to completely make stuff up as he goes.

"What is math?" is a question situated in a larger metaphysical arena. If we say "it is just symbolic manipulation," we are then led to ask: "what are symbols? And: "why do we manipulate them?"

The "why" here leads right to physics, and the natural sciences more broadly, because a big part of the "why" seems to involve how our symbolic systems have an extremely useful correspondence to how the "physical world" is.

Explanations that just posit mathematics as "a social practice," "an activity," etc. are really non-explanations IMHO. No one denies that mathematics is a product of human culture engaged in by humans. But the question "why do we do this?" leads right to questions about "how the world is" which tend to include physics and metaphysics.

To the extent that we use mathematics to understand the world, our understanding of mathematics also seems to underpin our very notions of "what our lives and our world are."

My question would instead be: "why physics over metaphysics, semiotics, information theory, computer science, or biology?"

These all seem equally relevant. To my mind it has to do with a certain sort of view of naturalism and physics' role in the sciences, one that, if not "reductive," at least tends towards the ideas fostered by reductionism.

If there are any practicing or retired mathematicians reading these threads I wish you would speak up. I would ask my old colleagues what they think of these philosophical discussions, but they are pretty much all gone to greener pastures.

I have a friend who is a math PhD. I have never really had a chance to discuss this sort of thing in depth, but I have asked him before if he though mathematics was something created or discovered. He said "created" but not with any great deal of confidence and waffled on that a bit.

It just so happens, nowadays, most scientists are not natural philosophers — which is a shame, but a consequence of the expansion of scholarship and accessibilisation of knowledge.

The "why" here leads right to physics, and the natural sciences more broadly, because a big part of the "why" seems to involve how our symbolic systems have an extremely useful correspondence to how the "physical world" is. — Count Timothy von Icarus

Aren't these symbolic systems of mathematics extremely useful in the US elections too? Isn't counting the votes quite essential in free and fair elections?

But the question "why do we do this?" leads right to questions about "how the world is" which tend to include physics and metaphysics. — Count Timothy von Icarus

And that's why reporters ask metaphysical questions from cosmologists or quantum-physicists and not from philosophers, who actually could be far more knowledgeable about metaphysical questions.

Yes, I totally understand the arguments of mathematics being an essential tool for physics and physics is an inspiration to create new mathematics and this all leads to reductionism of physics and math.

However, why do we stop there? Or to put it in another way, why then the rejection of what is quite important to us, the society and the World humans have built for themselves and which is studied by the humanities/social sciences in academia?

Let's remember topic of the thread and the idea that there's non-computable mathematics: that many true mathematical statements aren't provable or computable. How do we get to those things that are not computable, not provable? As discussed here in the OP and then later in the discussion of Lawvere's theorem in Category theory, many of these theorems showing the limitations of mathematics have self-reference and diagonalization in their argument. Negative self-refence seems to be a limit for computation.

Now, just ask yourself: We base a lot of our actions on past history. And we also try to learn from our past mistakes even as a collective, that we don't the same mistakes as in the past. Wouldn't that be perfectly modeled by negative self-reference? If so, then could you then argue that historians don't explain history by computing functions because their field of study falls into non-computable mathematics? Without computabilty, the only thing might be left is a narrative explanation of what happened.

And please understand, my argument is that indeed everything is mathematical, when we want to be logical.

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

Then the matter at issue is what constitutes a distinct individual, in order that we say that there is five of them. And this is a product of the way that we sense things. We sense things as having a separation from their environment, as distinct objects, particulars.

Fewer differences. — Tarskian

But the simulation is completely different. By the conditions of your example, it is digital, a numerical representation. How are numbers similar to the world which is represented? The number "2" is in no way similar to two separate objects.

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

This still does not make sense to me, it gives no real meaning to "perfect" You are saying that what was first described as two, the abstract world and its simulation, are really just one, because the simulation is "perfect". But then there really is no simulation, just the one "perfect" abstraction. So all you are saying is that to be an abstraction is to be perfect. So all abstractions are perfect, ideal, as being one and the same as themselves.

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

Now you're using "equivalent to the structure", and before you said the perfect map "is" the structure it maps. This is saying two different things. When we say it "is" that, we allow no difference, but to say it is "equivalent" allows for a world of difference. In my example above, "2" is completely different from the two things it represents, but it is equivalent.

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

You already said, "the perfect map of an abstract world is the abstract world itself". If it "is" the thing then it is truly identical. But now you take that back and claim they are not truly identical. If they are not truly identical then we need to account for the difference between them. You say they are "isomorphic" and that implies that they have the same form. So how could the abstraction and the model of the abstraction have the very same form, yet be different? A difference is always a difference of form. And since they are both abstractions there is no "matter" here to account for the proposed difference. Therefore we end up with contradiction. They are not truly identical so there must be a difference between them. The difference must be a difference of form. Therefore they cannot be isomorphic.

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. — Tarskian

This is where the problem is, "essentially identical" is an oxymoron. "Identical" means the same, but you degrade "identical" to say "essentially identical", such that it can no longer mean "the same" any more, because "essentially identical really means different. All you are really saying is that it is the same but different, which is contradictory.

Abstraction are never truly unique. — Tarskian

I totally agree, but the problem comes when we try to say that an abstraction, which is never truly unique, has an identity, just like a thing which is unique does. That is the case when you say "A perfect map of an abstract world is the abstract world itself". You have given identity, uniqueness, to the abstract world, to allow that there is a "perfect" map of it. Only if the abstraction is truly unique could there be a perfect map of it. If it is not truly unique, as you admit here, then the map could equally be a map of a number of different abstractions. This would mean that it is ambiguous, and less than perfect, by that fact.

I have a friend who is a math PhD. I have never really had a chance to discuss this sort of thing in depth, but I have asked him before if he though mathematics was something created or discovered. He said "created" but not with any great deal of confidence and waffled on that a bit — Count Timothy von Icarus

I'm guessing, typical. Philosophical speculations distract from True mathematics. :cool:

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

True or False?: The Earth is a planet. Answer: True (by virtue of classification)

True or False?: The square of the hypotenuse in a right triangle equals the sum of the squares of the two sides. Answer: True (by virtue of proof)

True or False?: The Continuum Hypothesis is true. Answer: Well, let's see . . . .

True or False?: The Earth is a planet. Answer: True (by virtue of classification)

True or False?: The square of the hypotenuse in a right triangle equals the sum of the squares of the two sides. Answer: True (by virtue of proof)

True or False?: The Continuum Hypothesis is true. Answer: Well, let's see . . . . — jgill

Interesting to hear you arguing against the concept of truth. Well moral relativism is the ethos of the age, I suppose.

Interesting to hear you arguing against the concept of truth — fishfry

Ordinarily I would not give it much thought, but this thread seems to focus on math truth beyond virtue of proof. You seem to know what that is all about. Can you provide a very simple definition of this sort of truth in math? I suppose the definition of a triangle is truth without proof. Truth by definition. But what makes a string of symbols true? Model theory? I thought I understood a parallel idea when I quoted the group theory example from StackExchange, but I guess not. Are axioms true by virtue of definitions?

Isn’t there a sense in which an arithmetical proof offers a kind of certainty by virtue of the fact that it is ‘the terminus of explanation’? If any school child learning arithmetic asks ‘but why does ‘2 and 2 equal 4’?, the answer surely must be that ‘it just is’. It can’t be, and doesn’t need to be, explained further. ‘4’ is the terminus of explanation. I think what impressed the ancients is that this provides a kind of certainty which rarely avails in the world of human affairs, where everything is hedged about by conditions, by ifs and buts and ‘it depends’. Here you have a kind of pristine certainty that is close to unconditional. Furthermore, this certainty can be leveraged to great effect in the building of structures, the estimation of value, and so on and so forth. Not hard to see the charisma of arithmetic if you see it like that.

So maybe, in some sense, the demand that mathematics itself be explained is a bit like the child’s question. Mathematics, after all, is the source of a considerable number of explanations, not something that itself needs explaining. I’m reminded of the concluding paragraph of Wigner’s ‘ode to mathematics’:

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. — The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Eugene Wigner

Ordinarily I would not give it much thought, but this thread seems to focus on math truth beyond virtue of proof. You seem to know what that is all about. Can you provide a very simple definition of this sort of truth in math? — jgill

Noson Yanofsky's paper, "True but unprovable", is about arithmetical truth, also called "true arithmetic":

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

In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers.This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms.

The signature of Peano arithmetic includes the addition, multiplication, and successor function symbols, the equality and less-than relation symbols, and a constant symbol for 0. The (well-formed) formulas of the language of first-order arithmetic are built up from these symbols together with the logical symbols in the usual manner of first-order logic.

The structure $\mathcal {N}$ is defined to be a model of Peano arithmetic as follows.

- The domain of discourse is the set ℕ of natural numbers,
- The symbol 0 is interpreted as the number 0,
- The function symbols are interpreted as the usual arithmetical operations on ℕ,
- The equality and less-than relation symbols are interpreted as the usual equality and order relation on ℕ.

This structure is known as the standard model or intended interpretation of first-order arithmetic.

A sentence in the language of first-order arithmetic is said to be true in $\mathcal {N}$ if it is true in the structure just defined. The notation $\mathcal {N}$ ⊨ φ is used to indicate that the sentence φ is true in $\mathcal {N}$.

True arithmetic is defined to be the set of all sentences in the language of first-order arithmetic that are true in $\mathcal {N}$, written Th($\mathcal {N}$). This set is, equivalently, the (complete) theory of the structure $\mathcal {N}$.

I have used the term "mathematical truth" instead of "arithmetical truth" because alternative foundational theories of mathematics such as set theory have large fragments that are bi-interpretable with arithmetic and therefore have the same properties.

Yanofsky points out that only a very small part of Th($\mathcal {N}$), i.e. arithmetical truth, is provable. The remainder of Th($\mathcal {N}$) is unpredictable and chaotic. Most of Th($\mathcal {N}$) is even ineffable.

Furthermore, this certainty can be leveraged to great effect in the building of structures, the estimation of value, and so on and so forth. — Wayfarer

The fact that it has leveragability in the material world, means that there is something more to it than "it just is". It is useful.

So maybe, in some sense, the demand that mathematics itself be explained is a bit like the child’s question. Mathematics, after all, is the source of a considerable number of explanations, not something that itself needs explaining. — Wayfarer

The explanation needs to take a different tact, one which addresses the usefulness which we observe. That's why Peirce was led into pragmaticism. Notice in my exchange with @Tarskian above, I was quickly led to ask what makes one theory "better" than another. Tarskian claimed the "perfect" model of an abstraction is one which is identical with the abstraction which it models. However, this is clearly incorrect if we consider what actually works in practise. In practise, what makes one specific model of an abstraction better than another is some principle of usefulness, and this is not at all a principle of similarity. That is reflected in the fact that the symbol often has no similarity to the thing symbolize ("2" in my example, is not similar to the idea of two).

Notice in my exchange with Tarskian above, I was quickly led to ask what makes one theory "better" than another. Tarskian claimed the "perfect" model of an abstraction is one which is identical with the abstraction which it models. However, this is clearly incorrect if we consider what actually works in practise. In practise, what makes one specific model of an abstraction better than another is some principle of usefulness, and this is not at all a principle of similarity. — Metaphysician Undercover

My initial interpretation of the term "better" was "more faithful", but indeed, this doesn't necessarily make an abstraction more useful. That does indeed depend on what you are going to use it for.

Wikipedia and article:

A sentence in the language of first-order arithmetic is said to be true in N if it is true in the structure just defined.

A sentence in the language of first-order arithmetic is said to be true in N {\displaystyle {\mathcal {N}}} if it is true in the structure just defined

It looks like passing the buck to me. The word "true" in mathematics appears to be a kind of primitive when used outside of "true by virtue of proof". However, the statement of Goldbach"s Conjecture from Wikipedia:

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.

might very well be true in the common sense of the word, even if possibly unprovable. But one cannot actually assert it is true - only that it might be.

Yanofsky points out that only a very small part of Th(N), i.e. arithmetical truth, is provable. The remainder of Th(N) is unpredictable and chaotic. Most of Th(N) is even ineffable. — Tarskian

With a formal system with Peano Arithmetic we already get the results of Gödel's incompleteness. Hence this has been shown earlier than Yanofsky's paper. Yet do notice that Presburger Arithmetic is complete.

So what's the thing with multiplication?

So what's the thing with multiplication? — ssu

Skolem Arithmetic only has multiplication (no addition) and is also complete. The problem occurs when you try to add both addition and multiplication.

Ok, so what's the interesting thing with having both addition and multiplication?

Ok, so what's the interesting thing with having both addition and multiplication? — ssu

That is a bit of a mystery. Any simplification to Robinson's arithmetic will make it complete: https://en.wikipedia.org/wiki/Robinson_arithmetic . It just turns out to be like that when you do it.

Indeed that's interesting. With Robinson arithmetic you rule out mathematic induction and the axiom schema. But you do have the successor function, addition and multiplication...and that seems to be all it takes for incompleteness.

On the other hand, for instance Presburger arithmetic is complete. But then:

Presburger arithmetic is designed to be complete and decidable. Therefore, it cannot formalize concepts such as divisibility or primality, or, more generally, any number concept leading to multiplication of variables.

If anyone has something more to say about this and why this is so, I'll definitely want to hear from you.

Ordinarily I would not give it much thought, but this thread seems to focus on math truth beyond virtue of proof. You seem to know what that is all about. Can you provide a very simple definition of this sort of truth in math? I suppose the definition of a triangle is truth without proof. Truth by definition. But what makes a string of symbols true? Model theory? I thought I understood a parallel idea when I quoted the group theory example from StackExchange, but I guess not. Are axioms true by virtue of definitions? — jgill

Ok I'll do my best.

When we manipulate symbols, we use syntax rules. There's no "meaning" associated with the symbols except the meaning in our minds. The symbol manipulations are entirely mechanical, they could be worked out by a computer program. In fact there's much contemporary research on computers doing proofs. Mathematicians are starting to use https://en.wikipedia.org/wiki/Proof_assistant]proof assistants and proof formalizer software. It's a big field, going on ten or twenty years now.

But ok, there's the mechanical symbol manipulation. Syntax.

Now we want to talk about semantics, or meaning. So we cook up, if we can, a model or interpretation of the symbols. The variables range over such and so set. The operation symbols mean such and so. For example, we might have the formal axioms of integer arithmetic, say the ring axioms. And then there is a model, the set of integers.

Now -- and this is actually a very deep point, I do not pretend to begin to understand the nuances -- things are said to be either true or false in the model.

So truth and falsity, semantic concepts, are always relative to a particular model. The integers and the integers mod 5 both satisfy the same ring axioms, but 1 +1 + 1 + 1 + 1 = 0 is false in the integers; and true in the integers mod 5.

That's what we mean by truth. Mathematical truth is always:

Axioms plus an interpretation.

If I say, "Our planet has one moon," as a purely syntactic entity, it's a valid sentence. But it's neither true nor false.

If I interprete "Our planet" as Earth, it's true. If I interpret it as Jupiter, it's false.

So. Given a collection of axioms; and an interpretation of the axioms, which is (1) a domain over which the symbolic variables range; and (2) a mapping from the symbols to objects in the model.

When you do that, then each valid sentence in the language of the axioms is then either true or false in the model. (It could be independent, too, but we're not concerned with that here).

Now the paper in the OP makes the point that there are more mathematical truths than there are symbol strings to express them. So most mathematical truths don't have proofs we can write down. In fact we can't even express most mathematical truth.

This is already long enough so let me know if this is the answer you were looking for. In the end it's related to Tarski's truth thing and Godel's incompleteness theorem and Turing's Halting problem -- though I recently learned that in fact he did NOT talk about what we call the Halting problem in his famous 1936 paper. So all this stuff was "in the air" in the 1930s. A guy named Chaitin came along later and recast all of this in terms of information theory. How "incompressible" a string is as a measure of its randomness. Using those ideas he proved Godel's incompleteness theorems from a different point of view.

From that, you can show that mathematical truths are essentially random. True for no reason than we could ever write down. That's what all this is about.

But you must understand, none of this is of the least importance to the vast majority of working mathematicians. You're not missing anything.

So truth and falsity, semantic concepts, are always relative to a particular model. The integers and the integers mod 5 both satisfy the same ring axioms, but 1 +1 + 1 + 1 + 1 = 0 is false in the integers; and true in the integers mod 5.

That's what we mean by truth. Mathematical truth is always:

Axioms plus an interpretation. — fishfry

Thank you. This is similar to the group theory example. It makes more sense now.

Mathematicians are starting to use https://en.wikipedia.org/wiki/Proof_assistant , proof assistants and proof formalizer software. It's a big field, going on ten or twenty years now — fishfry

My favorite game on the internet is guessing the number of page views per day for math and other topics. I guessed 126 here, whereas it is 111. Close, but no cigar.

Thank you. This is similar to the group theory example. It makes more sense now. — jgill

Glad that helped.

My favorite game on the internet is guessing the number of page views per day for math and other topics. I guessed 126 here, whereas it is 111. Close, but no cigar. — jgill

That's interesting. Which page views? I think you've mentioned in the past that you look at papers written or something like that.

"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. — Tarskian

The law of identity, the indiscernibilty of identicals, and the identity of indiscernibles are different. With a semantics for '=' such that '=' is interpreted as the identity relation on the universe, the first two hold.

'w+7 = w+7' is true in every model. For any term 'T', 'T = T' is true in every model.

set theory [has] large fragments that are bi-interpretable with arithmetic — Tarskian

No, a fragment of set theory with also the negation of the axiom of infinity is bi-interpretable with PA. I pointed that out previously.

Can you provide a very simple definition of this sort of truth in math? [...] Model theory? — jgill

Yes, there is a mathematical definition of 'true in a model'.

Skolem Arithmetic only has multiplication (no addition) and is also complete. — Tarskian

For perspective, keep in mind that Skolem arithmetic and Presburger arithmetic are not fully analagous, since Skolem arithmetic has more detailed axioms about its operation symbol.

so what's the interesting thing with having both addition and multiplication? — ssu

For starters, the system represents all the primitive and recursive functions, and is incomplete.

Mathematical truth is always:

Axioms plus an interpretation. — fishfry

The truth of a sentence is per interpretation, not per axioms.

Some sentences are true in all models. Some sentences are true in no models. Some sentences are true in some models and not true in other models.

Axioms are sentences. Some axioms are true in all models (those are logical axioms). Some axioms are true in no models (those are logically false axioms, hence inconsistent, axioms). Some axioms are true in some models and not true in other models (those are typically mathematical axioms).

The key relationship between axioms and truth is: Every model in which the axioms are true is a model in which the theorems of the axioms are true. And every set of axioms induces the class of all and only those models in which the axioms are true.

Ok, so what's the interesting thing with having both addition and multiplication?
— ssu

[...] Any simplification to Robinson's arithmetic will make it complete — Tarskian

Robinson arithmetic is incomplete.

Presburger arithmetic is designed to be complete and decidable. Therefore, it cannot formalize concepts such as divisibility or primality, or, more generally, any number concept leading to multiplication of variables.

If anyone has something more to say about this and why this is so — ssu

We know it is so because having both addition and multiplication entails incompleteness, so, since Presburger arithmetic is complete, it can't define multiplication.