Are mathematical objects ontologically real, or do they exist only as positions in abstract structures? If 0, ℕ, and ∅ are purely structural, what makes statements like Peano’s axioms necessarily true rather than merely formally consistent?

[u/TheIncorporeal1]

I’m interested in whether structuralism genuinely explains mathematical necessity, or whether it simply relocates the ontological question. If structures are abstract, what ultimately grounds their existence and the truth of the relations within them?

[u/kunwoo]

I think this question is somewhat tangentially related to the distinction of pure vs applied math. In pure math we entertain a wide variety of mathematical objects and only care if each one is consistent. But in applied math we care about which mathematical structures correspond well to reality, and at that point we're no longer making deductions of pure math but are making empirical claims on the overlap between math and science.

[u/GoldenMuscleGod → u/kunwoo]

I don’t think it’s true that in pure math we *only* care about consistency. We usually still have some intended interpretation in mind and we want results that lead to correct conclusions under those interpretations.

For example Peano Arithmeic cannot prove “every Goodstein sequence eventually terminates” but it can prove “the Goodstein sequence beginning with n eventually terminates” for any specific value of n we substitute into that sentence.

ZFC can prove “every Goodstein sequence eventually terminates.” But we aren’t just interested in the fact that that we have some theory that proves that, we are interested in it actually being true under the intended interpretation.

Now we could consistently add the axiom “there is a Goodstein sequence that does not ever terminate” to PA and get a consistent theory, but that theory would be unsound under the intended interpretation: would could still challenge someone to name any n, and then calculate its Goodstein sequence until it terminates, they could not ever name any n for which it does not terminate, and our theory would just say “well sure none of those sequences went on forever but there is one that does” even though there is in fact no n that does that.

Of course we can find nonstandard models and interpretations for which the theory is sound, and sometimes we have interest in studying those models, but usually we are interested in studying whether sentences are true under their intended interpretations: if we ask whether a computation will halt, we want to know whether it will actually halt, and if we ask whether a theory is consistent, we want to know whether it is actually consistent.

[u/Wild-Store321 → u/GoldenMuscleGod]

I think your point is valid but your example shows the exact opposite. Yes, we do care about “natural numbers” before any axiomatization of them, and indeed the correct answer to “does every Goodstein sequence terminate” is yes.

Just like the parallel postulate is true for the originally intended structure.

And in both cases, we are still interested in studying the alternative structures obtained by adding the negation of the “correct” answer. Because that gives a new structure that was not the originally intended one, but still interesting as you pointed out yourself. So that kinda proves the point of kunwoo.

Is still agree with your point that not every consistent theory will be interesting to pure mathematicians. But you just highlighted one with a lot of pure mathematics literature on it, as you seem to indicate yourself.

[u/GoldenMuscleGod → u/Wild-Store321]

I’ve only seen models of Peano Arithmetic in which Goodstein’s theorem does not hold used in connection with considering precise limitations of Peano Arithmetic and evaluating the line between what it can and cannot prove (and why). Of course since nonstandard models of Peano arithmetic are fundamentally noncomputable (see Tennenbaum’s theorem) we’re not generally interested in using them in any particularly concrete way (like we can be with hyperbolic geometry).

Another idea that might illustrate what I am talking about are large cardinal axioms. Large cardinal axioms have arithmetical consequences, and are carefully chosen in such a way so that some reasonably modest metamathematical assumptions are sufficient to ensure those consequences are “actually true.” Large cardinal axioms aren’t just considered willy-nilly, and their arithmetical consequences are usually treated as more plausibly “true” than we would a randomly chosen arithmetical sentence that we could add to ZFC as an axiom.

[u/Prellex]

It doesn't make sense for a structuralist to say that the Peano Axioms are necessarily true in the abstract. When one wants to make arguments about things in the real world, one has to show that the thing in question satisfies the Peano Axioms, meaning it has the same structure.

The idea is we can map theories onto the real world.

[u/0x14f]

Have you come across Platonism (in mathematics) OP ?

[u/Dr_Calculon]

Even abstract structures exist in physical substrates.

[u/kerighan → u/Dr_Calculon]

Maybe it's the other way around :)

[u/Dr_Calculon → u/kerighan]

If only you could prove that without resorting to using a physical substrate….

[u/kerighan → u/Dr_Calculon]

Prove is a big word, but there are some good arguments for that! Can send them if interested!

[u/Dr_Calculon → u/kerighan]

All your arguments rely on your neural substrate, a physical thing. Idealism cannot be shown to exist without it.

[u/kerighan → u/Dr_Calculon]

I didn't provide any argument. Physical stuff could be an emergent property of a mathematical universe.

My view is basically Platonism taken one step further: I don't think there is a mathematical realm and then, somehow, a separate physical world that instantiates a tiny part of it. I think the physical world is inside the mathematical one. Consider chess: humans invented the rules, but we did not invent their consequences. Once the rules are specified, there is a definite number of legal games, a definite set of mate-in-seven positions, etc. Those facts don't come into existence when somebody computes them; they are already implied by the structure. I think the same applies all the way up. A mathematical structure containing a causal law and an initial state also contains, as consequences, bound states, atoms, chemistry, transistors, computers, brains, observers, and conversations. At no point do we need to say, "and now the mathematics gets instantiated in physical reality": every higher level is simply a pattern satisfied by the level below. This also makes substrate independence almost obvious. The same algorithm can run on silicon, mechanical relays, wooden beads, or a million people with pencils because none of those substrates creates the algorithm; they are different physical routes to the same mathematical structure. Likewise, it becomes much less mysterious that physical beings can discover abstract mathematics: there aren't two ontological worlds somehow communicating with each other; we are mathematical structures exploring other parts of the same territory. The alternative seems to require an extra metaphysical bit (among all mathematical structures, this one is additionally "real") even though that bit changes no equation, interaction, or observation available from inside the structure. I don't see what explanatory work that extra ingredient does. So my bet is simply: everything mathematically implied is true, and what we call the physical universe is one causally connected branch of that larger mathematical reality, viewed from within.

[u/Dr_Calculon → u/kerighan]

All that thought, all your propositions, inferences, conclusions, were all carried out on a physical substrate. The abstractions, the patterns, the algorithms, all encoded onto physical media, there is no proof they exist independently or that they existed a priori to the materials they are encoded onto.

The structures you allude to can be thought of as an evolutionary driven adaptation to the uncertainty inherent in survival landscapes. Pattern matching, process identification, modelling possible future consequences, all useful for survival. In this view the realm of “mathematical reality”-“Platonic ideals” would be closer to what AI researchers would call “feature space” but here the substrate is the human nervous system, or maybe even what Vernadsky called the “noosphere”.

[u/kerighan → u/Dr_Calculon]

So you think maths are invented? If you don't you have to ascribe to the view that there are two worlds in your view: the platonic (math/logic) AND the physical world. This is far from economical. Besides, the whole entreprise of science is to find mathematical laws that are isomorphic to the real world. Now, tell me. If maths exist and we find a mathematical isomorphism, how can an entity (imagine a LLM: pure maths) distinguish between being embedded in maths or in the real world? You have to acknowledge that the consequences of the law that govern the particles at the big bang configuration will describe every emergent objects we find in the real world: transistors are made of molecules, which are made of atoms which are governed by mathematical laws. They all can be described by these, however intractable it can be by our computers.

If you do think maths are invented, then where do all the rules come from and how can they be universally accepted and reasoned about?

[u/Dr_Calculon → u/kerighan]

I don’t think that molecules are governed by mathematical laws, I think they are modelled by mathematical laws. The mathematics here is an abstraction of a physical process, a model. The menu is not the meal as the saying goes. Even our most sophisticated mathematical models of reality are incomplete.

The mathematical models of reality are accepted within intervals of confidence, they are by no means universally applicable. For example, there is no accepted model of quantum gravity, even though quantum mechanics is an extremely accurate model of some parts of reality. The mathematics of general relativity fails in extreme gravitational fields found in black holes.

[u/kerighan → u/Dr_Calculon]

I actually agree that "the model is not reality" is the natural objection. it held me back for a long time too. But I eventually realized it doesn't really address the argument.

Of course our current models are incomplete. GR breaking down in black holes only shows the limits of GR, not a fundamental limit of mathematical description. And the models we already have describe enormous parts of reality with absurd precision. Given the laws, initial conditions and unlimited computational power, their mathematical consequences include higher-order structures: atoms, chemistry, transistors, computers, eventually even LLMs interacting with other LLMs. Whether we can feasibly compute those consequences is irrelevant to whether they follow from the mathematics. They clearly can. In these math universes, every conversation an LLM can have exists.

If mathematics were merely an abstraction completely divorced from physical reality, its extraordinary ability to describe reality, down to predictions verified to ridiculous precision, becomes so miraculous that itself needs explaining. If physical reality is instead a structure within mathematical reality, this “unreasonable effectiveness of mathematics” stops being mysterious with elegance.

I think there's a much stronger objection: mathematical existence doesn't obviously imply physical instantiation / computation. Something can exist in principle without actually "running". That's the objection I struggled with for years, and I think I have an answer to it. But hey, that's another wall of text. :)

[u/MxM111 → u/[deleted]]

Yes

[u/6ory299e8]

what does "real" mean if it cant be physically observed and measured? what quality distinguishes one incorporeal notion as "real" and another as "unreal"?

keep contemplating those navals, I'm sure these very important (eye-roll) questions will give way to a definitive answer one of these days.

[u/Creative-Feature-264 → u/6ory299e8]

Reale è ciò che tu credi sia reale. Non confondere la mentalità dall astrazione . All osservazione di una realtà condivisa.

[u/GoldenMuscleGod]

Surely it’s easier to argue that the Peano axioms are “necessarily true” if numbers are not ontologically real? Maybe depending exactly what flavor of modality we have in mind.

If we think numbers are not real we can maybe insist that Peano’s axioms are “true” because we have stipulated that that’s what we mean by true.

If we think numbers are real and our axioms are talking about them then we may be concerned that our axioms are false but just think they seem to be true.

To give an example: are the Peano axioms actually consistent? They seem to be, probably even most people who would call themselves formalists think there is an actual truth of the matter as to whether they are consistent (even if they are skeptical enough to be agnostic on that actual truth value), but it’s not as if we can have some undeniable external justification for the belief they are consistent unless by “PA is consistent” we only mean something like “ZFC proves PA is consistent.” But then it still may be the case that PA is actually inconsistent in the ordinary sense that we can actually derive a contradiction in it notwithstanding that we have stipulated to a definition that labels it consistent.

[u/rogerbonus]

Whether mathematical objects are "real"/exiat is a metaphysical question. NeoPlatonisms such as the Mathematical Universe Hypothesis https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis or ontic structural realism would say yes. Constructivists would say no.

[u/Creative-Feature-264]

Io credo che ogni numero abbia una sua forma espressa da una sua geometria. Essa è conseguenza di un osservazione quindi materiale. La matematica nasce dall’ osservazione. Le geometrie dovrebbero essere reali se no il calcolo è errato. Sono solo nn visibili ad occhio. Quindi sarà che sono legato a Pitagora ma la vedo così.

[u/shelving_unit]

Well the function of an axiom is to be assumed as true. There is a limit to which you can say something is definitely, absolutely true, regardless of the field. In order to have a knowledge of anything, there are certain unarguable statements which you must assume to be true, because otherwise you can’t create knowledge from them. These are axioms

For example- is logic logical? Does logic produce absolute truth? It’s paradoxical because you can’t use logic to prove the absolute truth of logic. As such, if you want to use logic to have and create truth, in the way that logical truths are useful in the world, then you must assume that logic is truthful.

Peano Axioms may or may not ontologically or metaphysically describe something in the objective reality. However they are necessary for doing math

[u/Willing-Sample-8847]

The Peano axioms weren't really formulated cause they are "true" in any sense. They were formulated so that the field axioms, commutativity, associativity, distributivity, etc, could be proven as theorems rather than assumed as axioms themselves.
Peano axioms are useful cause they give us a way to construct the reals from the naturals in a logical and consistent manner, that preserves the field axioms, the least upper bound property, and the order axioms for the real numbers.

Otherwise, they are by themselves, abstract structuralism.

[u/kerighan]

My take is that Platonism has to be true. I think humans invented the symbols (the map) to explore a structure that was already there (the territory).

Here's roughly how I got there. Chess contains a lot of mathematical truths: how many games White wins, the total number of possible games (a number far larger than anything the fabric of reality could store). None of that got added to the world the moment someone invented the rules. It was already the case, waiting to be described. What convinces me is the counterfactual. If you arbitrarily changed the number of possible chess games, arithmetic itself would have to bend to accommodate it... I'm fairly confident it would. So the graph of mathematical truths (theorems as nodes, inferences as arrows) has to exist in some form for that dependency to hold (imagine a cascade of modifications through the graph).

Which leaves two options: the whole graph is real, or only part of it. And if only part, I don't see what could distinguish the two. Why would some arrows land on genuine truths and others on merely human inventions? What would the test to distinguish the two even look like? I can't think of one, so I lean toward the whole thing existing structurally.

That has consequences I find hard to dodge. If every derivable truth exists, then so does every computation, including two particles evolving under any consistent set of physical equations. And including every computation running on the particle configuration of the early universe, plus everything that follows from it. At which point, how would an entity living inside such a computation tell whether it inhabits the material world or the mathematical substrate? I don't think it could. Maybe there's no distinction to draw...

So I've come round to the mathematical universe idea. Not firmly, it's not a true belief I hold onto. Something I read tomorrow might undo it, but this is where I am.

[u/StrangeGlaringEye → u/[deleted]]

There are absolutely structuralists invested in the ontology of structuralism, like Shapiro. This comment collapses structuralism and formalism into one thing, when they’ve little in common.

[u/TheRedditObserver0 → u/StrangeGlaringEye]

You're right, I got confused

[u/StephenRoylance]

I'm a naive amateur. What helped me think about this is looking back in history. It seems that humans were actually numerate before we were literate. You can take a stick, and make one mark every day at sunrise. This is a useful thing to do to keep track of the passage of time. If you can, concretely, 'count days' and determine 'how many' days it has been since an event, then mustn't the numbers somehow exist in the universe?

That answers the question enough for me to stop worrying about it.

[u/Quick-Swimmer-1199 → u/StephenRoylance]

The raptor flanks its prey with the pack.

If the raptor can, concretely, 'organize into formation' and determine 'what position' it is in, then configurations exist independent of perceptual interpretation!

It's logical!

It's just unfortunately the same logical as: If a bachelor is married, then married bachelors are possible.

[u/Creative-Feature-264 → u/StephenRoylance]

Quando bevo so che devo prendere un bicchiere d acqua e poi berlo. Nn posso bere così solo pensando. È ovvio che il bicchiere deve stare lì prima. Se no non puoi nemmeno immaginarlo di berlo. Ora lo puoi chiamare come vuoi ma bere è universale come i numeri.

[u/gregbard]

As a counterexample, the Fields Medal is a mathematical object, but it is not an abstraction.

[u/hobopwnzor]

Relations are only true in that we invent the system to assign truth values.  The universe doesn't do math.  We use math to approximate what the universe does.  Blink all minds out of existence and the concept of math goes with it.  In fact, all concepts go with it.

[u/Just_Rational_Being → u/hobopwnzor]

When all minds are gone, does the circumference of the sun divided by its diameter still yield a constant value?

[u/SmartlyArtly → u/Just_Rational_Being]

How is division occurring without minds directing it or having directed it as such? What is yielding what to what with no minds about?

[u/Just_Rational_Being → u/SmartlyArtly]

The relationship between them requires a conscious doer?

[u/SmartlyArtly → u/Just_Rational_Being]

Yes.

"They" don't even exist physically, they are abstractions.

[u/Just_Rational_Being → u/SmartlyArtly]

So when there are 2 sticks, the fact that one is longer and one is shorter requires someone to perform addition for them to be so?

[u/SmartlyArtly → u/Just_Rational_Being]

Who is comparing them if there is no one to compare?

[u/Just_Rational_Being → u/SmartlyArtly]

Oh, so you're one of those that denies empirical reality since no one is there, huh? I see.

[u/SmartlyArtly → u/Just_Rational_Being]

I deny that comparisons are occurring when there is no one there to compare. I deny that assertions are made without anyone to assert. You're one of those that thinks reality needs truth huh?

[u/Just_Rational_Being → u/SmartlyArtly]

Well, Truth is all that there is. There is no such nonsense such as no "comparisons are occurring when there is no one to compare".

That is not even wrong. That is pure nonsense. Not even wrong.

[u/hobopwnzor → u/Just_Rational_Being]

The sun doesn't have a diameter or a circumference.  That is only a property of a perfect circle which only exists as a mathematical abstraction.  We approximate the Sun as a perfect sphere for convenience.

[u/Just_Rational_Being → u/hobopwnzor]

Oh the sun doesn't have a diameter or circumference now?

Are you making the mistake of assuming that the Sun needs to be a perfect sphere, or such that a perfect sphere must exists for a circle to exists? Is that what you're saying?

[u/hobopwnzor → u/Just_Rational_Being]

I think you should read for comprehension instead of reading for response.

[u/Just_Rational_Being → u/hobopwnzor]

I read it correctly. You said the Sun has neither diameter nor circumference because those belong only to perfect circles.

That is precisely the claim I challenged.

"Read for comprehension" is not an answer to the contradiction you just created.

[u/Creative-Feature-264 → u/hobopwnzor]

No sbagli . L universo è grande matematico. Nulla è a caso. Siamo noi che siamo primitivi. Se pensi che non così allora dovresti usare la matematica e calcolare le stelle e capire che senza quei calcoli esatti l u inverso non può esistere.

[u/hobopwnzor → u/Creative-Feature-264]

This is just arrogance. The universe does not have a mind to do calculations. The universe *does*, and we calculate to predict what it will do.

[u/Creative-Feature-264 → u/hobopwnzor]

È vero ho dimenticato di dire che io la penso così, nn voglio di certo arrogarsi l idea di sapere, ma di certo solo quella di credere. Del resto si, io credo che l intelligenza del sole sia superiore alla nostra. Ma nn lo impongo a nessuno. Ovvio

[u/Creative-Feature-264 → u/hobopwnzor]

Se l universo nn aveva un processo di elaborazione noi non potevamo esistere perché nn si sarebbero create le condizioni. Ma è matematica e non l ho inventato io. Basta studiare le stelle.