1

This question is related. It asks: "Can truths about the natural numbers vary across possible worlds?". One comment says: "Well, no, if they use same definitions and axioms about natural numbers.".

However, with the same definitions and axioms about natural numbers, in different models of ZFC different things can be proved about them. This suggests the possibility that our natural numbers might be non-standard, and perhaps some of the statements we prove about them are not true in every possible world. Somehow this seems intuitively absurd, and feels like any number we can reach physically must be standard, but this intuition seems unjustified because in a non-standard world there is no way to distinguish standard from non-standard numbers.

Since this question seems to have been misunderstood, i will try to give an illustration. Imagine we are somehow able to create a machine with infinite memory and capable of supertasks. Perhaps we can then simulate our world there but with a twist: the natural numbers in this world are now non-standard. The people inside this world will then performs actions that they will believe are finite but to us outside are infinite. In particular they may be able to prove in peano arithmetic some statements that we can't. Is it possible that we are actually inside such a world?

6
  • 1
    Definitions and axioms about natural numbers that we intuitively "use" are better described not by the first order ZFC but by the second order arithmetic. It does not have non-standard models, there is only one "full" model, the intended arithmetic. And one cannot prove anything "in a model", let alone different things in different models. Proofs are made in theories, not models, what can be proved in ZFC is exactly what is true in all of its different models. This is Gödel's Completeness Theorem.
    – Conifold
    Commented Oct 27, 2022 at 21:41
  • @Conifold Second order arithmetic is just moving the issue around. ZFC can still have different models. Second order arithmetic just guarantees that the natural numbers agree with the natural numbers with the model of ZFC we are using, but nothing guarantees that those are standard in any absolute sense. Commented Oct 28, 2022 at 11:08
  • I think a math or logic that can vary is no use whatsoever. What is the idea of suggesting it?
    – Scott Rowe
    Commented Oct 28, 2022 at 23:34
  • @ScottRowe Because we might not be able to make sure it can't vary. Commented Oct 29, 2022 at 10:44
  • 1
    You might find mathoverflow.net/questions/284747/… insightful
    – TomKern
    Commented Nov 4, 2022 at 7:44

4 Answers 4

1

I argue here that number lines are an abstraction of indistinguishable objects imagined to undergo continuous sequential translations The Unreasonable Ineffectiveness of Mathematics in most sciences (imaginary numbers can be thought of as additionally allowing rotation as a transformation, which is equivalent to phase)

From Noether's theorem we know that saying a thing remains constant under specific transformations is equivalent to saying there is a conservation law for it of a given kind. If an interaction will occur in the same way regardless of where or when, then momentum and energy are conserved respectively (eg if it's time-dependent energy may not be conserved). So there is a deep connection between geometry, symmetry operations, and numberlines, that leads to the power of math in the physical sciences.

Can things be otherwise? Yes, for beings that experience different dimensions, and different conservation laws. It's thought life is possible in 2D. The possible interactions in higher dimensions would likely make chemistry in higher dimensions problematic, but it could be out there. Note, this would only impact 'natural numbers' in these circumstances, what the most obvious or intuitive abstractions are.

I would relate the expanded idea of what numberlines are from Ring Theory in Abstract Algebra, to identifying possible geometries. For instance, you can only expand imaginary numbers to quaternions and octonions, and that's it, which seems to indicate something deep about rotation, and has impacted the chosen dimensions of M-theory, the elaboration of String Theory.

We can focus on different abstractions, different sets and ways to relate them. But I would argue numbers and their power ultimately come from our intersubjectively-shared experiences of spacetime geometry, of symmetries under transformation. We can understand 2D math, a higher dimensional mind could clearly understand our math, so I think there is something fundamentally unifying there about natural numbers that all beings can come to understand, even if their intuitions vary.

1
  • 1
    Right, someone who talks about a math we can't directly relate to is either babbling or beyond our comprehension.
    – Scott Rowe
    Commented Nov 1, 2022 at 10:10
1

So it seems that we want to reconcile two claims: 1) there is no possible world in which the properties of the natural numbers, as fixed by the axioms of arithmetic and set theory, are any different, and 2) there are other conceivable sets of axioms according to which the properties would be different.

The best answer here is probably to say that the true axioms of arithmetic and set theory are metaphysically necessary. Not physically necessary, and not logically necessary, but metaphysically necessary. If you'd like an explanation of the difference between these different types of necessity, see the SEP article on modal epistemology, which provides a nice summary. Their article on the varieties of modality is also good.

By saying that the true axioms are metaphysically necessary, we can say that there is no possible world in which they are different, whilst also acknowledging that they are not logically necessary (since by Godel's second incompleteness theorem their consistency cannot be proved). This seems to allow us to have our cake and eat it, so to speak. Indeed, some philosophers have used the axioms of arithmetic as an argument for the existence of metaphysical necessity; Alex Pruss (a professor of philosophy at Baylor, who also has a PhD in mathematics) has written something interesting to this effect on his blog.enter link description here

1
  • My question is not about different sets of axioms. It is about the possibility that even using the same axioms, different models of the natural numbers in the metatheory can lead to different provable statements. Commented Oct 28, 2022 at 14:48
0

Somehow this seems intuitively absurd, and feels like any number we can reach physically must be standard, but this intuition seems unjustified because in a non-standard world there is no way to distinguish standard from non-standard numbers.

It's only true in first-order logic that we can't distinguish non-standard objects from standard ones. Propositions like "For every set of numbers..." do not fall within first-order logic and are capable of distinguishing standard from nonstandard objects. But this distinction becomes vacuous when the sets are limited in size. For example, if only sets with up to 100 elements are allowed, then any such set can be encoded into a single natural number. Since we are limited to finite experience with real-world collections of pebbles, words, and so on, there is no way in real-world experience to enact this kind of situation where a certain type of truth is hidden from first-order logic but exposed to higher-order logic.

1
  • Isn't higher order logic simply hiding the issue? Sure, we can prove in ZFC that there is a unique up to isomorphism model of second order arithmetic, but then we have the problem that there are different models of ZFC where these natural numbers differ from each other anyway. Commented Oct 28, 2022 at 14:58
0

Natural numbers (or any numbers, for that matter) are cognitive constructs; they do not exist outside of conscious minds. Numbers arise because conscious minds create categories of (ostensibly) independent objects and begin counting objects that exist within these categories. Without a conscious mind there are not even (say) stones, much less five or seven stones.

Categories are a mental abstraction; counting is an abstraction from that abstraction; mathematical laws are an abstraction of the abstraction of that abstraction...

In any possible world where (a) consciousness exists, and (b) consciousness abstracts categories, math will be the same, because math is nothing more than the progressive abstraction, systemization, and generalization of category operations.I cannot (personally) imagine a consciousness that that creates math without a concept of categories (our measurement without identification); if you can, then there is a possible world with a potentially alternative math.

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .