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Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than 2. The question was, is this a logically necessary truth or some kind of different kind of necessity such as broad logical necessity/metaphysical necessity?

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    All modern theorems of Mathematics follow the same format. A proof of a theorem is a finite sequence of implications, φ_1 → ... → φ_nwhere Φ_1 is an Axiom, and φ_n is the theorem we are proving. The whole sequence is called a derivation sequence. If there exists a derivation sequence which is a true statement ( by logical necessity) for a given φ_n then φ_n is a theorem of the theory T, where T is the collection of starting Axioms. Notice that as φ_1 is an axiom, if we have a derivation sequence as a true statement, by finitely applications of Modus Ponens φ_n is true. Nov 19, 2023 at 18:36
  • Note: The above is all while working within some formal language of mathematics, often some first order logic, with added in symbols Nov 19, 2023 at 18:38
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    Fermat's Last Theorem depends, at least, on the principle of induction for positive integers, which is not considered to be part of logic, so it is not logically necessary. It is likely mathematically necessary when one adds some basic arithmetic to logic. Most mathematicians believe that FLT can be derived in Peano arithmetic (or even some weaker arithmetic), although no such derivation is currently known, see MathOverflow.
    – Conifold
    Nov 20, 2023 at 6:15
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    Can't you ask the exact same thing about anything mathematical truth? What makes Fermat's last theorem special? (In case you're out of the loop, while it was an famous unsolved problem, it was proven 2 decades ago.)
    – NotThatGuy
    Nov 20, 2023 at 10:09

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Strictly speaking it can be proved using a combination of the mainstay mathematical axioms, logical rules, and syntax. Mathematics is not known to be “reducible” to logic, so this can’t be a logical necessity. Any mathematical proof that relies on set membership at some step, as an example, can’t be reduced to logic because elementhood is not logical. I’m not sure if FLT uses elementhood specifically (that’s just one example) but as a whole mathematics can’t be reduced to logic.

Many platonists (mathematics is about objects and those objects are abstract) will probably say it is a metaphysical necessity.

Others less sure about the ontology or with different ontologies will be less likely to say metaphysically necessary, such as Elaine Landry who says mathematics is unlike philosophy and pretty absent metaphysics. (As I understand her)

It’s certainly a mathematical necessity.

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The proof of Fermat’s last theorem, i.e. of the theorem of Wiles, was very challenging. But concerning its state as a mathematical theorem, Wiles’ theorem does not differ from all other mathematical theorems.

Theorems in mathematics follow from the definitions and the axioms by logical conclusions. Insofar the theorems are logical necessary. – I avoid the term “metaphysical necessary”, because it does not provide more clarity, on the contrary.

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Mathematics has a set of axioms (statements that we call "true", and that cannot be constructed from other axioms), and a set of rules that allow us to construct more statements from these axioms; anything that can be constructed from these axioms by these rules is called "true".

We would hope that for every statement S, either S itself or the statement not(S) would be true, but not both. Gödel showed that this is not the case; every set of axioms and rules is necessarily either incomplete (there are statements S where neither S nor not (S) can be proven; this situation is slightly unsatisfactory) or contradictory (there is a statement S so that both S and not(S) can be proven; this is absolutely fatal because using S, any statement whatsoever can be proven, so the whole system is completely worthless). Any statement or proof whatsoever implicitely includes "assuming our system of axioms and rules is not contradictory...".

Statements S where either S or not(S) can be proven are called "decidable". There are undecidable statements (unless our axiom system is contradictory). There are even statements that are proven to be undecidable. For a long time, it wasn't known whether Fermat's Last Theorem was decidable or undecidable. Interestingly finding an example where a^n + b^n = c^n would make FLT not only false, but also decidable, so if it was undecidable there would be no counter example possible, so it wouldn't be mathematically true, but for mere mortals it would be true.

As it turned out, because Wiles proved it, the problem was both decidable and true. That's plain mathematics. It HAD to be true because it followed from the axioms by applying the rules, just because the rules to apply were very very difficult to find, that doesn't make a difference.

On another level, if you replaced a^n, b^n, c^n with "a number close to a^n, b^n, c^n chosen at random that behaves in a way similar to powers", then a^n + b^n = c^n could be true for some numbers; for example if you interpret "close to c^n" as "between (c - 1/2)^n and (c + 1/2)^n". However, if you picked these random numbers, and looked for a solution and didn't find them quickly, you could reasonably say "it is not impossible that these numbers exist, but it is very very extremely unlikely".

There's a fact that for example a^3 + b^3 = c^3 + k has no solutions if k = 0 (part of FLT and quite hard to prove itself), it has no solutions if k = 4 modulo 9 or 5 modulo 9 (the proof is quite trivial), and it most likely has an infinite number of solutions for all other values k - which can be very hard to find, so for many small k there are no known solutions. So there seems to be no necessity that a^3 + b^3 ≠ c^3 until you proved it. For n = 4 there are not even solutions of a^4 + b^4 = c^2. For n >= 5 the existence of solutions always was heuristically very unlikely. Before Wiles' proof it was proven there would be no solutions with fewer than 300,000 digits - so clearly no way to find a solution if it exists.

Again, it is just a (very, very hard) mathematical proof. With no deeper reasoning or justification behind it.

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    Upvoted, but: "We would hope that for every statement S, either S itself or the statement not(S) would be true, but not both. Gödel showed that this is not the case" << I disagree. Gödel showed that there exist statements that cannot be proven, not that there exists statements that are neither true nor false.
    – Stef
    Nov 20, 2023 at 12:17
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Excellent question, and one in which there may not be a mutual exclusion.

Obviously, if a theorem is true, it is logically necessary in respect to the axioms of the system. That's the easy half of answering. As to whether or not there is some sort of metaphysical necessity at play in regards to a theorem, I guess the question to answer first is 'Metaphysically necessity in regards to what?' A simpler example might be the Pythagorean theorem.

The Pythagorean theorem expresses a relationship. Specifically, between the length of the legs and the hypotenuse of a right triangle. One can start by saying that unlike the triangle inequality, there's no intuition that guides one's sense of the truth of the claim. Even in the geometrical proof, looking at the areas of the squares doesn't in some way inform one's intuitions in an obvious manner. So, is it in some sense necessary that the sum of the squares of the legs is equal to the square of the hypotenuse? I would say yes along a broader line of reasoning as such:

Argument: We know from the triangle inequality, that the three sides have a relationship, and therefore, if the three sides of a triangle have a relationship (lots of them, in fact, as trigonometry shows), then it seems that the areas delineated by their squares should as well. Functions built from one relationship should in essence demonstrate another relationship.

But what about Fermat's Last Theorem? If we carry over that reasoning from the triangle inequality to cubes built of the sides of the right triangle, we should expect again there to be some sort of deep result showing a relationship of the cubes. Remember, with n=1, we have the trivial case of the group defined by addition on the naturals in which the relation applies to any a,b, and c. In a plane, we see that only some of the naturals satisfy, a,b, and c. It seems likely then that even fewer or no naturals may satisfy a,b, and c if we extend constructions into the third dimension.

But is there a more conclusive logic to claim it is in some sense metaphysically necessary? Here's the crux of the problem. What does it mean to be metaphysically necessary in the first place? Does my reasoning count? If not, is there some reasoning that does? I would say that the most reasonable way to answer is simply to determine if there is some sort of metaphysical explanation (SEP) that carries the metaphysical logic. I'm not sure that can be provided for Fermat's Last Theorem, but if it can, then one can argue there is metaphysical necessity.

This is why I think the notion of metaphysical necessity is a source of disagreement among thinkers. I suspect there is no canonical response to your question, and that arguments of broad logic are presented on a case-by-case basis.

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  • If its logical necessary won't it mean its denial is a contradiction? But is denying mathematical theorems a contradiction?
    – Vihan
    Nov 19, 2023 at 17:37
  • Logical necessity simply means that given premises, a conclusion is true. In mathematics, one's premises are axioms. Thus, you can deny any axiom you want to deny the truth of a theorem. Now, there are very PRACTICAL reasons why we choose axioms, for instance they produce results that assist us in the world, but the truth doesn't come from the logical necessity as much as it does from the choices of the axiom. There is no privileged set of axioms, and in set theory, for instance, some people argue over them. en.wikipedia.org/wiki/…
    – J D
    Nov 19, 2023 at 19:31
  • Mathematical truths are conventional truths, because it is convention to pick axioms. The axioms that are picked are conducive to producing results in a discipline, and there are many disciplines. Consider that Euclidean and non-Euclidean geometries have different axioms and therefore different theorems. Is one geometry really "better" than another inherently? No. Spherical geometrical theorems are good for working with spheres and planar geometrical theories are good for working on planes. Modern physics doesn't use the latter at all anymore. No contradictions. Better models.
    – J D
    Nov 19, 2023 at 19:34
  • Our mathematical models of things are always normative, and what we construct affects what we believe, plain and simple. Math isn't a pathway to an ultimate reality. It's just a tool to build machines for prediction.
    – J D
    Nov 19, 2023 at 19:36

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