Therefore (she argued) if we assume a counter-example to FLT and then, via some long and complicated chain of reasoning, arive at a contradiction, how do we know that this contradiction is essentially a consequence of the assumed counter-example? Could it not be that we have just exposed a contradiction inherent in arithmetic itself? In other words, if arithmetic itself is inconsistent, then proof by contradiction loses its persuasiveness.
On one level, this kind of objection can easily be vitiated by simply prefacing every theorem with the words "If our formalization of arithmetic is consistent, then...". Indeed, for short simple proofs by contradiction we can strenghten the theorem by reducing this antecedent condition to something like "If arithmetic is consistent over this small set of operations, then...". We can be confident that the contradiction really is directly related to our special assumption, because it's highly implausible that our formalization of arithmetic could exhibit a contradiction over a very short chain of implication.
However, with long proofs of great subtlety, extending over multiple papers by multiple authors, and involving the interaction of many different branches and facets of mathematics, how would we really distinguish between a subtle contradiction resulting from one specific false assumption vs a subtle contradiction inherent in the fabric of arithmetic itself?
If MVS objects to the proof of FLT on this basis, then she objects to almost all other proofs as well. Very few axiomatic systems are known to be consistent.
"It's very easy to prove the consistency of arithmetic. Just work in ZFC."I hope somebody points out that this is a joke...
In view of this, it isn't clear how "working in ZFC" resolves the issue. There is no complete and absolute proof of the consistency of arithmetic, so every arithmetical proof is subject to doubt. (By focusing on arithmetic I don't mean to imply that other branches of mathematics are exempt from doubt. Hermann Weyl, commenting on Godel's work, said that "God exists because mathematics is undoubtedly consistent, and the devil exists because we cannot prove the consistency".)
As Morris Kline said in his book "Mathematics and the Loss of Certainty":
"Godel's result on consistency says that we cannot prove consistency in any approach to mathematics by safe logical principles..."Similarly, in John Stillwell's book "Mathematics and its History" we find
"If S is any system that includes PA, then Con(S) [the consistency of S] cannot be proved in S, if S is consistent." [Godel's 2nd theorem]I've received email suggesting that the contemplation of inconsistency in our formal arithmetic is tantamount to a renounciation of reason itself, i.e., if our concept of natural numbers is inconsistent then we must be incapable of rational thought, and any further considerations are pointless. This attitude is reminiscent of the aprehensions mathematicians once felt regarding "completed infinities". "We recoil in horror", as Hermite said, believing that the introduction of actual infinities could lead only to nonsense and sophistry. Of course, it turned out that we are quite capable of reasoning in the presence of infinities. Similarly, I believe reason can survive even the presence of contradiction in our formal systems.
This belief may be partly due to my unorthodox view of formal systems, which I think should be seen not as unordered ("random access") sets of syllogisms, but as structured spaces, with each layer of implicated objects representing a region, and the implications representing connections between different regions. The space may even possess a kind of metric, although "distances" are not necessarily commutative. For example, the implicative distance from an integer to its prime factors is greater than the implicative distance from those primes to their product.
According to this view a formal system does not degenerate into complete nonsense simply because at some point it contains a contradiction. A system may be "locally" consistent even if it is not globally consistent. To give a crude example, suppose we augment our normal axioms and definitions of arithmetic with the statement that a positive integer n is prime if and only if 2^n - 2 is divisible by n. This axiom conflicts with our existing definition of a prime, but the first occurrence of a conflict is 341. Thus, over a limited range of natural numbers the axiom system possesses "local consistentency".
Suppose I then substitute a stronger axiom by saying n is a prime iff f(r^n) = 0 (mod n) where r is any root of f(x) = x^5 - x^3 - 2x^2 + 1. With this system I might go quite some time without encountering a contradiction. When I finally do bump into a contradiction (e.g., 2258745004684033) I could simply substitute an even stronger axiom. In fact, I can easily specify an axiom of this kind for which the smallest actual exception is far beyond anyone's (present) ability to find, and for which we have no theoretical proof that any exception even exists. Thus, there is no direct proof of inconsistency. I might then, with enough imagination, develop a plausible (e.g., as plausible as Banach-Tarski) non-finitistic system within which I can actually prove that my arithmetic is consistent. In fact, it might actually BE consistent. But I would have no more justification to claim absolute certainty than with our present arithmetic.
As to the basic premise that we have no absolute proof of the consistency of arithmetic, here are a few other people's thoughts on the subject:
"A meta-mathematical proof of the consistency of arithmetic is not excluded by...Goedel's analysis. In point of fact, meta-mathematical proofs of the consistency of arithmetic have been constructed, notably by Gerhard Gentzen, a member of the Hilbert school, in 1936. But such proofs are in a sense pointless if, as can be demonstrated, they employ rules of inference whose own consistency is as much open to doubt as is the formal consistency of arithmetic itself. Thus, Gentzen used the so-called "principle of transfinite mathematical induction" in his proof. But the principle in effect stipulates that a formula is derivable from an infinite class of premises. Its use therefore requires the employment of nonfinitistic meta - mathematical notions, and so raises once more the question which Hilbert's original program was intended to resolve."-Ernest Nagel and James Newman
"Goedel showed that...if anyone finds a proof that arithmetic is consistent, then it isn't!"-Ian Stewart
"...Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions."-Carl Boyer
"An absolute consistency proof is one that does not assume the consistency of some other system...what Goedel did was show that there must be "undecidable" statements within any [formal system]... and that consistency is one of those undecidable propositions. In other words, the consistency of an all-embracing formal system can neither be proved nor disproved within the formal system."
-Edna Kramer
"Do I contradict myself? Very well then, I contradict myself.
I am large, I contain multitudes."-Walt Whitman