Why did Fermat believe he had `a truly marvellous demonstration' of FLT?

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Conventional wisdom dictates that proofs of mathematical propositions should be treated as necessary, and sufficient, for entailing `significant' mathematical truths only if the proofs are expressed in a---minimally, deemed consistent---formal mathematical theory in terms of: * Axioms/Axiom schemas * Rules of Deduction * Definitions * Lemmas * Theorems * Corollaries. Whilst Andrew Wiles' proof of Fermat's Last Theorem FLT, which appeals essentially to geometrical properties of real and complex numbers, can be treated as meeting this criteria, it nevertheless leaves two questions unanswered: (i) Why is x^n +y^n = z^n solvable only for n < 3 if x, y, z, n are natural numbers? (ii) What technique might Fermat have used that led him to, even if only briefly, believe he had `a truly marvellous demonstration' of FLT? Prevailing post-Wiles wisdom---leaving (i) essentially unaddressed---dismisses Fermat's claim as a conjecture without a plausible proof of FLT. However, we posit that providing evidence-based answers to both queries is necessary not only for treating FLT as significant, but also for understanding why FLT can be treated as a true arithmetical proposition. We thus argue that proving a theorem formally from explicit, and implicit, premises/axioms using rules of deduction---as currently accepted---is a meaningless game, of little scientific value, in the absence of evidence that has already established---unambiguously---why the premises/axioms and rules of deduction can be treated, and categorically communicated, as pre-formal truths in Marcus Pantsar's sense. Consequently, only evidence-based, pre-formal, truth can entail formal provability; and the formal proof of any significant mathematical theorem cannot entail its pre-formal truth as evidence-based. It can only identify the explicit/implicit premises that have been used to evidence the, already established, pre-formal truth of a mathematical proposition. Hence visualising and understanding the evidence-based, pre-formal, truth of a mathematical proposition is the only raison d'etre for subsequently seeking a formal proof of the proposition within a formal mathematical language (whether first-order or second order set theory, arithmetic, geometry, etc.) By this yardstick Andrew Wiles' proof of FLT fails to meet the required, evidence-based, criteria for entailing a true arithmetical proposition. Moreover, we offer two scenarios as to why/how Fermat could have laconically concluded in his recorded marginal noting that FLT is a true arithmetical proposition---even though he either did not (or could not to his own satisfaction) succeed in cogently evidencing, and recording, why FLT can be treated as an evidence-based, pre-formal, arithmetical truth (presumably without appeal to properties of real and complex numbers). It is primarily such a putative, unrecorded, evidence-based reasoning underlying Fermat's laconic assertion which this investigation seeks to reconstruct; and to justify by appeal to a plausible resolution of some philosophical ambiguities concerning the relation between evidence-based, pre-formal, truth and formal provability.
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Archival date: 2021-06-07
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