Does Water Obey Fermat's Last Theorem?

Does Water Obey Fermat’s Last Theorem? Could Fermat’s Unrecorded ‘Proof ’ FUP of FLT ‘validate’ string theories? Bhupinder Singh Anand✯ May 18, 2024 1. Fermat’s Last Theorem Fermat’s Last1 Theorem FLT states that no three positive integers x, y, z satisfy the equation xn +y n = z n for any integer value of n greater than 2. However, we shall consider here only the equivalent form of FLT2 , which states that no three positive integers x, y, z satisfy the equation xp + y p = z p for any prime p greater than 2. 1.A. Does the volume of water obey Fermat’s Last Theorem? Indisputably, given any two cubic water tanks T1 and T2 of sides t1 and t2 where t1 , t2 ∈ N,3 we can always construct a cubic water tank T3 of side t3 ∈ R whose volume equals that of T1 + T2 . The question arises: Query 1. Can we further claim that the volume Z of water in tank T3 , when completely full, equals the sum of the water volumes X in tanks T1 , and Y in T2 when the latter too are completely full? To appreciate the significance of Query 1, we note that—under our current mathematical representation of water as uniquely identified by a distinctive molecular structure—each of the cubic water tanks T1 , T2 and T3 can only hold a finite number of molecules of water when completely full. We shall see why this entails that we cannot assume X = t31 , Y = t32 , Z = t33 . For the purposes of this argument, we shall not distinguish between the two chemical forms of water molecules4 , but treat them as discrete objects occupying identical ‘effective volumes’ in the mathematical model under consideration; where we further disregard differences due to pressure, temperature, or other extraneous factors that do not affect the property of ‘discreteness’ which constitutes the focus our argumentation. Moreover, we shall tentatively assume that the volume ‘effectively occupied’ in a continuous 3-D Euclidean space by an individual molecule of water can be represented mathematically as ( kp )p , where k ∈ N, p is an unspecified, but specifiable, prime, and kp ∈ Q is treated as a fundamental constant ✯ Email: [email protected] For the significance of the adjective ‘Last’, see ➜2. 2 “To prove Fermat’s Last Theorem for all values of n, one merely has to prove it for the prime values of n. All other cases are merely multiples of the prime cases and would be proved implicitly.” . . . Singh: [2], p.99. See also [1], p.448, THEOREM 0.5. Suppose that up + v p + wp = 0 with u, v, w ∈ Q and p ≥ 3, then uvw = 0. 3 We denote the domain of the natural numbers by N; that of the rationals by Q; and that of the real numbers by R. 4 See https://en.wikipedia.org/wiki/Water. 1 1. 2 Fermat’s Last Theorem of nature (akin, for instance, to the reduced Planck constant ℏ5 ), which could further be treated as denoting a unit of discreteness for any dimensions associated with a water molecule6 . It would then immediately follow that if k, m, n, x, y ∈ N, p is a prime and x, y are co-prime7 , then we can always find x, y ∈ N such that: k p p (a) The total volume X of a set of m = ( px k ) water molecules, each of volume ( p ) , is p x ; k p p p (b) The total volume Y of a set of n = ( py k ) water molecules, each of volume ( p ) , is y ; and k p p (c) The total volume Z of the set of m + n = ( pz k ) water molecules, each of volume ( p ) , p is z ; whence: (d) (X + Y = Z) ↔ (xp + y p = z p ); if, and only if, z ∈ N and k = 1, or z ∈ N and k = p (i.e., the equivalence holds in either case). py p p Reason: If p is a prime, k ∈ N, and x, y ∈ N are co-prime, then (m = ( px k ) ), (n = ( k ) ), ((m + n) = pz p ( k ) ) ∈ N if, and only if, z ∈ N and k = 1, or z ∈ N and k = p (since no prime factor q of k > 1 can be a factor of both x and y in the latter case). Hence k = 2, since 32 + 42 = 52 if p = 2 in the latter case; which immediately entails FLT8 . Reason: If x, y, k ∈ N and x, y are co-prime, such that xp + y p = z p is solvable if, and only if, z ∈ N and k = 1, or z ∈ N and k = p, then p ̸= 2 in the latter case entails the contradiction that x2 +y 2 = z 2 is not solvable9 . We conclude that: Conclusion 1: In any mathematical model that admits molecular phenomena, we can consistently claim only that the volume Z of water in T3 , when completely full, equals the sum of the water volumes X in T1 and Y in T2 when the latter too are completely full if, and only if, (X +Y = Z) ↔ (xp +y p = z p for some prime p and x, y, z ∈ N only if p = 2). Moreover, even if we ignore ‘packing’ issues10 , and assume that the volume of each tank in Query 1 can be treated as the volume of the water it can contain when completely full, it would still follow from the above argumentation that: Conclusion 2: In any mathematical model that admits molecular phenomena, we cannot consistently claim that the volume of water in T3 , when completely full, equals the sum of the water volumes in T1 and T2 when the latter too are completely full. 1.B. Could Fermat’s Unrecorded ‘Proof ’ FUP of FLT ‘validate’ string theories? “The issue of the radius of the electron is a challenging problem of modern theoretical physics. The admission of the hypothesis of a finite radius of the electron is incompatible to the premises of the theory of relativity. On the other hand, a point-like electron (zero radius) generates serious mathematical difficulties due to the self-energy of the electron tending to infinity.[88] . Observation of a single electron in a Penning 5 Whose numerical value has been adopted as a rational number (i.e., in Q, albeit with a finite decimal representation) in the 2019 redefinition of the SI base units. See https://en.wikipedia.org/wiki/Planck constant. 6 Such as, for instance, its molar volume. See https://en.wikipedia.org/wiki/Water. 7 Two natural numbers are defined as co-prime if, and only if, they have no common factor except 1. 8 The significance of this is highlighted in ➜2., paragraphs (a) and (b). 9 Which, further, answers the question raised in ➜2., paragraph (ii). 10 As addressed by the Kepler Conjecture; see https://en.wikipedia.org/wiki/Kepler conjecture. B S Anand, Fermat’s ‘Lost’ Argument 3 trap suggests the upper limit of the particle’s radius to be 10−22 meters.[89] The upper bound of the electron radius of 10−18 meters[90] can be derived using the uncertainty relation in energy.” . . . https://en.wikipedia.org/wiki/Electron. Downloaded 11th May 2024. With hindsight, the above now suggests Fermat too might conceivably have argued that the smallest possible object in the universe must have a finite volume; and essentially soliloquized (see ➜2., paragraph (i)) that: FUP: In any mathematical model of a ‘universe’ Up , where p is an unspecified, but specifiable, prime; and a fundamental particle is not treated as a (problematic) point particle, but as a generalised p-D hypercube11 of side p2 and hypervolume ( p2 )p —where p2 is treated as a fundamental constant ∈ Q12 —we cannot have sets of fundamental particles X, Y, Z—with corresponding hypervolumes xp , y p , z p —and 2 p px p 2 p py p p py p pz p natural numbers ( px 2 ) , ( 2 ) , ( 2 ) , where x, y, z are co-prime, such that ( p ) ( 2 ) + ( p ) ( 2 ) = p p p p ( p2 )p ( pz 2 ) (i.e., x + y = z ) if p ̸= 2. Comment 1. We note that forbidding ‘point’ particles—as in FUP—is a premise of any string theory13 that admits fundamental particles as representable mathematically only by ‘open’ and ‘closed’ strings (which can be treated as ‘existing’ in a continuous, multi-dimensional, physical space, of which we can only directly experience three physical dimensions plus the passage of time). Moreover, although represented mathematically as ‘one-dimensional’ (open) or two-dimensional (closed), such strings can be viewed as implicitly admitting of an associated—even if only ‘notional’ or ‘virtual’—mathematical volume; unless a string is treated as having zero ‘cross-section’—which would, however, again entail a curious assumption akin to the forbidden assumption of ‘point’ particles! However, admitting a ‘notional’ or ‘virtual’ volume for a string entails that, as it propagates through spacetime, it not only ‘sweeps out a two-dimensional surface called its worldsheet’14 , but also a ‘notional’ or ‘virtual’ three-dimensional mathematical object that could be treated as its ‘worldwake’. Comment 2. Since xn + y n = z n , where x, y, z, n ∈ N, is solvable only for n = 1 (trivially) and n = 2, but not for n = 0 and n > 2, the hypothetical FUP suggests not only that fundamental particles cannot be represented mathematically as having 0 dimensions but, moreover, can only be represented mathematically as having dimensions 1 or 2, as is suggested by string theories that admit both ‘open’ (one-dimensional) or ‘closed’ (two-dimensional) strings. Comment 3. Since p2 = 1 in FUP, the natural number 1 could thus be treated as representing a physical constant15 of nature, which could further be treated as an ‘absolute’ unit of discreteness for the mathematical ‘volume’ of a string; which would suggest that the string could be treated mathematically as having an associated finite, ‘notional’ or ‘virtual’, mathematical ‘cross-section’. Moreover, the question—whose ambit lies beyond the scope of this article—arises: Query 2. Can treating natural phenomena as representable mathematically in a (continuous) continuum, which admits ‘point’ particles unqualifiedly as having zero, ‘notional’ or ‘virtual’, mathematical volume introduce unsuspected inconsistencies into the mathematical model? 11 cf., https://en.wikipedia.org/wiki/Hypercube. Akin to the reduced Planck constant ℏ; whose numerical value has been adopted as a rational number (i.e., in Q, albeit with a finite decimal representation) in the 2019 redefinition of the SI base units. 13 cf., https://en.wikipedia.org/wiki/String (physics). 14 cf., https://en.wikipedia.org/wiki/String (physics). 15 cf., https://en.wikipedia.org/wiki/Physical constant. 12 2. 4 2. The origin and epistemological status of Fermat’s Last Theorem The origin and epistemological status of Fermat’s Last Theorem Fermat’s Last Theorem FLT states that no three positive integers x, y, z satisfy the equation xn +y n = z n for any integer value of n greater than 2. FLT has been made famous, literally and literarily (see [2], p.73) beyond it’s innate challenge for mathematicians, by Pierre de Fermat’s posthumously revealed remarks, written around 1637 in the margin of his copy of Diophantus’ Arithmetica: “It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum of two like powers. . . . I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain”. . . . Singh: [2], p.66, An English translation of Fermat’s marginal noting in Latin. For 358 years, FLT remained unproven; until the arcane 108-page proof [1]—appealing inexplicably to geometrical properties of real and complex numbers in order to prove an essentially arithmetical problem over the natural numbers—was published in 1995 by Andrew Wiles in the Annals of Mathematics. It proved an equivalence between geometric properties of elliptic curves and, seemingly disparate, modular forms that could cogently be argued as entailing FLT from their explicit (and implicitly set-theoretical) premises. However, Wiles’ proof leaves two questions unaddressed: (i) What argument or technique might Fermat have used that led him to, even if only briefly, believe he had ‘a truly marvellous demonstration’ of FLT? “Wiles’s proof of Fermat’s Last Theorem relies on verifying a certain conjecture born in the 1950s. The argument exploits a series of mathematical techniques developed in the last decade, some of which were invented by Wiles himself. The proof is a masterpiece of modern mathematics, which leads to the inevitable conclusion that Wiles’s proof of the Last Theorem is not the same as Fermat’s. Fermat wrote that his proof would not fit into the margin of his copy of Diaphantus’s Arithmetica, and Wiles’s 100 pages of dense mathematics certainly fulfills this criterion, but surely the Frenchman did not invent modular forms, the Taniyama-Shimura conjecture, Galois groups, and the Kolyvagin-Flach method centuries before anyone else. If Fermat did not have Wiles’s proof, then what did he have?” . . . Singh: [2], p.307. (ii) Why is xn + y n = z n solvable only for n = 2?16 A curious feature of recorded, post-Fermat, attempts to prove FLT has been the, seemingly universal, focus on seeking a formal proof, and understanding, of only (as claimed by Fermat) why xn +y n = z n is unsolvable for both specific, and general, values of n > 2 when x, y, z, n ∈ N. Comment 4. ‘Curious’ since, for instance, if FLT is not provable in the first-order Peano Arithmetic PA, it would follow17 that no deterministic algorithm TM could, for any specified n > 2, evidence that xn + y n = z n is unsolvable. In which case, even if FLT can be evidenced as numeral-wise true 18 under a well-defined interpretation of PA over N, seeking to understand why xn + y n = z n is unsolvable generally 19 for all n > 2 may be futile. Instead, one could reasonably expect a better insight by seeking why xn + y n = z n is solvable for n = 2 (and trivially for n = 1), but not for n ̸= 2. 16 The Diophantine equation is, of course, trivially solvable for n = 1; and solvable for n = 2 by Pythagoras’ Theorem. By [3], Theorem 7.1 (Provability Theorem for PA): A PA formula [F(x)] is PA-provable if, and only if, [F(x)] is algorithmically computable as always true in N. 18 Defined formally as an algorithmically verifiable truth (see [3], Definition 1). 19 Defined in this context as entailing an algorithmically uncomputable truth (see [3], Definition 2). 17 B S Anand, Fermat’s ‘Lost’ Argument 5 The hypothetical, one-paragraph, elementary ‘reconstruction’ of Fermat’s Unrecorded ‘Proof’ FUP in ➜1.B., of what Fermat could, conceivably, have argued but found both: ❼ too long initially to record in the margin of his copy of Diophantus’ Arithmetica; and, ❼ not formally convincing enough as a proof of FLT on further reflection; now suggests an alternative to post-Wiles wisdom which: - despite the absence of any well-defined (finitary) interpretation of Wiles’ argument in [1] that would validate it as—at the very least—an algorithmically verifiable arithmetical truth (in the sense of [3], Definition 1); - dismisses Fermat’s claim of ‘a truly marvellous demonstration’ as unjustified ; - lacking even a plausible argument for treating FLT as a ‘pre-formal’ mathematical truth in Markus Pantsar’s sense in [4]. Comment 5. The significance of a plausible ‘pre-formal’ argument for FLT is that it would serve to ‘justify’ the search for not only a formal proof of theoretical conjectures such as FLT (as eventually evidenced by the success of Wiles’ relentless pursuit of the problem, reportedly over decades), but also for the further search for a ‘post-formal’, evidence-based, Tarskian interpretation that would establish the Theorem as a mathematical truth 20 ; one whose necessary finitarity (cf., [3], Theorem 7.1: Provability Theorem for PA) could well-define the various mathematical models sought, and relied upon as definitive, by the applied sciences (especially when a model appeals to a formal proof that cannot yet claim to be a mathematical truth). In other words: (a) Whereas Wiles’ arcane proof [1] considers only a mathematical interpretation (model) of FLT over continuous, 2-dimensional, objects (elliptic curves) and their Galois representations; concluding that constraints on scalar properties (genus and level of associated modular forms) associated with these objects, in a continuous 2-D Euclidean space, entail FLT as mathematically proven; (b) Query 1 in ➜1.A., and the hypothetical reconstruction of Fermat’s Unrecorded ‘Proof’ FUP of FLT in ➜1.B., consider physical interpretations (gedanken) of FLT over discrete, 3-dimensional, objects (finite sets of molecules/putative volumes associated with fundamental particles), which validate Wiles’ proof as an arithmetical truth by finitarily showing that, and why, constraints on scalar properties (volumes) associated with these mathematical objects in a continuous 3-D Euclidean space entail, and are entailed by, FLT. 3. Can Wiles’ proof be treated as an arithmetical ‘truth’ ? A question occasionally raised, but seemingly lacking the foundational appeal it merits (see [5]), arises: Query 3. Can Wiles’ proof be treated as an arithmetical ‘truth’ ? To place Query 3 and Michael Harris’ argumentation [5] in perspective, we first address the need for recognising the primacy of pre-formal reasoning—and of pre-formal ‘truth’ and ‘proof’—as argued cogently, and unequivocally, by Markus Pantsar in [4]: 20 The need for which is occasionally sought to be diminished; see, for instance, [5]. 3. 6 Can Wiles’ proof be treated as an arithmetical ‘truth’ ? “In this work I will argue that without any outer reference, mathematics as we know it could simply not be possible: it could not have developed, and it could not be learnt or practised. Sophisticated formal theories are the pinnacle of mathematics but, philosophically, they cannot be studied separately from all the non-formal background behind them. This way, what might seem like a completely formalist theory of mathematics turns out to be nothing of the sort. It could not have existed without a wide pre-formal background, which we will see when we examine mathematical practice in general.3 Formal systems are not of the self-standing type that extreme formalism seems to claim. My purpose in this work is to show that the formalist program uses the actual practice of mathematics as a ladder that they later discard. This by itself is of course perfectly acceptable, and it mirrors the way we strive for formal axiomatic systems in mathematics. What is not acceptable is how they refuse using the ladder. When it comes to the question of truth and proof, this could not be any more relevant. The deflationist truth of extreme formalism equates mathematical truth with formal proof. However, as we will see, that strategy requires that we take mathematics to concern only formal systems. Once we look at the wider picture, we see that outer criteria are needed to avoid arbitrariness. Theory choice must be explained, and this requires reference outside formal systems of mathematics. Philosophers have tried to explain this by a wide array of concepts—usefulness, assertability, consistency and conservativeness, to name a few—but ultimately none of them have been satisfactory. The only plausible way to answer the problem of theory choice, I will argue, is by appealing to truth. [3] What I refer to as pre-formal mathematics in this work is more often discussed as informal mathematics in literature. The choice of terminology here is based on two reasons. First, I want to stress the order in which our mathematical thinking develops. We initially grasp mathematics through informal concepts and only later acquire the corresponding formal tools. Second, the term “informal mathematics” seems to have an emerging non-philosophical meaning of mathematics in everyday life, as opposed to an academic pursuit—which is not at all the distinction that I am after here.” . . . Pantsar: [4], ➜1.1 General background. “(Extreme) Formalism: to say that a mathematical sentence is true involves no reference to any entity outside formal systems. Hence, a mathematical sentence is true in a formal system S if and only if it is provable in S, and mathematical truth cannot be discussed in any other context.” . . . Pantsar: [4], ➜2.4 Formalism/nominalism. The significance of, and need for, Pantsar’s explicit distinction between formal and pre-formal proofs of mathematical propositions is highlighted by Harris’ recent questioning, in [5], of the necessity for a foundational perspective that would justify why Wiles’ proof [1] of FLT may be treated putatively as a logically true arithmetical proposition: “After Wiles’ breakthrough, it became common to hear talk of a new “golden age” of mathematics, especially in number theory, the field in which the Fermat problem belongs. The methods introduced by Wiles and Taylor are now part of the toolkit of number theorists, who consider the FLT story closed. But number theorists were not the only ones electrified by this story. I was reminded of this unexpectedly in 2017 when, in the space of a few days, two logicians, speaking on two continents, alluded to ways of enhancing the proof of FLT—and reported how surprised some of their colleagues were that number theorists showed no interest in their ideas. The logicians spoke the languages of their respective specialties—set theory and theoretical computer science—in expressing these ideas. The suggestions they made were intrinsically valid and may someday give rise to new questions no less interesting than Fermat’s. Yet it was immediately clear to me that these questions are largely irrelevant to number theorists, and any suggestion that it might be otherwise reflects a deep misunderstanding of the nature of Wiles’ proof and of the goals of number theory as a whole. The roots of this misunderstanding can be found in the simplicity of FLT’s statement, which is responsible for much of its appeal: If n is any positive integer greater than 2, then it is impossible to find three positive numbers a, b and c such that an + bn = c n This sharply contrasts with what happens when n equals 2: Everyone who has studied Euclidean geometry will remember that 32 + 42 = 52 , that 52 + 122 = 132 , and so on (the list is infinite). Over the last few centuries, mathematicians repeatedly tried to explain this contrast, failing each time but leaving entire B S Anand, Fermat’s ‘Lost’ Argument 7 branches of mathematics in their wake. These branches include large areas of the modern number theory that Wiles drew on for his successful solution, as well as many of the fundamental ideas in every part of science touched by mathematics. Yet no one before Wiles could substantiate Fermat’s original claim.” . . . Harris: [5], Other publications, #21. Since the aim of this brief commentary is only to suggest why the foundational, cross-disciplinary, issues raised by Query 1, by ➜1.B., by Query 2, and by Query 3 might merit more serious consideration, we conclude by simply noting without further comment that, prima facie, Harris seems to hold the, epistemologically fragile, belief that ‘the simplicity of FLT’s statement’ and, presumably, the seeming straightforwardness of his concise outline (reproduced below) of the argument underlying Wiles’ proof—covering ‘large areas of the modern number theory that Wiles drew on for his successful solution, as well as many of the fundamental ideas in every part of science touched by mathematics’— should suffice for establishing FLT informally (seemingly also pre-formally in Pantsar’s sense) as a logically true arithmetical proposition that ‘could substantiate Fermat’s original claim’: “. . . Wiles’ proof, complicated as it is, has a simple underlying structure that is easy to convey to a lay audience. Suppose that, contrary to Fermat’s claim, there is a triple of positive integers a, b, c such that (A) ap + bp = c p for some odd prime number p (it’s enough to consider prime exponents). In 1985, Gerhard Frey pointed out that a, b and c could be rearranged into (B) a new equation, called an elliptic curve, with properties that were universally expected to be impossible. More precisely, it had long been known how to leverage such an elliptic curve into (C) a Galois representation, which is an infinite collection of equations that are related to the elliptic curve, and to each other, by precise rules. The links between these three steps were all well-understood in 1985. By that year, most number theorists were convinced—though proof would have to wait—that every Galois representation could be assigned, again by a precise rule, (D) a modular form, which is a kind of two-dimensional generalization of the familiar sine and cosine functions from trigonometry. The final link was provided when Ken Ribet confirmed a suggestion by Jean-Pierre Serre that the properties of the modular form entailed by the form of Frey’s elliptic curve implied the existence of (E) another modular form, this one of weight 2 and level 2. But there are no such forms. Therefore there is no modular form (D), no Galois representation (C), no equation (B), and no solution (A). The only thing left to do was to establish the missing link between (C) and (D), which mathematicians call the modularity conjecture. This missing link was the object of Wiles’ seven-year quest. It’s hard from our present vantage point to appreciate the audacity of his venture. Twenty years after Yutaka Taniyama and Goro Shimura, in the 1950s, first intimated the link between (B) and (D), via (C), mathematicians had grown convinced that this must be right. This was the hope expressed in a widely read paper by André Weil, which fit perfectly within the wildly influential Langlands program, named after the Canadian mathematician Robert P. Langlands. The connection was simply too good not to be true. But the modularity conjecture itself looked completely out of reach. Objects of type (C) and (D) were just too different.” . . . Harris: [5], Other publications, #21. 3. 8 Can Wiles’ proof be treated as an arithmetical ‘truth’ ? References [1] Andrew Wiles. 1995. Modular Elliptic Curves and Fermat’s Last Theorem. In Annals of Mathematics, Second Series, Volume 141, No. 3 (May, 1995), pp.443-551 (109 pages), Princeton University, Princeton, New Jersey, USA. doi:10.2307/2118559 http://math.stanford.edu/ lekheng/flt/wiles.pdf [2] Simon Singh. 1997. Fermat’s Last Theorem. Harper Perennial, 2005, Harper Collins, London, UK. [3] Bhupinder Singh Anand. 2016. The truth assignments that differentiate human reasoning from mechanistic reasoning: The evidence-based argument for Lucas’ Gödelian thesis. In Cognitive Systems Research. Volume 40, December 2016, 35-45. doi:10.1016/j.cogsys. 2016.02.004. [4] Markus Pantsar. 2009. Truth, Proof and Gödelian Arguments: A Defence of Tarskian Truth in Mathematics. In Eds. Marjaana Kopperi, Panu Raatikainen, Petri Ylikoski, and Bernt Österman, Philosophical Studies from the University of Helsinki 23, Department of Philosophy, University of Helsinki, Finland. https://helda.helsinki.fi/bitstream/handle/10138/19432/truthpro.pdf?sequence=2 [5] Michael Harris. 2019. Why the Proof of Fermat’s Last Theorem Doesn’t Need to Be Enhanced. In Quanta Magazine, June 2019. See Other publications, #21, on author’s web-page (Downloaded 15/05/2021). http://www.math.columbia.edu/ harris/website/publications