The Cosmic Ensemble1:Reflections on the Nature–Mathematics

SymbiosisJOSEPH ALMOG

Mathematics spellbinds us (philosophers). We sing its praises—“it is purepoetry,” says Kant—but at the same time, we treat it as a delphic stranger—

though mathematics asserts nothing but truths, we have no idea about what, viz.what are the objects making the truths true. One problem here is as old as the hills,and it simmers in the exchanges between Theaetetus and Plato. It is encapsulatedin the following impossibility thesis, which they both adhered to, the impossibility ofunity thesis:

(I) We cannot integrate in a single unitary realm (“nature”) the physicalentities of the cosmos and those of mathematics.

The claim does have an oracular feel to it. We all believe (I) is true, but at the sametime we cannot see—for mathematics pervades our ordinary lives—how (I) couldbe true. And so, we are led to ask: where does the impossibility come from?

A trenchant diagnosis—one I believe isolates the ur-problem—is offered ina dialog (across time) between Kurt Gödel and Charles Hermite. In closing his1951 Gibbs lecture On Some Basic Theorems in the Foundations of Mathematics,Kurt Gödel quotes from the eminent nineteenth-century French number theorist:

1. This is part I of a duo of papers. Part II will appear elsewhere. Before the reader plungesin, I would urge him to have a look at the last, autobiographical note (note 35). This may explain(if not attenuate) the hardships likely to accrue from the reading. For the philosopher, the workmay well seem overly technical, and for the mathematically inclined overly philosophical. Idedicate the paper to the unforgettable teacher, the late Serge Lang.

Midwest Studies in Philosophy, XXXI (2007)

© 2007 Copyright The AuthorsJournal compilation © 2007 Blackwell Publishing, Inc.

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There exists, unless I’m mistaken, an entire world consisting of the totality ofmathematical truths, which is accessible to us only through our intelligencejust as there exists the world of physical realities; each one is independent ofus, both of them divinely created.

So runs Gödel’s main text. In a footnote, he tells us that Hermite’s passage con-tinues as follows:

. . . and appear different only because of the weakness of our mind; but for amore powerful intelligence, they’re one and the same thing, whose synthesisis partially revealed in that marvelous correspondence between abstractmathematics on the one hand and astronomy and all branches of physics onthe other.

About this, Gödel comments,

So here, Hermite seems to turn towards Aristotelian realism. However, hedoes so only figuratively, since Platonism remains the only conception under-standable to the human mind.2

I am driven by Hermite’s integrative instinct (“they are one and the same thing”)but with one amendment, on which perhaps everything turns. Hermite fears that inapprehending truths, we—unlike Him—are subordinated to an epistemic dualism,an inevitable split between the senses and the intellect.This epistemic dualism soonbreeds a metaphysical dualism of two realms of being (“worlds”). My amendmentis that in the pertinent respect, we are just like Him. Apprehension of the nature–mathematics symbiosis (which I prefer to “synthesis” for a reason yet to emerge) isnot reserved to the sense-free unembodied pure intellects of angelic and divineminds. Even our “weak minds” can and indeed must—for mathematics is some-thing we live by—achieve cognition of the integrated ensemble.

SYNOPSIS OF PARTS I–II—HOW TO UNDO THE DUALISMS?

The project below—and I hope this does not sound overly naïve ormegalomaniac—is to consider manners of untying a complex knot we philosophershave bound ourselves by. I say “we philosophers” because the knot binds not thosewho practice mathematics but those who analyze-theorize the practice. This splitbetween the one who lives the phenomenon and the one who theorizes about it isreminiscent of another such self-inflicted dualism philosophical thinking-and-analysis has nurtured—the bond René Descartes called the mind–body union.

2. The quotations are from the last page (p. 323) of Gödel’s paper, “Some Basic Theorems inthe Foundations of Mathematics,” in The Collected Works of Kurt Gödel, vol. III, Oxford Univer-sity Press, 1995. I should like it noted that where the English translator makes Hermite say “anentire world consisting of the totality of mathematical truths,” the French uses the pregnant term“ensemble,” as in “tout un monde qui est l’ensemble de verités mathematiques.”

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Here too, once we are bound, we seek understanding by analysis of the knot.And analysis leads us either to (i) sublimation—a segregated realm totally inde-pendent of (immortal) minds or to (ii) reduction—all there is is purely material-nature, with minds gone by the board. In tow, we shackle ourselves again withan (I)-like impossibility result that precludes any third way of (mind–body)integration-without-reduction.

Reflection on the mind–body interface suggests the same order of causes asengendering the “crisis.” Each of us is first struck by the fundamentally differentways of knowing, in the first person, (1) that I am thinking versus (2) that I have twohands. Soon, we find ourselves segregating two correlative realms of being, one asublimated realm of immaterial (and destined for eternity and immortality) minds,the other the mundane realm of material bodies. In turn, we proclaim that the onlygenuine knowledge to be had—propositions we see clearly and distinctly—is of thesublimated realm—for example, that I am thinking. It is now incumbent on us toderive from such transparent premises, by stepwise deductions, in effect, a stepwisereduction, our mundane knowledge, for example, that I have two hands.

It is against this background of separatist sublimation and a doctrine ofdeductive–reductive foundationalism that Descartes offered Princess Elizabeth hisalternative model—symbiotic interdependence of (read with the hyphens) mind-and-body, familiar to us all from humdrum existence. Descartes encapsulates it tothe princess in this one sentence:

Finally, it is by relying on life and ordinary conversations, and by abstainingfrom meditating and studying things that exercise the imagination, that welearn how to conceive the union of mind and body . . . the notion of the unionthat each of us has inside himself without philosophizing: that he(she) is asingle person that has together a body and thought, which by nature are suchthat the thought can move the body and feel the accidents that happen to it.3

I would like to pursue a similar defusing method vis-à-vis mathematical dualism. Itis thus that I distinguish below between (1) a pre-philosophical (“sans philoso-pher”) mathematics as practiced in situ and (2) mathematics as studied in vitro byreconstructive philosophers.

The in vitro studies are dominated, right from the outset, as in the Gödel–Hermite exchange, by epistemological dualism. And twice over. First, we segregatebetween our knowledge of the physical world (“astronomy”)—knowledge aposteriori—and that of mathematics—knowledge purely a priori. Next, inside thesublimated domain of the a priori, to secure all of mundane practiced mathematicaltruths the mark of clear and distinct apriority, we launch an intra-mathematical

3. See The Philosophical Writings of René Descartes, edited by J. Cottingham et al., vol. III,letter to Princess Elizabeth, June 28, 1643. In the French original:“Et enfin, c’est en usant seulementde la vie et des conversations ordinaires, et en s’abstenant de méditer et d’étudier aux choses quiexercent l’imagination, qu’on apprend à concevoir l’union de l’âme et du corps . . . la notion del’union que chacun éprouve toujours en soi-même sans philosopher; à savoir qu’il est une seulepersonne, qui a ensemble un corps et une pensée, lesquels sont de telle nature que cette pensée peutmouvoir le corps, et sentir les accidents qui lui arrivent.”

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cleansing program. We are to recall here Frege’s regulative ideal of “deductionwithout gaps” or the Zermelo–Gödel deductions of all of practiced mathematics inthe universal characteristic of set theory. Such reconstructive programs are meantto unravel the many interrelated subject matters of mathematics as all logic/settheory in disguise. And therein—in this “deep meaning” exposing reductions—liesthe key to the transparent a priori knowledge of mathematics. However we actuallycame to know this or that theorem of Diophantine geometry or algebraic topology,however we discovered it, its ultimate justification, now with its real logical/settheoretic meaning unraveled, lies in a deduction of that “deep meaning” fromself-evident a priori truths. We are led to Figure 1.

So much for the various segregations in the picture of the in vitro philosopherof mathematics. In contrast, the mathematics practiced in situ, the mathematics welive by, is a mathematics without borders. To echo Descartes’ above reciprocitytheme to the Princess, each of us feels, sans philosopher, that there is a single naturewith both mathematics and physical things hanging together so that each dependson the other. We might call it, following Descartes, l’union vécu, the mathematics–nature union lived and experienced.4

This lived union leads us to pursue a single-realm mathematics–nature inte-gration. And thrice over. First, ur-structures of mathematics are grounded in—andessentially so, viz. owing their existence to—the infrastructure of cosmic nature(see, e.g., on Riemann surfaces in the next paragraph). Second, various branches ofmathematics (e.g., see below for the cases of number theory, complex analysis,algebraic geometry) are strongly interdependent. Third and last, mathematicallogic (set theory) is not a universal characteristic into which all of mathematics isto be reduced; logic and set theory are rather just more mathematics. To invert thereductionist’s motif—mathematics is not a branch of set theory (logic); set theory(logic) is a branch of mathematics. In all, Figure 2 emerges.

4. The French adjective “vécu” connotes both the English “lived” and “experienced.” I willharp both on the living-it and the experiencing of it.

2- The Higher Realm

Logic (Set Theory) [A priori]

Reduction Reduction

Arithmetic Analysis Topology Geometry

1- The Lower Realm

Cosmic Nature [A posteriori]

Figure 1. Separatist Dualism.

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TWO SYMBIOTIC THESES

As I start drawing the symbiotic nature–mathematics picture, let me articulatethe two guiding ideas I am after. The first concerns the mathematics/nature unionproper, the second our apprehension thereof. First, at the level of existence andtruth (“metaphysics”), I argue that mathematics exists in nature—not in anyother realm—and the material cosmos has already in it all the structures ofmathematics. In the case of the specific ur-mathematical kind of structure I men-tioned (Riemann surfaces), I submit the following symbiotic thesis: no materialcosmos without the complex plane and no complex plane without the materialcosmos.

I beg the professional philosophical reader not to rush and translate thisimmediately into questions about ontological categories (“Is JA saying that thecomplex numbers are objects? Or properties (modes) of material objects?Or . . .”). Putting questions of objecthood/properties on the back burner, I wantus to linger on just this claim of essential interdependence: no material cosmoswithout engendering mathematics and no mathematical structures without theengendering infrastructure of nature. I will encapsulate this conception of math-ematical notions and truths as engendered by cosmic structure as generativemathematics.

The second and perhaps more critical tenet I would like to propose concernsour cognition (thinking of) mathematical notions and, eventually, mathematicalepistemology (our knowledge and justification of mathematical truths). The thesisI am after is this: Our apprehension of mathematical notions is inseparable fromcosmic contact. And twice over: (1) Our mathematical cognition and knowledge is

Logic

Set theory

Generation Generation

Algebraic Geometry ! Analysis ! Topology

Generation Generation Generation

Complex Plane (Riemann Surfaces)

Generation

Cosmic Nature (Spatio-Temporal Infrastructure)

Figure 2. Integrative Unification.

348 Joseph Almog

inextricably linked to our cognition and knowledge of the material cosmos and (2)our cognition and knowledge of one mathematical domain (e.g., below, Diophan-tine equations) is inextricably linked to our cognition and knowledge of othermathematical domains, for example, complex analytic and group theoretic facts.Without lived human experience—of (1) the infrastructure of the cosmos and (2)other findings by fellow human mathematicians across human history, we could notcognize and know the mathematics we do cognize and do know. I will characterizethis thesis as evolutionary mathematical cognition.

Our second thesis should not be hurriedly forced into the philosophicallystandard exclusionary vocabulary of a priori/a posteriori knowledge (cognition).The notion of a priori/a posteriori is multiply ambiguous. Let us remind ourselvesof, for example, three famous glosses thereof—Descartes’ “known by the light of(our) nature,” “Kant’s “known independently of sense experience,” and Frege’s“knowledge justified by reasoning through absolutely general propositions.” Onthe first notion, mathematical cognition, as I will pursue it, may be taken as a priori,for it does emanate from our very cognitive nature, prior to and independent of anyspecific sense impressions we might have during our lifetime. But this is only tostress that the cognition is strictly dependent on our having the nature we have,itself a product of our living in nature and cognizing it so that cosmic contact withnature’s infrastructure is built into our cognitive system.

Moving on to the second gloss, it may well be that our mathematical cogni-tion is a priori in being quite independent of sensory experiences we are about tohave in our lifetime, although I will argue against this much as well—where we arein history, what communications impinge on us, what notions have emerged by thetime we live and think, will very much condition what cognitions we can attain andwhat knowledge we can come to have, let alone actually have.

Finally, relating to our third gloss, our mathematical knowledge is now dis-tinctly not a priori for it involves a deeply specific subject matter. Key to the caseI will discuss are fine-tuned generalizations about elliptic curves and modularforms, about finite abelian groups-generating curves, and about analytic functionson the complex plane, none of which is resolvable to purely general logical or settheoretic propositions.

I. THE COSMIC ENSEMBLE: INTRA-MATHEMATICAL SYMBIOSIS

The Roots of Dualism

Mathematical dualism (as early as Plato and Theaetetus and all the way to theabove Gödel–Hermite) is inextricably linked with mind–body dualism—it is ourconception of how we are cognitively structured and consequently how we appre-hend truths (e.g., as opposed to Him) that feeds into our segregating apart math-ematics and material (physical, cosmic) nature.

So let us linger, if only for a moment, on our own all-too-human dualism ofmind and body. We tend to teach our students the metaphysical dualism first, theseparation of two disjoint realms of existence—immaterial minds versus materialbodies. We then add on, as derivative, remarks on a consequent epistemological

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dualism—we know the bodies—our own or others’—by the senses, but we knowour own minds by sheer intellectual introspection (and not by the senses).5

So goes our common teaching. Earlier reflection on this hard case convincedme that the true chain of reasons in the human dualism case runs the other wayround. The primal dualism (in both senses of the adjective—temporal priority andfundamentality) is an epistemic dualism—the way we know our own psychologicalstates strikes us as totally—categorically—different from the way we know ourbody (and other bodies). To accommodate the split in modes of knowledge, we goon to posit correspondingly disjoint realms of objects to match the segregatedepistemological channels—a realm of bodies, to be known by the senses, versus arealm of (our own) minds, to be known by the sense-free intellect.6

In a similar vein, I see the chain of philosophical reasons running the otherway round with this new case of fundamental dualism—the mathematics versusnature segregation. Again, we teach the dualism as starting from an “obvious”bifurcation of ontological realms—the intra-cosmic versus the a-cosmic (or “pla-tonic”). We then go on to append matching cognizing channels, the senses (withwhich to know the cosmic) versus the intellect (with which to know the “other” or“platonic” realm). But, I would like to propose, it is rather the segregation apart ofthe cognitive channels that breeds the subsequent duality of realms of being.

The driving force—and I feel we are here driven rather than doing thedriving—behind our flight to a dualism of nature versus mathematics is a human-bound syndrome of desire to transcend what Hermite describes as our “weakcondition.” In confronting mathematics, we often speak of striving for under-standing but we are driven by a need—to control. And so, we sublimate thedesire into an epistemic regulative ideal. Be that as it may—a desire or anideal—we seek to transcend our (speaking with Hermite) “weak (human) con-dition” and attain a more perfect form of knowledge, a replica of His form ofknowledge (control). This leads us to isolate the senses as the impeding shackles.And it is thus that soon enough an epistemic dualism of senses versus intellect isengendered.

Recent technical philosophy—by which I understand the turning of thetheory of justification into a technical subject with its own internal architecture—transformed the quest for purely intellectual knowledge by introducing a technicalcorrelate notion—a priori knowledge. We transform the primal ur-desire forperfect knowledge into a technical program—show that each theorem of

5. I study mind–body Cartesian Dualism in my monograph What Am I?, Oxford UniversityPress, 2002.

6. I said we will only linger for a moment on our human dualism but let me just note that onmy reading of Descartes’ articulation (to the princess) of his “symbiotic way out,” what he urgesis a rethinking of our common idea (ironically often called “Cartesian”) that we (1) cognizeourselves and (2) know truths about ourselves, by the sheer intellect (recall the item to be knownis the full self, the human being Joseph Almog, not just this or that mental abstraction from JA).In a similar vein, the path followed below questions whether sheer reliance on the intellect is theway we (1) cognize the notions and (2) know (justify) the truths of mathematics. Another way ofsetting our question is this: Does the notion of cosmos-interaction-free (“sheer”) intellect makesense?

350 Joseph Almog

mathematical practice is justified a priori.This was Frege’s pronounced ideal and ithas become our regulative ideal.7

Why Foundationalism?

There is something of a puzzle in the automatic adoption of this regulative ideal.The puzzle is this. We adopt the ideal in spite of our exposure as high school orundergraduate (or higher yet) students to real practiced mathematics where justi-fication (including proof-giving) does not actually proceed according to founda-tionalist models. The experience each of us has with mathematical practice isrelegated to the level of the anecdotal—yes, indeed, that is how “they” (the

7. Where is this ideal coming from? I have no genuine acquaintance of ancient Greek textsto speculate so far back (although I have my doubts about repeated philosophical references toEuclid and, later, to Archimedes. Both seem to me rather experimental and anti-foundationalist intheir practice, as one would expect problem-solvers like them to be.About problem-solving Greekmathematics, see also R. P. Langlands, The Practice of Mathematics, lectures given at the Instituteof Advanced Study [and available in electronic form] 2000).

In postmedieval times, it is often said that a paradigm of this blueprint for perfect knowledgehad been provided by the writings of the great rationalists of the seventeenth century, for example,Descartes and Spinoza (I cannot speak of Leibniz’s work because I have not read it). On thistextbook rationalist model, it is said that knowledge of the self’s mental life, knowledge of God (ifsuch there be), and knowledge of morals, all domains where perfect knowledge had been deemedpossible, are to be based in the intellect only.And it is said that for the rationalists, mathematics hadcontinually served as the role model (e.g., both Descartes and Spinoza presented important ideasof theirs in “geometric” [axiomatic-seeming] form).

I must confess that I have not found this segregative rationalism in the seventeenth-centuryfigures (although interested in our ratio they surely were). Ironically, at least on my own reading,the role-model thinkers of this allegedly purely intellectual methodology, Descartes and Spinoza,were, each in his way, deeply committed naturalists, rather of the kind of Hermite, averse to anymultiplication of realms. For both Descartes and Spinoza, nature is one. They were also both“unromantic” about the source of our quest for perfect knowledge, tracing it back to our existentialsituation as mortally endangered beings prone to doubt and desire (a beautiful concise “Freudian”analysis is provided by Descartes late in Meditation III comparing the plight of man to God).I linger on this epistemological naturalism—unified in its approach to nature (includingmathematics)—in a forthcoming book called Cogito? (Oxford University Press, 2007).

I am not a historian of philosophy, but on my own subjective sense, encounters with dualisticepistemology multiply only in writings posterior to Kant (and as emphasized below and in Part II,Kant himself, with his beguiling focus on the form of spatio-temporal intuition, is not a paradigmof this dualism nor of foundationalism vis-à-vis mathematics). On my reading, the most shiningparadigm of the dualism is Frege, for example, in his (in my view, deeply unjustified) critique ofMill early in the Grundlagen, his insistence on a priori justification (“deduction without gaps”) foreach theorem and his crafting of a metaphysics of multiple realms to serve the dualistic episte-mology. Although Mill has been made the whipping boy of introduction classes in the philosophyof mathematics, I view his (true, not Frege-caricatured) practice-based and nonrevisionist view ofevidence accumulation as vindicated by the type of considerations developed below.

After I completed the present study and presented it orally in Europe, it was pointed out tome by the number theorist Michael Harris that similar observations about Mill are made by IanHacking in his “What Some Philosophers Have Learned from Mathematics,” British AcademyLectures 2000. Hacking’s paper pursues—in ways rather different from my own—other interestinganti-foundationalist themes arising from mathematical practice.

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mathematicians) actually did things on the blackboard or how I actually solved myhomework problem . . .

Autobiographical anecdotes about one’s practice provide no substitute fortheory. And so instead of recounting how I actually—in real history—came toknow (or be convinced of) this or that mathematical truth, we now focus on how Ior we in general, in principle we like to say, ought to justify this known truth.We distinguish the (historical) context of discovery from the (logical) context ofjustification.

A Case Study: Fermat’s Last Theorem

Let me use a ubiquitous (both mathematically and philosophically) case study tobring home the distinction—the (Frey–Serre–Ribet) Wiles–Taylor 1994 proof ofFermat’s last theorem (FLT).

Fermat’s theorem states that:

(FLT) for integers n , for all x, y, z nonzero integers x y zn n n> + "2

This is a simple statement that middle-school students have no problem under-standing. Of course, it had a long and famous history, in between 1637 and 1994, andthat makes it all the more a vivid and interesting case study. But my reasons forfocusing on it are different.8

8. Since FLT is our case study, I will say a word about what it says and about its actual contextof discovery. We need for its statement a few informal glosses of the pertinent notions. Andinformal they will be for we are philosophers (at least I am) and we address here a highly technicalset of results in number theory and so some level of generality (and genericity) is inevitable. Forlight but enlightening introductions to an amateur like the present writer, see Cox, “Introductionto Fermat’s Last Theorem,” American Mathematical Monthly 1 (1994): 3–14 or the very elegant C.Goldstein,“Le theoreme de Fermat,” La Recherche 263 (1994): 268–75 (in French). I here state thebare structure of what we need. I amplify on some technical notions in the appendix.

The main tool in the Wiles–Taylor 1994 proof is the theory of elliptic curves. An elliptic curveE over the field of rationals Q can be characterized as a set of solutions in Q of an equation ofthe form:

E: Y X aX bX c2 3 2= + + +

with a, b, c integers.The overall context of discovery of the proof is this.(i) The conjecture was known to be reducible to prime exponents and verified for p = 3(Euler).(ii) G. Frey argued in 1985 that if nonzero integers a, b, c greater or equal to 1 and n greateror equal to 5 verify an + bn = cn, then the associated elliptic curve

Y X(x a X b2 n n= # +)( )

is semistable (for the notions of semistable and conductor, see the appendix).(iii) A few years later, Ribet showed that an elliptic curve associated with a hypotheticalsolution of FLT cannot be modular (see the appendix for Serre–Ribet work).

352 Joseph Almog

Why Not the Homier 7 + 5 = 12?

It would have been more pleasant and pedagogically better to make the likes of therudimentary 5 + 7 = 12 our case study. And indeed for some of the features I’d liketo bring to the fore, 5 + 7 = 12 suffices. Not least among them are the following fourfeatures—henceforth the rudimentary quartet—true of our living by and convers-ing about 5 + 7 = 12:

(i) not by a logical/set theoretic object—the arithmetic fact in question is notmade to hold by relations among pure sets or (Frege’s) concept-extensions.

(ii) not by logical/set theoretic meaning—what we understand and communicateto one another in using this arithmetic sentence in language is not a settheoretic/logical meaning.

(iii) not by cosmos-free logical/set theoretic cognition—we need intra-cosmic contact to cognize (have in our mentality) the notions of 5, 7, 12,and +.9

(iv) In 1993–94, Wiles (with additions by Taylor) proved a restricted form of the 1955Taniyama–Shimura–Weil hypothesis, or in short the semistable modularity theorem (SMT)(for semistable curves):

Every semistable elliptic curve is modular.This proves FLT by reductio (the full modularity theorem was to be settled a few years later

[see below]).Finally, I mention here an earlier background fact. FLT is often stated by asking for solutions

in integers. But it is equivalent to the problem of finding rational values for A = x/z and B = y/z suchthat An + Bn = 1 (A, B > 0). So stated it becomes a geometric problem. We consider A, B as thecoordinates of points in the plane.We consider now the first quadrant represented by the equationAn + Bn = 1 (see figure below). For n = 2, we get the unit circle; for n $ •, the unit square; and for2 < n < •, with curves in between, we get the shaded critical region.

n = 2

1

1 %

n = &'

This region is filled with rational points, points such that both their coordinates are rational.FLT now becomes a geometric statement—the curves for n > 2 cut through the region withouthitting a single rational point. The Mordell conjecture–Faltings’ theorem already restricted thenumber of points hit by each curve to a mere finite one (1983). The Wiles–Taylor proof reduced itto 0.

9. In my own “doctrines,” we need cosmic contact (with material objects) for apprehendingeven the purely logical relation of “=.” I bracket away at the moment how we apprehend logical/settheoretic notions, pretending they are given to us for free, prior to and independent of cosmiccontact.

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(iv) not by logical/set theoretic epistemology—the justification for (evidence for,conviction in) the statement’s truth does not come from a proof in a perti-nent (logical/set theoretic) axiomatic system.10

Thus, in certain nontrivial ways, 5 + 7 = 12 would have sufficed to make some of ourpoints. Kant must have sensed something of the kind and used this simple exampleto illustrate various types of irreducibility of arithmetic. Of course, in his day, settheory was not the characteristica universalis. But as Gödel intimates (correctly inmy view), if Leibniz ever got word of it, he would have immediately adopted it asthe reductionist ur-grammar of any possible human thought and Kant would haveto tell him: no, no, 5 + 7 = 12 is no piece of set theory.

But alas, in the end, 5 + 7 = 12 does not suffice to make our case study. Forthere is a cluster of fundamental features of number theoretic practice, I will call itthe sevenfold spectrum, of which 5 + 7 = 12 is not illustrative. This segregation ofcharacteristic marks—separating the rudimentary from the advanced—should initself indicate that we are about to take issue with the common foundationalistclassifications. On the foundationalist view, “practicalities notwithstanding” as welike to say, “essentially” the two cases are on a par.

On this classical philosophical view, we take God’s point of view and armourselves with logical omniscience, so that we can factor out the differencebetween the practically simple reasoning to 5 + 7 = 12, as opposed to the complexreasoning leading to FLT. And we not only arm ourselves with the ability tofollow all logical consequences of the axioms (notions, definitions, etc.) that weactually do, as luck and history would have it, have in mind; we assume, stipulatereally, that each individual mathematician, all by him/herself, just like Him, can,at any point in time and space, have the full repertoire of axioms, notions,definitions, etc. from which the consequences are to be drawn; for example, itdoes not matter whether our mathematician is Diophantus, Fermat (1637) orWiles (1994). What is viewed as critical, on this (as we like to call it) rational

10. There exist of course such proofs of the translates (“logical [set theoretic] symboliza-tions”) of 5 + 7 = 12 both in the combinatorial sense of proof, in, for example, first order formalaxiomatization of arithmetic (e.g., first order Peano Arithmetic [PA]), and in the model theoreticsense of full logical consequence in the categorical second order axiomatization. My point is thatthe evidence, conviction, and ultimate justification for (1) the truth of 5 + 7 = 12 and (2) our beliefin it do not rest on such proofs or on its being a second-order model theoretic consequence of theaxioms.

Let me point out that from now on, when I shall speak of logical proof, I will assume as theunderlying system the full second-order logic and the pertinent (respectively, to arithmetic and settheory) axiomatic systems of second-order PA and second-order ZF(C). I include both strictlydeductive proofs in those systems as well as semantic consequence relations (an argument in thatlanguage that preserves truth in all models). No doubt, there are important differences betweenthe notion of combinatorial proof in a first-order system and the notion of combinatorial proof ina second-order system. Stronger yet, both differ from the (nonrecursively enumerable) notion ofsemantic (model theory) notion of consequence (either in the standard model of first-order PA orin full second-order PA). Still for all the differences, I urge below that (1) the fact that 5 + 7 = 12;(2) the meaning of this sentence when I use it; (3) the notions I cognize when I apprehend it; and(4) my justification for my belief in it and its truth, are all given neither by such combinatorialproofs nor by a semantic consequence-drawing from the second-order Peano-Dedekind axioms.This much I claim below directly and primarily for FLT. But I also claim it eventually for 5 + 7 = 12.

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reconstruction, is the following common fact: both 5 + 7 = 12 and FLT are (prooftheoretically and model theoretically) consequences of self-evident axioms. Onthis common foundationalist view, we assimilate the complex case of FLT to thesimple case of 5 + 7 = 12.11

Why the More Involved Case of FLT?

I follow quite a different methodology by starting with the complex case of FLT. Ifany assimilation is eventually to take place (and it will later, in part II), it is ratherthe other way around—even 7 + 5 = 12 presupposes more than meets the eye. Butat the moment, the main thing to take in is that when we start with FLT as aparadigm case of number theoretic practice, the profile that emerges is categori-cally different from that of 5 + 7 = 12.

For a contrast with the fully transparent meaning of 7 + 5 = 12, let me use anexample deeply embedded in the natural world, an example after which I wouldlike to model the case of FLT. In recent philosophical theories of meaning (“seman-tics”), philosophers—I am thinking chiefly of the work of Hilary Putnam—havepondered examples like the word “water.”12

One apparent aspect of our use of “water” is the availability of a certainkind of “transparent and qualitative information,” at least to those who everinteracted with water, regarding water’s appearance, for example, “tastes so andso, looks thus and such, runs in rivers, etc.” We may call this the apparent meaningof “water.” The apparent meaning is supposed to be available to every lay userof the word.13

However, as pointed out by Putnam, natural science goes beyond thisapparent meaning. Modern chemistry found out that what water is to what theliquid itself is: is hydrogen hydroxide (H2O). This was not known to Aristotle(our analog of Diophantus). Aristotle did use the same word as we do (or a

11. As should be obvious by now, my claim that 5 + 7 = 12 does not suffice to make ourcase—does not have the right profile—does not turn on its being part of a fragment of completearithmetic. From a certain deeply epistemological perspective centered on proof—whether com-binatorial (as in Hilbert) or logical inhalt-involving as in Frege—the threat might seem to comeonly with certain universal sentences over recursive relations, for there we get incompleteness(recall that Gödel’s sentences can be coded as a certain kind of Diophantine problems). But toreiterate: such unprovable (Gödel) sentences are not what I want to contrast with 7 + 5 = 12. It isthe very PA-provable FLT (and its simple Diophantine mates, see below) we will contrast with7 + 5 = 12.

12. See Hilary Putnam, “The Meaning of Meaning,” in his Mind, Language and Reality:Philosophical Papers, vol. 2 (Cambridge: Cambridge University Press, 1975), 215–71; and especiallyPutnam, “Is Semantics Possible?,” ibid., 139–52. These insights were amplified and deepened laterto points about the environment-dependence structure of our cognition by Tyler Burge; see, for aclear statement, Burge,“Other Bodies,” in Thought and Object, ed.Andrew Woodfield (New York:Oxford University Press, 1982), 97–120.

13. The terminology of apparent meaning is intended to connote both uses of the adjectivalmodifier—a meaning that involves surface information and a type of information that is onlyapparently the real meaning (with this last being not so apparent).

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Greek cognate), and I shall assume that he associated with it the same apparentmeaning. But the chemical fact was unavailable to him and may well still not beknown to the majority of the present users of the word, for example, when theyask for a glass of water.

Reflection on this example suggests that there are two appearance-transcendent layers of meaning to mark out. First, there is what-we-mean (with thisverb now taken in the active-transitive form, what stuff we target and pick out), viz.we mean with “Water” the stuff water, H2O. I would like to call this the embodiedmeaning of “water.” So we here transcend the internal, “in the head” of the layspeaker, apparent and transparent information. We break out of our mental cagesand embed ourselves in the embodied world, in particular, in the liquid stuff outthere. And now, after this embedding, we are ready for the second step. Havingmade H2O the embodied meaning of “water,” there are now structural character-istics of this embodied meaning awaiting to be revealed. Hydrogen and oxygenhave natural stable and radioactive isotopes and water molecules of massesroughly 18(H2

16O) to 22 (D218O) are expected to form. Pure water, H2O, has a

unique molecular structure. The O-H bond-lengths are 0.096 and the H-O-H angleis 104.5.This distinct geometry calls for a complex explanation.This second layer ofstructurally revelatory information about the embodied meaning I will call the finestructure embodied meaning.

In all, we separate: (1) the apparent meaning—what’s in every lay user’shead—(2) the embodied meaning, the basic stuff meant by the word—H2O—andfinally, (3) the fine structure embodied meaning unraveling the fine structure ofhydrogen hydroxide. Of course, many philosophers, aware of Putnam’s observa-tions, may well want to call the apparent level the meaning of the word “water” andcontrast this with the nature or essence of the worldly stuff proper. They wouldregard what I call the embodied meaning, let alone the fine structure meaning, asnot part of what we mean but part of what the stuff is. I should like it noted that thisseparation of semantics from metaphysics, what-we-mean from what-the-stuff-is,rests on turning a deaf ear to an ambiguity in the very phrase “what-we-mean,” forat least on one reading the phrase “what-we-mean” targets the whatness of thething meant. After all, what we mean by “water” is . . . water.

In any event, this two-tiered separation between what we mean and what thestuff itself is (the essence of water proper) is a version of the dualism I have beenlamenting—an us versus it duality. As the philosopher Van Quine put it (so well),“Meaning is what essence becomes when it is divorced from the object of referenceand wedded to the word.”

Putnam himself does not so divorce meaning and essence. He famously quips“meanings ain’t in the head” and relegates the apparent-transparent qualitativeinformation—what is in the lay head—to the status of mere “stereotypic informa-tion.” And he says just the right thing about what we do mean by “water”—wemean water (the stuff), the hydrogen hydroxide. But let me not pin too much hereon Putnam and so, in what follows, let me not hang on to this large tree. I will justsay for myself that what water means (or in the active transitive verb form, what wemean in using the word “water”) is the embodied meaning, further unraveled at thelevel of fine structure embodied meaning.

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Two Models: Apparent versus Embodied Meaning

So now we have two purported models, “7 + 5 = 12” and (say) “water is wet” (orbetter yet: “Water is a compound”). In the former, the meaning we see is themeaning we get, the apparent meaning is the full meaning. With the latter, there ismore to the meaning than meets the eye—the apparent and the embodied mean-ings differ. We may now try to classify new cases using the two models as ourparadigms.

In a moment, I will try to so classify FLT. But since I have harped on thetheme of dualism, let me mention an analog hard case lying at the very core ofdualism, a case that like FLT could (and did) go either way, be made to fit either ofour paradigms. Consider then my use of the first person pronoun “I.” What do Ithereby mean?

We could view me as meaning by “I” (what philosophers call) the transpar-ently available inner self, what many have called the “Cartesian Ego,” an immate-rial item seemingly so prominent in our first-person reflections (“meditations”)cogito (“je pense”) and in sum (“j’existe”). For example, the leading philosopher oflanguage of our age, Saul Kripke, has often looked for such a “purely qualitative”(his own phrase), epistemologically transparent meaning for the word “I.” He hasemphasized this purely qualitative meaning with his famous case of my use of “I amin pain” to bring home that what I mean is this inner item (and the qualitativefeeling of hurt), all allegedly available prior to and independently of any allusion tomy mind’s (self, ego) embodied connections, the physical body (in particular, brain)of Joseph Almog. Sure enough, JA has such a body. But this is a subsequent pointabout what JA is (or his essence if you will), not about what I mean by “I.” Wecrown here yet again a dualism between a transparent meaning and a hiddenessence.14

In contrast to this dualism, we may insist on not divorcing meaning andessence.We may (I surely would) answer my original query,“What do I mean whenusing ‘I’?” with: I mean this man—Joseph Almog, the fully embodied human being.And of course, once we have that item of the natural order, the man, as theembodied meaning of my use of “I,” we have, in turn, waiting to unfold a layer ofa fine structure embodied meaning. As we unravel what it is to be this man—howit-JA—originated from a particular sperm and egg and wasn’t brought by a stork orangels—and also unravel how this kind of thing—mankind—has emerged fromhigher primates, we unravel the fine structure of the embodied meaning—what Imean—the human being I mean—with my use of “I.”

Very well then:We can treat “I” dualistically and we can treat it unitarily. Letus go back to our number theoretic sentences. I see us as facing a similar choice

14. This kind of account is often called “Cartesian,” but in my view it was not the one putforward by that man, René Descartes, who championed the orthogonal picture about to beintroduced. (See my What Am I?). In any event, the adjective “Cartesian” is so embedded inphilosophical use that it seems futile to try to undo it. Saul Kripke is a paradigm of such a“Cartesian”: See his recent “The First Person,” address delivered at the CUNY Graduate CenterConference, Saul Kripke: Philosophy, Language and Logic, New York, 25 January 2006, availableat web.go.cuny.edu/philosophy/events/kripke.conference.html.

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with mathematical—and our specific simple example, Diophantine—sentences.Wecan think of their meaning on the apparent-meaning model of “7 + 5 = 12” or onthe model of the embedded-embodied meaning of “water” and “I.” I would like todevelop below the latter picture.

Diophantine Sentences—the Sevenfold Spectrum

The embedded account of Diophantine sentences involves a cluster of character-istic marks that I will call the sevenfold spectrum. The first batch of marks—four ofthem—concern the witnessing by FLT-like cases of the phenomenon of intra-mathematical integration; the second batch of marks reflects the phenomenon ofnature–mathematics integration. I will state the seven headers concisely, then go onto develop each in detail.

A. Intra-Mathematical Integration

(Intra-1) apparent meaning versus embodied meaning

There is more—from inside the theory of elliptic curves—to the embodied meaningof FLT than meets the eye—viz. than the visible meaning of the Diophantinestatement.15

(Intra-2) proof as fine structure-revelatory

There is more to the full meaning of FLT than even the just-mentioned algebraicmeaning; the proof of FLT—from the modularity theorem—is revelatory of hiddenstructural layers, features of FLT’s algebraic meaning—what, in the complex plane,is the analytic generative basis of the embodied meaning—elliptic curve-involving—meaning. I will call this the full fine-structure embodied meaning.16

(Intra-3) no algebra (arithmetic, logic) versus geometry dualism

Frege and Kant, who disagreed much about geometry and logic, did nonethelessagree on there being a deep difference between the types of meaning of arithmetic-algebraic statements and those of geometry. If one’s paradigm of the former is thelikes of 5 + 7 = 12 and the paradigm of the geometric is, for example, the likes of thePythagorean theorem, it is natural to let a dualism emerge—this one intra-mathematical—between the meaning (and basis of knowledge) of algebraic state-ments and those of (Euclidean) geometry. But if our paradigm is the slightly morecomplex Diophantine equations (e.g., like FLT or the problem of Pythagorean

15. As mentioned in the last, autobiographical note of this paper (note 35), after completingthe present work it was pointed out to me that the theme of looking for such a deeper complexmeaning appears (in quite a different form) both in A. Weil’s “Two Lectures on Number Theory”and especially McMullen’s “From the Dynamics of Surfaces to Rational Points on Curves.” See thelast note of this paper (note 35).

16. See the discussion that follows below on the generation of elliptic curves from notions ofthe complex plane (as reflected by the modularity theorem).

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triples [“congruent numbers”] mentioned below), the algebra/geometry dualismdisappears, as indeed witnessed by the emergence of a domain called Diophantinegeometry. Integer and rational solutions draw upon rings and fields from which toassemble the solutions but at the same time the system of polynomial equationsdescribes an algebraic variety, a geometric object. Linear and quadratic equationsin two variables express curves of genus zero and our key paradigm case—onwhich much below—elliptic curves, express curves of genus 1 (emerging, in tow, inthe complex plane, as Riemann surfaces).17

(Intra-4) a web of experimental conjectures unravels the fine-structure ofthe embodied meaning

As in the natural sciences, for example our “water” case above, to lay out the fullmeaning of a Diophantine equation like FLT, we bring its embodied meaningunder deep structure-unraveling conjectures regarding this type of object—ellipticcurve on the rationals.18

B. Trans-Mathematical Integration

(Trans-1) existence—cosmically generated meanings

The sheer existence of the full fine-structure embodied meaning of FLT (involvingas it does, for example, Riemann surfaces) rests on the existence of the complexplane, whose existence in turn rests on the existence of the infrastructure of thephysical universe.

(Trans-2) cognition—evolutionary emergence of notions

The notions required by the full fine-structure embodied meaning of FLT could nothave been cognized by us without (1) contact with a community of human math-ematicians across history and (2) a natural historical order behind their emergence,for example, Fermat could not have formulated the modularity theorem.

(Trans-3) proof-evolutionary and regressive emergence of proofs

Not only is the emergence of the notions and their cognition history-bound; evenwhen they have been fully cognized, the laying down of the conjectures needed forproving FLT is evolutionary and regressive. That is, the theorem to be provedprecedes, both in time and conceptually, the hypotheses from which it is eventuallyestablished. Conceptually, the process is regressive—the theorem precedes andsends us looking for the conjecture (its eventual ground), in the sense that we oftensearch deep generalizations about the key notions (e.g., elliptic curves) from whichto derive the target theorem.

17. See our discussion below of congruent numbers for another anti algebra/geometrydualism case study.

18. Examples discussed below concern the modularity conjecture, the Mordell conjecture(both now theorems) and the Swinnerton-Dyer/Birch conjecture.

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Unwinding the Sevenfold Spectrum of FLT

First, the embodied meaning of FLT is not the meaning that meets the eye. Inunraveling a “deeper” meaning I do not mean a reductive translation to somefavorite foundationalist universal characteristic. I target rather what mathematicalpractice reveals the sentence to mean in humdrum mathematics. The sentenceexpresses a claim in the theory of elliptic curves. This is surprising because on theface of it (the visible meaning) xn + yn ! zn is not a cubic. But as mentioned above,Frey has associated with a nontrivial solution of ap + bp = cp, with p equal to orgreater than 5, an elliptic curve given by the cubic Y2 = x(x - ap)(x + bp).

Let me linger on this point that I regard as the ground-zero fact of theintra-mathematical integrative picture that is drawn below—what the sentence,when put into “mathematical action,” really means; later and further discoveriesdue to proofs of Diophantine equations like FLT, about what I called its fullyfine-structure embodied meaning (expounding deep structural connections to ana-lytic and group theoretic features) emanate from this one ground fact—what thesentence’s embodied meaning is.

I have spoken of Fermat and his last theorem in which we consider any oldexponent n. For this Fermat thought he had a proof. It is often speculated that hemust have believed that his correct proof, in the particular cases of n = 3 and n = 4,somehow generalizes. About this, he was wrong. But already in the cases he didcontrol—x4 + y4 = z2 (which gave him the impossibility of FLT solutions for n = 4),we can see the basic idea that the Diophantine equation’s embodied meaning callsupon elliptic curves and their structural features. So we can see that the Diophan-tine equation has an elliptic curve meaning quite independent of subsequentfine-structure revelations (about which much more below) coming from eventualproofs. These last, to analogize with our “water” paradigm above, reveal very finestructure and thus the generative basis of the phenomenon (what engenders water[elliptic curves on Q]). But what I focus on at the moment is the primary embodiedmeaning of FLT and the fact, again in analogy with “water,” that it involves morethan the apparent meaning—calls upon elliptic curves on Q.19

Still lingering on this ground point—the embodied meaning invokes ellipticfunctions—let us recall another deceptively simple case, where what meets theeye—the apparent meaning—is not what does the mathematical action.

19. The points that follow, both historically about the man Fermat and, as it were withhindsight, on the elliptic curve algebraic-embodied meaning of the case he did control, are inspiredby A. Weil’s discussion “Two Lectures about Number Theory—Past and Present” and his discus-sion of Fermat’s method of infinite descent as part and parcel of assigning elliptic curves as themeanings of the equations. I have in mind pp. 95–96, where he discusses Euler’s articulation of thecase of n = 4 and n = 3 using de facto elliptic curve articulations. Then, Weil explains why we oughtto read (with hindsight) Fermat’s infinite descent as already processing the elliptic curve meaning.Indeed, I believe the point generalizes and even for the “easy” solvable cases of x2 + y2 = z2, weshould assign the elliptic curve embodied meaning. This is what Pythagoras talked about(meant)—even if it took a while to see that this is what he was meaning (just as H2O is what he wasmeaning in asking for a glass of water 2,500 years ago).

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A famous problem running back to the ancient Greeks concerns the areas ofright triangles.20 It is standardly called the problem of congruent numbers. Arational (non zero) number is congruent if it is the area of a right angle with threerational sides. The question whether a number is congruent can be correlated withthe following problem about elliptic curves. A rational number R is the area of aright triangle with rational sides and hypotenuse h iff (h/2)2 with R added orsubtracted are both rational squares. Thus R is a congruent number iff the follow-ing two equations

u Rv w2 2 2+ =(1)u Rv z2 2 2# =(2)

have a solution in integers (u,v,w,z) with v non-zero.These two hypersurfaces in theprojective plane P3 intersect in a smooth quartic in P3 which contains the point(1,0,1,1). The intersection is an elliptic curve over Q. The projection from (1,0,1,1)to the plane z = 0 gives an isomorphism with a cubic plane whose forms are:

E y x R x or also Ry x xR 2 2 2 2 3: = # = #

The points on the space curve with v = 0 correspond to the points where y = 0 andthe point at infinity on ER. When these points are reduced modulo p (for finitep-fields offering values for solutions), it turns out the rational solution points onER(Q) are precisely those of finite order. As will turn out in our discussion in amoment, a test whether ER(Q) is finite involves use of one of the deepest standingconjectures about elliptic curves, the Swinnerton-Dyer and Birch conjecture. Andso, on my reading, when Pythagoras wondered about the seemingly homey con-gruent number problem, his assertion concerned the deepest structural features ofelliptic curves.21

II. APPARENT VERSUS EMBODIED MEANING

The apparent meaning of “water”—its appearance—results from its embodiedmeaning, the chemical structure. It is because the molecular structure is what it isthat there results a certain appearance of this liquid. It is surely not the other wayaround—because something looks thusly, it has the H2O fabric. That it so lookshelps us approach the phenomenon—we track and identify the phenomenon,communicate to others its presence, etc. by using the apparent meaning. But thatwe can so identify water (using the appearance) is due to the determinationrunning the other way: It is the H2O structure that determines the appearance.

In like manner, it is the elliptic curve features that determine that for ngreater than 2, xn + yn ! zn. We use the Diophantine notation to track and identifythe elliptic curve. In the same vein the order of determination runs as follows—it

20. I follow here S. Lang’s Diophantine Geometry, 135–37. Thanks to slow and patient dis-cussions by the author.

21. On the Swinnerton-Dyer and Birch Conjecture, see below.

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is because the Frey elliptic curve could not have a certain feature (be modular andsemistable at once) that the visible Diophantine claim cannot be true. The use of“because” here is determinative: The impossibility of the Frey curve makes asolution for the equation impossible.

We could say of course that the association of the curve with the Diophantinesentence is accidental, merely representational or instrumental, in the way we canassociate with a given person a social security number. It would be silly to suggestthat this social security number makes the person what he is.

And that is exactly what is not the case with the elliptic curve “association”with FLT. We don’t just so “represent” the Diophantine claim. Here the ellipticcurve structure makes the Diophantine claim be what it is, in the way the molecularstructure of hydrogen/oxygen combination makes water be what it is. Our word“water” primarily tracks the underlying structure. The diophantine notation pri-marily tracks the elliptic curve structure.22

Two Corollaries of Embodied Meanings

In the case of “water” the primality of the embodied meaning—we mean by“water” H2O—leads to two fundamental characteristics. The first concernsthe metaphysics of the phenomenon, the second our epistemology, the kind ofknowledge we can have of the phenomenon. The metaphysical corollary inducesthe epistemological result. And so I start with it.

The metaphysical corollary concerns the constitutional unity underlying allnatural phenomena. I spoke of “water” (and water) but of course I could havespoken of “salt” and “gold”; in the same vein, we could have considered “whale”and “elm”; and, in turn, “river” and “planet” and “black hole”; and indeed, I havespoken of that most surprising of all natural elements, mankind, the embodiedmeaning of all our uses of “I.” What is at first blush so striking, what is so apparent,is the staggering diversity of natural phenomena; what later blushes reveal is thatthe diversity is underlined by symbiotic codependencies, not least among them adeeply unitarian fabric that goes into the composition of all the aforementionedphenomena.A black hole is many tons of underlying atoms, so is a water-pond and,in the end (and that’s why there is necessarily an end), I just am, as Tom Nagel putsit coldly, a hundred and sixty-five pounds of atoms. Let me call this lesson—fromreflection on the embedded-embodied meanings of “water” and “black hole” and“I”—the (constitutional) unity behind the diversity.

The epistemological thesis emanates from this non-apparent but nonethelessprimal constitutional unity. It concerns our inevitably trial-and-error, conjecturaland evolutionary apprehension of what phenomena like water and salt, whales andtrees, black holes and men are. The hidden natures (and in turn, the inducednontransparency of the nature-imbued meanings) make for a slow growth ofknowledge, generation after generation, building on cumulative trials and errors of

22. Like remarks apply to the Diophantine notation and the elliptic curve structure in thecase of the Congruent number problem. See below.

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previous ones. Our current apprehension of what water is was not available toParmenides or Aristotle or Descartes or Newton. Let us call this second thesis theevolutionary character of natural cognition.23

Diophantine Statements: the Unity behind the Diversity

With the water model in mind, let us have a closer look at statements like FLT andthe congruent number (CN) (Pythagorean) problem and what they teach us aboutthe meanings of Diophantine statements.24

First, let us look at the metaphysical front and the problem of constitutionalunity/diversity. Philosophy often ponders with some angst the alleged rift betweenalgebra and geometry. It is even said to students that the Greeks primed thegeometric and defined the algebraic in terms of it. It is then suggested Descartesinverted everything upside down and primed the algebraic, reducing things geo-metrical to the language of polynomial equations. For myself, I think a simplereflection on cases like FLT and CN exposes the rift as illusory. There never wasany rift—not in ancient Greece, not in Descartes, and not anywhere else. Geometryand algebra are rather like two sides of the same piece of paper—generatedsimultaneously and interwoven in one another.

We may put the symbiosis by speaking, but only for a moment, in thelanguage of Kronecker about God the creator, (“God created the integers, all therest is the work of man”): even for God, there was to be no algebra withoutgeometry nor geometry without algebra. Of course, as mentioned, 7 + 5 = 12 maynot quite bring this home. But FLT does.25

As already mentioned above, when FLT was introduced into our discussion(see note 8 above), critical to Wiles’s proof of FLT was the (semistable) modularitytheorem: (SMT) Every semistable elliptic curve over Q is modular.26

Reflection on the many forms of this theorem brings out just how interde-pendent are the following notions—the algebraic idea of elliptic curve E, theanalytic idea of L-function associated with (the growth of solutions mod p of) E,

23. By using the adjective “evolutionary” I target two separate features: the fact that the verynotions to be cognized (e.g., hydrogen atom) are becoming available to cognizing agents in a slowcumulative historical process; furthermore, the fact that our knowledge (of truths about them) iscumulative. We may articulate the two features by recalling the two senses of “know” (in French,German, etc.), for example, “kennen” versus “wissen.” Both our kennen of the notions, ouracquaintance with them, and our wissen of the truths, are cumulative.

24. I here follow two illustrative cases discussed by S. Lang, Survey of Diophantine Geometry,130–39. As mentioned, I return to this theme of intra-mathematical integration in Part II when Idiscuss a different type of example resting on interconnections between Riemann surfaces, Dessinsd’enfant and algebraic curves (but see below, for a bit of ankle, the remark about Belyi’s theorem).

25. One does not feel comfortable speaking for God but I should add that if what I am aboutto develop below makes sense, He was further constrained (than making Geometry-Algebra inone fell swoop)—He had to create cosmic space-time to engender geometry-algebra.

26. For the restriction to semistable curves (Frey’s curve is of the kind), see the appendix.(SMT) had to be combined with a result of Serre–Ribet to yield FLT. See the appendix. The manyforms of the modularity theorem (by which I now mean the full modularity theorem proved in fulla few years after Wiles) are reviewed in detail (and their interconnections laid out) in FredDiamond and Jerry Shurman, A First Course in Modular Forms (Springer, 2004).

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the hyperbolic geometry notion of modular curve, and the group theoretic notionof Galois representation associated with E (as with the modular curve). We haveanything but an algebra–geometry rift here; we are rather exposed to the manylives of E. An elliptic curve leads a double life as both algebraic and geometric.I explain.

The key conceptual (or “philosophical”) fact in reading the modularitytheorem is not the bidirectional correlation of elliptic curves over Q and modularforms. This is the mathematical ground fact; without it we wouldn’t have a furtherconceptual question I am about to raise. The conceptual query is this: In betweenthe elliptic curve and modular form, what, if anything, engenders what? Why are weso lucky to have such a deep correlation? Is it a cosmic accident or is there amechanism explaining the correlation as a generative relation?27

From Riemann Surfaces to Elliptic Curves

On my understanding of the theorem, all elliptic curves with rational invariants aregenerated from modular curves by appropriate holomorphic maps. The generationreveals both the modular curve and the engendered elliptic curve to be forms ofcompact Riemann surfaces. So, to linger for a moment at this elementary under-graduate “I am puzzled” level, if one wonders (as I at least often did) why a curveand a surface are so intimately correlated (let alone said by some to be the same),my speculative diagnostic is this: the ground notion is that of a (compact) Riemannsurface. We generate from it the elliptic curve by preserving in the generationprocess—by the use of certain types of maps—a bunch of critical properties.

In so speaking of the generative order, we must of course keep our discovery-order and the generative order apart. We may well discover the chemical (andfine-structure chemical) structure of H2O late in the game, having entered theinvestigation by way of the apparent meaning of“water.”But in the generative orderit is the fine structure chemistry that engenders the H2O molecules, which engender,in turn, the water-appearance. I would like to pursue a similar tack of explanation inour case.

There are various manners of explaining this generation process, differentlanguages with which to close the maps on the Riemann surfaces under groupactions that preserve the key structure—for example, the just mentioned detailedanalysis of modularity (see note 27). A First Course in Modular Forms reviews atleast five different languages-methods of describing the structure of such genera-tive maps. My own preferred explanation—the one that I grasp best—rests on asixth language, the language of dessin d’enfant, in my view the most vivid(“topological”—what Grothendieck describes as the one employing the most

27. Instead of “correlation,” we may speak technically of “parametrization” or “uniformiza-tion,” but the conceptual puzzle remains—what, if anything, is the “hidden mechanism” thatguarantees the successful parametrization of the curves? After all, the one-one correlation worksin both directions and one may well say: Why not generate the compact Riemann surface from thecurve? I argue below why the surfaces are the basic notion and we get from them, by abstraction-generation, the curves.

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passe-partout notions) way of explaining the algebra–geometry symbiosis. In par-ticular, this offers us the starkest way (known to me) of rethinking the ur-notionof compact Riemann surface as an algebraic curve. The rethinking rests on abridge theorem due to Belyi.

We start with the ur-idea of a Riemann surface. Riemann proved that aRiemann surface X is compact if and only if it is isomorphic to the Riemann surfaceof an algebraic curve f(x; y) = 0 for some polynomial f(x, y) in C[x, y].Those that arekey to the generation of our notion of curve are the polynomials with coefficientsin the field of algebraic numbers. Belyi’s result shows that the Riemann surfacescorresponding to such polynomials f(x, y) are those obtained from certain bipartitemaps. The bipartite maps are the reflection of the dessin’s actions. A bipartite mapM consists of a bipartite graph G embedded (without crossings) in a compact,connected, oriented surface X, so that the faces (connected components of X\G)are simply connected (thus, the dessin). One can describe M by a pair of permu-tations g0 and g1 of its edge-set E:The vertices can be colored black or white, so thateach edge joins a black and a white vertex; the orientation of X then determines acyclic ordering of the edges around each black or white vertex, and these are thedisjoint cycles of g0 and g1, respectively.

In a nutshell, what I propose is that the mystery of how a rational ellipticcurve can be, in its “other life,” a Riemann surface is that it is generated fromthe underlying surface.28

Let me reiterate the point just made, for it strikes me as the fundamentalpoint (or the basic conceptual conjecture) I am trying to make. The point was justmade using the language of bipartite maps and there are other languages to makeit (regarding “correlation” of L-functions or Galois representations, etc.) but at itsbottom it is a conceptual conjecture viz. that the “correlation” is not one of goodluck, a cosmic lucky break, but rather one in which we find that two seeminglydifferent kinds of objects (phenomena) are in truth two forms of the same object(phenomenon). The correlating equation is not a “lucky match” but rather theresult of our re-thinking the very objects we are talking about in new-deeper terms.Thus, if I may allow myself two simplified models of such “identifications” (“cor-relations”), we may state the conjecture that E = M, where E is the ranking byheights of the employees of a certain bank and M is their ranking by their respec-tive salaries. This strikes as an accidental correlation (pending astounding discov-eries about our innate disposition to favor in promotions tall people). At the otherend, we may consider identifications of the E = MC2 type, where we come tore-understand certain familiar notions in a deeper way and see that they are notreally two “separate” categories that happen to match but that, in fact, the one is a

28. More precisely, Belyi showed that a compact Riemann surface X is defined over Q if andonly if there is a Belyi function B from X to the Riemann sphere S = C U {•}, that is, a meromor-phic function on X which is unbranched over B\ {0, 1, •}. Of course, the “conceptual” fact justworked out—that the surface originates the curve (not the other way around)—is just a “back-ground” fact. The modularity theorem’s claim that modular curves parametrize the elliptic curveson Q is a very strong claim and that is where the complex mathematics comes in to establish howdeep invariant features of surfaces control the rational curves.

The Cosmic Ensemble 365

form of the other, an engendered form emerging from the other. Of course, thereare cases which may initially pass for “cosmic accidents” but which, upon a deepertracing of the mathematical structure used on both sides might well turn out notaccidental at all (e.g. famously, spacings between consecutive zeros of the Riemannzeta function align with spacings between consecutive energy levels of heavyatoms, a seemingly accidental alignment. But it may not be so when it turns out thatboth are forms of some eminently natural sequencing of eigenvalues associatedwith certain types of random matrices).

The claim proposed here is that the modularity theorem rests on a correla-tion of the E = MC2 type, a re-thinking of a familiar algebraic item—ellipticcurves—in deeper terms borrowed from complex analysis.We find ways of “param-etrizing”, better put informally by a metaphysician like myself in terms of “gener-ating”, the one form—elliptic curves—from the deeper form—the modular forms.The re-thinking runs like this. We consider elliptic curves over Q (up to isomor-phism over algebraic-numbers). And we now re-think them as compact Riemannsurfaces of genus one definable by polynomial equations with rational coefficients.The modularity theorem then explains to us the generative basis of the ellipticcurves: for each such surface S, there is a congruence subgroup G of SL(2, Z) (Z theintegers) and non constant analytic map over the complex upper half place H,G\H$S. It is in this map of G\H Into S that lies the generative basis of S and theelliptic curve it encodes.29

So much then for the embedded meaning of the Diophantine statement, theH2O of the cubic equation—microscoped, the elliptic curve turns out to be gener-ated from a complex analytic object, a compact Riemann surface. In sayingxn + yn ! zn, we, like it or not, aware of it or not, in effect, speak of compactRiemann surfaces in the very sense that in saying “water,” we, like it or not, awareof it or not, in effect, speak of hydrogen hydroxide.

And there is more structure, fine structure, to our embodied meaning, just asan H2O molecule packs a lot of fine structure underlying its hanging together.Whatgoes into the rational points of the elliptic curve hanging together?

I spoke of the elliptic curve and the set of its rational points as the H2O of theDiophantine equation. We want to microscope further the basis of the structure ofthese rational points. This involves a combination of various results (and conjec-tures) about the basis of elliptic curves. Let me give two fundamental examples ofsuch basis-facts regarding fine-structure.

Given the curve E over Q, the group of rational points E(Q) makes anAbelian group. A fundamental theorem regarding E(Q) due to Mordell back in1922 shows that E(Q) is finitely generated. Thus, even if E(Q) has infinitely manypoints, there is always a finite basis of points from which the full infinite set mightbe reached by finitely many additions.

29. An interesting detailed working out of such a generation of elliptic curves over Q frommodular forms is offered by Barry Mazur in “Number Theory as a Gadfly”, American Mathemati-cal Monthly 98, 1991, 593–610. He there shows, working inside the complex upper half plane H,how to derive the elliptic curve (thought of as Riemann surface) from (Mazur’s variant) of theholomorphic X0(N) modular forms mentioned in the appendix below.

366 Joseph Almog

Next, to understand better the group structure—and here we veer fromMordell’s theorem toward a second structural claim about E(Q)—we need to beable to calculate the rank of the curve, a measure of the size of the set of rationalpoints on E.

A key heuristic used here was to connect the absolute number of rationalpoint solutions to a count of solutions modulo p, at each prime p. Each such fieldat p is of course finite and involves at most finitely many solutions, thus computa-tionally a more tractable task. The heuristic idea was that an infinite absolutesolution set is witnessed in a large number of solutions mod p at each p. If we calln(p), the number of solutions mod p, the heuristic idea was that for many primes p,the ratio p/n(p) should be much less than 1. If so, when we take the product overall primes p, it should be to 0. Now, a fundamental conjecture about the finestructure of elliptic curves urges that the converse is also true: if the infinite productis 0, the solution count should be infinite. This conjecture is called the Swinnerton-Dyer/Birch conjecture.We may well describe it as the bridge to infinity conjecture.30

What am I reading into these fine structure results about elliptic functions?I said earlier that the embodied meaning of an FLT-like Diophantine equa-

tion is an elliptic curve over Q. On my reading, the algebraic object has a verycomplex underlying structure. We telescope this structure by unraveling the curveto be what it really is, a Riemann surface, whereby tools of complex analysis trackthe fine structure of this now revealed-to-be-complex analytic object. The ellipticcurve is no more isolated-ly algebraic (an “algebraic island”) than the modularcurve is distinctly geometric (a “geometric island”). Both are forms of Riemannsurfaces, which are neither algebraic nor geometric but algebraico–geometric orgeometric–algebraic or simply just what they are: Riemann surfaces, describable atonce by the languages of algebra (curves, Abelian groups, Galois representationsetc.), as by the languages of hyperbolic geometry and analysis.

The Epistemology of Diophatine Statements: the EvolutionaryCharacter of Cognition

As in the case of the fine structure of water, the fine structure of elliptic curves isanything but apparent. We made use here of various forms of the modularityconjecture (now theorem) for semistable elliptic curves, and, in the more general

30. I abstract here from the very complex problem of calculating such an infinite product (seebelow in this note).

Harping on our theme of integration of different domains, it must be noted that the keyinvolvement of such infinite products over all primes leads to the activation of analytic methods ofcalculating this infinite product derived from Dirichlet’s account of the density of primes inarithmetic progressions, methods extended by Hasse to the conjecture (now after the modularitytheorem work, theorem) that there exists an analytic L-function defined for every complexnumber s computing such infinite products. The Swinnerton-Dyer/Birch conjecture is that E(Q) isinfinite iff L(E,1) = 0. This would settle in particular whether there are infinitely many points onthe specific elliptic curve expressing the congruent number problem (whether there is a rational-sided right triangle with area d).

The Cosmic Ensemble 367

case (now also proved), for any elliptic curve over Q.31 We also mentioned inpassing the fundamental conjecture (not a theorem yet, although proved forspecial cases)—the bridge to infinity—of Birch and Swinnerton-Dyer.

What is striking in both cases is the conjectural-natural-science—rather thana priori self evident—profile of the investigations.32 Both conjectures mentioneddid not arise from reflection, even in the broadest sense, on the apparent meaningof the key notions involved or from antecedent axioms. Rather, the conjectureswere derived regressively as it were from theorems that needed proving. Onelooked for generalizations about elliptic curves that would explain what was sus-pected to be a true observation, sometimes supported by actual computer calcu-lations of large numbers.33 In both cases, restricted forms of the conjecture wereproven first, suggesting subsequent bold generalizations.34 Through and through,we do not have an a priori set of principles validated by inspection of the apparentmeaning from which we squeeze stronger and stronger theorems. Rather, throughand through, we have theorems awaiting proof, theorems-in-waiting. We unraveltheir embodied meaning—such and such elliptic functions are claimed to have norational points. To understand the notion involved—elliptic function—we investi-gate, painstakingly by trial and error, its fine structure. This sends us looking forunderlying mechanisms in a richer (if more complex) language—we understandthe H2O molecule by invoking atoms, that invoke protons and electrons (etc.), andwe look for complex mechanisms of bondage relating the atoms. In like manner, weborrow from the language of bipartite maps, as from the language of complexanalysis and L-functions and their definability over the whole complex plane andfrom the associated Galois representations, all in order to microscope the finestructure of the elliptic curve. We then conjecture about the borrowed notionsgeneralizations that suggest new test cases, new observations. This is how Freycould have contemplated a hypothesis—critical-test-case—for refutation: Look fora hypothetical semistable curve and check to see whether it is modular (bound bythe conjecture that they all are).

If this is not the profile of investigation manifested in conjectural-experimental natural sciences, what is?

31. This allows us to prove Fermat-like claims (but generalized), for example, a perfect cubecannot be written as a sum of two relatively prime nth powers n greater or equal to 3.

32. Another such anything-but-a priori conjecture that we have not discussed about Fermat-type equations is Mordell’s generalization of his 1922 result to curves of genus 2 and higher, provedby Faltings in 1983. See the above-mentioned paper of McMullen “From the Dynamics of Surfacesto Rational Points on Curves,” in the last footnote (note 35).

33. I have in mind statistical data about solutions for large primes p (working in mod p)amassed by computers by Swinnerton-Dyer and Birch.

34. E.g., the modularity theorem was proved for elliptic curves with complex multiplicationor, by Wiles, for semistable curves. In the Swinnerton-Dyer and Birch conjecture case, Coates andWiles (in 1976) proved that if E is a curve with complex multiplication and L(E,1) is not 0 then Ehas only a finite number of rational points, in the case of class number 1. In 1991, Rubin showedthat for elliptic curves defined over an imaginary quadratic field K with complex multiplication byK, if the L-series of the elliptic curve was not zero at s = 1, then the p-part of the Tate–Shafarevichgroup had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes p > 7.

368 Joseph Almog

INTERMEZZO

Let us take stock and see what, if anything, has been accomplished vis-à-vis thevarious “inevitable’ dualisms we started with. At the origin, there were two suchtypes of dualisms: intra-mathematical and trans-mathematical. In the first category,I mentioned the late nineteenth century and thus “modern,” vertical form ofdualism between practiced mathematics (arithmetic, algebra, geometry, analysis,etc.) and the foundations level of a universal characteristic, whether Frege’s Logicor modern set theory. I also pointed to a more ancient, horizontal as it were,dualism between “algebra” (and things calculational) and “geometry” (and thingsimaginable-visualizable). In the trans-mathematical dimension, I mention, follow-ing Hermite–Gödel, a metaphysical dualism of realms of existence, the materialcosmos versus a “whole world” (realm) of mathematical objects; and I connected it,again following Hermite–Gödel, with an epistemological dualism of modes ofapprehension—unlike God, we are doomed to live with separate (and segregable)modes of apprehension and, consequently, knowledge: the senses that inform usabout the material cosmos and our intellect, informing us about the “other,”mathematical realm. Let us ask then: Where do we stand on this quartet ofdualisms?

I have pursued the two intra-mathematical dualisms by examining a casestudy, elementary number theory, Diophantine equations. I tried to bring out howboth the foundational dualism between practice and ultimate foundations and,now inside the practice, between the algebraic and the geometric, are illusory.Diophantine geometry practice is as unitarian as could be. Intra-mathematicalintegration shines through by looking at such simple equations as FLT or thecongruent number case.

We have not yet broached in earnest—at least not head-on—the two trans-mathematical dualisms. But even here progress was made and we gained somechips that await deployment. On the metaphysical front, the as it were “ontology”of mathematics, we have isolated the primality (in both senses of “firstness” and“fundamentality”) of the complex plane and Riemann surfaces as ur-mathematicalcategories.And on the epistemological front, we took notice of the conjectural nona priori character of investigating what I called the embodied meanings (and theirfine structures). Indeed, I analogized such investigations with the unfolding of“water’s” embodied meaning and its fine structure.

Granted, we are not yet back with one unified world—the cosmos withnature-cum-mathematics forming one whole. And granted again, we are not yetfree of the senses-versus-thought dualism of modes of apprehension, that is, we arenot yet back with one synthetic mode of apprehending the cosmos, human imagi-nation. This is the task of the complementary Part II.35

35. An autobiographical word about the growth of the present project. In presenting theseideas orally, it was pointed out to me that a central theme—(1) there is much more to the meaningof (what is said by) a Diophantine equation than meets the eye and (2) that leads us to unravelintra-mathematical symbiosis between different “areas” of mathematics—has been pursued in twoother works. One is the work of Andre Weil, in particular his “Two Lectures on Number Theory-Past and Present”; see his Collected Papers, vol. III (New York: Springer, 1979), 279–302 (where

The Cosmic Ensemble 369

APPENDIX: SOME NOTIONS FROM THE FLT PROOF

Two propositions concern us here. Together they prove FLT.The first is the weak form of the modularity conjecture used by Wiles:

(SMT) Every semistable elliptic curve over Q is modular.

The second is the Frey-Serre-Ribet conditional:

(SR) If FLT is false, (SMT) is false.

I do not discuss the proofs of either, only the notions they mention. Expansions ofthe present glosses giving technical details may be found in Fernando Gouvea, “AMarvelous Proof,” American Mathematical Monthly 101 (3, 1994): 203–22. A thor-ough technical foundation is afforded by the aforementioned text A First Course inModular Forms. Here we offer an informal summary of the key notions.

Consider first (SMT). We are focused here on an elliptic curve E over Q. Asnoted, by a theorem of Mordell, the group of rational points E(Q) is finitelygenerated. Now, in seeking (the number of) rational solutions for E, it is natural towork mod p.A field Fp is finite and so is the group E(Fp). Expounding the structureof such groups (and through it that of the more elusive E(Q)) is the basicmethodology.

Now, for the reduction to mod p to work we must make sure E is still ellipticat the Fp’s. This calls for checking the discriminant of the curve (see note 22) andverifying it is still not zero at Fp. If at p, it is zero, p is called a bad reduction. Onealso needs to check that curves that are isomorphic over Q, do not get differentreductions modulo p.

This leads to a classification of types of reductions.To code the reduction type(there are three), the notion of a conductor N is introduced, with n(p) = o for goodreduction, n(p) = 1 for almost good (“multiplicative”) reduction and n(p) " 2 forbad (“additive”) reduction. The overall value of N is the product, for all p, of n(p).

historically much richer examples are dissected on top of the Fermat case). The theme is alreadybroached beautifully (and with much pedagogical patience) in his letter of 1940 to Simone Weil(pp. 244–55 in Vol. I of the Collected Papers, especially pp. 251–54, where he is working out hisalgebraic/Riemannian “dictionary” [“Rosetta stone”]). On quite another dimension but regardingthe very statement I focus on, FLT, Curtis McMullen argues very convincingly (and with acharming quote from Molière to cement his case) that what Fermat meant with his Diophantineequation is a claim in “arithmetic topology” (about the dynamics of complex surfaces). See “FromDynamics on Surfaces to Rational Points on Curves,” Bulletin (New Series) of the AmericanMathematical Society 37 (1999): 119–40.

I owe many thanks, over the years, to discussions with Bill Tait, Sten Lindstrom, co-teachingwith Tony Martin and, ten years ago, to hard questions from my searching student DominikSklenar (alas, now, lost to the subject). Recent co-teaching and co-thinking with John Carrieroabout themes of cosmology in Descartes’ early modern science was very formative. I also owemany thanks to the second-hand remarks of J. Oesterle and L. Schneps and to the first-handcomments from M. Harris, R. Langlands, B. Mazur, W. Goldring, N. Goldring and especially to thatinimitable teacher Serge Lang.

370 Joseph Almog

An elliptic curve is a semistable curve if its conductor is a square free number.In other words, its reduction is as good as could be (good reduction at almost allprimes, almost good reduction at the rest).

Now, as for modular functions and curves. A function f(z) on the upper halfplane (thinking here of H = {z = x + iy: y > 0}) is a modular function of level N (i) iff(z) is meromorphic, even at the “cusps” (on which more in a moment) and (ii) forall integers a,b,c,d with ad-bc = 1 and N|c, we get

faz bcz d

f(z)++( ) =

We consider transformations z $ (az + b) / (cz + d) of the upper half plane asso-ciated to the group of matrices:

(0 1( :N)a bc d

a, b, c, d Z, ad-bc N c= )*

+, ={ }%

The group acts on H with quotient H/G0(N). There is a compact Riemann surfaceX0(N) such that:

H/ N) X (N) {finite set of points}0 0( ( = #

The points are the cusps. X0(N) is a modular curve of level N.The modularity theorem now reads as follows. Consider an elliptic curve over

Q, y2 = Ax3 + Bx2 + Cx + D.The theorem asserts the existence of non-constant func-tions f(z), g(z) of the same level N such that:

F(z) Ag(z) Bg(z) Cg(z) D2 3 2= + + +

For more on the sense that X0(N) thus offers a parametrization (or better (Mazur):uniformization) of the elliptic curves as well as for a genuine development of thenotions involved, see A First Course in Modular Forms. For the role of X0(2) (andthe fact that there are no cusps of weight 2 at level 2) in showing the Frey curvecould not exist (because it is semistable but not modular), see Lang Survey ofDiophantine Geometry, 130–35.

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