=0A= =0A= [Note from moderator: This is the full text of the book review discussed in= November.]=0A= =0A= This looks especially interesting to category theorists. The commented out= =3D=0A= stuff was graphical, including a photo of Grothendieck.=0A= =0A= =0A= In the Mountains of Mathematics=0A= Jim Holt=0A= =0A= December 3, 2015 Issue=0A= =0A= Mathematics Without Apologies: Portrait of a Problematic Vocation=0A= by Michael Harris=0A= Princeton University Press, 438 pp., $29.95=0A= =0A= =0A= =0A= =0A= The irony of pure mathematics begetting crass commercialism is not lost=0A= on Michael Harris, whose Mathematics Without Apologies irreverently=0A= echoes Hardy=3DE2=3D80=3D99s classic title. Harris is a distinguished midd= le-age=3D=0A= d=0A= American mathematician who works in the gloriously pure stratosphere=0A= where algebra, geometry, and number theory meet. =3DE2=3D80=3D9CThe guiding= probl=3D=0A= em=0A= for the first part of my career,=3DE2=3D80=3D9D he writes, was =3DE2=3D80= =3D9Cthe Conje=3D=0A= cture of=0A= Birch and Swinnerton-Dyer,=3DE2=3D80=3D9D which =3DE2=3D80=3D9Cconcerns the= simplest cl=3D=0A= ass of=0A= polynomial equations=3DE2=3D80=3D94elliptic curves=3DE2=3D80=3D94for which = there is no =3D=0A= simple way to=0A= decide whether the number of solutions is finite or infinite.=3DE2=3D80=3D9= D One =3D=0A= such=0A= elliptic curve is y2 =3D3D x3 - 25x, which can be shown to have infinitely= =0A= many rational solutions (that is, solutions that are whole numbers or=0A= fractions); another is y2 =3D3D x3 - x, which has only finitely many=0A= solutions. Though elementary in appearance, elliptic curves turn out to=0A= have a deep structure that makes them endlessly interesting.=0A= =0A= Harris has spent much of his research career in Paris, and it shows:=0A= his book is full of Gallic intellectual playfulness, plus references to=0A= figures like Pierre Bourdieu, Issey Miyake, and Catherine Millet (=3DE2=3D8= 0=3D9C=3D=0A= the=0A= sexual Stakhanovite=3DE2=3D80=3D9D), and mention of the endless round of Pa= risian=0A= champagne receptions where =3DE2=3D80=3D9Cmathematical notes are compared f= or the=0A= first glass or two, after which conversation reverts to university=0A= politics and gossip.=3DE2=3D80=3D9D It is rambling, sardonic (the term =3DE= 2=3D80=3D9Cf=3D=0A= uck-you=0A= money=3DE2=3D80=3D9D appears in the index), and witty. It contains fascina= ting=0A= literary digressions, such as an analysis of the occult mathematical=0A= structure of Thomas Pynchon=3DE2=3D80=3D99s novels, and lovely little inter= ludes =3D=0A= on=0A= elementary math, inspired by Harris=3DE2=3D80=3D99s gallant attempt to expl= ain nu=3D=0A= mber=0A= theory to a British actress at a Manhattan dinner party.=0A= =0A= Starting with the simple definition of a prime number, he builds, bit by=0A= bit, to an explanation of the aforementioned Birch-Swinnerton-Dyer=0A= conjecture=3DE2=3D80=3D94which, at a press conference given in Paris in the= year =3D=0A= 2000=0A= by an international group of leading mathematicians, was declared to be=0A= one of seven =3DE2=3D80=3D9CMillennium Prize Problems=3DE2=3D80=3D9D whose = resolution w=3D=0A= ould be=0A= rewarded by a million-dollar prize. The book takes an intimate look at=0A= the deepest developments in contemporary mathematics, especially the=0A= visionary work of the recently deceased Alexander Grothendieck. And it=0A= successfully conveys what Harris calls the =3DE2=3D80=3D9Cpathos=3DE2=3D80= =3D9D of the=0A= mathematician=3DE2=3D80=3D99s calling.=0A= =0A= Harris is rudely skeptical of the usual justifications for pure=0A= mathematics: that it is beautiful, true, or even much good, at least in=0A= the utilitarian =3DE2=3D80=3D9CGolden Goose=3DE2=3D80=3D9D sense. =3DE2=3D8= 0=3D9CIt is not on=3D=0A= ly dishonest but also=0A= self-defeating to pretend that research in pure mathematics is motivated=0A= by potential applications,=3DE2=3D80=3D9D he writes. He notes that public-= key=0A= cryptography, by making the world safe for Amazon, has destroyed the=0A= corner bookstore (in America, not France, where online retailers are=0A= prohibited by law from offering free shipping on discounted books). And=0A= he displays an olympian scorn for the sudden popularity of =3DE2=3D80=3D9Cf= inance=0A= mathematics,=3DE2=3D80=3D9D which offers a path to derivative-fueled wealth= on Wa=3D=0A= ll=0A= Street:=0A= =0A= A colleague boasted that Columbia=3DE2=3D80=3D99s mathematical finance = progra=3D=0A= m was=0A= underwriting the lavish daily spreads of fresh fruit, cheese, and=0A= chocolate brownies, when other departments, including mine in Paris,=0A= were lucky to offer a few teabags and a handful of cookies to=0A= calorie-starved graduate students.=0A= =0A= Even at France=3DE2=3D80=3D99s elite =3DC3=3D89cole Polytechnique, 70 perce= nt of the=0A= mathematics students today aspire to a career in finance.=0A= =0A= Nor is Harris impressed with the claim, voiced by Hardy and so many=0A= others, that pure mathematics is justified by its beauty. When=0A= mathematicians talk about beauty, he tells us, what they really mean is=0A= pleasure. =3DE2=3D80=3D9COutside this relaxed field, it=3DE2=3D80=3D99s con= sidered poor=3D=0A= form to=0A= admit that we are motivated by pleasure,=3DE2=3D80=3D9D Harris writes. =3DE= 2=3D80=3D9CA=3D=0A= esthetics is=0A= a way of reconciling this motivation with the =3DE2=3D80=3D98lofty habit of= mind.=3D=0A= =3DE2=3D80=3D99=3DE2=3D80=3D9D=0A= =0A= Why should society pay for a small group of people to exercise their=0A= creative powers on something they enjoy? =3DE2=3D80=3D9CIf a government min= ister =3D=0A= asked=0A= me that question,=3DE2=3D80=3D9D Harris writes,=0A= =0A= I could claim that mathematicians, like other academics, are needed=0A= in the universities to teach a specific population of students the=0A= skills needed for the development of a technological society and to keep=0A= a somewhat broader population of students occupied with courses that=0A= serve to crush the dreams of superfluous applicants to particularly=0A= desirable professions (as freshman calculus used to be a formal=0A= requirement to enter medical school in the United States).=0A= =0A= Although physicians don=3DE2=3D80=3D99t really need calculus, Harris at lea= st con=3D=0A= cedes=0A= that engineers, economists, and inventory managers couldn=3DE2=3D80=3D99t g= et by=0A= without a fair amount of math, even if it is trivial math by his lights.=0A= =0A= Finally, there is the presumed value of mathematical truth. Since the=0A= ancient Greeks, mathematics has been taken as a paradigm of knowledge:=0A= certain, timeless, necessary. But knowledge of what? Do the truths=0A= discovered by mathematics describe an eternal and otherworldly realm of=0A= objects=3DE2=3D80=3D94perfect circles and so forth=3DE2=3D80=3D94that exist= quite indep=3D=0A= endently of=0A= the mathematicians who contemplate them? Or are mathematical objects=0A= actually human constructions, existing only in our minds? Or, more=0A= radically still, could it be that pure mathematics doesn=3DE2=3D80=3D99t re= ally=0A= describe any objects at all, that it is just an elaborate game of formal=0A= symbols played with pencil and paper?=0A= =0A= The question of what mathematics is really about is one that continues=0A= to vex philosophers, but it does not much worry Harris. Philosophers=0A= who concern themselves with the problems of mathematical existence and=0A= truth, he claims, typically pay little attention to what mathematicians=0A= actually do. He invidiously contrasts what he calls =3DE2=3D80=3D9Cphiloso= phy of=0A= Mathematics=3DE2=3D80=3D9D (with a capital M)=3DE2=3D80=3D94=3DE2=3D80=3D9C= a purely hypotheti=3D=0A= cal subject invented=0A= by philosophers=3DE2=3D80=3D9D=3DE2=3D80=3D94with =3DE2=3D80=3D9Cphilosophy= of (small-m) math=3D=0A= ematics,=3DE2=3D80=3D9D which takes=0A= as its starting point not a priori questions about epistemology and=0A= ontology, but rather the activity of working mathematicians.=0A= =0A= Here, Harris is being a little unfair. He fails to remark that the=0A= standard competing positions in the philosophy of mathematics were=0A= originally staked out not by philosophers, but by mathematicians=3DE2=3D80= =3D94in=3D=0A= deed,=0A= some of the greatest of the last century. It was David Hilbert=3DE2=3D80= =3D94a=0A= =3DE2=3D80=3D9Csupergiant,=3DE2=3D80=3D9D in Harris=3DE2=3D80=3D99s estimat= ion=3DE2=3D80=3D94who or=3D=0A= iginated =3DE2=3D80=3D9Cformalism,=3DE2=3D80=3D9D which=0A= views higher mathematics as a game played with formal symbols. Henri=0A= Poincar=3DC3=3DA9 (another =3DE2=3D80=3D9Csupergiant=3DE2=3D80=3D9D), Herma= nn Weyl, and L.E=3D=0A= .J. Brouwer were=0A= behind =3DE2=3D80=3D9Cintuitionism,=3DE2=3D80=3D9D according to which numbe= rs and other=3D=0A= mathematical=0A= objects are mind-dependent constructions. Bertrand Russell and Alfred=0A= North Whitehead took the position known as =3DE2=3D80=3D9Clogicism,=3DE2=3D= 80=3D9D ende=3D=0A= avoring to=0A= show in their massive Principia Mathematica that mathematics was really=0A= logic in disguise. And =3DE2=3D80=3D9Cplatonism=3DE2=3D80=3D9D=3DE2=3D80= =3D94the idea that m=3D=0A= athematics describes=0A= a perfect and eternal realm of mind-independent objects, like Plato=3DE2=3D= 80=3D=0A= =3D99s=0A= world of Forms=3DE2=3D80=3D94was championed by Kurt G=3DC3=3DB6del.=0A= =0A= All of these mathematical figures were passionately engaged in what=0A= Harris slights as philosophy of Mathematics-with-a-capital-M. The=0A= debate among them and their partisans was fierce in the 1920s, often=0A= spilling over into personal animus. And no wonder: mathematics at the=0A= time was undergoing a =3DE2=3D80=3D9Ccrisis=3DE2=3D80=3D9D that had resulte= d from a ser=3D=0A= ies of=0A= confidence-shaking developments, like the emergence of non-Euclidean=0A= geometries and the discovery of paradoxes in set theory. If the old=0A= ideal of certainty was to be salvaged, it was felt, mathematics had to=0A= be put on a new and secure foundation. At issue was the very way=0A= mathematics would be practiced: what types of proof would be accepted=0A= as valid, what uses of infinity would be permitted.=0A= =0A= For reasons both technical and philosophical, none of the competing=0A= foundational programs of the early twentieth century proved=0A= satisfactory. (G=3DC3=3DB6del=3DE2=3D80=3D99s =3DE2=3D80=3D9Cincompletenes= s theorems,=3DE2=3D=0A= =3D80=3D9D in particular,=0A= created insuperable problems both for Hilbert=3DE2=3D80=3D99s formalism and= for=0A= Russell and Whitehead=3DE2=3D80=3D99s logicism: they showed=3DE2=3D80=3D94= roughly spea=3D=0A= king=3DE2=3D80=3D94that the=0A= rules of Hilbert=3DE2=3D80=3D99s mathematical =3DE2=3D80=3D9Cgame=3DE2=3D80= =3D9D could never =3D=0A= be proved consistent,=0A= and that a logical system like that of Russell and Whitehead could never=0A= capture all mathematical truths.) The issues of mathematical existence=0A= and truth remain unresolved, and philosophers have continued to grapple=0A= with them, if inconclusively=3DE2=3D80=3D94as witness the frank title that = Hilary=0A= Putnam gave to a 1979 paper: =3DE2=3D80=3D9CPhilosophy of Mathematics: Wh= y Noth=3D=0A= ing=0A= Works.=3DE2=3D80=3D9D=0A= =0A= To Harris, this looks a bit vieux jeu. The sense of crisis in the=0A= profession, so acute less than a century ago, has receded; the old=0A= difficulties have been patched up or papered over. If you ask a=0A= contemporary mathematician to declare a philosophical party affiliation,=0A= the joke goes, you=3DE2=3D80=3D99ll hear =3DE2=3D80=3D9Cplatonist=3DE2=3D80= =3D9D on weekdays =3D=0A= and =3DE2=3D80=3D9Cformalist=3DE2=3D80=3D9D on=0A= Sundays: that is, when they=3DE2=3D80=3D99re working at mathematics, mathe= matici=3D=0A= ans=0A= regard it as being about a mind-independent reality; but when they=3DE2=3D8= 0=3D99=3D=0A= re in=0A= a reflective mood, many claim to believe that it=3DE2=3D80=3D99s just a mea= ningle=3D=0A= ss=0A= game played with formal symbols.=0A= =0A= Today, as Harris observes, paradigm shifts in mathematics have less to=0A= do with =3DE2=3D80=3D9Ccrisis=3DE2=3D80=3D9D and more to do with finding su= perior metho=3D=0A= ds. It used=0A= to be thought, for example, that all mathematics could be constructed=0A= out of sets. Starting with the simple idea of one thing being a member=0A= of another, set theory shows how structures of seemingly limitless=0A= complexity=3DE2=3D80=3D94number systems, geometrical spaces, a never-ending= hiera=3D=0A= rchy=0A= of infinities=3DE2=3D80=3D94can be built up out of the most modest material= s. Th=3D=0A= e=0A= number zero, for example, can be defined as the =3DE2=3D80=3D9Cempty set=3D= E2=3D80=3D9D=3D=0A= : that is,=0A= the set that has no members at all. The number one can then be defined=0A= as the set that contains one element=3DE2=3D80=3D94zero and nothing else. = Two, i=3D=0A= n=0A= turn, can be defined as the set that contains zero and one=3DE2=3D80=3D94an= d so o=3D=0A= n,=0A= with the set for each subsequent number containing the sets for all the=0A= previous numbers. Numbers, instead of being taken as basic, can thus be=0A= viewed as pure sets of increasingly intricate structure.=0A= =0A= % holt_2-120315.jpg Cath=3DC3=3DA9rine Goldstein =3DC2=3DA9 2015 Artists R= ights So=3D=0A= ciety=0A= % (ARS), New York/ADAGP, Paris A mathematical formula for love by Isidore= =0A= % Isou, 1988; from Michael Harris=3DE2=3D80=3D99s Mathematics Without Apol= ogies=0A= =0A= In the 1930s, a cabal of brilliant young Paris mathematicians, including=0A= Andr=3DC3=3DA9 Weil, resolved to make the house of mathematics more secure = by=0A= rebuilding it on the logical foundation of set theory. The project,=0A= under the collective nom de guerre =3DE2=3D80=3D9CBourbaki,=3DE2=3D80=3D9D = went on for =3D=0A= decades,=0A= resulting in one fat treatise after another. Among its consequences,=0A= crazily enough, was the advent of the =3DE2=3D80=3D9Cnew math=3DE2=3D80=3D9= D education =3D=0A= reforms back=0A= in the 1960s, which so befuddled American schoolchildren and their=0A= parents by replacing intuitive talk of numbers with the alien jargon of=0A= sets.=0A= =0A= Physicists talk about finding the =3DE2=3D80=3D9Ctheory of everything=3DE2= =3D80=3D9D; w=3D=0A= ell, set=0A= theory is so sweeping in its generality that it might appear to be (as=0A= Harris quips) =3DE2=3D80=3D9Cthe theory of theories of everything.=3DE2=3D8= 0=3D9D It ce=3D=0A= rtainly=0A= appeared that way to the members of Bourbaki. Yet a few decades after=0A= their program got underway, the extraordinary Alexander Grothendieck=0A= came into their midst and transcended it. In doing so, he created a new=0A= style of pure mathematics that proved as fruitful as it was dizzyingly=0A= abstract. Long before his death last November at the age of eighty-six=0A= in a remote hamlet in the Pyrenees, Grothendieck had come to be regarded=0A= as the greatest mathematician of the last half-century. As Harris=0A= observes, he likely qualifies as the =3DE2=3D80=3D9Cmost romantic,=3DE2=3D8= 0=3D9D too: =3D=0A= =3DE2=3D80=3D9Chis life=0A= story begs for fictional treatment.=3DE2=3D80=3D9D=0A= =0A= The raw facts are astounding enough. Alexander Grothendieck was born in=0A= Berlin in 1928 to parents who were both active anarchists. His father,=0A= a Russian Jew, took part in the 1905 uprising against the tsarist regime=0A= and the 1917 revolution. He escaped imprisonment under the Bolsheviks;=0A= clashed with Nazi thugs on the streets of Berlin; fought on the=0A= Republican side in the Spanish civil war (as did Grothendieck=3DE2=3D80=3D9= 9s mot=3D=0A= her);=0A= and was deported, after the fall of France, from Paris to Auschwitz to=0A= be murdered.=0A= =0A= His mother, a gentile from Hamburg, raised Grothendieck in the south of=0A= France. There the boy showed talent both for numbers and for boxing.=0A= After the war, he made his way to Paris to study mathematics under the=0A= great Henri Cartan. Following early teaching stints in S=3DC3=3DA3o Paulo,= =0A= Kansas, and Harvard, Grothendieck was invited in 1958 to join the=0A= Institut des Hautes =3DC3=3D89tudes Scientifiques, which had just been foun= ded =3D=0A= by=0A= a private businessman outside Paris in the woods of Bois-Marie. There=0A= Grothendieck spent the next dozen years astonishing his elite colleagues=0A= and younger disciples as he recreated the landscape of higher=0A= mathematics.=0A= =0A= % Grothendieck was a physically imposing man, shaven-headed and handsome, = =3D=0A= as charismatic as he was austere. A staunch pacifist and antimilitarist, he= =3D=0A= refused to go to Moscow in 1966 (where the International Congress of Mathe= =3D=0A= matics was being held) to accept the Fields Medal, the highest honor in mat= =3D=0A= hematics. He did, however, make a trip the next year to North Vietnam, wher= =3D=0A= e he lectured on pure mathematics in the jungle to students who had been ev= =3D=0A= acuated from Hanoi to escape the American bombing. He remained (by choice) = =3D=0A= stateless most of his life, had three children by a woman he married and tw= =3D=0A= o more out of wedlock, founded the radical ecology group Survivre et Vivre,= =3D=0A= and once got arrested for knocking down a couple of gendarmes at a politic= =3D=0A= al demonstration in Avignon.=0A= % holt_3-120315.jpg=0A= % Erika Ifang=0A= % Alexander Grothendieck, 1988=0A= =0A= Owing to his unyielding and sometimes paranoiac sense of integrity,=0A= Grothendieck ended up alienating himself from the French mathematical=0A= establishment. In the early 1990s he vanished into the Pyrenees=3DE2=3D80= =3D94wh=3D=0A= ere,=0A= it was reported by the handful of admirers who managed to track him=0A= down, he spent his remaining years subsisting on dandelion soup and=0A= meditating on how a malign metaphysical force was destroying the divine=0A= harmony of the world, possibly by slightly altering the speed of light.=0A= The local villagers were said to look after him.=0A= =0A= Grothendieck=3DE2=3D80=3D99s vision of mathematics led him to develop a new= =0A= language=3DE2=3D80=3D94it might even be called an =3DE2=3D80=3D9Cideology= =3DE2=3D80=3D9D=3DE2=3D=0A= =3D80=3D94in which hitherto=0A= unimaginable ideas could be expressed. He was the first, Harris=0A= observes (breaking into emphatic boldface), =3DE2=3D80=3D9Cto be guided by = the=0A= principle that knowing a mathematical object is tantamount to knowing=0A= its relations to all other objects of the same kind.=3DE2=3D80=3D9D In othe= r word=3D=0A= s, if=0A= you want to know the real nature of a mathematical object, don=3DE2=3D80=3D= 99t lo=3D=0A= ok=0A= inside of it, but see how it plays with its peers.=0A= =0A= Such a peer group of mathematical objects is called, in a deliberate nod=0A= to Aristotle and Kant, a =3DE2=3D80=3D9Ccategory.=3DE2=3D80=3D9D One catego= ry might con=3D=0A= sist of=0A= abstract surfaces. These surfaces play together, in the sense that=0A= there are natural ways of going back and forth between them that respect=0A= their general form. For example, if two surfaces have the same number=0A= of holes=3DE2=3D80=3D94like a donut and a coffee mug=3DE2=3D80=3D94one surf= ace can, mat=3D=0A= hematically,=0A= be smoothly transformed into the other.=0A= =0A= Another category might consist of all the different algebraic systems=0A= that have an operation akin to multiplication; these algebras too play=0A= together, in the sense that there are natural ways of going back and=0A= forth between them that respect their common multiplicative structure.=0A= Such structure-preserving back-and-forth relations among objects in the=0A= same category are called =3DE2=3D80=3D9Cmorphisms,=3DE2=3D80=3D9D or someti= mes=3DE2=3D80=3D94=3D=0A= to stress their=0A= abstract nature=3DE2=3D80=3D94=3DE2=3D80=3D9Carrows.=3DE2=3D80=3D9D They de= termine the overal=3D=0A= l shape of the play=0A= within a category.=0A= =0A= And here=3DE2=3D80=3D99s where it gets interesting: the play in one catego= ry=3DE2=3D=0A= =3D80=3D94say, the=0A= category of surfaces=3DE2=3D80=3D94might be subtly mimicked by the play in= =0A= another=3DE2=3D80=3D94say, the category of algebras. The two categories th= emselv=3D=0A= es=0A= can be seen to play together: there is a natural way of going back and=0A= forth between them, called a =3DE2=3D80=3D9Cfunctor.=3DE2=3D80=3D9D Armed w= ith such a f=3D=0A= unctor, one=0A= can reason quite generally about both categories, without getting bogged=0A= down in the particular details of each. It might also be noticed that,=0A= since categories play with one another, they themselves form a category:=0A= the category of categories.=0A= =0A= Category theory was invented in the 1940s by Saunders Mac Lane of the=0A= University of Chicago and Samuel Eilenberg of Columbia. At first it was=0A= regarded dubiously by many mathematicians, earning the nickname=0A= =3DE2=3D80=3D9Cabstract nonsense.=3DE2=3D80=3D9D How could such a rarefied = approach to =3D=0A= mathematics,=0A= in which nearly all its classical content seemed to be drained away,=0A= result in anything but sterility? Yet Grothendieck made it sing.=0A= Between 1958 and 1970, he used category theory to create novel=0A= structures of unexampled richness. Since then, the heady abstractions=0A= of category theory have also become useful in theoretical physics,=0A= computer science, logic, and philosophy.*=0A= =0A= The project undertaken by Grothendieck was one that began with=0A= Descartes: the unification of geometry and algebra. These have been=0A= likened to the yin and yang of mathematics: geometry is space, algebra=0A= is time; geometry is like painting, algebra is like music; and so on.=0A= Less fancifully, geometry is about form, algebra is about structure=3DE2=3D= 80=3D=0A= =3D94in=0A= particular, the structure that lurks within equations. And as Descartes=0A= showed with his invention of =3DE2=3D80=3D9CCartesian coordinates,=3DE2=3D8= 0=3D9D equat=3D=0A= ions can=0A= describe forms: x2 + y2 =3D3D 1, for example, describes a circle of radius= =0A= 1. So algebra and geometry turn out to be intimately related, exchanging=0A= what Andr=3DC3=3DA9 Weil called =3DE2=3D80=3D9Csubtle caresses.=3DE2=3D80= =3D9D=0A= =0A= In the 1940s, thanks to Weil=3DE2=3D80=3D99s insight, it became apparent th= at the=0A= dialectic between geometry and algebra was the key to resolving some of=0A= the most stubbornly enduring mysteries in mathematics. And it was=0A= Grothendieck=3DE2=3D80=3D99s labor that raised this dialectic to such a pit= ch of=0A= abstraction=3DE2=3D80=3D94one that was said to leave even the great Weil da= unted=3D=0A= =3DE2=3D80=3D94that=0A= a new understanding of these mysteries emerged. Grothendieck laid the=0A= groundwork for many of the greatest mathematical advances in recent=0A= decades, including the 1994 proof of Fermat=3DE2=3D80=3D99s Last Theorem=3D= E2=3D80=3D94=3D=0A= a magnificent=0A= intellectual achievement of zero practical or commercial interest.=0A= =0A= Harris, whose own impressive work is much indebted to Grothendieck, is=0A= eloquent in praising his =3DE2=3D80=3D9Cruthless=3DE2=3D80=3DA6 minimalism,= =3DE2=3D80=3D9D wh=3D=0A= ich extended to his=0A= contempt for money and his monkish wardrobe. I only wish that Harris=0A= had more conspicuously credited another figure in the modern=0A= transformation of mathematics, Emmy Noether. It was Noether, born in=0A= Bavaria in 1882, who largely created the abstract approach that inspired=0A= category theory. Yet as a woman in a male academic world, she was=0A= barred from holding a professorship in G=3DC3=3DB6ttingen, and the classici= sts=0A= and historians on the faculty even tried to block her from giving unpaid=0A= lectures=3DE2=3D80=3D94leading David Hilbert, the dean of German mathematic= s, to=0A= comment, =3DE2=3D80=3D9CI see no reason why her sex should be an impediment= to he=3D=0A= r=0A= appointment. After all, we are a university, not a bathhouse.=3DE2=3D80=3D= 9D Noe=3D=0A= ther,=0A= who was Jewish, fled to the United States when the Nazis took power,=0A= teaching at Bryn Mawr until her death from a sudden infection in 1935.=0A= =0A= The intellectual habit of grappling with a problem by ascending to=0A= higher and higher levels of generality came naturally to Emmy Noether,=0A= and it was shared by Grothendieck, who said that he liked to solve a=0A= problem not by the =3DE2=3D80=3D9Chammer-and-chisel method,=3DE2=3D80=3D9D = but by letti=3D=0A= ng a sea of=0A= abstraction rise to =3DE2=3D80=3D9Csubmerge and dissolve=3DE2=3D80=3D9D it.= In his vis=3D=0A= ion, the=0A= familiar things dealt with by mathematicians, like equations, functions,=0A= and even geometrical points, were reborn as vastly more complex and=0A= versatile structures. The old things turned out to be mere shadows=3DE2=3D= 80=3D=0A= =3D94or,=0A= as Grothendieck preferred to call them, =3DE2=3D80=3D9Cavatars=3DE2=3D80=3D= 9D=3DE2=3D80=3D94o=3D=0A= f the new. (An=0A= avatar is originally an earthly manifestation of a Hindu god; as Harris=0A= notes, =3DE2=3D80=3D9Ca taste for Indian=3DE2=3D80=3DA6metaphysics inflecte= d [the] term=3D=0A= inology=3DE2=3D80=3D9D of=0A= many French mathematicians.)=0A= =0A= Nor is this a one-off process. Each new abstraction is eventually=0A= revealed to be but an avatar of a still-higher abstraction. As Harris=0A= puts it, =3DE2=3D80=3D9Cthe available concepts are interpreted as the avata= rs of =3D=0A= the=0A= inaccessible concepts we are trying to grasp.=3DE2=3D80=3D9D With the grasp= ing of=0A= these new concepts, mathematics ascends a kind of =3DE2=3D80=3D9Cladder=3DE= 2=3D80=3D9D =3D=0A= of increasing=0A= abstraction.=0A= =0A= This, Harris says, is what philosophers should be paying attention to.=0A= =3DE2=3D80=3D9CIf you were to ask for a single characteristic of contempora= ry=0A= mathematics that cries out for philosophical analysis,=3DE2=3D80=3D9D he wr= ites, =3D=0A= =3DE2=3D80=3D9CI=0A= would advise you to practice climbing the categorical and avatar ladders=0A= in search of meaning, rather than searching for solid Foundations.=3DE2=3D8= 0=3D9D=3D=0A= And=0A= what lies at the top of this ladder? Perhaps, Harris suggests with=0A= playful seriousness, there is One Big Theorem from which all of=0A= mathematics ultimately flows=3DE2=3D80=3D94=3DE2=3D80=3D9Csomething on the = order of sam=3D=0A= sara =3D3D=0A= nirvana.=3DE2=3D80=3D9D But since there are infinitely many rungs to climb,= it is=0A= unattainable.=0A= =0A= Here, then, is the pathos of mathematics. Unlike theoretical physics,=0A= which can aspire to a =3DE2=3D80=3D9Cfinal theory=3DE2=3D80=3D9D that would= account for=3D=0A= all the=0A= forces and particles in the universe, pure mathematics must concede the=0A= futility of its own quest for ultimate truth. As Harris observes,=0A= =3DE2=3D80=3D9Cevery veil lifted only reveals another veil.=3DE2=3D80=3D9D = The mathemat=3D=0A= ician is=0A= doomed to what Andr=3DC3=3DA9 Weil called an endless cycle of =3DE2=3D80=3D= 9Cknowledg=3D=0A= e and=0A= indifference.=3DE2=3D80=3D9D=0A= =0A= But it could be worse. Thanks to G=3DC3=3DB6del=3DE2=3D80=3D99s second inc= ompletenes=3D=0A= s=0A= theorem=3DE2=3D80=3D94the one that says, roughly, that mathematics can neve= r prov=3D=0A= e its=0A= own consistency=3DE2=3D80=3D94mathematicians can=3DE2=3D80=3D99t be fully c= onfident tha=3D=0A= t the axioms=0A= underlying their enterprise do not harbor an as-yet-undiscovered logical=0A= contradiction. This possibility is =3DE2=3D80=3D9Cextremely unsettling for= any=0A= rational mind,=3DE2=3D80=3D9D declared the Russian-born mathematician (and = Fields=0A= medalist) Vladimir Voevodsky, in a speech on the occasion of the=0A= eightieth anniversary of the Institute for Advanced Study. Indeed, the=0A= discovery of such an inconsistency would be fatal to pure mathematics,=0A= at least as we know it today. The distinction between truth and=0A= falsehood would be breached, the ladder of avatars would come crashing=0A= down, and the One Big Theorem would take a truly terrible form: 0 =3D3D 1.= =0A= Yet, oddly enough, e-commerce and financial derivatives would be left=0A= untouched.=0A= =0A= * A charming and improbably successful attempt to explain category=0A= theory in culinary terms is made by the mathematician Eugenia Cheng in=0A= her new book How to Bake Pi: An Edible Exploration of the Mathematics=0A= of Mathematics (Basic Books, 2015). ?=0A= =0A= Sender: categories@mta.ca Precedence: bulk Reply-To: Michael Barr <barr@math.mcgill.ca>=0A= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]