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"At the ICM 2006 in Madrid I attended a lecture by Manin speaking about the different uses of mathematics, as models,\ntheories, and metaphors. Of all the lectures I attended at that congress, this was the one that stuck out to me. It was\nobviously not a technical talk, but a philosophical one in the best sense of the term, namely, fuelled not by\nprofessional pedantry, but by a deep personal curiosity expressed in a very original and captivating way. A year later,\na collection of Manin’s essays had been translated into English and handsomely published by the AMS under the title of\n‘Mathematics as Metaphor.’ I got the book, read it with delight, as I had read previous books by him as a young man,\nand in fact I wrote a review of it which was published in 2010 in the EMS Newsletter – incidentally, a fact I had\nalready forgotten this spring. However, I was alerted to it and learned that I had at its end expressed my regret that\nnot more of his essays were available to readers not knowing Russian. Now my wish has been granted. That a wider\ncollection had recently been published, I actually found out from Manin himself in what would turn out to be my last\ncommunication with him. I immediately got the book, published by the small French firm, Les Belles Letters, and thus\ncontaining French translations of his texts. The AMS version was about 200 pages, while this edition runs well over 500\npages, so one surmises that it is a very significant extension. On the other hand, a mere page count is a bit\nmisleading because the pages of the first edition are larger than those in the latter one and also the font size\nemployed is somewhat smaller. I estimate that the American edition sports about 3500 characters a page, and the French\nedition about 2000 characters, but still we are talking about a significant extension. My first intention was to single\nout what was new in the extended edition and concentrate on that, but I have decided to abandon that and treat it as\na whole, fully independent of my first review.\n"
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"We are talking about essays, not scientific articles, and there is of course a significant difference between the two.\nAn essay is, like the terminology indicates, an attempt. Namely, an attempt to come to grips with a subject in a\nnon-technical way using a meta-perspective. You should not write a scientific article if you are not an expert, but\nanyone is welcome to write an essay on any subject that occurs to them (they need not be published). In fact, any such\nattempt reminds me of the American diplomat George Kennan, who during his career wrote many dispatches from his various\npostings with scant hope that they would ever be read, but justifying his activity by claiming that he wrote in order to\ndiscover what he thought. This points to a crucial aspect of essay writing, namely, exploration. Karl Popper did not,\nunlike his colleagues in the Vienna Circle (disparaged by posterity as positivists) reject metaphysics, instead he was\nthinking of it as proto-science, potentially developing into one.\n"
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"As indicated, most essays in general may be ignored (which does not necessarily mean that writing them is a useless\nactivity); what makes Manin’s essays worth pondering is the originality of his mind and imagination, the precision of\nhis formulations, all supported by his wide culture, and the boundless curiosity which made this culture possible.\nEssays should be classified as literature, and thus subjected to the demanding criteria such writing invites.\nImagination requires obstacles to be circumvented in order to be properly stimulated; this is why, according to\nHilbert, mathematics requires more imagination than poetry, or, as claimed by the biographer Peter Acroyd, the writing\nof a biography requires more imagination than the writing of a novel. But in this general frame there are different\nkinds of imaginations, the iron-clad laws of logic typically lead to frustration, while writing essays and fiction\nleaves you more liberty. Arguments need not to be watertight as long as they are exciting, and inconvenient facts can\nbe ignored or simply made up, as typically in fiction; what matters are the ideas, which need not be technically\ndeveloped. Thus, I cannot resist speculating that the writing of essays (and poetry?) gave Manin a relief from the\nrigors of mathematical work, but this does not necessarily mean that it should be thought of as a mere diversion – on\nthe contrary, it was an essential component of his mathematical work, without which the latter may not have been\npossible. His essays are also more accessible to readers, provided they have the required temperament, than his purely\nmathematical work, although the charm of the latter derives much from being presented in an essayistic spirit (this is\nwhy the above-mentioned books made such an impression on my young mind)."
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"The point of an essay is not only to profit the writer but also to inform and inspire the reader, this is why it is\nvery hard for me not to elaborate on Manin’s essays,\nand to just present sober resumes; but then again, they are\npublished and available for everyone to read and engage with in their own ways, so I hope that my taking of liberties\ncan be excused as a kind of homage."
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"First, what is the nature of mathematics? This is a question that cannot be treated mathematically, but nevertheless\nmust at least to some degree engage every serious mathematician, and even influence the way and why they persist in\ntheir obsessions. Manin himself is puzzled why mathematics engages him so much, yet without this potential skepticism\nin any way dampening his enthusiasm for the subject. Now there is a vulgar idea of mathematics, prevalent not so much\namong the general public as among philosophers and physicists and other concerned academics. Mathematics is, according\nto this view, seen as a game; you set up some axioms as rules and then apply logic to it and grind away. From this it\ndoes not take much to conclude that mathematics is just a matter of symbolic manipulation, and although its concepts do\nnot have any real meaning (like vertices in a graph), it can still amazingly serve as a useful language and even tool\nin the study of the real world. The idea that mathematics is applied logic goes at least as far back as Frege and was\nfurther developed by his successors Russell and Wittgenstein. On the other hand, the American philosopher Charles\nPeirce claimed that the integers were more basic than logic, and that mathematicians had no need to study logic, they\nwere anyway able to instinctively draw the necessary conclusions, on which mathematics rests and develops. The emphasis\non logic has led to the dictum that mathematics is but a sequence of tautologies, which has been taken to heart by\nmany. Any idea that has spread successfully must have some truth to it, so it is admittedly true that a large part of a\nmathematician’s everyday work may amount to a ceaseless manipulation of symbols. Manin cites, not without approval,\nthe claim by Schopenhauer that when computation begins, thought ends. Mathematics is indeed a very special activity\nwhich delights in reasoning using long deductive chains and thereby coming up with true facts in a systematic way. We\nrecall Leibniz’s exhortation, stop arguing let us calculate, hoping there would be a verbal calculus which would resolve\nhuman problems as neatly as celestial ones (for which calculus was once invented). Manin insists that the logical\nstraight-jacket that mathematics is forced into is necessary – without it, it would degenerate, as anything to remain\nsolid has to be contained. It is the possibility of falsification, that allows things to grow purposefully by pruning\noff false leads.\nIt is also this that leads to the frustrations of mathematicians, by the presence of\nwhat which\ncannot be willed away. But for the serious mathematicians there is\nalso something else to mathematics without which they would never pursue it. Mathematics involves more than a random\nwalk in a logical configuration space. It requires thinking in a natural language, a thinking that is not in the nature\nof a computation in some generalized sense, but is meta-thinking whose mission is not to produce new facts, but to\ndistinguish between the interesting and the fruitful, of coming up with new ideas and strategies. Without this\nmeta-thinking mathematics would be a sterile subject indeed. In fact, what the serious mathematician aims for is the\nelusive goal of understanding, of seeing different pieces coming together, something which cannot be conveyed by mere\nmathematical formulations, just as little as ideas can be precisely formalized and expressed, at best only conveyed\nobliquely, and in this elusive vagueness lies their power. One important difference between a natural language and a\nformal artificial one is that the latter is precise, while the former is vague; as a result, the latter can be treated\nas a mathematical object. Being vague, natural languages have a recourse to forming metaphors, which, I never tire of\npointing out, should never be taken literally, as they then become merely silly; while metaphors in formal languages\nhave no choice but be taken literally. In a natural language nothing stops you from imagining the set of all sets (or\nthe wish to have all ones wishes granted), but in a formal, strict logical setting one is forced to make explicit the\ndifferent notions of ‘set’ involved and be forced to adopt a new word for one of them, such as ‘class.’ The Russell\nparadox does not affect natural languages, as they thrive on contradictions – in fact, languages evolved socially,\nmeaning in particular that expressing truth is not necessarily the main purpose, rather deception; which incidentally\nties up with Manin’s fascination with the ‘Trickster.’ Thus, the metaphorical idea of the diagonal argument\nwhen\napplied\n‘literally’\n(in the sense of rigidly logical) has interesting consequences. At the heart of Gödel’s\nargument, as Manin points out, is this partial embedding of the meta-language into a formal one on which it comments.\nIncidentally, there is much hype connected with Gödel’s theorem and Manin’s excellent presentation of it has as a\npurpose to demystify it. As he notes, the theorem has had marginal influence on mathematics as practiced."
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"What is mathematical intuition? Mathematical and logical concepts are anchored in a physical and hence tangible reality\nin the human mind. Numbers are in particular associated with the counting of physical objects, such as buttons and\nshells. One may talk about small numbers such as billions and trillions when they can so be concretely represented; but\nwith the advent of the positional system of representing numbers one was able and hence seduced to write down huge\nnumbers with millions of digits, numbers that in no way can be represented by the counting of physical objects of any\nkind, only of imagined objects of the mind, such as all possible books in Borges’ celebrated story. Let us call such\nnumbers, numbers of the second kind, which for all practical matters can serve as (countable) infinities. Then\nof course there are numbers of the third kind, represented by those which need a number of second kind to count their\ndigits, and we can proceed inductively, and the whole thing carries an uncanny analogue of Cantor’s hierarchy of\ninfinities, except there are of course no precise boundaries between them, but the idea remains (one could of course\nimpose precise demarcations, but that would be artificial and pointless). We are in the realm of natural language after\nall, where precision is not required. Of course, they are all finite, but even finite numbers can be unbelievably large\nand induce a sense of vertigo our usual congress with infinity does not involve. What is easier than suggesting\ninfinity by a sequence of dots ",
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"It is tempting to insert a slight digression here, touching upon Manin’s interest in Kolmogorov complexity. It is\ntrivial to write down numbers of any kind by using specialized notation (or more generalized inductively-defined\nfunctions), but the generic number of, say, the third kind cannot be physically represented in, say, decimal form,\nwhich is the type of form that in general is the most efficient. So in what sense can we get our hands on them? How\nmany ‘7’s are there in the decimal representation of a number of type ",
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"? Can any solution to this\nproblem be feasibly described in any other sense than by the question itself? Maybe an interesting example of a totally\nuninteresting question."
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"Metaphors are important for human thinking, and Manin brings up the notion of the Turing Machine and the influence it\nhad on logic. Yes, machines are tangible objects of the imagination and embody themselves logic in a palpable way –\nafter all, their parts are connected in long chains of causes and effects, like the deductive chains in logical\nreasoning. Classically, they were represented by the sophisticated machinery of a clockwork; nowadays, we have the\ncomputer, although its machinery is not so much exhibited in its hardware, of which most users are blissfully ignorant,\nbut in its software when the old tinkering with cogwheels has been replaced by letting the fingers dance on the\nkeyboard instead, through writing computer codes. As David Mumford has pointed out, a mathematical proof and a computer\nprogram have much in common. Indeed, the word ‘mechanical’ is what we use in describing a mindless manipulation of\nobjects subjected to inexorable laws outside our control."
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"Set theory was created by Cantor by taking infinities very literally as objects to be mathematically handled (but one\nmay argue that infinite convergent sums actually involve a literal, not only potential infinity, and go back to\nantiquity – just think of Zeno). The uncountability of the reals is something most of us encounter in our teens, and\nit is usually considered as something rather metaphysical, apart from\nmainstream mathematics. However, without the\nnegative aspect of the uncountability of the reals modern measure theory with its countable additivity would be\nimpossible. For it to work, the setting has to be uncountable, and that uncountability could indeed be seen as the\nmetaphysical setting of all those manipulations. It stands to reason that such a theory would have been developed\nsooner or later and then the uncountability of the reals would have been staring in our faces. Cantor’s hierarchy of\ninfinities met a lot of resistance when it appeared, Manin reminds us, and also a lot of skepticism as it was\ndeveloped. As it is based on human mathematical intuition involving the manipulation of physical objects, which has no\nlonger any relevance, that ordinary expectations would come to grief is not surprising. What could be more natural than\npicking one object each from a collection of non-empty sets, but the Axiom of Choice has very counterintuitive\nconsequences when applied in, say, an uncountable context, giving rise to the Banach–Tarski paradox, or the\nwell-ordering of the reals. The fate of the continuum hypothesis is a case in point, the physical intuition was that\nhere it was, a subset of the real line just in front of our eyes, it had to be true or not. But it turned out to be a\nquestion of mere convention, what rules are allowed or not in forming subsets. Thus, it degenerated to a formal game\nhaving no relation whatsoever to our conception of physical reality. The very notion of mathematical Platonism seemed\nto founder when exploring the transfinite world, where we seem at liberty to bend the rules at our discretion. ‘What did\nthe paradoxes and problems of set theory have to do with the solidity of a bridge?’ – Ulam rhetorically asked, as reported\nby Rota. Our sense of the solidity of mathematics seems to be connected to tangible models, such as physical space to\nclassical Euclidean geometry. The real line has\nfor us an almost physical existence. But when it comes to models for set\ntheory, the very notion of a set as a mental construct seems inextricable from a verbal description; but there is only\na countable infinitude of such, and hence the existence of countable models even for uncountable sets (where there are\ntwo notions of cardinality, one extrinsic, and one intrinsic). Naively we think of all subsets existing of, say, the\nreals, but from a strict logical and formal point of view, only those which in principle can be described. This\nthreatens, as noted, to indeed reduce mathematics to a game whose objects mean nothing (just like the chess pieces on a\nboard). On the other hand, a piece of mathematics considered as a game has nevertheless some content as a game, and we\ncan ask questions about it, such as its consistency, which we feel is a definite yes or no question, not contingent\nupon some axioms we introduce in the meta-game of investigation. According to Manin, it is as if we feel that the game\nitself, defined by its axiomatic rules, is a physical object, and systematically drawing all the conclusions is a\nphysical activity anchored in the real world, no matter how unfeasible in practice; just as concluding that a\nDiophantine equation must have a solution or not by making an almost physical thought experiment of an infinite search.\nManin’s attitude to set theory is pragmatic, as that of most mathematicians. He does not seem engaged in the classical\ncontroversies and refers to intuitionists and constructivists as somewhat neurotic. Set theory for Manin, like for most\nmathematicians, provides a convenient language of mathematics, as famously exemplified by Bourbaki. On a more\nexistential level, Manin’s attitude to mathematical Platonism is ambivalent; he has described it as psychologically\ninescapable and intellectually indefensible. What is really meant by that can only be speculated upon. He stresses that\nhis physically tangible intuition, especially when confirmed by mathematical applications to physics as a scientific\ndiscipline, makes him inclined to Platonism, an attitude made even more inescapable from his own experience as a\nmathematician, in particular when studying number theory; but as strong as those convictions may be, they are\nultimately based on subjective experience. Of course intellectually Platonism is not amenable to any formal proof, as\nlittle as proofs of the existence of God pursued by the scholastics (the concerns of whom seem uncannily similar to\nthose of set-theorists). But as Pascal famously\nnoted ‘Le cœur a ses raisons que la raison ne connaît point.’"
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"I would like to conclude this mathematical section with a nice toy example of Manin. Consider a finite set ",
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". Its algebra of functions is given by the Boolean polynomials\n",
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", thus any such polynomials can be written as\na sum of monomials which are naturally identified by the elements (vectors) ",
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"). Consider now the\npolynomial"
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"). What is the point of this formal\nalmost tautological game? Manin brings it up as a finite version of the Axiom of Choice: given a set of polynomials how\ndo we pick an element in each of the sets they define, or show that the polynomial\nis ",
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"? Given the polynomial in\ncanonical form (or any random form), this is not so easy in general: do we have to check all the elements of the vector\nspace? This also leads to a particular instance the P/NP problem, an instance which, according to Manin, is intractable\nat the time."
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"Now I have not touched upon the section of mathematics and physics, which is greatly expanded, nor upon the essays on\ngeneral topics from linguistics, Jungian psychology (of which Manin was charmed with many references in his works), art\nand poetry. Had I done so, the review would have been far too long, not only too long as it already is. Having thus\nfailed to do full justice to the book, I hope that I have at least inspired a few readers to consult the master\nhimself."
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"Yuri Manin, ",
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"Les Mathématiques comme Métaphore. Essais choisis"
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". Les Belles Lettres, 2021, 600 pages, Softcover ISBN 978-2-251-45172-5"
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"\nUlf\nPersson is a professor emeritus at Chalmers University of Technology (Göteborg, Sweden) and an editor of the EMS Magazine.\nHe obtained his PhD in mathematics at Harvard in 1975 with David Mumford as advisor. His mathematical publications\ndeal almost exclusively with surfaces. He is also interested in philosophy. He has published a\nbook on Popper (the prophet of falsification) and also articles on the subject as\nwell as on mathematical philosophy, in particular. He has also written on art, including philosophical and mathematical connections.\nAs a hobby he also writes essays on every book he has been reading for the last twenty years (",
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{
"type": "Link",
"target": "mailto:ulfp@chalmers.se",
"content": [
"ulfp@chalmers.se"
]
}
]
}
]
}