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Reuben Hersh: What is Mathematics, Really?
(Jonathan Cape: 1997)


“The working mathematician is a Platonist on weekdays, a formalist on weekends. On weekdays, when doing mathematics, he’s a Platonist, convinced he’s dealing with an objective reality whose properties he’s trying to determine. On weekends, if challenged to give a philosophical account of this reality, it’s easiest to pretend he doesn’t believe in it. He plays formalist, and pretends mathematics is a meaningless game.... Does it matter? Yes. Truth and meaning aren’t recondite technical terms. They concern anyone who use or teaches mathematics. Ignoring them leaves you captive to unexamined philosophical preconceptions. This has practical consequences.... ‘What’s interesting in mathematics?’ is an urgent question for anyone doing research, or hiring or promoting researchers, [yet] there’s no public discussion of this question. No vehicle for public discussion of it. No language or viewpoint that could be used for such a discussion. Not to say there should be agreed-on standards of what’s interesting. Precisely because tastes differ, we need discussion on taste. We have some common standards. That’s proved by our identity as one profession, and our agreement that certain feats in mathematics deserve the highest rewards. Bringing out those standards for analysis and controversy would be important philosophical work [but] our inability to sustain public discussion on values betrays philosophical unawareness and incompetence.... Ideas have consequences. What I think mathematics is affects how I present it...[as was shown by] the unfortunate importation into primary and secondary schools, during the 1960s, of set-theoretic notation and axiomatics. This wasn’t an inexplicable aberration. It was a predictable consequence of a philosophical doctrine: Mathematics is axiomatic systems expressed in set-theoretic language. Critics of formalism in high school say ‘This is the wrong thing to teach, and the wrong way to teach’ [but] such criticism leaves unchallenged the dogma that real mathematics is formal derivations from formally stated axioms. If this dogma rules, the critic of formalism is seen as asking for lower quality, to give students something ‘watered down’ rather than the ‘real thing.’ The fundamental question is ‘What is mathematics?’ Controversy about high school teaching can’t be resolved without controversy about the nature of mathematics.”
(Hersh, pp.39-41)

Undoubtedly the greatest division within our intellectual world lies between those disciplines within which mathematical methods form an essential component, and those that do not... And, with people educated in the latter generally making no serious effort to understand the former - despite the pretensions to critique of the “postmodern” Humanities - we are, as so often, left to our own devices re this truly essential task. Thankfully, however, there is a range of widely accessible (and often entertaining) writing by professional mathematicians today which can serve to show the rest of us how mathematical thinking works, and thus can help open up not only this field, but also all those others dependent upon this approach.

Of such works, I’ve found those of Reuben Hersh the most useful, addressing as they do the fundamentals via empirical and historical enquiries...and offering, as well, an enviably hard-nosed approach to social constructivism - a badly-needed corrective to the Humanities’ smugly ill-informed relativism. Some samples from his early chapter on criteria for theoretical judgement should make this point better than anything I can say for myself:


“Evaluate a body of thought according to its own goals and presuppositions. Understand it historically, in the sense of history of ideas. Pay attention to its consequences, theoretical and practical. Beneficial consequences don’t verify a doctrine. Harmful consequences don’t falsify it. But consequences are as important as plausibility, consistency, or explanatory power.... [And] the argument, ‘What I don’t know, ipso facto doesn’t matter’ isn’t new. Age hasn’t made it palatable.”
(Hersh, pp.24-5)

“One wants all three - truthfulness, precision, and simplicity. But, one usually can’t maximize at once goal A and goal B. If you’re not willing to pick one goal and ignore the others (maximum cash flow, for instance - reputation and legality be damned!) then you have to do some balancing or juggling. (Work on cash flow, but don’t actually go to jail.) [And] precision is easier to achieve in a simple situation than in a complex one. Some phenomena are inherently imprecise.... Simplicity goes with single-mindedness. Where several factors interact to give a complex result, simplicity can be created by ignoring all factors but one [and] different scholars may single out different factors. This kind of simplicity leads to fruitless controversy, like between Red Sox fans and White Sox fans. For example, both formalization and construction are central features of mathematics. But the philosophies of formalism and constructivism are long-standing rival schools. It would be more productive to see how formalization and construction interact than to choose one, and reject the other.... Putting simplicity and precision ahead of truthfulness is treating philosophy as a game, art pour l’art, like some exotic branch of algebra. Philosophy can be serious - no less than how and why to live. Being serious means putting truthfulness first. First get it right, then go for precision and simplicity. My first assumption about mathematics is: It’s something people do. An account of mathematics is unacceptable unless it’s compatible with what people do, especially what mathematicians do.”
(Hersh, p.30)

“Comprehensibility is valued by readers, not by all writers. Philosophy students think that, among professional philosophers, incomprehensibility gets ‘brownie points’ and comprehensibility gets demerits. Unworthy suspicions aside, it’s a question of comprehensibility to whom. What’s incomprehensible to you may be crystalline to the Heidegger expert. This book aims to be easily comprehensible to anyone. If some allusion is obscure, skip it. It’s inessential.”
(Hersh, p.29)

As should already be evident, Hersh is an insightful thinker, with an unusually terse yet clear and colloquial style. In some ways, I suspect, this may reflect the mathematical mindset, in which processual clarity is particularly evident, and there is no place for ornament? Irrespective, I find Hersh’s style a very effective one...and highly appropriate to his subject matter:


“Should the philosophy of mathematics be precise? ...Mathematics is precise; philosophy cannot be. Expecting philosophy of mathematics to be a branch of mathematics, with definitions and proofs, is like thinking philosophy of art can be a branch of art, with landscapes and still lives.... It happens that the creators of foundationist philosophy of mathematics were mathematicians (Hilbert, Brouwer) or mathematically trained (Husserl, Frege, Russell). This training may explain their bias. They sought to turn philosophical problems into mathematical problems, to make them precise. This bias was fruitful mathematically. Some of today’s mathematical logic descended from the search for mathematical solutions to philosophical problems. But, even though mathematically fruitful, it was philosophically misguided.”
(Hersh, p.29)

“I once wrote that mathematicians hate contradiction. That’s not accurate. We love it - like a duck hunter loves ducks. Nothing draws us to the chase like a contradiction in a famous theory.... Consistency is important. But, it’s less important than fruitfulness (inside and outside of mathematics), imaginative appeal, and linking new mathematical devices to old, respected problems. A contradiction can generally be fixed up, one way or another. As Bourbaki explained, ‘freedom from contradiction is attained in the process, not guaranteed in advance.’ ...In practice, we can’t always prove in advance the consistency of all possible deductions. Instead, we develop a technique for preserving partial consistency - absence of contradiction up to the latest set of results. In that way, we continue to forestall contradiction each time it raises its ugly head.... Mathematical buildings collapse - lose interest, are forgotten - not because of contradictions, but because their questions are no longer interesting, or because another theory answers them better.”
(Hersh, p.32)

“Teacher thinks she perceives other-worldly mathematics. Student is convinced teacher really does perceive other-worldly mathematics. No way does student believe he’s about to perceive other-worldly mathematics.”
(Hersh, p.238)

By refusing to accept the overly reified notions of mathematics that mainly prevail in philosophical circles - and which have helped ruin early mathematical education - Hersh allows us to approach mathematics as a human endeavour...in fact, he goes so far as to label his philosophy of mathematics “humanistic”, a marvellous dissent from avowedly “antihumanist” approaches within the Humanities itself. Equally in contrast with such approaches, too, is his plainspoken approach to issues of genuine complexity:


“Even without three years of graduate school, you can get a rough notion of modern mathematics. Here’s a mini-sketch of its method and matter. The method of mathematics is ‘conjecture and proof’. You come to an inherited network of concepts and facts, properties and connections, called a ‘theory.’ ...This presently existing theory is the result of a historic evolution. It is the cooperative and competitive work of generations of mathematicians, associated by friendship and rivalry, by mutual criticism and correction, as leaders and followers, mentors and protégés. Starting with the theory as you find it, you fill in gaps, connect to other theories, and spin out enlargements and continuations.... But you [don’t do this] in isolation...[for] no matter how isolated and self-sufficient a mathematician may be, the source and verification of his work goes back to the community of mathematicians.... Mathematical discovery rests on a validation known as ‘proof,’ the analogue of experiment in physical science. A proof is a conclusive argument that a proposed result follows from accepted theory. ‘Follows’ means the argument convinces qualified, skeptical mathematicians. Here, I am giving an overtly social definition of ‘proof.’ Such a definition is unconventional, yet it is plainly true to life. In logic texts and modern philosophy, ‘follows’ is often given a much stricter sense, the sense of mechanical computation. No one says the proofs that mathematicians write actually are checkable by machine. But it’s conventional to insist that there be no doubt they could be checked that way. Such lofty rigour isn’t found in all mathematics. From one speciality to another, from one mathematician to another, there’s variation in strictness of proof and applicability of results, [and] mathematics that stresses results above proof is often called ‘applied mathematics.’ ...A naive non-mathematician - perhaps a neo-Fregean analytic philosopher - looks into Euclid, or a more modern math text of formalist stripe, and observes that axioms come first. They’re right on page one. He or she understandably concludes that in mathematics, axioms come first. First your assumptions, then your conclusions, no? But anyone who has done mathematics knows what comes first - a problem. Mathematics is a vast network of interconnected problems and solutions...[and] sometimes a solution is a set of axioms!”
(Hersh, pp.5-6)

“When a piece of mathematics gets big and complicated, we may want to systematize and organize it, for esthetics and for convenience. The way we do that is to axiomatize it. Thus a new type of problem (or ‘meta-problem’) arises: ‘Given some specific mathematical subject, find an attractive set of axioms from which the facts of the subject can conveniently be derived.’ Any proposed axiom set is a proposed solution to this problem. The solution will not be unique. There’s a history of re-axiomatizations of Euclidean geometry, from Hilbert to Veblen to Birkhoff the Elder. [And, so] in developing and understanding a subject, the axioms come late. Then, in the formal presentations, they come early. Sometimes, someone tries to invent a new branch of mathematics by making up some axioms and going from there [but] such efforts rarely achieve recognition, or permanence. Examples, problems, and solutions come first. Later come axiom sets, on which the already existing theory can be ‘based.’ The view that mathematics is in essence derivations from axioms is backward. In fact, it’s wrong.”
(Hersh, p.6)

By focusing upon what mathematicians do - rather than viewing mathematics as an independent world of its own - Hersh succeeds in showing us that it is not at all as alien as many of us may think...merely much more reliable/reproducible than the rest of our social world. In fact, this - rather than any Platonic existence - is what defines a field as mathematical, as modern mathematics is an astonishingly variegated field today, literally overflowing with bizarre objects & strange relations of all sorts...


“So far, I’ve described mathematics by its methods. What about its content? The dictionary says math is the science of number and figure (‘figure’ meaning the shapes or figures of geometry). This definition might have been O.K. 200 years ago. Today, however, math includes the groups, rings, and fields of abstract algebra, the convergence structures of point-set topology, the random variables and martingales of probability and mathematical statistics, and much, much more. Mathematical Reviews lists 3,400 subfields of mathematics! No one could attempt even a brief presentation of all 3,400, let alone a philosophical investigation of them all. To identify a branch of study as part of mathematics, one is guided by its method, more than its content.”
(Hersh, p.7)

“What’s the nature of mathematical objects? The question is made difficult by a centuries-old assumption of Western philosophy: ‘There are two kinds of things in the world. What isn’t physical is mental; what isn’t mental is physical.’ Mental is individual consciousness...[and] physical is taking up space.... Frege showed that mathematical objects are are neither physical nor mental. He labelled them ‘abstract objects.’ What did he tell us about abstract objects? Only this: They’re neither physical nor mental. Are there other things besides numbers that aren’t mental or physical? Yes! Sonatas. Prices. Eviction notices. Declarations of war.... Platonist philosophy masks this social mode of existence, with a myth of ‘abstract concepts.’ ...Once created and communicated, mathematical objects are there.... We learn of them as external objects, with known properties and unknown properties. Of the unknown properties, there are some we can discover [and] some we can’t discover, even though they are our own creations.... Mathematical objects can have well-determined properties because mathematical problems can have well-determined answers. To explain this requires investigation, not speculation. The rough outline is visible to anyone who studies or teaches mathematics. To acquire the idea of counting, we handle coins or beans or pebbles. To acquire the idea of an angle, we draw lines that cross. In higher grades, mental pictures or simple calculations are reified...and become concrete bases for higher concepts. These shared activities - first physical manipulations, then paper and pencil calculations - have a common product - shared concepts.... The observable reality of mathematics is this: an evolving network of shared ideas with objective properties. These properties may be ascertained by many kinds of reasoning and argument.... They have the rigidity, the reproducibility, of physical science. They yield reproducible results, independent of particular investigators. Such kinds of ideas are important enough to have a name. Study of the lawful, predictable parts of the physical world has a name: ‘physics.’ Study of the lawful, predicable parts of the social-conceptual world also has a name: ‘mathematics.’”
(Hersh, pp.13-19)

Personally, I am convinced by this...even though one skilled mathematician of my acquaintance disliked the concept so much that he refused to finish reading the book! Yet, as Hersh compellingly argues, the other alternatives are genuinely incoherent - whereas this proposal is merely distasteful to minds predisposed to Platonic “solutions”. Still...Hersh is no diplomat, and he gives short shrift to some of the “big” (yet vacuous) questions which have traditionally occupied philosophers in this area - albeit he does offer a fascinating  (and rather mathematical) conjecture as to why philosophy (as opposed to history of ideas) so often makes a fool of itself with idealism:


“Some questions, which at first seem meaningful, are futile -to answer them neither possible nor necessary. Why are there rigid, reproducible concepts , such as number or circle? Why is there consciousness? Why is there a cosmos? We need not answer Kant’s question, ‘How is mathematics possible?’ any more than we need answer Heidegger’s question, ‘Why should anything exist?’ I haven’t heard about progress on either problem.... Ethnology, comparative history, developmental psychology, the development of non-Euclidean geometry, and general relativity all show that Euclidean geometry is not built into everyone’s mind/brain. We think about space in more than one way. We reject Kant’s answer. Must we still accept his question? ...This much is clear: Mathematics is possible. It’s the old saying, ‘What is happening can happen.’ ...Since Dedekind and Frege in the 1870s and 1880s, philosophy of mathematics has been stuck on a single problem - to find a solid foundation to which all mathematics can be reduced, a foundation to make mathematics indubitable, free of uncertainty, free of any possible contradiction.... That goal is now admitted to be unattainable. Yet, with the exception of a few mavericks, philosophers continue to see ‘foundation’ as the main interesting problem in philosophy of mathematics.... I have two concluding points: Point 1 is that mathematics is a social-historic reality. This is not controversial. All that Platonists, formalists, intuitionists, and others can say against it is that it’s irrelevant to their concept of philosophy. Point 2 is controversial: There’s no need to look for a hidden meaning or definition of mathematics beyond its social-historic-cultural meaning. Social-historic is all it needs to be.”
(Hersh, pp.20-3)

“Niagara Falls is the outlet of Lake Ontario. It’s been there for thousands of years. It’s popular for honeymoons. To a travel agent, it’s an object. But from the viewpoint of a droplet passing through, it’s a process.... Seen in the large, an object, felt in the small, a process.... Movies show vividly two opposite transformations:

A: Speeding up time turns an object into a process.
B: Slowing down time turns a process into an object

...In the social-cultural-historic  domain, the continuity between object and process is blatant, even though some institutions, beliefs, and practices seem eternal. All institutions change. If they change slowly, over centuries - slavery, piracy, royalty, private property, female subjection - they are thought of as objects. If they change daily - clothing fashions, stock market prices, opinion polls - they are thought of as processes.... High speed and lime-lapse photography show that the object-process polarities are ends of a continuum. Any phenomenon is seen as an object or a process, depending on the scale of time, the scale of distance, and human purposes.... In brief, an object is a slow process. A process is a speedy object.... Only in mathematics we think we have pure objects. There, it is thought, we find nothing but pure objects. Infinitely many of them! Could this thinking come from seeing mathematics in too short a time scale? Wouldn’t a view that encompassed centuries show mathematics evolving - a process?”
(Hersh, pp.78-80)

As I’ve suggested earlier in this review, Hersh’s version of social constructivism has little or nothing to do with those currently in favour in the academic Humanities, since he so evidently has no time for relativistic excesses, or jargon-clotted prose. However, as a mathematician, he undoubtedly encounters little of such foolishness amongst his peers, so he basically ignores it here...preferring to concentrate upon the versions of nonsense common in his profession, which he does to devastating effect. Moreover, these are hardly that alien to the Humanities, with both philosophical idealism and aesthetic formalism having v.long (and dubious) pedigrees:


“Two principal views of the nature of mathematics are prevalent among mathematicians - Platonism and formalism. Platonism is dominant, but its hard to talk about in public. Formalism feels more respectable, philosophically, but it’s almost impossible for a working mathematician to really believe it.... The formalist philosophy of mathematics is often condensed into a short slogan: ‘Mathematics is a meaningless game.’ ...Wittgenstein and some others seem to think that, since the making of rules doesn’t follow rules, then the rules are arbitrary. They could just as well be any way at all. This is a gross error. The rules of language, and of mathematics, are historically determined by the workings of society that evolve under pressure of the inner workings and interactions of social groups, and the physical and biological environment of earth. They are also simultaneously determined by the biological properties, especially the nervous systems, of individual humans. These biological properties and nervous systems have permitted us to evolve and survive on earth, so of course they reflect somehow the physical and biological properties of this planet. Complicated, certainly. Mysterious, no doubt. Arbitrary, no.... [Moreover,] rule-making tasks don’t follow rules...[but,] rules are made for a purpose. To be played, or accepted, or performed by people, they have to be playable or acceptable by people. Tradition, taste, judgement, and consensus matter. Eccentricities of individual rule-makers matter.... Some formalists in philosophy of mathematics say discovery is lawless - has no logic - while proof or justification is nothing but logic...[but] in real life, there are no totally rule-governed activities. Only more or less rule-governed ones, with more or less definite procedures for disputes.... Mathematics is, in part, a rule-governed game...[but] the notion of strictly following rules without any need for judgement is a fiction. It has its use and interest. It’s misleading to apply it literally to real life.”
(Hersh, pp.7-9)

“The mystery of mathematics is its objectivity, its seeming certainty or near-certainty, and its near-independence of persons, cultures, and historical epochs.... Platonism says mathematical objects are real, and independent of our knowledge. Space-filling curves, uncountably infinite sets, infinite-dimensional manifolds - all the members of the mathematical zoo - are definite objects,  with definite properties, known or unknown. These objects exist outside physical space and time. They were never created.... An inarticulate, half-conscious Platonism is nearly universal among mathematicians [since] research or problem-solving, even at an elementary level, generates a naive, uncritical Platonism. In math class class, everybody has to get the same answer. Except for a few laggards, they do all get the same answer. That’s what’s special about math. There are right answers. Not because that’s what Teacher wants us to believe. Right because they are right.... Yet most of this Platonism is half-hearted, shamefaced. We don’t ask, How does this immaterial realm...make contact with flesh and blood mathematicians. We refuse to face this embarrassment: Ideal entities independent of human consciousness violate the empiricism of modern science...yet most mathematicians and philosophers of mathematics continue to believe in an independent, immaterial abstract world - a remnant of Plato’s heaven, attenuated, purified, bleached, with all entities but the mathematical expelled...like the grin on Lewis Carroll’s Cheshire cat.”
(Hersh, pp.11-12)

“We can understand the working mathematician’s oscillation between formalism and Platonism, if we look at her work experiences and at the philosophical dogmas she inherited - Platonism and formalism. Both dogmas say mathematical truth must possess absolute certainty. Her own experience in mathematics, on the other hand, offers plenty of uncertainty.... The basis for Platonism is awareness that the problems and concepts of mathematics are independent of him as an individual. The roots of a polynomial are where they are, regardless of what he thinks or knows. It’s easy to imagine that this objectivity is outside human consciousness as a whole...[however,] once mysticism is left behind, once scientific skepticism is focused on it, Platonism is hard to maintain.... [And] formalism needs its own act of faith. How do we know that our latest theorem about diffusion on manifolds is formally deducible from Zermelo-Frankel set theory? No such formal deduction will ever be written down. If it were, [due to its length,] the likelihood of error would be greater than in the usual informal or semi-formal mathematical proof. Now another question: ‘How come these examples were known before their axioms were known? If a theorem is only a conclusion from axioms, then do you say Cauchy didn’t know Cauchy’s integral formula? Cantor didn’t know Cantor’s theorem? Formalism doesn’t work! Back to Platonism. We don’t quit doing mathematics, of course. Just quit thinking about it.”
(Hersh, pp.42-3)

Perhaps most intriguing to me, in this book, is the way Hersh makes very real sense of such quintessentially human aspects of mathematics as aesthetics, intuition, and error - situating mathematics firmly within the human world, where it (now) seems to belong. For, once we ditch unworkable idealism and formalism - and insist on seeing mathematics as a human process - it becomes much less forbidding, even if its workings (as such) still remain outside my range of skills. Yet, thanks to Hersh & many others, I now have some useful understanding of such skills, and of the viewpoints which structure them.

And this is no small thing...


“There’s an amazing consensus in mathematics as to what’s correct or accepted. But, just as important is what’s interesting, important, deep, or elegant. Unlike correctness, these criteria vary from person to person, speciality to speciality, decade to decade. They’re no more objective than esthetic judgements in art or music. Mathematicians want to believe in unity, universality, certainty, and objectivity, as Americans want to believe in the Constitution and free enterprise, or other nations in their Gracious Queen, or their Glorious Revolution. But while they believe, they know better. To become a professional, you must move from front to back. You get a more sophisticated attitude to myth. Backstage, the leading lady washes off powder and blusher. She’s seen with her everyday face. The front-back codependence makes it hopeless to understand the front while ignoring the back.... You can’t understand a restaurant meal if you’re unaware of the kitchen. Yet you can present yourself as a philosopher of mathematics, and be aware only of publications washed and ironed for public consumption.”
(Hersh, p.39)

“Accounting for intuitive ‘knowledge’ in mathematics is the basic problem of mathematical epistemology. What do we believe, and why do we believe it? To answer this question, we ask another question: what do we teach and how do we teach it? Or, what do we try to teach, and how do we find it necessary to teach it? We try to teach mathematical concepts, not formally (memorizing definitions) but intuitively - by examples, problems, developing an ability to think, which is the expression of having successfully internalized something. What? An intuitive mathematical idea. The fundamental intuition of the natural numbers is a shared concept, an idea held in common after manipulating coins, bricks, buttons, pebbles.... [This] intuition isn’t direct perception of something external. It’s the effect in the mind/brain of manipulating concrete objects - at a later stage, of making marks on paper, and still later, manipulating mental images.... [And] different people’s representations are always being rubbed against each other, to make sure they’re congruent.... The difficulty in seeing what intuition is arises because of the expectation that mathematics is infallible.”
(Hersh, pp.65-6)

“‘The mistakes of a great mathematician are worth more than the correctness of a mediocrity.’ I’ve heard those words more than once. Explicating this thought would tell us something about the nature of mathematics. For most academic philosophers of mathematics, this remark has nothing to do with mathematics or the philosophy of mathematics. Mathematics for them is indubitable - rigorous deduction from premises. If you make a mistake, your deduction wasn’t rigorous. By definition, then, it wasn’t mathematics! So the brilliant, fruitful mistakes of Newton, Euler, and Reimann, weren’t mathematics, and needn’t be considered by the philosopher of mathematics. Reimann’s incorrect statement of Dirichlet’s principle was corrected, implemented, and flowered into the calculus of variations. On the other hand, thousands of correct theorems are published every week. Most lead nowhere. A famous oversight of Euclid and his students (don’t call it a mistake) was neglecting the relation of ‘between-ness’ of points on a line. This relation was used implicitly by Euclid in 300 B.C. It was recognized explicitly by Moritz Pasch over 2,000 years later, in 1882. For two millennia, mathematicians and philosophers accepted reasoning that they later rejected. Can we be sure that we, unlike our predecessors, are not overlooking big gaps? We can’t. Our mathematics can’t be certain.”
(Hersh, pp.44-5)

“Is mathematics created or discovered? This old chestnut has been argued forever. The argument is a front in the eternal battle between Platonists and anti-Platonists.... Let’s not replow this well-trodden ground. Instead, let’s listen impartially for ‘create’ and ‘discover’ in non-philosophical mathematical conversation. Why do both words - ‘create’ and ‘discover’ - seem plausible? ...When we solve [any clearly and definitely formulated] problem, we say the solution is ‘found’ or ‘discovered.’ Not created - because the solution was already determined by the statement of the problem, and the known properties of the mathematical objects on which the solution depends.... But solving well-stated problems isn’t the only way mathematics advances. We must also invent concepts and create theories. Indeed, our greatest praise goes to those like Gauss, Reimann, Euler, who created new fields of mathematics....[which are] in part predetermined by existing knowledge, and in part a free creation.... When several mathematicians solve a well-stated problem, their answers are identical. They all discover that answer. But when they create theories to fulfil some need, their theories aren’t identical. They create different theories.... But then, after you invent a new theory, you must discover its properties, by solving precisely formulated mathematical questions.... [Moreover,] you may have to invent a new trick to discover the solution.... Is mathematics created or discovered? Both, in a dialectical interaction and alteration. This is not a compromise; it is a reinterpretation and synthesis.”
(Hersh, pp.73-5)

Reuben Hersh’s What is Mathematics, Really? is, to my mind, the best introduction to mathematical thinking for the rest of us...all those who cannot think mathematically and, in consequence, find many specialist disciplines forbidding, to say the least. And, as I noted earlier, it also serves to cast a very useful sidelight on a wealth of other issues - principally methodological - from a mathematical perspective, and in doing so demonstrates exactly why mathematics has always been associated with rigorous thought.

Even more valuably, it makes an extremely strong argument for seeing mathematics as a fundamentally human activity, thus aiding those of us who would build bridges between seemingly disparate disciplines, by demonstrating that such rigour as mathematics enables is not inhuman, after all...but, rather, exists on a continuum with other forms of collective knowledge. Such an awareness is doubly crucial today for, with the knowledge-driven proliferation of the sciences, and the self-imposed isolation of the Humanities, we are all too likely to forget Terence’s ancient aspiration...to forget that “nothing human is alien to me.”


“This book offers a radically different, unconventional answer to [its title question]. Repudiating Platonism and formalism, while recognizing the reasons that make them (alternately) seem plausible, I show that from the viewpoint of philosophy mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. I call this viewpoint ‘humanist.’ I use ‘humanism’ to include all philosophies that see mathematics as a human activity, a product, and a characteristic of human culture and society.... This book is a subversive attack on traditional philosophies of mathematics. Its radicalism applies to philosophy of mathematics, not mathematics itself. Mathematics comes first, then philosophizing about it, not the other way around. In attacking Platonism and formalism and neo-Fregeanism, I’m defending our right to do mathematics as we do.... Of course, it’s obvious common knowledge that mathematics is a human activity, carried out in society and developing historically. These simple observations are usually considered irrelevant to the philosophical question, what is mathematics? But without the social historical context, the problems of the philosophy of mathematics are intractable. In that context, they are subject to reasonable description and analysis.”
(Hersh, pp.xi-ii)


John Henry Calvinist