What Is Mathematics, Really?

by Reuben Hersh

Most philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Reuben Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of his book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Descartes, and Spinoza, to Bertrand Russell, David Hilbert, and Rudolph Carnap--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, and Lakatos.
What is Mathematics, Really? reflects an insider's view of mathematical life, and will be hotly debated by anyone with an interest in mathematics or the philosophy of science.

Genres: Mathematics, Philosophy, Science, Nonfiction, History, Education, Logic

368 pages, Paperback

First published August 21, 1997

About the author

Reuben Hersh was an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics.

Reviews

[BlackOxford · Rating 5 out of 5]

Peace in Troubled Times

For many, including myself, mathematics is comforting. In an era of fake news, worldwide illness, and economic uncertainty, mathematics provides proof of another reality which is harmonious, universal, and eternal. Or so it would seem.

In fact mathematics, like all literature, is none of these things. Mathematics is, of course, a human artefact. It is a language which consists of a vocabulary, a grammar, and a community which employs these enthusiastically. Arguably, mathematics is the most refined language ever produced.

Or rather, set of languages. There are apparently some 3400 recognised branches of mathematics. Many of these have their peculiar dialects which are unintelligible to members of other mathematical communities. At least some have never been translated.

Hersh identifies two historical schools of thought which have dominated popular as well as professional discussion of mathematics: Platonists and Formalists. Platonists consider mathematics as a kind of religion. Numbers, they believe, exist independently of human thought about them. They constitute the basic fabric of the universe and determine its orderliness and predictability. For them, mathematics is reality.

Formalists dismiss this quasi-spiritual view. Their opinion is that mathematics is a game, the rules of which are entirely arbitrary. If Platonists are the religious enthusiasts of mathematics, Formalists are the agnostic clergy who have lost the certainty of belief but continue to exercise their ritualistic duties regardless.

Hersh dislikes both Platonists and Formalists. His credible claim is that mathematics developed and continues to develop because it is useful. And it’s usefulness varies so that what mathematics means and how it develops also varies continuously. There is no fixed ‘mathematical method’ by which good mathematics can be distinguished from bad. There are just mathematicians talking among themselves.

This fact - that mathematics emerges from its adherents discussing mathematics - may appear a truism. What else could be happening? But the recognition that mathematics emerges from a restricted community is an important insight. The usefulness of mathematics is not that of engineers or architects or astrophysicists or people filing tax returns.

These and other ‘users’ of mathematics eventually benefit from the products of mathematical discussions in their own work but they are not mathematicians. We may tolerate mathematicians among us because of what their work allows the rest of us to do; but mathematicians could care less. It’s not why they do mathematics.

The practical (or in their minds pedestrian) usefulness of the work of mathematicians does not concern them. Even a brief exposure to number theory, for example, is sufficient to convince most outside the mathematical community (or even outside the community of number theorists) that the things mathematicians are concerned about are essentially trivial. The strange and often captivating relationships among numbers are simply alien to practical experience. The non-mathematician can only ask ‘Why bother?’.

And the answer to this question must be the same as it is to the issue of literature in general. There is no reason for mathematics other than itself. Mathematics is a form of highly refined, esoteric poetry. Its form and subject matter is not to everyone’s taste. But neither is the Iliad, or The Wasteland, or Finnegans Wake. It takes considerable linguistic skill and aesthetic fortitude to comprehend the content of mathematical poetry. Success in such an endeavour is, as usual, its own reward.

[Lyle · Rating 3 out of 5]

In the later chapters, it turned more or less into a survey of what different philosophers have to say about math. Which is nice, because you can't just turn up that stuff with a Google search; it's a useful survey.

But, the book loses the thrust in building an argument that it had earlier. And many of the passages seem redundant. Also, the author attempts to put all philosophers in a modern American political framework: who is on the "left" and the "right". It's a stretch. He decides that those on the left tends to be the ones he agrees with about math. It all has a pseudo-scientific air about it; he's counting up points in an arbitrary fashion.

Finally, in a act reminiscent of Senator John McCain, he labels all the philosophers he agrees with "mavericks".

I'm not sure he proved anything in this book, but I think it's a great source of material for thinking about the subject. And I tend to agree with a lot of his conclusions, even if I think he took shortcuts to get there.

[Paulo · Rating 2 out of 5]

This book is about philosophy of mathematics, not mathematics itself. Hersh shows that from the viewpoint of philosophy, mathematics must be understood as a human activity, part of human culture, historically evolved and intelligible only in a social context. He calls that "humanism", and is opposed to platonism and formalism.
Besides, the book neither report classroom experiments nor make suggestions for classroom practice, though the author considers it can assist mathematics teachers and educators by helping them to understand what mathematics is. In fact, he says «what's teh connection between philosophy of mathematics and teaching of mathematics? Each influences the other. The teaching of mathematics should affect the philosophy of mathematics [...] A philosophy that obscures the teachability of mathematics is unacceptable».
He also claims that a defect of the book is neglect of non-Western mathematics. And it is. No african, arabic, indian or chinese authors are mentioned here.
Mathematics is all about solving problems. It's what comes first. Here Reuben Hersch discusses, among other things, the principles of mathematical proof, the mainstream philosophies of mathematics and the opinions of the main philosphers in history, whether mathematical objects are real or not (mathematics, invention or discovery?).
Whatsoever interest I may have had in this philosophycal approach to mathematics, I think I will stick with mathematics itself. I acknowledge this as a good book, but there is too much philosophy for me, hence the two stars.

[Maurizio Codogno · Rating 4 out of 5]

Premessa necessaria: questo non è un libro di matematica. È di filosofia della matematica, che è una cosa completamente diversa, anche se non necessariamente più comprensibile... La fregatura è che sono pochi i matematici che fanno filosofia, e ancora meno i filosofi che hanno fatto matematica. L'autore è un matematico, ma nel libro è soprattutto in contrasto con i suoi colleghi che a suo dire sono sostanzialmente platonici (gli enti matematici esistono "da qualche altra parte", basta solo trovarli) o formalisti (la matematica non ha significato reale, sono tutte manipolazioni di simboli), mentre lui propende per un'interpretazione che definisce umanistica-aristotelica (la matematica è un'attività umana). Il libro è diviso in tre parti. La prima, dove Hersh mostra le varie tendenze, l'ho trovata favolosa; la seconda, dove passa in rassegna i vari filosofi che hanno trattato di matematica, sarebbe stata migliore se Hersh non avesse voluto metterci sempre il becco; l'ultima contiene approfondimenti matematici per il lettore interessato, e deve essere stata un duro colpo per la traduttrice Rosalba Giomi, che ha anche sbagliato qua e là una formula. Alla fine insomma va bene solo per chi è interessato alla materia.

[william ellison · Rating 4 out of 5]

(?)

Hersh sets out to define the parameters of a philosophy of maths, his best answer being a socio-historiic-cultural context, resolving the Platonist / formalist split, the ethereal real zone of Platonism being replaced by the collective human mind/brain.
The delight of the book though is the later sections on the history of philosophy of maths where he namechecks many's a classic; and the section on basic principles of maths - I say basic, but the piece on Godel' s incompleteness theorem ...?!

[briz · Rating 3 out of 5]

A nice, dense philosophy book, written with lots of wit. This was NOT a book that you can zone out on, and - indeed - it required Full Attention to reap its benefits. This meant that I read it in tiny bursts and, if before bed, would basically immediately fall asleep. This would be a good book to sneak into a meditation retreat, or to have on a long flight/train ride, or anywhere else it's easy to focus and be alert.

I was looking for something that talked about the kinda default way everyone thinks about math as being inherently Platonic - that is, that we think of math as something True, beyond human fallibility and human society even. This book argues firmly against that: Hersh establishes the "mainstream" philosophy of math as being Platonist, but then discusses his "humanist" belief in math as a socio-cultural construct. The book is a combination of sly jokes/author personality (which I usually dislike, but here really enjoyed - he was funny!), a philosophical discussion using impressively simple examples, and then a survey of philosophers - from Plato onwards - and where they fell on the mainstream/"magical math from the eternal cosmos" vs. humanist/"math is just a game and shorthand" spectrum. He also interestingly framed the mainstream vs. humanist philosophy of math as correlating strongly with the right-wing vs. left-wing beliefs of the philosophers themselves (!) and - even more surprisingly - with the fixed mindset/elitism vs. growth mindset/math education is shitty paradigm. That last dichotomy was the most surprising - and the most eye-opening for me! It definitely explained my own fraught relationship with math (brief bio below).

So I came into the book firmly on the Platonist side - I'm not Platonist on anything else, but I had subconsciously swallowed the mainstream attitudes whole: math is magical truth from the cosmos, something we discover rather than invent, and something that you can either see or not. Or, as my favorite Leonardo da Vinci quote says: there are those who see, there are those who see when they are shown, and there are those that do not see. Magical math!

Hersh dismantles this argument by noting the ways that math is a social convention, collaboratively and imperfectly invented using impossible-to-totally-logically-underpin proofs written by real, living, fallible mathematicians who mostly take it on faith, and that many previously-held mathematical truths (Euclidean space) have been dismantled or revised. I think this book would have been a bunch more enjoyable for actual pure/academic mathematicians, since he drops a bunch of stuff I had absolutely zero knowledge of - very exotic stuff! But his general points he makes concisely and with great wit. I also really enjoyed the survey of philosophers: I loved the short bios with sarcastic, lively interjections (boy, he really has it in for Wittgenstein, haha!), and I came away from this book with a list of people I keep meaning to read more about: John Stuart Mill (u r a hero, jsm), John von Neumann (also such a hero, wow), Karl Popper (world 3!!!! WORLD 3!! what a brilliant idea), Bertrand Russell.

Okay, small feminist aside on the surprising way this philosophical argument impacts the mindset/math education stuff: part of the mainstream "magical Platonic math" mentality is a strong elitism. If math is something that's baked into reality's DNA (and, again, not a tool we invented to understand reality's DNA), but if it's the actual - ahem - SOURCE CODE - then there are people who can read this code/see these truths, and those that can't. It matters less how you teach it, since there's just One True Math and "smart" people eventually get there one way or another. This, of course, perpetuates all manner of bad things: like how only white dudes have the True (Math) Sight. If you permit me the indulgence, I realized recently that my "math bio" is like a textbook case of stereotype threat:
- In high school, I considered myself "bad at math", but scored a higher math SAT score than my (male) friend - who went on to major in mechanical engineering. :/
- In college, I minored in math and adored every class. When I asked my calc 2 prof if I should apply to an applied math master's programs (WHAT FUN!), he discouraged me. ://
- In grad school, I struggled through those goddamn Lagrangians and assumed the struggle was because I was "bad at math" (and not because grad school is grad school).
- In one of my first jobs, I figured I'd never learn to use Stata (the statistical programming language we used) well, because I was "bad at coding" and "bad at math" and didn't "think logically". (I ended up writing the Stata tutorials for staff training, lecturing on it and TAing it.)
- Now I'm a data scientist, and I math it up for ~40 hrs/week with GREAT PLEASURE and yet I STILL constantly have imposter syndrome and assume I "have no natural talent" for math and will "never be that good at it", but am grateful that my joy/passion for doing it at least keeps me from totally sucking.
I mean, LOOK AT THAT. It's INSANE. I have literally mathed with pleasure near-constantly since I was 15, and have meanwhile never recognized it as a core part of my identity, but instead felt always like an interloper. I mean, DAMN.

Anyway, so there you go. Definitely recommended, especially if you like epistemology.

[Joseph Schrock · Rating 4 out of 5]

The book being reviewed here, “What is Mathematics, Really?”, is engagingly written. I found the literary style to be highly palatable. However, I do not concur with the author’s philosophy of mathematics. Admittedly, he was a professional mathematician, while I’m a mere amateur mathematician and amateur philosopher. Nevertheless, the stance I take on the philosophy of mathematics is not idiosyncratic. I am essentially Platonist in my worldview – with respect to the ultimate nature of mathematics. Platonism in mathematics is the most widely held view of modern mathematicians – or, so I believe. This puts me in favorable company among professional mathematicians.

Rueben Hersh’s philosophy of mathematics is “humanist-social-historical”. He stated that his favorite philosopher in college was David Hume. There’s little wonder that Hersh is a left-leaning humanist with an (apparently) atheistic worldview. On pages 248-249, Hersh states the following: “…Mathematics is another particular, special social-historical phenomenon. Its most salient special feature is the uniquely high consensus it attains.”

My reply to this assertion is this: Is that consensus not because mathematics asymptotically approaches the objectively existing perfect and infallible mathematics? Note that the study of physics leads physicists deeper and deeper into a better, more precise, and valid philosophy of physics. Newton’s theories of space and time were “corrected” by Einstein’s deeper insights into the true nature of space and time. In like manner, mathematicians learn more and more about the true nature of mathematical realities, even as physicists learn more about the true nature of mass-energy and space-time.

Overall, I give Hersh’s book moderately high marks, notwithstanding his unpalatable atheist-humanist-social philosophy that’s espoused in his engaging book.

[Dan DalMonte · Rating 4 out of 5]

This is an interesting book. Hersh weighs in on a debate about the nature of mathematical objects. There is Platonism, which is the view that mathematical objects are timeless and eternal entities existing independently of any person or society, and which we discover as engage in mathematics. There are formalists, who thinks that mathematical language is just a game we play and the words in it lack any objective reference. Hersh puts forth a social-historical theory of mathematics, which makes it a product of human culture. It is not timelessly true, some Fregean abstract object in the mind of God. But nor is it just a matter of individual opinion, since a culture can establish norms that regulate the application and significance of mathematics within that culture.
Hersh thinks that mathematics will be easier to teach once we root it in lived practice. But, I cannot accept his thesis. Does his thesis entail that 2 + 2 does not equal 4 independently of a certain culture. So, if my culture dies off, does that mean that 2 + 2 does not equal four. I believe in the Tao, so to speak--there is just an intelligibility in the universe that does not change with culture. I think trying to integrate students into mathematics as humanistic discipline is actually more difficult than helping them to discover an abstract structure of the universe.

[Alexi Parizeau · Rating 5 out of 5]

Before reading this book I first spent a long time studying its predecessor, the book "What is Mathematics?". And while they both turned out to be uniquely mind blowing, Hersh' book in particular has changed my entire world view; it's a rare and powerful book that can do that. In brief, this book is the user's manual for Mathematics: The Game. And holyshitbawls mathematics is the most hardcore game out there.

(Note, before reading this I also read "Finite and Infinite Games", which is a sort of manifesto about hardcore (infinite) games like mathematics, so it really coloured my reading of "What is Mathematics, Really?")

[Jonathan Mckay · Rating 4 out of 5]

Ideal Formulas

In America, math is religion. Its creed might be mostly separate from human affairs, but its method of teaching and acquisition is that of a dogma handed from a higher being. Indeed, mathematics got its Western start with the Pythagoreans, a Greek cult worshipping formulas that carry their name today. Math then became enmeshed in the western philosophical/religious mainstream via Plato and Christianity, fitting nicely into concepts that exist in a world apart from our reality entirely (like heaven!). In a typical classroom, math is taught like the commandments. Why does it work? Nobody has the answer and few ask the question. This is platonic idealism at its finest, and I wonder how many of maths practitioners understand the beliefs they transmit.

A competing school of thought claims that math has no meaning, as it is simply string manipulation following internally consistent rules. Essentially every operation of math is a rearrangement of the same truth that could be immediately apparent to higher minds.

Hersch disagrees with both the platonic and formalist interpretations of math. He offers that math is merely a human construct built into our intersubjective reality. This puts math on par with language, government, currency or even religion itself. Specifically mathematics is a language that can be used to describe abstract phenomena with a precision that other languages cannot match, and exhibits the pared down knowledge base of all practitioners that have previously 'spoken' in the language.

At first glance, this is not only compelling, but mindblowing to think about the Platonic assumptions behind our society's blind recitation of mathmatical formulas. Putting math into the same category as language helps understand how we learn it, as well as the natural human capacity to understand and expand the field. Furthermore, it allows us to sidestep the woo-woo beliefs necessitated by attempting to formalize tacit Platonism. It also avoids the trap of formalism, insisting that mathmatics has *no* meaning outside of itself.

After all, if we start using wavelengths, cosine, sine, and milimetters to describe the color violet, is it truly different in nature to simply using the word violet? We don't have physics figured out, and keep discovering that every paridigm seemingly able to explain the world is indeed only a mental model i.e. a metapheor i.e. not reality i.e. it the math was a coarse description of something completely different. Math is only our most precise metaphor, using symbols to describe concepts that while imaginary (like i), serve as crutches for us to make out what we can of the world.

Yet reading through, a doubt persisted in my mind: why is there such a special relationship between math and physics? Is this fundamentally the same as any other human construct? Hersh never dives in, to my regret.

This book is a math-lite investigation of the philosophy of math, and while I have a broader view now, I'm still missing the complete picture. I'll want to read some more mainstream philosophy of math to come to a conclusion I firmly believe.

90th book of 2022

[Paige McLoughlin · Rating 5 out of 5]

I read this book over a decade ago. I really enjoyed Hersh's exploration of this topic. I have a fascination with the topic of the philosophy of mathematics, Cantorism, and the foundational crisis of mathematics in the early twentieth century (computers are an indirect spin-off from this crisis.) I dunno I have a drive to keep everything as secular as possible but also have a love of the numinous that combination probably made the platonist positions in mathematics very appealing to me. Hersh is not sympathetic to the platonist position. He is a good writer and I enjoyed his book as for his philosophy about mathematics I don't share it but it is a free country. Very good book though.

[E. · Rating 4 out of 5]

The book is clearly-written, well-researched, and provocative. At times Hersh's expositions are clear, illustrative, and convincing; at times, they can be very rushed and, in my opinion, amateurish. The beginning (a conversation with his daughter that definitely did not happen as presented) and the end (where he attempts to claim Platonism is related to political conservatism) really do his more mature arguments an unfortunate disservice.

The first half of the book is conceptual, and its evidence is more anecdotal. I don't think he develops a systematic philosophy of mathematics here, rather than sketching out some common points of contention and elaborating on his stance. The second half is an ambitious rewriting of the history of the philosophy of mathematics that moves too fast because it aims to cover too many thinkers and mathematicians. His readings of some philosophers are questionable. I am also not a fan of quotations that last pages only to be followed by a short paragraph of Hersh's own reading.

Overall, I think Hersh does a good job sketching out what is at stake in the conflict between the Platonic and humanist approaches to the philosophy of mathematics. To the practicing mathematician, the biggest thing at stake is perhaps pedagogy, which Hersh unfortunately only briefly discusses. It is unclear what exactly he is trying to accomplish with the book, but besides being such an introduction to the conflict and a reference for writings by included philosophers, it leaves much to be desired.

[Stefan · Rating 4 out of 5]

This is my second reading of this book. It is a classic from Professor Hersh who passed away last year. I'm not sure if I finished it the first time since it is very dense with concepts and historical information. This time around, I read it completely and carefully, including the end notes. It is a good read for anyone interested in the history and philosophy of mathematics. It was full of thought provoking ideas and covered quite a lot of ground.

I would like to have rated it higher but the pacing and some aspects of the book structure affected my enjoyment and made some parts tedious to get through. The main challenge I found in reading the book was that the middle (historical) section wandered around and only gave weak support to Hersh's proposed innovation in thinking about mathematics.

In part 1 (5 chapters), Hersh discusses why the philosophy of math matters, and clearly states his philosophical position. Hersh also provides a list of criteria that he believes a successful philosophy of mathematics must satisfy.

However, instead of continuing on to a coherent account of how his proposed humanist view addresses the ontological and epistemological concerns of mathematics, the book takes a very long (7 Chapter) historical excursion into the alternatives.

This comes in the form a "hit-parade" of all the major philosophers of mathematics from Pythagoras and Plato to the working philosophers contemporaneous with the book. Hersh's ideas are obscured by very long quotations from other philosophers with only brief interjections. Some of these quotes span multiple pages making it unclear which thoughts are his.

Hersh finally summarizes his position in a single-chapter section (Part 3). He briefly grades a humanist approach against the criteria that were previously stated in part one. He then engages into some speculation into the social and political factors that affect the formation of mathematical philosophy. This part felt very rushed and confusing. The book didn't seem to draw a firm conclusion. It just stopped and proceeded to the end matter.

The extensive technical notes were enjoyable but somewhat incongruent with the preceding parts. Comprising ~20% of the total page count, it is longer than the typical technical end notes that I have seen in other math books for general audiences. This was a welcome addition. However, as should be expected, the breadth of topics in the endnotes far exceeds what could be given a thorough treatment. Entire books can and have been written on the topics touched upon. Overall, the appendix was more than adequate for the purpose of technical clarification of issues raised in the main text.

In conclusion, I thought this was a worthwhile read. Since Hersh wrote a few other books and papers on this subject subsequent to the publication of this book I will probably seek those out and read them as well. I am interested in finding out more about how his thoughts evolved over time.

[Sean · Rating 3 out of 5]

That hurt my brain some, not actually being a mathematician of any kind. But then this isn't math, it's the philosophy of math, and Hersh's argument that mathematics is a social-historical-cultural pursuit, in opposition to Platonism and formalism, the two most popular math philosophies going. Knowing little of this world, I'd say he makes a good case. As a reading experience, it gets a bit bogged down in the second half as Hersh covers the history of western philosophers who've had anything to say about math and what they argued for. Interesting, but not much in the way of flow. His writing is at least clear and to the point, but yeah... this one takes some serious concentration to get through.

[Tiantian Chen · Rating 4 out of 5]

I started reading this book after I took my first math class at temple. Professor Datskovsky recommended Hersh to me. I finished the first chapter that summer then I preceded the following summer which is now, in 2017. I would like to find meaning in what I am doing. As a math graduate student, solving problems can be much less fun if there is no meaning of what I am doing. In the book, Hersh claims mathematics is not an eternal world. It is related to social activity. I can't agree more! Mathematics flourishes when people communicate with each other. No wonder there is always rooms for discussion in math conference. If only speakers and audiences, it would be boring and only certain people can talk mathematics. As a leftist graduate student, this book is a wonderful guideline of philosophy of mathematics to me. I don't like describing math in words. I think numbers and symbols would be much clear and straight forward. That is the disadvantage of the book.

[James · Rating 4 out of 5]

It's brilliant when it's accessible, but gets a bit obscure when the philosophy becomes esoteric. There are pages and sections that are lost on me as I try to milk the meaning out of them, but obviously this is my fault, not the author's. Said that, I cannot help but think that a different editor might offer a different structure that would have made this book a little bit more accessible. Still, I enjoyed it very much.

[Jake · Rating 4 out of 5]

A fantastic book for thinking about math. What it is (and isnt), where it come from, what we do with it, what it can do for us ect

This was enjoyable, and it makes me want to get further into the field, but know this. The guy is very convinced of his anti - Platonist perspective. He doesnt do Platonism much investigation and rather works out somewhat more of a sociologist of math than our typical metaphysics man like penrose

[Samuel · Rating 4 out of 5]

Reviews historical views on "foundations". Introduces Hersh's humanism which is further detailed in "Experiencing Mathematics".

[Paige McLoughlin · Rating 5 out of 5]

Really interesting take on Mathematics and the philosophy of mathematics in the wake of the Foundational crisis. Touches on questions platonism (is math invented or discovered), infinity, and many dualities of math like finite/infinite, object/process, invention/discovery, etc. It also brings up an interesting distinction of Goffman's theatricality of front room/backstage spaces in many professions including mathematics. Fun stuff.

[Alison Scott]

[Not a review yet!]

This is the first book that I've tried to read from the Open Library (Internet Archive)'s one-hour borrowing system. This should fit well with my 15 minute reading chunks, I think.

This is the book that we found in my daughter's accumulated junk after she failed to return it to her undergraduate professor. I have sent it back with apologies after concluding I could not possibly read this in print without a magnifying glass.

[Westin Williams · Rating 3 out of 5]

I enjoyed Part I of the book, which was about what the author thinks mathematics really is. However, Part II, which is over 140 pages on the history of the philosophy of mathematics, left my eyes glazed over. I guess it's important to include in this kind of book, but it was very dull and not as instructive as Part I.

[Arthur Drury · Rating 1 out of 5]

Hersh is a mathematician, not a philosopher. This isn't a very good philosophy book.

Warning: Apparently Hirsh simply doesn't understand the importance or interest or value of Frege's work.

[Ferris Mx · Rating 3 out of 5]

Not devoid of interest, but I think the presentation could have been more clearly sequenced.

[David Nathans · Rating 2 out of 5]

The biggest single issue with Hersh’s account is his inability to distinguish between Mathematics epistemological status, and its ontological one.
A good example of this is his account of classical logic, where he claims, accurately, that according to most logicians, once the axioms have been set out, all the possible consequences of a system have already been determined. He then proceeds to argue that this is actually not the case, as logicians have to actually go through the theory to determine those consequences! Of course this is just confusing the epistemological aspect, (How logicians come to know those consequences), with the ontological (What those logical consequences actually are), It does not follow from the fact that logicians have to discover those consequences that they are actually creating them as they go along any more than it follows from the fact that photons were not discovered until 1905, that prior to then, photons did not exist. The irony that Hersh’s position is very similar to Berkeleyan Idealism is not lost on me. Hersh recognizes that his argument is very liable to degenerate into mere relativism, that everything in mathematics is entirely arbitrary, so he attempts to stem the tide by arguing in favor of cultural determinism, or that mathematics actually isn’t arbitrary, because culture and society dictate it. This is to put it mildly, highly unconvincing, first because it seems entirely unfalsifiable, and it could reasonably be used to argue that nothing is arbitrary because of cultural determinism. A second objection is that many things do in fact vary from society to society, and that mathematics is in fact one of the few things that seems not to.
Surely, if it was nothing more than a socio-cultural construct, we would see many different mathematics floating around, some of which at least, should be entirely incompatible with one another. The fact that there are no such variant forms of mathematics is devastating to Hersh’s position, but it is never really addressed.

[Glenn Kriger]

A great book for a philosophical nerd.