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, 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, 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 nonWestern 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, 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 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 umanisticaaristotelica 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, A great book for a philosophical nerd, 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 socialhistoricalcultural 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, 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 authors philosophy of mathematics, Admittedly, he was a professional mathematician, while Im 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 Hershs philosophy of mathematics is “humanistsocialhistorical”, He stated that his favorite philosopher in college was David Hume, Theres little wonder that Hersh is a leftleaning humanist with an apparently atheistic worldview, On pages, Hersh states the following: “Mathematics is another particular, special socialhistorical 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.
Newtons theories of space and time were “corrected” by Einsteins 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 massenergy and spacetime.
Overall, I give Hershs book moderately high marks, notwithstanding his unpalatable atheisthumanistsocial philosophy thats espoused in his engaging book.
Hersh sets out to define the parameters of a philosophy of maths, his best answer being a sociohistoriiccultural 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.
. . ! 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 socialhistorical 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 thatdoes not equalindependently of a certain culture, So, if my culture dies off, does that mean thatdoes not equal four, I believe in the Tao, so to speakthere 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.
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 spinoff 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. 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 somerecognised 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 quasispiritual 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 its 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.
Its 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 nonmathematician 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 everyones 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, .
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Reuben Hersh