Why Is There Philosophy of Mathematics At All? by Ian Hacking (PDF)

Description

This truly philosophical book takes us back to fundamentals – the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as ‘what makes mathematics mathematics?’, ‘where did proof come from and how did it evolve?’, and ‘how did the distinction between pure and applied mathematics come into being?’ In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that proof and other forms of mathematical exploration continue to be living, evolving practices – responsive to new technologies, yet embedded in permanent (and astonishing) facts about human beings. It distinguishes several distinct types of application of mathematics, and shows how each leads to a different philosophical conundrum. Here is a remarkable body of new philosophical thinking about proofs, applications, and other mathematical activities.

User’s Reviews

Editorial Reviews: Review “Hacking does not restrict himself to the foundations of mathematics, but dares to cover both the breadth and the depth of mathematical philosophy.” Literary Review of Canada”… readable, presented in easily digestible chunks, clearly explained, and just a lot of fun …” Danny Yee’s Book Reviews”Show[s] non-specialists … the sort of distinctive contribution to science and maths that a brilliant, very well-informed, philosopher can bring … I thoroughly recommend this book.” Alan Weir, The Times Literary Supplement”Hacking has composed a great overview of our understanding of mathematics and of the historical turning points and philosophical basics.” Peeter Müürsepp, Mathematical Reviews Book Description Hacking explores how mathematics became possible for the human race, and how it ensured our status as the dominant species. About the Author Ian Hacking is a retired professor of Collège de France, Chair of Philosophy and History of Scientific Concepts, and retired University Professor of Philosophy at the University of Toronto. His most recent books include The Social Construction of What? (1999), An Introduction to Probability and Inductive Logic (Cambridge University Press, 2001), The Emergence of Probability (Cambridge University Press, 2006), Scientific Reason (2009) and Exercises in Analysis (Cambridge University Press, 2009). Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Ian Hacking was one of my first professors of philosophy, and I am partial both to his somewhat breezy style of writing and the fact that he essentially knows everyone in his field. His essential argument that the legitimacy of ideas about mathematics can only be demonstrated through mathematics will strike some as tautological, but the background argument–that there is no difference between “pure” and “applied” mathematics, and that–except by its effects–we don’t really know if mathematics is “there” is characteristically quite stunning. Hacking sets his argument against Plato’s idea of the Forms–which would mean that of course mathematics is there, if in a form we cannot yet fully recognize–but it’s clear that Hacking does not buy this.

⭐Philosophers, mathematicians and scientists have been debating the nature of mathematical objects for at least twenty five hundred years. It’s refreshing then that Dr. Hacking chose not to provide yet another philosophical theory but instead focused on elucidating the various positions taken in this perennial argument.Several pivotal questions get asked again and again. Are mathematical objects the creation of the human mind or are they discovered? Why does math so accurately model the world? What is the difference, if any, between applied and pure mathematics?If these kind of questions interest you then Dr. Hacking’s book is a good choice. I personally found the creation/discovery parts fascinating but was somewhat bored by the difference between pure and applied mathematics. But that’s just a matter of taste.Well-written and relatively easy to read given the subject matter, I recommend the book to anyone interested in such questions. As always, they may not get resolved but the careful reader will grow in understanding the enigmas behind the truths of mathematics.

⭐The purpose of this book is to expose the reader to the hidden world of the philosophy of mathematics (hidden in my opinion), and in that light it does an excellent job. Hacking brings the subject alive by making it feel like a large community full of thought and dedication. There are a ton of obscure references in the book, many of which spin off into equally obscure discussion tangents, but all the while staying within the subject of the philosophy of mathematics, and making it feel very lively and diverse as a result.Judging by the other reviews the book may not be for everyone, but as someone fairly new to philosophy of mathematics I love it. If you’re feeling a little hesitant because of the other reviews I would suggest downloading the kindle sample (if you can). If you enjoy the sample, I think you will enjoy the rest of the book as well.

⭐Interesting subject. Lots of references. When the author says he has no position on some point, you feel the points he DOES take a position must be just a few pages away. It never comes. He writes as if his own position is irrelevant which is odd for an author.

⭐This book is a relation particularly interesting about the actual questions of the mathematics, The author connects the phylosophy with the mathematics, so that the traditional arguments have same importance respect the modern results.Geometry, physics and logics are all examined with attention, and those facts are seen particularly in relation with the more important phylosophers.They are Kant, Pierce, Wittengstein, Plato, who have been valid for their contribute to the science world. Shortly this book gives a model for learning particular informations, but also it is a lecture made with pleasure, because the style is surely elegant and the analysis is sufficiently complex to understand it.

⭐Perchance too personal? Hackings Meditations on Witgnesteins Lectures on the Foundations of Mathematics. A good read, useful tips for further reading, but the book itself changed nothing for me.

⭐eBook format review:Amazon please listen!! Add a separate rating method for the eBook presentation and / or print edition production: binding, paper, printing and so on.Books need to be reviewed in terms of content and increasingly due to variable quality of eBook production separately in terms of format / production quality.I have purchased the PDF eBook version of this book elsewhere since I prefer “print replica” eBooks which I read on an iPad. As far as I can tell from the Kindle sample that I have downloaded, the Table of Contents is appropriately hierarchical and the rest of the content appears quite in line with the PDF version. This is unusual for technical material but this book is not heavy on equations and so on. Generally, mathematical texts are not at all set well in Kindle (or ePub). I disagree with Dabs’ rating and comments on the Kindle version.Content review:This is a pleasant read that surveys the breadth of the history of ideas about what mathematics is up to the present. Langlands and Grothendieck as well as Gowers and the economist von Mises and Kant all make appearances and insert their comments on the nature of mathematics as a human activity and its relation to the world around us.I have thus far skimmed the contents and dipped in here and there so I’ll return with a more detail review after I complete reading. I just wanted to put my comments in regarding Amazon’s broken review system.

⭐This is a very self-indulgent book. It’s full of references to the author’s own works, and when quoting others he always mentions if at all possible his personal connexion to them, so that one has the impression of permanent name-dropping. He has obviously done a lot of reading about mathematics, both popularizations and obiter dicta of famous mathematicians about their subject, but his grasp of logic and foundations looks a bit shaky: he says (p. 23) that Gödel thought that by adding new axioms we could get a complete set theory (as though Gödel had forgotten his own incompleteness theorem!), suggests (p. 33) that some might doubt the proof that there is no uneven dissection of the cube because it’s proved by reductio (but it’s a negative claim and nobody objects to reductio to negative conclusions) and announces that Kripke semantics provides the key to understanding intuitionist logic (apparently unaware that it only provides a *classical* semantics for intuitionist logic, not an intuitionist one). These slips wouldn’t matter if there were philosophical meat in the book, but there isn’t. After Hacking himself the most quoted author is Wittgenstein, but no attempt is made to construct a philosophy of mathematics out of Wittgenstinian ideas (as for example was done by Michael Dummett); on the contrary, all Hacking seems to have taken from Wittgenstein is the idea that you can write a book of philosophy via a motley association of ideas without any structured argument whatever. He vaguely says that he’s a logicist, i.e. one who thinks mathematics reduces to logic, without saying what counts as logic or how the reduction is supposed to go. He announces that modern philosophy of mathematics makes the mistake of presupposing a semantics in terms of reference and truth, to which he opposes the slogan (and he admits it’s no more than that) that meaning is use, without giving any hint of what semantics this might lead to, or how that would affect the philosophy of mathematics. The moral is, one shouldn’t write a book on a subject on which one has nothing to say.

⭐I am a PhD candidate in engineering, not mathematics. I think this book contains too many names of philosophers, assuming readers familiarity to various -isms. That makes this book not so friendly to amateurs like me. However, I did get a bit of interesting ideas and some practices of mind from the book.

⭐I hesitated to write a review of this book as I am neither a philosopher nor a mathematician. However, as the broad outlines of the arguments are accessible to someone who, like myself, has no more than high-school math, I feel justified in sharing the excitement and pleasure I experienced in reading Why is there a Philosophy of Mathematics at all? The book explores two themes:1. The nature of mathematical proof2. The applicability of mathematics (obviously to physics and economics, but also, as Hacking points out, across fields within mathematics itself).The answer to the question Why is there a Philosophy of Mathematics is, I take it, that proof and applicability are issues that both mathematicians and philosophers feel compelled to return to again and again (notwithstanding the many practitioners who do not feel thus compelled). In one sense, the answer is psychological: the astonishment generated by these two aspects of mathematics continually draws mathematicians and philosophers alike into a discussion that is, not itself, mathematical. It is, for lack of a better home, philosophical.A historical perspective informs all the arguments. To summarize the thesis of the book: “A central aspect of the Ancient answer to the why question is none other than Proof. A central aspect of the Enlightenment answer is one notion of applying mathematics, namely Kant’s (p 83).” Indeed, the entire book can be read as an essay in historical epistemology. And it is not history in a crude historicist sense: there is a full recognition that the actual course of historical development is the product of complex contingencies. One interesting example he cites is the successful development of mathematics in ancient China without the Greek (Euclidean) notion of proof. Thus raising the question of whether the latter is fundamental to foundations, or simply a cultural artifact.Early on in the book the author suggests two different conceptions of proof that he terms the Cartesian and the Leibnizian. The former is associated with the sense of an “aha” moment and the concept of apodeictic certainty in the Critique of Pure Reason. The latter is associated with an algorithmic approach in which the steps are too numerous to be comprehended in a single “aha” moment. I have over-simplified. Nonetheless, one of the many fascinating parts of this work is the way Ian Hacking illustrates the manner in which the “gold standard” of proof is itself subject to the contingencies of history (though, for every practitioner, it must stand outside of history).Ian Hacking teases out three different ways in which we can explore the concept of applicability. First, the applicability of one branch of mathematics to another; secondly, the application of mathematics to physics (with references to Wigner’s famous essay: The Unreasonable Effectiveness of Mathematics in the Natural Sciences) and, lastly, mathematics and engineering. The latter is used as a further illustration of the complexity of historical contingencies (as in: compare and contrast Göttingen and the École Polytechnique).In a work on the philosophy of mathematics, the question of the the putative existence of mathematical objects (platonism vs nominalism ) is unavoidable. Ian Hacking treats the question seriously and describes the major fault lines among modern mathematicians, taking care to reveal the complexities and subtleties of the question (among the archetypal representatives: Tim Gowers and Alain Connes). He uses a quotation twice (from Robert Langlands) to illustrate the ambiguities implicit in the way mathematicians approach the question. Speaking of the notion that “mathematics, and not only its basic concepts, exists independently of us. This is a notion that is hard to credit, but hard for a professional mathematician to do without.” (quoted on pages 41 and 256). Although the initial provisional answer to the question why is, Proof and Applicability, the perennial arguments over ontology suggest that they also form part of the answer. The author is well aware that the ontological question is probably irrelevant to the practice of mathematics…but, nonetheless, it keeps on reoccurring (moths to a light!)There is something about the rapid, dense style of this book that is infectious: it effectively conveys the author’s enthusiasms, interests and perplexities. If one test of a good book is that it inspires readers to explore further, then Why is there Philosophy of Mathematics at all? deserves the highest possible rating. And the book contains an excellent bibliography.

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