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Philosophy of mathematics – a reading list (2020)
- oxff 4y agoULTRAFINITISM and CONSTRUCTIVISM gang gang
- Koshkin 4y agoThis is great, this shows that philosophy of mathematics has a long history and continues to be an active area of analytical thought. (Judging by recent discussions, I believe that much of the HN community desperately needs some education in this area.)
- joe_the_user 4y agoModern philosophy of mathematics is still in part shaped by debates starting well over a century ago, springing from the work of Frege and Russell, from Hilbert’s alternative response to the “crisis in foundations”, and from the impact of Gödel’s work on the logicist and Hibertian programmes. I wonder what the author thinks of Van Plato, The Great Formal Machinery Works and other works on the history of the foundational mathematics. One of the things that stands out in the book is that when notions of mathematical logic and foundations of arithmetic were being formulated by Frege and Grassman in the 19th century, neither the notation nor the concept of proof as a mechanical process existed and the process of creating theories about proof processes also involved laying down the concept of proof and creating tractable notations for it (Frege's original notation quickly becomes incomprehensible as expressions grow, for example). Principia Mathematica is notable for creating modern notation despite it's failure to be a complete foundation of mathematics.
- fan_of_yoinked 4y agoThis is great - I don't know if a full survey is in the cards for me, but The Search for Certainty grabbed my eye. This reminds me of this guide https://www.susanrigetti.com/physics https://www.susanrigetti.com/physics for physics, and she has one for self teaching math and philosophy as well. It makes me curious to see a similar reading list put together for computer science - the history and theory of computing, or the kinds of things you might generally study in a Comp Sci program (as opposed to practical skills/how to types of reading)
- dr_dshiv 4y agoMeh. No mention of Pythagoras or any Platonism. Max Tegmark, Karl Popper and Roger Penrose are the three best known for promoting the Pythagorean-Platonic idea that mathematics precedes matter. Because that seriously freaks some people out—they can’t even deal with the idea. But, fairly basic, that triangles are transcendent and would exist in any civilization, in any galaxy? Matter has never produced a perfect sphere, but spheres are nevertheless truly real—- right?
- Koshkin 4y agoIndeed! Abstractions deal with commonalities, and those do exist. ("There exists something in common between the three horses and the three apples you want to treat the horses with.")
- colechristensen 4y agoSo much of it comes down to what exactly you mean by exist. Which ultimately ends up being a boring disagreement. People can have bigger or smaller definitions of what it means and then have passionate disagreements with each other about what fits inside which are ultimately about nothing but how big a person prefers their definition.
- Koshkin 4y agoSure, there are many ways in which something may not exist; but the existence usually demonstrates itself in a pretty straightforward way, like with the horse who bites you if you show her that you have no more apples left.
- ogogmad 4y agoLook up intuitionistic logic and constructivism to see why existence can have a weaker or stronger meaning depending on the foundations of mathematics.
- colechristensen 4y ago
- mberning 4y agoIt’s kind of disappointing that history, philosophy, and mathematics are not taught together. When you try to piece these things together yourself you start to realize how much context you miss out on when studying them in isolation.
- adamnemecek 4y agoThe most important idea in math (and philosophy of math) is the idea of fixed points (aka spectra, diagonalizations, embedding, invariants, braids). By fixed point I mean something like the "Lawvere's fixed point theorem". Fixed points give rise to “meaning” (it’s really that general). https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theorem https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theore... I have a linkdump on this https://github.com/adamnemecek/adjoint https://github.com/adamnemecek/adjoint I also have a discord https://discord.gg/mr9TAhpyBW https://discord.gg/mr9TAhpyBW
- openfuture 4y agoConsider matrix bridge for those of us who will never use discord?
- Koshkin 4y agoThere goes my evening...
- morelisp 4y agoSmells like ⸣ to me.
- adamnemecek 4y agoWhich part?
- morelisp 4y agoThe part where you take a basic mathematical concept and claim it solves all "meaning".
- adamnemecek 4y agoIt’s not a basic concept. If you think it is you legit don’t understand it.
- raphlinus 4y ago
- sbdaman 4y agoEdit: replies are right!
- silent_cal 4y agoI would venture to say that one of the most difficult philosophy books of all time is not a good intro to anything, lol
- Koshkin 4y agoNot if you read it in German
- deleted 4y ago[deleted]
- mjh2539 4y agoThis would be considered an incredibly opaque (and...particular?) introduction to the philosophy of mathematics.
- sbdaman 4y agoFair enough.
- steve_john 4y ago
- Koshkin 4y agoWhy just quote Wikipedia?
- Jtsummers 4y agoThat's pretty much all their comments. Direct quotes from something like Wikipedia, the article, or a paraphrased version of a statement from the article.
- steve_john 4y agojust to convey the right info.
- openfuture 4y agoThe Dialogical Roots of Deduction[0] is an excellent book that I was recommended by my math professor after making a comment about how mathematics are a form of persuasion, not a religious truth. It was refreshing to read something I could agree with so easily. That being said, I think a lot of philosophy of mathematics is not as insightful unless you also study mathematics. It is easy to misunderstand or try to apply theorems incorrectly. 0: https://www.cambridge.org/core/books/dialogical-roots-of-deduction/ADAD1844B5F559ECA15EE175690B612D https://www.cambridge.org/core/books/dialogical-roots-of-ded...
- throwaway81523 4y agoThe stuff on that list all seems rather historical. I have no clue as to what is going on in the field in the current century, and the list doesn't seem to help much. This was better: https://www.andrew.cmu.edu/user/avigad/Papers/PhilMath.pdf https://www.andrew.cmu.edu/user/avigad/Papers/PhilMath.pdf Philosophy as an MO search keyword also finds interesting posts: https://mathoverflow.net/search?q=philosophy https://mathoverflow.net/search?q=philosophy
- zmgsabst 4y agoI think the past 100 years are: “Algebra and geometry are the same?… Oh, our whole field is blind people describing elephants.” https://en.wikipedia.org/wiki/Blind_men_and_an_elephant https://en.wikipedia.org/wiki/Blind_men_and_an_elephant
- Koshkin 4y agoA.k.a. Algebraic Topology.
- zmgsabst 4y agoYeah — but we’re still digesting the impacts. Eg, you see that algebra and geometry “reappear” in things like computing — and the different perspectives of difference equations (geometry) versus typed statements (algebra). Or, in things like category theory giving a framework where we can “abstract” arguments from different (type) theories that are the same “shape” — which has connected a number of fields.
- steppi 4y agoWorking through all of Peter Smith’s suggested books from the original post should give one a solid understanding of much of what’s been going on in field up until fairly recently. Every recommended book in Smith’s list from the original post up to and including Shapiro’s The Oxford Handbook of Philosophy of Mathematics and Logic also appears in Avigad’s suggested readings from your link, several under the section on contemporary developments. I think it’s a plus that Smiths list is more focused. Smith’s list also contains more recent works that do not appear in Avigad’s list.
- valyagolev 4y agoSynthetic Philosophy of Contemporary Mathematics by Fernando Zalamea for something very lively, contemporary and more continental in spirit
- jesuslop 4y agoZalamea also has a serious guide to Grothendiek work (in Spanish for the moment) at http://matematicas.unex.es/~navarro/res/zalamea.pdf http://matematicas.unex.es/~navarro/res/zalamea.pdf, that he introduces lucidly in live lectures here "https://www.youtube.com/results?search_query=cienciastv+mini+curso+grothendiek https://www.youtube.com/results?search_query=cienciastv+mini..."
- gone35 4y agoDisagree, but opinions differ on Zalamea.
- flatline 4y agoProofs and Refutations was a great, sometimes challenging, and often funny read.
- 725686 4y ago“Philosophy of science is about as useful to scientists as ornithology is to birds.” ― Richard Feynman Came to my mind.
- danielam 4y agoRead in a straightforward manner, Feynman simply didn’t understand the value of philosophy, though he did inevitably and unwittingly tread across philosophical territory. In some sense, we all philosophize to one degree or another. We can either acknowledge that and learn to do it well, or we can do it poorly.
- practal 4y agoSo according to you, Feynman did it poorly? Seems to me then being a poor philosopher is a good thing.
- ngcc_hk 4y agomay be to some individuals like him but not to all. Bell’s theorem came into mind. If you think philosophy not the academic one but the one about what you are doing and what assumption you made, you cannot just shut up and calculate. The standard model has to stagnate for 50 years. May be a re-think of the fundamental can drive new innovation.
- yakubin 4y agoOr we can acknowledge that and aim at restraint, stopping ourselves whenever we catch ourselves doing it, recognising that it's a bug in our software. I've spent most of my life reading philosophy. I've been clean for several years and I'm glad (it's no small feat for me). I recognise now that it was nothing but a bug in my mind. To younger people, I'd advise to focus on the actual mathematics and science, because that's where true knowledge lies. Solution to talking nonsense is not talking better nonsense, but recognising that one speaks nonsense and should better be silent.
- 4y ago
- user3939382 4y agoOne philosophical lesson I’ve learned from mathematics is that some problems objectively, probably have no solution.
- jonahx 4y agoThis list is incomplete without Wittgenstein's "Remarks on the Foundations of Mathematics".
- alanbernstein 4y agoWell at least it's consistent
- TuringTest 4y agoYou, sir, made me laugh
- morelisp 4y agoLike the recommendation of Kant previously in the replies, it would be a miserable and incoherent first step for anyone.
- danielam 4y agoJames Franklin’s “An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure” is worth mentioning.
- graycat 4y agoWhen I was a math major in college and in some of the years later, I went through such questions and responses. If you like digging into the issues and work of G. Cantor, B. Russell, J. von Neumann, K. Godel, P. Cohen, etc., fine. But here is a shorter response: Q. Is the square root of 2 rational? A. Rational means the ratio of two whole numbers. So, suppose p and q are whole numbers and we have (p/q)^2 = 2 then we have p^2 = 2(q^2) So that the number of factors of 2 on the left side is even while on the right side it is odd. But 2 is a prime number and we have a theorem the fundamental theorem of arithmetic, that for each positive whole number there is only way to factor that number as a product of primes. So, the square root of 2 is not rational. Done. This argument illustrates the philosophy of mathematics. Here, never mentioned an "ism". Opinions can vary, but my opinion from my background in pure/applied math is that this little argument about the square root of 2 is about all there is that is solid and needed about the philosophy of math. Opinions can vary!
- ngcc_hk 4y agoThe original proof I heard in high school just assume no common factor between p and q i.e. we can eliminate all factors with common factors for x being rational as m/n to p/q. I miss any argument involving any fundamental theorem. The issue is that it is a contradiction, as it is obvious p and q have 2 as their common factor. This is a proof by contradiction and if logic right, the assumption they are rational is wrong. The historical original by p school (as they are Greek) involving geometry not algebra.
- bmitc 4y agoSome interesting entries I know about: * Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being by George Lakoff and Rafael Nunez * The Mathematician's Mind by Jacques Hadamard * Mathematics Form and Function by Saunders MacLane * On the Brink of Paradox: Highlights from the Intersection of Philosophy and Mathematics by Agustin Rayo The preface and chapter 1 of A Book of Abstract Algebra by Charles C. Pinter has an excellent discussion of a certain perspective of mathematics. Hermann Weyl wrote a lot about this stuff too.
- Eupraxias 4y agoNeophyte speaking - I love it, but so much in this area feels deeply established. Who are the rebels, the provacateurs, and what are their arguments? If I wish to 'know thine enemy', who would I read? For example: - who questions the foundational axioms of mathematical structuralism these days? - who argues from a comparative analysis of logic and field theory?
- keiferski 4y agoIt's not an answer to your question, but you may be interested in Alfred North Whitehead. He was initially trained as a mathematician and co-wrote some important works, but eventually shifted to religion and developed what is called "process philosophy." > Beginning in the late 1910s and early 1920s, Whitehead gradually turned his attention from mathematics to philosophy of science, and finally to metaphysics. He developed a comprehensive metaphysical system which radically departed from most of Western philosophy. Whitehead argued that reality consists of processes rather than material objects, and that processes are best defined by their relations with other processes, thus rejecting the theory that reality is fundamentally constructed by bits of matter that exist independently of one another. Today Whitehead's philosophical works – particularly Process and Reality – are regarded as the foundational texts of process philosophy. https://en.wikipedia.org/wiki/Alfred_North_Whitehead https://en.wikipedia.org/wiki/Alfred_North_Whitehead Alain Badiou also wrote some books on mathematics. He is definitely outside of the "official canon" in analytic philosophy of mathematics. His math-related work seems to have been controversial among more traditional mathematicians and philosophers: https://en.wikipedia.org/wiki/Alain_Badiou#Being_and_Event https://en.wikipedia.org/wiki/Alain_Badiou#Being_and_Event
- gone35 4y agoBadiou is a user, not a philosopher of mathematics (he himself would agree). Being and Event —neither volume of the now trilogy— is certainly not about mathematics, nor philosophy of mathematics. Rather, self-avowedly, it is a work in the service of radical politics.
- behel 4y agoThere are people like the late Edward Nelson who represent radical formalism: https://web.math.princeton.edu/~nelson/papers/rome.pdf https://web.math.princeton.edu/~nelson/papers/rome.pdf and question the consistency of the commonly used axiomatic systems like peano arithmetic: https://www.lesswrong.com/posts/gsvQSpeDHKXxjXwuM/edward-nelson-claims-proof-of-inconsistency-in-peano https://www.lesswrong.com/posts/gsvQSpeDHKXxjXwuM/edward-nel... (Although his inconsistency proof of PA turned out to be flawed.) One should also mention Wittgensteins remarks: https://plato.stanford.edu/entries/wittgenstein-mathematics/ https://plato.stanford.edu/entries/wittgenstein-mathematics/ And in general thoughts going into the direction of ultrafinitism are quite provocative. See for example D. Zeilbergs opinions: https://sites.math.rutgers.edu/~zeilberg/OPINIONS.html https://sites.math.rutgers.edu/~zeilberg/OPINIONS.html
- practal 4y agoHere is my pretty unphilosophical take on the philosophy of mathematics: https://obua.com/publications/philosophy-of-abstraction-logic/2/ https://obua.com/publications/philosophy-of-abstraction-logi... Why should you read it? Because it introduces and explains the best logic known to man, and except me, nobody else really knows about it.
- ngcc_hk 4y agoI love this area. And unlike physics (Bell theorem aside), I hope p of maths would be useful. Or is it a distraction. Just “calculate” is suffice as it is, well, maths?
- frodetb 4y agoThis is wonderful! Just what I need. I keep returning to this one YT series by Frederic Schuller, "Lectures on Geometrical Anatomy of Theoretical Physics"[1], where he goes through the mathematical foundations of theoretical physics from the very bottom. He starts with propositional and predicate logic, presents the ZF axioms of set theory, and then boom, you're off to the races! Recently, I posted a comment in a questions thread on /r/math asking where I could get a more thorough treatment of the philosophy of mathematics, as these lectures can get a bit arm-wavey, but the responses I got were disappointingly dismissive. One guy said all this philosophy was "not interesting" and considered the foundational stuff to be "just a construction". I get that it may not be directly useful for current research to a mathematician trying to get published, but common! [1] https://www.youtube.com/watch?v=V49i_LM8B0E&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic https://www.youtube.com/watch?v=V49i_LM8B0E&list=PLPH7f_7Zlz...
- croh 4y agoI would highly recommend 'Mathematics: Its Content, Methods and Meaning'
- gexaha 4y agoSome more recent books in this area: David Corfield "Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy" Fernando Zalamea "Synthetic Philosophy of Contemporary Mathematics" (though the book is a bit convoluted in the language it uses) also recent translation from French: Albert Lautman "Mathematics, Ideas and the Physical Real" also a book that uses category theory for purposes of philosophy: Rocco Gangle "Diagrammatic Immanence: Category Theory and Philosophy" I also would guess, that nowadays such a book should include chapters about automatic theorem provers, but don't know them
- gone35 4y agoFor a guide with some wonderfully lucid essays, see the masterful Timothy Gowers (et al., eds) The Princeton Companion to Mathematics (Princeton UP, 2008). Utterly disagree. It borders scholarly malpractice.