[HN Gopher] Philosophy of Mathematics - A Reading List (2020)
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Philosophy of Mathematics - A Reading List (2020)
Author : keiferski
Score : 89 points
Date : 2022-09-20 20:00 UTC (2 hours ago)
(HTM) web link (www.logicmatters.net)
(TXT) w3m dump (www.logicmatters.net)
| steve_john wrote:
| Koshkin wrote:
| Why just quote Wikipedia?
| Jtsummers wrote:
| That's pretty much all their comments. Direct quotes from
| something like Wikipedia, the article, or a paraphrased
| version of a statement from the article.
| mberning wrote:
| It'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.
| Koshkin wrote:
| It's not unusual for math books to include historical notes.
| For example, _Elements de mathematique_ by N.Bourbaki includes
| an entire companion volume, _Elements d 'histoire des
| mathematiques_.
| ouid wrote:
| bourbaki is unusual among math textbooks, to say the least.
| openfuture wrote:
| The 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-
| ded...
| throwaway81523 wrote:
| The 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
|
| Philosophy as an MO search keyword also finds interesting posts:
| https://mathoverflow.net/search?q=philosophy
| steppi wrote:
| Working 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, many under the section on contemporary
| developments. Smith's list also contains more recent works that
| do not appear in Avigad's list.
| zmgsabst wrote:
| I 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
| adamnemecek wrote:
| Koshkin wrote:
| There goes my evening...
| morelisp wrote:
| Smells like [?] to me.
| adamnemecek wrote:
| Which part?
| morelisp wrote:
| The part where you take a basic mathematical concept and
| claim it solves all "meaning".
| adamnemecek wrote:
| It's not a basic concept. If you think it is you legit
| don't understand it.
| openfuture wrote:
| Consider matrix bridge for those of us who will never use
| discord?
| Koshkin wrote:
| This 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 wrote:
| _Modern 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 Godel'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.
| valyagolev wrote:
| Synthetic Philosophy of Contemporary Mathematics by Fernando
| Zalamea
|
| for something very lively, contemporary and more continental in
| spirit
| fan_of_yoinked wrote:
| This 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 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 wrote:
| Meh. 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?
| silent_cal wrote:
| Transcendent triangles... hm...
| criddell wrote:
| > Matter has never produced a perfect sphere, but spheres are
| nevertheless truly real--- right?
|
| But doesn't Tegmark say matter _is_ mathematics? Or is that
| your point?
| chestervonwinch wrote:
| parent is saying mathematics is the abstract base class.
| they're in the source, but you never see instances of them at
| runtime.
| sbdaman wrote:
| Edit: nevermind, this was overly critical.
| goatlover wrote:
| Platonism is a major school of thought in the philosophy of
| mathematics.
| virissimo wrote:
| Not only that, but most philosophers who specialize in the
| philosophy of mathematics are Platonists: https://philpaper
| s.org/surveys/results.pl?affil=Target+facul.... IME, this
| is also true of working mathematicians, but much less true
| of physicists.
| [deleted]
| EpiMath wrote:
| good comment. This seems to come up more in number theory than
| in foundations/philosophy of mathematics, but I agree is has an
| important place. Not just triangles, but natural numbers having
| a fundamental place in reality ( e.g. integral numbers of
| dimensions, degrees of equations at the foundations of physics,
| etc., etc. Daniel Shanks has a list of about 60 of these
| "arguments" for Pythagorean interpretation of numbers )
| anthk wrote:
| Triangles' concept yes, in any of them, as long as you have two
| dimensions.
| tomrod wrote:
| You can have triangles in higher dimensions, embedded in
| manifolds or subspaces.
|
| You can have super pathological triangles in 1 and 0
| dimensions. Those aren't terribly interesting though.
| steppi wrote:
| There's an entire chapter devoted to Plato in Stewart Shapiro's
| _Thinking About Mathematics_ , the first book the author
| recommends. I think it's pretty reasonable to recommend people
| start with an accessible contemporary survey rather than diving
| directly into Plato's dialogues or any other particular primary
| source. As far as I'm aware, Pythagoras left no written works
| and would thus be unlikely to appear on a reading list.
| Koshkin wrote:
| Indeed! 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.
| ")
| Barrin92 wrote:
| the _notion of commonality_ exists in the head of the
| observer because it 's a useful fiction. (even 'the horse'
| itself is). But that is a subjective form of existence, what
| Jakob von Uexkull called one's _" Umwelt"_ (the world as it
| presents itself to you). There's no reason to believe it
| precedes matter, or that triangles exist in a world without
| anyone to conceive of them.
| colechristensen wrote:
| So 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 wrote:
| Sure, 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.
| colechristensen wrote:
| Then you're painting yourself as one who has a certain
| definition of _exist_ and can 't imagine other
| definitions who would participate in such discussions
| unwittingly about differences in definition rather than
| the subject matter.
| Koshkin wrote:
| But it is _existence_ that is _the_ subject matter.
| jonnybgood wrote:
| As another comment said. What do you mean by exist? It
| requires a little more rigour here. Using your example: What
| does it mean for three horses and three apples to exist?
|
| The better question: What does it mean for three (of
| anything) to exist? Why not one thing, one thing, one thing?
|
| The definition of existence you appear to be using is the
| physical proximity of those objects. But even that can get
| quite hairy. If there is two apples within inches of each
| other and another apple 100 feet away, are there three apples
| or two? The answer to this question depends on what you mean
| by the existence of three apples.
| sbdaman wrote:
| Edit: replies are right!
| mjh2539 wrote:
| This would be considered an incredibly opaque
| (and...particular?) introduction to the philosophy of
| mathematics.
| sbdaman wrote:
| Fair enough.
| silent_cal wrote:
| I would venture to say that one of the most difficult
| philosophy books of all time is not a good intro to anything,
| lol
| Koshkin wrote:
| Not if you read it in German
| [deleted]
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