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Is it even worth studying mathematical philosophy?

  1. Dec 29, 2016 #1
    Most of it isn't helping so much. :sorry: im convinced all "philosophy of xyz" where xyz is a subject, economics or physics is just trashing the subjects foundations and trying to prove it isnt sound.
     
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  3. Dec 29, 2016 #2

    micromass

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    If you ask whether something is worth it, then you have to have some kind of goal in mind. So what's your goal?
     
  4. Dec 29, 2016 #3
    I love it, but it's more of a weird hobby. I have experienced what you are talking about though. I think what helped was to study all of it in historical context. See what questions are being asked and how those questions are changing over time. It's not all bad news.

    -Dave K
     
  5. Dec 29, 2016 #4
    \

    My goal was to try and learn a little bit about mathematics(which I now realise is vague) Im not good at it and I use micromass guide to analysis right now to learn when I have time. I want to learn proofs and stuffs so Im going through book of proof and how to prove it along with some analysis texbooks. I wanted to contact you because you offered self study help but I like your guides so i use them instead haha.

    edit: new goal was to learn proofs and analysis

    edit: I have too many books on the computer and in hardcopy. its hard to keep disciplined :sorry:
     
    Last edited: Dec 29, 2016
  6. Dec 29, 2016 #5
    Well to be honest I want to major in math and I thought logic and analysis, and a little bit of philosophy (to learn them) was the best prepation?
     
  7. Dec 29, 2016 #6
    Actually I think I understand the issue.

    See, the thing is, there there's foundations and then there's f o u n d a t i o n s .

    Does that help? :-p

    By the first I mean the foundations of mathematics that are helpful for being able to communicate about math and do proofs. Logic, set theory, and so forth. This is the kind of stuff that you will find in an "intro/bridge to advanced/abstract" mathematics type book or course or sometimes in a "chapter 0" of a math textbook where the "basics" are covered.

    The other type of foundations is going even deeper into what numbers really are, completeness, incompleteness, etc. Usually this will be under the title "mathematical logic."

    There's a little bit of overlap. The axiom of choice (AOC), zorns lemma, ZFC set theory (as opposed to other types) are things you will learn in a course on mathematical logic but which will also be explained in a less detailed way in a bridge type course. You can write proofs just fine if you don't have a deep understanding of all these things, but you may run into areas where you should know that you are invoking AOC and why for a long time people thought this was a big deal.

    I THINK what you want to learn is the type of foundations you will actually need for studying math.

    Where to learn it?

    I personally loved going through the "bridge to abstract math" type books and just learning about set theory, logic, proofs, etc. I enjoyed https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995.

    Some people find approach this dry, or curmudgeonly exclaim how that when they were in school, (which they walked to uphill in the snow for 10 miles to get to) they didn't have books or classes like this, and they had to learn it the hard way in an analysis course.

    As I mentioned earlier, many textbooks have a chapter 0 or dedicate the first chapter to the foundational stuff that will be needed later in the course. I've found this especially with graduate textbooks like Munkre's topology. It's considered "The stuff you should know before studying this." Hungerford's Algebra also has a great first chapter.

    Extremely hard. I couldn't really do it without specific classes looming.

    -Dave K
     
    Last edited by a moderator: May 8, 2017
  8. Dec 29, 2016 #7
    I have a :sorry:habit of downloading 10000s of ebooks...but i have bought every book I really liked, thanks for the post, it cleared up pretty much most of the issue.
     
  9. Dec 29, 2016 #8
    wow they must be geniuses. I am no genius, hence I though studying the philosophy of math would have helped, but all it makes you into is a pretentious person. So, I will stop. I have learned enough anyway ?:):confused:
     
  10. Dec 29, 2016 #9
    I think I ran into the same thing. Nobody explains this stuff to you when you start studying, and you don't know what half of your classes are about until you're done taking them. (And even then...)

    Not to derail your thread, but will you be able to start classes soon?

    -Dave
     
  11. Dec 29, 2016 #10
    I have 5 months, have to drop out of engineering. Already studied business and econ for one semester before switching. and now switching again :frown: luckily the uni allows one to declare major after first year so I can have ome more time to decide. I dont feel as if I have any talent for this stuff but I realise that you have to work hard, which I try.
     
  12. Dec 29, 2016 #11
    Math education up to this point has worked basically by weeding out people who may have an interest in mathematics but do not catch on that fast. So if they survived without a foundations class, good for them, but I wouldn't have.

    I believe the phrase is "insufferable know-it-all." Seriously though, philosophy and foundations is a fascinating subject, but if you want to study it you might want to leave it for later unless you feel a strong inclination to go in that direction.

    -Dave K
     
  13. Dec 29, 2016 #12
    the thing is I dont know anything, I know there are 14 year olds on here who know more than me haha. Yes, thanks.
     
  14. Dec 29, 2016 #13
    Screw talent. Work hard. If it is feasible, take fewer math classes per semester until you have "learned how to learn" math.
     
  15. Dec 29, 2016 #14

    Stephen Tashi

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    It isn't clear what you mean by "mathematical philosophy". On the one hand you might be the subjects that are called "foundations of mathematics", such as advanced treatments of mathematical logic, various ways to axiomatize set theory. On the other hand, you might mean ill-posed questions that pop into your mind like "What is infinity?", "Is 0.999... = 1 ?", etc.

    Until you have a good grasp of the formal structure of mathematics, pondering things like "What is infinity" won't help you learn mathematics. Before such questions are useful in mathematics, you need to understand mathematics to the point where you question becomes "What specifically do I mean when I say "What is infinity?" ?". You must understand the role of definitions in formal mathematics.

    Studying the history of thought can involve questions like "What was Aristotle's concept of infinity?" or "What was Issac Newtons concepts of infinity?". But often you won't be able to use the historical concepts of something to do modern mathematics because historical concepts also tend to be vague concepts.

    As to studying the foundations of mathematics, there are degrees of understanding proofs and degrees of using it when doing mathematics. It we look at how a typical college course in analysis, algebra, or topology is taught (even at the graduate level), the proofs that are covered don't require a thorough knowledge of advanced topics in the foundations of mathematics (such as the various types of set theory). They only require a "working knowledge" of topics in mathematical logic.

    If you tend to tackle an e-book every time its title looks interesting then your self-education won't resemble the more specialized and directed education of someone taking an organized sequence of courses that is provided by a university's lists of "prerequisites" for courses. As far as covering the material in a standard college curriculum, you won't make the same progress (per unit time) as students whose studies are restricted and organized.
     
  16. Dec 29, 2016 #15
    You are right, as I said, I dont know anything in all honesty. I also don't think most applications of math or solving of mathematical problems or development of theories would require someone to know the difference between naive set theory and ZF axioms, or even peano arithmetic or finitism logicism and others (i honestly forgot :sorry: etc. So, this pursuit is kind of detremental to acquiring real prowess in mathematics.
     
  17. Dec 29, 2016 #16
    ...at the current time.
     
  18. Dec 31, 2016 #17
    You all mean e.g. "Philisophy of Mathematics" (not mathematical philosophy). In general "philosophy of xyz" are not trivial or useless subjects. First of all you learn how to think and you get into some deep conceptual (sometimes even formalistic and logical analytical) understanding and grasp (e.g. of the theories). That way you can more easily know when you or someone is wrong or correct. Also you can better categorize and identify the structure in thinking and of the theories, as sometines we can be very wrong without even knowing or noticing.
    [For example, ~ "you can be the best [precision] "cutter" in the world, and cut perfectly [down] a tree! ... but what if you realize that you shouldn't have cut the tree [down] in the first place! ..." (famous approx. quote - I don't recall whose ... at this moment) ]

    That's why philisophy is needed ... . It's not just trashing ... , it's also orientation etc. ...
    But it doesn't of course substitute physics or math etc.
     
  19. Dec 31, 2016 #18
    Good examples are e.g. Geometry (which inevitably connects to physics) ...
    And set theory, with e.g. Russell's paradoxes (e.g. set of all sets etc.) ... [that not even the best mathematicians can solve ...].
     
    Last edited: Dec 31, 2016
  20. Dec 31, 2016 #19
    Eh? I've pretty much never seen a book of that sort. Can you cite some titles? Mostly the books I've come across that deal in "the philosophy of xyz" are the opposite; they try to establish foundations, not tear them down.
    As an older adult in need of math remediation, mostly I'm re-learning stuff like arithmetic, fractions, and algebra - later trig, powers, logs, etc. if I last that long. Right now I'm working through Gelfand's Algebra and enjoying it. I also have read & enjoyed the first two or three chapters of Tim Gowers Mathematics: A Very Short Introduction in which he introduces abstraction; I will probably go back & re-read once I'm further along in the Algebra book; there is more overlap than I would have guessed given the description offered by Gowers's publisher: "A concise explanation of the differences between advanced mathematics and what we learn at school."

    History of math/science books are interesting too, if well written, and will often touch upon "philosophy" in the sense of how social agreements about science a.k.a. natural philosophy and math have changed over the centuries. I have just started The Invention of Science by David Wooton; it's a brick-like ultra-scholarly tome and may wind up being too much for me; but the first few chapters have been very interesting. I must confess I do get a kick out of books like this that don't so much tear down foundations as bypass conventional narratives in favor of something more interesting. A couple of months ago I browsed through physicist Steven Weinberg's https://www.amazon.com/Explain-Worl...id=1483221565&sr=8-1&keywords=Steven+Weinberg, which is full of his pugnacious & interesting opinions on the vast gulf separating ancient thinkers (mostly Greeks and Arabs) from the first genuine scientists to emerge (starting with Tycho as I remember); and now there is Wooten, who is not a scientist but "an historian" yet obviously finds science of great interest; and who is going to bury me alive in endnotes if I'm not careful.
     
    Last edited by a moderator: May 8, 2017
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