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So I've been interested in philosophies of different sorts in the past, but nowadays I've reduced philosophy only to topics in philosophy of science, because the other branches are so inconclusive and often useless in practice.

But if I was to read about some philosophies regarding mathematics, I might actually learn something new that would improve my thinking towards studying and reading mathematics.

Some of the topics that particularly interest me are:

  • The relationship between the real world and mathematical objects.

  • The philosophy of mathematical systems (that is, e.g. is logically exact mathematics good or bad (and why?) compared to more relaxed systems of logic)

  • The relationship of mathematics to other sciences (but I don't want to read any hocuspocus here, but something that's as rigid as natural scientific thought and as practical as real world natural scientific research).

Also, I'm not interested in reading philosophical thoughts that much as reasoning that's inherently tied to mathematical research. I.e. I don't want to read opinions about mathematics, but philosophy that's related to mathematical research. I for example am not sure whether some "schools of thought" (formalism, logicism, ...) are that useful per se, but the ideas that they have regarding the improvement of mathematics could be more interesting than some dogmatic theses or idealistic theories (e.g. "mathematics is an universal language").

N. F. Taussig
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mavavilj
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  • Would highly recommend Bertrand Russel's Introduction to Mathematical Philosophy as a start to gain some exposure first. – Harry Alli Oct 01 '17 at 07:41
  • @HarryAlli Why do you think it's a good book? – mavavilj Oct 01 '17 at 07:41
  • Well it starts off from the real foundation of maths, adopting a very abstract approach. It begins with Peano's definitions of number and what a number really is. Then it progresses into the abstract notions of infinity and limits. I guess what I really enjoyed about it was that it really deepened my understanding and provided me with a different outlook of maths. – Harry Alli Oct 01 '17 at 07:46
  • @HarryAlli What do you mean by "real foundation"? I perceive that the current formal mathematics is just a thing of these times, but I'm not sure if it'll stay. So I wouldn't call the formalization of mathematics that begun in the early 20th century to be "real foundation", because it's possible that it could change. There's no definition for "real mathematics" that I know or would agree on. – mavavilj Oct 01 '17 at 07:49
  • Of course if you mean that it's real in the sense that it has a good degree of being in accord to e.g. other sciences and the nature, then in this sense it could be considered a good foundation, but I don't know about "real". – mavavilj Oct 01 '17 at 07:51
  • By foundation, I meant the vague principles that the first people used to define the most elementary constructs in maths. So how we defined maths before we knew what maths was. – Harry Alli Oct 01 '17 at 07:53
  • Even if I don't always agree with him, I advice to let apart Russel and read Morris Kline https://en.wikipedia.org/wiki/Morris_Kline#Books The one I liked most is "Mathematics: The Loss of Certainty" (I read it twice) but a classic it's definitely "Mathematical Thought From Ancient to Modern Times" – Raffaele Oct 01 '17 at 11:11

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