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I made this question because I've seen so many mathematicians who were also philosophers and vice versa; also from the way that mathematicians can build arguments in non-mathematical contexts, which suffers from a strong bias of the so called "exact" science.

What can be said about the distance between philosophical methodology, that is, the modus operandi through which philosophers justify their hypothesi, to the way that mathematical knowledge is structured? Is there any gain for a philosopher to acquire some base in mathematics, and for mathematicias who gather some pjilosophy knowledge and methods?

  • The laws of logic are part and parcel of maths. – Neil Meyer Jul 18 '15 at 8:48
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(Background: I'm a PhD student in logic, so I'm a little bias, but I've done a fair amount of work in both mathematics and philosophy.)

I'd probably argue that the philosophy-mathematics divide is narrower than most non-philosophers/non-mathematicians realize, but wide enough to be importantly different.

They certainly overlap, at least ideally, insofar as they both stress the use of systematic, critical thinking. Both put emphasis on arguments and on making distinctions to help elucidate problems. They also overlap in practice, insofar as they both tend to fall short of this ideal. Many mathematical proofs are given informally, invoking intuitions and leaving out a number of the details ("left as an exercise"); in a similar fashion, many philosophical arguments are not stated/stateable in premise-conclusion form, and so their analysis is not always as systematic as one would hope.

But their methodologies are importantly different. Philosophers, for instance, are very eager to argue over the foundations, and often debate about the most fundamental aspects of their field. Mathematicians, by contrast, work in a more cumulative fashion, often using the work of other mathematicians as a springboard for more elaborate proofs rather than questioning the foundations of their field. That's not to say mathematicians don't ever disagree, or that philosophers don't use the work of previous philosophers as springboards. But when mathematicians disagree, it's often presupposing there is an objective way of settling the matter (though not always). Furthermore, when philosophers build off of other philosophers' theories, it's usually involves modifying (sometimes very fundamental) aspects of the original theory.

With regards to whether a philosopher should learn some mathematics, or whether a mathematician should learn some philosophy, I would say almost unqualifiedly in both cases yes. They overlap enough that they should be aware of at least some of the basics in each field. I'd probably say that about most fields though: most fields can benefit from studying philosophy and mathematics. Whether philosophers have something in particular to gain from mathematics largely depends on what area of philosophy you're talking about, and even then it may be less about acquiring a certain methodology and more about the content of the mathematics in question (some branches of mathematics could, for instance, contain theorems that have relevance to certain philosophical positions). Similarly, mathematicians certainly have something to gain from philosophy, but studying Nietzsche is probably not directly going to make you a better mathematician (though it may make you a better person).

  • Re: "Many mathematical proofs are given informally, invoking intuitions and leaving out a number of the details ('left as an exercise')." This may be the case in many textbooks, but all of the theorems discussed by mathematicians have been proven rigorously. This came about from an intense period of questioning the foundations of mathematics that occurred in the early 20th century. See plato.stanford.edu/entries/hilbert-program See also, Foundations of Analysis by Edmund Landau. Statements that have not been proven are hypotheses. – Andrew Jul 18 '15 at 7:12
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I think the relationship is only one way. I believe that mathematics can have a huge influence and does have an influence in philosophy. The mathematics in the uncertainty principle has very heavy philosophical underpinnings as does the work by Georg Cantor on infinities, and if you have read the history of Georg Cantor and his journey down that rabbit hole you will know that it affected him to the point of suicide. One might argue he already had suicidal tendencies but his writing on his progress show a very psychologically desperate human having seen ,what seems to him a nightmare in absolute truth. I dont think i need to explore the effect of the uncertainty principle on philosophy but i think we can all agree that is it very large.

I dont think we can say the same for philosophy. I dont think mathematics is influenced in much by the ideas of marx or the non ideas of existentialism ( see what i did there ;) ... maybe but i dont think it is vital or even close to any sort of necessary.

So how close is philosophical thinking to mathematical thinking? I think they are on the opposite sides of the spectrum but with that said i think mathematics can influence psychology. This question itself is of a philosophical nature of which pure mathematics has no concern for. When one starts to talk about mathematical proof or abstract concepts in mathematical theories they have already entered into philosophy.

Mathematical Thinking as i understand it is more about the search for absolute truth in numbers whereas philosophy is a search for an interpretation of truth in areas of the mind and society where absolution becomes very relative and factors of determination of ones absolution is subjective to an infinity of probabilities.

I think it takes a different type of thinking to become a master of one of these art forms. And again i think an overlay could only occur in one direction , a mathematician would have a far easier time in conversation with a philosopher but i do not think a philosopher could easily understand a conversation of any major publication in mathematics.

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There are few examples of persons active in both disciplines mathematics and philosophy. The two most eminent are Leibniz and Russell.

Kant - in reply to Leibniz - published one essay on a certain topic from mechanics. He also has been thinking on the origin of the planetary system. But because physics is not the topic of the present question I will not go into the details.

Instead I want to emphasize the differences between mathematics and philosophy and between the methods employed in each field.

  1. Mathematics, at least pure mathematics, is a game. It does not need any link to real-world problems. Nevertheless, sometimes the results of some of these games, i.e. certain mathematical theories like differential geometry or Hilbert space theory, apply to physics. Hence the results of some mathematical games can be used to explain real-world phenomena.

  2. Like every game mathematics has strong and clear-cut definitions and rules.

  3. The general content of any mathematical game is to invent interesting mathematical propositions and to find out proofs for them. Mathematics is one of only two sciences which is able to prove its statements.

  4. In general, the experts agree whether a proof is correct or not.

On the other hand, philosophers

  1. do not consider their profession a game

  2. consider topics with a much deeper scope than the one-layer problems from mathematics

  3. strive to answer fundamental real-world questions, often those where science cannot offer any answer

  4. do not handle these questions on the basis of a formal language with clear-cut definitions

  5. present arguments which are questioned by their companion philosophers.

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I have (to me) a very interesting example of combining mathematical thinking with philosophical thinking, and even to some degree, with mistical thinking as well:

It is the book "Infinity and the Mind" by Rudy Rucket. There you will find how

  • The concept of infinity
  • Set theory
  • Formal systems
  • Computation
  • Proof & mathematical truth

are approached with or taken as a ground for ( or vice versa)

  • The one vs. many problem
  • Absolute
  • Mind
  • Soul
  • Mistical knowledge
  • Intuiton
  • Reality
  • Platonic existence

etc.

I hope the book will convince you how mathematical thinking is not too distant from philosophical one, as it convinced me that, it really is the case, at least for many common questions and problems.

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First of all, I would like to indicate that I think this nearness or parallelism is only apparent in analytic philosophy which is not the whole of philosophy. But secondly, I would claim that analytic philosophy made better in proportion to which it is clear and rigorous (which are sometimes competing virtues), just as in mathematics. Still, as in each discipline, these are not the only virtues.

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Mathematics overlaps with philosophy mainly in the Anglo-American tradition. In Continental philosophy it is more influenced by literature and theology.

Of course there are important exceptions.

Plato approved of mathematicians but also warned that its grasp is small. But he also disapproved of poets.

Its certainly true that philosophers can do philosophy without mathematics.

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I always wonder, why this kind of question, "capturing" mathematics as if it is an existence the origin of which is not from human beings, but something alienating object facing to human beings, from time to time appear frequently. I feel like people favor to dwell in more metaphysical side rather than materialistic side.

From Critique of Hegel’s Philosophy in General

||XXXII| The absolute idea, the abstract idea, which...

(The man estranged from himself is also the thinker estranged from his essence – that is, from the natural and human essence. His thoughts are therefore fixed mental forms dwelling outside nature and man. Hegel has locked up all these fixed mental forms together in his logic, interpreting each of them first as negation – that is, as an alienation of human thought – and then as negation of the negation – that is, as a superseding of this alienation, as a real expression of human thought. But as this still takes place within the confines of the estrangement, this negation of the negation is in part the restoring of these fixed forms in their estrangement; in part a stopping at the last act – the act of self-reference in alienation – as the true mode of being of these fixed mental forms; * –

Even Hegel's idea, human thoughts, aka, science or mathematics etc, etc, according to Marx, need to be aufhebened, citing another line from the source,

||XXXIII|

Nature as nature – that is to say, insofar as it is still sensuously distinguished from that secret sense hidden within it – nature isolated, distinguished from these abstractions is nothing – a nothing proving itself to be nothing – is devoid of sense, or has only the sense of being an externality which has to be annulled.

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