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As it was discussed, for instance, in answers to this MSE question, formal mathematical proofs are deductive in nature. The question I am interested in is:

What is the nature of mathematics at the discovery stage (that is, discovery of new conjectures, new concepts, new proofs)? For instance, is it deductive, or inductive, or something else? Since answers to the question are likely to be too long for MSE, what are references to books or papers addressing this question?

A similar question was asked here, at Mathoverflow, but, I believe, answers at MSE-level, addressed to a more general audience than just research mathematicians would be also useful.

Moishe Kohan
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iwab
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    There is a difference between how new theorems are proven and how mathematicians come up with new theorems or proofs. – Moishe Kohan Oct 24 '21 at 12:37
  • @MoisheKohan and how do we prove a new theorem in math? Natural sciences would now do it with targeted questions to nature (experiments), what do mathematicians do? – iwab Oct 24 '21 at 12:42
  • This is a very complex question, depending on the meaning of the word "prove": Do you mean formal proofs or do you mean the process of discovery of a proof? – Moishe Kohan Oct 24 '21 at 12:44
  • unfortunately I don't know this myself, how can you check if a new theorem is valid? For that you first have to discover a proof and then show it formally I think. What I wonder is that a proof has to be based on something, scientific proofs are based on the answer of nature itself. What is the basis of mathematical proofs? – iwab Oct 24 '21 at 12:49
  • I see, so it seems that you are interested in the process of "creating/discovering" new math as opposed to the nature of formal proofs. For the latter, I suggest taking a look at this question (and answers). For the former, start by reading Poincare's thoughts on the matter which you can find, e.g. here. – Moishe Kohan Oct 24 '21 at 12:57
  • Also, take a look at Alon Amit's answer here. – Moishe Kohan Oct 24 '21 at 14:01
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    Yes it is a deductive science and No, new "mathematical facts" are conjectured/intuited before than proved. – Mauro ALLEGRANZA Oct 24 '21 at 15:24

1 Answers1

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At its core, the question you are asking (when properly formulated) is interesting, difficult and poorly understood.

The first issue, as discussed in the comments, is that one has to differentiate between different stages:

  1. How do mathematicians come up with conjectures or guesses what's true? (Before something is proven, it is not called a theorem but a conjecture.) What plays the role of "nature" or "experimental evidence" in mathematics, when compared to other sciences?

  2. How do mathematicians come up with proofs of their (or somebody else's!) conjectures?

  3. What's the nature of a formal mathematical proof?

  4. How do mathematicians explain their proofs to others or/and convince others that their proofs are correct?

Only stage 3 is deductive. See, for instance, this question and answers.

There are no definitive answers for 1, 2 and 4. Poincare was very interested in 1 and 2 and discussed these (based on his own experience) in his "Reflections on Mathematical Creation". One can say that part of this process is induction, part is deduction. But, overall, the dichotomy deduction/induction is utterly inadequate here.

Also, Alon Amit's answer to a similar Quora's question here is quite good.

As for "what plays the role of nature", the brief answer is:

a. The rest of mathematics, serving as "useful analogy."

b. Heuristics: Look at a simplified form of the question and see it is helpful (regarding conjectures or proofs). If you are lucky, you get the right heuristics.

c. Checking special cases and doing some computer-aided experimentations or calculations. (For instance, Riemann tried some extensive calculations, by hand, when thinking about Riemann Hypothesis.)

d. Other branches of science, especially physics.

Also, take a look at the article by Bill Thurston On proof and progress in mathematics. Thurston responds to an article by Jaffe and Quinn here which is also worth reading.

Thurston also addresses the question (4) I did not even touch here, the one of communication of proofs to other mathematicians. The ratio of "deductive" to "intuitive/informal/etc" here depends heavily on personalities involved.

Lastly, a brief answer to your title question "Is mathematics a deductive science?" is:

(i) "Yes," regarding formal proofs.

(ii) "Not purely deductive," regarding the "discovery stage."

(iii) "It depends," on the "communication stage."

Moishe Kohan
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