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In an extensive anthropological joint research project concerning the necessities in the development of life and civilisation my group is concerned with mathematics. This forum seems to be extremely well-suited to answer the following question:

Given the case that a civilisation independent of mankind is or will be existing. How would their mathematics look like and why?

Some teasers: Integers from simple counting to the Binomial Theorem, Geometry including Pythagoras' Theorem, Probability Theory, and Analysis seem to be dictated by practical requirements. Research on Prime Numbers may be a necessary by-product of calculating with integers. What about more advanced concepts like Grothendieck universes or Category Theory? Can even somewhat be said about branches that we don't yet have exploited ourselves?

(This question should be made wiki but I could not find the button. Seems that something has changed during the recent year.)

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closed as primarily opinion-based by quid, Stefan Kohl, Yoav Kallus, Felipe Voloch, Lucia Oct 9 at 21:08

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

This question is too broad, even for community wiki; it is also not answerable. I'm not sure of another appropriate forum for this kind of question, but I don't think it fits mathoverflow. Maybe – Eric Tressler Jul 5 '13 at 7:23
Even the ancient greeks had a big problem with the concept of numbers, so even very smart and mentally "very near to us" minds would probaby not use a concept of numbers. The other observation is that a look into discussions in "High IQ"-groups in the internet show an amazingly low level of understanding the concept of science/mathematics (e.g. all those quarrels about Fermat_Wiles' theorem, relativity, etc.). so, even under bright, curious, high ed. humans the basic concept of math in nonexistent. – Thomas Riepe Jul 5 '13 at 7:31
For the record, my reason for voting to close was that answers would be too "opinion-based". Cf. Nagel's "What is it like to be a bat?" – Yemon Choi Jul 5 '13 at 8:43
A shame that this interesting question has been closed. It is conceivable that a form of life might not include individuals -- an intelligent fluid layer in an ocean or a star perhaps -- would such life appreciate the integers which seem so natural to us as a result of our discrete nature? – J.J. Green Jul 5 '13 at 18:45
I think it is Mumford who once observed that the best evidence for the existence of extraterrestrial intelligence was Deligne doing math. – Olivier Jul 6 '13 at 10:55

2 Answers 2

Here is an interesting paper which addresses this question:

MR0940484 (90c:00017) Ruelle, David(F-IHES) Is our mathematics natural? The case of equilibrium statistical mechanics. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 1, 259–268.

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See also his "Conversations on Mathematics with a Visitor from Outer Space"… – Alexey Ustinov Nov 5 at 2:37

I would think this question about alien mathematics would need to be addressed in connection with the question of alien physics, in view of Wigner's "unreasonable effectiveness of mathematics in the natural sciences". Since the physics is likely to be the same in the entire Universe, that would drive a civilization towards similar mathematics.

See also: Alien mathematics: is Pi universal? by Ian Stewart.

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