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We have all heard the argument : a lot of mathematics that was thought to be useless, abstract constructions with no links to the real world ended up being of use, like some arithmetic is useful in crypto. Some people say that the same applies to all of mathematics.

My question is : do you buy this ? More precisely, and to make this into a question that I hope fits the requirements of this website :

Do you have an example of a mathematical theory (i.e. not an isolated theorem, but a coherent set of mathematical concepts and theorems) that you believe will be of no use, ever, to let's say engineers or physicists or non-mathematician scientists in general ?

By of use I mean that it is a mathematical object so relevant to a field or a model that non-mathematicians have to think in terms of this mathematical theory, OR that it is a crucial ingredient to the mathematical proof of some other useful mathematical result. Giving one proof among other, more simple ones is not sufficient. And a theory which language can be used to describe certain models but gives no significant insight or power of prediction doesn't count.

A few examples :

  • hamiltonian systems are a good way to model most mecanical systems : useful
  • Fourier transforms are a useful tool in numerous calculations : useful
  • not being an expert in mathematical physics I can't give a precise example, but I would classify homology theory in useful because it is such a powerful mathematical tool that I'm sure it is a necessary ingredient to something with real-life applications, or will be
  • tilings : applications in chemistry, including recently one with Penrose's aperiodic tilings : useful
  • p-adic analysis : to my knowledge not useful.

Let me stress the "to my knowledge", as I know next to nothing about p-adic analysis and what may or may not be its applications.

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I'm sorry, which statement ? I am just asking a question... –  Glougloubarbaki Jan 15 '14 at 21:45
I would disagree that these links provide applications of p-adic analysis in the sense I required. Some applications to number theory and complex dynamics, OK, but are those results themselves important for a physicist or an engineer ? would they be useful in the sense that a physicist or an engineer would have to learn p-adic analysis because of them ? –  Glougloubarbaki Jan 15 '14 at 21:58
From one of those answers, the "about this journal" description here is a pretty compelling list. –  Chris Culter Jan 15 '14 at 22:03
we can't prove your affirmation even there is physics which eventally will worth nothing is a pity :( for example i can not ge t a job because my math is unesefull :D vixra.org/author/jose_javier_garcia_moreta –  Jose Garcia Jan 15 '14 at 22:11

1 Answer 1

up vote 4 down vote accepted

I think "all math is eventually useful" must be false... There is certainly math that gets abandoned when better things come along, as with cylindrical algebras (as I understand their history). There are things like first-order mereotopology which have seen little development outside philosophy departments (and not much in them either). Arguably (and I say this with much sadness), NF will never catch on as a serious set theory so much as a source of odd model theory; probably likewise with NFU (which is even less deserving of such a fate).

Not that I think uselessness is a bad thing, of course.

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In fact the finest mathematics is often useless, whereas second rate mathematics (numerical math, statistics, probability) is very useful but intrinsically of no value whatsoever –  user88576 Jan 16 '14 at 0:28
@KarlKäfer while the sentiment is more or less shared by most pure mathematician, the formulation is a bit harsh... also there definitely are some beautiful math in probability (even if you haven't been that far yourself) –  Glougloubarbaki Jan 16 '14 at 18:52
thank you for your answer. I did not know about these examples –  Glougloubarbaki Jan 17 '14 at 14:06

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