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.

  • $\begingroup$ I'm sorry, which statement ? I am just asking a question... $\endgroup$
    – Albert
    Jan 15, 2014 at 21:45
  • $\begingroup$ For applications of homology theory and p-adic analysis, see math.stackexchange.com/questions/13627/…, math.stackexchange.com/questions/214883/…, math.stackexchange.com/questions/122345/…, mathoverflow.net/questions/84320/… $\endgroup$ Jan 15, 2014 at 21:51
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    $\begingroup$ 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 ? $\endgroup$
    – Albert
    Jan 15, 2014 at 21:58
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    $\begingroup$ From one of those answers, the "about this journal" description here is a pretty compelling list. $\endgroup$ Jan 15, 2014 at 22:03
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    $\begingroup$ Assume that some area of math has no use. Then it could be used as an answer to this question. Contradiction. $\endgroup$
    – user1712
    May 9, 2015 at 22:40

3 Answers 3


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.

  • 4
    $\begingroup$ 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 $\endgroup$
    – user88576
    Jan 16, 2014 at 0:28
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    $\begingroup$ @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) $\endgroup$
    – Albert
    Jan 16, 2014 at 18:52
  • $\begingroup$ thank you for your answer. I did not know about these examples $\endgroup$
    – Albert
    Jan 17, 2014 at 14:06

"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 ?"

Take Laplace Transform (LT) and Fourier Transform (FT) for examples. Both are used widely in engineering and physics. Both use infinity. Infinity is not there in nature and in engineering. Therefore these theories are false and cannot or should not be used. A large number is never an approximation to infinity. If you replace infinity, by a large number, then the characteristics of both LT and FT will change. The result of applications will become false.

For example, finite LT will not have poles any more. Therefore by using LT we are corrupting engineering by introducing poles. Similarly, finite FT will change capacity theorem, and uncertainty principle (UP). Finite FT will remove uncertainty from quantum mechanics (QM). Take a look at chapter one on Truth and another chapter on QM in the book at the blog site https://theoryofsouls.wordpress.com/

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    $\begingroup$ Focusing on the LT/FT divergence point, "bona fide" divergences vanish when you truncate, but their effects linger in the extreme size of the numbers that appear in the truncated integrals, provided the truncation is at a sufficiently large point. This is part of why Weierstrass' definition of limit is so powerful in relating the theory back to science and engineering. Also, this question is almost two years old now and has an accepted answer. $\endgroup$
    – Ian
    Oct 7, 2015 at 15:46

I think the Collatz problem and the twin prime conjecture have no applications. I think study of them is entirely for fun.

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    $\begingroup$ It's not very practical, but the Collatz conjecture can be used to multiply large integers. $\endgroup$
    – DaBler
    May 16, 2020 at 12:29
  • $\begingroup$ The theories that people built to solve/attack those questions is most definitely useful though. It's true that the questions themselves may be useless, but the actual mathematics that people do about those questions is far from useless. $\endgroup$
    – D.R.
    Mar 20, 2023 at 7:27
  • $\begingroup$ Probably all of number theory has applications. Think cryptography and such. $\endgroup$
    – mick
    Aug 9, 2023 at 18:34

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