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In the past a lot of math was motivated by practical applications from real life; that's how geometry started for example. Some other areas were developed when trying to solve problems that already existed in math, say groups via Galois theory for roots of a polynomial.

I wonder if any math theory started somewhat out of the blue, say someone just decided to study some objects and their relations out of pure curiosity, even though those things were not really related to the existing math at that moment.

As an example, a person just might have thought whether one can write 9 first integers in a square such that every row and column sum up to the same number, and that's how the mathematics of magic squares happened to start.

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  • $\begingroup$ Yes, this has sometimes been done. But in most of those cases, no one else found it interesting enough to work on it themselves. So that person remains an isolated researcher. As long as (good) journals will still publish the work, he/she can continue working on it. $\endgroup$
    – Gerald Edgar
    Apr 5, 2022 at 16:51
  • $\begingroup$ The answer to your concrete question is "no". An interesting new branch of math cannot start from nowhere simply because it won't be interesting. From nowhere one can get a new idea to solve an old problem or create a new math model of an existing practical situation. That happens all the time. One can also make a lot of computations and discover new unusual patterns for some old objects (prime numbers, or geometric figures, for example). But the objects should be really interesting. $\endgroup$
    – markvs
    Apr 6, 2022 at 2:13
  • $\begingroup$ @GeraldEdgar I completely agree, that's why I'm interested in those examples that are not most of those cases and actually happened to be interesting to others as well. Most likely, something started as an idle curiosity and out of a sudden happened to be appealing to others - e.g. unexpected connections to another area popped up. $\endgroup$
    – Ilya
    Apr 6, 2022 at 11:26
  • $\begingroup$ Btw, the example of magic squares came to me exactly when I've recalled a non-English website on math, where there was a person who claimed to have some break-through results in this area. This person published their results only in their own language, and claimed that if someone is interested, they should learn this language to learn about the results (rather than publishing them in English). So magic squares certainly serves as a bad example to my case, I just used it as something close to what I have in mind. $\endgroup$
    – Ilya
    Apr 6, 2022 at 11:29
  • $\begingroup$ @markvs I tend to disagree: you may start studying something out of curiosity, but meet interesting and challenging theory there. I just thought that Diophantine equations would sound as a completely random and artificial exercise to me, not to mention the whole hype about prime numbers (at least before the cryptography came into play), yet number theory was a popular topic for quite some time to say the least. $\endgroup$
    – Ilya
    Apr 6, 2022 at 11:33

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What you describe is closely related to what is called recreational mathematics. One could make the case that games, puzzles, and doing things just for fun are just as much a part of "real life" as anything else, but when people say "practical applications from real life," they usually tacitly exclude recreation.

It is not easy to come up with mathematical problems that are neither "too hard" nor "too easy." Recreational mathematics has a tendency to fall into one of these two categories, and hence rarely develops into what we would call a "branch of math." For example, the question of the existence of Lychrel numbers was probably first "randomly" asked out of curiosity by someone with no practical application in mind. Although there has been quite a bit of effort devoted to Lychrel numbers, the subject is "too hard" in the sense that it seems difficult if not impossible to make significant theoretical progress on it. So Lychrel numbers are not what one would call a "branch of math."

The surreal numbers are perhaps an exception in that they originated in recreational study of the game of Go, but have turned out to be a rich subject of mathematical study. Origami has its roots in recreation, or at least in art, and its mathematical study has turned out to be surprisingly rich, with applications in engineering.

EDIT: Another interesting case study is graph theory. Its origins can be traced to the seven bridges of Königsberg and the four-color problem—questions that were driven primarily by curiosity rather than practical necessity. There is no doubt that graph theory is now a major branch of mathematics. However, one could argue that its development into a major branch of mathematics was driven by the rise of computers and the many applications of graph theory to computer science and electrical engineering.

It's also worth mentioning that even in well-established areas of mathematics, new subfields often arise not because of perceived applications, but because of questions that mathematicians find interesting to study. If you're looking for examples of areas of mathematics that were born from "curiosity-driven" rather than "applications-driven" reasons, then there are too many to list.

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  • $\begingroup$ I'd gladly accept your answer if you were kind enough to include a couple of examples of those areas (of truly serious math) that were started curiosity-driven, rather than application-driven. I'd still consider bridges and game of Go as somewhat application-driven cases, whereas a four-color problem does certainly constitute to a rather random curiosity imho. $\endgroup$
    – Ilya
    Apr 6, 2022 at 11:23
  • $\begingroup$ @Ilya You gave some examples yourself. Much of number theory originated with no "practical applications" in mind. In the late 20th century, cryptography emerged as an application of number theory, but prior to that, most number theory was pursued without thought of applications. G. H. Hardy famously argued that most of pure mathematics, especially number theory, is devoid of practical applications (but not devoid of value). Hardy's essay is controversial but he is right that much pure math is not motivated by applications. $\endgroup$
    – Timothy Chow
    Apr 6, 2022 at 12:22
  • $\begingroup$ Yeah, I've realized that just now, did not expect you to read the comments to the OP :) number theory certainly qualifies and perhaps is a benchmark. Yet, I would not say that much of the rest of pure math developed that way, so a couple of examples would be good $\endgroup$
    – Ilya
    Apr 6, 2022 at 13:08
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    $\begingroup$ @Ilya If you look at the achievements for which the Fields Medal has been awarded, most of them have no "practical applications to real life." The Poincare conjecture, Grothendieck's revolution in algebraic geometry, forcing in set theory, the Burnside problem—all of these questions were raised and pursued without regard to applications outside pure mathematics. Yes, centuries ago, when people first considered spheres and Cartesian coordinates and sets and groups, there was some connection to the "real world," but the modern developments lack that. $\endgroup$
    – Timothy Chow
    Apr 6, 2022 at 15:24
  • $\begingroup$ I can track the Poincare conjecture back to study of sphere vs torus etc, so I would not say it does not have any real world roots, as opposed to say number theory. Perhaps the latter is the only example of a field that started out of pure curiosity. Anyways, since we did not seem to understand each other about which examples are needed, I think it is not fair to keep your answer without being accepted! $\endgroup$
    – Ilya
    Apr 7, 2022 at 11:37

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