26

I am looking examples of a mathematical theory (i.e. a body of knowledge, with its own definitions, results, principles etc., i.e., its own language) that is completely inactive or forgotten by today.

For example, it seems to me that sphere geometry is largely inactive today (although I am sure someone in the comments will correct, that, actually, somewhere there is a very active seminar on sphere geometry going on).

I would like such examples of theories to be fairly obscure, yet still readable, if I were to devote time to them. In this sense:

  1. Bonus points of the books or papers where the theory is explained cannot be found on books.google.com.
  2. Bonus points if this theory is not mentioned in the Princeton Companion to Mathematics.
  3. Bonus points if the books are available only in languages that are not English.
  4. Bonus points if there exist carefully written textbooks explaining the theory (no matter how old the books is).

It is OK if the theory is subsumed by a more general theory that is active today, as long as today's theory uses a different "language" (I will let you decide how exactly you define what "language" is). For example, if there was a lot of theory to explicitly solve certain types of equations in the 17th century, for various tricky cases, but that is all supplanted today by, say, a numeric approach, than that would be fine by me.

I have read this post, as well as these notes on lost mathematics, but they don't quite fit the bill.

LSpice
  • 11,423
user7088941
  • 395
  • 18
    Branches of mathematics don't die, they just ascend to higher generality. – Gro-Tsen Jul 20 '23 at 22:25
  • 2
    Joke aside, what often happens is that either the branch gets subsumed under a more general branch or language, or else it is no longer under active research because it is believed that there are no longer any truly interesting results to discover. But even in those cases, they never disappear into oblivion. Even, say, descriptive geometry and screw theory aren't entirely forgotten by mathematicians (and certainly not entirely forgotten by other people). – Gro-Tsen Jul 20 '23 at 22:30
  • 5
    @Gro-Tsen This could be a case of survival bias - since the only theories one still knows about today, are the ones that were fruitful enough to ascend to higher generality and therefore are remembered. Those unfruitful theories were erased from the memories of most. (But screw theory -funny name- is an example that could come close to what I'm interested in, but it is not obscure enough, the language doesn't seem to really have change significantly; and it is definitely not forgotten: https://golem.ph.utexas.edu/category/2019/10/screw_theory.html) – user7088941 Jul 20 '23 at 23:42
  • 4
    Re, I am reminded of the paradoxically titled question Do you know important theorems that remain unknown?. – LSpice Jul 21 '23 at 00:46
  • 1
    Spherical trigonometry was discussed in another MO question: Is spherical trigonometry a dead research area? – Timothy Chow Jul 21 '23 at 03:58
  • Aristotelian logic? See also https://math.stackexchange.com/questions/192626/what-mathematical-ideas-concepts-became-obsolete-due-to-technological-progress – Gerry Myerson Jul 21 '23 at 10:38
  • "All The Methods I Learned In My Mathematics Degree Became Obsolete In My Lifetime" https://www.huffpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_b_58693ef9e4b014e7c72ee248 – Gerry Myerson Jul 21 '23 at 10:41
  • 2
    I'd suggest investigating the mathematics underlying obsolete technology or theories. For example, optimizing the programming of early electronic computers, developing a mechanical marine chronometer, Ptolemaic astronomy, the caloric theory of heat, etc. – Timothy Chow Jul 21 '23 at 11:58
  • 1
    It seems due to the moderator strike big list questions like this can’t be changed to CW – Benjamin Steinberg Jul 21 '23 at 12:21
  • 1
    @BenjaminSteinberg What moderator strike is going on? Turning my question into a CW would be very good. – user7088941 Jul 21 '23 at 13:55
  • The link you provided contains a lot of nice mathematics, but unfortunately falls out of scope regarding my question. That MO question contains rather known theorems but whose assumptions have been weakened (e.g. the Monotone Convergence Theorem, which is currently highest voted, that doesn't require monotonicity), or theorem from well-known areas of mathematics that were not known for quite a while (Zabreiko's lemma). I, on the other hand, am more interested in entire (niche) areas of mathematics that are unknown to this day. – user7088941 Jul 21 '23 at 13:59
  • 4
    @user7088941 "Inactive" is probably more promising than "unknown." If something is truly unknown then nobody reading this will know about it – Timothy Chow Jul 21 '23 at 14:15
  • Look at meta about the strike – Benjamin Steinberg Jul 21 '23 at 16:23
  • 1
    Does the 19th century theory of determinants count? – Steven Landsburg Jul 21 '23 at 17:43
  • @StevenLandsburg You mean the sort of thing that's in Muir's Treatise on the Theory of Determinants? I've seen that book cited in modern literature, e.g., in Krattenthaler's Advanced determinant calculus. – Timothy Chow Jul 22 '23 at 12:22
  • 1
    @TimothyChow : Yes, I had Muir in mind. I know it still gets cited, but I believe (though cannot prove) that it contains much that is essentially forgotten. – Steven Landsburg Jul 22 '23 at 15:19
  • 1
    Does sangaku meet your criteria? – Somos Jul 22 '23 at 19:34
  • 2
    Calculus of finite differences? (Part of it has been absorbed by numerical analysis; part by combinatorics.) – Ira Gessel Jul 24 '23 at 19:41
  • This question is obviously different from the question that it's supposedly a duplicate of, so I'm voting to reopen. Overly aggressive closing of questions that aren't really duplicates has ruined Stack Overflow, and I don't want to see the same thing happen here. – arsmath Jul 29 '23 at 17:02
  • 1
    @arsmath, you could post to https://meta.mathoverflow.net/questions/223/requests-for-reopen-and-undelete-votes-for-closed-and-deleted-questions – Gerry Myerson Jul 30 '23 at 01:34

1 Answers1

20

The most difficult condition to satisfy is 1, since Google book "knows" most of books. All other conditions are easy to satisfy, and many examples can be given, but I will give only one.

Princeton Companion in fact does not mention MOST of mathematics, if you count by volume of output (books and papers). It only mentions what is "most important" on the view of the authors. And this view is inevitably subjective.

Here is an example of a theory which satisfies most of your criteria. It is called "Series of exponentials", and deals with representation of analytic functions in the form of $$\sum_{n=0}^\infty a_ne^{\lambda_nz},$$ where $a_n$ and $\lambda_n$ are arbitrary COMPLEX numbers. The theories with real or pure imaginary $\lambda_n$ are much better known.

The main developers of this theory were A. F. Leont'ev and his collaborators and students. Leont'ev published 5 books on the subject, all of them exist only in Russian (were never translated to any other languages). Google books only lists the titles. All his books are carefully written, and the smallest one:

MR0753827 Leontʹev, A. F. \cyr Tselye funktsii. Ryady èksponent. (Russian) [Entire functions. Series of exponentials] "Nauka'', Moscow, 1983. 176 pp.

is a good introduction to the subject. So this area satisfies all your criteria, except possibly 1.

Still I would not call it "obscure" or "inactive". There is a group of mathematicians, mostly in Russia which still pursues the subject, and some of their results are quite deep.

On the other hand, there are "obscure" and "inactive" areas which do not satisfy any of your conditions 1-4. An example is "arithmetic of probability distributions". The book which describes the state of the art is

MR0428382 Linnik, Ju. V.; Ostrovsʹkiĭ, Ĭ. V. Decomposition of random variables and vectors. Translated from the Russian. American Mathematical Society, Providence, R.I., 1977.

It contains many deep and beautiful results, but for some reasons this direction of research is inactive since the end of 1980s. And my experience shows that even knowledge of these results, even among specialists in statistics, gradually disappears.

Remark. Princeton Companion contains some interesting articles, but its claim that it gives a perspective of the WHOLE mathematics can be misleading and harmful for a young mathematician.

It does not even mention many large and vigorously developing modern areas of research. A survey of ALL mathematics of reasonable length is probably impossible at this time. The multi-volume German Encyclopedia of Mathematical Sciences published in the early 20 century comes closer to it, but this was in the early 20th century and very many volumes were required.

Alexandre Eremenko
  • 88,753