# Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians

I would like to ask a question inspired by the title of a book by Sir Roger Penrose ([1]). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics.

I'm going to focus the post (and modulate my genuine idea) about an aspect that I think can be discussed here from an historical and mathematical point of view, according to the following:

Question. I would like to know what are examples of remarkable achievements (in your research subject or another that you know) that arose against the general view/work of the mathematical community since the year 1900 up to the year 1975. Refer the literature if you need it. Many thanks.

An example is the mention that the author of [2] (as I interpret it) about Lennart Carleson and a conjecture due to Lusin in the second paragraph of page 671 (the article is in Spanish).

Your answer can refer to (for the research of pure or applied mathematics, and mathematical physics) unexpected proofs of old unsolved problems, surprising examples or counterexamples, approaches or mathematical methods that defied the contemporary (ordinary, mainstream) approaches, incredible modulizations solving difficult problems,... all these in the context of the question that is: the proponents/teams of these solutions and ideas swam against the work of the contemporary mathematics that they knew at the time.

*You can refer to the literature for the statements of the theorems, examples, methods,... if you need it. Also from my side it is welcome if you want to add some of your own historical remarks about the mathematical context concerning the answer that you provide us: that's historical remarks (if there is some philosophical issue also) emphasizing why the novelty work of the mathematician that you evoke was swimming against the tide of the contemporary ideas of those years.

## References:

[1] Roger Penrose, Fashion, Faith, and Fantasy in the New Physics of the Universe, Princeton University Press (2016).

[2] Javier Duoandikoetxea, 200 años de convergencia de las series de Fourier, La Gaceta de la Real Sociedad Matematica Española, Vol. 10, Nº 3, (2007), pages 651-677.

• I'm not sure I really understand the question (especially because the "example" is not explained in the post); but: Paul Cohen's work on the continuum hypothesis probably qualifies. In his own words, at the time he developed the technique of forcing, he "had the feeling that people thought the problem [i.e., the Continuum Hypothesis] was hopeless, since there was no new way of constructing models of set theory. Indeed, they thought you had to be slightly crazy even to think about the problem." Apr 11 at 19:25
• Swimming against the tide is more or less the definition of world-class mathematics, either by pushing some known tools beyond what was thought possible or by inventing/discovering unexpected tools. Apr 11 at 20:32
• @RolandBacher, some world-class math goes with the tide! E.g. Wiles's proof of FLT in 1993-95 followed a path suggested by Taniyama-Shimura-Weil and used work on "the main conjecture of Iwasawa theory" from 1984, plus work for which Ken Ribet had received the Fermat prize in 1989. So rather than going against the trend, Wiles represented its triumph. Apr 11 at 20:51
• We can take this question in two different directions: "proved something remarkable following an approach that was thought to be hopeless" and "proved something that was widely believed to be false". Apr 11 at 21:50
• Godel’s Incompleteness theorems seem like the canonical answer to this. I know of no more unexpected result in the history of mathematics. Apr 12 at 18:28

After mathematicians had been been taught for decades that a consistent theory of the calculus based on infinitesimals was impossible, Abraham Robinson was certainly swimming against the tide when he proved otherwise.

Robinson, A. (1961): Non-standard analysis, Indagationes Mathematicae 23, pp. 432-440.

Robinson, A. (1966): Non-standard Analysis, North-Holland Publishing Company, Amsterdam.

• And Jerome Keisler's calculus text (and "instructor's manual" for it). Apr 12 at 2:28

This is a well-known example, but it is Georg Cantor. While ultimately many of his ideas have become part of the bedrock of mathematics, taught to every undergraduate maths student, he faced opposition in his day.

Joseph Dauben's biography gives a thorough explanation of the opposition his ideas faced and how he responded. For example, Cantor's original proof of the uncountability of the reals appears as a lemma in the innocuously titled "On a property of the collection of all real algebraic numbers", downplaying the true novelty of the result to sneak it into a journal edited by Kronecker.

• This is an excellent example but did any of the relevant work happen between 1900 and 1975? Apr 11 at 21:50
• @WillSawin, no, the criticism he received and the loss of his son led to poor mental health and no substantial math after 1900; see Wikipedia (en.wikipedia.org/wiki/Georg_Cantor) Apr 11 at 22:00
• "Cantor suffered his first known bout of depression in May 1884. Criticism weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker, [e.g.] 'I don't know when I shall return to my scientific work. I limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness.' Soon after [his 1899] hospitalization, Cantor's youngest son Rudolph died suddenly...and this tragedy drained Cantor of much of his passion for mathematics." Apr 11 at 22:00
• Cantor is a definite 1st by large in this list, with his set theory. It was such a straightforward violation of the ancient taboos in mathematics/philosophy. It induced reactions in the non-cantorial mathematics such as intuitionism and category theory, quite remarkable in themselves. Nothing comparable can be observed even wrt NSA that would be my 2nd (very distant from the 1st) example. Apr 12 at 10:52
• Weierstrass (in a letter to Mittag-Leffler IIRC) wrote that he had previously believed there to be one kind of infinity. Cantor's work convinced him otherwise. At least some did side with Cantor, even in his lifetime. Apr 12 at 20:50

In the first decades of the 20th century, $$p$$-adic analysis (or valuation theory more generally) was regarded by many as rather exotic. After Hensel's work there was a steady development by Strassmann, Kürschák, Krasner, et al., and $$p$$-adic numbers were the direct inspiration for Steinitz's major paper that turned field theory into a new part of abstract algebra, but for a long time it was not considered fully mainstream mathematics. Bourbaki has some discussion about this in the Historical Notes to their text on commutative algebra.

If I should list an example of a major result that was proved by these methods that had not been proved before by other methods, I'd pick Dwork's proof of the rationality of the zeta-function of an arbitrary algebraic variety over a finite field. He did this without smoothness assumptions, which went beyond Weil's original conjectures at least as far as rationality is concerned.

Another example in number theory might be the Artin reciprocity law, although only for a few years. Artin said that when he discussed it with other people while he was trying to prove it, they laughed at him and said it couldn't possibly be true. But after he proved his reciprocity law, it was recognized as the central result in class field theory.

Herbert Busemann's work on differential geometry is a good example. See his many characterizations of Euclidean space -- without any requirement of differentiability! They are all from his 1955 masterpiece, The Geometry of Geodesics.

The general trend of geometry was and mostly is focused on differentiable manifolds, following Hassler Whitney's 1936 definition, and often specializing to Riemannian geometry. The small group of people interested in non-Riemannian metric geometry, like Hanno Rund in 1959, mostly wrote about The Differential Geometry of Finsler Spaces. (If you doubt the smallness of the non-Riemannian trend, consider: Can you define a Riemannian manifold? And have you ever learned the definition of a Finsler manifold?)

Against the differentiable trend and the Riemannian trend that are often too pervasive to notice, Busemann took an axiomatic and metric approach, in the spirit of David Hilbert's 1899 Foundations of Geometry. He got the above characterizations of Euclidean spaces; he wrote about areas in those spaces; along the way he defined the Busemann function for which his name most often comes up today.

But Busemann's work is little known, despite his clear 1955 presentation, and convenient 1970 collection of Recent Synthetic Differential Geometry. This is why I've written so much about Busemann here, without being in a math department. When someone asked about an old-school approach like this, I answered but the question was still closed as unclear! And I've often commented to clear up confusions, or distinguish Lawvere's thing with the same name, or respond to a recent graduate's comment that "differential geometry is all about manifolds?"

If you want to learn about Busemann's approach, you can start with the Geometry of Geodesics, or the links here. As I've said elsewhere: If you think "the weaker notion is the appropriate notion", you're likely to enjoy Busemann a lot. And, by comparison with coordinatized Finsler geometry, approaches like Busemann's make it easier to generalize geometry well.

• Tissot ellipses are to a Riemannian metric as Tissot non-ellipses are to Finsler metrics, right? Apr 12 at 21:46
• @AkivaWeinberger, that’s a good image but the question is: which possible non-ellipses and non-ellipsoids are associated with Finsler metrics? Apr 12 at 23:54
• You may have meant to link directly to your answer giving Busemann's characterisations, perhaps not just to the question about characterisations (in any spirit). Apr 13 at 21:34
• @LSpice, I like linking to that question to emphasize the naturalness of the question that Busemann was answering. Apr 14 at 2:26

I think a good example would be the first rigorous construction of an interacting quantum field theory by Edward Nelson using methods from probability theory.

That probabilistic approach had been attempted before, and many mathematicians had written it off as impossible, said Eric Carlen, a professor of mathematics at Rutgers University who studied under Nelson before receiving his Ph.D. from Princeton in 1984. Nelson’s colleague Arthur Wightman, a renowned mathematical physicist and Princeton’s Thomas D. Jones Professor Emeritus, introduced him to the problem.

Nelson was told that the sort of approach he was taking had been tried and failed, but “Ed trusted his intuition over all the experts in the field and he was right,” Carlen said. “To bring forth a really new idea and bring it forth in a real polished way requires tremendous effort, concentration and focus. When he got a new idea he really followed it through to the end. And because of that, his ideas have really gone on to have a life beyond their original application.”

• What was the trend in this area before Nelson? Advances usually have new ideas and are in that sense against the tide, but for a satisfying answer here I'd want to know about the prior trends. Apr 11 at 20:33
• Well I guess the prior trend would be anything except probability theory: PDE theory, spectral theory, the theory of operator algebras, and what not. Perhaps a comparison of the review articles on the matter by Glimm-Jaffe, Wightman and Nelson in Proceedings of Symposia in Pure Mathematics, Vol. XXIII, 1973, would give an idea of how original Nelson's ideas were at the time. Apr 11 at 22:55

There are various examples of certain areas of mathematics being regarded as sterile and disconnected from the rest of mathematics, but nevertheless being pursued doggedly by some researchers, and eventually finding unexpected applications in other areas of mathematics.

Computability theory, especially the study of computably enumerable sets, is an example. Though it has been pursued ever since the mid-twentieth century, computability theory is even today regarded by most mathematicians as an arcane subject with little connection with the rest of mathematics. However, as explained by Soare in his paper, Computability theory and differential geometry (non-paywalled version here), computability theory has found unexpected applications in the work of Nabutovsky and Weinberger in differential geometry.

Another possible example is the use of model theory to attack number-theoretic problems. The Ax–Kochen theorem dates from 1965, but my impression is that for a long time, most number theorists did not think that model theory was a particularly powerful or insightful tool. That skepticism has presumably evaporated with (for example) the success of o-minimality and the spectacular proof of the André–Oort conjecture.

These examples involve an overlap among interests of mathematicians and researchers in cross-disciplinary fields--physics, chemistry, and cognitive science.

The acceptance of group theory in theoretical physics in the early decades of the XXth century was not as straightforward as one may suppose.

Excerpt from pg. 77 of

In 1929 Slater published his influential paper “Theory of Complex Spectra” on Physical Review, showing that all results can be derived by using the simplest mathematics only [Slater 1929]. Slater discovered that by introducing the electronic spin at the very beginning of a calculation an enormous simplification could be achieved. He successfully obtained expressions for the energies of spectroscopic terms replacing considerations involving the group representation by considerations involving the three operators of angular momentum satisfying simple commutation relations.

“As soon as this paper became known, it was obvious that a great many other physicists were as disgusted as I had been with the group-theoretical approach to the problem. As I heard later, there were remarks made such as ‘Slater has slain the Gruppenpest’. I believe that no other piece of work I have done was so universally popular” [Sternberg 1994, xi]. Here Slater is recalling a Goethe’s Faust parody which was staged in April 1932 at Copenhagen, on the occasion of the tenth anniversary of Bohr’s Institute. In a short scene of this memorable play, the Blegdamsvejen Faust, Slater was seen to kill a dragon representing group symmetry: “Das indizbeschuppte Vieh (der Gruppendrache), er starb an Antisymmetrie.” (The beast covered with indices scales (the group dragon) it died of antisymmetry.) Trace of this general rejection still remains in George Gamow’s original drawings [Gamow 1966].

Another example (albeit not quite as initially mathematically well-developed/delineated as the first) is provided in the theory of oscillating reactions in nonequilibrium thermodynamics by the history of the Belousov-Zhabotinskii reaction. See also "An Analysis of the Belousov-Zhabotinskii Reaction" by Casey R. Gray and "The Prehistory of the Belousov-Zhabotinsky Oscillator" by A. T. Winfree.

A third, again not as initially mathematically well-developed as the first, is in the prehistory of neural networks as presented in "A revival of Turing's connectionist ideas: Exploring unorganized machines" by Teuscher and Sanchez (see also McCulloch & Pitts Publish the First Mathematical Model of a Neural Network). In addition, neural-network models (NNMs) didn't become mainstream until well after 1970, so Turing was far ahead of his times as well as McCulloch and Pitts. One reason for the delay in acceptance of NNMs is the historical competition between the logicist's camp and the connectionist/neural-network camp in AI/cognitive science with their distinct mathematical/computational approaches as sketched in "The Improbable Machine" by Jeremey Campbell (see also "The Mind's New Science: A History of the Cognitive Revolution" by Howard Gardner).

• But then again, Turing believed that telepathy was real, so he also straddled many approaches to the reality of the mind :-) Apr 13 at 0:44
• Years before. It appears in his article introducing the Turing test; he says "Perhaps this psycho-kinesis might cause the machine to guess right more often than would be expected..." and that "to put the competitors into a ‘telepathy-proof room’ would satisfy all requirements [of dealing with this issue]". Apr 13 at 5:33
• @LSpice, I agree. Read the article for the context. (Of course, if you are doing permutations of the rows/columns of a matrix or addition of the reals you can couch the operations in group theoretical terms as well.) Apr 13 at 15:37
• @LSpice, I typically don't include links that are behind paywalls, especially if they can be found at other sites via google search or at your favorite freedom-of-information site (long live that hacker ethic). Apr 13 at 22:12
• Ehrenfest, for example, writes in a letter to van der Waerden ''a real group plague [Gruppenpest] has broken out in our physical journals''. Apr 14 at 9:38