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Mathematics in the 20th century, by Michael Atiyah [pdf] (2002)

by practal on 2/9/25, 8:54 AM with 18 comments

• by phwlarxoc on 2/9/25, 1:11 PM

  The most stunning point of this paper is the paradoxical benefit of deliberately assuming some kind of "blindness" in one's own thinking as a mathematician: in reckoning algebraically we proceed eyes closed so to say. What we look at is neither the real world nor our own mind but abstract signs on paper. That is the algebraic, formal", "symbolic" way of thinking.

  Atiyah has this tradition start with Leibniz, and it marks exactly his opposition to Newton, the latter being mainly interested in physics and therefore restraining math by its grounding in the real world, whereas Leibniz would have understood the formal nature of the discipline. The antagonism re-emerges in the 20th century with Poincaré-Arnold on one side and Hilbert-Bourbaki on the other.

  The point has been aptly made in the polemics of Brouwer against Hilbertian formalism, by saying that for the formalist mathematical exactness is basically grounded in paper: "Op de vraag, waar die wiskundige exactheid dan wel bestaat, antwoorden beide partijen verschillend; de intuitionist zegt: In het menschelijk intellect, de formalist: Op het papier", see Hermann Weyl, Philosophie der Mathematik und Naturwissenschaft, 1927, p.49.

  I guess a very large majority of people would still think that math is the rational, systematic account of what is ("real world"), but Atiyah seems to say that from an inner-mathematical perspective, the purely formal conception of mathematics prevailed. Algebra was the "Faustian offer" handed over to mathematicians: in exchange for the formidable machine of symbolic reasoning, we would have to sacrifice the meaning of what we are dealing with, at leat temporarily.

• by practal on 2/9/25, 8:57 AM

  Lots of interesting thoughts and insights in that article. I find it especially interesting to relate Geometry to Space, and Algebra to Time.

• by Jun8 on 2/9/25, 3:27 PM

  This lecture achieves the astonishing feat of being very accessible to non-mathematicians while being deeply insightful. The global vs local distinction mentioned, although not the same, brought to mind Dyson’s distinction between “birds” and “frogs”: https://www.ams.org/notices/200902/rtx090200212p.pdf

• by jll29 on 2/9/25, 3:47 PM

  Here is Sir Atiyah's Mathematics Geneaology entry: https://genealogy.math.ndsu.nodak.edu/id.php?id=30949

  Receiving the RSE Fellowship with a handshake from him was one of my highlights during my beautiful time in Edinburgh. May he rest in peace.

• by zyklu5 on 2/9/25, 5:05 PM

  Atiyah is truly one of the giants of modern mathematics. I remember long ago I struggled through a reading course of his and Bott's Yang-Mills paper in graduate school. Like many great works of math it too had that paradoxical characteristic of transforming seemingly 'non-mathematics' into mathematics* by reversing the usual direction of application of one to the other, in this case, from physics to math. It would start a whole movement that'll produce much of modern geometries greatest hits like Donaldson's (his student) theorem in 4 manifolds to Witten's great papers.

  * A reason I think modern LLM architecture as they currently stand with their underlying attention mechanisms will not produce interesting new mathematics. A few other ideas are going to be needed.

• by bluenose69 on 2/9/25, 11:11 AM

  An RIS entry:

      TY  - JOUR
      TI  - Mathematics in the 20th century
      AU  - Atiyah, Michael
      T2  - Bulletin of the London Mathematical Society
      AB  - A survey is given of several key themes that have characterised mathematics in the 20th century. The impact of physics is also discussed, and some speculations are made about possible developments in the 21st century.
      DA  - 2002/01//
      PY  - 2002
      DO  - 10.1112/S0024609301008566
      DP  - DOI.org (Crossref)
      VL  - 34
      IS  - 1
      SP  - 1
      EP  - 15
      J2  - Bull. Lond. Math. Soc.
      LA  - en
      SN  - 0024-6093, 1469-2120
      UR  - https://www.cambridge.org/core/product/identifier/S0024609301008566/type/journal_article
      Y2  - 2025/02/09/11:08:49
      ER  -