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.
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.