FOM: Einstein, Godel, physics/math culture

[Harvey Friedman]

Reply to Black Fri, 14 Jan 2000.

I don't think there is a real contradiction between what Black is saying
and what I said.

>I think Harvey's last posting (I don't say Harvey himself, because I
>suspect that on reflection he will agree with what I am about to say) may
>underestimate Einstein's importance for the general intellectual interest
>of mathematics.

I am aware that Einstein's use of classical mathematical ideas greatly
influenced the subsequent development of (many aspects of) differential
geometry. However, Einstein was a consumer of classical 19th century
mathematics, which he molded to fit his single minded purpose of reworking
the foundations of physics at a deeply fundamental level.

In fact, the basic approach to intellectual life that Einstein represents
is very different from the basic approach to intellectual life that the
core mathematicians of the 20th century represent. Einstein starts with
problems of very wide general intellectual interest and uses and molds
whatever mathematical tools he finds that he needs.

Godel also starts with problems of very wide general intellectual interest
and uses and molds whatever mathematical tools he finds that he needs. In
particular, whatever number theory he needs in order to prove his
incompleteness theorems for Peano Arithmetic. Specifically, he uses the
Chinese remainder theorem - even more classical than the mathematics
Einstein uses. I would not call the reception of Godel's incompleteness
theorems any kind of special success for number theory.

>It's well-known that if you ask Joe Public to name a 20th-century
>mathematician, Joe Public will say Einstein, even though Einstein wasn't
>really a mathematician at all. But there is a way in which Joe Public is
>right.

This may be approximately right. Perhaps if you ask "Einstein a physicist
or mathematician, pick exactly one that is the best answer" you will get a
majority for physicist.

But putting this aside, what Joe Public approximately knows that Einstein
did is all physics, or perhaps engineering outgrowths of physics.  The
famous equation E = mc^2 is physics.

>One of the major differences between, say, 18th-century mathematics and
>20th-century mathematics is that modern mathematicians investigate
>structures which are quite independent of our 'intuitive' picture of the
>world. The point at which this starts is surely non-euclidean geometry, but
>I suspect that the key person in this development is not Gauss, Bolyai or
>Lobachevsky but rather (for all sorts of reasons) Riemann.

19th century people. A different age, where mathematicians and mathematics
was somewhat different.

The main point is that Einstein's "man of the century" represents 20th
century physics, not 20th century mathematics.

>Einstein's use of Riemannian geometry in general relativity provided
>something where if not Joe Public at least Joe-Educated-Public could see
>how something abstract, counterintuitive and non-vizualizable but logically
>coherent could be put to real work.

And the same is true of the computer revolution. Abstract models of
computation eventually get put to real work.

>Let me quote a historian who is certainly not a mathematician but seems to
>me to get this exactly right, Eric Hobsbawm, who on pp. 253-4 of _The Age
>of Capital_, talking about 19th-century science, says:
>
>'The strange, abstract and logically fantastic world of the mathematicians
>remained somewhat isolated both from the general and the scientific public,
>perhaps more so than before, since its main contact with both, physics
>(through physical technology) appeared at this stage to have less use for
>its most advanced and adventurous abstractions than in the great days of
>the construction of celestial mechanics. The calculus, without which the
>achievements of engineering and communications of the period would have
>been impossible, was now far behind the moving frontier of mathematics.
>This was perhaps best represented by the greatest mathematician of the
>period, Georg Bernhard Riemann (1826-66), whose university teacher's thesis
>of 1854 'On the hypotheses which underlie geometry' (published 1868) can no
>more be omitted from a discussion of nineteenth-century science than
>Newton's _Principia_ can from that of the seventeenth century. [...] Yet
>these and other highly original developments did not come into their own
>until the new revolutionary age of physics which began at the end of the
>century.'

19th century people. A different age, where mathematicians and mathematics
was somewhat different.

The main point is that Einstein's "man of the century" represents 20th
century physics, not 20th century mathematics.

>If Hobsbawm is right - and I think he is - then Einstein's use of
>Riemannian geometry could reasonably be identified as the point at which
>abstract modern mathematics (already developed by others, of course) became
>popular.

What does "modern" mean? In any case, Einstein's "man of the century"
represents 20th century physics and early 20th century physics culture, not
20th century mathematics or 20th century mathematics culture.