FOM: Time Mag, Einstein, Godel, Foundational Studies

[Vaughan Pratt]

From: Harvey Friedman <friedman at math.ohio-state.edu>
>>  To an extent, I agree with Professor Friedman here.  But note that
>>  Einstein has been declared ``Person of the Century'' by one of
>>  these
>>  magazines, and it is his use of non-Euclidean Geometry and other
>>  higher-level ``core mathematics'' that won him that ``title''.
>This is not a reasonable way of stating how Einstein is "man of the
>century". He won it for penetrating, spectacular insights into fundamental
>issues in physics. The fact that there was some use of classical
>mathematics does not distinguish Einstein from any of a number of other
>great and not so great scientists in physics and many other disciplines.

I have to agree with the content of Harvey's observation, namely that
Einstein's highly original application of new mathematics to physics
probably played only a bit part compared with the physics itself.  But I
must take exception to his "some use of classical mathematics," which from
the vantage points of both mathematics and mathematical physics grossly
understates the mathematical side of Einstein's contributions to physics.

It's a very good question who single-handedly and famously applied
practically unknown mathematics to reshape our perception of the shape
of the universe in as revolutionary way as did Einstein.  The "absolute
differential calculus" of Ricci and Levi-Civita, which generalized
the Euclidean coordinatization of tensor algebra to locally Euclidean
curvilinear coordinates in n-dimensional manifolds, was not merely
unmotivated but unknown to most mathematicians let alone physicists when
Einstein applied it so effectively to general relativity.  There is no
sense in which this could be considered "classical mathematics" at the
time Einstein applied it.

Heisenberg based his invention of modern quantum mechanics on matrices,
which were just as unknown to physicists at that time (1925).  However it
was Born and Jordan who really spearheaded that effort, pointing it out
to Heisenberg in the first place as well as writing a parallel paper that
showed the right way to handle matrices in that application.  So while
this was also spectacular it was more of a team effort.  As was quantum
mechanics itself, in which the full story, namely the duality of matrix
mechanics and wave mechanics, was due initially to Schrodinger and then
developed in much greater and more satisfactory depth by von Neumann
in terms of Hilbert space.  Prior to both Schrodinger and von Neumann,
complementarity had been heavily promoted by Bohr, and so even though
each of Schrodinger's and von Neumann's contributions to the elucidation
of complementarity was spectacular, again this story was a team effort.

In contrast Einstein worked out his whole application of the tensor
calculus from start to finish on his own, and made a tremendous
splash in the public's imagination in doing so.  That, I think, is
what makes Einstein so special for both physicists and mathematicians,
whose judgement the public has always had to rely on in assessing the
technical contributions of famous scientists.

Quantum gravity, especially string theory, has been in the public's eye
a lot lately as both consumer of and motivator for very new mathematics,
but there has been no single step in this development that is half as
dramatic as Einstein's.  Furthermore the developments have been slower
and less spectacular and convincing to date than relativity, and even
more of a team effort than the other examples of the importation of
brand new mathematics or its actual creation by physicists.

To match the *mathematical* originality of Einstein's contributions to
physics we really have to leave this century and turn to Gauss, Fourier,
Hamilton, Newton, Galileo, Archimedes, and so on.

Of these the only one that seems to me to really measure up to Einstein's
combination of brand new mathematics with mind-grabbing results is Newton,
who if not the inventor of the differential calculus was certainly the
first to apply it to changing spectacularly our understanding of the role
of gravity in determining the orbits of the planets and their satellites,
which today every bright high school graduate understands and views as
perfectly natural.  And even there Kepler had developed part of that
picture before Newton; Einstein had no Kepler to show him the way, his
application of the tensor calculus to general relativity was without
any doubt whatsoever 100% his, and mind-blowingly so.  Einstein is a
viable candidate for person not just of the century but the millennium,
though he comes up against some stiff competition from the likes of Bach,
Beethoven, and Da Vinci.

Of course this is just my opinion, and Harvey has his, and so on.
To put the question I've taken us to be addressing, who applied
(new?) mathematics most spectacularly to physics, on the same footing
as the question to which Einstein was Time's answer, we'd have to put
the question to a vote.  Unlike Time's question, this one is not for
the public but for those who appreciate the application of mathematics.

Vaughan Pratt