FOM: Einstein, Godel, Turing, Hardy

[Harvey Friedman]

The recent FOM postings discussing or touching on Einstein as "man of the
century" include Insall Sun, 9 Jan 2000 19:29, Friedman Thu, 13 Jan 2000
23:09, Black Fri, 14 Jan 2000 16:20, Friedman Fri, 14 Jan 2000 12:19, Pratt
Fri, 14 Jan 2000 11:20, Pratt Fri, 14 Jan 2000 13:28, and Steiner Sat, 15
Jan 2000 19:24.

In my view, the most important aspect of what Einstein did (at least with
respect to relativity theory) is as foundations. He took on issues of great
g.i.i. (general intellectual interest), analyzed them with great
imagination and clarity, and formulated his theories in a sufficiently
clear and convincing way to the physics community. He used whatever
mathematical tools were appropriate. The fact that his theories were so
experimentally confirmable adds greatly to the excitement of influence of
his work.

His is an example of the speical effectiveness of the foundational approach
to science. The same is true of other celebrated figures, especially Godel
and Turing.

To consider Einstein's general theory of relativity as an example of pure
mathematics is to trivialize the essence of his achievment. By the same
token, to consider Godel's theorems or Turing's machines as examples of
pure mathematics is also to trivialize the essences of their achievments.

I am quite familiar with the counterproductive process of taking work in
f.o.m. and stripping away the essence, and considering it merely as a
contribution to mathematics or mathematical logic. This happens all the
time, but is a completely misguided approach that does not respect the
essence of what is achieved.

The foundational approach and foundational aspects of what Einstein,
Turing, and Godel did completely dwarf any mathematics that is involved.
The mathematics involved is completely trivial (even if it is highly
nontrivial) compared to the foundational essence of what they did.

The foundational essence of what these people did can be conveyed with a
minimal amount of mathematics. The mathematics gets more involved when one
wants to give full and complete formulations of the basic ideas -
formulations that need to be given for the proper further development of
the ideas by professional scientists.

As far as spectacular and effective applications of mathematics to physical
science is concerned, I am under the impression that one more commonly
thinks of Newton, Gauss, and Eule, than Einstein.

Steiner Sat, 15 Jan 2000 19:24:

>On Einstein and mathematics, I think it highly relevant that G.H. Hardy,
>a twentieth-century pure mathematician, states in his "A Mathematician's
>Apology" as an example of pure mathematics, Einstein's theory of
>relativity.

I think that this is highly irrelevant (smile). Sure, there is pure
mathematics in it, but that is a *relatively* pedestrian aspect.

>As most readers of this list will remember, the major
>criterion for a theory to be mathematics (according to Hardy) is
>aesthetic.

This is not a useful view of mathematics, in that only the tiniest portion
of the educated public will appreciate this aspect of mathematics. It also
is too ill defined (at present) to be generally useful for the evaluation
of and development of mathematics. Adherence to this view causes great
difficulties for mathematicians.

>He predicted that pure mathematics has no military
>applications (Macpherson in the Millenium Conference pointed out that
>even Hardy's field , number theory, has classified theorems today).

I think it is worthwhile to put this matter of number theory in
cryptography into perspective. From a general intellectual viewpoint, the
important advance in modern cryptography was perhaps foundational. Making
sense of things like public key cryptography and the idea of hidden
information in some mathematical setting is perhaps the major advance.

Number theory (prime factorization, etcetera) is the first mathematical
vehicle for reasonably efficiently implementing this advance. However, it
is not satisfactory from many points of view, and there is a lot of
feverish current research to replace the number theoretic approach with
other approaches. Let us not forget what the main problem is:

"...the ultimate in cryptography -- a truly invincible mathematically
proven method for protecting private computer information from unwanted
viewing," said Ashok Chandra, manager of computer science at IBM's Almaden
lab.

This problem has not been solved. But it is believed that one must get away
from the number theoretic based approach. For some new approach being
developed, see
http://www.research.ibm.com/resources/magazine/1997/issue_2/results297.html
http://www.ibm.com/security/html/prcryptosys.html
ftp://ftp.eccc.uni-trier.de/pub/eccc/reports/1996/TR96-065/index.html
ftp://ftp.eccc.uni-trier.de/pub/eccc/reports/1996/TR96-065/Paper.ps

My point is not that pure mathematics has no military applications. Maybe
yes, maybe no, since few people know to what extent number theoretic
cryptography is in military use (that's classified!). Most people guess
that it is not in military use. And if some pure mathematics has
applications in a certain practical context, it may not in the future (or
some other kind of pure or applied mathematics may take its place, perhaps
trivial mathematics).

In fact, elementary mathematics of various kinds not only have military
applications but also applications everywhere - even in the grocerty store
check out counter.

My point is that the great intellectual events of the 20th century are
judged by standards that are quite different from the way mathematicians
judge their own work - and from the way mathematicians judge f.o.m. The
foundational approach to intellectual life - where issues of great g.i.i.
are met head on with creative imagination and clarity - has consistently
been the approach that has yielded the greatest mathematical and scientific
events of at least the 20th century. I am sure that it will continue to be
one of the most effective approaches in the next century, and will continue
to be far more effective than the approach to intellectual life that is
standard in either the mathematical or philosophical communities (of
course, these communities generally do not claim to be aiming for century
level mathematical/scientific events; especially not the philosophical
community). Of course, some of the greatest scientific events of the 21st
century are likely to come out of enormous research lab efforts,
particularly in the biological sciences, where the role of g.i.i. may or
may not be apparent.