FOM: The Mathematical Nature of Einstein's Contributions to Twentieth Century Science

[Matt Insall]

In an earlier post, I suggested that I would argue that, according to one
``definition'' of Mathematics presented in this forum, Einstein's major
contributions to twentieth century science are mathematical in nature.  In
this post, I intend to present such an argument.  The definition to which I
refer is the following(posted by Professor Mycielski, and referred to on
12/22/99 by Professor Sazonov):


     Mathematics is a kind of *formal engineering*, that is engineering
     of (or by means of, or in terms of) formal systems serving as
     "mechanical devices" accelerating and making powerful the human
     thought and intuition (about anything - abstract or real objects or
     whatever we could imagine and discuss).


The theories of relativity (both special and general relativity), and of
quantum mechanics (also greatly influenced by Einstein's work), are
formalizable `systems serving as "mechanical devices" accelerating and
making powerful the human thought and intuition (about anything - abstract
or real objects or whatever we could imagine and discuss)', and, with the
advent of quantum logic and with the axiomatic approaches to the
mathematical apparatus used to expound these theories, are becoming more
``formalized'' all the time.    The fact that Einstein's work is considered
a part of ``foundational studies'', as Professor Friedman so aptly put it,
does not diminish the Mathematical flavour of Einstein's work (or that of
his contemporaries and his and their academic descendants).  On the
contrary, since a significant portion of Mathematics *is the* foundations of
Physics, the very fact that the relativistic and quantum mechanics are
``foundational'' puts them in the realm of Mathematics.  It seems to me that
the less Mathematical parts of Einstein's work was the (certainly
nontrivial) recognition that appropriate experimental results which were at
that time new and disturbing to the contemporary theory of light could be
taken as axioms in a new theory of light that would then explain other
significant experimental results of that time.  I have no doubt that when
the theory of relativity has been around long enough, it will be
axiomatized, as is geometry, both euclidean and non-euclidean.  This will
make it a ``formal system'', and therefore, a bit of Mathematics.  I do not
agree, though, with the contention that this classification of Einstein's
work as ``Mathematical'' in any way trivializes his major accomplishments.
For I do not see the need to separate Mathematics from the rest of human
existence quite so dramatically:  To a great extent, Mathematics is a human
endeavour to better understand our universe, whether the part of our
universe in question is physical or psychological, or is metaphysical or is
mainly the universe of our systems of reasoning.  The rejection of FOM by
some ``core mathematicians'' not withstanding, FOM is, IMHO, a part of
Mathematics, which may, as Professor Friedman seems to have suggested,
eventually not reside in Mathematics Departments, but in departments devoted
to ``foundational studies''.  It is, in fact, a beautiful and significant
part of Mathematics, and so, as has been pointed out in this forum, should
be supported by the rest of the Mathematics community.




 Name: Matt Insall
 Position: Associate Professor of Mathematics
 Institution: University of Missouri - Rolla
 Research interest: Foundations of Mathematics
 More information: http://www.umr.edu/~insall