[FOM] From Hilbert

Harvey Friedman friedman at math.ohio-state.edu
Thu Jan 22 21:37:08 EST 2004

It is interesting to see just how Hilbert viewed foundations of physical
science in 1900. This is from his list of problems. See, e.g.,


Below, I omitted certain citations - HMF.

6. Mathematical treatment of the axioms of physics

The investigations on the foundations of geometry suggest the problem: To
treat in the same manner, by means of axioms, those physical sciences in
which mathematics plays an important part; in the first rank are the theory
of probabilities and mechanics.

As to the axioms of the theory of probabilities, it seems to me desirable
that their logical investigation should be accompanied by a rigorous and
satisfactory development of the method of mean values in mathematical
physics, and in particular in the kinetic theory of gases.

Important investigations by physicists on the foundations of mechanics are
at hand; I refer to the writings of Mach, Hertz, Boltzmann and Volkmann. It
is therefore very desirable that the discussion of the foundations of
mechanics be taken up by mathematicians also. Thus Boltzmann's work on the
principles of mechanics suggests the problem of developing mathematically
the limiting processes, there merely indicated, which lead from the
atomistic view to the laws of motion of continua. Conversely one might try
to derive the laws of the motion of rigid bodies by a limiting process from
a system of axioms depending upon the idea of continuously varying
conditions of a material filling all space continuously, these conditions
being defined by parameters. For the question as to the equivalence of
different systems of axioms is always of great theoretical interest.

If geometry is to serve as a model for the treatment of physical axioms, we
shall try first by a small number of axioms to include as large a class as
possible of physical phenomena, and then by adjoining new axioms to arrive
gradually at the more special theories. At the same time Lie's a principle
of subdivision can perhaps be derived from profound theory of infinite
transformation groups. The mathematician will have also to take account not
only of those theories coming near to reality, but also, as in geometry, of
all logically possible theories. He must be always alert to obtain a
complete survey of all conclusions derivable from the system of axioms

Further, the mathematician has the duty to test exactly in each instance
whether the new axioms are compatible with the previous ones. The physicist,
as his theories develop, often finds himself forced by the results of his
experiments to make new hypotheses, while he depends, with respect to the
compatibility of the new hypotheses with the old axioms, solely upon these
experiments or upon a certain physical intuition, a practice which in the
rigorously logical building up of a theory is not admissible. The desired
proof of the compatibility of all assumptions seems to me also of
importance, because the effort to obtain such proof always forces us most
effectually to an exact formulation of the axioms.

So far we have considered only questions concerning the foundations of the
mathematical sciences. Indeed, the study of the foundations of a science is
always particularly attractive, and the testing of these foundations will
always be among the foremost problems of the investigator. Weierstrass once
said, "The final object always to be kept in mind is to arrive at a correct
understanding of the foundations of the science. ... But to make any
progress in the sciences the study of particular problems is, of course,
indispensable." In fact, a thorough understanding of its special theories is
necessary to the successful treatment of the foundations of the science.
Only that architect is in the position to lay a sure foundation for a
structure who knows its purpose thoroughly and in detail. So we turn now to
the special problems of the separate branches of mathematics and consider
first arithmetic and algebra.


The following quotation from Hilbert's Introduction to Hilbert's problem
list is also very interesting.

An old French mathematician said: "A mathematical theory is not to be
considered complete until you have made it so clear that you can explain it
to the first man whom you meet on the street." This clearness and ease of
comprehension, here insisted on for a mathematical theory, I should still
more demand for a mathematical problem if it is to be perfect; for what is
clear and easily comprehended attracts, the complicated repels us.

Harvey Friedman

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