Physics at a mature level is a mathematical discipline

Hilbert's view of physics from a mathematician's
perspective becomes quite explicit in remarks he made regarding the
relationship between physics and geometry. Hilbert regarded geometry as a
genuine branch of mathematics. But, originally, geometry was a natural
science. Only it was no longer subject to experimental examination and had
become mathematized, arithmetized and eventually axiomatized. For Hilbert,
this development is not only an account of the factual historical
development but also of the proper advancement of science, an advancement
which should be furthered wherever possible.

Thus, as early as 1894, in a lecture on geometry which
he gave while still in Königsberg, Hilbert wrote

Geometry is a science which essentially has developed
to such a state that all its facts may be derived by logical deduction
from previous ones. [1].

Later in this lecture, in the course of discussing the
axiomatic foundations of geometry, he presents the axiom of parallels and
discusses the alternatives of Euclidean, hyperbolic and parabolic
geometries. In this context he remarks

Now also all other sciences are to be treated
following the model of geometry, first of all mechanics, but then also
optics and electricity theory. [2].

This is 1894. Hilbert presents very similar remarks in a
lecture course on Euclidean geometry given in winter 1898/99. There
Hilbert characterizes geometry as

a natural science but of such a kind that its theory
may be called a perfected one which, as it were, provides a model for
the theoretical treatment of other natural sciences. [3].

In the same semester Hilbert also lectured on
mechanics. This was Hilbert's first lecture course dealing with physics
proper. In the introduction to this course, Hilbert again characterized
geometry as a mathematical science which used to be a natural science.
Hilbert then explained that mechanics was not yet acceptable as a
mathematical science because it wasn't yet perfectly clear what should be
selected for mechanics as the fundamental axioms.

Also in mechanics the basic facts are accepted by all
physicists. But the arrangement of the basic concepts nevertheless is
subject to the changes in viewpoint. The structure is also far more
complicated [than that of geometry]; even deciding what is simpler is
something which depends on further discoveries. Hence, even today,
mechanics cannot yet be called a purely mathematical discipline, at
least not to the extent that geometry is. [4].

Given this state of affairs, Hilbert continues:

We must strive for it to become [a mathematical
science]. We must extend the range of pure mathematics further and
further, not only in our own mathematical interest but also for the sake
of science as such. [5].

In 1900, in a lecture delivered before the
International Congress of Mathematicians at Paris, Hilbert again outlined
his program for axiomatizing physics with the intent of putting it on the same level
as axiomatized geometry. Hilbert's challenge to axiomatize all
of physics was merely number 6 out of 23 extraordinarily difficult,
important, yet unsolved mathematical problems at that time.

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.
...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
assumed. [6].

Hilbert's sixth problem remains unsolved but many
prominent physicists and mathematicians regard Hilbert's program for the
axiomatization of physics as the highest and purest form of science ever
conceptualized by the human mind. The importance of axiomatizing physics
is obvious to mathematicians:

In the history of physics, ideas that were once seen
to be fundamental, general, and inescapable parts of the theoretical
framework are sometimes later seen to be consequent, special, and but
one possibility among many in a yet more general theoretical framework.
... Examples are the earth-centered picture of the solar system, the
Newtonian notion of time, the exact status of the laws of
thermodynamics, the Euclidean laws of spatial geometry, and classical
determinism. In view of this history, it is appropriate to ask of any
current theory "which ideas are truly fundamental and which are
`excess baggage'." [7].

The reductionist approach -- explaining physical
phenomena in terms of simple, mathematically precise, quantities -- has
been extraordinarily successful in almost all areas of physics. It goes
against everything we have learned about nature to propose a theory in
which complicated macroscopic objects, whose precise definition must
ultimately be arbitrary, are fundamental quantities. [8].

A great
physical theory is not mature until it has been put in a precise
mathematical form, and it is often only in such a mature form that it
admits clear answers to conceptual problems. [9].

Our
experience hitherto justifies us in believing that nature is the
realization of the simplest conceivable mathematical ideas. I am
convinced that we can discover by means of pure mathematical
constructions the concepts and the laws connecting them with each
other, which furnish the key to the understanding of natural
phenomena. Experience may suggest the appropriate mathematical
concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the
sole criterion of the physical utility of a mathematical construction.
But the creative principle resides in mathematics. In a
certain sense, therefore I hold it true that pure thought can grasp
reality, as the ancients dreamed. [10].

7. J.B. Hartle, Classical physics and Hamiltonian quantum mechanics
as relics of the Big Bang, Physica Scripta T 36 (1991), 228-236.

8. A. Kent, Against many-worlds interpretations, Int. J. Mod.
Phys. 5 (1990), 1745-1762.

9. A. S. Wightman, Hilbert's sixth problem:
mathematical treatment of the axioms of physics, in: Proc. Sympos.
Pure Math., Vol. 28, AMS, 1976, pp. 147-220.

10.
Einstein, 1954, Ideas and Opinions, quoted from Schweber,
"Einstein and Oppenheimer: the meaning of genius." (This quote, according
to Schweber, was an observation by Einstein on Hilbert's program on the axiomatization of physics.)