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 that mathematicians axiomatize 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].

Sources

1-5. Tilman Sauer, The Relativity of Discovery: Hilbert's First Note on the Foundations of Physics. 

6. David Hilbert, Lecture delivered before the International Congress of Mathematicians, Paris France, 1900.

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.

The Axiomatization of Physics - Step 1