The Algorithmization of Physics: Math Between Science and Engineering - Artificial Intelligence and Symbolic Computation - page 3

Judging from our present understanding, it may seem obvious that one should
proceed in this way. Looking at the historical development, however, we perceive
a movement of increasing abstraction that has not stopped at the point indicated
above [9]. We can see four distinct stages in this abstraction process:

1. Concrete Algebra (Weber, Fricke): Virtually any ring considered was a sub-
ring of the integers or of the complex polynomials or of some algebraic num-
ber field (similar for other domains). Various “identical” results were proved
separately for different instances.
2. Abstract Algebra (Noether, van der Waerden): Rings are described by their
axioms; the classical domains mentioned above are subsumed as examples.
All the proofs are now done once, within “ring theory”.
3. Universal Algebra (Graetzer, Cohn): Classes of algebraic structures are con-
sidered collectively; rings are just one instance of an algebraic structure.
Results like the homomorphism theorem can be proved for generic algebraic
structures that specialize to rings, groups, and the like.
4. Category Theory (MacLane, Eilenberg): Categories are any collection of ob-
jects (like algebraic or non-algebraic structures) connected through arrows
(like homomorphisms in algebraic structures). The objects need not have a
set-theoretic texture (as the carriers of structures have).

The role of mathematics as reasoning in abstract models becomes very clear
in the process of algebraization: the mathematical models are now specified pre-
cisely by way of axioms and we need no longer rely on having the same intuition
about them. Let me detail this by looking at one of the most fundamental struc-
tures used in physics - the notion of the continuum , which provides a scale for
measuring virtually all physical quantities. Its axiomatization as the complete
ordered field of reals needed centuries of focused mathematical research culmi-
nating in the categoricity result. Proceeding with computer algebra, we would
now strip off the topological aspects (the “complete ordered” part) from the
algebraic ones (the “field” part) and then study its computable subfields (finite
extensions of the rationals).
If one models physical quantities by real numbers, analyzing their mutual
dependence amounts to studying real functions (real analysis), and the natural
laws governing them are written as differential equations. Their application to
specific situations is controlled by adjusting some data like various parameters
and initial/boundary conditions. Since the latter are the most frequent data in
physical problems [10], boundary value problems (BVPs) will serve as a fitting
key example in the last section of this presentation.

3 Algorithmization: The Realization of Computing

In order to actually “compute” the solution of a problem in an abstract model,
algebraization alone is not enough. We have already observed this in the above
example of the continuum: The field of real numbers is regarded as an algebraic
domain, but it is clearly uncomputable because of its uncountable carrier. In