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The Algorithmization of Physics:

Math Between Science and Engineering

Markus Rosenkranz

Johann Radon Institute for Computational and Applied Mathematics

Austrian Academy of Sciences, A-4040 Linz, Austria

Markus.Rosenkranz@oeaw.ac.at

Abstract.
I give a concise description of my personal view on symbolic

computation, its place within mathematics and its relation to algebra.

This view is exemplified by a recent result from my own research: a new

symbolic solution method for linear two-point boundary value problems.

The essential features of this method are discussed with regard to a

potentially novel line of research in symbolic computation.

1

Physics: The Source and Target of Math

What is the
nature of mathematics
? Over the centuries, philosophers and math-

ematicians have proposed various different answers to this elusive and intriguing

question. Any reasonable attempt to systematically analyze these answers is a

major epistemological endeavor. The goal of this presentation is more modest:

I want to give you a personal (partial) answer to the question posed above, an

answer that highlights some aspects of mathematics that I consider crucial from

the perspective of symbolic computation. At the end of my presentation, I will

substantiate my view by a recent example from my own research.

According to [4], humankind has cultivated the art of
rational problem solving

in a fundamental three-step rhythm:

1.
Observation:
The problem of the real world is specified by extracting relevant

data in an abstract model.

2.
Reasoning:
The model problem is solved by suitable reasoning steps, carried

out solely in the abstract model.

3.
Action:
The model solution is applied in the real world by effectuating the

desired result.

In this view,
mathematics
is not limited to any particular objects like numbers

or figures; it is simply “reasoning in abstract models” (item number 2 in the

enumeration above). For highlighting its place in the overall picture, let us take

up the example of physics - of course, one can make similar observations for

other disciplines like chemistry, biology, economics or psychology.

We can see physics as a
natural science
that deals with observations about

matter and energy (item number 1 in the three-step rhythm). In doing so, it