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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
