The Issue of Experiment in Mathematics
ABSTRACT: The issue of the status of mathematical knowledge a priori or a posteriori has been repeatedly considered by the philosophy of mathematics. At present, the development of computer technology and their enhancement of the everyday work of mathematicians have set a new light on the problem. It seems that a computer performs two main functions in mathematics: it carries out numerical calculations and it presents new areas of research. Thanks to cooperation with the computer, a mathematician can gather different data and facts concerning the issue of interest. Moreover, he or she can carry out different "tests" with the aid of a computer. For instance, one can study strange attractors, chaotic dynamics, and fractal sets. By this we may talk about a specific experimentation in mathematics. The use of this kind of testing in mathematical research results in describing it as an experimental science. The goal of the paper is to attempt to answer the questions: does mathematics really transform into experimental or quasi-experimental science and does mathematics vary from axiomatic-deductive science into empirical science?
For thirty years the computer has been used by mathematicians to solve some problems. Automatic proving of theorems, proofs obtained with the aid of the computer for the theorems whose traditional proofs are not known (e.g. the four colour problem), using computer graphics, observations of different systems behaviour with parameters changed, solving differential equations, integration — these are only a few possibilities of computer application in mathematics. Using the computer created new work conditions for a mathematician, at the same time bringing about several questions concerning a method of practising mathematics. One of them is the question about the essence of and role of the so called computer experiments. In a wider perspective issues related to experimenting are linked with one of the fundamental problems for the philosophy of mathematics, which is the question whether mathematical knowledge is a priori or a posteriori.
I shall try to show the issues concerning the role of computer experiments in mathematics taking examples from the theories of fractal sets and deterministic chaos. These theories have been developed very quickly in the recent years and there have been used to describe a lot of phenomena and processes which so far could not be researched by means of mathematics. The choice of those theories is not accidental. Their development is greatly caused by the use of computers for research in mathematics.
The examples of sets, that at the moment are called fractal sets, have been known for years, and sensitivity to initial conditions of certain mechanic systems, whose behaviours are described by Hamilton's equations, had already been noticed at the end of previous century by H.Poincarw (1892). Those had been treated, however, as "curiosities" of little...