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 severa...
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...objects. Because there can be shown an analogy between mathematics and natural sciences. Physical objects are recognized in the process of our experiencing materialistic reality. The experiment in natural sciences can be defined as a dialogue between the learning subject and the nature, which exists objectively. If we treat the experiment in mathematics in similar way, then there has to be two interlocutors: a mathematician and the field of mathematical objects, subjected to its own rules independent on the researcher's will.
Notes
(1) B.Mandelbrot in the context of using computer graphics states that: "The eye deserves to be made an integral part of the process of scientific thought" ("Opinions", Fractals 1(1993)1, p.120).
(2) Those examples are quoted by G.Polya in "Mathematics and Plausible Reasoning", vol. I, Princeton-New Jersey 1954, p. 90-100, 168.
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