The Issue of Experiment in Mathematics
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 importance for the development of mathematics, placed on the margin of the main stream. The interest with the issues, that arise while studying fractal sets or non-linear dynamical systems, has become possible due to computer technology. Calculations promptly carried out, which was virtually impossible without using computer, created new conditions for studying fractal sets and chaotic dynamics. That is why the theories of fractals and deterministic chaos constitute perfect examples of computers application in mathematics for without experiments conducted on digital machines they would not have probably been developed at all.
Now, I shall turn to present examples of computer use for studying chaotic dynamics and fractal sets. Let us consider the mapping f (x)=ax(1-x) of the closed interval [0, 1] into itself with the parameter a that belongs to the interval [0, 4]. For any x that belongs to the interval [0, 1] let us create a sequence (x , x , x ,...x ,...), where x =f (x ). This sequence is called the orbit of x .
Depending on the value of the parameter a we can observe different qualitative behaviour of the orbit of x . This behaviour can be easily observed on the computer screen. If 0<a<1, then for any x the orbit of x is convergent to 0. If 1<a<3, then orbits are convergent to the fixed point of f (then we say that the fixed point is itself attracting). Essential quality changes in the behaviour of orbits follow when the value of the parameter a excesses 3. In point a=3 there is the period-doubling bifurcation, which means that the fixed point stops attracting orbits, while periodic orbit of period two appears. This periodic orbit is attracting. For a=1+r6 there is again the period-doubling bifurcation: the attracting periodic orbit of period four appears. There exists an infinite sequence of bifurcation points: a , a , a ,... convergent to a =3,56995... At each point a a new periodic orbit of period 2 arises.
If the parameter a increases a the period 2 orbits disappear and orbits begin to behave chaotically. There is the phenomenon of sensitivity to initial conditions, described as deterministic chaos: a slight change of the initial point x can cause dispersal of orbits after several or even few iterations. Also even slight change of the parameter itself causes quality changes in the dynamics of the whole system. The phenomenon of deterministic chaos can be easily observed on a computer screen. The same qualities as those possessed by logistic mapping are typical for other functions.
Fixed points of mappings, stable periodic orbits are examples of attractors, that is such invariant sets to which all nearby orbits converge. The examples of the so called strange attractors can be supplied by the system of iterated contracted mappings. For certain systems of such mappings the strange attractor has a very complicated structure whose complexity as well as beauty can be seen on the computer screen.
Strange attractors from the above mentioned example are also the examples of the so called fractal sets. Another example of the fractal set is studied by B.Mandelbrot (1980) with the use of numeric methods. The Mandelbrot set appears quite naturally while studying the behaviour of the iteration of mapping F (z)=z +c, where z and c are complex numbers. Let z =0, z =F (z ). The Mandelbrot set is the set of those parameters c for which the sequence(z ) is bounded. This set turns out to possess a very complex structure and to study it the computer is very helpful.
It is worth stressing that a lot of physical, biological, social and economic processes can be modelled by such dynamical systems in which deterministic chaos appears. The results of computer simulations of such phenomena also supply information useful for a mathematician.
Thus, the computer "produces" fractal sets, strange attractors, allows for observing the behaviour of chaotic dynamics. Thanks to it we can witness characteristic features of examined sets , as well as, which is particularly important for the development of mathematical theories, gather information on "occurrences" taking place in the field of mathematical objects and observe new areas of research.
Thereby with the aid of the computer a mathematician does what iscolloquially understood by experimenting. Then the question arises: whether using the computer has changed the so far used mathematical practice. More and more often there can be heard an opinion according to which mathematics has become an experimental science, but it seems that the gist of the way in which mathematics is used has not changed, to support this view we can quote the following arguments.
The drawing in geometry has always played an important role. There are commonly known graphical proofs of theorems, for instance Pythagorean theorem. It is enough to look closer at the drawing to "see" the truth of the theorem. In fields of mathematics other than geometry with the aid of various drawings, diagrams, sketches, etc., people have tried to give graphic presentation of certain situations or inspire given intuitions. An interesting example of the theory using diagrams is the theory of category. Physical models play a similar role to this of drawings. Also intuition shaped by various factors is of enormous importance. For instance intuitions, which were created by applying the infinitesimal calculus, particulary in mechanics, had a great influence on shaping the basic notions of analysis. The set theory was also initially developed by Cantor on the basis of intuition of the notion of set. Checking the behaviour of chosen mathematical objects in particular examples often led to formulating true theorems (as it later appeared), although their deductive proofs were not known. For instance Cartesius calculating fields of given figures with the same perimeter stated that a circle has the largest area. According to him such an iductive proof was enough. Similarly, Euler checked in many cases the correctness of formulas for the sum of divisors of the given natural number. Examples of the influence of various experiments on the development of mathematical theories can be multiplied.
As it seems, then, clarity of various models and drawings,mathematical intuition and intuitions shaped under the influence of interaction between mathematical theories and their applications have always been present in mathematical practice. In this context computer experimenting does not differ in substance from what has happened in mathematical research so far. The difference is only quantitative: now we have far greater calculation possibilities.
Due to the co-operation with the computer mathematician acquires new information on the field that is of interest to him. This information can both suggest formulating the theorem and its proof. Computer results supply examples or counterexamples for given theorems or hypotheses. All this shows the similarities between the role of experiment in mathematics and the role of experiment in natural sciences. That where the similarities end, however. The results of experiments in natural sciences fulfil an essential function in the phase of justifying the theory. Each scientific theory is continually confronted with the results of experiments and observations, although it happened that certain theories had waited for a long time to get experimental data that would agree with theoretical anticipations. It is not enough for the theory to be logically consistent; it is the result of the experiment that decides whether a scientific theory is accepted or rejected. That is why scientific theories are permanently corrected, they can undergo changes affected by new experimental data. In mathematics nothing like this happens. If there is a correctly carried out proof of a theorem then there cannot be any other experiments whose results would be inconsistent with the thesis of this theorem. If mathematical theory is consistent, then there cannot occur results inconsistent with this theory that would make a mathematician to reject this theory. Formulating scientific theory in axiomatic form and a consistency of such a theory do not protect its results against incompatibility of experimental or observational results.
In works on fractal sets or chaotic dynamics the results of computer experiments only illustrate a given problem. They do not, however, justify the genuineness of presented theorems. Deductive proof still remains the final justification. Axiomatic-deductive method is still the only method in mathematics. Obviously we cannot exclude the possibility of arising of basic changes in the method of practising mathematics in future.
In the phase of discovering new knowledge the experiment in mathematics fulfils a role similar to this in natural sciences. While the stages of justification and presentation of completed products differ from each other in the above mentioned sciences. The results of experiments in mathematics in this phase play a secondary role, they mainly serve to illustrate a given issue. Obviously a lot of facts obtained with the aid of computer have not had a deductive proof so far and for the time being we must be satisfied with their experimental justification. It does not mean, however, treating those results in the same way as "ordinary" mathematical theorems or that any search for deductive proofs should be stopped. Understanding and explaining of what we observe on the computer screen is done by formulating and proving proper mathematical theorem. For instance, the so called structural universality was discovered and "proved" numerically. Then J.Guckenheimer obtained precise mathematical basics for it.
Then my answer to the question that was asked at the beginning is as follows: mathematical knowledge is a priori in this aspect that tojustify the truthfulness of the theorem the only necessary thing is to correctly carry out a proof within a given theory. The fact that mathematics seems to be an a priori knowledge do not mean that the development of mathematical knowledge, new discoveries in this field are made in the precise axiomatic-deductive form. Just the opposite, they are conditioned by various factors, among which the results of different experiments, also carried out by means of computer, play a very important role. The process of creating mathematics, including the search for deductive proof, is stimulated by present experience and the intuition of a mathematician. So it can be said that genetically in the process of creating mathematics is an a posteriori knowledge.
The possibility of carrying out experiments in mathematics seems to be an advantage of the opinion in the philosophy of mathematics which claims objective, that is independent on mathematician's mind, existence of the world of mathematical 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.
(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.