Evert W. Beth

Strict Demonstration and Heuristic Procedures

We shall see in Section 26 that our information about mathematical thought, and about creative thought in particular, is very limited. Yet this information, deriving for the most part from introspection, must be interpreted and used with extreme caution, not only because it is offered to us by mathematicians who are not necessarily psychologists and who, furthermore, can hardly be considered as disinterested observers, but above all because the few truly creative mathematicians who have wished to give a more or less detailed account of their innermost experiences do not apparently manifest one single type of thought activity.

Some Convergences between Formal and Genetic Analyses

In this chapter I want to give one or two examples of the convergence between genetic and axiomatic investigations, since such examples can show how certain general results of formal analysis are psychologically explicable if based on what we know of the subject’s activities.

Epistemological Problems with Logical and Psychological Relevance

We should now like to draw certain conclusions about general epistemological problems from these reflections on the psychology of mathematics, taking epistemology in the sense of Chapter VII, Section 42, including ontological problems which imply the comparison of logical analyses with genetic data.

Mathematical Reasoning Cannot be Analysed by Traditional Syllogistics

We may today state as an established fact that mathematical reasoning as it would be found, for example, in a modern version of Euclid’s Elements cannot be expressed as a succession of Aristotelian syllogisms.

Intuitive Structures and Formalised Mathematics

We have seen that there exists a kind of “pre-established harmony” between pure mathematical thought, the deductive method and Platonism; the alliance between the three is so stable that it is difficult to consider it only as the result of a fortuitous historical grouping.

The Psychological Interpretation of Mathematical Reasoning

To show Mill’s radical empiricism, I shall first of all quote his discussion of the principle of contradiction, which provides a good example of his method.1

The Psychological Problems of “Pure” Thought

One of the reasons why a certain number of logicians and mathematicians hold aloof from or sometimes mistrust psychology is that, according to their conception of it, genetic analysis is held to be relevant only to “intuitive” thought, which alone is considered as “natural”. Formalisation being only the prerogative of a small elite (as opposed to the majority of other people, all capable of “intuition”) then appears as “artificial” if not as going “against human nature” (Pasch) in a similar sense in which, before scientific sociology, social institutions were considered with Rousseau as outside nature (freely set up by contracts) and the individual alone was considered as “natural”. But on the one hand, the thought of a small elite is at least as interesting, if not more so, than that of the majority for the psychology of the development of human thought. On the other hand, as the object of genetic studies is not introspective consciousness but the mechanism of the successive constructions which lead to the adult state, we must examine closely, before we can come to a decision about it, whether the passage from intuitive thought to axiomatisation is not prepared by the preceding development; and especially whether the gap thus bridged is so great that it is not comparable to the yet very large gap separating the baby’s sensory-motor activities from the hypothetico-deductive thought of the normal adolescent in our society, possessing merely a certificate of primary education.

Lessons of the History of the Relations between Logic and Psychology

It is an instructive fact for epistomology in general that the deductive sciences arose long before the experimental sciences. Even if mathematics passed through an empirical phase (Egyptian mathematics, which moreover was a technique rather than an enquiry having a truly scientific objective), it reached a much higher level of elaboration with the Greeks than did their physics. Whilst the Elements of Euclid provided a model of axiomatic deduction which over a long period was considered as complete, Greeks physics only consisted of a systematisation of the data of common sense (Aristotle’s physics), or in very partial results expressed in a deductive and non-experimental manner (Archimedes’ statics) or again in diverse attempts at celestial mechanics foreign to true experimentation. We had to wait for the 17th century (in spite of several precursors at the end of the Middle Ages and during the Renaissance) for a physics which had a methodological autonomy comparable to that which it exhibits today.

The Logicist Tradition

I have published elsewhere more detailed studies of Aristotle’s theory of the sciences, which allows me to limit myself here to a concise exposition of what is important in the present context.

“Thinking Machines” and Mathematical Thought

Amongst the objections to the formalisation of logic and mathematics, one of the most common consists in asserting that such a formalisation would reduce logical and mathematical thought to purely mechanical operations, and would thus allow the construction of a “thinking machine” capable of replacing the logician and the mathematician. The acceptance of the possibility of such a replacement would force us to deny all originality to logical and mathematical thought, and it would thus be incompatible with our experience according to which the solution of mathematical problems, in particular, requires original thought.