Mathematical thinking (English)

Mathematical thinking: Universal content and the individual’s ability to experience mathematical laws

 

How can something experienced in individual consciousness have a universal character independent of this consciousness? This is the fundamental problem that is to be solved for the proof of the objective existence of mathematical laws. To solve this, both direct and indirect methods have been suggested. By indirect methods it is a matter of proving that without the acceptance of the reality of mathematical laws a meaningful and elegant science which is as plausible as possible to the human intellect would not be possible. Such indispensability arguments ultimately lead to hypothetical realism, a sort of myth about the reality of specified entities which in this sense cannot be distinguished from other myths, legends or creeds.

By direct methods for the proof of the reality of mathematical laws, it is a matter of analyzing the immediate manner of experiencing these laws. Experience is part of individual consciousness. It is thus only accessible to introspection and for this reason has so far been rejected by many authors as suspect, unclear or unscientific. From the apparent failure of all attempts by means of introspection to come to objective results, in contrast to subjective enlightenment, it is almost exclusively the indirect method that is still taken seriously. In this essay it will be shown that the possibilities of the direct method are in no way exhausted or sufficiently researched – not to mention the fact that a consistent scientific consciousness can never and must never be satisfied with mere, albeit rationally-based, belief in a myth.

Before positive proof of the reality of mathematical concepts can be tackled, a few prejudices must be cleared out of the way.

First prejudice: The content and the process of mathematical thinking arise from convention. – The origins of conventions are not necessarily of a conventional nature: a ‘convention’ established for the first time cannot arise from an agreement, because it is initially known to nobody but the subject who establishes it. If it is possible however for this subject who establishes the convention to have an unconventional approach to thought, then is not clear why this should not be possible for other subjects too. In addition, agreements between people, which are communicated explicitly, inexplicitly or otherwise, require individual insight into or assent to the meaning of the agreement. Otherwise, in passing on conventions, one is merely dealing with blind faith or obedience.

Second prejudice: The subjective experience of mathematical thinking (introspection, intuition, inspiration etc) is of an inexpressible nature and thus lies outside science. – Here is a confusion of thinking with communication, or rather the muddling of the content of thinking and the expression of this content in a language. In order to think, one neither has to talk to oneself nor communicate with oneself in any other way. In addition, the meaning of linguistic expressions cannot ultimately be inferred from a language; the investigation of the meaning always stops with the individual insight into the meaning of the expressions of the (natural) language. Therefore, if what cannot be expressed in language cannot be exactly understood, then ultimately the source of knowledge of scientific investigation would be removed and thus science would only be able to be established through extra-scientific personal experiences.

Third prejudice: The experience of mathematical thinking belongs exclusively to the subject. It has no significance beyond the subject. – The determination of the subjective character of the experience of mathematical thinking occurs through the subject himself and results from the experience of his own activity which is connected with this experience as well as from the fact that only I myself experience directly what I think and no other person has an immediate part in

my unspoken thinking. But this only means that the activity as well as the consciousness of the thought content belong to the subject; however this yields nothing about the constitution of the content. Here there often exists a further prejudice:

Fourth prejudice The subject produces the content of mathematical thought. – Not a single direct observation based on mathematical thinking has so far been advanced for this hypothesis. All phenomena which apparently support it concern the consciousness of contents, but not the contents themselves.

Fifth prejudice: The contents of mathematical thinking are determined through the structure of the psycho-physiological cognitive apparatus. – For immediate confirmation of this thesis it must be shown that for establishing and deducing mathematical laws the structural principles of the cognitive apparatus must of necessity be explicitly enlisted. In the direct experience of mathematical thinking (not: the formal-symbolical representation of this thinking) there are however no grounds for such an incompleteness or dependence of mathematics in principle. In addition, all arguments for the dependence of mathematical thought contents on the structure of the cognitive apparatus concern the consciousness of the contents, not these contents themselves. Finally there is the evident incompatibility and diversity of the contents of consciousness of mathematical thinking and the results of observation obtained by means of investigation of the cognitive apparatus.

For a deeper insight into the structures of argument used here we shall introduce the distinction between proper and improper hypotheses. A hypothesis (model, theory, structure) with respect to a realm of facts is improper, when there are observations lying immediately inside this realm which justify the hypothesis. There must not merely exist inferences for confirmation of the hypothesis. A classical example for an improper hypothesis is the following statement: The period of swing of a freely swinging pendulum is dependent on the length of the pendulum.

A hypothesis with respect to a realm of facts is proper, when there are no immediate observations within this realm which justify the hypothesis. There exist only methods of inference which, from the factual material available, suggest the existence of something which is not itself part of this material. Any indirect method for the confirmation of realism is an example of this.

In the following investigation, strict attention will be paid to whether we are dealing with proper or improper hypotheses. This is of fundamental significance, because we are not dealing with the investigation of any arbitrary object, but with something which plays a fundamental role in all scientific activity, namely thinking, in particular in its strict form of mathematical thinking.

Excerpt from: Mathematics as a spiritual science. Philosophical investigations into the significance of mathematics with reference to Plato, Goethe and Steiner. Newsletter of the Science Group of the Anthroposophical Society in Great Britain, Articles Supplement, Vol. 1, September 1995, pp. 18-37.