Mathematical intuition (English)

Mathematical intuition

Mathematical intuition must first be distinguished from idea, taken here in its usual sense as something that occurs to the individual mind (having an idea etc.). In previous sections we used this word with different connotations following Platonic tradition. Idea in its most common sense means, however, a content which is given to the thinking subject without him having contributed himself directly with his own conscious activity to the process of this content being given. Such ideas certainly play an important part in the life of a mathematician, but they come ‚by chance‘ and are not subject to the control of individual consciousness. As a rule however, a fruitful idea is preceded by an intensive occupation with mathematical contents in the ’neighbourhood‘ of the contents of the idea. Furthermore, following the idea there is the task of finding the actual evidence, i.e. the concrete pattern and detailed interrelating of the contents of the idea with contents already known, for instance, axioms and theorems derivable from them.

What should be understood here by mathematical intuition are only those phases of the mathematical work by which the mathematician has a complete clarity and overview of his actions, i.e. where he knows exactly his point of departure and how he reached the contents he is actually thinking about. This implies no devaluation of other phases of mathematical thinking (heuristics, ideas, analogies, games etc), but these are of a preparatory nature and are not determinative for the ultimately intuitive insight.

Mathematical intuition is bound by two conditions: One concerns the purity of the content produced in thought and the other the manner of its production. By purity of content we understand the complete freeing of mathematical thinking from concrete examples from the world perceived by the senses. Thus, in section 1 it was not a case of any particular circle existing anywhere, but of the principles which govern and constitute all circles.

The manner of production is concerned with the degree of comprehensibility and clarity of the insight into the inner necessity of a thought content being dependent on the extent to which the subject participates in the thinking process. We can comprehend completely only that which we ourselves bring about, bring into existence. Everything given without the subject’s own activity is initially a problem for the attentive subject. In mathematical intuition, no content is given to the thinking subject without his having produced it. However, this does not mean that mathematical thinking itself produces its content (see previous section). Rather it means not only following every step of the process, but also performing these steps autonomously.

Inside mathematical intuition, two realms of experience can be distinguished from one another: one concerns the subject’s activity (see following section) and the other the constitution of the content.

Within the process of mathematical intuition, three properties can be distinguished as regards the contents of mathematical thinking, i.e. the contents of mathematical concepts, here also called laws. These properties play a fundamental role in the judgement of the constitution, i.e. of the ontological make-up, of these contents. Attention has been drawn above to one of these properties, namely inner necessity and complete comprehensibility. Another concerns the unchangeability or invariability of the laws by the thinking subject. The laws offer a (passive) resistance to a corresponding test and cannot in their content be either changed or arbitrarily linked with other laws. Put metaphorically, mathematical thinking is ‚guided‘ by the laws in maintaining its state of intuition – like someone’s hand consciously feeling a marble relief. The relief does not press the hand, but it does not allow itself to be changed by it. Every apparently successful alteration of a law leads either to a new one or is confined merely to the concrete relationship of the subject to the thought contents. So-called extensions of concepts or conceptual generalisations (eg. of the laws of multiplication) are not variations of a concept as such, but an expression of a different perspective of the thinking subject to the corresponding realm of laws.

The independent and self-supporting character of mathematical laws is revealed in mathematical intuition. They are, in fact, in the sense of section 3 invariants of the operations of individual mathematical thinking. A structural principle higher than the operations carried out by the individual subject forms their basis. This is the universal principle of mathematical intuition used, indeed, by all mathematicians, but which none of them own privately.

Here the question arises as to whether mathematical laws are invariant only relative to the thinking subject, or whether they are generally (absolutely) invariant. The invariance of laws means that their content cannot be subjected to a change by another being or by themselves. The invariance of laws implies their unchangeability or invariability, but the reverse does not hold true.

It must first be established that there is no experienceable, i.e. not only proper hypothetical, basis for the assumption of a variance or a changeableness of mathematical laws. What changes is at most the individual grasp of or the consciousness of these laws, but not the laws themselves.

The understandable psychological resistance to the invariance of laws is not primarily directed at mathematical laws, but at the acceptance of unchanging laws in general. This appears to be confirmed by so-called everyday experience. But here we do not make it sufficiently clear to ourselves that the acceptance in principle of a variance or changeableness of all laws has the consequence that there must be one or more ’super laws‘ which do not change and which exhibit with each concretely demonstrable change the structures which remain invariant (the invariants). For, given that law A transforms to law B, i.e. that A is changed in that it becomes B, the question arises: On the basis of which property can B be determined as coming from A? This is only possible when there is a predicate C which is common to both A and B, whereby B, as something still connected with A in some way, can be recognised as related with A. For this however, C must show an invariant property relative to the transformation of A to B, i.e. cannot be subject to change. Therefore the principle C is unchanging and A and B thus do not belong to the realm of laws.

It could be objected that here we are dealing with a proof of only relative variance or unchangeableness, but not one of absolute unchangeableness. That is not however the case, because the assertion behind this objection that all is relative is, taken in the absolute sense, necessarily self-contradictory.

From this it follows that the realm of change is not to be established in the realm of laws, but in the realm of phenomena, i.e. the place where these laws operate or take effect. The situation here is totally analogous to the relationship of abstract groups to the elements of their domain of possible transformations. The operations of the groups concern only these elements or sets of such elements (see section 3). – Sometimes the objection arises here that there might also be ‚flexible‘ or ‚living‘ concepts. Following the above discussion, this cannot mean a self-changeableness of concepts, but a flexible or living perspective of the thinking subject relative to the self-determined and unchanging contents of laws.

To conclude this section, attention should be drawn to the fact that it is not in the nature of the principle of mathematical intuition that it can only be used on mathematical contents. It cannot be denied from the outset that there are also concepts lying outside the domain of mathematics which can be manifested in the form of mathematical intuition.

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.