Max Dehn on Mathematical abilities in humans

In 1932 Dehn wrote the 16-page essay Das Mathematische im Menschen (Mathematical ability in humans) which he published in Scientia, an Italian journal produced in Bologna by Federigo Enriques. We give below an English translation of the first few paragraphs of the article:

Mathematical ability in humans

The following could also be the introduction to a history of mathematics. It is an attempt to link the science of mathematics with the whole of human activity, whereby in many cases the limited space available here will force us to restrict our comments to short allusions. At first, we will outline the roots of mathematics in the human soul. We will then show how the historical development of mathematics reflects the general human condition. Our considerations might have a certain value regarding the ever so important question of the meaning of order for the human mind. in some passages we will also touch upon the problem of the creation and the impact of works of art, although we will of course be considering only a very narrow aspect of this problem.

1. In the coming pages, rhythm will be understood as being the succession and accentuation of identical or regular changing things. We will attempt to demonstrate how the activity of a human being creating rhythm is related to the work of a mathematician. In our considerations, we will distinguish different kinds of rhythm.

One-dimensional rhythm (simple music). linear, one dimensional rhythm manifests itself, for instance, in drumbeats that are strongly and less strongly repeated at equal intervals, in the beat and in the accompanying dance. When, as ethnologists tell us, groups of people in certain tribes perform several distinct rhythms or drumbeats at the same time, they create a situation that already comes close to mathematical activity. The delight in the regular recurrence of moments when the accentuated beats, the main strong beats of the various rhythms, coincide corresponds to the delight of the beginner number theorist when he realises how the sequences of multiples of various numbers are embedded in each other. the two emotions arise from the same basic characteristic of the human soul.

Moving from this simplest of musical activities on to more complicated constructs, we might come to the imitation, where various themes following each other in several voices become intertwined.

Human beings have known for centuries that there is a link between music and mathematics. However, our considerations are not to be confused with mathematical descriptions of acoustic conditions. The Pythagoreans, for instance, made the connection between physiological phenomena such as tonal relationships and the length of strings. This is fundamentally different from what will be examined here. Nevertheless, they may indeed have had a certain idea of the common source of musical and geometrical emotions that we have just described.

2. Higher dimensional rhythm (architecture, ornamentics, symmetry, adaptation).

JOC/EFR January 2014

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