# Max Dehn on Space, Time and Number in Aristotle

In 1936, Max Dehn published

Aristotle was not a productive mathematician, and in all his works there are only very few individual facts relating to pure or applied mathematics. However, he dedicated considerable attention to the topic of space and time as continua, at first on their properties on "a small scale". These reflections are part of topology. Naturally, the combination of the two continua, i.e. movement, plays a major role. The consideration of movement as a relation between space and time leads to the first reflections on the theory of functions. In deriving time measurement from movement, the concept of number take on a significant role. We see that Aristotle indeed felt the difficulties of the transition from number to general quantity, though of course without having more than an inkling that this transition must lead through infinite numerical sequences.

Mathematics is of interest to many people. First, there is pleasure in specific individual things such as special combinations of numbers or individual geometrical shapes such as the regular bodies; then, people take delight in the more general properties of numbers and geometrical forms, such as decomposition laws or constructions. And finally, it gives the mature mathematician great satisfaction to methodically examine large systems of propositions, to see the wonderful architecture of entire disciplines rise as a result of stringent combination from simple fundamentals to heights inaccessible to direct observation. This is especially the case when the mathematician himself is personally involved in the construction, when perhaps there are entire parts of the structure that would not have come to light without his thinking.

In all of this, what is essential is the ability to delight in the consummate form. Considerations involving the elements of these forms, however, for instance the essence of the integer or of space and its parts, do not have this primal appeal, and although an understanding of these elements is the foundation of any mathematical activity, mathematicians have never paid more than a little attention to them. Irresistibly drawn by the appeal of creativity, they have tended to focus their attention only on accomplishing the goal, on striving for perfection. Philosophers, on the other hand, principally strive to become increasingly aware of who they are and what they do, and this tendency leads many of them to also delve into the foundations of mathematics, often even before having a sufficient notion of the structure which towers above these foundations. There are therefore only very few philosophers who have contributed anything of particular value to foundational research. However, these few have been extremely important for the process of making mathematical activity more aware. And this process is absolutely necessary for ensuring that this creativity remains fruitful on an ongoing basis.

It seems to me that Aristotle's work contains very important ideas on the foundations of mathematics. we shall discuss these in the coming pages. The work of mathematicians has evolved through very different stages of awareness in the course of the millennia. In pre-Greek times, there seems to have been no concept of formulated proof of results, at least, as far as we know today, and it is only in our time that the axiomatic method and the concept of rigorous proof has gradually come into being, i.e. the precise description of what had until now been a more or less unconscious advance in one's thoughts. the concept of complete induction for instance, which is so important for this description of though processes, did not exist at all in ancient times.

Pre-Greek mathematics had primitive knowledge about integers and simple geometrical forms like points, straight lines, planes, etc. The axioms underlying them are imperfectly described in Euclid's Elements. It is only in the past hundred year that they have been discovered and sufficiently well established to allow the axiomatic method to be applied. In Euclid's works, for instance, there is no mention whatsoever of the fundamental properties of the order of points on a straight line or of points in a plane in relation to a straight line etc., which are used everywhere but always left unmentioned or at least not formulated. In Aristotle, however, we find the attempt to describe the fundamental facts of order. He subsequently discusses in detail the properties of the continuum, both of space and time, which also have been examined by other philosophers. It is astounding that at this point in Aristotle one lso finds an attempt at functional reflections. I would like to give an account of these Aristotelian considerations in the first three paragraphs.

Aristotle's ideas and approaches had little influence on Greek mathematics. The theory of the continuum, set theory and topology, are not yet one hundred years old, and the theory of functions did not begin until the 17th century. And yet Aristotle has indirectly become very important for the dynamic development of these disciplines through Nicolaus Cusanus (15th century), Barrow (17th century) and many other mathematicians, possibly through Bolzano (19th century).

The theory of the continuum was followed by research on absolute space. We shall see below that the Aristotelian approach is, strangely enough, highly compatible with the most modern physical ideas.

*Raum, Zeit, Zahl bei Aristoteles vom mathematischen Standpunkt aus*(Space, time and number in Aristotle from a mathematical perspective) in*Scientia*. We give below a translation of the first two pages of the First Part of this 16-page article:**First Part**Aristotle was not a productive mathematician, and in all his works there are only very few individual facts relating to pure or applied mathematics. However, he dedicated considerable attention to the topic of space and time as continua, at first on their properties on "a small scale". These reflections are part of topology. Naturally, the combination of the two continua, i.e. movement, plays a major role. The consideration of movement as a relation between space and time leads to the first reflections on the theory of functions. In deriving time measurement from movement, the concept of number take on a significant role. We see that Aristotle indeed felt the difficulties of the transition from number to general quantity, though of course without having more than an inkling that this transition must lead through infinite numerical sequences.

**Introduction**Mathematics is of interest to many people. First, there is pleasure in specific individual things such as special combinations of numbers or individual geometrical shapes such as the regular bodies; then, people take delight in the more general properties of numbers and geometrical forms, such as decomposition laws or constructions. And finally, it gives the mature mathematician great satisfaction to methodically examine large systems of propositions, to see the wonderful architecture of entire disciplines rise as a result of stringent combination from simple fundamentals to heights inaccessible to direct observation. This is especially the case when the mathematician himself is personally involved in the construction, when perhaps there are entire parts of the structure that would not have come to light without his thinking.

In all of this, what is essential is the ability to delight in the consummate form. Considerations involving the elements of these forms, however, for instance the essence of the integer or of space and its parts, do not have this primal appeal, and although an understanding of these elements is the foundation of any mathematical activity, mathematicians have never paid more than a little attention to them. Irresistibly drawn by the appeal of creativity, they have tended to focus their attention only on accomplishing the goal, on striving for perfection. Philosophers, on the other hand, principally strive to become increasingly aware of who they are and what they do, and this tendency leads many of them to also delve into the foundations of mathematics, often even before having a sufficient notion of the structure which towers above these foundations. There are therefore only very few philosophers who have contributed anything of particular value to foundational research. However, these few have been extremely important for the process of making mathematical activity more aware. And this process is absolutely necessary for ensuring that this creativity remains fruitful on an ongoing basis.

It seems to me that Aristotle's work contains very important ideas on the foundations of mathematics. we shall discuss these in the coming pages. The work of mathematicians has evolved through very different stages of awareness in the course of the millennia. In pre-Greek times, there seems to have been no concept of formulated proof of results, at least, as far as we know today, and it is only in our time that the axiomatic method and the concept of rigorous proof has gradually come into being, i.e. the precise description of what had until now been a more or less unconscious advance in one's thoughts. the concept of complete induction for instance, which is so important for this description of though processes, did not exist at all in ancient times.

Pre-Greek mathematics had primitive knowledge about integers and simple geometrical forms like points, straight lines, planes, etc. The axioms underlying them are imperfectly described in Euclid's Elements. It is only in the past hundred year that they have been discovered and sufficiently well established to allow the axiomatic method to be applied. In Euclid's works, for instance, there is no mention whatsoever of the fundamental properties of the order of points on a straight line or of points in a plane in relation to a straight line etc., which are used everywhere but always left unmentioned or at least not formulated. In Aristotle, however, we find the attempt to describe the fundamental facts of order. He subsequently discusses in detail the properties of the continuum, both of space and time, which also have been examined by other philosophers. It is astounding that at this point in Aristotle one lso finds an attempt at functional reflections. I would like to give an account of these Aristotelian considerations in the first three paragraphs.

Aristotle's ideas and approaches had little influence on Greek mathematics. The theory of the continuum, set theory and topology, are not yet one hundred years old, and the theory of functions did not begin until the 17th century. And yet Aristotle has indirectly become very important for the dynamic development of these disciplines through Nicolaus Cusanus (15th century), Barrow (17th century) and many other mathematicians, possibly through Bolzano (19th century).

The theory of the continuum was followed by research on absolute space. We shall see below that the Aristotelian approach is, strangely enough, highly compatible with the most modern physical ideas.

Last Updated January 2014