The influence of worldview on the first foundational crisis in mathematics

Abstract

The aim of this study is to determine what the ontic status of mathematical entities (sometimes called mathematicals) is in the three main phases of ancient Greek philosophy: the Presocratic stage, Plato and his early Academy, and Aristotle. It turns out that the basic assumptions that a person makes about the structure of reality (which are also collectively known as the person's worldview) have an impact on what his view is on the philosophy of mathematics. This will also be examined throughout. In order to specify exactly what aspect of the philosophy of mathematics we are interested in, it is maybe a good idea to first have a look at what aspects of the philosophy of mathematics can be discerned in this time period. The following questions were treated by Plato and some of the other early Greek philosophers (Wedberg, 1955:9-10): 1. Where are mathematicals located in a classification of the universe? 2. How are mathematicals generated? 3. Are mathematicals Ideas? 4. What relationship is there between mathematicals and the external world? 5. How should mathematics be done? Although each of the other points are certainly important and have an influence on how we answer the first question, the point of this study is to answer the first question. We will certainly at times in this study briefly look at some of the other questions in terms of how they help us to answer the question of the ontic status of mathematical entities. For instance, if the answer to the fifth question includes saying that observation (naively understood) is part of the methodology of mathematics, then clearly our answer to the first question could not possibly be that mathematical entities have no perceptible existence. It turns out that asking what the ontic status of mathematical entities is, in the context of Greek philosophy, immediately leads to the necessity of classifying the different kinds of mathematical existence. The ancient Greeks did not think that numbers and geometrical configurations had the same kind of existence. In fact, three phases can be discerned in how they approached the existence of arithmetical entities (sometimes called arithmeticals) and geometrical entities (sometimes called geometricals). These are as follows: A. Thales (and probably also the later Milesians) thought of arithmeticals and geometricals as having coordinate places in the ontic scheme: neither could numbers be reduced to figures, nor could figures be reduced to numbers. B. The Pythagoreans thought of arithmeticals as subordinate to geometricals, that is, numbers were just special kinds of geometrical arrangements.

C. From Plato onwards, there is a coordinated effort to place arithmeticals and geometricals on an equal footing, with equal ontic status. Both numbers and figures are fundamental kinds of existences, and neither could be reduced to the other. Plato himself seems to have been still very undecided about this issue. It will be the aim of this study to flesh out the details of these claims and to suggest why the only reasonable solution of the problem of the ontic grounding of mathematics is to say that both arithmetical and geometrical entities have a fundamental kind of existence.