Mathematics and Philosophy

Daniel Parrochia

• Publisher: Wiley-ISTE
• Year: 2018
• Print ISBN: 9781786302090
• eBook ISBN: 9781119528074
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

This book, which studies the links between mathematics and philosophy, highlights a reversal. Initially, the (Greek) philosophers were also mathematicians (geometers). Their vision of the world stemmed from their research in this field (rational and irrational numbers, problem of duplicating the cube, trisection of the angle...). Subsequently, mathematicians freed themselves from philosophy (with Analysis, differential Calculus, Algebra, Topology, etc.), but their researches continued to inspire philosophers (Descartes, Leibniz, Hegel, Husserl, etc.). However, from a certain level of complexity, the mathematicians themselves became philosophers (a movement that begins with Wronsky and Clifford, and continues until Grothendieck).

PART 1
The Contribution of Mathematician–Philosophers

Introduction to Part 1

In antiquity, a period when science was both knowledge and wisdom, there was no real distinction between a philosopher and a seeker of learning, that is, a person who loved knowledge or loved wisdom. Thus, people studied and manipulated both concepts and quantities, which could be discrete (and, therefore, could be expressed in whole numbers) or continuous (segments, surfaces, etc.). In Greece, as in virtually any society, the only numbers known from the beginning were whole numbers. However, the existence of division imposed the use of other numbers (fractions or fractional numbers) both to translate the form as well as the results of this operation. Initially, therefore, fractions were only ratios between whole numbers¹.

It was the Pythagoreans who first created the theory of whole numbers and the relations between whole numbers, where they would sometimes find equalities (called proportions or medieties). But, as they would very soon discover, other quantities exist that cannot be expressed using these numbers. For example, the Pythagoreans would explore a spectacular and intriguing geometric quantity: the diagonal of a square.

Everyone knows what a square with a given side a is. The area of the square, S, is obtained by taking the product of one side by another. In this case, S = a × a = a². The Pythagoreans were interested in the diagonal of the square as they were trying to solve a particular problem, that of doubling a square. In other words: how to construct a square whose area is double that of a square of a given side (a problem evoked in Plato’s Meno). The response, as it is well known, is that we construct the square that is double the original square with diagonal d. But the question is: how is the length of this diagonal expressed?

The Pythagoreans knew of a theorem, which we usually attribute to their leader, Pythagoras, but which is undoubtedly much older. The theorem states that, in an orthogonal triangle (that is, a right triangle), the square of the hypotenuse (the diagonal) is equal to the sum of the squares of the two sides of the right angle. If we apply this theorem to the square we considered above, we obtain:

From this, it is easy to observe that d cannot be a whole number.

If we take a = 1, then d² = 2. Thus, the number d is necessarily larger than 1, because if d was equal to 1, d² would also be equal to 1. However, d must also be smaller than 2, because if d was equal to 2, then d² would be equal to 4. This number, d, therefore, lies strictly between 1 and 2. However, there is no whole number between 1 and 2. Thus, d is not a whole number.

In addition, as we will see further (see Chapter 1), we also prove that d cannot be a fraction or, as we say today, a “rational” number.

Here, we highlight the quantities that the Pythagoreans would, for lack of a better alternative, define negatively. They called these quantities irrational (aloga, in Greek), that is, “without ratio”. The discovery of these incommensurable quantities or numbers would have large philosophical consequences and would require Plato, in particular, to completely rethink his philosophy.

Finally, as mathematics progressed, it was seen that certain numbers are the solutions to algebraic equations but others could never be the solutions to equations of this kind. These numbers, which are not algebraic (such as π or e, for example) would be called “transcendental”. They also brought specific problems with various philosophical consequences.

Greek geometry asked other crucial questions, such as those concerning the doubling of a cube, the trisection of an angle (Chapter 2), or again the squaring of a circle. However, it found itself limited when it came to those constructions that could not be carried out using a scale and compass and which would not be truly resolved until the invention of analytical methods.

The squaring of a circle especially (Chapter 3) (i.e. how to relate the area of a circle and that of a square) would bring with it reflections on the infinite, the differences between a line segment and a portion of a curve, the contradictions linked to the finite and ...

Table of contents

• Cover
• Table of Contents
• Introduction
• PART: 1 The Contribution of Mathematician–Philosophers
• PART: 2 Mathematics Becomes More Powerful
• PART: 3 Significant Advances
• PART: 4 The Advent of Mathematician-Philosophers
• Conclusion
• Bibliography
• Index
• End User License Agreement