Philosophy is not descended from heaven. It does not follow a completely autonomous line of thought or a mode of speculation that is unknown to this world. Experience has shown us that the problems, concepts and theories of philosophy are born out of a certain economic and political context, in close conjunction with sources of knowledge that fall within positive learning and practices. It is within these sites that philosophy normally discovers the inductive elements for its thinking. This is where, as they say, it finds life. A little historical context, therefore, often makes it possible to reconstitute these elements that may sometimes leap off the surface of a text but always inform its internal working. All we have to do is identify them. Thus, metaphysics, from Plato to Husserl and beyond, has largely benefited from advances made in an essential field of knowledge: mathematics. Any progress and revolution in this discipline has always provided philosophy with not only schools of thought, but also tools and instruments of thinking.

This is why we will study here the link between philosophy and the discipline of mathematics, which is today an immense reservoir of extremely refined structures with multiple interconnections. We will examine the vicissitudes of this relationship through history. But the central question will be that of the knowledge that today can be drawn from this discipline, which has lately become so powerful and complex that it often and in large part soars out of reach of the knowledge and understanding of the philosopher. How can contemporary mathematics serve today’s philosophy? This is the real question that this book explores, being neither entirely an history of philosophy, nor an history of the sciences, and even less so that of epistemology.

We will not study science, its methods and laws, its evaluation or its status in the field of knowledge. We will simply ask how this science can still be of use to philosophers today in building a new vision of the world, and what this might be. A reader who is a philosopher will, therefore, certainly be asked to invert their thinking and reject their usual methods. Rather than placing scientific knowledge entre parentheses and embarking on a quest for a hypothetical other knowledge, assumed to be more remarkable, more native or more radical (the method called the “phenomenological method”), we prefer suspending judgment, using the epoché (reduction) method for phenomenology itself and sticking to the only effective knowledge that truly makes up reason (or, at any rate, a considerable part of reason): mathematical knowledge. This knowledge contains within itself the most remarkable developments and transformations not only of thought, but also of the world. This knowledge, by itself, has the capacity of constructing, in a methodical and reflective manner, the basic conceptual architecture needed to create worldviews. It would seem that philosophers have long forgotten this elementary humility that consists of beginning only with which is proven, instead of developing, through a blind adherence to empiricism, theories and dogma that lasted only a season, failing the test of time, their weaknesses revealed over the course of history.

In doing this, we follow in the footsteps of thinkers who are more or less forgotten today, but who kept repeating exactly what we say here. Gaston Milhaud, for example, had already noted this remarkable influence. In the opening lesson of a course taught at Montpellier in 1908–1909, which was then published in the Revue Philosophique and reprinted in one of his books [MIL 11, pp. 21–22], we find the following text:

“My intention is to bind myself to certain essential characteristics of mathematical thought and, above all, to study the repercussions it has had on the concepts and doctrines of philosophers and even on the most general tendencies of the human mind.

How can we doubt that these repercussions have been significant when history shows us mathematical speculations and philosophical reflections often united in the same mind; when so often, from the Pythagoreans to thinkers such as Descartes, Leibniz, Kant and Renouvier (to speak only of the dead), some fundamental doctrines, at least, have been based on the idea of mathematics; when on all sides and in all times we see the seeds of not only critical views, but even systems that weigh in on the most difficult and obscure metaphysical problems and which reveal especially, through the justifications offered by the authors, a sort of vertigo born out of the manipulation of or just coming into contact with the speculations of geometricians? The excitation in a thinker’s mind, far from being an accident in the history of ideas, appears to us as a continuous and almost universal fact”.

A few years later, in 1912, Léon Brunschvicg published Les étapes de la philosophie mathématique (Stages in Mathematical Philosophy), a book in which, as Jean-Toussaint Desanti noted in his preface to the 1981 reprint, it clearly appears that mathematics informed philosophy¹. In this book, hailed by Borel as “one of the most powerful attempts by any philosopher to assimilate a discipline as vast as mathematical science”, we can already see, as Desanti recalls, that “the slow emergence of forms of mathematical intelligibility provided the reader with a grid through which to interpret the history of different philosophies”. [BRU 81, p. VII]. The fact remains, of course, that these two effects were secondary to Brunschvicg’s chief project: to give an account of mathematical discourse itself in its operational kernels, where the forms of construction of intelligible objects take place and where the activity of judgment (which he found so important) chiefly manifests itself, along with the dynamism inherent to the human intellect.

Admittedly, today mathematics is no longer accepted as truth in itself. Shaken to its foundations and now seen as being multivarious, if not uncertain², it has seen its relevance diminish further of late. Knowing that 95% of truths are not demonstrable within our current systems and that the more complex a formula the more random it is³, we may well wonder as to the philosophical interest of the discipline. And so, Brunschvig’s concluding remark, according to which, “the free and fertile work of thought dates back to the time when mathematics gave man the true norm for the truth” [BRU 81, p. 577], may well make us smile. His Spinozian inspiration seems quite passé now and the lazy philosopher will delight in stepping into the breach.

Nonetheless, not even recent masters – Jules Vuillemin, Gilles-Gaston Granger, Roshdi Rashed – who dedicated a large part of their work to mathematical thought and its philosophical consequences, have gone down this path. If they are often close, it is in the sense that their work generally looks at measuring the influence, or even truly the impact, of mathematics on philosophy⁴. We will thus content ourselves with modestly following in their path. This book will thus undoubtedly follow a counter current. However, it joins certain observations made by contemporary mathematicians in the wake of Bachelard. “The truth is that science enriches and renews philosophy more than the other way around”⁵, as Jean-Paul Delahaye wrote in the early 2000s [DEL 00, p. 95]. In addition, we do not seek to lay out a pointless culture. We only aim to communicate the essential. That is, in the teacher’s experience, what is most easily lost or forgotten. The majority of this book will thus redemonstrate that philosophical reason, while it has undoubtedly been subject to multiple inflexions over its history, can only be constructed by looking at the corresponding advances made in science, and especially the discipline that contains the major victories of the sciences: mathematics. From the Pythagoreans to the post-modern philosophers, nothing of any importance has ever been conceived of without this near-constant reference.

Implementing philosophy today assumes an awareness of this creative trajectory. Once this is done, there are, of course, still some evident problems: if we believe in our schema, then should today’s philosophy follow the same inspiration as the philosophies of earlier ages? Is it possible for today’s philosophy to escape the biases that burden ancient systematic thinkers without denying its own nature? What definitive form must philosophy take today? These are but a few of the many questions that surround this reflection, which is, in our view, constantly inspired by mathematics. History has shown us that the true philosophers have not always been those who stirred up radical ideas, political criticisms or those short-sighted moralists who, today, many consider great philosophers. This is chiefly due to their lack of knowledge of science as well as the echo-chamber created by the media around the most insignificant things, which pushes the media itself to discuss nothing but this phenomenon. However, the existence of real facts and strong movements, generally ignored by the media buzz, leads us to think that things of true importance are happening elsewhere. Philosophy, with all due respect to Voltaire, used to be something quite different. And, for those who are serious, this remains an undertaking that goes well beyond what we find today in journals and magazines.

A note on the notations used here: When we speak of the mathematics of antiquity, the Middle Ages or the Classical Age – in brief, the mathematics of the past! – we will use present-day notations to ensure clarity. However, it must be understood that the symbols that we will use to designate the usual arithmetic operations have only existed in their current usage for about three centuries [BRU 00, p. 57]. It was at the beginning of the 17th Century, for example, that the “plus” sign, (+) (a deformation of the “and” sign (&)) and the “minus” sign (–) began to be widely used. These symbols are likely to have appeared in Italy in 1480; however, at that time it was more common to write “piu” and “meno”, with “piu” often being shortened to “pp”. In the 16th Century (1545, to be exact), a certain Michael Stiffel (1487–1567) denoted multiplication by a capital M. Then, in 1591, the algebraist François Viète (1540–1603), a specialist in codes who used to transcribe Henri IV’s secret messages, replaced this sign by “in”. The present-day use of the cross (×) was only introduced in 1632, by William Oughtred (1574–1660), a clergyman with a passion for mathematics. The notation for the period (.) owes itself to Leibniz (1646–1716), who used it for the first time in 1698. He also generalized the use of the “equal to” sign (=). This was used by Robert Recorde (1510–1558) in 1557 but was later often written as the Latin word (œqualitur) or, as used by Descartes and many of his contemporaries, was abridged to a backward “alpha”. While the notation for the square root appeared on Babylonian tablets dating back to 1800 or 1600 B.C. (see Figure I.1), its representation in the form we know today, dates back no earlier than the 17th Century. Its use in earlier mathematics is, thus, only a simplification and has no historical value.

Finally, this book is not a history of mathematics, but rather a study of the impact of mathematical ideas on representations in Western Philosophy over time, with the aim of highlighting teachings that we can use today.

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 the possibility of overcoming these contradictions in the infinite. All these speculations, as we will see, sparked off the reflection of Nicholas of Cusa.

The rational approximations of π – notably those given by Archimedes – would mobilize trigonometric functions, which were also used in astronomy to calculate certain unknown distances using known distances. And the birth of financial mathematics, linked to the growth of capital, would play an important role in the discovery of the logarithmic function and its inverse, the exponential function.

The progressive extension of calculations would then lead mathematicians to create new numbers. For example, from the Middle Ages onwards we have seen that a second-degree equation of the type ax² + bx + c = 0 only admits real numbers as the solutions if the quantity b² – 4ac (the discriminant) is positive or nul. But what happens when b² – 4ac is negative? For a long time, it was stated that the equation would have no solution. But then a subterfuge was invented that would make it possible to find non-real solutions to this equation. A new set of numbers was created for this purpose – they were first called “imaginary” numbers and later “complex” numbers. These numbers are solutions to second-degree equations with a negative discriminant.

Euler formulated an early law for the unification of mathematics by positing an equation that related the three fundamental mathematical constants: π, e and i (this last being the fundamental symbol of the imaginary numbers). These numbers, which would later find application in the representation of periodic functions associated with physical flux, would be the origin of a new representation of the world, where energy seemed to be able to replace matter. Bergsonian philosophy, as we will see, resulted from such an error.

Mathematicians have, over time, also invented many other types of numbers: for example, ideal numbers (Kummer) or again the p-adic numbers (Hensel). We will not discuss them here as they are not very well known to non-mathematicians and thus to the best of our knowledge have not yet inspired any philosophy.