Mathematics and Philosophy

By Daniel Parrochia

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).

• Language: English
• Publisher: Wiley
• Release date: May 24, 2018
• ISBN: 9781119528074

Book preview

Introduction

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 around5, 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.

Figure I.1. The YBC 7289 tablet (source: Yale Babylonian Collection)

1 J.-T. Desanti, preface to L. Brunschvicg [BRU 81, p. VI].

2 See the title of the book by Kline [KLI 93].

3 Toward the end of Chapter 7, we will be able to return to this and comment on the results obtained notably from the work of Gregory Chaitin.

4 See, for example, Rashed [RAS 91]. G.-G. Granger has sometimes highlighted the reverse, as is the case with Leibniz, where the philosophical principle of continuity determines different aspects of his mathematics. But this, in his own words, is an exceptional phenomenon [GRA 86].

5 Further on in this book (pp. 95–104), the author lists different important philosophical consequences of the progress of Kolmogorov’s theory of complexity and, notably, the definition of the randomness of a string as algorithmic incompressibility, which resulted in: (1) a new understanding of Gödel’s theorems of incompleteness; (2) an objective conception of physical entropy (Zurek); (3) Chaitin’s Omega number and the assurance of coherence in the theory of measurement; (4) a new understanding of scientific induction, of Bayes’ rule and Occam’s razor; (5) the distinction between random complexity and organized complexity (Bennett). A final epistemologically non-negligible consequence is the famous law propounded by Kreinovich and Longpré, according to which if a mathematical result is potentially useful, then it is not possible for it to have a complex proof. It would seem to result from this that which is complex is potentially useless, a result which many long-winded philosophers would do well to contemplate (see [KRE 00] and [LI 97]).

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 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.

1 Of course, as soon as there were real numbers, it was possible to think of relations between them. At that moment, then, there would also be relations between irrational or even transcendental numbers.

1

Irrational Quantities

In traditional philosophy, what is the fundamental philosophical operation? It is that which consists of constructing a miniature image of the world, as complete as possible, and whose aim is to capture the essential of the real world. The benefit of this approach is patently obvious: simplify to understand better. That is, taking everything together, we reduce so that we can retain more. We are, moreover, pushed to carry out such a task, which proves itself to be highly useful, for obvious reasons:

– there is, among other things, a vital necessity to know the relative importance of each thing, our own situation in the world as well as the place that is ours;

– let us also note that such a project is democratic;

– finally, everyone has the right to know who they are and where they are: the very procedure that allows this, in accordance with its objective, could only have emerged in a context that was conducive to its appearance (Ancient Greece).

In any event, this consists of factorizing all that is perceived into a certain number of classes and then, for each group being considered, to choose one or more distinct representatives1. The dimensional reduction, if it is correct, then becomes heuristic and leads to an undeformed model of the real world. But there are many ways of operating this concentration and, contrary to the old adage, in philosophy, alas, less, is not always more.

What then must be prioritized and preserved from the range of phenomena? From this colorful variety, these often garish shades of existence that were made up, in the time of the Ancient Greeks, by people, places and even the monuments around them – from all this, the earliest philosophers, the disciples of Pythagoras, only wished to retain the number.

For them, the number was an able replacement for any thing as, with things and numbers in the same stratification, it was easy to substitute one for the other. Numbers themselves could be contained within the first four (the Mystic Tetrad) and, consequently, the entire universe can thus be contained within the beginning of the numerical series (1 + 2 + 3 + 4 = 10)2. How economical! Not only is the world condensed into symbols, but all we need is 4 of these to cover the full spectrum. To these thinkers, who were still naive, the alpha and omega in the real world were nothing but numbers and relations (logoi) between numbers. Reason itself, which is nothing but science being exercised, identified with these relations. It also bears the same name (logos). Reason, therefore, is essentially proportion. At the time, reason fell within the limits of the Pythagorean theory of medieties3. And, perhaps the thing that is most difficult for us 21st Century people to understand, it is nothing else.

1.1. The appearance of irrationals or the end of the Pythagorean dream

Naturally, this kind of a perspective, rather too radical, would not be tenable in the long run. A simple mathematical problem, the doubling of a square4, brought up the first irrational, which we have named square root of two ( ) and which, geometrically, can be identified with the diagonal of the square on which is constructed the square that is the double of a square with side one. As we have seen, the Pythagoreans named such quantities a-loga, that is, strictly speaking, without ratio or incommensurables. It is easy to demonstrate why – or, as the Greeks really called it the number whose product with itself gives 2 (as they did not know the expression – is not rational. Any rational must be of the form p/q, an irreducible fraction. But the property of an irreducible fraction is that its numerator and denominator cannot both be even (if they were, we could of course further reduce this by dividing it by 2). Let us thus posit = p/q is irreducible. We then have p² = 2q², which signifies that p² is even, and therefore, p is even. Let us then posit that p = 2n and substitute this value in the equation. We obtain 4n² = 2q², that is q² = 2n², thus q² is even, which signifies that q is even. We thus have a contradiction and is not rational.

1.2. The first philosophical impact

We find many echoes of this discovery in Greek philosophy, especially in Platonic thought. In his weighty tome on Mathematical Philosophy [BRU 93], Léon Brunschvicg included the following observation:

"In Plato’s Dialogues, there is more than one hint that the discovery of irrationals is not alien to the Platonic doctrine of science. In the introduction to Thaetetus, the dialogue that would mark the first degrees of analysis that went from perceptible appearance to truth, Plato recalled the writings of his tutor, Theodore5, who established the irrationality of , , etc. and pursued the search for irrational square roots up to 6. In book VII of The Laws, he deplores, as a crime against the nation, that young Greeks were left ignorant (as he was left ignorant) of the distinction between commensurable quantities themselves and incommensurable quantities7, a distinction that he used as the basis for the ‘humanities’. Above all, the example of Meno must be highlighted: the problem, one of the simplest of those that could arise after the discovery of incommensurability, consists of determining the length of the side of a square that would be double that of another square with a surface of four feet. What is significant is the objective of this example: it was to prove the Reminiscence Theory of Knowledge. The Platonic Socrates introduces a slave who, it was claimed, without any direct learning, and using solely the effect of natural light which revealed itself, could find the veritable solution to the problem8. The first responses of the slave were borrowed from the framework of pure arithmetic: the square with double the area seems to have a side with double the length. But, double the length would be 4 and thus the doubled area would be 16. The side of the square would, thus, be greater than 2 and smaller than 4, that is, 3. But this response, which exhausts the truly numerical imagination, is still inexact: the square with a side of three feet would have an area of 9 feet. Socrates, thus, proposes an exclusively geometric reflection.

"Let the square be ABCD (Figure 1.1)9; we can juxtapose this with three equal squares so as to obtain the quadruple area AEGF. Taking the diagonals BC, CI, IH and HB, we divide into two each of these four areas, equal to a primitive square. The square BCIH is, therefore, double the primitive square; the side whose length would be equal to is the line that the Sophists call the diameter: it is from the diameter, thus, that the doubled area is formed"10.

Figure 1.1. Meno’s square

Plato’s theory of science and, in particular, the idea of reminiscence are thus not anchored in mythology, as we believe only too often (appealing to the Myths is, as always, simply a pedagogical or psychagogic tactic Plato uses to express himself) but instead it is anchored in a rationality that is fundamental to the human mind; in the cognitive abilities of the mind which are expressed here, precisely, in the fundamental movement which, at the time of this crisis or near-crisis of the irrationals11, quite suddenly saw the growth of an extension to the concept of the number.

1.3. Consequences of the discovery of irrationals

The Pythagorean discovery led to several important philosophical consequences.

1.3.1. The end of the eternal return

The end of everything is numbers was not simply the rejection of the Pythagorean hypothesis according to which only whole and rational numbers existed. This idea had quite concrete consequences. It brought about a general modification of the cosmological representation of the world, especially the representation of time. Indeed, as Charles Mugler once wrote, it brought about a veritable cosmic drama. It marked the collapse of the Pythagorean concept of circular time where the revolutions of different heavenly bodies, assumed to be expressed only in whole numbers, would give rise to the calculation of a lowest common multiple (LCM). They thus led to the Pythagorean concept of the Grand Year, at the end of which period it was assumed that the heavenly bodies had returned to their initial position and that life on earth, which depended on them, would recommence completely.12 However, the end of this periodic cosmology and the presence of possible disorder in the celestial world were not the only consequences of the appearance of irrationals.13

1.3.2. Abandoning the golden ratio

Plato had already noted in classical Greek architecture that there were some distortions between the apparent and real proportions in certain monuments, even reproaching architects for having used falsehoods to get people the truth. The problem only worsened, as the Greek aesthetic shifted over time not only to an excess of refinement and mannerism but toward a renouncement of reserve, sobriety and equilibrium in favor of expressing a certain dramatic tension, a certain pathos or hubris of despair as in the famous group of Laocoon. This is a sculpture that presents people in a state of agony, muscles taut and bulging eyes with despair in their eyes, as they are defeated by serpents. With this sculpture, which dates from after the 2nd Century B.C., the Apollonian order was overthrown by much more troubling and tormented representations, which would, one day in the future, attract the German Romantics14. This is nothing but another consequence of the collapse of the existing order brought about by the undeniable existence of the irrational in mathematics.

1.3.3. The problem of disorder in medicine, morals and politics

As we know, Greek wisdom also recommended nothing in excess. Stobaeus, before Plato, already judged that there must be proportion in the soul and in the city: Once the rational count is found, revolts will die down and amity increase (Flor. IV, I, 139). Alcmaeon, although he certainly lacked the means of experimentally taking measurements, tried to prove the equality of forces in the body [VOI 06, pp. 11–78]. According to a fragment gathered by Diels-Kranz, "Alcmaeon said that what maintains health is the balancing of forces (tèn isônomian tôn dunaméôn), humid, dry, cold, heat, bitterness, sweet and others, the domination of any one of them (en autois monarchian) causing disease; this was because the domination of a single is corrupting… health is the combination of qualities in the correct proportion (tèn summetron tôn oiôn krasin)"15.

With the appearance of the irrationals, the end of Greek life was, in the long run, programmed. Nothing would be in proportion anymore: not in the cosmos, nor in the human soul, nor in the city. The potential presence of a destabilizing element, introducing the incommensurable (irrational movement, unrestrained passion, tyranny, etc.) would, every time, threaten to bring about a rapid downfall.

It was, therefore, useful to put up the defenses and fight back – starting with the field of mathematics itself.

1.4. Possible solutions

How does one rid oneself of these irrationals? Squaring them, that is, turning aloga into dunamei monon rêta, is an easy solution. However, this will, obviously, change the value of the numbers. To remain faithful to the data given in the problem, we will try to find approximation formulas that would make it possible to turn the irrationals into rationals.

One of the procedures, while it may not have been entirely known to the Pythagoreans, was, nonetheless, anticipated by them. This was the procedure of continuous fractions, which seems to have been explicitly introduced by the Hindu mathematician Aryabhata (550–476 AD). This is written (using the modern forms) as:

Today, we can easily obtain this formula in the following manner. Because 1 < we first posit, as the first approximation:

[1.1]

From this, we then find the value for a. Hence:

[1.2]

However, since:

upon replacing in expression [1.2] by its value, we also immediately have:

[1.3]

By then substituting this value for a in expression [1.1] and then in the successive expressions, wherever a appears, we obtain the desired formula.

It appears that we cannot find an explicit trace for continuous fractions in Greek mathematics earlier than the work of Aristarchus (3rd Century B.C.) and Heron (1st Century A.D.). However, Paul-Henri Michel was able to suggest that the procedures to dimidiate unity and other approximations that Thomas L. Heath was able to report16 contributed to anticipating them.

Jules Vuillemin, following in their path, noted more recently that the Pythagoreans made use of infinite sets in their polygonal number tables and in the definitions of progressions [VUI 01, p. 11]. Reflecting on the use of these procedures to demonstrate the irrationality of , especially in the algorithms called Theon’s algorithms and alternate division, he observed that we could thus easily deduce the laws of continuous fractions from these [VUI 01, p. 71]. The Pythagorean use of triangular number tables also meant that it was not necessary to explicitly know these algorithms, but to know them, only through certain properties of their approximation [VUI 01, p. 71].

Despite their lack of resolution, according to Vuillemin himself, these procedures seem to have served as models for the Platonic method of division. In his own words, again, "while logical rigor is lacking in this initial recourse to finite sets, and while these difficulties inherent to continuous fractions also affect their rudiments, let us remember that the chief obstacle Greek mathematics came up against is the idea of the real number and we will see Theodore conceiving of roots of natural, non-squared whole numbers as the limits of the infinite series of rational approximationx [VUI 01, p. 106].

1.5. A famous example: the golden number

Among the ratios that the Pythagoreans loved, one could pass for a clever compromise: this was a ratio and, at the same time, corresponded to an irrational quantity. This ratio is defined in the following manner. We posit:

But this is equal to:

[1.4]

We then posit:

And equation [1.4] becomes:

One of these two solutions (the positive solution) to this new equation (which would, moreover, find many applications in the field of aesthetics, notably architecture) has been the subject of reams of writing over history17. The solutions are classically obtained as follows. The discriminant of the equation is:

which gives, as the roots,

kʹ is conventionally called the golden number or the golden section. We ordinarily designate this by the letter ϕ.

We then observe that kʹ.kʺ = –1, and that kʹ + kʺ = 1. From this, we then can write:

Hence, the rational approximation of the golden number:

By developing the successive approximations of ϕ, we then have the set of ratios formed by the numbers belonging to a famous mathematical series18:

Knowing that the above ratios may be denoted in a generic manner by , we can demonstrate that:

1.6. Plato and the dichotomic processes

The processed called the dimidiation of unity, a characteristic of the golden number, finds an obvious parallel in language with the dichotomic processes, which Plato would use increasingly often in his dialogues. In fact, Plato compared ideas to numbers and a wrong calculation to an identification error (Thaetetus, 199c). And like the Pythagoreans, who saw a simple correspondence between numbers and things, Plato believed for a long time that it was possible to establish a simple correspondence between the intelligible and the tangible, ideas and their referents in the world. This belief, however, threw up many difficulties, as can be seen in the conversation between the young Socrates and Parmenides in the eponymous dialogue.

The Parmenides, and then the Sophist, acknowledged this failure and the later dialogues raised other objections. Thus, the Philebus showed that certain ideas, such as those of pleasure, which had multiple variables and that could only be attained through excess or failure, could be difficult to identify with intangible, full or whole realities. Irrational numbers are like this: they cannot be used in clearly defined relations and are not commensurable.

To approach them, we must have a process analogous to the dimidiation of unity. The processes that play this kind of a role are dichotomic processes (or procedures for