Chapter One

The Greek Conceptions of Unity

Many of the Greek thinkers were inspired by monism. They aspired to explain phenomena on the basis of a single principle. Our senses may well suggest a world made of an infinite variety of things and phenomena, but for the purpose of trying to understand it, we have no choice but to introduce some semblance of order, in other words, some underlying unity. Natural elements, these philosophers believed, are not to be understood as isolated pieces disconnected from the rest. Instead, they must be amenable to integration into a broader explicative framework that gives them both meaning and functionality. Hence the conviction that, hidden deep within reality, there must exist some principle or fundamental substance from which all things can be derived. Out of this came a number of doctrines promoting one primordial element or another, the nature of which was not even necessarily inert or material.

The justification for starting this analysis by reviewing a few doctrines originating with the Greek thinkers is that they are the ones who planted in physics the seed of its eventual power. Yet, the Greeks do not have the corner on monism. For instance, the school of energeticism, promoted by Wilhelm Ostwald (1853-1932) and others, was a bona fide monistic metaphysics based on rejecting the concept of atom. The doctrine was rooted in a belief in the absolute primacy of energy as the agent governing the physical world in that it made it possible to understand, organize, and unify all observable phenomena. "Matter is an invention," wrote Ostwald, "and an imperfect one at that, which we forged to describe what is permanent in all vicissitudes. The effective reality, the only one that has an effect on us, is energy.... Nature as a whole appears to us in the form of spatially and temporally varying energy, distributed in space and time. We become aware of that energy only to the extent that it is transmitted to our bodies, and particularly to the sensory organs whose function is to detect it." According to him, the triumphant expression of such a message was embodied in the principles of thermodynamics, especially the second law, which states that the entropy of a system can only increase.

The Greeks looked at nature from a certain distance, although with profound admiration. Their intent was not to found a science in the mold of modern physics. The idea of dominating matter in order to transform it was alien to them. They perceived nature as the stuff of life, generating and confronting beings, evolving and changing endlessly, exploding into myriad forms, and unanswerable to logic and identification. These early thinkers were primarily concerned with observing and understanding the phenomenon of growth, ubiquitous, seemingly universal, not only in the vegetal and animal worlds, but in the realm of minerals as well, which in those days were believed to be subterreanean fruits growing in the belly of the earth and deposited on its surface through some continuous and unobtrusive process of percolation. Hence the etymology of the Greek word phusis , which is derived from the verb phuein , meaning "to grow." The Greeks were simply looking for the primeval element of this natural growth, the fertile seed serving as the nutritive principle of life and of the entire cosmos. How was the world around us created, and how does it go on existing?

Water, the Primordial Stuff

Thales of Miletus (624-548 b.c.) held that water was the primordial element. "Water is the material cause of all things," he affirmed. It nourishes plants, quenches man's thirst, and is home to the fish that feed him. By falling from the sky and collecting at the bottom of wells, water irrigates fields and makes everything grow. Of all the known substances, water is the one that can take on the most varied forms: ice, snow, steam, cloud, liquid, not to mention rocks, which the Greeks believed to be made of frozen water (the word "crystal" comes from the Greek krustallos , which means "ice"). In river deltas, it appears as though it turns to earth. Elsewhere, it seems to spring forth from it. Water flows, disperses, soaks, transforms, infiltrates, fertilizes the soil, and carries ships. It is the liquid link that unites the whole with itself, saving it from being nothing more than a collection of things alien to one another. Besides, are air and fire not merely exhalations of water? It is universally recognized today that water is indeed an essential condition for life. Madison Avenue understands this fact quite well when it routinely associates water (preferably the bottled kind) with vitality.

Thales believed that the earth floats on water, having precipitated out on its surface much like silt precipitates in estuaries. With a little imagination, one can view this concept as the distant precursor of the theory of continental drift proposed at the beginning of our century by Alfred Wegener (1880-1930) who, upon noticing that the coasts of Africa and Brazil fit like pieces of a jigsaw puzzle, speculated that the two continents must have slowly drifted apart by floating on a liquid support. Great ideas never completely die out: Water is an idea that refuses to fade away. Nietzsche acknowledged in his Philosophy in the Tragic Age of the Greeks the momentous impact of the "gigantic generalization" carried out by the greatest philosopher to come out of Ionia: "Thales saw the unity of Being, and when he set out to describe it, he spoke of water."

Indeed, for the first time ever, a single and all-encompassing explanation had been put forth to account for everything. This economical approach was a departure from a long-standing tradition that relied on a multiplicity of arbitrary causes and ad hoc explanations. This in no way means that prior explanations were all incoherent. Myths had their own logic and fit together with a certain plausibility. In his Philosophy of Symbolic Forms , Ernst Cassirer showed how even a universe full of myths is invariably characterized by a chain of interlocking steps. Such a chain may be "independent of the laws of empirical thought," but it conforms to laws nonetheless and reveals "a kind of original and autonomous structure," that is to say, a "theory," provided that the meaning of the word is generalized. For instance, Hesiod recounted in his Theogony how an initially chaotic world was gradually unified through the sovereign power of Zeus: "At the very beginning Chaos and the Gaping Abyss were born, but next came Gaia the Earth with her broad bosom...." It all seems to mesh together seamlessly, even with a certain elegance. But by invoking so many different causes and divinities, myths ended up creating a narrative more than an explanation. They provided a description rather than a system. The interconnection, orderly as it may be, of too many causes ends up looking like randomness.

Thales expressed for the very first time three ideas that were to prove seminal. First, he raised the question of the material cause of all things. Second, he insisted that the answer not rely on mythology and folklore or on gods that were evidently far too anthropomorphic. Lastly, he postulated that in the end it must be possible to reduce everything to a fundamental substance of a more profound nature than any other. Thales and his followers legitimized the will to frame what amounted to scattered bits of knowledge into a rational and coherent system freed from bondage to the occult. Myths and fables were gradually discredited and came to be replaced by the reasoning of philosophers. Neither the multiple creations of Gaia nor the authoritarian interventions of Zeus could continue to be satisfactory explanations. With his primordial water, Thales had ushered in the era of the rational mind.

Air, Fire, and Earth

In the sixth century B.C., Anaximenes (ca. 550-480 B.C.) propounded yet another cosmogony founded on the idea that the principle of the universe is air. Before exhaling our last breath, do we not indeed spend our entire lives drawing in this life-sustaining stuff that is all around us, according to a rhythm that orchestrates the interplay between the external world and ourselves? According to Anaximenes, the whole world is suffused with a vital flow that brings life to its various forms and coordinates its many parts. Our soul is a form of air that sustains us. The Earth, thin and delicate, is carried like an autumn leaf by the currents of an invisible atmosphere. The wind, created by circulating air, fills the sails of ships. By condensing, it forms clouds that can themselves condense further to bring rain and even create stones. By rarefying, air turns to fire. And what about snow, one might ask? "It is a breath of air imprisoned in humidity." At the cost of a few contortions, monism can come up with an answer for everything.

Anaximenes's greatest merit is to have shown that once a particular primordial element is chosen, it is possible to deduce all other known components of the world. It was not all that easy, even with all sorts of metaphors, to reduce all known phenomena to arguments involving nothing more than air. In Anaximenes's defense, it took no fewer than twenty-four centuries to finally establish that air--as well as water--is actually a composite mixture, which disqualifies it from the title of primordial element.

At about the same time, Heraclitus of Ephesus (ca. 550-480 B.C.) picked fire as the element explaining all phenomena in the universe: "All things are exchanged against fire, and fire against all things." Fire lights the world and rules it. By condensing, it generates water; and by condensing even further, it becomes earth. Fire is the principle of heat and life; it is the flame that feeds every passion, and which all lovers pledge to each other. For Heraclitus, it was all quite simple: "Lightning governs the universe," which itself is nothing but an eternal fire that will never be completely extinguished. Destined to a worldwide blaze and final conflagration, it will consume itself, not to disappear forever, but to rediscover in this ultimate inferno the principle that will enable it to be reborn from its own ashes.

The world, conceived of as a cyclical brazier, would evolve according to a recurring pattern, fire serving as the unifying link between successive phases. Some will undoubtedly detect in such a picture the seed of the doctrine of eternal rebirth, which would prove so appealing to the Stoics, and which Nietzsche would later describe as the only thought with a chance of defeating nihilism. Others might even make a parallel between this concept and a particular version of the big bang model, which holds that the universe is subject to a series of expansions and contractions, in which the transition between successive cycles is characterized by enormously high densities and temperatures.

The concept of becoming occupies a central place in Heraclitus's philosophy, to the point of implying a categorical rejection of the notion of being, so dear to Parmenides. Fire becomes the paradigm of things that evolve. No form of permanence can ever fix it. If fire fascinates us so, it is because not a single one of its flames can ever be captured. "I cannot imagine anything more fleeting and alive than fire ...," wrote Jean de la Fontaine (1621-1695) in one of his celebrated fables.

Questions of unity invariably bring up the issue of completedness. How is it possible to reduce every possible phenomenon to a matter of mere flames? To resolve this dilemma--confronted by any monistic philosophy--Heraclitus came up with an ingenious idea. He asserted that a "struggle between opposites" presides over nature. The "electricity" between antipodes supposedly constitutes the true unity of the world, which would be at once one and multiple. As the driving force of becoming, it is to be viewed as principle and law. "Sea water, to take one example, is both very pure and very impure; it is a nutrient for fish; but for men, it is undrinkable and harmful." Nature, asserted Heraclitus, manages to sublimate antagonism into harmony, much like music created by the lyre is rooted in the tension between strings and wood. "Opposites are brought into consonance and a beautiful harmony is engendered by what is contrary." Heraclitus repeatedly emphasized that all things originate in conflict and amount to accommodations between contrasts; precarious equilibria between opposing forces, manifesting themselves in endless change. "We must realize that discord is common to all things, that justice results from struggle and that all things are born and disappear through strife."

One could argue that many aspects of modern physics indeed appear to conform to Heraclitus's doctrine. It often seems that mutual opposites are fated to be the source of conceptual developments, as if no progress were conceivable without an infusion of dichotomy. The concept of quantum complementarity, for instance (to which we will return later), is directly inspired by such a notion. It asserts that waves and particles are two descriptions--logically incompatible, but inextricably intertwined just the same--of quantum reality. Pushing the metaphor a bit further, other examples of interplay between opposites can be found in the world in the infinitesimally small as well. A photon, for instance, can be created by the annihilation of an electron and a positron, and in the reverse process, a photon can disappear to generate an electron-positron pair. The symmetry between matter and antimatter could also be construed as reflecting the unity of opposites. More generally, any discourse exalting the virtues of paradoxes is never too far removed from Heraclitus's philosophy.

Yet, the living world beats physics by miles when it comes to demonstrating the power of opposites. In one of his books, Arthur Schopenhauer describes the strange case of an unusual animal called the Australian ant-bulldog: "When you cut it in half, a battle ensues between head and tail; the head clasps the tail, which frantically uses its sting to try to evade the head's bite. The fight can last half an hour, until death follows, unless other ants carry away the two pieces." Inasmuch as the story is true, the ant-bulldog, belligerent to its very core, illustrates the ferocious traits the dialectic of the One and the multiple can assume when it leaves the province of pure philosophy.

Werner Heisenberg (1901-1976), one of the founding fathers of quantum physics, observed in Physics and Philosophy that in some regards modern physics has much in common with Heraclitus's doctrine: "If we replace the word `fire' by the word `energy,' we can almost repeat his statements word for word from our modern point of view. Energy is in fact the substance from which all elementary particles, all atoms and therefore all things are made, and energy is that which moves.... Energy can be changed into motion, into heat, into light, and into tension. Energy may be called the fundamental cause for all change in the world."

Plato, on the other hand, took a dim view of those who, like Heraclitus, engaged in game playing with opposites, with no apparent justification. Such a proclivity was, in his opinion, the mark of superficial thinkers and novices in the art of argumentation. The combination of antipodes tells us nothing, he argued; it forms a net so loose as to let virtually everything through. That is why Plato consistently and relentlessly ridiculed anyone who claimed that the One is multiple, and the multiple One, or that the Being is the non-being, and vice versa. "But to show that `the same' is other in some way no matter what, and `the other' is the same, and the big small, and the similar dissimilar, and in this way to take pleasure in always putting forward the contraries in one's speeches, this is not a simply true examination, and it shows as well that it is the fresh offspring of someone who just now is getting his hands on the things which are." That did not prevent Plato from considering dialectics the most important part of any philosophical endeavor!

In summary, water, air, and fire were in turn viewed by the early Greek thinkers as playing a primordial role in the organization and evolution of the world. Empedocles of Agrigentum (490-435 B.C.) brought some modicum of agreement among these various doctrines before he threw himself, according to legend, into the crater of Mount Etna to prove his reputation as a god. He maintained that all of our earthly world could be explained on the basis of four elements of equal importance--water, air, fire, and earth (which Xenophanes had in the meantime proposed as the primordial stuff)--and two principles that united or dissociated them--love and hatred. Introducing several primordial materials, no matter how few, offered much greater flexibility because of the possibility of their various combinations.

The preeminence of these four elements was to permeate the cosmologies of the Middle Ages and the Renaissance. For example, the affinities between natural elements and corporeal humors made it natural to regard man as part of the world and, at the same time, the world as a human organism. Moreover, it is not too far-fetched to consider this quartet of primeval materials, made up of elements endowed with qualities that enable them to interact, as presaging our modern views on the structure of matter. Admittedly, the fundamental constituents recognized by modern science bear little resemblance to water, air, earth, or fire. Instead, we now have six quarks and six leptons, grouped in three families with similar structures. And, perhaps sadly for the poetry of things, it is no longer love and hatred that cause them to interact, but four fundamental forces.

The Concept of Fundamental Substance

Early on, the Athenian philosopher Anaxagoras (610-547 B.C.) had proposed the intellect as principle and organizing force of the universe. Conceived of as a subtle substance, it would render all things liable to a mechanistic explanation. Anaxagoras distinguished two kinds of beings. First, the elements proper, which are actually mixtures because "in everything is a part of all other things," and, second, our reasoning (the so-called nous ), which is the architect of the world. Nous stands by itself, pure and unadulterated. Faced with the chaos of the elements, it restored order to it. With his doctrine of the organizing nous , Anaxagoras was to exert a decisive influence, notably on Socrates.

During the same period, Anaximander (610-546 B.C.), a pupil of Thales's, rejected water, or any other known substance for that matter, as primordial stuff. He could not accept that fire could come from water, when every ordinary observation shows these two elements to be rather incompatible. Since no element is privileged, it is impossible to consider any one of them more suitable than any other as substratum. In order to get around this difficulty, Anaximander taught that the primordial stuff was of a different essence. It had to be infinite, eternal, and ageless, so that it may embrace the world in its entirety. Only something far richer and more all-encompassing than anything else could possibly guarantee the equilibrium and harmony of the universe. In the end, this primordial substance was just an abstraction, the nature of which was more ethereal than any material substance, and the existence of which carried within itself the potentiality for the existence of all the others. Anaximander gave it the name apeiron , which means "boundless." He described it in terms of adjectives traditionally reserved for god-like entities, such as uncreated, immortal, all-encompassing, and so forth. It seems that the One, with a capital O, often tends to bear an uncanny resemblance to God.

That the world can be structured on an abstract principle is an idea that science would take note of and would ultimately use to great benefit. It constitutes a premonition of the fruitful role to be played by the notion of the infinite in developing the science of mathematics. It also heralds the concept of a universe encircling totality, the introduction of which, in the seventeenth century, would essentially launch physics on its modern course and ensure from the outset its unifying potential.

Anaximander explained how the apeiron contains in its midst the entire concrete future of things. It is the effective cause of all that is born, matures, and dies off, the ultimate origin of all individuals, who may segregate themselves from it, but to which they all end up returning at the time of their final dissolution. The apeiron transforms itself to give us all the substances we are familiar with. In the only extant portion of his work entitled On Nature , Anaximander wrote: "The apeiron is the principle of all things that exist.... That from which all things derive their existence is also what they return to upon their destruction, in accordance with the order of necessity. Things mutually justify their own existence and correct their injustices according to the flow of time." Evidently, Anaximander thought of existence as something of a loss, a renunciation of a primitive source. Nietzsche, perhaps Anaximander's most enthusiastic exegete, would later declare that "all becoming is an emancipation guilty of forsaking the eternal Being, an iniquity that must be atoned for with death." Upon their birth, things cast themselves off the original unity in order to fulfill their own destiny. But in doing so, they commit an impious act for which, in the interest of justice, the ultimate punishment must be exacted. To be born becomes tantamount to dying.

For Anaximander, the struggle never ends between warm and cold, fire and water, moist and dry, as they take turns predominating. One form disintegrates into another but never remains static for long. Any victory of one over another is but transitory. All the parts keep changing, while only the whole remains immutable (centuries later, by establishing that energy is globally conserved even though it can assume multiple forms, physics will adopt a very similar view). Anaximander thus places being and becoming on opposite sides of the equation. The primordial stuff, infinite and ageless, forms an undifferentiated being that degenerates into various forms destined to struggle endlessly with one another. Multiplicity results from falling into individuality, and the very process of becoming is to be considered a degradation of being.

The Negation of the Multiple and of Motion

The problem of the connections between the One and the multiple brings us inescapably to Parmenides (544-460 B.C.). He declared quite simply: "Being is, non-being is not." Such a statement implicitly rejects the multiple and becoming, both of which Parmenides equated with non-being. Since they did not exist, any speculation about them could only lead to contradictions, hence to errors: "What does not exist cannot be known or even described; for what can be thought of and what can exist are one and the same." Only the One, necessary, eternal, and unengendered, exists, and there is no such thing as becoming or disappearance. By the same logic, Parmenides also denied the existence of void, which he assimilated with non-being. He further rejected change and even movement as pure illusions, which would earn him the wrath of Aristotle. In Parmenides' mind, any change contravenes the innate propensity of the mind to accept identity and permanence.

The lesson was not lost on physics, even in its most modern form. Does the perenniality of its laws, supposedly immutable and invariant, not imply the eradication of time? The moment we deal with processes that evidently have a past history and have evolved, it is only to try to discern substance and forms, rules and laws that are themselves independent of time. The very goal of physics is to understand what changes in terms of what is permanent. It accomplishes this by establishing laws that have been set free of the dictates of time, even though these same laws may well describe time-dependent phenomena.

Admittedly, physics does not really have much of a choice. How could anyone ever construct a theory starting from fleeting concepts? What would physical laws look like if they were based on notions applicable one moment and not the next? How could unity be preserved if concepts were constantly changing? Science rests implicitly on the postulate that the relations implied by laws are steady and constant. If so, what, if anything, can we know about how laws might evolve with time? The question happens to be a crucial one in cosmology, which deals with a universe in expansion. It seems reasonable to follow Henri Poincaré's lead in decreeing that "laws not be functions of time, even if facts were to subsequently prove us wrong and force us to broaden these same laws, and possibly even amend them." From a pragmatic standpoint, though, the challenge is to find the most economical strategy, which is probably to assume that what changes with time is not the laws themselves but the "universal" constants that enter into them.

Atomism

Atomism developed in parallel with monistic doctrines. The theory was born on the shores of the Mediterranean. It aimed to answer certain questions about the beginning and end of the cosmos, about the unity and diversity of material objects, and about permanence and change. Gaston Bachelard (1884-1962) would later call it a "doctrine of diminutive objects," or a "metaphysics of dust." The theory was originally developed by Leucippus (fifth century B.C.), a contemporary of Zeno of Elea, and later amplified by Democritus of Abdera (ca. 460-370 B.C.), a contemporary of Socrates. It explains the world by resorting to atoms, entities without sense qualities, which the intellect alone is capable of comprehending, and to void. These tiny units of matter are, in a sense, being's last refuge from reduction to its primordial constituents. They are undissociable ( a-tomos ), eternal, filled, solid, and unlimited in numbers.

In this picture, the qualities we might today refer to as secondary (such as color, odor, flavor) have but a subjective existence. Only the geometrical properties of atoms (size, shape, position) are real. When in rapid motion, atoms undergo mutual collisions that result in the formation of aggregates distinguishable by the shape, order, and position of the constitutive atoms, rather like words are composed of letters. According to Democritus, who wanted "to feel in the world as if he were in a brightly lit room" (to use Nietzsche's phrase), the entire world--including the heavens, the earth, animals, and even the soul--was made of these aggregates and nothing else. Final causes were specifically excluded. In the process, Democritus relegated the gods to the cloakroom, since no intervention on their part was required to explain the universe. The fear they once inspired in us being groundless, all our superstitions could be safely discarded.

It is no doubt unnecessary to underscore the success enjoyed by this doctrine, at least in its modern incarnation. Yet, it was not until the early part of the twentieth century that the atom itself became the object of direct studies in physics. Its entrance onto the stage may have been long delayed (it remained dormant for twenty-five centuries!), but it came in with a bang. It triggered nothing less than the quantum revolution. It quickly became apparent that the atom was in fact quite a complex object, devoid of almost every incipient quality Democritus and his followers had ascribed to it. Between antiquity, when it was merely an abstract idea, and the modern era, when it became an object of scientific study, the atom has inspired extraordinary intellectual jousts spanning two millennia. The enduring attention it has commanded among the best minds seems to vindicate Bergson's view that human intelligence, "shackled as it is by geometry," cannot break free of the issue of the divisibility of matter.

Indeed, most great thinkers--be they theologians, philosophers, or scientists--have taken an active part in this great debate, as if atomism had been a powerful magnet for human thought: Epicurus (341-270 B.C.) invented the concept of clinamen , "the tiny swerving of the atoms at no fixed place and no fixed point of time"; Aristotle rejected atomism outright on the grounds that void itself could not exist; Descartes did likewise; Gassendi resurrected the idea of elementary entity; Giordano Bruno added to it spiritualistic undertones; Leibniz initially believed in atoms and void because "that is the most satisfactory picture for our imagination," but he later realized that it is impossible "to find the principles of a genuine unity solely in matter or in what remains inert, since under these conditions everything is but an infinite collection of separate parts"; Hegel too had a negative opinion of corpuscular theories; Schopenhauer condemned the reductionism they imposed. Not to mention Kant, Nietzsche, Maxwell, Marx, Comte, and many others. We cannot list them all.

The theory of atoms would ultimately lead to the great unifications of twentieth-century physics.

The Plurality of the One

Greek philosophy gave rise to a host of metaphysical entities, both visible and invisible. In spite of their individual characteristics, these entities had at least one property in common: Each one of them, constructed and promoted for its unique power of synthesis and for its ontological merit, purported to account all by itself for the great diversity of the perceptible world. The atom, to take a specific example, was touted as the substance underlying all phenomena, causing effects that are observable all around us, and the real permanence hidden in changing appearances. It is a principle truly able to yield a synthesis of the physical world, and much more. All the way to its most recent developments, physics would repeatedly draw from this reservoir of images and concepts.

The pre-Socratic philosophers had foreseen the plurality of the concept of unity. Accordingly, they envisioned the One either as a principle rooted in numbers or as a principle generating and organizing the world. The systems they proposed offered a vast array of choices, including pluralism and mobility (Heraclitus), pluralism and divisibility (Democritus), unity without diversity (Parmenides and the Eleatics), and even diversity without unity (Cratylus, a disciple of Heraclitus, believed that the flux of existing things is so overwhelming that reason is powerless to apprehend it).

Yet, for all the achievements of the Greek philosophers intent on constructing an overall description of the world, it is easy to forget the criticisms that such proposals immediately provoked. Plato, for instance, wrote about the deep sense of disillusionment his teacher Socrates experienced upon reading Anaxagoras. Socrates lamented:

I was glad to think that I had found in Anaxagoras a teacher about the cause of things after my own heart, and that he would tell me, first, whether the earth is flat or round, and then would explain why it is so of necessity, saying which is better, and that it was better to be so. If he said it was in the middle of the universe, he would go on to show that it was better for it to be in the middle.... This wonderful hope was dashed as I went on reading and saw that the man made no use of Mind, nor gave it any responsibility for the management of things, but mentioned as causes air and ether and water and many other strange things.

Socrates came to the conclusion that scholars in the philosophy of nature (in a broad sense, this means physics) often pretended to know "the cause of all things" when in fact they were a long way from having all the answers. "Imagine not being able to distinguish the real cause from that without which the cause would not be able to act as a cause. It is what the majority appear to do, like people groping in the dark; they call it a cause, thus giving it a name that does not belong to it. That is why one man surrounds the earth with a vortex to make the heavens keep it in place; another makes the air support it like a wide lid." Aside from the Sophists, Socrates was the first to sound a warning against the pretensions of the human mind and to bring back down to earth philosophers blinded by premature speculations.

Blaise Pascal, too, was convinced that human knowledge is incapable of comprehending the first principles, let alone the unity that supposedly enfolds them. He shared Socrates' wholesome caution and skepticism: "Strangely enough, they [men] wanted to know the principles of things and go on from there to know everything, inspired by a presumption as infinite as their object. For there can be no doubt that such a plan could not be conceived without infinite presumption or a capacity as infinite as that of nature." The comment might serve as a healthy reminder to those who even today claim they can explain, with a couple of ideas, what the world is made of, where it comes from, where it is going, how it works, and why it is the way it is!

Unity through Numbers

The reflections of the Pythagoreans on numbers around the fifth century B.C. also center on the issue of unity. Starting with the concept of the One, they sketched the outline of a coherent model of the world. If things are assimilated with numbers, or so they claimed, then everything can be derived from the One, and the very affirmation of the multiple becomes a basis for the notion of unity. Since each number amounts to unity multiplied, the process of multiplication itself can in principle encompass all numbers as well as all things. The multiple lives hidden and folded into the One. Moreover, if numbers are the reason for everything, it is but a short extra step to attribute to them a moral value. And so, justice came to be represented by the numbers 4 or 9, which are squares and evoke perfect harmony.

From the Pythagoreans' point of view, it is quite simply the genesis of numbers that reveals the laws of the universe and of human thought, not to mention those of music or the structure of constellations. "Everything there is to know has a number. Without numbers, we cannot understand or know anything," declared Philolaus in the fifth century B.C. Numbers, invested with considerable symbolic power, are the repositories of unity for two reasons. First, because they constitute a utilitarian and operational concept with which everything else can be thought of through extension and multiplication. And second, because their very nature suggests that they can be assimilated with the fundamental unit, in other words, with the One in the arithmetical sense.

The Pythagoreans' concepts have not fallen into oblivion. The idea that numbers have a real existence of their own, with nature simply reflecting indirectly their properties, has enjoyed an enduring popularity. The fascination elicited by Pythagoreanism had fertile repercussions on the development of arithmetic. It gave credence to the idea that truth is inherently connected with harmony, and it inspired a geometrical vision of the world based on arithmetical principles. It continued to evolve under various forms right down to the time of Galileo, Newton, and even Maxwell, Einstein, and Dirac. In fact, both Einstein and Dirac felt that the aesthetic mathematical appeal of a physical theory was not just to please the mind. It was also an indication--indeed perhaps the best there is--of its validity.

Mathematical reasoning has the ability to venture beyond facts. In a 1952 letter to Maurice Solovine, Einstein clarified the distinction he made between "constructive theories" and "theories of principle." The former seek to "uncover true laws through an effort of elaboration" ( durch ... konstruktive Bemühungen ); they derive their value solely from empirical confirmation. The latter aim to "discover some general formal principle"; they derive their value from experience, obviously, but most of all from their "internal perfection" ( innere Vollkommenheit ). By virtue of their intrinsic properties, they enjoy empirical validity as well.

In his Spencer Lecture, delivered at Oxford in 1933, Einstein stressed the importance to be accorded to formal beauty: "Experience can of course guide us in our choice of serviceable mathematical concepts; it cannot possibly be the source from which they are derived; experience of course remains the sole criterion of the serviceability of a mathematical construction for physics, but the truly creative principle resides in mathematics. In a certain sense, therefore, I hold it to be true that pure thought is competent to comprehend the real, as the ancients dreamed."

Born in antiquity, the concept that the universe is founded on numbers was thriving in the Middle Ages, as Bernard Ribémont pointed out: "Numbers are given a symbolic dimension and endowed with original properties participating in the creation of the universe and matter." So much so that "through practice and the study of numbers and their combinations, man is called upon to try to rediscover the properties of the world and nature."

Christianity had long embraced such concepts and claimed them as its own. For instance, God's mastery of things mathematical is extolled in the Old Testament: "You [the creator] have arranged all things by measure and numbers and weight."

The twelfth century even witnessed the advent of a "Christian arithmology," based explicitly on the reality of numbers and oriented toward the reading of sacred texts. It differed from traditional arithmetic in that, instead of establishing formulas, it sought to extract from numbers as much information as it possibly could in terms of symbolic correspondences. Its position was that "the more substance is extracted from numbers, the better the chances to understand the significance of those numbers, even if outside the domain of mathematics," as Bernard Ribémont reminds us. For instance, the number 3, being a prime number and therefore "undissodable," corresponded to a perfection strongly evocative of a divinity. It came to represent the mystery of the divine trinity, itself associated with the heavens. The number 2, on the other hand, obviously less perfect since it is divisible by two, symbolized the duality of soul and body and stood for the flaws besetting the earth.

As can easily be appreciated from this simple example, such an approach has very little to do with mathematics, except for borrowing a few selected members of the infinite sequence of integers. Proceeding by analogies, correspondences, and other potentially risky associations, by mixing symbolism, arithmetic, astronomy, sacred texts, and natural laws, this type of doctrine hardly qualifies as a rigorous science. More aesthetic than effectual, it fell in disfavor after the genuine science of algebra, which reenergized the tradition of the early Greek mathematicians (such as Aristotle, Euclides, and Archimedes), began to make inroads in the twelfth century of our era, notably with the work of the Arab mathematician al-Khwarizmi. Neither the symbolism of numbers nor the philosophical and religious implications of their presumed "power" were able to compete with an authentic mathematics based on systematic numerical manipulations. The mysticism of numbers, even though it managed to introduce a limited amount of order in some concepts, provided not much more than a vague and arbitrary form of numerology.

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