a dove illustration of life
The path entanglements are so numerous and so fast that it is impossible to follow them and the noted path is always infinitely simpler and shorter than the real one. Likewise, the apparent average speed of a grain during a given time (quotient of displacement by time) varies wildly in size and direction without any limit when the time of observation decreases [...] Of course, one cannot fix a tangent at any point of the trajectory either, even in the crudest way. This is one of the cases in which we cannot help thinking of those continuous functions that do not admit any derivative, which we would wrongly regard as mere mathematical curiosities, since nature can suggest them as well as derivative functions


Related Topics

The Eleatic school
translated by Alison Quayle

So far (1) – admittedly not without bias, and with some modern and personal interpretations – I have discussed three different approaches to the production, recall or discovery of Knowledge. The Milesians invented coherent explanatory models that resembled astronomical and meteorological phenomena. The Pythagoreans regarded the immanent wisdom of Number as more important (2), seeing it as the creator of the Law (Λόγος gr-flag) that governed the world. The "inspired" authors spoke prophetically of a knowledge inherited either from Truth itself, or from the manifestations of Truth, or even from the memory of their own past. They all had to deal with the problem of the relationship between the real world and what we perceive of it. Empedocles probably went furthest into the question of how we can approach the Truth through our sensory organs alone, but he did not draw any general conclusions about the intrinsic nature of things. It is the school of Parmenides, himself heir to Pythagoras and to a certain extent to Xenophanes, that deserves the credit for looking deeper into this duality: what can we say about the real world, and the way it appears, if we maintain that wherever we choose to look at them from, the Laws of the Universe must remain unchanged?

Discussion of the arithmetical properties of the world began in Western Greece, with the development of the esoteric school of Pythagoras and his successors. This research, rooted in the ancient knowledge of Babylon and Egypt, saw its success spread as its applications aroused interest. They included establishing the structure of musical harmony; a reasoned approach that led to the creation of metal alloys and of metallurgy; the rules of proportion in painting, sculpture and architecture; and the demonstration of biological cycles, which were associated with meteorological and astronomical cycles. Found throughout this part of the Greek world before the teachings of Empedocles, this philosophy was based on the theory that privileged prophets should speak of the truth they had learnt to a small elite. This was in complete contrast with the methods of the Ionian physicists, who had produced hypotheses and models. What was original about Parmenides and his successors was the way they revealed what their initiation into the mysteries of the Truth had shown them, by highlighting the paradoxes inherent in the approaches of their predecessors and contemporaries, especially the Pythagoreans.

Parmenides was born around 515 BCE in the town of Elea (founded in about 540 by Ionians who had fled from the conquering Persians). He composed a verse text, On Nature, to explain his thought to the public (3). It is both a cosmology and an epistemology (4), each dependent on the other, and it describes the paths of knowledge as well as its aim. There were two paths, the Way of Truth (ὀδός ἀληθειης gr-flag ) and the Way of Opinion (ὀδός δοξών gr-flag), also known as the Way of Belief. Truth represented the intrinsic state of nature, an objective state, completely independent of the observer; and Opinion represented our perception of nature, or the way we rationalised it (5).

First of all, Parmenides set out to demonstrate, by logical reasoning, the contradictions our senses necessarily led us into. By definition, truth is not contradictory, and if a paradox appeared at some point, it was not that the truth was different, but that our senses were misleading us. Every being – and as we will see, Parmenides believed there was a single, immobile Being (τὸ ἐόν gr-flag, also translated as what is) –  resulted from the fusion of a material structure and a motive force (ψυχή gr-flag) that gave it life (ζωή gr-flag), spirit (νοῦς gr-flag), and thought (φρόνησις gr-flag).

For Parmenides, truth could be reduced to the following reasoning, which had innumerable consequences:

(i) What is, is; what is not, is not.
(ii) What is is eternal, uncreated (ἀγένητον gr-flag) and imperishable (ἀνώλεθρον gr-flag).
(iii) What is is complete (οὐλομελες gr-flag: whole, without separate parts), immobile (ἀτρεμές gr-flag) and without end (ἀτέλεστον gr-flag).
(iv) What is is eternally present: it has no past and no future.
(v) What is is One and continuous.

These properties of Being or what is derived from the following simple logical chain:

1. What is cannot not be.
2. Beginning or ending presupposes non-existence before or after, and this is impossible by virtue of 1.
3. Movement is also impossible, as it would require what is to appear where it was not before, and to disappear from where it was.
4. The same reasoning that applied to space also applied to time, therefore what is had neither past nor future.
5. What is is indivisible and homogeneous (ὅμοιον gr-flag) otherwise it would lack one thing at one point and another thing somewhere else. If it lacked something, it would lack everything, also by virtue of 1 (in fact there would be local Non-Being, which is impossible).
6. Consequently what is is continuous and One, and the void does not exist.

Parmenides did not go beyond these simple consequences, and in particular he visualised Being or what is as having limits (since it possessed all attributes, including limit), and he identified it with the sphere - not simply Anaximander's sphere, which must naturally be in something, but a sphere in its pure form, similar to the immutable sphere of space-time (making due allowance for the conceptual anachronism).

It is the same, and it rests in the self-same place, 
abiding in itself. And thus it remains constant in its place; for hard necessity (ἀνάγκη gr-flag) keeps it in the bonds of the limit (πείρας gr-flag) that holds it fast on every side.
Furthermore, since there was an ultimate limit, Being was finite:
It is complete on every side, like the mass of a rounded sphere,
equally poised from the centre in every direction;
This placed it in explicit contrast with Anaximander's primary element, which was infinite, and generated intrinsic change, whereas Parmenides' Being resulted in eternal permanence. This, then, was the Way of the Truth, and yet everyday observation proved that it was possible to see things differently:
For this reason all these things are but names 
which mortals have given, 
believing them to be true (6) 
coming into being and passing away, being and not being, change of place and alteration of bright colour.
This was the Way of Opinion, which could only lead to error, but was inescapable :
While Being is, it teaches Non-Being, 
To eternity she opposes perpetuity, birth and death,
To permanence, change,
To immobility, movement, to the global, the local,
To the present the past and the future,
To unity, variety,
To the homogeneous the heterogeneous and to the continuous the discontinuous.

Our senses were deceived, and taught us to imagine the universe as fundamentally discontinuous, and the world as made up of distinct objects or elements with particular qualities. This resulted in a description of the world made up partly of truth and partly of opinion, quite similar to that of the Milesians (especially Anaximander); an understanding of biology impregnated with Pythagorism ("On the right, boys; on the left, girls"); and an emphasis on a principle of attraction (here, ῎Ερως  gr-flag) that harked back to Empedocles.

Thus the major discovery of the leader of the Eleatic School is the following: discontinuity is an intrinsic characteristic of our possibilities of perception, observation and measurement, it therefore underlies all our representations of the world and it is therefore impossible for us to free ourselves from it. As soon as we start to question discontinuity by reasoning (along the lines that claiming the existence of non-Being is a contradictory proposition) we come to a dead end. So since reality cannot be contradictory, why should it not be seen as continuous rather than discontinuous? The apparent paradoxes thrown up by this essential continuity would then be no more than a sign of our biased position as observers, necessarily constrained to see the world only in flashes, separate, disjointed fragments, giving us the illusion of discontinuity. And it is the human act of naming things that produces our scientific knowledge, together with the inevitable error due to the opinion that necessarily goes with that act.

This leaves Parmenides rigorously opposed to the intrinsic discontinuity of the theory of Number dear to the Pythagorean school, and also to the Heraclitean law of eternal change. His main position was that something permanent existed, something that did not change, and served as a reference for our reasoning about the world. Zeno, his successor, went on to demonstrate more clearly what did not stand up to scrutiny in the idea that the world was essentially discontinuous.

Like Parmenides, Zeno came from Elea, and he was his master's zealous disciple. Almost nothing is known of his life (except that he probably died violently, for political reasons), and much of his work is also unknown. He seems to have devoted himself to a detailed critique of the concepts of the discontinuous used by disciples of the Croton school. Like all the Eleatics, he was a heterodox Pythagorean, and he was criticized by those who accused him of breaking the rule of secrecy. In his defence, Zeno claimed that his manuscript had been stolen and published before he could decide whether the contents should be divulged. Zeno did not write a cosmogony, and his essential contribution was the method he invented - the reductio ad absurdum. Very little of it remains: only three intact fragments, striking illustrations of his reasoning technique, and six themes, developed and unfortunately considerably altered by Aristotle, who was more interested in making Zeno look ridiculous than in understanding his point of view.

Most people will recognise Zeno's problems on the nature of movement, but few are aware of how the issue arose. Pythagoreans of the Croton school claimed that everything is composed of distinct, measurable entities, and that the world can be therefore be reduced to the Finite. Those who pursued this thought furthest imagined that these entities could be relationships between whole numbers (7), arranged in a way that reflected a real discontinuous structure of both space and time. It was not until after Zeno's demonstrations, and after Anaxagoras had highlighted the principle of infinite divisibility, that Archytas used the movement of lines to create geometric forms, which were implicitly continuous. Zeno's argument involved two stages, setting out first a hypothesis in accordance with Pythagorean principles, then a chain of logical reasoning that used absurdity to show that the hypothesis did not hold. There were two different currents of Pythagorean thought. Some visualised both time and space as measured in units of the kind used in arithmetic, while others imagined the units as infinitely small, but not able to be added up ad infinitum. So Zeno put forward separate arguments to refute each of the two approaches. This is why his arguments feature two kinds of space (infinitely divisible and not infinitely divisible) and two kinds of time (infinitely divisible and not infinitely divisible). Combining them two by two, he had to visualize four ways of seeing the world, all of which were incompatible with the idea of motion, despite the fact that motion is routinely observed. Aristotle reconstructed three of them, in broad outline:

a. The "dichotomy" (or "racetrack") paradox (space infinitely divisible, time not infinitely divisible). If a moving object is to reach its goal, it must first reach the halfway point. In the same way, it must first reach the halfway point of the remaining half, and so on. Because space can be infinitely divided into rational entities, this situation is infinitely repeated. Given the discontinuous structure of space and time, each stage must take a minimum time, corresponding to the smallest hypothetically existing finite unit of time. So the moving object will take an infinite time (the sum of an infinite number of the smallest possible finite elements) to reach its goal. This contradicts observable reality, so if motion exists, time and space cannot have the discontinuous structure initially postulated.

b. Achilles and the tortoise (space infinitely divisible, and time infinitely divisible). Achilles and the tortoise have to cover the same ground, and the tortoise starts first. When Achilles starts, he must cover at least the ground already covered by the tortoise to catch him; but by the time he reaches this point, the tortoise will already have got a little way further, so the same problem arises again, and so on ad infinitum. If we accept that time and space are fundamentally discontinuous, we must accept that the sum of an infinite number of units of time is not finite, even if they are infinitely small, so Achilles cannot catch up with the tortoise (8). Once again, this contradicts what we observe, so the postulate of discontinuity must be rejected, along with the corresponding hypotheses about the fine structure of space and time. 

c. The arrow paradox (space not infinitely divisible, and time infinitely divisible). If space is made up of distinct, measurable units, a moving arrow must be able to jump from one interval to the next (9). The arrow occupies a space equal to itself, and since it cannot be in two places at once (as space is supposed to be made up of discrete entities) it must be at rest, so at every moment in time during its movement, the arrow is immobile. Since time is infinitely divisible, this means that at two separate moments, one after the other, the arrow must be at the same place, which contradicts the hypothesis of motion. These conjectures about the structure of time and space must therefore be rejected.

d. The moving body paradox (space not infinitely divisible, time not infinitely divisible). Here Zeno visualised a group of equal masses moving in opposite directions from the two ends of a stadium, past regularly spaced markers which corresponded to distinct units of space. A paradox immediately appears: because the units of time are also finite, the apparent speed of the moving bodies in relation to each other is the same as the speed of the same bodies in relation to the fixed markers, which clearly contradicts observed reality. So this particular way of structuring time and space seems no more reasonable than the others.

Two further arguments have come down to us. One plays on the chain of reasoning:

If every being must be in a place, then since this place is itself a being, it must be in a place, and so on. This proves that every being is identical to its place. This must be one of the earliest uses of the geometric character of logic, and I will come back to it.  

The final argument, regarded by some as puerile, is a classic example of how important it is to define concepts precisely before constructing an argument. If a certain quantity of grain makes a sound when it falls, there must be a relationship between the mass and the sound. Consequently a single grain or a tiny piece of a grain should make a sound when it falls. If on the other hand it is accepted that there is no relationship between the sound and the mass, there must be a minimal quantity that falls without making any sound. Adding a tiny quantity to this should be enough to make a sound appear, which seems absurd for reasons of symmetry (10).

A clear conclusion can be drawn from Zeno's four paradoxes. It is impossible to account for motion using any hypothesis that involves a discontinuity of space and time. So since these hypotheses do not fit reality, space and time must be seen as continuous. The complete fragments that have come down to us apply this reductio ad absurdum method to the structure of space and the "units" that make it up, and once again Zeno comes to the conclusion that the notion of discontinuity is inherently incorrect.

However, although the absurdity of discontinuity justified Parmenides' view of the world, this was not without paradoxes of its own, which have been described thanks to Melissus of Samos, a third member of the Eleatic school.

From the shores of Asia Minor, Greek thought had developed and spread to the west in Sicily and to the south in Italy. It seems apt that the school was to return to Samos, close to those shores, and already famous as the birthplace of Pythagoras. The last master of the Eleatic school, and probably its most profound thinker, was an admiral from Samos, who inflicted a crushing defeat on Pericles' fleet in 441-440 BCE.  Such an insult to the power of Athens, then at its height, could not go unpunished, and the following year Samos was captured and Melissus disappeared (he may have been killed). More seriously, his memory was systematically eradicated and soon nothing was left of his writings - On Nature and On Being - while the doxographers spent their time ridiculing him, even more than Parmenides. Aristotle stands out among the chorus of critics because of his violent animosity towards this original thinker, so far removed from Aristotelian dualism (the Eleatic Being - animate, of course - was One, and above all Pythagorean), who haunted the discourse of the Lyceum, and attributed to the Finite the characteristics of perfection.

Melissus did adopt Parmenides' arguments on the rational necessity of thinking that there exists something (Being) that does not change. However, to eliminate a surprising weakness in Parmenides' theory - the spatial finiteness of Being - he added an appropriate argument. It was similar to that later put forward by Archytas (see the Pythagoreans), which led to the conclusion that Being was unlimited, but Melissus went much further. His reasoning demonstrated that the two infinities, that of time and that of space, must necessarily be connected within the definition of that-which-does-not-change, thus introducing a homogeneity into the properties of the two fundamental notions of time and space. His approach was to be sidelined for more than two thousand years after Aristotle's definitive statement that finiteness is one of the essential attributes of the harmony of space. And yet Plato, no less famous and influential than Aristotle, adopted Melissus' way of thinking about Being, and acknowledged that the idea had originated with the Eleatic School.  But the dualism of his thought, incompatible with the strict monism of the Eleatics, meant the school was doomed to be forgotten. Melissus attributed the spherical nature of Being, its perfection, homogeneity and immobility to space-time. He took up the arguments of Parmenides, and this time took their logic to its conclusion: nothing was born from nothing, implying that time itself was not finite, but everything that was born had a beginning and an end. So it could be conjectured that what was not born had no limits (11). What was infinite in time was therefore also infinite in space, so Being became an absolute invariant, attained through reasoning, and essential to reasoning, since it was the only point of reference that enabled cause to be distinguished from effect, action from reaction. Clearly, observing things does not provide evidence of the immutability of Being, and no object or phenomenon taken by itself could be eternal - everything would have an end. For nothing that was not totality itself could be eternal. So the invariant principle was the All that remained unchanged in quality and quantity:

And it is impossible for its order to change, 
for the order existing before does not perish, nor does another which did not exist come into being; and since nothing is added to it
or subtracted from it or made different, how could any of the things that are change their order? for if it becomes different,
it is necessary that being should not be homogeneous, but that which was before must perish,
and that which was not must come into existence.
The overall principle of invariance therefore applied to the whole ensemble of things, which could then be seen as continuous and homogeneous (or "all alike", for reasons of symmetry). Because of our position as observers - and here again Melissus was cleverly ahead of his time - our senses could only give us an image of this ensemble, which was not the real world itself. But our reason tends to subject each individual thing to the same reasoning that led to Being, giving each thing an identity, the organised sum of its qualities. This ought to give objects permanence, and yet we can see that they are constantly changing, so how can we be certain that the objects we see today are the same ones we were looking at yesterday? For if a multiplicity of things existed it would be necessary that these things should be just such as I say the one is.

In this conflict between reason and observation, Melissus chose the way of reason, which states that the senses err. A principle of conservation exists, beyond everyday observation, and beyond changing objects. We must always try to follow the way of truth and to discover the underlying principle of invariance. In fact, Melissus had taken up the arguments of Parmenides and Zeno to try to show that Being was immobile, without realising that the question of movement became meaningless when it was developed in time and space. At this stage it might look as if the Eleatic school would open up new avenues of knowledge. But history decided otherwise. Samos was defeated by Athens, and the name of Melissus, once conqueror of the city of Pallas, was brought down. Plato praised the greatness of Eleatic thought, but Anaxagoras had already shown the way forward, along the path of dualism, which Plato followed, and the Academy after him. Aristotle's attacks a few years later were more serious. Obsessed with the way his own thought had developed, he was unable to understand the answer to the paradox of permanence and change. And so the space-time imagined by the admiral from Samos sank into oblivion.

1: Besides the sources already quoted, here my commentary draws on the remarkable study by J. Zafiropoulo, L'Ecole Eléate ed. Les Belles Lettres 1950. (back to text)

2: In fact, of small numbers. As with all kinds of numerical symbolism, it rarely goes further than the number of days in the lunar or solar cycles, and the geometric properties of polyhedrons only hold for small numbers. (back to text)

3: There is a certain incompatibility between the use of verse and Parmenides' desire to educate the public. It is difficult to reconcile the constraints of poetry, particularly rhythmic constraints, with the precision required by the subjects of his proofs. It is probably a hangover from the esoteric Pythagorean tradition, in which nothing was to be divulged, much less written down. The disciples of Pythagoras had to learn everything by heart, and it is well known that verse makes this easier, especially when the texts are very long. The prologue of Parmenides' poem clearly recalls the author's initiation by the Goddess (Truth). (back to text)

4: Epistemology will be discussed later. (back to text)

5: These two Ways represented two modes of reaching knowledge: a subjective mode in which the truth was perceived directly, thanks to the animate element in us all (ψυχή  gr-flag ), which communicated directly with the soul of the world; and an objective mode, which saw just the shape of things and disregarded what drove them, and which produced opinions about the world. (back to text)

6: Nominalism is still a relevant issue today, as in Jorge Luis Borges' poem El Golem:

"Si (como el griego afirma en el Cratilo)
El nombre es arquetipo de la cosa,
En las letras de rosa esta la rosa
Y todo el Nilo en la palabra Nilo. (...)
Yes, (like the Greek says in the Cratilo)
the name is an archetype of the thing,
in the letters of rose is the rose
and all the Nile in the word Nile. 
(Translation Xomalin G. Peralta, 1998)

Are Borges and his poem at the enigmatic heart of the famous book by Umberto Eco? Pristina rosa stat nomine nomina nuda tenemus? Knowing that the central character of The Name of the Rose, Jorge de Burgos, is Borges, there is no doubt, to me, that the real origin of the last sentence of the book is the poem I quote: it is often the case that one does not remember a source of one's own writing ! Bien sûr qu'un texte échappe à son auteur ‘ of course a text escapes it's author's grasp.

In his postscript to The Name of the Rose Eco says:

"Since the publication of The Name of the Rose I have received a number of letters from readers who want to know the meaning of the final Latin hexameter, and why this hexameter inspired the book's title. I answer that the verse is from De contemptu mundi by Bernard of Morlay, a twelfth-century Benedictine, whose poem is a variation on the "ubi sunt" theme (most familiar in Villon's later "Mais ou sont les neiges d'antan"). But to the usual topos (the great of yesteryear, the once-famous cities, the lovely princesses: everything disappears into the void), Bernard adds that all these departed things leave (only, or at least) pure names behind them. I remember that Abelard used the example of the sentence "Nulla rosa est" to demonstrate how language can speak of both the nonexistent and the destroyed. And having said this, I leave the reader to arrive at his own conclusions."

Elsewhere he has said:

"An author who has entitled his book The Name of the Rose must be ready to face manifold interpretations of his title. As an empirical author (Reflections, p.3 ) I wrote that I chose that title just to set the reader free: "the rose is a figure so rich in meanings that by now it has any meaning left: Dante's mystic rose, and go lovely rose, the Wars of the Roses, rose thou art sick, too many rings around Rosie, a rose by any other name, a rose is a rose is a rose is a rose, the Rosicrucians..." Moreover someone has discovered that some early manuscripts of De Contemptu Mundi of Bernard de Morlay, from which I borrowed the hexameter "stat rosa pristina nomine, nomina nuda tenemus", read "stat Roma pristina nomine" — which after all is more coherent with the rest of the poem, which speaks of the lost Babylon. Thus the title of my novel, had I come across another version of Morlay's poem, could have been The Name of Rome (thus acquiring fascist overtones).
But the text reads The Name of the Rose and I understand now how difficult it was to stop the infinite series of connotations that word elicits. Probably I wanted to open the possible readings so much as to make each of them irrelevant, and a result I have produced an inexorable series of interpretations. But the text is there, and the empirical author has to remain silent." (back to text)

7 : Rational numbers, therefore. (back to text)

8: Mathematical definitions of the infinite did not yet exist, and it was not possible to conceive that Σ 1/2n was a finite number. In any case, allocating a minimal value to each interval of time does produce an infinite sum.(back to text)

9: This paradox now has a perfectly serious formal analogue: for instance we accept unquestioningly the idea of an electron being "promoted", jumping from one "orbital" to a higher orbital, after absorbing a photon. (back to text)

10: This argument contains the germ of a whole realm of thought on the continuous, and rediscovered two thousand years later it was to lead to a certain definition of the continuous in mathematics. It also contains an algebraic reasoning on the extrema, which is still very effective today. Finally, problems connected with symmetry are still up to date. They can be found in all the laws that govern the Universe; it is usually considered that when space or time are transformed, symmetry is maintained, and that every action involves a reaction. It sometimes happens that to preserve other characteristics of the model, it is necessary to postulate a spontaneous violation of symmetry. In this case it would be better to bear Zeno's argument in mind (but this is rarely done!), and ask whether the entire theory should be reconsidered, rather than introducing a postulate of this kind. However the Eleatic reasoning involves a kind of equilibrium of forces present which imposes symmetry. It goes without saying that in contrast, in other circumstances symmetry is very difficult to maintain, and breaks down spontaneously. If we should fall from a peak, we can only fall to one side or the other, not both at once. (back to text)

11: This reasoning is not a syllogism, and is thus open to Aristotle's sarcasm. However, as J. Zafiropoulo points out, it should be remembered that the rules of syllogism - which Aristotle established - had not yet been put forward in Melissus' day. (back to text)

Note on the translations from Greek:

The translations from Parmenides are by John Burnet.  The translations from Melissus are by Arthur Fairbanks: http://history.hanover.edu/texts/presoc/Melissos.html

Fifth chapter: The atomists