The Avatars of the Tortoise


There is a concept which is the corruptor and the dazzler of the others. I do not speak of Evil, whose limited empire is ethics; I speak of the infinite. I once longed to compile its mobile history. The numerous Hydra (paludal monster which comes to be a prefiguration or an emblem of geometric progressions) would give convenient horror to its portico; it would be crowned by the sordid nightmares of Kafka and its central chapters would recognize the conjectures of that remote German cardinal – Nicholas of Krebs, Nicholas of Cusa – who in the circumference saw a polygon of an infinite number of angles and wrote that an infinite line would be a straight one, it would be a triangle, and it would be a circle and it would be a sphere (De docta ignorantia, I, 13). Five, seven years of metaphysical, theological, mathematical learning would capacitate me (maybe) to plan decorously that book. Useless to add that life prohibits me that hope, let alone that adverb.
To that illusory Biography of the Infinite belong in a certain way these pages. Their purpose is to register certain avatars of Zeno’s second paradox.
Let us remember, now, that paradox.
Achilles runs ten times lighter than the tortoise and he gives it an advantage of ten meters. Achilles runs those ten meters, the tortoise runs one; Achilles runs that meter, the tortoise runs a decimeter; Achilles runs that decimeter, the tortoise runs a centimeter; Achilles runs that centimeter, the tortoise runs a millimeter; Achilles Lightfeet runs that millimeter, the tortoise a tenth of a millimeter and so on fourth infinitely, without reaching it… Such is the habitual version. Wilhelm Capelle (Die Vorsokratiker, 1935, p. 179) translates Aristotle’s original text: “Zeno’s second argument is the so-called Achilles. He reasons that the slowest will not be reached by the fastest, as the chaser has to go through the place the chased one just evacuated, so the slower one always has a determinate advantage.” The problem doesn’t change, as it can be seen; but I would like to know the name of the poet who provided him with a hero and a tortoise. To those magical competitors and the series

10 + 1 + 1/10 + 1/100 + 1/1000 + 1/10000

the argument owes its diffusion. Almost no one remembers the one which antecedes it -[the one with the track]-, although its mechanism is identical. Movement is impossible (argues Zeno) as the mobile must go through the middle to get to the end, and before that the middle of the middle, and before that the middle of the middle of the middle, and before that…[FOOTNOTE 1]
We owe to Aristotle’s pen the communication and first refutation of those arguments. He refutes them with a perhaps disdainful brevity, but its memory inspires him the famous argument of the third man against the platonic doctrine. That doctrine wants to demonstrate that two individuals which have common attributes (for example two men) are mere temporary appearances of an eternal archetype. Aristotle interrogates whether the many men and the Man – the temporary individuals and the Archetype- have common attributes. It is notoriously so; they have general attributes of humanity. In that case, affirms Aristotle, it will be necessary to postulate another archetype which covers them all and a fourth afterwards… Patricio de Azcárate, in a note within his translation of Metaphysics, attributes to a disciple of Aristotle this presentation: “If what is affirmed of many things is a separate being, distinct from the things which are being affirmed of (and this is what the Platonists intend), it is necessary for there to be a third man. It is a denomination which is applied to the individuals and the idea. There is, then, a third man distinct from the particular men and the idea. There is at the same time a fourth which will be in the same relation to this one and with the idea of the particular men; afterwards a fifth and so on fourth ad infinitum” We postulate two individuals, a and b, which integrate the genre c. We will have, then

a + b = c

But also, according to Aristotle:

a + b + c = d
a + b + c + d = e
a + b + c + d + e = f…

Strictly speaking, two individuals are not required: the individual and the genre suffice to determine the third man which Aristotle denounces. Zeno of Elea recurs to infinite regression against locomotion and numbers; his refuter, against universal forms. [FOOTNOTE 2]

The next of Zeno’s avatars which my disorderly notes register is Agripa, the skeptic. He denies that something can be proved, for all proof requires a previous proof (Hypotyposes, I, 166). Sextus Empiricus argues similarly that definitions are vain, as one would need to define each of the voices used and, then, define the definition (Hypotyposes, II, 207). One thousand seven hundred years later, Byron, in the dedication of Don Juan, will write of Coleridge: “I wish he would explain His Explanation.”
Hitherto, the regressus in infinitum has served to deny; Saint Thomas Aquinas recurs to it (Summa Theologica, 1, 2, 3) to affirm that there is a God. He referred that there is nothing in the universe which does not have an efficient cause and that cause, clearly, is the effect of a previous cause. The world is an endless enchainment of causes and each cause is an effect. Each state proceeds from the one anterior and determines the subsequent one, but the general series could have not been, as the terms which form it are conditional, that is to say, random. However, the world is; from that we can infer a non-contingent first cause which will be the divinity. Such is the cosmological proof; Aristotle and Plato prefigure it; Leibniz rediscovers it. [FOOTNOTE 3]
Herman Lotze appeals to the regressus to not comprehend that an alteration of object A can produce an alteration of object B. He reasons that if A and B are independent, to postulate an influx of A over B is to postulate a third element C, an element which to operate over B will require a fourth element D, which won’t be able to operate without E, which won’t be able to operate without F… To elude that multiplication of chimeras, he resolves that in the world there is a single object: an infinite and absolute substance comparable to Spinoza’s God. Transitive causes are reduced to immanent causes; facts, to manifestations or modes of cosmic substance. [FOOTNOTE 4]
Analogous, but even more alarming, is the case of F. H. Bradley. This thinker (Appearance and Reality, 1897, p. 19-34) does not limit himself to combat causal relation; he denies all relations. He asks if a relation is related to its terms. They answer him yes and he infers that this is admitting the existence of another two relations, and then another two. In the axiom “the part is smaller than the whole” he does not perceive two terms and the relation “smaller than”, he perceives three (“part”, “smaller than”, “whole”) whose connection implies another two relations, and so on fourth ad infinitum. In the judgment “Juan is mortal” he perceives three inconjugable concepts (the third one is the conjunction) which we will not finish uniting. He transforms all concepts into incommunicated objects, solidified. To refute him is to contaminate oneself with unreality.
Lotze inserts the Zeno’s periodic abysses between cause and effect; Bradley, between subject and predicate, between subject and its attributes when he doesn’t; Lewis Carroll (Mind, fourth volume, p. 278), between the second premise of the syllogism and the conclusion. He narrates an endless dialogue, whose speakers are Achilles and the tortoise. Having reached the end of their interminable race, the two athletes converse calmly about geometry. They study this clear reasoning:

a) Things that are equal to the same are equal to each other.
b) The two sides of this triangle are things that equal to the same.
z) The two sides of this triangle are equal to each other.

The tortoise accepts the premises a and b, but denies that they justify the conclusion. He gets Achilles to insert a hypothetical proposition.

a) Things that are equal to the same are equal to each other.
b) The two sides of this triangle are things that equal to the same.
c) If a and b are true, z is true.
z) The two sides of this triangle are equal to each other.

With this brief clarification, the tortoise accepts the validity of a, b and c, but not of z. Achilles, indignant, inserts:

d) If a and b and c are true, z is true.

Carroll observes that the Greek’s paradox implies an infinite series of distances which lessen and that in the one proposed by him distances increase.
A final example, perhaps the most elegant of all, but also the one which differs the least from Zeno. William James (Some Problems of Philosophy, 1911, p. 182) denies that fourteen minutes can pass, because before that it is obligatory for seven to have passed, and before seven, three minutes and a half, and before three and a half, one minute and three-quarters, and thus until the end, until the invisible end, by tenuous labyrinths of time.
Descartes, Hobbes, Leibniz, Mill, Ranouvier, Georg Cantor, Gomperz, Russell and Bergson have formulated explanations- not always inexplicable and vain- of Zeno’s paradox. (I have registered some) There abound as well, as the reader has verified, its applications. The historical ones do not exhaust: the vertiginous regressus in infinitum is perhaps applicable to all subjects. To aesthetics: such verse moves us for such motive, such motive for another motive… To the problem of knowledge: to know is to recognize, but it is necessary to have known in order to recognize, but to know is to recognize… How to judge that dialectic? Is it a legitimate instrument of inquiry or just a bad habit?
It is venturesome to think that a coordination of words (philosophies are not but this) can resemble the universe very much. It is also venturesome to think that of those distinguished coordinations, one- at least in an infinitesimal way- does not resemble it a little more than some other. I have examined the ones which enjoy certain credit; I dare to assure that only in the one Schopenhauer formulated I have found a few features of the universe. According to that doctrine, the world is a fabrication of the will. Art- always- requires visible unrealities. It is sufficient to cite one: the metaphorical or numerous or carefully casual diction of the speakers in a play… Let us admit what all idealists admit: the hallucinatory character of the world. Let us do what no idealist has done: let’s search for unrealities which confirm that character. We will find them, I believe, in Kant’s antimonies and Zeno’s dialectic.
“The greatest wizard- memorably writes Novalis- would be the one which enchanted himself to the point where he takes his own illusions as autonomous apparitions. Wouldn’t that be our case?” I conjecture that so it is. We (the undivided divinity which operates in us) have dreamed the world. We have dreamed it resistant, mysterious, visible, ubiquitous in space and firm in time; but we have consented in its architecture tenuous and eternal gaps of unreasonableness to know that it is false.

FOOTNOTES

1 - A century later, Chinese sophist Hui Tzu reasons that a cane which they cut in half every day, is interminable (H. A. Giles, Chuang Tzu, 1889, p. 453).
2 - In Paramenides- whose Zenonian character is irrecusable- Plato invents an argument very similar to demonstrate that one is really many. If one exists, it participates in being; thus, there are two parts in it, that are the being and the one, but each of those parts is one and is, so that it encloses another two, which also enclose another two: infinitely. Russell (Introduction to Mathematical Philosophy, 1919, p. 138) substitutes Plato’s geometric procession with an arithmetic procession. If one exists, it participates in being; but since being and one are different, two exists; but since being and two are different, three exists, etc. Chuang Tzu (Waley: Three Ways of Thought in Ancient China, p. 25) recurs to the same interminable regressus against the monists who declared that the Ten Thousand Things (the Universe) are a single one. In the meantime – he argues- the cosmic unity and the declaration of that unity are already two things: those two and the declaration of their duality are three; those three and the declaration of their trinity are four… Russell thinks that the vagueness of the term being is enough to invalidate the reasoning. He adds that numbers do not exist, that they are mere logical fictions.
3 - An echo of that proof, now dead, resounds in the first verse of Paradiso: “La gloria de Colvi che tutto move.”
4 - I follow James’ exposition (A Pluralistic Universe, 1909, p. 55-60). Cf. Wentscher: Fechner und Lotze, p. 166-171.