Labyrinths (26 page)

Avatars of the Tortoise

There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite. I once longed to compile its mobile history. The numerous Hydra (the swamp monster which amounts to a prefiguration or emblem of geometric progressions) would lend convenient horror to its portico; it would be crowned by the sordid nightmares of Kafka and its central chapters would not ignore the conjectures of that remote German cardinal ― Nicholas of Krebs, Nicholas of Cusa ― who saw in the circumference of the circle a polygon with an infinite number of sides and wrote that an infinite line would be a straight line, a triangle, a circle and a sphere
(De docta ignorantia,
I, 13). Five or seven years of metaphysical, theological and mathematical apprenticeship would allow me (perhaps) to plan decorously such a book. It is useless to add that life forbids me that hope and even that adverb.

The following pages in some way belong to that illusory
Biography of the Infinite.
Their purpose is to register certain avatars of the second paradox of Zeno.

Let us recall, now, that paradox.

Achilles runs ten times faster than the tortoise and gives the animal a headstart of ten meters. Achilles runs those ten meters, the tortoise 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, a millimeter; Fleet-footed Achilles, the millimeter, the tortoise, a tenth of a millimeter, and so on to infinity, without the tortoise ever being overtaken. . . Such is the customary version. Wilhelm Capelle
(Die Vorsokratiker,
1935, page 178) translates the original text by Aristotle: "The second argument of Zeno is the one known by the name of Achilles. He reasons that the slowest will never be overtaken by the swiftest, since the pursuer has to pass through the place the pursued has just left, so that the slowest will always have a certain advantage." The problem does not change, as you can see; but I would like to know the name of the poet who provided it with a hero and a tortoise. To those magical competitors and to the series

the argument owes its fame. Almost no one recalls the one preceding it ― the one about the track ―, though its mechanism is identical. Movement is impossible (argues Zeno) for the moving object must cover half of the distance in order to reach its destination, and before reaching the half, half of the half, and before half of the half, half of the half of the half, and before. . .
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We owe to the pen of Aristotle the communication and first refutation of these arguments. He refutes them with a perhaps disdainful brevity, but their recollection served as an inspiration for his famous
argument of the third man
against the Platonic doctrine. This doctrine tries to demonstrate that two individuals who have common attributes (for example, two men) are mere temporal appearances of an eternal archetype. Aristotle asks if the many men and the Man ― the temporal individuals and the archetype ― have attributes in common. It is obvious that they do: the general attributes of humanity. In that case, maintains Aristotle, one would have to postulate
another
archetype to include them all, and then a fourth. . . Patricio de Azcárate, in a note to his translation of the
Metaphysics,
attributes this presentation of the problem to one of Aristotle's disciples: "If what is affirmed of many things is at the same time a separate being, different from the things about which the affirmation is made (and this is what the Platonists pretend), it is necessary that there be a third man.
Man
is a denomination applicable to individuals and the idea. There is, then, a third man separate and different from individual men and the idea. There is at the same time a fourth man who stands in the same relationship to the third and to the idea and individual men; then a fifth and so on to infinity." Let us postulate two individuals,
a
and
b,
who make up the generic type
c.
We would then have:

a + b = c

But also, according to Aristotle:

a + b + c = d

a + b + c + d = e

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

Rigorously speaking, two individuals are not necessary: it is enough to have one individual and the generic type in order to determine the
third man
denounced by Aristotle. Zeno of Elea resorts to the idea of infinite regression against movement and number; his refuter, against the idea of universal forms.
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The next avatar of Zeno my disorderly notes register is Agrippa the skeptic. He denies that anything can be proven, since every proof requires a previous proof
(Hypotyposes,
I, 166). Sextus Empiricus argues in a parallel manner that definitions are in vain, since one will have to define each of the words used and then define the definition
(Hypotyposes,
II, 207). One thousand six hundred years later, Byron, in the dedication to
Don Juan,
will write of Coleridge: "I wish he would explain his Explanation."

So far, the
regressus in infinitum
has served to negate; Saint Thomas Aquinas resorts to it
(Summa theologica,
I, 2, 3) in order to affirm that God exists. He points out that there is nothing in the universe without an effective cause and that this cause, of course, is the effect of another prior cause. The world is an interminable chain of causes and each cause is also an effect. Each state derives from a previous one and determines the following, but the whole series could have not existed, since its terms are conditional, i.e., fortuitous. However, the world
does
exist; from this we may infer a noncontingent first cause, which would be the Divinity. Such is the cosmological proof; it is prefigured by Aristotle and Plato; later Leibniz rediscovers it.
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Hermann Lotze has recourse to the
regressus
in order not to understand 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 influence of A on B is to postulate a third element C, an element which in order to affect B will require a fourth element D, which cannot work its effect without E, which cannot work its effect without F. . . In order to elude this multiplication of chimeras, he resolves that in the world there is one sole object: an infinite and absolute substance, comparable to the God of Spinoza. Transitive causes are reduced to immanent causes; phenomena, to manifestations or modalities of the cosmic substance.
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Analogous, but even more alarming, is the case of F. H. Bradley. This thinker
(Appearance and Reality,
1897, pages 19-34) does not limit himself to combatting the relation of cause; he denies all relations. He asks if a relation is related to its terms. The answer is yes and he infers that this amounts to admitting the existence of two other relations, and then of two more. In the axiom "the part is less than the whole" he does not perceive two terms and the relation "less than"; he perceives three ("part," "less than," "whole") whose linking implies two more relations, and so on to infinity. In the statement "John is mortal," he perceives three invariable concepts (the third is the copula) which we can never bring together. He transforms all concepts into incommunicable, solidified objects. To refute him is to become contaminated with unreality.

Lotze inserts Zeno's periodic chasms between the cause and the effect; Bradley, between the subject and the predicate, if not between the subject and its attributes; Lewis Carroll
(Mind,
volume four, page 278), between the second premise of the syllogism and the conclusion. He relates an endless dialogue, whose interlocutors are Achilles and the tortoise. Having now reached the end of their interminable race, the two athletes calmly converse about geometry. They study this lucid reasoning:

      a) Two things equal to a third are equal to one another.

      b) The two sides of this triangle are equal to MN.

      c) The two sides of this triangle are equal to one another.

The tortoise accepts the premises
a
and
b,
but denies that they justify the conclusion. He has Achilles interpolate a hypothetical proposition:

      a) Two things equal to a third are equal to one another.

      b) The two sides of this triangle are equal to MN.

      c) If
a
and
b
are valid,
z
is valid.

      z) The two sides of this triangle are equal to one another.

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

      d) if
a
,
b
and
c
are valid,
z
is valid.

And then, now with a certain resignation:

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

Carroll observes that the Greek's paradox involves an infinite series of distances which diminish, whereas in his, the distances grow.

One final example, perhaps the most elegant of all, but also the one differing least from Zeno. William James
(Some Problems of Philosophy,
1911, page 182) denies that fourteen minutes can pass, because first it is necessary for seven to pass, and before the seven, three and a half, and before the three and a half, a minute and three quarters, and so on until the end, the invisible end, through tenuous labyrinths of time.

Descartes, Hobbes, Leibniz, Mill, Renouvier, Georg Cantor, Gomperz, Russell and Bergson have formulated explanations ― not always inexplicable and vain in nature ― of the paradox of the tortoise. (I have registered some of them in my book
Discusión,
1932, pages 151-161). Applications abound as well, as the reader has seen. The historical applications do not exhaust its possibilities: the vertiginous
regressus in infinitum
is perhaps applicable to all subjects. To aesthetics: such and such a verse moves us for such and such a reason, such and such a reason for such and such a reason. . . To the problem of knowledge: cognition is recognition, but it is necessary to have known in order to recognize, but cognition is recognition. . . How can we evaluate this dialectic? Is it a legimate instrument of investigation or only a bad habit?

It is venturesome to think that a coordination of words (philosophies are nothing more than that) can resemble the universe very much. It is also venturesome to think that of all these illustrious coordinations, one of them ― at least in an infinitesimal way ― does not resemble the universe a bit more than the others. I have examined those which enjoy certain prestige; I venture to affirm that only in the one formulated by Schopenhauer have I recognized some trait of the universe. According to this doctrine, the world is a fabrication of the will. Art ― always ― requires visible unrealities. Let it suffice for me to mention one: the metaphorical or numerous or carefully accidental diction of the interlocutors in a drama. . . Let us admit what all idealists admit: the hallucinatory nature of the world. Let us do what no idealist has done: seek unrealities which confirm that nature. We shall find them, I believe, in the antinomies of Kant and in the dialectic of Zeno.

"The greatest magician (Novalis has memorably written) would be the one who would cast over himself a spell so complete that he would take his own phantasmagorias as autonomous appearances. Would not this be our case?" I conjecture that this is so. We (the undivided divinity operating within us) have dreamt the world. We have dreamt it as firm, mysterious, visible, ubiquitous in space and durable in time; but in its architecture we have allowed tenuous and eternal crevices of unreason which tell us it is false.

Translated by J. E. I.

The Mirror of Enigmas

      The idea that the Sacred Scriptures have (aside from their literal value) a symbolic value is ancient and not irrational: it is found in Philo of Alexandria, in the Cabalists, in Swedenborg. Since the events related in the Scriptures are true (God is Truth, Truth cannot lie, etc.), we should admit that men, in acting out those events, blindly represent a secret drama determined and premeditated by God. Going from this to the thought that the history of the universe ― and in it our lives and the most tenuous detail of our lives ― has an incalculable, symbolical value, is a reasonable step. Many have taken that step; no one so astonishingly as Léon Bloy. (In the psychological fragments by Novalis and in that volume of Machen's autobiography called
The London Adventure
there is a similar hypothesis: that the outer world ― forms, temperatures, the moon ― is a language we humans have forgotten or which we can scarcely distinguish. . . It is also declared by De Quincey:
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"Even the articulate or brutal sounds of the globe must be all so many languages and ciphers that somewhere have their corresponding keys ― have their own grammar and syntax; and thus the least things in the universe must be secret mirrors to the greatest.")

A verse from St. Paul (I Corinthians, 13:12) inspired Léon Bloy.
Videmus nunc per speculum in aenigmate: tune autem facie ad faciem. Nunc cognosco ex parte: tunc autem cognoscam sicut et cognitus sum.
Torres Amat has miserably translated: "At present we do not see God except as in a mirror and beneath dark images; but later we shall see him face to face. I only know him now imperfectly; but later I shall know him in a clear vision, in the same way that I know myself." 49 words do the work of 22; it is impossible to be more languid and verbose. Cipriano de Valera is more faithful: "Now we see in a mirror, in darkness; but later we shall see face to face. Now I know in part; but later I shall know as I am known." Torres Amat opines that the verse refers to our vision of the divinity; Cipriano de Valera (and Léon Bloy), to our general vision of things.

So far as I know, Bloy never gave his conjecture a definitive form. Throughout his fragmentary work (in which there abound, as everyone knows, lamentations and insults) there are different versions and facets. Here are a few that I have rescued from the clamorous pages of
Le mendiant ingrat, Le Vieux de la Montagne
and
L'invendable.
I do not believe I have exhausted them: I hope that some specialist in Léon Bloy (I am not one) may complete and rectify them.

The first is from June 1894. I translate it as follows: "The statement by St. Paul:
Videmus nunc per speculum in aenigmate
would be a skylight through which one might submerge himself in the true Abyss, which is the soul of man. The terrifying immensity of the firmament's abysses is an illusion, an external reflection of
our own
abysses, perceived 'in a mirror.' We should invert our eyes and practice a sublime astronomy in the infinitude of our hearts, for which God was willing to die. . . If we see the Milky Way, it is because it
actually exists in our souls."

The second is from November of the same year. "I recall one of my oldest ideas. The Czar is the leader and spiritual father of a hundred fifty million men. An atrocious responsibility which is only apparent. Perhaps he is not responsible to God, but rather to a few human beings. If the poor of his empire are oppressed during his reign, if immense catastrophies result from that reign, who knows if the servant charged with shining his boots is not the real and sole person guilty? In the mysterious dispositions of the Profundity, who is really Czar, who is king, who can boast of being a mere servant?"

The third is from a letter written in December. "Everything is a symbol, even the most piercing pain. We are dreamers who shout in our sleep. We do not know whether the things afflicting us are the secret beginning of our ulterior happiness or not. We now see, St. Paul maintains,
per speculum in aenigmate,
literally: 'in an enigma by means of a mirror' and we shall not see in any other way until the coming of the One who is all in flames and who must teach us all things."

The fourth is from May 1904.
"Per speculum in aenigmate,
says St. Paul. We see everything backwards. When we believe we give, we receive, etc. Then (a beloved, anguished soul tells me) we are in Heaven and God suffers on earth."

The fifth is from May 1908. "A terrifying idea of Jeanne's, about the text
Per speculum.
The pleasures of this world would be the torments of Hell, seen backwards, in a mirror."

The sixth is from 1912. It is each of the pages of
L'Âme de Napoléon,
a book whose purpose is to decipher the symbol Napoleon, considered as the precursor of another hero ― man and symbol as well ― who is hidden in the future. It is sufficient for me to cite two passages. One: "Every man is on earth to symbolize something he is ignorant of and to realize a particle or a mountain of the invisible materials that will serve to build the City of God." The other: "There is no human being on earth capable of declaring with certitude who he is. No one knows what he has come into this world to do, what his acts correspond to, his sentiments, his ideas, or what his real name is, his enduring Name in the register of Light. . . History is an immense liturgical text where the iotas and the dots are worth no less than the entire verses or chapters, but the importance of one and the other is indeterminable and profoundly hidden."

The foregoing paragraphs will perhaps seem to the reader mere gratuities by Bloy. So far as I know, he never took care to reason them out. I venture to judge them verisimilar and perhaps inevitable within the Christian doctrine. Bloy (I repeat) did no more than apply to the whole of Creation the method which the Jewish Cabalists applied to the Scriptures. They thought that a work dictated by the Holy Spirit was an absolute text: in other words, a text in which the collaboration of chance was calculable as zero. This portentous premise of a book impenetrable to contingency, of a book which is a mechanism of infinite purposes, moved them to permute the scriptural words, add up the numerical value of the letters, consider their form, observe the small letters and capitals, seek acrostics and anagrams and perform other exegetical rigors which it is not difficult to ridicule. Their excuse is that nothing can be contingent in the work of an infinite mind.
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Léon Bloy postulates this hieroglyphical character ― this character of a divine writing, of an angelic cryptography ― at all moments and in all beings on earth. The superstitious person believes he can decipher this organic writing: thirteen guests form the symbol of death; a yellow opal, that of misfortune.

It is doubtful that the world has a meaning; it is even more doubtful that it has a double or triple meaning, the unbeliever will observe. I understand that this is so; but I understand that the hieroglyphical world postulated by Bloy is the one which best befits the dignity of the theologian's intellectual God.

No man knows who he is,
affirmed Léon Bloy. No one could illustrate that intimate ignorance better than he. He believed himself a rigorous Catholic and he was a continuer of the Cabalists, a secret brother of Swedenborg and Blake: heresiarchs.

Translated by J. E. I.