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Topic: Matheology § 280
Replies: 8   Last Post: Jun 12, 2013 12:51 AM

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Scott Berg

Posts: 2,111
Registered: 12/12/04
Re: Matheology � 280
Posted: Jun 6, 2013 11:04 AM
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"WM" <> wrote in message
>Matheology § 280
>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.
>[Jorge Luis Borges: "Los avatares de la tortuga", translated by by
>BombaMolotov: "The Avatars of the Tortoise"]
>Regards, WM

your beef is with Newton, not these other guys you refrence all the time
total artical below (nope, its not math, some entertainment value)

The Avatars of the Tortoise

by ~BombaMolotov, May 28, 2008, 4:11:23 PM
Literature / Prose / Non-Fiction / Academic Essays
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
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.

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
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.


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.
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.

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