From

Archimedes’ mathematical treatises, originally conceived in the 2nd century BCE, derived from a prayer book of the Orthodox Church; a

The media, always eager for progress, concluded that the history of mathematics had been rewritten. Its hypothesis was based on the fact that Archimedes, according to one of the recovered fragments, conceived the volume of a sphere as a result of the infinite sum of all its circles: an application of the mathematical concept of infinity to Euclidean geometry. Isaac Newton reformulated it without knowledge of Archimedes’ text: a force is an infinite geometric progression, useful when calculating the curved trajectory of bodies in motion.

Archimedes’ commentary deals, on the other hand, with the problem of Achilles and the turtle, proposed in the fifth century BCE by Zenon of Elea. This tells us to imagine a line between two points, A and B.

It is along such a line that Achilles undertakes the race with the turtle. As everyone knows, in a gesture of gallantry, the Greek hero gives the turtle half the land, that is, up to the point on the line I shall call AB/2. Achilles doubles the turtle in speed. However, when Achilles, ‘that of the light feet’, reaches the point AB/2, the turtle has moved to point AB/4. Without despair Achilles moves to point AB/4, but by then the turtle has reached point AB/8 and when Achilles reaches point AB/8 the turtle will be at point AB/16, then at AB/32, at AB/64 and so on, infinitely. In mathematical terms Achilles will never be able to overcome the turtle.

Since Aristotle every philosopher and conspicuous mathematician has tried to clarify this aporia; Hume wrote that the units of the physical world are indivisible, but he hesitated when he gave an account of the reasons of such indivisibility. Contemporary encyclopedias, delving into Hume's reasoning, proclaim that a line is not composed of infinite points, but finite points, so densely compressed that the distance between one and another is non-existent. This explanation, which some disciple of Plato had already formulated, is illusory: the assumptions that there are indivisible units in the physical world, and that the distance between two points will disappear at a given moment, does not satisfy our understanding. Physicists formulate their response based on neutrinos, the smallest particles they have detected in the universe. Their answer, being useful, is, nonetheless, inconsistent. Although they suppose that the most miniscule particle of the universe will disappear at a certain moment to become energy, our understanding will--at a conceptual level, divide such particle ad infinitum.

What I propose instead therefore is both a mathematical and an ontological solution. The first postulate is elucidated by a dialogue of

Returning to the application of Archimedes, a sphere is the sum of an infinite number of circles, as these are conceived from the sphere, but an infinite number of circles does not necessarily constitute a sphere. In the same way, if we count from one to two, we do not need to count 1.1, 1.11, 1.111, 1.1111, 1.11111, and so on. Each of these numbers, however, will always refer to the numbers one and two, while all are decimals created from two whole numbers. Zenon upsets the hierarchy of the units by correlating a decimal number with an integer.

Our perception of reality relies on pieces of eternity, rather than on eternity itself. A daily proof of this certainty is rendered daily by cinema. In order to watch a film, we only require images that last sixteenth of a second. Just to be above our limit of perception, current film and digital TV display from 24 to 30 frames per second. Viewers, in fact, are not able to see the difference between a sequence projected first at 1000 frames per second and then at a rate of 24 frames per second.

The mathematical problem becomes clear. Not so the epistemological one. Zeno invites our understanding to reproduce infinitely this mathematical progression. The space is divided into new segments, tirelessly. If our reasoning corresponded to the concrete space, the turtle would retain its advantage

It seems that the problem would be solved if we reduced the infinite to a mere formality of our understanding. Kant assumed this position and preached that time and space are not concrete entities, but universal categories of our intuition. His purpose, sufficiently appreciated, was to conduct our speculations towards the finite and immediate phenomena, renouncing the metaphysical inferences

Zeno’s problem occurs in the finite universe we perceive. Infinity is defined and manifested in our understanding. A child who learns to count will abandon his passion for counting once he discovers that one hundred years will not be enough to pronounce all numbers up to 63072900000. Similarly, readers abandon the career between Achilles and the turtle as soon as they identify Zeno’s mathematical progression: B/1:2.

Locke defines infinity as repetition. Repetition, indeed, is born as soon as an idea loses its mystery, that is, when it becomes static. In mathematics, it is enough to enunciate a series; once done we limit its execution. Thus, we accept the coexistence of the infinite within the limits of our understanding: moreover, we accept the coexistence of a plurality of infinities. The infinite within the infinite with new mathematical and geometrical progressions that multiply in aeternum.

Spinoza defined truth from this eternal flow: thus, he selected a criterion among an infinite number of thoughts: the idea of Peter, which differs from Peter in itself, and of the idea of Peter’s idea, and of the idea of idea of Peter’s idea, etc., is the truth about Peter. Spinoza renames the truth

Since Descartes, sciences have appropriated metaphysical concerns. Until a few months ago it was proclaimed, following Einstein, that space was a sphere contracted on itself, ready to repeat itself infinitely. This hypothesis, refuted by the fact that the Universe’s edge has not been discovered yet, would not account for a hyperspace or plane capable of receiving such contracted sphere. I would dare to postulate, from Zeno, that the coexistence of the finite within the infinite defines our reality.

One of the proofs of immortality, Descartes wrote, is permanency; the fact that we continue to be the same all the time. The consciousness of living in the infinite was the true mystery of the universe for Descartes, and even more, for Pascal:

If our understanding abandons the infinite as soon as it is apprehended, as soon as it discovers its systematic repetition, life is discovered as a transition over infinite universes. Montaigne agrees with this judgment when he points out that life is but passing. Mystics advocate detachment for the sake of a dynamic existence: living afraid is being afraid of the transitions of Being. You walk on a straight line where it’s not possible to cling forever to a given point.

Consider the influence of plurality in the psyche, where the most ingrained pleasures or fears cling ferociously to repetitions. In the last pages of the

Coleridge, expressing his worries, writes:

Indian philosophy denies time, and feeds from present. One of the beggars in Samuel Beckett’s play moans that our present is wrecked between past and future. Memory, fear and hope are inconsistent, indeed. They are static states that cause pain or satisfaction. Time flows with indolence, while our remembrances, hopes and fears are before or ahead of an infinite we haven’t learnt.

By wanting to refute movement, Zeno of Elea proved the eternity of time and space. Yet Zeno must also be remembered as an accomplished comedian, forcing his commentators forever to consider the athletic aspirations of turtles.

*The Philosopher,*Volume CVI No. 2 Autumn 2018
ZENO,

ACHILLES, THE TURTLE

ACHILLES, THE TURTLE

AND THE

DEMONSTRATION OF INFINITY

By Hugo Noël Santander-Ferreira

*In any case, the concept of infinity is not infinite...*Aristotle

*, Metaphysics*994b, 28

Archimedes’ mathematical treatises, originally conceived in the 2nd century BCE, derived from a prayer book of the Orthodox Church; a

*palimpsests*, or reused parchment dating from the 9th century. The manuscript was stolen from a Greek monastery in Constantinople in 1922; Loius Siriex, a French citizen, bought it and decorated it with pious and spurious illuminations. After his death his heirs sought a buyer: his patience was rewarded in New York.The media, always eager for progress, concluded that the history of mathematics had been rewritten. Its hypothesis was based on the fact that Archimedes, according to one of the recovered fragments, conceived the volume of a sphere as a result of the infinite sum of all its circles: an application of the mathematical concept of infinity to Euclidean geometry. Isaac Newton reformulated it without knowledge of Archimedes’ text: a force is an infinite geometric progression, useful when calculating the curved trajectory of bodies in motion.

Archimedes’ commentary deals, on the other hand, with the problem of Achilles and the turtle, proposed in the fifth century BCE by Zenon of Elea. This tells us to imagine a line between two points, A and B.

It is along such a line that Achilles undertakes the race with the turtle. As everyone knows, in a gesture of gallantry, the Greek hero gives the turtle half the land, that is, up to the point on the line I shall call AB/2. Achilles doubles the turtle in speed. However, when Achilles, ‘that of the light feet’, reaches the point AB/2, the turtle has moved to point AB/4. Without despair Achilles moves to point AB/4, but by then the turtle has reached point AB/8 and when Achilles reaches point AB/8 the turtle will be at point AB/16, then at AB/32, at AB/64 and so on, infinitely. In mathematical terms Achilles will never be able to overcome the turtle.

Since Aristotle every philosopher and conspicuous mathematician has tried to clarify this aporia; Hume wrote that the units of the physical world are indivisible, but he hesitated when he gave an account of the reasons of such indivisibility. Contemporary encyclopedias, delving into Hume's reasoning, proclaim that a line is not composed of infinite points, but finite points, so densely compressed that the distance between one and another is non-existent. This explanation, which some disciple of Plato had already formulated, is illusory: the assumptions that there are indivisible units in the physical world, and that the distance between two points will disappear at a given moment, does not satisfy our understanding. Physicists formulate their response based on neutrinos, the smallest particles they have detected in the universe. Their answer, being useful, is, nonetheless, inconsistent. Although they suppose that the most miniscule particle of the universe will disappear at a certain moment to become energy, our understanding will--at a conceptual level, divide such particle ad infinitum.

What I propose instead therefore is both a mathematical and an ontological solution. The first postulate is elucidated by a dialogue of

*La Leçon*, a theatre play by Eugéne Ionesco:TEACHER-. How many units are between three and four? Or between four and three, if you prefer?

Achilles surpasses the turtle because AB/4 is not a superior unit to AB/2, but another kind of unit, a unit incompatible with AB/4. Achilles measures the terrain and places the turtle in AB/2. During the competition neither Achilles nor the spectators stop to consider the multitude of units that derive from AB/2. Zeno, on the other hand, leads us to fabricate and to brood about the sub-units of AB/2. AB/4 is senselessly equated with AB/2, and from AB/4 we produce another kind of unit: AB/8, and from there AB/16. Zeno's fallacy is to assume that AB/4 is an essential component of AB/2, and not a by-product.ALUMNA- There are no units between three and four, professor. The four comes right after the three. There is absolutely nothing between three and four!

Returning to the application of Archimedes, a sphere is the sum of an infinite number of circles, as these are conceived from the sphere, but an infinite number of circles does not necessarily constitute a sphere. In the same way, if we count from one to two, we do not need to count 1.1, 1.11, 1.111, 1.1111, 1.11111, and so on. Each of these numbers, however, will always refer to the numbers one and two, while all are decimals created from two whole numbers. Zenon upsets the hierarchy of the units by correlating a decimal number with an integer.

Our perception of reality relies on pieces of eternity, rather than on eternity itself. A daily proof of this certainty is rendered daily by cinema. In order to watch a film, we only require images that last sixteenth of a second. Just to be above our limit of perception, current film and digital TV display from 24 to 30 frames per second. Viewers, in fact, are not able to see the difference between a sequence projected first at 1000 frames per second and then at a rate of 24 frames per second.

The mathematical problem becomes clear. Not so the epistemological one. Zeno invites our understanding to reproduce infinitely this mathematical progression. The space is divided into new segments, tirelessly. If our reasoning corresponded to the concrete space, the turtle would retain its advantage

*ad infinitum*and the race would have no end.It seems that the problem would be solved if we reduced the infinite to a mere formality of our understanding. Kant assumed this position and preached that time and space are not concrete entities, but universal categories of our intuition. His purpose, sufficiently appreciated, was to conduct our speculations towards the finite and immediate phenomena, renouncing the metaphysical inferences

*à la*Plato. Unfortunately Kant, in an effort to give coherence to his thought, subordinated reality to understanding. We do not know the object itself, but only its phenomenological manifestation. Kant’s contradiction is obvious, as Hegel pointed out: Kant assures us that we cannot discuss the existence of objects outside the limits of our reason, yet he immediately goes on to postulate the existence of such objects.Zeno’s problem occurs in the finite universe we perceive. Infinity is defined and manifested in our understanding. A child who learns to count will abandon his passion for counting once he discovers that one hundred years will not be enough to pronounce all numbers up to 63072900000. Similarly, readers abandon the career between Achilles and the turtle as soon as they identify Zeno’s mathematical progression: B/1:2.

Locke defines infinity as repetition. Repetition, indeed, is born as soon as an idea loses its mystery, that is, when it becomes static. In mathematics, it is enough to enunciate a series; once done we limit its execution. Thus, we accept the coexistence of the infinite within the limits of our understanding: moreover, we accept the coexistence of a plurality of infinities. The infinite within the infinite with new mathematical and geometrical progressions that multiply in aeternum.

Spinoza defined truth from this eternal flow: thus, he selected a criterion among an infinite number of thoughts: the idea of Peter, which differs from Peter in itself, and of the idea of Peter’s idea, and of the idea of idea of Peter’s idea, etc., is the truth about Peter. Spinoza renames the truth

*certainty*, since it is addressed at a certain time. Truth in itself is incomprehensible or infinite.Since Descartes, sciences have appropriated metaphysical concerns. Until a few months ago it was proclaimed, following Einstein, that space was a sphere contracted on itself, ready to repeat itself infinitely. This hypothesis, refuted by the fact that the Universe’s edge has not been discovered yet, would not account for a hyperspace or plane capable of receiving such contracted sphere. I would dare to postulate, from Zeno, that the coexistence of the finite within the infinite defines our reality.

One of the proofs of immortality, Descartes wrote, is permanency; the fact that we continue to be the same all the time. The consciousness of living in the infinite was the true mystery of the universe for Descartes, and even more, for Pascal:

‘There are herbs on the earth: we observe them... And under this herb, hair, and within these hairs there are microscopic animals, but then nothing? Oh, presumptuous!’Achilles and the turtle walk on a space open to infinity without disturbances.

If our understanding abandons the infinite as soon as it is apprehended, as soon as it discovers its systematic repetition, life is discovered as a transition over infinite universes. Montaigne agrees with this judgment when he points out that life is but passing. Mystics advocate detachment for the sake of a dynamic existence: living afraid is being afraid of the transitions of Being. You walk on a straight line where it’s not possible to cling forever to a given point.

Consider the influence of plurality in the psyche, where the most ingrained pleasures or fears cling ferociously to repetitions. In the last pages of the

*The Book of the Knight Zifar,*a family delights eternally drinking and celebrating; in*Pedro Paramo,*the souls of the dead groan without rest. The texts of both authors evidences the anxiety or the fear of repetition. Dante created a paradise where the just enjoy the flowing infinite: a continuous ecstasy, exalted by a music and a sublime light. He also conceived of an inverted mountain where the condemned suffer the repeated infinite. They endure their favourite vices. The anguish of these Dantesque sinners doesn’t emanate from their pain, but rather from the belief of living a single infinity. They fear that their pain will be continuous. The hell of Dante, as Schopenhauer would point out, corresponds to life, where a vice is defined as the fall of the mind in a painful infinite. Excesses enslave their victims.Coleridge, expressing his worries, writes:

The anguish of everyday life, like that of nightmares, emanates from the incomprehension of plurality. Suicide doesn’t support a painful and static thought, because it confronts the infinite in time. But anguish, although inevitable, is irrelevant. In Donne’s words:Spent daily in the poison of sad thoughts

Although we produce reality from our senses, we perceive the coexistence of the infinite. Reality is a point that our understanding constructs in eternity.Those rays, so strong and reverent, why should you think them?

I eclipse them and I blot them when I blink

Indian philosophy denies time, and feeds from present. One of the beggars in Samuel Beckett’s play moans that our present is wrecked between past and future. Memory, fear and hope are inconsistent, indeed. They are static states that cause pain or satisfaction. Time flows with indolence, while our remembrances, hopes and fears are before or ahead of an infinite we haven’t learnt.

By wanting to refute movement, Zeno of Elea proved the eternity of time and space. Yet Zeno must also be remembered as an accomplished comedian, forcing his commentators forever to consider the athletic aspirations of turtles.

*About the author:*Hugo Santander is a writer and producer, known amongst other things, for*Hamlet Unbound*(2012)

*Address for correspondence:*Hsantand@yahoo.co.uk
Thank you, Hugo, for your interesting and informative discussion of a complex topic.

ReplyDeleteMy struggle is even envisioning what spatial, temporal, or numerical infinities actually are, in terms that are concrete, not ethereally abstract. I suspect I’m not alone. Definitions of infinity often strike me as circular and tautological. Expressions like ‘no beyondness’ generally complicate, not explicate. As do commonplace terms, I believe, like ‘boundless’, ‘immeasurable’, ‘endless’, and ‘uncountable’. For me, putative infinities, qualified infinities, infinities based in paradoxes, infinities based in thought experiments, and all other varieties of infinities seem unsatisfyingly manufactured rather than incontrovertible. (Though good grist for poets and artists.)

In many cases, I’m hard put to think of stuff, conditions, relations, and events being infinite without their having been refuted by someone in some fashion at some time: infinite multiverses, the density at the center of a black hole, Cantor’s infinite set of numbers, the arrow of time and of entropy, Zeno’s infinitely divisible line, Hilbert’s infinite hotel paradox. And so on (whew, I almost slipped and said ‘ad infinitum’). Few, if any, of these examples are categorical. Not all, if any, help us get to that feel-good eureka moment of indubitably comprehending the concept. At various times and with different degrees of success, most (all?) supposed infinities have fallen under the raised-eyebrow scrutiny of philosophers, scientists, and mathematicians. Frankly, I’m not sure to what measure.

As an indulgence in a hypothetical, let’s assume ‘infinity’ happens to have the feature of looking backward as well as forward. (Contrary to the ‘finite past–infinite future’ model.) That is, our being able to temporally and spatially leap back before the Big Bang — if ‘before the Big Bang’ even makes sense. I wonder if the result of that backward-looking feature would ultimately lead to ‘nothingness’ — to be clear, not entailing properties like quantum fluctuations flitting in and out, but categorical nothingness littered with its own assortment of snags. Two conundrums for the price of one: infinity and nothingness.

All rather head-scratching, I find. But thank you, Hugo, for putting the topic center stage, for us to think about further.