"Magnitudes are infinitely divisible and therefore lack 'atoms'" (Euclid. Elements).
"Infinity is either by addition or by division" (Aristotle).
"Zeno's paradoxes have had a lasting impact through attempts, from Aristotle to the present day, to answer the problems they
they pose" (John Palmer)
The Problem of Space and Time
Zeno, of Elea −today Lucania, in southern Italy− came to establish numerous paradoxes or aporias (difficulties). None of his writings have survived, although it is known that he wrote a book that Proclus said contained 40 paradoxes. But only four have come down to us, which Aristotle relates in his Physics and which Plato refers to in Parmenides, one of his Dialogues.
Zeno's paradoxes are related to motion, to the relationship between space and time. The first two deal with the indefinite division of space. The third deals with the division of time into timeless instants. The fourth deals with discrete time. Zeno intended to show that motion is impossible (or leads to inconsistency) by appealing to the notion of infinity: the infinite divisibility of space, and time as infinite successions of timeless instants. Zeno wanted to highlight the contradictions underlying our conceptions of space and time.
Zeno intended to defend the doctrine of his teacher Parmenides −the most influential of the pre-Socratic philosophers−, who held a monistic philosophy:
All is one. Everything that exists forms a whole, an indivisible unity, both spatially and temporally. Plurality does not exist, change does not exist, the only real thing is Being.
Being is innate, imperishable, immutable and indivisible.
Change (and motion in particular) presupposes non-Being, so that all motion is illusory because it is meaningless to speak of non-Being. There is only one substance: Being, which is unique and immutable. Multiplicity is an illusion. Reality is a whole that cannot be divided.
Reality, the universe, can have neither origin nor end. The Whole is infinite because if it were finite, it would border on the void.
All philosophical and scientific systems that postulate principles of conservation (of matter-energy, of the quantity of motion, etc.) are heirs of the philosophy of Parmenides.
The Four Paradoxes of Zeno
The Paradox of Dichotomy
When we have to travel a certain distance, we must reach the midpoint between the origin and the destination, then the midpoint between the new position reached and the destination, and so on in an infinite recursive process that never ends. Therefore, motion as displacement between one point and another is impossible because it implies an infinite number of stages that cannot be completed in a finite way. In this argument the distance between origin and destination is irrelevant, so motion is impossible even if the distance to be traveled is very small.
Rudy Rucker [1995] gives an original version of this paradox: it is impossible to leave a room because to leave one must travel the average distance between where one is and the door, then the average distance between the new position and the door, and so on.
The dichotomy problem is expressed as the infinite sum
1/2 + 1/22 + 1/23+ 1/24 + ... ,
whose limit is 1. Zeno wondered what was the first term of this series if considered in the opposite direction.
Nicolas of Oresme (1323-1382) showed graphically that this dichotomous archetypal series tended to 1:
Oresme Diagram
The paradox of Achilles and the tortoise
Achilles and the tortoise set off simultaneously in the same direction. Achilles runs 10 times faster than the tortoise and gives him a 10-meter head start. When Achilles has run 10 meters, the tortoise will have run 1 meter. When Achilles has run 1 meter, the tortoise will have run 1 decimeter. When Achilles has gone 1 decimeter, the tortoise will have gone 1 centimeter. And so on and so forth. Therefore, since it involves infinitely many steps, Achilles will never be able to catch up with the turtle, even if he comes infinitely close to it.
Actually, this paradox is a generalization of the dichotomy paradox when the goal or destination is mobile and its speed is less than that of the pursuer.
The paradox of the arrow
When an arrow is shot at a target, the arrow at a given instant cannot be in motion because motion implies a period of time and an instant is conceived as a point lacking duration. It follows that the arrow is immobile at each instant, so it does not move and cannot reach the target. And what is true for the arrow is true for everything. Therefore, nothing moves. Movement does not exist, it is impossible. It is the same conclusion as that of the paradox of dichotomy, but with the argument of time as a succession of timeless instants.
The paradox of the stadium
Time is assumed to consist of indivisible discrete units. The discrete unit is symbolized by τ. We consider three equal bodies (A, B and C) of the same length l aligned initially (time 0). At the next instant (time τ), A moves one unit equal to the length l, B remains stationary and C moves one unit l to the left. Relative to B, A is shifted one unit in time τ. Relative to C, A has shifted two units. Since A would need 2τ to move two units relative to C, the absurdity follows that τ=2τ.
The 3 bodies of the stadium (before and after the time τ
Valuation of Zeno and his paradoxes
The 4 paradoxes of Zeno of Elea have had a profound influence on the development of mathematics.
The paradox of dichotomy is considered one of the most important contributions ever made to the concept of infinity.
The paradox of Achilles and the tortoise is the best known of Zeno's paradoxes. It is an archetypal paradox that is embedded in our collective imagination, on the same level as the Pythagorean theorem or the irrationality of √2.
The paradox of the arrow immerses us in the inability to capture the instant. The instant has the same conceptual nature as the geometric point.
The paradox of the stadium contains the germ of relativity.
According to Aristotle, Zeno was the originator of paradoxical thinking and dialectics. Zeno is also considered the forerunner of the method of reductio ad absurdum.
"The arguments of Zeno of Elea have provided, in one way or another, a foundation for almost all the theories of space, time and infinitesimalism that have been constructed since then until today" (Bertrand Russell).
According to Bertrand Russell and Adolf Fraenkel, after 2400 years of explanatory attempts, Zeno's paradoxes have still not been clarified. Russell is well known. Fraenkel is known mainly for his work in axiomatic set theory, in which he tried to eliminate the paradoxes and improving Zermelo's axiomatic system. The axioms of set theory are called "Zermelo-Fraenkel axioms" (ZF). If the axiom of choice is considered, they are referenced as ZFC.
MENTAL and Zeno's Paradoxes
The Paradox of Achilles and the Tortoise
It is often claimed that Achilles catches up with the tortoise because although the number of intervals is infinite, the sum D is finite, following the reasoning below:
But the solution to the paradox of Achilles and the tortoise is based on distinguishing between the logical and physical levels:
At the logical level, Achilles never catches up with the tortoise because time is not involved. It is a logical level because it involves or uses condition.
At the physical level, Achilles reaches the tortoise because the space-time lines of the two intersect precisely at distance D.
In general, if d is the advantage Achilles gives the tortoise, vt is the speed of the tortoise, vA>vt is the Achilles velocity, and r is the ratio of vt to vA, Achilles reaches the tortoise at distance D and in time T:
T = d/(vA−vt)
D = vAT = d/(1−r)
In the case d = 10 and r = 1/10, we have:
D = 10/(1−1/10) = 100/9.
Note that D only depends on d and r. It does not depend on the absolute velocities of Achilles and the tortoise, but on the ratio of the two. When vt=0, we are in the dichotomous paradox and D=d. Instead, the time T is a function of d and of the difference between the speeds of Achilles and the tortoise.
The crossing of the two lines of space-time.
Space and time as physical and mathematical abstractions
The idea of space is one of the most important concepts in Western culture, both philosophically and scientifically, and as such arose in ancient Greece. Space is an abstraction:
It is a "continent" that contains the physical objects, the things perceptible by the senses. But space itself is not a physical object, nor is it perceptible by the senses.
It houses all kinds of objects. Every physical object occupies a certain space and is in a certain place. What does not occupy space or is not somewhere, has no physical existence.
Space is a geometric type of concept because it involves the notions of position and distance.
Space is a concept close to Being because it possesses some characteristics in common with Being:
It is immutable, it does not transform and is indestructible. It houses changing objects, perceptible by the senses, but space itself does not change. We can say that space is a primary archetype.
It has no properties. It only has them when it houses physical objects and geometrical relations are established between them and when geometrical relations are established between the parts of physical objects.
Space is what grounds all physical existence, what gives nature to everything that exists at the physical level, because only that which occupies a place and a position exists.
It is continuous, it does not jump, it permeates everything.
There are two types of space: physical space and abstract or mathematical space. Mathematical space transcends physical space and connects with our consciousness, acting as the foundation of everything around us. It makes reality intelligible because it is the common element of everything. Abstract space has existence in a higher realm, beyond the very existence of the objects it contains. The mathematical world is not the same as the world of the senses. Space, time and motion perceived by our senses (or our scientific instruments) are not coextensive with the mathematical concepts that have the same names. "The mathematician must admit that the symbolic world he creates is not identical with the world of the senses" (Tobias Dantzig).
Space, as such, is indefinable, like every primary archetype. In the 17th century there were several attempts to define space exactly. For Newton, space is absolute and immutable. For Kant it is an a priori concept.
It is infinitely extensive. It has no limits.
It contains itself, it is similar to itself. It is fractal and holographic. Each part of space has the same properties as the total space.
In MENTAL, space and time go together. At the set level, its components (expressions) occupy the same abstract "place" and exist simultaneously in abstract time. At the sequence level, its components have successive abstract spatial and temporal positions.
The points have no extension, so they have no physical existence, since they occupy no place in space. But they have mathematical existence, they are mathematical entities.
Space and time are concepts that have common properties. If space is divisible, then time is also divisible, or the dichotomy of space implies the dichotomy of time.
To suppose that time is infinitely divisible is equivalent to representing time as a geometrical line in which duration is identified with extension, and instant with point. This is the first step in the geometrization of mechanics.
Motion is just a correspondence between position and time, that is, a function. When we have the function e = f(t), t is just another mathematical variable, without temporal meaning. Mathematical motion is an infinite succession of (instantaneous) rest states. In this sense, mathematics can be considered a branch of statics.
The conception of motion consisting of states without motion (without duration) is like considering a segment consisting of points without extension. The problem of time instants is analogous to that of the decomposition of a segment into points. The point is an abstraction. Galileo already discovered that a small circle had the same number of points as a large circle, since a radial correspondence can be established between a point of the small circle and a point of the large one.
We can distinguish between qualitative (or logical or abstract) time and quantitative (or physical or concrete) time.
The impossibility of motion argued by Zeno refers to the fact that at the deep level, there is neither space nor time, so there is no motion either. The deep level is of a logical-mathematical type, which connects with the Self, the soul, where there is neither space nor time.
The solution to all paradoxes is based on distinguishing between the deep and the superficial. Parmenides was right from the deep point of view.
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