Zeno's paradoxes

Zeno's paradoxes of plurality (5th century BC) include in addition to the more well-known Zeno's paradoxes of motion of the paradoxes of Zeno of Elea.

The three paradoxes of plurality are preserved in a commentary by the Byzantine philosopher Simplicius on the physics of Aristotle. In fact, Simplicius, who lived about a thousand years after Zeno, the only source which quoted extensively Zenon literally. Simplicius seems to have possessed Zeno's work in the original. In the opinion of Simplicius all paradoxes is the fact that they served the defense of Zeno's friend and teacher Parmenides against its critics. This important pre-Socratic philosophers, which is how his pupil Zeno the Eleatic attributed, is facing a non-uniform, the change in underlying world of perception, an indivisible, eternal and immutable Being in a didactic poem. According to popular, but not unproblematic interpretation of the difficult to access didactic poem Parmenides took a strict metaphysical monism, according to which movement and divisibility only be an illusion.

Zeno tried to show that Parmenides ' position, although paradoxical anmute, but the opposite, namely the idea that there are both much, as well as the possibility of movement, leads to contradictions and thus indirectly confirmed Parmenides. Of the nine paradoxes obtained from a total of probably forty, three deal specifically with the inconsistency of the ideas of multiplicity and continuity: the argument of the density, the argument of the finite size and the argument of the whole division. The group of motion paradoxes, Achilles and the tortoise, division paradox, Arrow 's paradox is concerned in contrast to the sub-problem of the impossibility of motion.

In contrast to the paradoxes of motion no single name has prevailed in the reception of the paradoxes of plurality; ever the importance of the preserved Greek text is much less clear than those of other authors indirectly traditional motion paradoxes.

Their significance for the mathematics and philosophy of the Greek contemporaries and their subsequent influence are judged differently. The influence on the consequential restriction of Aristotle and Euclid to potential infinities, which was only resolved with the work of Georg Cantor, is not conclusively assess.

More recently, initiated by work of Adolf Grünbaum, the paradox of the complete division of labor is bestowed become new attention to the basic mathematical research.

Argument of the density

The argument of the density is quoted by Simplicius in his commentary on Aristotle's Physics ':

"If there are so many things, so it must necessarily give just as much as things really exist, no more, no less. But if there is so much things as there are just so they are limited [ in number ].

If there is much, as the being [ in number ] is unlimited. For between individual things are always other and between those others. And thus the being is unlimited. "

The argument could be the underlying concept that distinct things when they are not separated by a third thing, One, are associated with a rejection of the idea of ​​empty space. The contradiction occurs because a certain finite number of things draws the existence of an unlimited, infinite number of things by themselves.

Argument to the finite size

The argument of the finite size has also been narrated in part by Simplicius ' commentary. First Zenon shows that if there is much that this can not have a size. (Up here Simplicius summary only, without citing the evidence. Below he then quotes verbatim. ) Zenon argued then that something that had no size, is just Nothing. In a third step, he concludes

Every one of its parts, "Is [ multiplicity ] is present, must have a certain size and thickness and distance ( apechein ) from the other have. And the same may be said of the past before that part. Even this will naturally have size and there is another lie before him. So the same is true once and for all. Because there is no such part thereof [ the whole ] will form the outermost border, and never will be the one without relation to the other be. So if there are many things, they must necessarily be both small and great: small to nullity, great to infinity ".

The interpretations of this argument are inconsistent. According to popular interpretation, where apechein ( ἀπέχειν ) is translated as be separated from each other by distance - as in the above translation of Diels - the argument is to be understood thus: things if they are distinct, separated, and then you have something between them lie. This something is different from the above two things, so it must again - ad infinitum a thing separate them. In this interpretation, the paradox is generally been dismissed as fallacy.

Others disagree with this interpretation, look at the context and understand apechein (Synonym for proechein ( προέχειν ) ) refer to the position of parts of a subdivision. The key point is replaced in the translation of Vlastos the form

"So if [ many ] exist, each [ existing ] must have some size and bulk and some [part of each ] must lie beyond ( ' apechein ') another [part of the same existent ]. And the same reasoning [' logo '] holds of the Projecting [part ]: for this too will have some size and some [ part] of it wants to project. Now to say this once is as good as saying it forever. For no such [part -that is, no part Resulting from this continuing subdivision ] wants to be the load nor wants one [ part] ever exist not [ similarly ] related to [ did is, Projecting from] another. [ ... ] THUS, if there are many, They Must be Both small and great. "

Here are two interpretations are distinguished according to Abraham: The division on the edge and the division through and through.

Division at the edge

This describes Vlastos with the following picture: Imagine a rod before, divide it into two equal parts, take the right part, and share it again, and so on ad infinitum. The resulting parts can be added according to the laws of the exhaustion or the limit very well, according to the arithmetic concept of the geometric series. In this interpretation, Zeno uses an analogue argument as in two of his motion paradoxes, the division of paradox and the paradox of Achilles and the tortoise.

Division through and through

The situation is different when the procedure is reapplied to the division of all such parts, when the rod is divided by and by in items. How exactly Zenon comes to the opposition in the last sentence is not clear, even under these assumptions.

A modern, well-meaning interpretation of dividend is the procedure analogous to the nested intervals. Dividing the interval [0, 1], [0, 1/2] and [1/ 2, 1], and the resulting parts, in turn, ad infinitum, to obtain chains of intervals, each of which can be smaller by half, for example,. Suppose the interval is divided through and through, so formed any of these chains.

The amount of the resulting chain is uncountable: In every chain of nested intervals a point can only lie. ( Assume that there are two points which are contained in each of the intervals, then there is a positive distance between them, but there are in the nest of intervals an interval as has a shorter length, so it can not contain both points. ) The summation therefore can not be solved by means of the limit form a ( countable ) infinite series as Achilles and the tortoise.

In this form the thought experiment is very similar to the argument of the whole division.

The argument of the complete division

The argument can be found in Aristotle in De generatione et corruptione and very similar in Simplicius, it has of Porphyry. Simplicius writes it to unlike Porphyry Zenon, it is also similar to the third version of the argument of finitude. Aristotle does not mention Zenon in connection with this thought. Regardless of the interpretation of the traditional paradox of Simplicius, Aristotle's example of Zeno's thought is influenced. Starting from the idea that a line was through and through ( Pantei ) divided into infinitely many things, argued Aristotle:

"What then, Will Remain? A magnitude? No: that is impossible, since then there will be something not divided, Whereas ex hypothesis the body which divisible through and through. But if it be neither a body nor did Admitted a magnitude Will Remain, and yet division is to take place, the Constituents of the body will be Either points (ie without magnitude ) or absolutely nothing. "

The only way for Aristotle, is not to be a line than the sum of its points and a consequent rejection aktualen divisibility of the line in infinitely many things. According to Aristotle, this was the argument which had made the introduction of atomic sizes ( atomism ) are required.

Zeno Maßparadox

As Zeno 's paradox of measure is called a relevance to the mathematics of the present synthesis of the problem of complete division. In the illustration according Skyrms:

Suppose that a line segment can be divided by and by into infinitely many different but similar parts, with identical means that they have the same length. Especially for them the concept of length is explained sense. Applies an axiom of unrestricted additivity - the length of the whole is the sum of its parts, even if an infinite number of parts in the game are -, we obtain a contradiction as follows:

According to the axiom of Euxodos then the length of the parts either a positive number, or it is and according to their sum either or both a contradiction to the finite, but different from 0 length of the line segment.

In the integration and measure theory now the axiom of unrestricted additivity is replaced by a tighter formulation, in contrast to Aristotle's solution or the way out of the atomists.

Giuseppe Peano and Camille Jordan defined the length of a line or set of points on the number line as the common limit of two approaches: less than any covering of the set with a finite number of disjoint intervals, greater than any exhaustion of the amount of such - and get the content function, a well-defined, finite additive set function, the Jordan content. The Zeno paradox is avoided, at the cost that no longer has a lot of content; even the set of irrational numbers in the unit interval is not jordan measurable.

Later, Émile Borel and Henri Lebesgue showed when she founded the measure theory that a theory of length can be defined also for lot of features that satisfies the stronger requirement of countable additivity (stronger than finite additivity, weaker than full additivity ). This approach brought important benefits in the first place, the positive consequence that under this concept, the most typically occurring, though not all quantities are measurable, the set of irrational numbers in the unit interval.