Zeno's paradoxes of multiplicity

from Wikipedia, the free encyclopedia

Zeno's paradoxes of multiplicity (5th century BC) belong to the paradoxes of Zeno of Elea alongside the more well-known Zenonic paradoxes of movement .

The three paradoxes of multiplicity are handed down in a commentary by the Byzantine philosopher Simplikios on the physics of Aristotle. In fact, Simplikios, who lived about a millennium after Zeno, is the only source that Zeno quotes verbatim in detail. Simplikios seems to have possessed Zenon's work in the original. Simplikios is convinced that all paradoxes have in common that they served the defense of Zenon's friend and teacher Parmenides against his critics. In a didactic poem, this important pre-Socratics, who, like his pupil Zeno, is assigned to the Eleates , contrasts an inconsistent, changing world of perception with an indivisible, eternal and unchangeable being. According to a widespread, but not unproblematic, interpretation of the difficult to access didactic poem, Parmenides represented a strict metaphysical monism, according to which movement and divisibility are merely an illusion.

Zeno tried to prove that Parmenides' position may seem paradoxical, but the opposite, namely the idea that there is both much and the possibility of movement, leads to contradictions and thus indirectly confirms Parmenides . Of the nine preserved paradoxes, out of a total of forty, three deal specifically with the contradictory nature of the notions of plurality and continuity: the argument of density , the argument of finite size, and the argument of complete division . The group of movement paradoxes, Achilles and the turtle , division paradox , arrow paradox , in contrast, deals with the sub-problem of the impossibility of movement.

In contrast to the paradoxes of movement, no uniform designation has established itself in the reception of the paradoxes of plurality; In general, the meaning of the preserved Greek text is much more unclear than the movement paradoxes passed down indirectly by other authors.

Their importance for the mathematics and philosophy of the Greek contemporaries and their later influence are judged differently. The influence on the consequential limitation of Aristotle and Euclid to potential infinities, which was only dissolved with the work of Georg Cantor , cannot be conclusively assessed.

More recently, initiated by the work of Adolf Grünbaum , the paradox of complete division has received new attention in basic mathematical research.

Argument of density

The argument of density is quoted by Simplikios in his commentary on Aristotle's physics :

“If there is much, there must necessarily be just as many things as are actually there, no more, no less. But if there are as many things as there are, they are limited [in number].

If there is much, then that which is [in number] is unlimited. Because between the individual things there are always others and between them there are still others. And thus beings are unlimited. "

- Simpl., Phys. 140 (29), from: The fragments of the pre-Socratics . Greek and German by Hermann Diels. 1. Volume, Berlin 1922, pp. 173-175.

The argument could be based on the idea that different things, if they are not separated by something third, are one, combined with a rejection of the idea of ​​empty space. The contradiction occurs because a certain, finite number of things entails the existence of an unlimited, infinite number of things.

Finite size argument

The argument of finite size has also been passed on in part through Simplikios' commentary. First, Zeno shows that if there is much, it cannot be great. (Up to this point, Simplikios merely summarizes without citing the evidence. In the following he then quotes verbatim.) Zeno then argues that something that has no size is nothing. In a third step he concludes

“If there is [multiplicity], then each of its individual parts must have a certain size and thickness and distance (apechein) from the other. And the same can be said of the part lying before that. Of course, this one will also be large and there will be someone else in front of him. So the same is true once and for all. For no such part of the same will constitute the outermost limit, and one will never be unrelated to the other. So if there are many things, they must necessarily be small and big at the same time: small to nothing, big to infinity. "

- Simpl., Phys. 140 (34), from: The fragments of the pre-Socratics. Greek and German by Hermann Diels. 1. Volume, Berlin 1922, pp. 173-175.

The interpretations of this argument are inconsistent. According to a widespread interpretation where apechein (ἀπιειν) is translated as being separated from one another by a distance - as in the above translation by Diels - the argument is to be understood as follows: Things, if they are different, are separated, and then something must be between them lie. This something is different from the above two things, so again - ad infinitum one thing must separate them. In this interpretation, the paradox has generally been rejected as a fallacy.

Others contradict this interpretation with a view to the context and understand apechein (synonym for proechein (προέχειν)) refer to the location of parts of a subdivision. The key point is given the form in the translation of Vlastos

“So if [many] exist, each [existent] 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 ['logos'] holds of the projecting [part]: for this too will have some size and some [part] of it will 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] will be the last nor will one [part] ever exist not [similarly] related to [that is, projecting from] another. [...] Thus, if there are many, they must be both small and great. "

Here, according to Abraham, a distinction is made between two interpretations: the division at the edge and the division through and through.

Division on the edge

Vlastos describes this with the following picture: Imagine a stick, divide it into two equal parts, take the right part, and divide it again, and so on ad infinitum . The resulting parts can very well be added according to the laws of exhaustion or the limit value, according to the arithmetic concept of the geometric series . In this interpretation, Zenon uses an argument analogous to that in two of his movement paradoxes, the paradox of division and the paradox of Achilles and the turtle.

Division through and through

The situation is different when the procedure of splitting is applied again to all the resulting parts, when the bar is divided through and through into individual parts. How exactly Zeno comes to the contradiction in the last sentence is not clear even under these assumptions.

A modern, well-meaning interpretation understands the process as analogous to interval nesting . If you divide the interval [0, 1] into [0, 1/2] and [1/2, 1] and the resulting parts in turn, ad infinitum, you get chains of intervals which are each halved, for example . Assume that the interval is divided through and through, i.e. every possible chain is formed.

The number of resulting chains is uncountable : in each chain of nested intervals there can only be one point. (Assuming there are two points that are contained in each of the intervals, then there is a positive distance between them , but there is an interval in the interval nesting which is shorter than , i.e. cannot contain both points.) The As with Achilles and the turtle, summation cannot be solved by means of the limit value formation of a (countable) infinite series.

In this form, the thought experiment is very similar to the full division argument.

The argument of full division

The argument is found in Aristotle in De generatione et corruptione and very similarly in Simplikios, who got it from Porphyrios . In contrast to Porphyrios, Simplikios ascribes it to Zenon; it also resembles the third version of the argument of finitude. However, Aristotle does not mention Zeno in connection with this thought. Regardless of the interpretation of the paradox handed down by Simplikios, Aristotle's example is influenced by Zenonic ideas. Based on the idea that a line is divided through and through (pantei) into an infinite number, Aristotle argues:

“What, then, will remain? A magnitude? No: that is impossible, since then there will be something not divided, whereas ex hypothesis the body was divisible through and through. But if it be admitted that neither a body nor a magnitude will remain, and yet division is to take place, the constituents of the body will either be points (ie without magnitude) or absolutely nothing. "

- (GC I 2, 316a15-317a18) In: On the Generation and Corruption, Aristotle. Book I, translation HH Joachim.

The only way out for Aristotle is not to understand a line as the sum of its points and to consistently reject the actual divisibility of the line into an infinite number. According to Aristotle, this was the argument that made the introduction of atomic quantities ( atomism ) necessary.

Zeno's paradox of dimensions

As Zenon's paradox of measure is relevant for mathematics contemporary synthesis of the problem of complete division is called. In the representation after Skyrms:

Assume that a line can be divided through and through into an infinite number of different, but similar parts, whereby similar means that they have the same length. In particular, the concept of length is explained meaningfully for them. If an axiom of unrestricted additivity applies - the length of the whole is the sum of its parts, even if there are an infinite number of parts involved - one gets a contradiction as follows:

According to the axiom of Euxodos , the length of the parts is either a positive number , or it is and its sum accordingly either or , both a contradiction to the finite length of the line, but which is different from 0.

In integration and measure theory, the axiom of unrestricted additivity is now replaced by a narrower 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 value of two approximations - smaller than any coverage of the set with a finite number of disjoint intervals, greater than any exhaustion of the set with such - and get the content function, a well-defined, finitely additive set function, the Jordan content . The Zenonic Paradox is prevented at the cost that not every quantity has any content; even the set of irrational numbers in the unit interval cannot be measured in Jordan.

Later on, Émile Borel and Henri Lebesgue showed , when they founded the theory of measure , that a theory of length can also be defined for set functions that fulfills the stronger requirement of countable additivity (stronger than finite additivity, weaker than unrestricted additivity). This approach brought important advantages, in the first place the positive consequence that under this concept most of the typically occurring, if not all, quantities become measurable, including the set of irrational numbers in the unit interval.

literature

  • Gerhard Köhler: Zenon of Elea. Studies on the 'arguments against multiplicity' and on the so-called 'argument of place'. (= Contributions to antiquity. 330). De Gruyter, Berlin / Boston 2014, ISBN 978-3-11-036292-3 .
  • Brian Skyrms: Zeno's Paradox of Measure. In: Robert Sonné Cohen, Larry Laudan (Ed.): Physics, Philosophy and Psychoanalysis: Essays in Honor of Adolf Grünbaum. Reidel, Dordrecht 1983, ISBN 90-277-1533-5 , pp. 223-254.
  • Gregory Vlastos, Daniel W. Graham: Studies in Greek Philosophy: The Presocratics. Volume 1. Princeton University Press, 1995, ISBN 0-691-01937-1 .

Web links

Individual evidence

  1. ↑ Nick Huggett: Zeno's Paradoxes . In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy . Winter 2010 edition.
  2. ^ Gregory Vlastos, Daniel W. Graham: Studies in Greek Philosophy: The Presocratics. Volume 1. Princeton University Press, 1995, ISBN 0-691-01937-1 , p. 243.
  3. ^ Adolf Grünbaum: Modern Science and the Refutation of the Paradoxes of Zeno. In: Wesley C. Salmon (Ed.): Zeno's Paradoxes. Bobbs-Merrill, Indianapolis 1955.
  4. ↑ Nick Huggett: Zeno's Paradoxes . In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy . Winter 2010 edition.
  5. ^ Kurt von Fritz: Zenon from Elea. P. 3.
  6. ^ Gregory Vlastos, Daniel W. Graham: Studies in Greek Philosophy: The Presocratics. Volume 1. Princeton University Press, 1995, ISBN 0-691-01937-1 , p. 243.
  7. ↑ Karin Verelst: Zeno's Paradoxes. A cardinal problem. 1. On Zenonian Plurality. In: Proceedings of the First International Symposium of Cognition, Logic and Communication. University of Latvia Press, Riga, p. 5. (pdf; 433 kB)
  8. ^ Gregory Vlastos, Daniel W. Graham: Studies in Greek Philosophy: The Presocratics. Volume 1. Princeton University Press, 1995, ISBN 0-691-01937-1 .
  9. ^ Brian Skyrms: Zeno's Paradox of Measure. In: Cohen, Laudan (Ed.): Physics, Philosophy and Psychoanalysis: Essays in Honor of Adolf Grünbaum. Pp. 223-254.
  10. ^ GEL Owen: Zeno and the Mathematicians. In: Proceedings of the Aristotelian Society, New Series, Vol. 58 (1957-1958), pp. 199-222, Blackwell Publishing on behalf of The Aristotelian Society.
  11. ^ Gregory Vlastos, Daniel W. Graham: Studies in Greek Philosophy: The Presocratics. Volume 1. Princeton University Press, 1995, ISBN 0-691-01937-1 , p. 230.
  12. ^ Brian Skyrms: Zeno's Paradox of Measure. In: Cohen, Laudan (Ed.): Physics, Philosophy and Psychoanalysis: Essays in Honor of Adolf Grünbaum. Pp. 223-254.