On the Origin of Indirect Proof
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概要
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Walter Burkert who is heavily skeptical to the achievements of Pythagoras or Pythagoreans in the history of Greek mathematics rejects in his famous book to give a positive evaluation for Pythagoras' or the Pythagorean's roles in the development of Greek mathematics and suggests another perspective of the development of Greek thinking: "deductive mathematics as well as logic took rise from Parmenides' ontological thinking about eon" i. e. "in discussing Being i. e. his ontological thinking, he discovered the independence of thought and the formal schematism." Examined analytically, his thesis implies the following very strong thesis on Parmenidean logic: No pre-Parmenideans discovered the independence of thought and every deductive mathematics as well as logic took rise from Parmenides' ontological thinking. But, what is then the identity of Eleatic's "dedutive" or "mathematical" logic? In the chapter VI PYTHAGOREAN NUMBER THEORY Burkert uses the term "mathematical logic" as an almost substitutable term for "deductive logic". By the term "mathematical logic" we immediately imagine something algebraic. But, this is certainly not that which Burkert means by the term "mathematical logic". Any algebraic logic is irrelevant to Eleatics, since according to Burkert "in Parmenides fragment there is no word that points toward the field of mathematics" and "Zeno in no way shows his dependence on mathematics." Here, we must be confronted with a paradoxical situation which necessitates us to speak about a "mathematical logic" which is entirely irrelevant to "the field of mathematics." But, then, why should Eleatic logic be called "mathematical logic"? The answer to this question according to Burkert appears very simple. For Burkert, I think, it is self-evident that Eleatic's logic = deductive logic = mathematical logic = logic utilizing indirect proof. Thus, Burkert says: "The attempt at purely logical argumentation, a systematic progression from one thought to another, and the advancement of proofs and conclusions in conscious contradiction to the evidence of the senses make their first appearance in Parmenides... The connection of geometry, and especially that of Hippocrates of Chios, with the logic of the Eleatics is obvious.... Zeno's methods of proof, the reductio ad absurdum and the regressus in infinitum, are basic to all the proofs about irrationality" (pp.424-425). Thus we may conclude that the issue of Burkert's perspective of the development of Greek mathematics depends on whether the thesis concerning Eleatic monopoly of indirect proof holds or not. That is, if we can show that someone of pre-Parmenideans consciously utilize a certain type of the indirect proof in his philosophical or ontological argumentation, Burkert's view will break down. Likewise, if we can verify that in Parmenides' fragments there are some words that points toward the field of mathematics or that Zeno in some way shows his dependence onmathematics, we may reject Burkert's perspective of the development of Greek mathematics. In this paper mainly I would like to deal with the former task and make clear that the deductive character typically seen in Greek thinking is not hallmarked by Parmenides for the first time, but exhibits its symtom ab initio from the time of Anaximander. Parmenides has forged this trend into his own thought pattern explicitly, especially into his principle of the identity of that what can be thought and can be (frag.3). Formulating this principle, Parmenides became the socalled "founder of logic", but it was never the case that in addition to Anaximander's case other paradigmatic thought patterns for his logic were not already there within his reach. I mean, the one case is Xenophanes' thought pattern and another case is explicitly detected in fragments of Heraclitus. And both thinkers were never alien in the formulation of his original thinking.
- 桃山学院大学の論文
- 1991-12-30