Plato's famous student

Aristotle(384-322) was Plato's most famous student and rival. He studied at Plato's academy in Athens from the age of 18 until Plato's death in 347 (20 years total). He was the private teacher of Price Alexander, who later became a  successful king conquering the Mediterranean world. Once Alexander took over the throne, Aristotle became his trusted advisor. Later, like Plato, Aristotle founded his own school in Athens, the Lyceum, where he spent the rest of his life writing and teaching. Aristotle introduce written examination. In regards to his studies, Aristotle studied 3 related topics in mathematics: incommensurable, infinity(potential and actual infinity) and continuity.

He believed that logical arguments should be constructed out of syllogisms. Syllogism are "statements that are taken as true and certain other statements that are then necessarily true"(pg 43 Katz). In a modern way, they are the if-then statement students learn in an introduction proof class. An example the author gives in the book is 'if all monkeys are primates, and all primates are mammals, then it follows that all monkeys are mammals." (pg 43)

Aristotle distinguished between a postulates and an axioms. To a modern students they are just facts to memorize but he regarded a postulate as "basic truths that are peculiar to each particular science" and an axiom as truths that are common to all.

Numbers and Magnitude was also clarified by Aristotle. Magnitude(which is continuous) is "divisible into divisibles that are infinitely divisible while a number(which he claimed is discrete) are made up indivisible units.

Two things are in succession if there is none of their kind in between them. Aristotle claimed things are continuous "when they touch and when 'the touching limits of each become one and the same'" (pg 45 Katz)

Although Aristotle emphasized syllogism Greek mathematicians chose to use other forms to build their logical argument. Chrysippus of Soli(280-206) was one of the more notable people that studied the the basic forms of argument used to construct mathematical proof. The basic rules of inference, which is what Chrysippus called the forms of arguments are based on propositions, statements that can be either true or false.

Rules of inference:

1. Modus ponens: If p, then q. [ex: p. Therefore q]
2. Modus tollens: If p then q.   [ex: not q. Therefor no p]
3. Hypothetical syllogism: If p, then q. If q, then r. Therefore if p, then r.
4. Alternative syllogism: p or q [ex: Not p. Therefore q.]

Unanswered question:

1. Is there an advantage of using the rules of inference over syllogisms since many Greek mathematics did not use the later?
2. Did Aristotle use syllogism himself?

Evaluation:
Interesting: 7
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complexity: 5