[Math Lair] Aristotle

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[Picture of bust of Aristotle]

Aristotle (384–322 B.C.) was one of the greatest philosophers of all time. He was not primarily a mathematician, but ancient Greek mathematics had a strong influence on his works, and he made significant contributions in the field of deductive logic.

Aristotle was born in 384 B.C. at Stagirus in Macedonia. At the age of 17 Aristotle entered Plato's school at Athens and remained there until Plato's death in 347 B.C., first as a student and later as an independent researcher. He then worked on the island of Assos along with a group of other philosophers for a few years. In 343, Philip of Macedon invited Aristotle to Macedon; tradition states that he acted as tutor to Alexander (who would become Alexander the Great) during this time. In 335 Aristotle returned to Athens and established his own school, the Lyceum, which competed with the Academy but pursued a wider range of subjects. He stayed there until 323, when he retired to Chalcis. He died the following year.

Aristotle's works that he had published during his lifetime are lost; the 30 or so works that remain today are lecture notes and other rough work that his followers polished after his death.

Most of Aristotle's writings on logic can be found in a collection of six books known as the Organon. The six books in this collection are Categories, On Interpretation, Prior Analytics, Posterior Analytics, Topics and On Sophistical Refutations (the last of which deals with logical fallacies). There is some mention of logic in Aristotle's other works. For example, in Metaphysics, book 4, he describes two of the three classical laws of thought.

An illustration of three trees According to Aristotle, if we take "three trees" and remove all properties relating to "trees", we are left with the properties of "three".
Also in Metaphysics, Aristotle considers the philosophy of mathematics. Aristotle asserts that mathematical entities are not substances. He disagrees with Plato's view that of the existence of mathematical objects in a formal world, because this view doesn't explain how numbers connect to objects in the real world. Aristotle's view is perhaps best illustrated by an example. As an example, take three trees, such as those in the illustration on the right. If we take "three trees" and remove all of the properties relating to "trees", we are left with the properties of "three". Numbers are physical objects, but a rather abstract type of objects.