Lecture 9. Aristotle and his ProfoundInfluence

Figure 9.1 Aristotle (384-322 B.C.)

Aristotle Unlike Socrates and Plato, Aristotle was not originally from Athens. Aristotle’sfather was a doctor who lived in Macedon north of Greece. When Aristotle was 18, he leftto study at Plato’s Academy until Plato’s death, but during this time Aristotle was neverchosen to be among its leaders. Soon after Plato’s death, Aristotle left Athens and went toMacedon to be the tutor of the young prince Alexander, who grew up to be Alexander theGreat.

Alexander may not have been particularly interested in learning anything from Aristotle,but they did become friends. When Alexander grew up and became the king, Aristotle wentback to Athens and opened his own school there, the Lyceum, to compete with Plato’sAcademy. Both schools were successful for hundreds years.

Aristotle wrote on mechanics, physics, mathematics, logic, meteorology, botany, psychol-ogy, zoology, ethics, literature, metaphysics, economics, and many other fields.

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Aristotle’s influence on mathematics There is no single book written on mathematicsby Aristotle but discussions of the subject occur in a variety of other books or other occasions.He said in Metaphysica 1-5 that

the so-called Pythagoreans, who were the first to take up mathematics, not onlyadvanced this subject, but saturated with it, they fancied that the principles ofmathematics were the principles of all things.

and that

The mathematical sciences particularly exhibit order, symmetry, and limitation;and these are the greatest forms of the beautiful.

His many points of view have had a profound influence on philosophical and mathematicalthinking and have had a large impact on general science up to now.

• Aristotle viewed the sciences as falling into three types —– theoretical (seeking truth),productive (e.g., arts), and practical (seeking regular human actions). In the theoret-ical category, mathematics is the most exact.

• Aristotle thought that mathematics ideas with abstract concepts, which are derivedfrom properties of physical bodies.

• Aristotle believed that a definition tells us what a thing is but does not assert that thething exists. So giving a definition is not enough, one must also show a thing describedby the definition exists. Leibniz gave a example: one can define a regular polyhedronwith ten faces, but such a figure does not exist. The method of proving existence thatAristotle and Euclid adopted was construction: all mathematical concepts must beconstructed to establish their existence.

Figure 9.2 Regular polyhedron

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• Aristotle distinguished between axioms (truths common to all sciences) and postulates(acceptable first principles for any one science). Some Aristotelian versions were takenup by Euclid.

• Aristotle discussed the fundamental problem of how points and lines can be related.

• Aristotle discussed infinity.

• A major achievement of Aristotle was the founding of the science of logic. His ba-sic principles of logic —– the law of contradiction, which asserts that a propositioncannot be both true and false, and the law of exclude middle, which maintains thata proposition must be either true or false —– are the heart of the indirect methodof mathematical proof. Aristotelian logic remained unchanged until the nineteenthcentury.

• Aristotle thought that logical arguments should follow syllogisms. By syllogisms wemean that one proposition (the conclusion) is inferred from two others (the premises)of a certain form:

Major premise: A general statement.

Minor premise: A specific statement.

Conclusion: based on the two premises.

For example,

Major premise: No geese are felines.

Minor premise: Some birds are geese.

Conclusion: Some birds are not felines.

• In mathematics, he emphasized deductive proof as the sole basis for establishing facts.

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Figure 9.3 Aristotle was the tutor of the young prince Alexander

Great Writer As a great writer, Aristotle’s writings presented a systematic account ofhis views. Aristotle’s systematic treatises may be grouped in several divisions.

• Logic:

– Categories (10 classifications of terms).

– On Interpretation (propositions, truth, modality).

– Prior Analytics (syllogistic logic).

– Posterior Analytics (scientific method and syllogism).

– Topics (rules for effective arguments and debate).

– On Sophistical Refutations (informal fallacies).

• Physical works:

– Physics (explains change, motion, void, time).

– On the Heavens (structure of heaven, earth, elements).

– On Generation (through combining material constituents).

– Meteorologics (origin of comets, weather, disasters).

• Psychological works:

– On the Soul (explains faculties, senses, mind, imagination).

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– On Memory, Reminiscence, Dreams, and Prophesying.

• Natural history:

– History of Animals (physical/mental qualities, habits).

– On the parts of Animals.

– On the Movement of Animals.

– On the Progression of Animals.

– On the Generation of Animals.

– Minor treatises.

– Problem.

• Philosophical works:

– Metaphysics (substance, cause, form, potentiality).

– Nicomachean Ethics (soul, happiness, virtue, friendship).

– Eudemain Ethics.

– Magna Moralia.

– Politics (best states, utopias, constitutions, revolutions).

– Rhetoric (elements of forensic and political debate).

– Poetics (tragedy, epic poetry).

Aristotle’s mistake Aristotle wrote too many books on many different fields, and dueto the limit of scientific knowledge at his time, he sometimes made mistakes. For example,Aristotle thought that stones, apples, and other heavy objects fall down because they seektheir natural place (i.e., at the center of the Earth). As they approached the ground,Aristotle argued, these bodies increased their speed because they were happy to returnhome. 1 Aristotle also claimed that heavier objects fall faster, and the falling speed isdirectly propotional to the weight of an object. More than one thousand year late, Galileo(1564-1642) found this theory was wrong. Here is Galileo’s logic thought. Suppose youtie together two objects, which have different weights. On one hand, if Aristotles law wascorrect, the combined object should fall even faster than any of the two objects; on the other

1Is God a mathematician? Mario Livio, Simon & Schuster Paperbacks, New York-London-Toronto-Sydney, 2010, p. 42.

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hand, since the lighter object would slow down the heavier one, the combined object shouldfall at some intermediate speed. This gives to a contradiction.

Figure 9.4 Aristotle

Remark To close this lecture, we mention Autolycus of Pitane, an astronomer andgeometer, who lived around 310 B.C. He was not a member of Plato’s or Aristotle’s schools,though he did teach one of the leaders who succeeded Plato. Of the three books he wrote,the On the Moving Sphere presupposes theorems of spherical geometry which must havebeen known to the Greeks of that time. The form of the book On the Moving Spheres issignificant.

• Letters denote points on diagrams.

• The propositions are logically ordered.

• Each proposition is stated generally.

• Explicit reference with the figures.

• The proof of each proposition is given.

This is the style Euclid uses later.2

2Morris Kline, Mathematical Thought from Ancient to Modern Times, volume 1, New York Oxford,Oxford University Press, 1972, p.54.

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