Aristotle, Детальна інформація

Barnes writes:-

He was a bit of a dandy, wearing rings on his fingers and cutting his hair fashionably short. He suffered from poor digestion, and is said to have been spindle-shanked. He was a good speaker, lucid in his lectures, persuasive in conversation; and he had a mordant wit. His enemies, who were numerous, made him out to be arrogant and overbearing. ... As a man he was, I suspect, admirable rather than amiable.

We have commented above on the disputes among modern scholars as to whether Aristotle wrote the treatises now assigned to him. We do know that his work falls into two distinct parts, namely works which he published during his lifetime and are now lost (although some fragments survive in quotations in works by others), and the collection of writings which have come down to us and were not published by Aristotle in his lifetime. We can say with certainty that Aristotle never intended these 30 works which fill over 2000 printed pages to be published. They are certainly lecture notes from the courses given at the Lyceum either being, as most scholars believe, the work of Aristotle, or of later lecturers. Of course it is distinctly possible that they are notes of courses originally given by Aristotle but later added to by other lecturers after Aristotle's death.

The works were first published in about 60 BC by Andronicus of Rhodes, the last head of the Lyceum. Certainly:-

The form, titles, and order of Aristotle's texts that are studied today were given to them by Andronicus almost three centuries after the philosopher's death, and the long history of commentary upon them began at this stage.

What do these works contain? There are important works on logic. Aristotle believed that logic was not a science but rather had to be treated before the study of every branch of knowledge. Aristotle's name for logic was "analytics", the term logic being introduced by Xenocrates working at the Academy. Aristotle believed that logic must be applied to the sciences:-

The sciences - at any rate the theoretical sciences - are to be axiomatised. What, then, are their axioms to be? What conditions must a proposition satisfy to count as an axiom? again, what form will the derivations within each science take? By what rules will theorems be deduced from axioms? Those are among the questions which Aristotle poses in his logical writings, and in particular in the works known as Prior and Posterior Analytics.

In fact in Prior Analytics Aristotle proposed the now famous Aristotelian syllogistic, a form of argument consisting of two premises and a conclusion. His example is:-

(i) Every Greek is a person.

(ii) Every person is mortal.

(iii) Every Greek is mortal.

Aristotle was not the first to suggest axiom systems. Plato had made the bold suggestion that there might be a single axiom system to embrace all knowledge. Aristotle went for the somewhat more possible suggestion of an axiom system for each science. Notice that Euclid and his axiom system for geometry came after Aristotle.

Another topic to which Aristotle made major contributions was natural philosophy or rather physics by today's terminology. (I [EFR] show my age and the traditional nature of St Andrews University if I remark that in the 1960s a pass in 'General Natural Philosophy' formed part of my degree.) Aristotle looks at matter, change, movement, space, position, and time. He also made contributions to the study of astronomy where in particular he studied comets, geography with an examination of features such as rivers), chemistry where he was interested in processes such as burning, as well as meteorology and the study of rainbows.

As well as important works on zoology and psychology, Aristotle wrote his famous work on metaphysics. This, according to Aristotle, studies:-

... the most general or abstract features of reality and the principles that have universal validity. ... metaphysics studies whatever must be true of all existent things just insofar as they exist, [and] it studies the general conditions which any existing thing must satisfy.

Although Aristotle does not appear to have made any new discoveries in mathematics, he is important in the development of mathematics. As Heath explains in :-

The importance of a proper understanding of the mathematics in Aristotle lies principally in the fact that most of his illustrations of scientific method are taken from mathematics.

Clearly Aristotle had a thorough grasp of elementary mathematics and believed mathematics to have great importance as one of three theoretical sciences. However, it is fair to say that he did not agree with Plato, who elevated mathematics to such a prominent place of study that there was little room for the range of sciences studied by Aristotle. The other two theoretical sciences, Aristotle claimed, were (using modern terminology) philosophy and theoretical physics.

 Heath notes in the introduction to  some of the mathematics referred to by Aristotle in his works:-

... Aristotle was aware of the important discoveries of Eudoxus which affected profoundly the exposition of the Elements by Euclid. One allusion clearly shows that Aristotle knew of Eudoxus's great Theory of Proportion which was expounded by Euclid in his Book V, and recognised the importance of it. Another passage recalls the fundamental assumption on which Eudoxus based his ' method of exhaustion' for measuring areas and volumes; and, of course, Aristotle was familiar with the system of concentric spheres by which Eudoxus and Callippus accounted theoretically for the independent motions of the sun, moon, and planets. ...

The incommensurable is mentioned over and over again, but the case mentioned is that of the diagonal of a square in relation to its side; there is no allusion to the extension of the theory to other cases by Theodorus and Theaetetus...

 Heath  also mentions the mathematics which Aristotle, perhaps surprising, does not refer to. There is:-

... no allusion to conic sections, to the doubling of the cube, or to the trisection of an angle. The problem of squaring the circle is mentioned in connection with the attempts of Antiphon, Bryson, and Hippocrates to solve it; but there is nothing about the curve of Hippias ...

While Heath  discusses the many mathematical references in Aristotle, the book attempts to construct (or reconstruct) a work on Aristotle's view of the philosophy of mathematics. As Apostle writes in:-

... numerous passages on mathematics are distributed throughout the works we possess and indicate a definite philosophy of mathematics, so that an attempt to construct or reconstruct that philosophy with a fairly high degree of accuracy is possible.

We end our discussion with an illustration of Aristotle's ideas of 'continuous' and 'infinite' in mathematics. Heath  explains Aristotle's idea that 'continuous':-

... could not be made up of indivisible parts; the continuous is that in which the boundary or limit between two consecutive parts, where they touch, is one and the same...

As to the infinite Aristotle believed that it did not actually exist but only potentially exists. Aristotle writes in Physics (see for example ):-

But my argument does not anyhow rob mathematicians of their study, although it denies the existence of the infinite in the sense of actual existence as something increased to such an extent that it cannot be gone through; for, as it is, they do not need the infinite or use it, but only require that the finite straight line shall be as long as they please. ... Hence it will make no difference to them for the purpose of proofs.