History of Western Philosophy - Bertrand Russell [468]
The origin of this philosophy is in the achievements of mathematicians who set to work to purge their subject of fallacies and slipshod reasoning. The great mathematicians of the seventeenth century were optimistic and anxious for quick results; consequently they left the foundations of analytical geometry and the infinitesimal calculus insecure. Leibniz believed in actual infinitesimals, but although this belief suited his metaphysics it had no sound basis in mathematics. Weierstrass, soon after the middle of the nineteenth century, showed how to establish the calculus without infinitesimals, and thus at last made it logically secure. Next came Georg Cantor, who developed the theory of continuity and infinite number. 'Continuity' had been, until he defined it, a vague word, convenient for philosophers like Hegel, who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word, and showed that continuity, as he defined it, was the concept needed by mathematicians and physicists. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated.
Cantor also overcame the long-standing logical puzzles about infinite number. Take the series of whole numbers from 1 onwards; how many of them are there? Clearly the number is not finite. Up to a thousand, there are a thousand numbers; up to a million, a million. Whatever finite number you mention, there are evidently more numbers than that, because from 1 up to the number in question there are just that number of numbers, and then there are others that are greater. The number of finite whole numbers must, therefore, be an infinite number. But now comes a curious fact: The number of even numbers must be the same as the number of all whole numbers. Consider the two rows:
1, 2, 3, 4, 5, 6, …
2, 4, 6, 8, 10, 12, …
There is one entry in the lower row for every one in the top row; therefore the number of terms in the two rows must be the same, although the lower row consists of only half the terms in the top row. Leibniz, who noticed this, thought it a contradiction, and concluded that, though there are infinite collections, there are no infinite numbers. Georg Cantor, on the contrary, boldly denied that it is a contradiction. He was right; it is only an oddity.
Georg Cantor defined an 'infinite' collection as one which has parts containing as many terms as the whole collection contains. On this basis he was able to build up a most interesting mathematical theory of infinite numbers, thereby taking into the realm of exact logic a whole region formerly given over to mysticism and confusion.
The next man of importance was Frege, who published his first work in 1879, and his definition of 'number' in 1884; but, in spite of the epoch-making nature of his discoveries, he remained wholly without recognition until I drew attention to him in 1903. It is remarkable that, before Frege, every definition of number that had been suggested contained elementary logical blunders. It was customary to identify 'number' with 'plurality'. But an instance of 'number' is a particular number, say 3, and an instance of 3 is a particular triad. The triad is a plurality, but the class of all triads—which Frege identified with the number 3—is a plurality of pluralities, and number in general, of which 3 is an instance, is a plurality of pluralities of pluralities. The elementary grammatical mistake of confounding this with the simple plurality of a given triad made the whole philosophy of number, before Frege, a tissue of nonsense in the strictest