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.
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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 sense of the term “nonsense.”
From Frege’s work it followed that arithmetic, and pure mathematics generally, is nothing but a prolongation of deductive logic. This disproved Kant’s theory that arithmetical propositions are “synthetic” and involve a reference to time. The development of pure mathematics from logic was set forth in detail in Principia Mathematica, by Whitehead and myself.
It gradually became clear that a great part of philosophy can be reduced to something that may be called “syntax,” though the word has to be used in a somewhat wider sense than has hitherto been customary. Some men, notably Carnap, have advanced the theory that all philosophical problems are really syntactical, and that, when errors in syntax are avoided, a philosophical problem is thereby either solved or shown to be insoluble. I think this is an overstatement, but
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there can be no doubt that the utility of philosophical syntax in relation to traditional problems is very great.
I will illustrate its utility by a brief explanation of what is called the theory of descriptions. By a “description” I mean a phrase such as “The present President of the United States,” in which a person or thing is designated, not by name, but by some property which is supposed or known to be peculiar to him or it. Such phrases had given a lot of trouble. Suppose I say “The golden mountain does not exist,” and suppose you ask “What is it that does not exist?” It would seem that, if I say “It is the golden mountain,” I am attributing some sort of existence to it. Obviously I am not making the same statement as if I said, “The round square does not exist.” This seemed to imply that the golden mountain is one thing and the round square is another, although neither exists. The theory of descriptions was designed to meet this and other difficulties.
According to this theory, when a statement containing a phrase of the form “the so-and-so” is rightly analysed, the phrase “the so-andso” disappears. For example, take the statement “Scott was the author of Waverley.” The theory interprets this statement as saying:
“One and only one man wrote Waverley, and that man was Scott.” Or, more fully:
“There is an entity c such that the statement ‘x wrote Waverley’ is true if x is c and false otherwise; moreover c is Scott.”
The first part of this, before the word “moreover,” is defined as meaning: “The author of Waverley exists (or existed or will exist).” Thus “The golden mountain does not exist” means:
“There is no entity c such that ‘x is golden and mountainous is true when x is c, but not otherwise.”
With this definition the puzzle as to what is meant when we say “The golden mountain does not exist” disappears.
“Existence,” according to this theory, can only be asserted of descriptions. We can say “The author of Waverley exists,” but to say “Scott exists” is bad grammar, or rather bad syntax. This clears up two millennia of muddle-headedness about “existence,” beginning with Plato Theaetetus.
One result of the work we have been considering is to dethrone mathematics from the lofty place that it has occupied since Pythagoras and Plato, and to destroy the presumption against empiricism which
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has been derived from it. Mathematical knowledge, it is true, is not obtained by induction from experience; our reason for believing that 2 and 2 are 4 is not that we have so often found, by observation, that one couple and another couple together make a quartet. In this sense, mathematical knowledge is still not empirical. But it is also not a priori knowledge about the world. It is, in fact, merely verbal knowledge. “3” means “2 + 1,” and “4” means “3 + 1.” Hence it follows (though the proof is long) that “4” means the same as “2 + 2.” Thus mathematical knowledge ceases to be mysterious. It is all of the same nature as the “great truth” that there are three feet in a yard.
Physics, as well as pure mathematics, has supplied material for the philosophy of logical analysis. This has occurred especially through the theory of relativity and quantum mechanics.
What is important to the philosopher in the theory of relativity is the substitution of space-time for space and time. Common sense thinks of the physical world as composed of “things” which persist through a certain period of time and move in space. Philosophy and physics developed the notion of “thing” into that of “material substance,” and thought of material substance as consisting of particles, each very small, and each persisting throughout all time. Einstein substituted events for particles; each event had to each other a relation called “interval,” which could be analysed in various ways into a timeelement and a space-element. The choice between these various ways was arbitrary, and no one of them was theoretically preferable to any other. Given two events A and B, in different regions, it might happen that according to one convention they were simultaneous, according to another A was earlier than B, and according to yet another B was earlier than A. No physical facts correspond to these different conventions.
From all this it seems to follow that events, not particles, must be the “stuff” of physics. What has been thought of as a particle will have to be thought of as a series of events. The series of events that replaces a particle has certain important physical properties, and therefore demands our attention; but it has no more substantiality than any other series of events that we might arbitrarily single out. Thus “matter” is not part of the ultimate material of the world, but merely a convenient way of collecting events into bundles.
Quantum theory reinforces this conclusion, but its chief philosophi-
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cal importance is that it regards physical phenomena as possibly discontinuous. It suggests that, in an atom (interpreted as above), a certain state of affairs persists for a
