History of Western Philosophy - Bertrand Russell [469]
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, and Carnap now agrees, that this is an overstatement, but 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-and-so' 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's 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 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