METAPHYSICS choice what properties the content of a sign has But these properties are properties of the content, not of the sign itself, and hence, according to the formalist, they will not be properties of the number What we cannot is to give things properties merely by definition In the Grundgesetze Frege uses against the formalists the kind of argument that Wyclif used against the nominalists of the Middle Ages.3 One cannot by pure definition magically conjure into a thing a property that in fact it does not possess—save that of now being called by the name with which one has named it That an oval figure produced on paper with ink should by a definition acquire the property of yielding one when added to one, I can only regard as a scientific superstition One could just as well by a pure definition make a lazy pupil diligent (BLA 11) For Frege, not only numbers but functions too were mind-independent realities Consider an expression such as 2x2 ỵ x This expression splits into two parts, a sign for an argument and an expression for a function In the expressions (2 12 ) ỵ (2 42 ) þ (2 Â 52 ) þ we can recognize the same function occurring over and over again, but with different arguments, namely 1, 4, and The content that is common to these expressions is what the function is It can be represented by 2( )2 ỵ ( ), that is, by what is left of 2x2 ỵ x’ if we leave the xs out The argument is not part of the function, rather it combines with the function to make a complete whole A function must be distinguished from its value for a particular argument: the value of a mathematical function is always a number, as the number is the value of our function for the argument 1, so that (2 12 ) ỵ names the number A function itself, unlike the numbers that are its arguments and its values, is something incomplete, or ‘unsaturated’ as Frege calls it That is why it is best represented, symbolically, by a sign containing gaps In itself, it is not a sign but a reality lying behind the sign It was not only in mathematics that Frege was a resolute realist He extended the notion of function in such a way that all concepts of any kind See vol II, pp 152–3 179