134 IV. STRUCTURAL FACTORS IN A LANGUAGES
become psycho-logicians also, or that psycho-logicians must study mathematics.
For, let us take a formula which exemplifies mathematics at its best; namely, one and one make two (1 + 1=2). We see clearly that this human product involves a threefold relation: between the man who made it,
let us say, Smith, and the black-on-white marks (A), between these marks and Brown, and between Brown and Smith. This last relationship is the only important one. The marks (A) are only auxiliary and are meaningless by 'themselves'. They would never occur if there were no Smiths to make them, and would be of no value if there were no understanding Browns to use and to appreciate them. It is true that when we take into account this threefold relation the analysis becomes more difficult and must involve a revision of the foundations of mathematics. Although it is impossible to attempt in this book a deeper analysis of these problems in a general way, yet this behaviouristic attitude follows the rejection of the 'is' of identity, and is applied all through this work.
The notion of 'function' has played a very great role in the development of modern science, and is structurally and semantically fundamental. This notion was apparently first introduced into mathematical literature by Descartes. Leibnitz introduced the term. The notion of a 'function' is based on that of a variable. In mathematics, a variable is used as an oo-valued symbol that can represent any one of a series of numerical elements.
It is useful to enlarge the mathematical meaning of a variable to include any oo-valued symbol of which the value is not determined. The various determinations which may be assigned to the variable we call the value of the variable. It is important to realize that a mathematical variable does not vary or change in itself, but can take any value within its range. If a particular value is selected for a variable, then this value, and, therefore, the variable, becomes fixed - a one-valued constant. In the use of these terms, we should take into account the behaviour of the mathematizer. His 'x' is like a container, into which he may pour any or many liquids; but once the selection has been made, the content of the container is one or a constant. So 'change' is not inherent in a variable; it is due only to the volition of the mathematizer, who can change one value for another. Thus, the value changes by quanta, in definite lots, according to the pleasure of the operator. This