[Undated]: α. Cogitations

The elements to which the student at the Pons Asinorum says goodbye [Euclid 1.5] have their counterpart in other mathematical situations. Thus there are the elements that are combined in a particular way to form the notation employed in determinants. But even when the mathematician does not employ the term, there are situations in which it is possible to see that what in fact could with justification be called ‘elements’ are being combined according to certain rules to produce formulas. These elements seem to vary considerably; it may be interesting to collect them together for investigation.

There are numbers; there are letters of the alphabet; there are points, lines, circles, angles; there are signs such as <, +, –; there are letters of the Greek alphabet, capital and lower case. All without exception are written, and though the marks that the writer makes have names – such as ‘greater than’, ‘plus’, ‘minus’, ‘alpha’, ‘beta’ – in fact mathematicians do not appear to use these signs for talking in the way that words are used in conversation. It must be supposed, therefore, that the object in synthesizing these elements is different from the object in verbalization, though the synthesizing process seems similar.

Why do not mathematicians speak mathematics? Is it that a string of mathematical formulas cannot be made to say, “It's a nice day”? Is the vocabulary not big enough? No: it must obviously be that its primary purpose is not conversational, although it is clear that one of the functions of mathematics is public-ation.

We shall have to consider three points: (1) the nature of the elements, (2) the nature of the methods employed to bring them together, and (3) the nature of the objects that are created by the syntheses of the elements (Poincaré, Science and Method, p. 30; The Thirteen Books of Euclid's Elements, I.1, I.5, I.47).

It may appear that I am suggesting that synthesis must be associated simply with a bringing together of the elements according to some kno