going-on which we call the world and which is admittedly not words. Several interesting statements can be made about mathematics considered as a language. First of all, mathematics appears as a form of human behaviour, as genuine a human activity as eating or walking, a function in which the human nervous system plays a very serious part. Second, from an empirical point of view a curious question arises: why, of all forms of human behaviour, has mathematizing proved to be at each historical period the most excellent human activity, producing results of such enormous importance and unexpected validity as not to be comparable with any other musings of man ? Briefly, it may be said that the secret of this importance and the validity of mathematics lie in the mathematical method and structure, which the mathematizing Smith, Brown, and Jones have used - we may even say, were forced to use. It is not necessary to assume that the mathematicians were 'superior' men. We will see later that mathematics is not a very superior activity of the 'human mind', but it is perhaps the easiest, or simplest activity; and, therefore, it has been possible to produce a structurally perfect product of this simple kind.
The understanding and proper evaluation of what has been said about the structure and method of mathematics will play a serious semantic role all through this work, and, therefore, it becomes necessary to enlarge upon the subject. We shall have to divide the abstractions we make into two classes: (1) objective or physical abstractions, which include our daily-life notions; and (2) mathematical abstractions, at present taken from pure mathematics, in a restricted sense, and later generalized. As an example of a mathematical abstraction, we may take a mathematical circle. A circle is defined as the locus of all points in a plane at equal distance from a point called the centre. If we enquire whether or not there is such an actual thing as a circle, some readers may be surprised to find that a mathematical circle must be considered a pure fiction, having nowhere any objective existence. In our definition of a mathematical circle, all particulars were included, and whatever we may find about this mathematical circle later on will be strictly dependent on this definition, and no new characteristics, not already included in the definition, will ever appear. We see, here, that mathematical abstractions are characterized by the fact that they have all particulars included.
If, on the other hand, we draw an objective 'circle' on a blackboard or on a piece of paper, simple reflection will show that what we have drawn is not a mathematical circle, but a ring. It has colour, temperature, thickness of our chalk or pencil mark,. When we draw a 'circle', it is no longer a mathematical circle with all particulars included in the definition,