It may seem remarkable that this independence of demonstra- tion from subject-matter, which is so fundamental that its absence in a demonstration can mean nothing but a flaw or an omission, should have taken so long to assert itself in mathe- matics; but the reason is simple. The older mathematics is full of unanalyzed assumptions. It is only very recently that mathe- maticians have succeeded in making their hypotheses explicit, though they have been trying to do so since the time of Euclid. Their ultimate success has naturally initiated most important reactions in philosophy. For (a), all proof is formal, and the philosopher may therefore abandon the hopeless task of con- structing a theory of " non-formal " proof. And (b), the modern mathematician separates the mathematical matter from the logical form, and requires logic to give an analysis of the latter. This has inevitably brought about a complete " renaissance " of logical studies, as the Aristotelian logic was entirely unequal to the task, on account both of its unsoundness on many points and of its total omission of relations.
Mathematics thus comes to appear as a beautiful logical exer- cise, which consists in developing the implications of various sets of formal premises (i.e. premises where the terms, other than logical, are variable, or symbols without assigned meanings). It may be pointed out that the motive for the choice of these premises, as well as for drawing certain consequences prefera- bly to others, must lie in the region of possible meanings which mathematics itself ignores. Yet there is also a sense of what is formally important and interesting, quite apart from any sub- ject-matter; and this sense, akin to the aesthetic sense, is often what suggests lines of development, and even modifications of the sets of axioms hitherto adopted. Thus Weierstrass says truly that the mathematician is a kind of poet.
But it is clear that some of the possible and indefinite mean- ings of the mathematical terms and axioms, namely their ordi- nary meanings, are of fundamental importance in the fabric of the world. What are these meanings, and how do we know that they satisfy this or that set of axioms? Not even the " pure " mathematician can wholly ignore this question; for the compatibility (or independence) of two given formal premises can be proved only by discovering some meaning which makes both premises true (or one true, and the other false).
Modern research has brought to light a fact, the possibility of which ha'd escaped all previous philosophy, namely, that the ordinary meanings of geometry and arithmetic are of totally different natures, the forme/ being as entirely empirical as dynamics, while the latter is a priori. We shall say nothing here of the ordinary spatial meaning of geometry, as this meaning belongs to physics, and owes no part of its substance to pure rea- son. 1 But we may note that an arithmetical translation, or rather a variety of such translations, can be found for geometrical axioms. Thus all questions as to the compatibility and independence of these axioms can be treated in a purely arithmetical form.
We now pass to the ordinary meanings of the symbols of arith- metic (see 2.523). Negative, fractional, irrational, and com- plex numbers are often regarded as entities whose existence is postulated in order that certain problems should not remain insoluble. But it is possible to point out certain logical combi- nations of integers which possess all the advantages of these hypothetical entities.
Thus, n can be the relation of x to x-\-n, while +re is the con- verse relation. Again, the rational m/n can be the relation of x to y which holds when nx=my. Take now all such rationals arranged in a series by order of magnitude (which is easily denned), and cut this series in two parts, any term of the lower part being inferi- or to any term of the higher part: then the lower parts of all pos- sible cuts or sections can be taken as the real numbers. Irrational numbers correspond to those " sections " in which neither part has either a first or a last term. Finally, a complex number may be regarded as an ordered couple of real numbers. It is important to realize that the integer n,+n, n/l, the real n and n+o.i are entities of different structures, and that addition and multiplication, applied
'See on this subject the philosophical works of Henri Poincare; Our Knowledge of the External World, by B. Russell (ch. iv.) ; The Principles of Natural Knowledge and The Concept of Nature, by A. N. Whitehead.
to different sets of entities, are different operations. The problem is to define these operations for each " extension " of number in such a way that the special properties of the new numbers result from the definitions, and that, at the same time, those among the new numbers which correspond to the old numbers (as, e.g., n/l to ) retain all the properties of the latter.
All arithmetic thus reduces itself to the arithmetic of the natural integers; and this has been shown by G. Peano to be deducible from five premises, in which " number " means " natural integer." These are: (i) o is a number; (2) the suc- cessor of any number is a number; (3) no two numbers have the same successor; (4) o is not the successor of any number; (5) any property which belongs to o, and also to the successor of any number which has the property, belongs to all numbers. Three non-logical expressions occur in these premises, namely, num- ber, o, and successor. It was discovered independently by G. Frege and B. Russell that the ordinary meanings of all three expressions can be defined in terms of those very logical notions which are the constituents of all proof. There is no need to insist upon the importance of this reduction of number to logic. It required an elaborate analysis of the fundamental concepts of logic; it had to meet considerable technical difficulties; and the chain of definitions which lead from logic- to arithmetic is in consequence very complex. We can give but a rough sketch.
It may perhaps first be noted that the logical nature of the fundamental notions of arithmetic need not come to us as a complete surprise. For it is clear that, since each concept has a determinate number of instances, numbers are as universal as concepts. Then the connexion between the number o and the logical notion of negation is obvious; and that between the num- ber r and the logical notion of identity hardly less so. Finally, the fundamental arithmetic operation of addition might well have as its kernel the logical operation and. But let us pass to the actual definitions of the logical theory of number I- DEFINITION i : A relation is said to be one-one when, if x has the relation in question to y, no other term x' has the same relation to y, and x does not have the same relation to any term y' other than y.
DEFINITION 2 : One class is said to be SIMILAR to, OR TO HAVE THE SAME NUMBER AS, another, when there is a one-one relation of which the one class is the domain, while the other is the converse domain (the domain of a relation being the class of those terms that have the relation to some term or other, its converse domain the class of those terms to which some term or other has the relation).
Now we think of a number as of a common property of a group of similar classes. But it is not clear that there is such a property, over and above the relation of similarity running through the group, just as it is not clear that there is a property of direction common to all the members of a group of parallels, over and above the relation of parallelism. A given direction need be nothing more than the group of all parallels to a given line; similarly, a given number need be nothing more than the class of all classes which have that number that is to say, which are similar to any one member of the class. We accordingly adopt:
DEFINITION 3 : A number is the class of all those classes that are similar to (or have the same number as) a given class.
This definition is sufficient, at any rate, for all arithmetic purposes. But it will be noticed that it renders necessary to postulate the existence of instances of every number, in order to obtain an orderly arithmetic. This postulate is " the axiom of infinity."
The number of Definition 3 applies to all classes, i.e. to all concepts; it includes both finite and infinite integers. To obtain the special properties of the finite integers, we must restrict the definition to them; and to do that, we need a logical definition of finitude. A " finite " number is a number of which the " principle of mathematical induction " is true. More strictly, if we adopt I- DEFINITION 4 : ois the number consisting of those classes that have no members;
and (roughly) :
DEFINITION 5 : The successor of the number of a class a is the num- ber of the class consisting of a together with x where x is any term not belonging to a ;
then we say:
