Foundations of Mathematics

But apart from this difficulty about the consistency of the axioms on which depends, as we have seen, the proof of the con sistency of the axioms of all other branches of mathematics, there is a further reason for being dissatisfied with the axiomatic treat ment of the natural numbers. For if we adopt it, we shall be mean ing by "numbers" any things that satisfy these axioms ; whereas, in fact, there seems to be one definite meaning of number which is of peculiar importance and of which we should expect the mathematician to give an account. For instance, if I say that I have 2 pennies in my pocket, I there use "2" in a definite sense which we all understand. I do not mean by it merely something satisfying certain axioms, because it is easy to see that any axioms satisfied by the series of natural numbers must also be satisfied by, for instance, the numbers from 1 oo onwards ; so that a purely axiomatic treatment of number would not enable us to distinguish between having 2 pennies in our pocket and having 102. Nor is it open to the mathematician to put this distinction aside as be longing to physics or some other branch of science, because he himself needs it in his own mathematical work, since he not only deals with the numbers as things about which he is talking, i.e., as substantives, but also uses them as adjectives in just the ordi nary sense. When he says, for instance, that a quadratic equation has two roots, the two in "two roots" is the same two as that in "two pennies," and it is essential to realize the difference between it and 102, a difference which depends on the individual natures of the numbers and not merely on the axioms they satisfy. (See NUMBER.) Mathematics and Logic.—We are therefore led to investigate cardinal numbers—the kind of numbers that answer the question "How many?"—and by discovering what these are, we shall be able to prove the consistency of Peano axioms. The cardinal numbers bring us to logic; they belong to the terms which we use in any sort of reasoning, such as "all," "some," "not," "or," "class" and "relation"; and Frege has shown that they could in fact be defined in terms of these simpler logical notions, so that to give a clear account of the cardinal numbers and provide a basis from which to deduce their properties, we must make an investigation of formal logic.

This is also indispensable for another reason; we have so far discussed the axioms from which geometry, for instance, can be deduced, but said nothing about the methods of deduction, such as the principle of reductio ad absurdum, which belong to formal logic, but are as much presupposed in the validity of geometry as are the axioms themselves. These principles of deduction can be set out as propositions, containing no notions except the purely logical ones referred to above ; and they can then themselves be made the subject of logical deductions. The propositions of formal logic can, in fact, all be deduced from a small number of primitive propositions, using only two or three particularly simple prin ciples of deduction, e.g., if p is true and p implies q, then q is true.

This leads to a complete merging of mathematics in formal logic; all mathematical propositions can be stated in purely logical terms and deduced from the primitive propositions of logic (terms such as "point" and "plane" being, as explained above, simply replaced by variables). The cardinal numbers can also be

defined in purely logical terms and their properties can be similarly established. It can incidentally be shown that they satisfy Peano's axioms (provided we assume what is called the Axiom of Infinity). But it is no longer necessary to give these axioms, or those for real numbers, a fundamental place in our system of analysis. It is simpler to define the real numbers as definite entities con structed from the cardinal numbers in a definite way, and not merely to regard them as any things satisfying certain axioms.

The Paradoxes.

This reduction of mathematics to formal logic was first projected by Frege and was actually carried out by Whitehead and Russell in their Principia Mathematica (191o). It was, however, faced with various difficulties which were not alto gether overcome, so that no system has been provided which finds general acceptance. These difficulties arise from the so called Paradoxes of the Theory of Aggregates, which are cases in which it seems possible to give logical or mathematical proofs leading to each of two contradictory conclusions, a result which is intolerable to the mathematician as it overthrows his science altogether. Many instances of this seem to be mere quibbles, but more complicated ones cannot easily be distinguished from ordinary mathematics. An easy illustration is provided by the following example, non-mathematical. The word "short" is a short word, but the word "long" is not a long word. This suggests a division of adjectives according as they do or do not have the property which they connote. Words like "short" which apply to themselves let us call autological ; and words like "long" which do not apply to themselves let us call heterological. Now suppose we put the question, "Is the word `heterological' a heterological word?" Then we at once obtain contradictory answers. For if it is heterological, that means that it does not apply to itself, i.e., that it is not heterological ; but if it is not heterological, then it does apply to itself, i.e., it is heterological.

In order to escape these contradictions Whitehead and Russell invented the Theory of Types, of which the essential idea is that a sentence which is perfectly grammatical English may yet be literally nonsense. To say that a class, e.g., the class of things other than men, is a member of itself (i.e., not a man) appears to be a truism, but on the Theory of Types it is really nonsense. To say that Socrates is a man is sense, but to say that a class whether of men or not-men is or is not a man is sheer nonsense, for the class is of a different logical type from Socrates and the same predicate cannot significantly be applied to more than one type of subject. The contradictions arise from sinning against the rules of logical grammar by confusing logical types.

Unfortunately, in order to escape some of the contradictions such as the one about "heterological" above, Whitehead and Russell were obliged to go further than this and introduce dis tinctions and restrictions which had the effect of invalidating some important types of mathematical argument, especially that known as Dedekindian section; to avoid this consequence they introduced an assumption, known as the Axiom of Reducibility, which is generally considered unplausible and unsatisfactory.