Symbolic Logic and the Integer

SYMBOLIC LOGIC AND THE INTEGER Symbolic logic attempts to reduce the so-called logical processes of thought to explicit objective form in such wise that logical reasoning may be comparable in its mechanical precision and objective quality with a game such as chess. Here the paper on which the symbolic propositions are written corresponds to the chessboard, and the symbols correspond to the chessmen. There is given an initial set of "true" propositions which corresponds to the initial position of the chessmen with which we start.

Now this logical game and its rules, as ordinarily conceived, do not have any reference to number or even to the integers. It would of course be possible to insert at the outset a given set of true propositions (i.e., postulates) sufficient to characterize the set of integers. It would then be possible in accordance with the strictly prescribed logical processes to define the fractions. In developing the ordinary number system, it would be necessary to deal with an infinite ordered sequence of fractions.

Our method of approach to the ordinary integers was itself of a logical character, since it defined the integers in terms of the notion of one-to-one correspondence of classes, which is a purely logical one. Thus the fundamental and interesting question is sug gested: in a symbolic logic not containing initial propositions which state the existence of the integers, might it not be possible to define the integers in purely logical terms? The kind of defini tion that has been considered by Frege, Russell and Whitehead may be stated as follows in ordinary language : an integer is the class of all classes in one-to-one correspondence with some speci fied finite class. The one-to-one correspondence between two classes is a relation R between the elements of the classes which may be expressed in the form : for each element x of the first class there exists one and only one element y of the second class such that x R y; likewise for each y of the second class there exists one and only one x of the first class such that x R y.

The attempt to devise a satisfactory system of symbolic logic is beset with many difficulties. In particular it must be able to deal with certain paradoxes which have been encountered in the theory of infinite classes. A theory of types of proposition has been devised by Whitehead and Russell with this purpose in view. At the present time there is no general agreement among mathe maticians as to the proper foundation of symbolic logic ; but on the basis of the work of Whitehead and Russell it seems clear that if symbolic logic can be given a definite form, it will prove to be possible to define the positive integers in purely logical terms.

Modular and Hypercomplex Numbers.

There are many generalizations and modifications of number besides those re ferred to above. In particular there are the modular and Galois number fields in which there are only a finite number of marks (numbers) and in which all the usual formal laws are satisfied (see NUMBERS, THEORY OF). There are also various hypercomplex number systems such as quaternions (see QUATERNIONS) in which there are more than the two "fundamental units" such as I and i of the ordinary complex number system. In these hyper complex number systems, however, the commutative law of multi plication does not hold.

Transfinite Numbers.

The above-mentioned directions of generalization are technical in character. Of a much more funda mental nature is the extension of the notion of the integer as the mark of a finite class so as to apply to infinite classes. These transfinite numbers, discovered by G. Cantor, have attracted much interest and have led to certain paradoxes to which we cannot do more than allude. An infinite class has the property that it can be put into one-to-one correspondence with part of itself. This was illustrated above by means of the infinite sequence (see NUMBER SEQUENCES) of integers I, 2, 3, . . . ; but from any infinite class we may remove such an infinite sequence.