The logical definition of numbers, though it seems an essential support to the theory of infinite numbers, was in fact discovered independently and by a different man. The theory of infinite numbers--that is to say, the arithmetical as opposed to the logical part of the theory--was discovered by Georg Cantor, and published by him in 1882-3.[51] The definition of number was discovered about the same time by a man whose great genius has not received the recognition it deserves--I mean Gottlob Frege of Jena. His first work, _Begriffsschrift_, published in 1879, contained the very important theory of hereditary properties in a series to which I alluded in connection with inductiveness. His definition of number is contained in his second work, published in 1884, and entitled _Die Grundlagen der Arithmetik, eine logisch-mathematische Untersuchung über den Begriff der Zahl_.[52] It is with this book that the logical theory of arithmetic begins, and it will repay us to consider Frege's analysis in some detail. [51] In his _Grundlagen einer allgemeinen Mannichfaltigkeitslehre_ and in articles in _Acta Mathematica_, vol. ii. [52] The definition of number contained in this book, and elaborated in the _Grundgesetze der Arithmetik_ (vol. i., 1893; vol. ii., 1903), was rediscovered by me in ignorance of Frege's work. I wish to state as emphatically as possible--what seems still often ignored--that his discovery antedated mine by eighteen years. Frege begins by noting the increased desire for logical strictness in mathematical demonstrations which distinguishes modern mathematicians from their predecessors, and points out that this must lead to a critical investigation of the definition of number. He proceeds to show the inadequacy of previous philosophical theories, especially of the "synthetic _a priori_" theory of Kant and the empirical theory of Mill. This brings him to the question: What kind of object is it that number can properly be ascribed to? He points out that physical things may be regarded as one or many: for example, if a tree has a thousand leaves, they may be taken altogether as constituting its foliage, which would count as one, not as a thousand; and _one_ pair of boots is the same object as _two_ boots. It follows that physical things are not the subjects of which number is properly predicated; for when we have discovered the proper subjects, the number to be ascribed must be unambiguous. This leads to a discussion of the very prevalent view that number is really something psychological and subjective, a view which Frege emphatically rejects. "Number," he says, "is as little an object of psychology or an outcome of psychical processes as the North Sea.... The botanist wishes to state something which is just as much a fact when he gives the number of petals in a flower as when he gives its colour. The one depends as little as the other upon our caprice. There is