ABSTRACT
8.1. The mathematics of logic and the logic of mathematics. Mathematical Logic has two sides. One, in its roots in the past (we say this because all sciences have a tendency to become self-conscious and, if necessary for their best development, to let the material that was their first business get dropped from view), is the science of the deduction form of reasoning, this science using ideas and operations like those in mathematics and copying mathematics in being system-making. This side, then, is the mathematics of logic. It was in this that Boole and others interested in the algebra of logic were working. This side has two levels: the logic of deduction itself in the form of systems covering statement connections, classes, relations and statements with variables; and the second-order theories of those systems, whose birth came nearly 50 years after that of those first-order systems. The other side of Mathematical Logic is the logic of mathematics. Here there are two chief sorts of questions to which one has given attention, certain points of these being in the same line of direction. One sort of question is that to be answered by the metamathematics of any given branch of mathematics—of arithmetic, Euclid’s geometry, group theory or whatever it may be. The second sort of question is about the bases of mathematics; for example, is it possible to have a system with the property of consistency in which the ideas used are enough and have the right properties so that, firstly, definitions of all the ideas of the common mathematics may be given from these and only these ideas, and, secondly, all the theorems in the common mathematics may be got as theorems from that system stretched to have new definitions resting on the ideas in the old ones? From the time of Schroeder’s death (1902) till now, most work in Mathematical Logic has been in connection with the logic of mathematics, and in 8.2-12.6 we will say something about this work. The reading of this coming discussion will have to be done very slowly if the material is going to be taken in. Because we have little space and because it is possible to get a clear view of what has been produced by the newer work in Mathematical Logic only if one goes through the long stretch of expert details, we will be limiting ourselves to outlining some of the chief directions which the later developments in Mathematical Logic have had.
