The logicist foundations of mathematics

The problem of the logical and epistemological foundations of mathematics has not yet been completely solved. This problem vitally concerns both mathematicians and philosophers, for any uncertainty in the foundations of the “most certain of all the sciences” is extremely disconcerting. Of the various attempts already made to solve the problem none can be said to have resolved every difficulty. These efforts, the leading ideas of which will be presented in these three papers, have taken essentially three directions: Logicism, the chief proponent of which is Russell; Intuitionism, advocated by Brouwer; and Hilbert's Formalism.

Since I wish to draw you a rough sketch of the salient features of the logicist construction of mathematics, I think I should not only point out those areas in which the logicist program has been completely or at least partly successful but also call attention to the difficulties peculiar to this approach. One of the most important questions for the foundations of mathematics is that of the relation between mathematics and logic. Logicism is the thesis that mathematics is reducible to logic, hence nothing but a part of logic. Frege was the first to espouse this view (1884). In their great work, Principia Mathematica, the English mathematicians A. N. Whitehead and B. Russell produced a systematization of logic from which they constructed mathematics.

We will split the logicist thesis into two parts for separate discussion:

1. The concepts of mathematics can be derived from logical concepts through explicit definitions.

2. […]