For a lucid account intended for a wide readership see Russell's Introduction to Mathematical Philosophy . For greater detail on Russell's position see The Principles of Mathematics , of which the first edition was published while Russell was still struggling for a satisfactory way of avoiding inconsistency in his subsequent masterwork with Whitehead Principia Mathematica . The traces are not covered over in the second edition, which therefore gives a sense of history in the making. Not about mathematical discovery. Only about mathematical theory.
Modern mathematics is founded in set theory.
The existence of sets is not logically necessary. The truths of mathematics are therefore contingent. Logical Systems Specific criticisms of the logical systems devised by Frege and Russell are now more of historical interest than in arguing against Logicism as a current doctrine. Putting aside the defects of these particular systems the most telling argument which remains against logicism is probably that the ontological premises necessary to carry through the development of mathematics cannot be shown to be principles of logic. It is consistent to add to first order logic an axiom asserting that there is only one individual, and it cannot therefore be logically necessary that there are more than one.