Originally Posted by All Systems Go
I found his History of Western Philosophy to be very concise and very readable, yet this book was printed in 1919 and I don't know to what extent mathematics has changed or developed in the ensuing 89 years.
I haven't read this particular book and I don't really know much about it but here goes anyway.
Russell, with Alfred North Whitehead, was trying to prove the internal consistancy of mathematics using as few axioms as possible. Using set theory and building on the earlier work of Frege, he found some paradoxes (mostly of self-reference I think) and therefore failed. This seemed to be a popular pasttime, I think David Hilbert was trying the same thing with geometry.
Later, in the 1930s, Paul Gödel proved that the task was impossible with his incompleteness theorem. I think this came as a bit of a shock to mathematicians.
The was a ressurgance of multi-valued logic in the 60s, this time dubbed fuzzy logic by Lotfi Zadeh. This does away with the law of excluded middle, which I believe to be the cause of Russell's paradoxes. Later work by Bart Kosko introduces the concept of fuzzy containment, which I think should provide an escape from the paradox, but I haven't seen anything directly say this. It might just be a cop-out of saying 'don't know' or 'maybe'.
I don't know if fuzzy logic had a big impact on maths as a whole but it did on me personally, I really should read more on it.
I'd be interested to know more about the book and whether you agree with the things it says.