A large part of the Search for Mathematical Roots focuses on Russell and the development of Principia (PM, hereafter). I found these chapters, which roughly comprise the latter half of the book, to be quite helpful since I’m less familiar with Russell than Frege and Wittgenstein and the chapters do a good job of explaining the influence of PM. In an interesting bit of trivia, Grattan-Guinness says that its name is a nod, not to Newton’s book, but to Moore’s Principia Ethica, Moore being a huge influence on Russell’s philosophical development. Grattan-Guinness makes Russell out to be heavily influenced by Peano. In the historical narrative, Russell’s interests seem to change along the same lines that Peano’s do. An exception to this is that Russell maintains that math is a part of logic whereas Peano thinks they merely overlap. Peano even wrote a paper on “the” in which he gave the same principles for its meaning that Russell did in “On Denoting.” According to Grattan-Guinness, Russell seems to have been familiar with that paper, at least reading it once, but seems to have forgotten about it.

There are two sections devoted to Wittgenstein in the midst of the Russell chapters. The first is on Russell’s early engagement with Wittgenstein, while he was writing the Theory of Knowledge. This section is exceedingly short, for a few reasons. One is that Grattan-Guinness wants to focus on the history as it relates to math and logic more than to epistemology and a lot of Wittgenstein’s criticisms were directed towards the epistemology of that book. Grattan-Guinness does point out that Wittgenstein, not being a logicist, had problems with Russell’s logicism infecting his logic. This was an oddly sharp point about Wittgenstein, since the rest of the Wittgenstein commentary is somewhat lame. The other section on Wittgenstein is an overview of the TLP that stretches over 4 or so pages. This is surprisingly long given the apparently minor role it plays in the story. The interpretation of the TLP presented would not be helpful to anyone not already familiar with the book. He presents Wittgenstein as a “logical monist” without explaining this term. The rest of the exposition of the TLP is unhelpful. Its main purpose seems to be to set the stage for talking about things that Grattan-Guinness thinks Russell and Carnap got right with respect to logic. Oddly, later on Grattan-Guinness says that Quine is a logical monist, possibly in the same sense that Wittgenstein was, although this is hard to discern (and almost certainly false) since it isn’t explained adequately later on either.

One gem of these chapters came out of the section on Russell’s interactions with Norbert Wiener. Wiener came up with an early version of the modern definition of the ordered pair, a definition absent from PM. This was interesting since Grattan-Guinness points out, quite nicely, all the things that were absent from PM, such as a definition of ordered pair in the modern sense. The gem is Wiener’s thesis. He wrote on the differences between the algebraic and mathematical traditions of logic, the latter being that of Frege and PM. (The name, “mathematical logic” seems to me to be not altogether happy, since the algebraic tradition was similarly mathematical. Perhaps it stems from the formulation of parts of mathematics in the axiomatic form of the logic.) His focus was on Schroeder and PM. He never published it because of a cold reception by Russell, not even a survey article. Grattan-Guinness wrote a summary article that I’m interested in tracking down, this topic being a recently developed interest of mine.

There is a large chunk of the book dedicated to investigating the influence of and reactions to PM. It seems that many early reactions to it were largely negative. Many people in the kantian tradition of logic were fairly critical of ex falso. It was nice to see a glimpse of the pre-history of relevance logic in the book. C.I. Lewis was pretty critical of it on this score as well. Many critics were skeptical of the philosophical value of all the axiom chopping that goes on in PM. Sounding quite similar to things one hears, in some circles, about contemporary logic, they thought that development in PM was an interesting mathematical exercise that didn’t end up illuminating key philosophical topics. The logicism of the book was also heavily criticized, on the bases of the presence of the axioms of infinity and reducibility. Grattan-Guinness points out a criticism that was not made. Large parts of mathematics were not treated in PM, or even sketched, so that it was not clear how or whether those parts could be given a logicist treatment. On the one hand, this seems unfair, since there is only so much that Russell and Whitehead could’ve done. They showed how to cast a lot of math in the system of PM. On the other hand, the lack of a treatment of some of the mathematics of their time is surely a failing. Presumably one would expect them to respond that, in principle, the rest of the math could be accommodated like the stuff they’ve already covered.