There are several interviews in Quine in Dialogue that are worth reading. A 1978 interview with Magee has a bit that made me stop.

I’ve been reading some of Quine in Dialogue. While the interviews are often light, Quine does say some illuminating things about his philosophical views and their development. I’ll post on that later.

I want to write a brief post on the Search for Mathematical Roots with some concluding thoughts on it. There is a lot in the book I didn’t talk about in my other posts on the book. I haven’t touched on the development of set theory in the book. I also haven’t talked at all about the disputes between the so-called part-whole theorists and set theorists. Before I get to the concluding thoughts, I want to talk about the role of the syllogism and the senses of logic in the book. Read the rest of this entry »

There was a lot in the Search on Russell, so I will continue my notes mainly on Russell. Read the rest of this entry »

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.

In my previous post on Search for Mathematical Roots, I mentioned that one of the things that Frege complained about, according to Grattan-Guinness, was the overloading of symbols. I want to expand on that briefly in this post. I said that the same thing was done in proof theory. It seems that this is not completely accurate. The situation, from what I can tell from the Search for Mathematical Roots, was that often logicians, especially in the algebraic tradition, would use one symbol to designate many different things. For example, some would use ‘1’ to designate the universe of discourse as well as the truth-value true. The equals sign ‘=’ would do double duty as identity among elements as well as equivalence between propositions. It would also get used to indicate a definition. This would not be so bad, but often there would not be any clear way to tell what sense a sign would be used in, sometimes appearing in one sentence in multiple ways. Things get hairier when quantifiers are added, since the variables quantified over would sometimes be propositions and sometimes not. Read the rest of this entry »

In Hodges’s model theory book, there is a proof that sort of surprised me. The proof is for lemma 9.1.5 and runs as follows.

A Skolem theory is axiomatized by a set of ∀_{1} sentences and modulo the theory, every formula is equivalent to a quantifier-free formula. Now quote Lemma 9.1.3 and Theorem 9.1.1.

What surprised me is the last sentence. It is more like a recipe, telling you what to do. Some other proofs in the book have this character to a small degree, but this one stood out a little. The proof is fine, but, unlike many other proofs, this one seems to encourage the reader to do the proof as well. Is this sort of “more active” style of writing proofs common? I don’t think I’ve come across it much at all in the things I read. I’d expect the last line of the proof to go: “The result then follows from Lemma 9.1.3 and Theorem 9.1.1.”

I’m reading Grattan-Guinness’s The Search for Mathematical Roots. There is a lot of philosophically interesting material in the book although a decent amount of his commentary on it is not particularly illuminating. Nonetheless, he gives a pretty good sense of the development of certain trends and the development of some concepts. In particular, the development of the algebraic tradition in logic is helpful, especially alongside the first chapter of Badesa’s book. He doesn’t put as fine a point on it as I’d like though. The presentation of the development of Russell’s logicism and his split from his neo-hegelian upbringing is well done. I’m going to write up some notes on the book, which will be spread over a few posts. In this one, I’ll focus on a few sections from the middle of the book. Read the rest of this entry »

I’m slightly late with this, but I’ll go ahead with it anyway. A few posts ago I said that I’d come up with some reflections on the term. This is more for my benefit than for the benefit of others, but someone might find it interesting. Read the rest of this entry »

Reading the Revision Theory of Truth has given me an idea that I’m trying to work out. This post is a sketch of a rough version of it. The idea is that circular definitions motivate the need for a sharper conception of content on an inferentialist picture, possibly other pictures of content too. It might have a connection to harmony as well, although that thread sort of drops out. The conclusion is somewhat programmatic, owing to the holidays hitting. Read the rest of this entry »

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