Tuesday, January 13, 2009

Words and other things v2.0

After a couple of years using Blogger, I'm tired of the lack of some key blogging functionality. I've decided it is time for a change. I'm moving over to a Wordpress blog. The new URL is inferential.wordpress.com. Unfortunately, indexical.wordpress was taken. I do, however, have the domain indexical.net, which will redirect to the new blog.

All of my old posts and all of the comments have transferred to the wordpress blog. Unfortunately the links on the author names in the comments didn't transfer. There are also some self-links in the old posts that still refer to the blogspot address. I hope to fix those in the near future. Other than that, things seem to be up and running at the new place, so please update your RSS readers and join me at my new place.

[Edit: Please update your blogrolls as well.]

This blog will remain up although I've shut down new comments.

Sunday, January 11, 2009

Yet more notes on the Search

There was a lot in the Search on Russell, so I will continue my notes mainly on Russell. Read more

One of surprising parts of the book was in the treatment of the meta-/object language distinction. Grattan-Guinness attributes the distinction to Russell. He didn't make it explicitly, although Grattan-Guinness thinks Russell all but said it. The passage that he cites in defense of this is in Russell's introduction to the Tractatus. In talking about the saying/showing distinction, Russell says something about there being an infinity of languages, each language talking about what can show or say in some other language. His comments may have inspired the distinction, although this shouldn't be hard to verify. Did any of the people who initially made the distinction explicit cite the Russell to this effect? Interpreting the Russell in the way Grattan-Guinness proposes seems a little weak, especially given the status of the distinction, nonexistent, in PM. Bernays seems to be cited as the first to make the distinction explicit when he distinguished axioms from rules of inference in 1918. Regardless, Grattan-Guinness takes Russell's comment to undercut the saying/showing distinction in the Tractatus. Carnap, and Goedel elsewhere presumably, similarly undercut the distinction, according to Grattan-Guinness, with the arithmetization of syntax as presented in Logical Syntax. While the book is not focused on Tractatus interpretation, it is unfortunate that the case is presented as so definitively closed on this score.

There was a good, albeit brief, section on Polish logicians post-PM. One of the noteworthy parts was on Lesniewski. He picked up on some of the problems in PM, such as the status of ⊦ in '⊦p'. Russell wanted to follow Frege in viewing it as an assertion sign. Lesniewski proposed abandoning assertion, and related notions, which led to some difficulties in the philosophy of the logic of PM. Of course, as has been pointed out in comments here, the notion of assertion in logic has been resurrected by constructive type theorists. I'm not clear on how the type theorsists' view of assertion connects to what is roughly found in PM. I'd like to get clearer on it to see if there is something from Frege/PM that can be salvaged once one has the meta-/object language distinction. Grattan-Guinness does a reasonably good job of conveying what a big deal the distinction was in the development of logic. Grattan-Guinness presents an ample selection of claims from PM which cry out for the distinction.

Grattan-Guinness makes some odd comments about Russell in places. The oddest of which is the following from p. 443: "The emphasis on extensionality [in PM 2nd ed.] hardly fitts well with a logicism in which, for example, non-denumerability is central." This seems odd to me because I'm not sure what non-denumerability has to do with extensionality. They seem to be orthogonal. Similarly odd comments about extensionality were made in the context of discussing the Tractatus. I didn't note any, but I expect there were some made in presenting Quine's extensional revamping of PM for his Ph.D. thesis.

The discussion of post-PM logic is somewhat short, since it is mainly to tie up loose ends from the PM period. Carnap and Quine are treated very swiftly. If one listened only Grattan-Guinness, one might think that the main contribution of Carnap was to write Logical Syntax and realize what a mistake it was. Quine was passed over, mainly talking about how his book Mathematical Logic failed to talk about the incompleteness theorems. For some reason, the incompleteness theorems were called the incompletability theorems, an apt name that I've never come across before. Goedel was given a short and rightfully glowing treatment. The discussion of the incompleteness theorems was a bit shallow, since Grattan-Guinness's main point concerning them was that they refuted Hilbert's program. He speculated that no one at the conference at which Goedel presented the first theorem grasped its significance. This is, from what I'm told, false, since von Neumann was there and apparently suggested to Goedel that he look for an arithmetic statement of the appropriate form, suggesting he did have an idea of what was going on. (I don't have a reference for this handy, but I could probably track it down.)

Friday, January 09, 2009

Even more notes on the Search

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. Read more

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.

Thursday, January 08, 2009

More notes on the Search

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 more

How is this different than is the case in proof theory? In proof theory symbols are often overloaded in the sense that, in a sequent calculus, the symbol on the left means something different than on the right. Two examples are the comma, behaving conjunctively on the left and disjunctively on the right, and the empty sequence, acting as the true on the left and the false on the right. The difference between proof theory and the situation in the 19th century is that the overloading in proof theory is systematic. One can easily figure out what is going on based on its context. Not so with the algebraists. Grattan-Guinness, and Badesa, indicate different points in proofs in which a symbol slides from, say, a propositional interpretation to one that doesn't have a clear propositional meaning. Often times, it seems, the logicians themselves did not make the distinctions and did not notice that they were slipping between the distinct senses. The overloading in proof theory is benign and useful whereas in the case of the 19th century algebraists it was confusing and sometimes hampered understanding.

Other examples are the signs for membership and set-inclusion, also things Frege harped on. Some logicians used membership and inclusion interchangeably. One reason Grattan-Guinness gives is that they were taking membership to be along the lines of the part-whole relationship of mereology, though he didn't use the word "mereology." While for some, this slip was due to unclarity about the concepts involved, this particular instance of overloading wasn't seen as bad. Peano apparently was guilty of it, and Quine, in an article on Peano's contributions to logic, said that he was right do so. One gets an interesting set theory, one Quine liked, if one doesn't distinguish an element from its singleton, and so treating membership the same as single element inclusion.

Wednesday, January 07, 2009

Do the proof

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."

Saturday, January 03, 2009

Notes on the Search for Mathematical Roots

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 more

Frege gets stuck in the middle of the chapter on concurrent developments in math and logic, along with Husserl and Hilbert. Grattan-Guinness is not terribly sympathetic to Frege. He wants to distinguish Frege the mathematician from Frege', the philosopher of language and mathematics that is by his lights mainly a product of 20th century philosophical commentators. He points out Frege's disagreements with some of his contemporaries, such as Cantor and Thomae. The description of the Frege-Hilbert-Korselt exchange was not particularly detailed and he made Hilbert out to be the clear victor.

Two things that Grattan-Guinness repeatedly mentions Frege as objecting to were bad definitional practice, i.e. implicit definitions via axiom systems, and unclear use of symbolism, i.e. overloading symbols. Frege preferred explicit definitions, single biconditionals, and, apparently, wrote a lot about people that cited something as a definition but was not of that form. He did not seem to require that definitional extensions be conservative. From what Grattan-Guinness says, there is no mention of eliminability, although that should follow from the explicit definition. I don't remember if Frege anywhere commented on that. I want to say more on the overloading of symbols at another time. It was a practice that was rampant in the early development of algebraic logic, although it is still widespread in proof theory today. It seemed to lead to more confusion back then, possibly because the concepts involved were ill-understood and less clearly formulated.

One puzzling thing was that Grattan-Guinness made it sound like Frege was against non-Euclidean geometries for some reason and this formed the basis of his criticisms of Hilbert. I'm not sure about the interpretation of Frege's criticisms, but, from what I've heard, one of Frege's theses was on a non-standard geometry of some sort. This makes it difficult to see why Grattan-Guinness portrays Frege the way that he does. One thing, worth drawing attention to, that Grattan-Guinness does is to point out the limits to the scope of Frege's logicism. He says it is limited to arithmetic, possibly meant to encompass the rational and real numbers, and not at all encompassing geometry, probability or most other areas of math.

There was a section on Husserl focusing on his mathematical roots, transition to phenomenology, and exchange with Frege. This section was hard to follow, although I was quite hopeful. I'm not sure the difficulty of this section was due to Grattan-Guinness or due to Husserl. The latter wouldn't be surprising. As near as I can tell, Husserl said nothing interesting about arithmetic.

There is a chapter on Peano, his influence and his followers. The thing that stuck out the most for me from this chapter was on the formulation of arithmetic. Grattan-Guinness lists Peano's axioms for arithmetic and notes that the induction axiom is first-order. This fact garnered a couple of sentences in the book but not much more. I was surprised by this since I had been taught that while the formulation we use is a first-order schema, Peano's original version was a second-order axiom. This appears to be false. I wonder what the origin of the "Peano's axiom was second-order" view is.