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.

No comments: