Saturday, April 05, 2008

Carnap and foundations

Things are rather busy around here between the end of term, prospectives visiting, and me trying to finish up my model theory work from last semester. I'm having some trouble posting because of this. Some ideas are bouncing around, but I haven't yet posted them. Here's a short bit on one of them.

Several philosophers have leveled objections against parts of Carnap's logical syntax project. Among these include Goedel, Kleene, Friedman, and Beth. Their objections focus on the Carnap's views on foundational issues in the philosophy of math. This comes out especially in the Goedel and Beth objections (the latter of which I want to discuss in more detail later). The Friedman (primarily in his "Tolerance and analyticity in Carnap's philosophy of mathematics") and Kleene objections find tension between the principle of tolerance and foundational issues. All of these objections share a common move, and it is this theme that some of the responses, such as Ricketts's and Goldfarb's, want to dispute. The move is to saddle Carnap's logical syntax project with more interest and involvement in foundational issues than Carnap had.

It sounds weird, in a way, to deny that Carnap had foundational aims. He talks about logicism a lot and seems to adopt that thesis. He wrote a pamphlet entitled "Foundations of Math and Logic." Nonetheless, there is something to the response of denying these foundational aims to Carnap. Carnap's logicism is different from Frege's and Russell's. Carnap doesn't try to reduce math to logic in Frege's sense. I'm not sure why exactly he is a logicist. He wants the mathematical part of any language to be valid or contra-valid, but this is not a reduction to logic. The pamphlet mentioned does not really address foundational issues in the way that Frege or Brouwer did. The parts that talk about foundations are surprisingly short. They seemed to be more interested in dismissing the questions than resolving them.

I wasn't looking for Carnap's discussion of foundational issues in Logical Syntax when I read it However, I was looking for the discussion of foundational issues in "Foundations." From the little bit that was there, it seemed consistent to deny that Carnap wanted to engage in the foundational debates of his time. He rather wanted to sweep them aside with the invocation of his principle of tolerance. I haven't gone back to LSL to check this, but I'm expecting to see a similar pattern there. Are there any places in LSL that are particularly hard to read as anything but Carnap engaging in foundational debates?

It might help to clarify what I mean by "foundational debates." An example would be the disagreement between the intuitionists and the classical mathematicians. The intuitionists, roughly, wanted to reject certain forms of mathematical reasoning and the results that followed from them. The classical mathematicians wanted to keep all of classical mathematics since they regarded it as coherent and correct. Classical math provided more tools, and possibly essential ones, for scientific inquiry. The intuitionists provided arguments that classical math was incoherent (or bad or...) and the classical mathematicians provided responses. By denying that Carnap was engaging in foundational debates, one is denying that Carnap wanted to provide arguments against intuitionism and for classical math. One is denying that he wanted to adjudicate the dispute and settle who had the better arguments. Instead, he invoked the principle of tolerance to sidestep the issues entirely. This is not to say that he doesn't have sympathies. He preferred classical math, at least from LSL. This was not the product of a settled foundational debate though.

2 comments:

Greg said...

A couple of small thoughts.

I think you are certainly right to say that Carnap was not really interested in picking sides in the intuitionist vs. classicist debate (and others like them). But he was (I think) interested in remaining neutral between them, and I think Friedman's saying that Carnap doesn't achieve his aim of neutrality. Friedman could be wrong, but saying Carnap does not engage in foundational debates of the kind you describe does not show Friedman is misguided, I think.

Also: the principle of tolerance is weakened somewhat (I think) in "Foundations", compared to LSL. Carnap says (section 12) that if words are assumed to have some meaning, then we CANNOT build whatever calculus we like. (If we start with a calculus, and specify meanings later, then we are completely free a la LSL.)

keep up the posting on Carnapia!

Shawn said...

I didn't know that the principle of tolerance was weakened in "Foundations." I'm trying to work out some things as regards tolerance.

The line of response I saw Ricketts as pushing in response to Friedman was that Carnap wasn't aiming for neutrality among the different foundational positions. I think Friedman is right that if Carnap was aiming at that, then he failed. Requiring both sides to lay out their positions in accordance with the principle of tolerance requires resources that the intuitionist doesn't use. Suggesting that both sides adopt tolerance, using a language expressive enough to describe the calculi of both sides, is a way of slipping the stronger logic in through the back door, so to speak. I definitely need to go into this in more detail. Maybe I can work out a post on it soon.

More Carnapia to follow.