Monday, September 29, 2008

Two minutes

Here are some more thoughts on Wittgenstein on the foundations of math. For those interested, be sure to check out the prose interpretation of Wittgenstein on math and games at Logic Matters. In part 6 of the Remarks on the Foundations of Math, Wittgenstein presents several thought experiments to probe a cluster of notions, calculation, proof and rule. One of these is two-minute England in section 34.

Two-minute England is described in the following way. God creates a country in the middle of the wilderness that is physically just like England. The caveat is that it only exists for two minutes. Everything in this new country looks like stuff in England. In particular, one sees some people doing stuff that exactly mimics what English mathematicians do when they do math. This person is the one to focus on. Wittgenstein asks, "Ought we to say that this two-minute-man is calculating? Could we for example not imagine a past and a continuation of these two minutes, which would make us call the process something quite different?"

The questions, I take it, indicate doubt that we must take the two-minute-man as calculating. We can, but it is not compulsory. This is because there is no reason, given what we've observed, to think that he must be calculating. In this case there is no fact about it. We might be tempted to attribute calculation to the two-minute-man because we fill out his story with some events leading up to and following this that lead us to think that he is calculating. These events don't happen, since he only exists for two minutes. There is no wider context for this person that settles whether they were calculating, or scribbling, or regurgitating symbols seen elsewhere.

The point of this thought experiment is to present some evidence that calculation is not identifiable with any bit of mere behavior. Connecting this with other sections of part 6, the behavior only becomes calculation when it is connected up with some appropriate purposes or situated in a normative context in which it is appropriate to talk about correct or incorrect calculation. Wittgenstein's focus is to argue that various mathematical notions are like this.

This passage is preceded by one in which Wittgenstein says, "In order to describe the phenomenon of language, one must describe a practice, not something that happens once, no matter of what kind." The two-minute England thought experiment is intended to illustrate this point. I'm not sure that there is anything in the two-minute-man's life that would let us embed it in a practice of some sort.

Connecting the thought experiment up in a nice way with what precedes it requires fleshing out the notion of a practice, which I can't do. There are scattered remarks on that idea in the Remarks, which I haven't begun to put together. Despite this, it seems to me that two-minute England fits together better with what comes earlier and later in this part of the Remarks, namely that calculation isn't just a matter of behavior. This needs to be connected with the wider concern of what it means to follow a rule in order to make it a bit clearer, I think. Incidentally, I think that the rule-following discussion in the Remarks is more accessible than the discussion in the Philosophical Investigations.


Duck said...

I haven't read the Remarks (on my list), but in PI at least the rule-following discussion is supposed to link back to the stuff in §§89-133 about our tendency to assume philosophical "requirements" and the connection between that and the "hardness of the logical must" – which makes that discussion a bit less focused on rule-following in particular. As I remember it that is.

That's why it's so weird to read Kripke saying "wow, look everyone, there's some stuff in PI *before* the private language argument which is clearly related to it!". To which one wants to respond: keep going back – or even (hold onto your hat) start at the beginning of the book and go forward.

Kenny said...

This sounds like Davidson's "swampman", who just emerges one day from the swamp fully formed as a result of some coincidence involving a lightning bolt. He has brain states that are identical to those of people who speak English, and sounds like he's speaking English, and has apparent memories of the past, but can't really be talking about things or remembering things because this is all just a coincidence rather than having the appropriate causal connections.

Shawn said...

I think the Remarks material antedates the PI stuff. The "hardness of the logical must" is certainly in there.

This may be odd, but I haven't read the Kripke stuff since reading PI. I read it before. That description does sound weird though.

It is like swampman a bit. I'm not that familiar with the argument that Davidson wants to use swampman for. What Wittgenstein is arguing here seems to be fairly modest, that behavior of a certain sort isn't sufficient for calculation. This is a sustained theme in part 6. Two-minute England just seemed to me to be a fairly convincing instance. I'm tempted to say there is a difference but I'd have to look at the Davidson to verify this hunch.

I wonder if Davidson was familiar with this. He read some Wittgenstein.