[FOM] Wittgenstein Inspired Skepticism

[tchow]

Thomas Klimpel wrote:

> I have the impression that the tacit consensus is rather that Turing
> machines successfully nail down what is meant by rule following.

I won't quibble with that.

> However, this doesn't mean that there would be a consensus that this
> solves all problems related to language and meaning. And I cannot see
> why mathematics should be free of vagueness iff Kripkenstein is silly.
> The ontological commitments of mathematics (and set theory) seem to go
> further than the mere existence of Turing machines (or the existence
> of a bedrock non-vague system of syntactic rules).

Certainly, what you say about going beyond Turing machines is true, but 
what I was implicitly referring to was the tendency of mathematicians to 
retreat to Turing machines if you push their backs to a wall.  That is, 
if we were to challenge a mathematician with the examples of vagueness 
that some other FOMers have mentioned, the mathematician would probably 
say something like this, "Well, yes, there is some vagueness, but that 
involves just the informal part of doing mathematics in practice.  In 
the end, the permanent results of mathematical research are encapsulated 
in formal proofs of theorems, and there is no vagueness there."  
Vagueness and strong ontological commitments are tolerated because there 
is a tacit understanding that there is a common bedrock to which we can 
all retreat if disagreements threaten to become severe.  (It is not 
unlike physicists' reaction to probing questions about quantum 
mechanics---they will, if pressed, point out that everyone agrees on how 
the calculations should go and what the experimental predictions of the 
theory are, even if there is disagreement about Copenhagen or Everett or 
pilot waves or whatever.)

> If it would really show that, than it would indeed be a major
> achievement. But then it should be clarified how it is different from
> the skeptic denying any possibility to communicate meaning at all.

As I said, I believe that there is a spectrum or continuum of commitment 
levels.  The skepticism that you're referring to here is targeted at an 
even more fundamental level.  What's different about Kripkenstein is 
that it's targeted precisely at the level (i.e., rule-following) that 
mathematicians instinctively retreat to, as I explained above.  That is, 
Kripkenstein argues that even if you accept the possibility of certain 
basic communication capabilities, there is still a barrier to surmount 
if you want to be able to follow rules.

Tim Chow