[FOM] vagueness in mathematics?

[Timothy Y. Chow]

Harvey Friedman wrote:

> The above formulation of WIS, which is about as good very brief 
> formulation that I am familiar with, is itself a (dogmatic) claim that 
> if we follow a given rule then we get a given result.
[...]
> So WIS seems to me to be compelling us to think in a way that is 
> essentially the same as the way that WIS is criticizing.

This is not a flaw in WIS.  It is merely an observation that WIS is a kind 
of reductio ad absurdum argument.  Assume towards a contradiction that it 
makes sense to talk about rules, so that we are free to use rules in our 
discussion.  Then conclude that rules don't make sense, QED.

Henning Basold wrote:
> In this latter view, the sequences that can be expressed depend of
> course of what processes we allow (c.f. lawless vs. lawlike vs.
> computable). But surely, there are systems that allow us to specify a
> process that represents a sequence precisely and without any ambiguity
> (Stream Coalgebras; Stream Differential Equations by Jan Rutten;
> Corecursion Schemes ? la Hagino, Mendler, and others; Copattern Calculus
> by Abel et al.; etc.).

The point of the skeptical argument is precisely to challenge the "surely" 
here.  The skeptical argument can be applied to even the simplest possible 
rule, e.g., the constant sequence 1,1,1,1,1,1,1,1,...  Suppose I tell the 
Martian that the rule is that this sequence is the constant sequence, with 
every member of the sequence being exactly the same as all the others. 
The Martian nods and, when asked to produce more terms of the sequence to 
confirm understanding, generates 1's obediently for a while and then, 
inexplicably to us, puts down a 0.  When challenged, the Martian looks 
surprised and befuddled.  Didn't we say that all the terms of the sequence 
are identical?  Well, then, that rule obviously implies that there should 
be a 0 at that point in the sequence.  Why are we putting something at 
that point in the sequence that is not identical to all the others?  The 
discussion then moves towards what words such as "identical" and "all" 
mean.  But if we didn't have any luck agreeing on the constant sequence 
then the prospects are dim for agreeing on what these words mean.  In the 
Martian's mind, the Earthling "identical" means something like, "identical 
up to the 251st term and then you have to introduce an exception" (cf. 
Goodman's grue paradox).

Tim