[FOM] Re: Shapiro on natural and formal languages

[Timothy Y. Chow]

On Thu, 2 Dec 2004, Jeremy Clark wrote:
> Am I alone in thinking that *neither* of these examples are "visually
> obvious"?

Your criticisms of these examples are good, and are similar to Joe 
Shipman's criticisms.  I think Shipman's original examples, and maybe
my most recent examples, are better.

> I am very skeptical about this debate concerning "visual" proof. I think 
> we tend to assume that the whole process of visualisation is much 
> simpler than it really is, that there *is* a distinctive process called 
> visualisation (roughly: making pictures - perhaps moving pictures - in 
> the brain and then looking at them) that goes on in the mind. It seems 
> to us that this is what goes on, but I doubt if that is exactly the 
> case. The truth is surely that we don't really know what we do when we 
> think about mathematics, not as much as we think we do at any rate. Is 
> Daniel Dennett widely read amongst FOM subscribers I wonder?

I read his "Consciousness Explained."  But what exactly is your objection 
here?  That there is no such thing as a visual proof?  Well, if "proof" 
here refers to the sorts of things that are labeled as such in 
mathematical publications, then there is no doubt that there are visual 
proofs.  Perhaps the objection is that in such cases, the word "proof" is 
a misnomer, and that all such "visual proofs" are not really proofs at 
all, but only heuristic suggestions of how a "real" proof might go? 
Well, it seems to me that the same objection can be levied against 
non-visual proofs too, because there are published proofs that are not 
visual in any way but that use only high-level concepts that are not at 
all trivial to formalize.  But your suspicion seems specific to visual 
proofs in particular, not to "sketchy" proofs in general---i.e., a visual 
proof, even a detailed one, is suspect in a qualitatively different way 
from a non-visual proof (sketchy or not), right?  If so, I'm familiar with 
this point of view, but it seems to me to be an unwarranted prejudice. 
For example, your mention of Dennett would seem to apply to any kind of 
informal proof, not just visual proofs.

In any case, Joe Shipman was also asking for examples where there is no 
question that the visual proof can be formalized, but where the visual
proof seems to give some kind of intuitive understanding that the formal
proof doesn't.  It hardly seems controversial that such things exist,
if we add the proviso that the "intuitive understanding" in question
need only be available to the proper subset of people who have a strong
visual intuition.  (It may be that the readership of this list is biased
towards people who like formal, or "algebraic," proofs, and against those
who comfortably visualize geometric objects in 4 or 5 dimensions.)

Tim