[FOM] Re: Shapiro on natural and formal languages

[Jeremy Clark]

On Nov 30, 2004, at 2:25 pm, Timothy Y. Chow wrote:

> Joe Shipman clarified what he was asking for; to test my 
> understanding, let me suggest two candidates:
>
> 1. Every finite graph can be embedded in R^3 without crossings.
>
> 2. The trefoil knot cannot be unknotted in R^3.
>
> Both of these are visually obvious, and it seems any kind of 
> formalization is likely to lose something in the translation.  Do 
> these then qualify?
>

Am I alone in thinking that *neither* of these examples are "visually 
obvious"? I doubt if anyone can visualise a finite graph of more than a 
few dozen nodes. I can visualise the induction step in proving 1. 
easily enough, but a proof by induction is already something other than 
entirely visual. (How many steps do I have to visualise?) As for 2., 
all I can do visually is try to unknot it in my imagination, give up 
after a while, and leave it at that. Hardly a proof.How could one 
possibly visualise the *non-existence* of a mathematical object (like 
the unknotting of the trefoil)? Perhaps I have some sort of brain 
damage. Wouldn't surprise me.

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?

Jeremy Clark