categories: RE: Categories and cognition

[S.J.Vickers@open.ac.uk]

Posina Venkata Rayudu wrote: 
> Pictures, by themselves, don't support propositions
> that take yes-or-no truth-values. A string (a
> concatenation) of images cannot replace well-formed
> formulae, assertion.
> 
> I dare not speculate as to whether non-Boolean
> truth-value object of topos has any bearing on
> Wittgenstein's 'proof against pictures in thinking.' 

I would say this.

It is often useful to work in the so-called geometric logic: not the full
intuitionistic internal logic of toposes but that fragment of it that is
stable under pullback along geometric morphisms. For instance, it is closely
related to continuity, and reasoning geometrically can give automatic
continuity proofs.

Geometric logic does not have negation as a logical connective, and
consequently you approach a proposition asking not "Is this true or false?",
but "What truth do I find in this?" (A concrete example is a computer
program that is taking a long time to complete. It may have gone into an
infinite loop, but you will never discover this. The question "Does this
terminate or not?" is not the useful one in practice; instead you ask "Have
I got a result yet?")

Somehow (and I don't think I can be any less vague here) this approach to
truth feels a more appropriate one for visual imagery, and even for much
verbal imagery, such as that of poetry, philosophy or religion. Is the story
of the Good Samaritan true or false? Quite probably false: it never happened
exactly as stated. But that misses the point that you can still find truth
in it.

Steve Vickers.