[FOM] Re: Shapiro on natural and formal languages
henriknordmark@mac.com
henriknordmark at mac.com
Wed Dec 1 07:00:04 EST 2004
On Nov 30, 2004, at 2:25 PM, Timothy Y. Chow wrote:
> 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?
I think these are great examples, especially number 2.
Our understanding that the trefoil knot cannot be unknotted in
3-dimensional space does not seem to stem from any kind of linguistic
reasoning, but rather from some sort of visual/sensory-motor
understanding of 3-dimensional space that we acquire in childhood.
In fact, I would not be surprised if chimpanzees could display some
sort of understanding of this fact even though they might not be able
to verbalize this understanding, nor create a formal system that
justifies this fact based on their spatio-temporal intuitions.
> This strikes me as an example of something that is as obviously true
> as any standard axiom.
I agree.
In fact, any formalization of 3-dimensional space that made this fact
false would probably be discarded as violating our intuitions of
3-dimensional space.
[Danger Sign: This is exactly the type of argument that people were
using to undermine the development of Non-Euclidean Geometry]
Although, I think the trefoil example demonstrates the existence of
non-linguistic understanding. I am still a bit skeptical about the
existence of non-linguistic reasoning. To me the very connotation of
the word 'reasoning' implies a rather strong relationship to language,
but perhaps I am being too narrow minded.
When we visualize a trefoil knot and we realize that there is no way of
unknotting it, do we really go through some sort of reasoning process?
I don't know... I can visualize moving the knot in different directions
and failing to unknot it no matter how I try to move it around without
ever having to make use of language. Does this count as non-linguistic
reasoning? Perhaps it does.
Henrik.
Henrik Nordmark
Institute for Logic, Language and Computation
Universiteit van Amsterdam
www.homepage.mac.com/henriknordmark
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