[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 Nordmark
Institute for Logic, Language and Computation
Universiteit van Amsterdam

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