|To:||Avril Styrman <Avril.Styrman@xxxxxxxxxxx>|
|From:||Pat Hayes <phayes@xxxxxxx>|
|Date:||Fri, 1 Feb 2008 09:26:25 -0600|
At 4:10 PM +0200 2/1/08, Avril Styrman wrote:
Quoting Pat Hayes <phayes@xxxxxxx>:
I see the self-reference here:
There is no point in continuing this conversation. You really should find something out about the topic you are talking about before giving us your views about it. At the very least, it will give you some guidance as to how to phrase your thoughts so that they can be understood by others.
PS. To illustrate, allow me to analyze one paragraph:
1+1=2 is the same as understanding
the difference between X and XX.
No, its not. Is 1*1=1 the same as understanding that difference? After all, this has two '1's in it, just like your XX. The addition equation uses three signs - '+', '=' and '2' - which are not in the X/XX example, and its comprehension depends upon knowing what they mean, which is a lot more than being able to recognize the difference between one thing and two (many insects can do the latter, but none of them know much about numerals). You are right in claiming that the ability to recognize the difference between one thing and two things is very basic in psychology, and indeed appears to be genetic - it can be found in infants only a few months old. But that ability is not the antecedent of a proof.
Because it is so
fundamental, it cannot be proved.
That does not follow. It all depends on what one takes to be so fundamental that all else will be derived from it. This is to some extent arbitrary, of course, but the accepted 'gold standard' is one of the formal set theories, typically ZFC.
If one starts to
prove it somehow, it is clear the the one uses the
It is clear, perhaps, that in order to understand any proof one uses this ability. One also uses other abilities, such as being able to perceive symbolic representations in text or speech. But that is a psychological observation, and has no bearing on the structure or properties of the proof itself, or on what 'provable' means, or on what can or cannot be proved. Computers have no psychology, but can both generate and check proofs.
It does not prove anything; and even as an informal argument, it does not lead one to that conclusion. The conclusion is false, anyway, as I have said: this simple fact can be proved in any formal arithmetic, and without using the fact that X differs from XX as any kind of antecedent.
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