[Haskell-cafe] Functional progr., images, laziness and alltherest

[Brian Hulley]

Piotr Kalinowski wrote:
> On 22/06/06, Brian Hulley <brianh at metamilk.com> wrote:
> ...
>> This doesn't mean that these contradictions reflect reality - just
>> that maths hasn't yet reached a true understanding of reality imho.
>
> Well, I for instance believe that contradiction IS the true nature of
> reality... ;)

Perhaps I should have said: are the *particular* contradictions found in 
maths relevant to reality? (Obviously they are in the sense that maths as an 
artifact of human endeavour is just as much part of reality as anything else 
and reflects aspects of the current and past human condition in the way it's 
formulated, and in turn possibly influences how we perceive reality...)

>
>>
>> For example, why do people accept that infinity == infinity + 1 ?
>> Surely this expression is just ill-typed. infinity can't be a number.
>
> This equation is just a shortcut, so I can't see how can it be
> ill-typed. It means something like: if you add one element to an
> infinite list, will it be longer?

What does your intuition say about this?

> I found the explanation in terms of
> defining bijection between both lists very appealing (along with a
> "metaphor" of taking one element at a time from both lists and never
> being left with one of the lists empty, which was demonstrated here as
> well).

But this explanation might just be vapid sophistry. Do you *really* want to 
trust it? Especially when the explanation makes use of the physical notion 
of "taking one element at a time from both lists". Can we really rely on 
intuitions about taking an element from an infinite list when we are trying 
to prove something counter-intuitive about adding an element to an infinite 
list?

>
> Seems I don't understand your problem with infinity after all :)

Just that there is a conflict with intuition no matter which option you 
choose: if I think that the list would be longer, I have to reject any proof 
to the contrary, but then my intuitions about valid proof are confounded, 
whereas if I accept the proof, my intuition about physical objects is 
confounded: if the list doesn't get longer, then where *is* the thing I 
added to it? Did it just disappear?
So for these reasons I find that infinity is a troublesome concept.

Regards, Brian.
-- 
Logic empowers us and Love gives us purpose.
Yet still phantoms restless for eras long past,
congealed in the present in unthought forms,
strive mightily unseen to destroy us.

http://www.metamilk.com