Re: How to do magic with infinity

From: Han de Bruijn (Han.deBruijn_at_DTO.TUDelft.NL)
Date: 10/28/04 

Date: Thu, 28 Oct 2004 09:43:06 +0200

Frank Piron wrote:

> Am Wed, 27 Oct 2004 10:05:05 +0200 schrieb Han de Bruijn
> <Han.deBruijn@DTO.TUDelft.NL>:
>
>>
>> Frank Piron wrote:
>>
>>> To take the point a bit more serious we should consider the
>>> following levels of "non-existence" for a natural number:
>>
>>
>> Hehehehe ... Sigh !
>> Thank you Frank, for jumping in here.
>> Finally someone who wants to take the point a bit more serious.
>>
>> Does a natural number exist if I have a little program to make it
>> as soon as I feel the need, but not earlier ?
>>
>> program tomorrow;
>> var
>> k : integer;
>> begin
>> Write('9');
>> for k := 1 to 1000 do
>> begin
>> Write(Round(10*random));
>
>
> Oops! random is not a number theoretical function, but a real world
> event. So the program tomorrow outputs a "story" (in fact a tiny
> part of our universes history) rather than a specific natural number.
>
>> end;
>> Writeln;
>> end.
>
>
> The question should be: Does a natural number exist if the shortest
> (Turing) program which produces n is too long to fit into the universe?
>
I showed the output of the program to my colleagues and everybody agreed
with me that it is a (big) natural number. Nobody felt the need to say
that it is just the *picture* of a natural number. If I hadn't shown you
the program, you wouldn't even have noticed that it's just the printout
of a random sequence of digits. If I had presented you a thousand digits
of Pi instead, say the 1000'th until the 2000'th digit, you wouldn't
have noticed the difference between those and my random sequence.

So what are you talking about ? I stay with my point that my output IS
actually a natural number. I can write it up. I can define additions and
multiplications with such numbers. They DO fit into my computer's memory
(which is BTW part of the universe ! And, please , don't rely on those
old fashioned & slow computing beasts called Turing machines :-)

Han de Bruijn