Le 16-mai-06, à 17:31, Tom Caylor a écrit :
>
> Bruno Marchal wrote:
>>
>> Now I think I should train you with diagonalization. I give you an
>> exercise: write a program which, if executed, will stop on the biggest
>> possible natural number. Fairy tale version: you meet a fairy who
>> propose you a wish. You ask to be immortal but the fairy replies that
>> she has only finite power. So she can make you living as long as you
>> wish, but she asks precisely how long. It is up too you to describe
>> precisely how long you want to live by writing a program naming that
>> big (but finite) number. You have a limited amount of paper to write
>> your answer, but the fairy is kind enough to give you a little more if
>> you ask.
>> You can ask the question to very little children. The cutest answer I
>> got was "7 + 7 + 7 + 7 + 7" (by a six year old). Why seven? It was the
>> age of his elder brother!
>>
>> Hint: try to generate an infinite set S of more and more growing and
>> (computable) functions, and then try to diagonalize it. S can be
>> {addition, multiplication, exponentiation, .... (?)....}. More hints
>> and answers later. I let you think a little bit before. (Alas it looks
>> I will be more busy in may than I thought because my (math) students
>> want supplementary lessons this year ...).
>>
>> Hope this can help; feel free to make *any* comments.
>>
>> Remember that if all this is too technical, you can also just read
>> Plotinus and the (neo)platonist which, accepting comp or weaker form
>> of
>> Pythagorism, do have a tremendous advance on most materialist of
>> today
>> ... I think it could even provide more light on the practical death
>> issue. The role of G and G* is just to get the math correct for some
>> notion of quantifying the 1-person probabilities.
>>
>> Bruno
>>
>> (*)SANE paper html:
>> http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.htm
>> SANE paper pdf:
>> http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.pdf
>>
>> http://iridia.ulb.ac.be/~marchal/
>
> In keeping with the incremental interactive process, here is a first
> guess. You simply start naming off the natural numbers in order.
> After naming each number you say, "That's not the largest possible
> natural number", or "That's not how long I want to live." This
> statement seems to play the role of diagonalization.
But it is not a finite process. The fairy asks you to give a well
defined number, in a finite time.
> The process I've
> just described can be defined with a finite number of symbols (I just
> did it). Thus, in a way you can say I've just "named" the largest
> natural number.
You have just given a procedure for building a bigger number from any
number. The function which send n on n+1 does that trick. But the
fairy asks you for a number, not a function.
>
> First question: Is this the same as Douglas Hoftstadter's supernatural
> numbers (in his book Godel, Escher, Bach)?
I have read that quite good book, but I don't have it under the hand,
and I don't think the big number problem is related to its supernatural
numbers.
> It seems the only way to
> really understand his book is to read it cover-to-cover (because of all
> the acronyms and his defining ideas with stories, etc.). I wish I
> would have read it cover-to-cover when I was young and had lots of time
> on my hands (and lots of spare brain cells).... or may I can just start
> reading it cover-to-cover now and simply ask the fairy for more
> (quality) time as I need it.
Hofstadter wrote a good book, yes, but on the pedagogical side it does
not help so much by diluting the proof of Godel's theorem in many
interesting themes (Bach, Escher, AI, etc.).
>
> Second question: When we switch over from natural numbers to "length
> of life", it seems we need to specify "units of time" in order for the
> specification of length of life to have any meaning.
You are right. Let us take *years".
> This crosses us
> over into the realm of meaning. Length of life has no meaning apart
> from an assignment of meaning or quality to the events that make up
> life. There seems to be some kind of diagonalization going on here (or
> perhaps transcendence, independent from any diagonalization argument).
> What good is MWI "immortality" (or any kind of "immortality") if the
> infinite sum of (units of time) * (quality or meaning) adds up to some
> finite number? Is it really immortality? Life is more than existence.
In the big number problem, immortality is not proposed by the fairy,
what is proposed is just a long but finite life.
Here too the quality is important. To stay a very long time awake in a
coffin is not pleasant. Also, to stay alive for a very long period
makes almost no sense if your brain is limited in space (bounded finite
machine eventually cycle when running a long time. Do you see why?).
The big number problem has been tackled by Archimedes. He got the
number 10^63. This is remarkable if you recall the very bad notation
for number used at that time. Today 10^63, although very big (the
universe seems to exist since less than 10^17 sec and I think that
number has shrinked recently) seems rather little. It is little than a
Googol (10^100) and a Googolplex is far bigger: 10^[a googol]. But
really we will end up with far bigger (but still finite) number once we
will diagonalise on set of growing functions (soon).
Meanwhile just a few questions to help me. They are hints for the
problem too. Are you familiar with the following "recursive" program
for computing the factorial function?
fact(0) = 1
fact (n) = n * fact(n - 1)
Could you compute "fact 5", from that program? Could you find a similar
recursive definition (program) for multiplication (assuming your
machine already know how to add)?
Could you define exponentiation from multiplication in a similar way?
Could you find a function which would grow more quickly than
exponentiation and which would be defined from exponentiation like
exponentiation is defined from multiplication? Could you generalize all
this and define a sequence of more and more growing functions. Could
you then diagonalise effectively (= writing a program who does the
diagonalization) that sequence of growing functions so as to get a
function which grows more quickly than any such one in the preceding
sequence?
I have no precise idea of your background, so ask if there is anything
unclear. Anyone can ask.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Thu May 18 2006 - 09:59:49 PDT