> Simo Kilponen wrote:
> But would 'Type III God' be possible and would that need "alternative
> mathematics' ? Perhaps you philosophers know can such a thing exist and
> what would it be? A set of theorems impossible to prove with theorems from
> another set?
M. Pour-El and I. Richards [Computability in Analysis and Physics, Springer,
Berlin, 1989] have proven that even though solutions of the "wave equation"
behave deterministically, there exist computable initial data (initial conditions)
with the weird property that for a later (computable) time the value of the "field"
is "non-computable" (non-computable evolution). Randomness in physics
correspond to mathematical uncomputability.
N.C.A. da Costa and F.A. Doria [Undecidability and Incompleteness in Classical
Mechanics, International Journal Theoret. Physics, 30 - 1991, p. 1041] have proven
that the problem whether a given Hamiltonian can be integrated, by quadratures,
is "undecidable" (in the sense of the incompleteness theorem).
Now the nature of God (type III) involves the problem of infinity, the problem of
undecidability (which has nothing to do with "truth"), and the problem of randomness.
P. Davies [The Mind of God, Penguin, 1992] wrote that one day J.A. Wheeler
was the subject in the game of the "20 Question". He started asking simple
questions, like "Is it big?" et cetera. Then he guessed "Is it a cloud?". And the
answer was "Yes!" and came out in a burst of laughing. Backstage: no word
had been actually chosen, all the answers to his questions came out randomly,
just keeping consistent with previous answers and Wheeler's questions.
This may be an "alternative mathematics".
J. L. Borges ["Argumentum Ornithologicum", 1952]
"I close my eyes and see birds in flight.
The vision lasts for a second or less;
I do not know how many birds I have seen.
Was it a definite or indefinite number?
This problem is also a question about the existence of God.
If God exists, the number is definite because God knows how many birds I saw.
If God does not exist, the number is indefinite because nobody was able to count the birds.
In that case (let us assume) I saw less than ten birds and more than just one,
but I did not see nine, eight, seven, six, five, four, three or two birds.
I saw a number between one and ten that is neither nine nor eight, nor seven, nor six, nor five etc.
We cannot imagine this total number; therefore, God exists."
[Unfortunately those "uncountable"numbers actually do exist,
Thoralf Skolem called them the "non standard numbers", in 1934]
This archive was generated by hypermail 2b30 : Mon May 28 2001 - 09:59:43 MDT