[FOM] Could spacetime be discrete?

José Félix Costa fgc at math.ist.utl.pt
Wed Jan 18 12:16:07 EST 2006

Wrt the «Discrete Universe» and to some (interesting!) comments by Apostolos
Syropoulos and others about «avoidance» of hypercomputation, I would like to
emerge from a few proofs I've been handling these days, to say the following
(that I consider fundamental for current discussion).

>>> Pour-El and Richards have «shown» that the wave equation has no
computable solution for some computable, yet not enough smooth, boundary.
This result was considered by Sir Roger Penrose meaningless for his purpose
of tackling a new non-computable Physics. But the remarkable argumentations
in this forum towards a discrete universe will remove Pour-El
and Richards' results and Penrose's ambition from our physical universe.

A curious thing is the following: recently, about 2 years ago, Klaus
Weihrauch proved that Pour-El and Richards' uncomputabilities were due to a
ill-defined concept of computability over the reals. In fact, working with
another definition of computability over Sobolev spaces, Weihrauch was able
to proof that anomalities found by Pour-El and Richards in physical models
due to a non-suitable definition of «computability».

This facts nothing have to due with non-standard physical systems capable of

>>> Warren Smith proved that some physical models are indeed simulable by a
computer: Einstein's general relativity, quantum theory, quantum
relativistic theory... But he also proved that Newtonian gravitation theory
(in the form of the n-body problem) and Navier-Stokes equation are not
simulable by a computer.

In his proofs, he used the physical models without qualitative analysis: the
pure mathematical equations.

Some models are simulable, some are not.

Again, these facts nothing have to due with non-standard physical systems
capable of hypercomputation.

Algorithmic Physics is a «discipline» in itself.
In some sense, physical theories that encode the solution of halting problem
(like Newtonian gravitation theory and Navier-Stokes equation) can be
refuted by Church-Turing postulate --- a computational neo-Popperian tool
(isn't it astonishing?).

Thus, it is not by virtue of the continuum that we have unsimulable
phenomena or hypercomputation within physical theories.

I think this short message might help the discussion.

J. Felix Costa
Departamento de Matematica
Instituto Superior Tecnico
Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL
tel:      351 - 21 - 841 71 45
fax:     351 - 21 - 841 75 98
e-mail:   fgc at math.ist.utl.pt
www:    http://fgc.math.ist.utl.pt/jfc.htm

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