In the words of
Tegmark, let’s assume that the physical world is
completely mathematical; and everything that exists mathematically exists
I have been
thinking along these lines since my days at university - where it occurred to
me that any alternative is mystical.
However, the problem
remains to explain induction - ie the predictability
of the universe. Why is it
that the laws of physics can be depended on when looking into the future, if
we are merely a “mathematical construction” - like a simulation running on a
computer. It seems to me that in
the ensemble of all possible computer simulations (with no limits on the
complexity of the “laws”) the ones that remain well behaved after any given
time step in the simulation have measure zero.
Given the “source
code” for the simulation of our universe, it would seem to be possible to add
some extra instructions that test for a certain condition to be met in order
to tamper with the simulation.
It would seem likely that there will exist simulations that match our
own up to a certain point in time, but then diverge. Eg it is
possible for a simulation to have a rule that an object will suddenly manifest
itself at a particular time and place. The simulated conscious beings
in such a universe would be surprised to find that induction fails at the
moment the simulation diverges.
In other words, at each time step in a simulation the
state vector can take different paths according to slightly different software
programs with special cases that only trigger at that moment in time. It seems that a universe will
continually split into vast numbers of child universes, in a manner
reminiscent of the MWI. However
there is a crucial difference – most of these spin off universes will have
bizarre things happen. It is
difficult to see how a computable system can be tamper proof. How can a past which has been well
behaved prevent strange things from happening in the
In the thread “a possible paradox”, there was talk
about a vanishingly small number of “magical”
universes where strange things happen.
However, it seems to me that the bigger risk is that a “normal”
universe like ours will be the atypical in the
A possible argument is to invoke the anthropic principle – and suggest that our universe is
predictable in order for SAS’s to evolve and
perceive that predictability. However, that predictability only needs to be a
trick – played on the inhabitants for long enough to develop intelligence.
There is no reason why the trick
needs to continue to be played.
I suggest that the requirement of a tamper-proof
physics is an extremely powerful principle. For example, we deduce that
SAS’s only exist in mathematical systems that aren’t
computable. In particular
our Universe is not computable.
- which is what
Penrose has been saying.
I have assumed
that non-computability coincides with being tamper-proof but this is far from
clear. For example, it is
conceivable that the Universe is a Turing machine running an infinite
computation (cf Tipler’s
Omega point), and “awareness” only emerges in the totality of this infinite
computation. Perhaps our
awareness is a manifestation of advanced waves sent backwards in time from the
I think it’s
important to distinguish between an underlying mathematical system, and the
formal system that tries to describe it. I think this is a crucial
distinction. For example,
the real number system can be defined uniquely by a finite set of axioms. Uniqueness is (formally)
provable - in the sense that it can be shown that an isomorphism exists
between all systems that satisfy the axioms. However the real numbers
are uncountably infinite - and therefore are very
poorly understood using formal mathematics - which is limited to only a countably infinite set of statements about them. So formal mathematics should be
regarded as an imperfect and coarse tool which only gives us limited
understanding of a complicated beast!
This is after all what Godel’s incompleteness
theorem tells us.
It is not
surprising that a computer will never exhibit awareness - because it is merely
using the techniques of formal mathematics, and not tapping into the “good