Cardinality issues block and confuse discussions of NKS and the significance of universality.
Some QM fans stump for more than Turing computation as physically real based on the notion that
access to continuum cardinalities can break the limited fetters of countable computables.
Others stick to Turing computables but make more than practical use of its own countable
infinity and span more than anyone will ever span, and pretend things are the same or have
been reduced to one another, sub species infinitum. We even see men educated in computational
complexity theory speak as though anything exponential corresponds to uncountables while
anything polynomial corresponds to countably infinities. Against all these loose associations,
it is necessary to insist forcefully that finite does not mean small. Even finite computation
exceeds all realizable grasp. Against cosmologists dreaming of towers of continuum
infinities and microscopists strident for infinitessimal distinction, both as the supposed
origin of limitations on knowledge or uncertainty in the external world, I must insist that
even denying all such claims a purely finite, discrete, and computable universe has nothing
simple about it. The operative cause of a limit on exhaustive knowledge is not the
hypothetical presence of infinite cardinals of any description, but follows simply and directly
from the term “universe”, and our existing minimal knowledge of its scope and complexity, and
is there already even if every such infinity is denied. People need to give up the equation
between “finite” and “simple”; it is a mere mistake. And if this is established it is
already enough to show that no appeal to experienced limits of knowledge can count as evidence
of any kind, for a real existence for any such hypothetical infinities.
I present a mesh to cover the practical universe and allow for its possible laws and regularities. The
smallest spatial distinction we think can matter for physical behaviors is the planck length.
But I allow for underlying generators below that scale, 100 orders of magnitude smaller. We
experience 3 large scale spatial dimensions, but some theories employ more; I’ll allow for
10,000. We differentiate a zoo of a few score particles, themselves understood as quanta of
fields, on that space. I’ll allow a million, no need to skimp. The smallest unit of time
we think likely to have physical meaning is the planck time, but that’s too long. Slice the
time domain in a manner unique at every single location as specified above, into time units
100 orders of magnitude shorter than the planck time. We can see tens of billions of light
years in any direction, but extend this outward 1000 fold, and allow for every location from
which a light signal might enter a forward position on our light cone 800,000 billion years
in the future, as the spatial extent we care about. Now let us consider every possible field
value for every possible hypothetical field quanta at each such location – quanta to the power
of the number of locations. Those are states. Now let us consider their transitions one
after another, not as compressed by some definite law, but the pure power set, any state can
go to any other state, as a purely formal and one-off transition rule. Allow this transition
to be multivalued and indeterministic, such that the same exact prior state can go to
literally any other distinct subsequent, or otherwise put, multiple by the number of time
instances at each location as though they are all independent. Any regularity actually seen
is a strict compression on this possibility space. Now add elaborate running commentary on
all events as they happen, in 3000 billion billion languages, surrounding the physical text.
All independent and voluminous, billions of times larger than all human thought to date, about
each infinitessimal instance. Don’t worry about where or how the last is instantiated, let
it float above reality in a platonic mathematical realm.
I am still measure zero in the integers. I can fit pure occasionalism in that mesh. I can
fit any degree of apparent indeterminism you can imagine. I can fit all possible physical
theories, true approximate or completely false. But it is all discrete and computable and
moreover, finite. All the realized computations of all the physically realized intelligences
in the history of the slice of the universe observable by us and all of our descendents or
successors for hundred of thousands of billions of years, along with all their aides or
computational devices, cannot begin to span that possibility space – but it is strictly
finite. So, what is it I am supposed to detect operationally, that I can’t fit into a theory
within that mesh, or above it? Notice, I didn’t even posit determinism let alone locality or
the truth of any given theory. It is enough if I can characterize a state by millions of values
at each of an astronomical number of locations. If supposedly I can’t, then no operational
theory is possible period. If any operational theory is possible, it will be strictly less
fine or exhaustive than the thought-experiment mesh given, and strictly more determined or
restrictive, as to transitions that actually occur according to that theory. I further note
that the mesh given is already completely intractable computationally, not because it is
formally noncomputable or has halting problems, let alone because of higher order infinities.
No, it is intractable already, for all finite intelligences and anything they will ever know,
without a single countable infinity. Naturally this does not preclude the possibility of
tractable, even more finite models. But it does show that intractability arises for reasons
of pure scale, within finitude.
Finite simply doesn’t mean small, nor simple.