Why Ockham's Razor?
Russell K. Standish
High Performance Computing Support Unit
University of New South Wales
Sydney, 2052
Australia
R.Standish@unsw.edu.au
http://parallel.hpc.unsw.edu.au/rks
Abstract: In this paper, I show why in an ensemble theory of the universe, we should be inhabiting one of the elements of that ensemble with least information content that satisfies the anthropic principle. This explains the effectiveness of aesthetic principles such as
Ockham's razor in predicting usefulness of scientific theories. I also show, with a couple of reasonable assumptions about the phenomenon of consciousness, that quantum mechanics is the most general linear theory satisfying the anthropic principle.
Wigner[12]
once remarked on ``the unreasonable effectiveness of mathematics'',
encapsulating in one phrase the mystery of why the scientific enterprise is
so successful. There is an aesthetic principle at large, whereby scientific
theories are chosen according to their beauty, or simplicity. These then
must be tested by experiment  the surprising thing is that the aesthetic
quality of a theory is often a good predictor of that theory's explanatory
and predictive power. This situation is summed up by William de Ockham
``Entities should not be multiplied unnecessarily'' known as Ockham's Razor.
We start our search into an explanation of this mystery with the anthropic
principle[1].
This is normally cast into either a weak form (that physical reality must be
consistent with our existence as conscious, selfaware entities) or a strong
form (that physical reality is the way it is because of our
existence as conscious, selfaware entities). The anthropic principle is
remarkable in that it generates significant constraints on the form of the
universe[1,9].
The two main explanations for this are the Divine Creator explanation
(the universe was created deliberately by God to have properties sufficient
to support intelligent life), or the Ensemble explanation[9]
(that there is a set, or ensemble, of different universes, differing in
details such as physical parameters, constants and even laws, however, we
are only aware of such universes that are consistent with our existence). In
the Ensemble explanation, the strong and weak formulations of the anthropic
principle are equivalent.
Tegmark introduces an ensemble theory based on the idea that every
selfconsistent mathematical structure be accorded the ontological status of
physical existence. He then goes on to categorize mathematical
structures that have been discovered thus far (by humans), and argues that
this set should be largely universal, in that all selfaware entities should
be able to uncover at least the most basic of these mathematical structures,
and that it is unlikely we have overlooked any equally basic mathematical
structures.
An alternative ensemble approach is that of Schmidhuber's[8]
 the ``Great Programmer''. This states that all possible halting programs
of a universal Turing machine have physical existence. Some of these
programs' outputs will contain selfaware substructures  these are the
programs deemed interesting by the anthropic principle. Note that there is
no need for the UTM to actually exist, nor is there any need to specify
which UTM is to be used  a program that is meaningful on UTM_{1}
can be executed on UTM_{2} by prepending it with another program
that describes UTM_{1} in terms of UTM_{2}'s instructions,
then executing the individual program. Since the set of programs (finite
length bitstrings) is isomorphic to the set of whole numbers ,
an enumeration of
is sufficient to generate the ensemble that contains our universe. The
information content of this complete set is precisely zero, as no bits are
specified. This has been called the ``zero information principle''.
In this paper, we adopt the Schmidhuber ensemble as containing all
possible descriptions of all possible universes, whilst remaining agnostic
on the issue of whether this is all there is.^{1}
Each selfconsistent mathematical structure (member of the Tegmark ensemble)
is completely described by a finite set of symbols, and a countable set of
axioms encoded in those symbols, and a set of rules (logic) describing how
one mathematical statement may be converted into another.^{2}
These axioms may be encoded as a bitstring, and the rules encoded as a
program of a UTM that enumerates all possible theorems derived from the
axioms, so each member of the Tegmark ensemble may be mapped onto a
Schmidhuber one.^{3}.
The Tegmark ensemble must be contained within the Schmidhuber one.
An alternative connection between the two ensembles is that the
Schmidhuber ensemble is a selfconsistent mathematical structure, and is
therefore an element of the Tegmark one. However, all this implies is that
one element of the ensemble may in fact generate the complete ensemble
again, a point made by Schmidhuber in that the ``Great Programmer'' exists
many times, over and over in a recursive manner within his ensemble. This is
now clearly true also of the Tegmark ensemble.
The natural measure induced on the ensemble of bitstrings is the uniform
one, i.e. no bitstring is favoured over any other. This leads to a problem
in that longer strings are far more numerous than shorter strings, so we
would conclude that we should expect to see an infinitely complex universe.
However, we should recognise that under a UTM, some strings encode for
identical programs as other strings, so one should equivalence class the
strings. In particular, strings where the bits after some bit number n
are ``don't care'' bits, are in fact equivalence classes of all strings that
share the first n bits in common. One can see that the size of the
equivalence class drops off exponentially with the amount of information
encoded by the string. Under a UTM, the amount of information is not
necessarily equal to the length of the string, as some of the bits may be
redundant. The sum

(1) 
where p means the length of p, gives the size of the
equivalence class of all halting programs generating the same output s
under the UTM U. This measure distribution is known as a universal
prior, or alternatively a SolomonoffLevin distribution[6].
We assume the selfsampling assumption[5,2],
essentially that we expect to find ourselves in one of the universes with
greatest measure, subject to the constraints of the anthropic principle.
This implies we should find ourselves in one of the simplest possible
universes capable of supporting selfaware substructures (SASes). This is
the origin of physical law  why we live in a mathematical, as opposed to a
magical universe. This is why aesthetic principles, and Ockham's razor in
particular are so successful at predicting good scientific theories. This
might also be called the ``minimum information principle''.
There is the issue of what UTM U should be chosen. Schmidhuber
sweeps this issue under the carpet stating that the universal priors differ
only by a constant factor due to the compiler theorem, along the lines of
where P_{UV} is the universal prior of the compiler
that interprets U's instruction set in terms of V's. The
inequality is there because there are possibly native Vcode programs
that compute s as well. Inverting the symmetric relationship yields:
The trouble with this argument, is that it allows for the possibility that:
So our expectation of whether we're in universe s_{1} or s_{2}depends
on whether we chose V or U for the interpreting UTM.
There may well be some way of resolving this problem that leads to an
absolute measure over all bitstrings. However, it turns out that an absolute
measure is not required to explain features we observe. A SAS is an
information processing entity, and may well be capable of universal
computation (certainly homo sapiens seems capable of universal
computation). Therefore, the only interpreter (UTM) that is relevant to the
measure that determines which universe a SAS appears in is the SAS itself.
We should expect to find ourselves in a universe with one of the simplest
underlying structures, according to our own information processing
abilities. This does not preclude the fact that other more complex universes
(by our own perspective) may be the simplest such universe according to the
selfaware inhabitants of that universe. This is the bootstrap principle
writ large.
An important criticism leveled at ensemble theories is what John Leslie
calls the failure of induction[4,
§4.69]. If all possible universes exist, then what is to say that our
orderly, wellbehaved universe won't suddenly start to behave in a
disordered fashion, such that most inductive predictions would fail in them.
This problem has also been called the White Rabbit paradox[7],
presumably in a literary reference to Lewis Carrol.
This sort of issue is addressed by consideration of measure. We should
not worry about the universe running off the rails, provided it is extremely
unlikely to do so. Note that Leslie uses the term range to mean
what we mean by measure. At first consideration, it would appear
that there are vastly more ways for a universe to act strangely, than for it
to stay on the straight and narrow, hence the paradox.
However, things are not what they seem. Consider an observer looking at
the world around it. Up until the time in question, the world behaves
according to the dictates of a small number of equations, hence its
description is a fairly short bitstring of length n. Next suppose an
irreducibly bizarre event happens. Now, lets be quite clear about this.
We're not talking about some minute, barely observable phenomenon  eg an
electron being somewhere it shouldn't  and we're not talking about a
phenomenon that might be described by adding new physical laws, as in the
explanation of the procession of Mercury by General Relativity. We're
talking about undeniable, macroscopic violations of physical law, for
instance the coalescing of air molecules to form a fire breathing dragon.
Such an event will have a large description, m bits, that will resist
compression.
Consider the expanded space of all bitstrings of length n+m,
sharing a common nlength prefix encoding the laws of physics that
describe the world up until the bizarre event. The observer is a finite
state machine in general, so there are a finite variety of these events that
can be recognised by the observer. In general, the mbit strings will
perceived as random noise by the observer, with a comparative minority being
recognised as vaguely like something (as in Rorshach plots, or shapes in
clouds), and a vastly rarer number having the convincing fidelity necessary
to sustain a belief that the miracle in fact happened.
Thus the initial presumption that law breaking events will outnumber the
law abiding ones is shown to be false. On the contrary, they will be
extremely rare in comparison.
In the previous sections, I demonstrate that members of the Tegmark
ensemble are the most compressible, and have highest measure amongst all
members of the Schmidhuber ensemble. In this section, I ask the question of
what is the most general (i.e. minimum information content) description of
an ensemble containing selfaware substructures.
There are a number of assumptions that need to be stated up front. The
first relates to the nature of consciousness, as referred to by the
Anthropic Principle. We have already stated that the conscious entity must
be performing some kind of information processing, so as to interpret the
universe. Human beings are capable of universal computation and perhaps all
forms of consciousness must be capable of universal computation.
The ability to compute requires a time dimension in which to compute. The
only mathematical structures in the Tegmark ensemble capable of being
observed from within must have a time dimension in which that observation is
interpreted. Denote the state of an ensemble by .
The most general form of evolution of this state is given by:

(2) 
Some people may think that discreteness of the world's description (ie of
the Schmidhuber bitstring) must imply a corresponding discreteness in the
dimensions of the world. This is not true. Between any two points on a
continuum, there are an infinite number of points that can be described by a
finite string  the set of rational numbers being an obvious, but by no
means exhaustive example. Continuous systems may be made to operate in a
discrete way, electronic logic circuits being an obvious example. Therefore,
the assumption of discreteness of time is actually a specialisation (thus of
lower measure according to the Universal Prior) relative to it being
continuous.
The conscious observer is responsible, under the Anthropic Principle, for
converting the potential into actual, for creating the observed information
from the zero information of the ensemble. This can be modeled by a
partitioning for each observable
,
where a indexes the allowable range of potential observable values
corresponding to A, and
is the measure associated with (
).
The
will also, in turn, be solutions to equation (2).
Secondly, we assume that the generally accepted axioms of set theory and
probability theory hold. Whilst the properties of sets are well known, we
outline here the Kolmogorov probability axioms[6]:
 (A1)
 If A and B are events, then so is the intersection
,
the union
and the difference AB.
 (A2)
 The sample space S is an event, called the certain
event, and the empty set
is an event, called the impossible event.
 (A3)
 To each event E,
denotes the probability of that event.
 (A4)
 P(S)=1.
 (A5)
 If
,
then
.
 (A6)
 For a decreasing sequence
of events with
,
we have
.
Consider now the projection operator
,
acting on a state ,
Vbeing an all universes ensemble, to produce
, where
is an outcome of an observation. We have not at this stage assumed that
is linear. Define addition for two distinct outcomes a and b
as follows:

(3) 
from which it follows that
These results extend to continuous sets by replacing the discrete sums by
integration over the sets with uniform measure. Here, as elsewhere, we use
to denote sum or integral respectively as the index variable a is
discrete of continuous.
Let the state
be a "reference state'', corresponding to the certain event. It encodes
information about the whole ensemble. Denote the probability of a set of
outcomes
by
.
Clearly
by virtue of (A4). Also, by virtue of equation (5),

(7) 
Consider the possibility that A and B can be identical.
Equation (7)
may be written:

(8) 
Thus, the set V naturally extends by means of the addition operator
defined by equation (3)
to include all linear combinations of observed states, at minimum over the
natural numbers. If
, then
may exceed unity, so clearly
is not necessarily a possible observed outcome. How should we interpret
these new nonphysical states? The answer lies in considering more than one
observer. The expression
must be the measure associated with a observers seeing outcome A
and b observers seeing outcome B. Since in general in the
multiverse, the number of distinct observers is uncountably infinite, the
coefficients may be drawn from a measure distribution, instead of the
natural numbers. The most general measure distributions are complex,
therefore the coefficients, in general are complex[3].
We can comprehend easily what a positive measure means, but what about
complex measures? What does it mean to have an observer with measure 1? It
turns out that these nonpositive measures correspond to observers who chose
to examine observables that do not commute with our current observable A.
For example if A were the observation of an electron's spin along the
z axis, then the states
and
give identical outcomes as far as A is concerned. However, for
another observer choosing to observe the spin along the x axis, the
two states have opposite outcomes. This is the most general way of
partitioning the Multiverse amongst observers, and we expect to observe the
most general mathematical structures compatible with our existence.
The probability function P can be used to define an inner product
as follows. Our reference state
can be expressed as a sum over the projected states
.
Let
be
the linear span of this basis set. Then,
,
such that
and
, the inner
product
is defined by

(9) 
It is straightforward to show that this definition has the usual
properties of an inner product, and that
is normalized (
).
The measures
are given by
where
is normalised.
Until now, we haven't used axiom (A6). Consider a sequence of sets of
outcomes
,
and denote by
the unique maximal subset (possibly empty), such that
.
Then the difference
is well
defined, and so
By axiom (A6),

(12) 
so
is a Cauchy sequence that converges to
. Hence V
is complete under the inner product (9).
It follows that V^{*} is complete also, and is therefore a Hilbert
space.
Finally, axiom (A3) constrains the form of the evolution operator . Since we suppose that
is also a solution of eq 2
(ie that the act of observation does not change the physics of the system), must be linear. The
certain event must have probability of 1 at all times, so
i.e. is i times a
Hermitian operator.
Weinberg[10,11]
experimented with a possible nonlinear generalisation of quantum mechanics,
however found great difficulty in producing a theory that satisfied
causality. This is probably due to the nonlinear terms mixing up the
partitioning
over time. It is usually supposed that causality[9],
at least to a certain level of approximation, is a requirement for a
selfaware substructure to exist. It is therefore interesting, that
relatively mild assumptions about the nature of SASes, as well as the usual
interpretations of probability and measure theory lead to a linear theory
with the properties we know of as quantum mechanics. Thus we have a reversal
of the usual ontological status between Quantum Mechanics and the Many
Worlds Interpretation.
 1
 J. D. Barrow and F. J. Tipler.
The Anthropic Cosmological Principle.
Clarendon, Oxford, 1986.
 2
 B. Carter.
The anthropic principle and its implications for biological evolution.
Phil. Trans. Roy. Soc. Lond., A310:347363, 1983.
 3
 D. L. Cohn.
Measure Theory.
Birkhäuser, Boston, 1980.
 4
 J. Leslie.
Universes.
Routledge, New York, 1989.
 5
 J. Leslie.
The End of the World.
Routledge, London, 1996.
 6
 M. Li and P. Vitányi.
An Introduction to Kolmogorov Complexity and its Applications.
Springer, New York, 2nd edition, 1997.
 7
 B. Marchal.
Conscience et mécanisme.
Technical Report TR/IRIDIA/95, Brussels University, 1995.
 8
 J. Schmidhuber.
A computer scientist's view of life, the universe and everything.
In C. Freska, M. Jantzen, and R. Valk, editors, Foundations of
Computer Science: PotentialTheoryCognition, volume 1337 of Lecture
Notes in Computer Science, pages 201208. Springer, Berlin, 1997.
 9
 M. Tegmark.
Is "the theory of everything" merely the ultimate ensemble
theory.
Annals of Physics, 270:151, 1998.
 10
 S. Weinberg.
Testing quantum mechanics.
Annals of Physics, 194:336386, 1989.
 11
 S. Weinberg.
Dreams of a Final Theory.
Pantheon, New York, 1992.
 12
 E. P. Wigner.
Symmetries and Reflections.
MIT Press, Cambridge, 1967.
I would like to thank the following people from the ``Everything'' email
discussion list for many varied and illuminating discussions on this and
related topics: Wei Dai, Hal Finney, Gilles Henri, James Higgo, George Levy,
Alastair Malcolm, Christopher Maloney, Jaques Mallah, Bruno Marchal and
Jürgen Schmidhuber.
In particular, the solution presented here to the White Rabbit paradox
was developed during an email exchange between myself and Alistair Malcolm
during July 1999, archived on the everything list (http://www.escribe.com/science/theory).
Alistair's version of this solution may be found on his web site at http://www.physica.freeserve.co.uk/p101.htm.
Journal Home Page
© Journal of Theoretics, Inc. 2001 (Note:
all submissions become the property of the Journal) 