a personal journey to natural theology
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The natural religion project
A new theology
A commentary on the Summa
The theology company
An essay on the divinity of money
Submitted to the Banking Law Association in
competition for prizes to be awarded for the best papers on important
issues of current banking law practice of relevance to Australia
and/or New Zealand, may 1992
In questions of science the authority of a thousand is
not worth the humble reasoning of a single individual. [Galileo Galilei (1632), quoted in Misner, C W
et al, Gravitation, W H Freeman, San Francisco
1973 p 38.]
Contents
Summary
Introduction
I A FRAMEWORK OF IDEAS
Theology
Science and Method
An old model of god
Cantor
Abstraction
The Limits to Mathematics
Revelation
Metamathematics
Completeness and Computability
Metaphysics
The Hilbert Oscillator
Knowledge
Parmenides
Physics
Cosmological Principle
Testing
How Universal is the Universe?
Locality and Uncertainty
Relativity
Quantum Mechanics
Computation
Conservation of Energy
The Principle of Least Action
II BANKING
LAW
Law
The Rule of Law
Power
Money, good and work
Banking
Tax
Consumer Sovereignty
Discussion
Conclusion
Summary
1 Western theological tradition
conceives of god as a personal being other than the universe who
creates and controls our lives. Our political system still (rather
tenuously) recognises god as a source of law through royal assent and
the use of oaths. Since god is believed to be other than the
universe, it cannot be observed and so cannot be the subject of
science as we now understand it. Our information about god comes from
ancient texts revealed to chosen prophets and interpreted by churches
with equally ancient roots.
2 The rise of science questioned
revelation and the churches as sources of truth, but they have
remained in existence because science still lacks the power to ask or
answer the fundamental questions of life and death that concern
theology.
3 Here I outline a new scientific
theology whose model of god derives not from ancient text but from
the mathematical theory of text and communication itself. I propose
that this model describes the universe of our experience, which is
therefore fittingly called god.
4 I then interpret this model using
elements of current physical theory. These ideas are then applied to
money.
5 The movement of money is an
abstract representation of the the activity of society as a whole,
just as the flow of momentum in spacetime is an abstract
representation of the physical universe. My hypothesis is that proper
understanding and political control of public cashflows is necessary
and sufficient to obtain peaceful civilisation.
(Back)
Introduction
6 This essay is a broad survey of
ideas I hope to develop and publish one day in a more formal idiom.
7 The global failure of
authoritarian socialist systems in competition with capitalism
suggests that liberal democracy and the market are components of the
ideal economic system, whatever that may be.
8 On the other hand currently
acceptable political distortions to the market system show that the
market alone is not considered sufficient to deliver peace and
economic wellbeing to everybody. Should we distort the market, or is
there another way of achieving economic stability, growth and
distributive justice?
9 This essay is an attempt to
reexamine the theoretical foundations of civilisation to see if it
is possible to develop a more powerful model of peace and economic
welfare. Civilisation teaches us how to live with people we do not
know. (Back)
I A FRAMEWORK OF IDEAS
Theology
10 My starting point is theological.
Briefly, theology is about god. God is the source of law. Law is the
foundation of civilisation.
11 Let us
say that god is the invisible or mysterious explanation of the
existence and activity of the universe. Traditional arguments
for the existence of god depend upon the premise that the observable
universe is not self explanatory. Thus Aristotle, in his
Metaphysics, was led to his first unmoved mover because he
insisted that nothing could move itself, yet the world moves.
(Aristotle XII: vivii)
12 Atheists say god is unnecessary
because what we see is self explanatory. Yet we find that scientific
explanations are possible and very valuable although not themselves
observable. Our most powerful theories, quantum mechanics and general
relativity, explain the flow of events using mathematical functions
which are manipulated according to certain mathematical and physical
rules.
13 We use these functions to model
the world because they provide order among myriad data, predict the
outcome of particular experiments and have much intrinsic beauty and
harmony. We feel that they correspond to something real that controls
the course of events and the outcome of experiments. On the other
hand, scientific discipline requires that we assert no more than we
can measure, and that we remain open to new models which may replace
the familiar formalism with something else.
14 The existence of mysterious
(nonobservable) explanation, and the failure of ancient texts to
give a consistent explanation of the mystery of our existence
together suggest that there is room for a new theology. The real
possibility of a scientific theology is here explored by producing a
candidate model of god (MoG) for evaluation.
(Back)
Science and
method
15 Misner, Thorne and Wheeler claim
that in physics
that view is out of date
which used to say, "Define your terms before you proceed." All the
laws and theories of physics ... have this deep and subtle character,
that they both define the concepts they use ... and make statements
about these concepts. Contrariwise, the absence of some body of
theory, law, and principle deprives one of the means properly to
define or even use concepts. Any forward step in human knowledge is
truly creative in this sense: that theory, concept, law, and method
of measurement  forever inseparable  are born into the world in
union. (Misner, 71)
16 I feel that this statement
applies to all of science, including theology. Each new birth has a
past and arises out of problems in the old theory. A history of these
problems is a history of science. Each is expressed and solved in the
language of the day, but adds new meaning to old words. As modern
physics has reinterpreted such ancient terms as physics, action,
energy, momentum, space and time, a new theology
would add new meaning to theology, god, creation, revelation,
grace and the thousands of other words in the theological
lexicon.
17 If we judge by output, the
scientific enterprise is devoted to producing scientific
literature . Science translates the language of nature into
text readable by (at least some) humans.
18 Science is not the only human
activity that reduces experience to symbolic form. The courts produce
textual representations of events, and less formal producers of text
are journalism, all the arts and the ceaseless everyday business of
communication through all the forms of language and symbolism known
to humanity.
19 This suggests that formal
arrangements of symbols can successfully represent or model nature.
We have a promising model when the observable consequences of
manipulating the text according to its formal rules are found to
match the observable outputs of natural process. If nature can be
modelled by a formal system, it seems reasonable to assume that
nature itself is in some way a formal system.
20 If this is so, it is plausible
that mathematical results that apply in formal systems qua
formal systems will apply also in nature. We should therefore expect
to find in nature properties and limitations akin to the limitations
to formal systems discovered by Goedel, Turing and others.
21 The
scientific process is asymmetrical. A good theory must be open to
test by observation, that is falsifiable.
(Popper, 33) While a model may be rejected on
the grounds of one false prediction, its success does not exclude the
possible existence of another model with the same observable input
and output. This situation arises from our ignorance of the
complexity of nature.
22 We need
not be so ignorant of formal systems of our own construction. If we
can arrange that there are only two possibilities, say by dividing
our whole universe of discourse into p and notp, we
can use the inconsistency of one possibility to prove the truth of
the other. This approach is the basis of nonconstructive
or indirect proof in mathematics. This form of proof, often called
reductio ad absurdum, was known to Aristotle and was used well
before his time to show that the square root of two is
irrational. (Kramer, 27)
23 A
similar form of argument is essential to theology where it is known
as the negative way or via negativa. As Aquinas puts it:
... since ... we cannot know what god is, but what
it is not, we cannot talk about how god is but rather how it is not.
... It is possible to show how god is not by removing from it things
which are not proper such as composition, motion and other similar
attributes. (Summa Theologiae I q I a 2)
(Aquinas)
24 Falsifiability,
nonconstructive proof and the via negativa are a
common thread which suggests a route to a scientific mathematical
theology. (Back)
An old model of
god
25 The details of the Catholic god
derive as much from ancient Greek thought as from the Bible. The
Greeks made a deep study of the relationship between invariance
and motion which is modelled by the mathematical
relationship between discrete and continuous.
26 The study of continuity was a
mature science in Zeno's day, 2400 years ago. Greek science was not
so compartmentalised as things are today, and the study of nature
(physics) and the study of knowledge (metaphysics)
were considered complementary.
27 The philosophers sought certain
knowledge, that is knowledge not subject to revision or change. To
remain true, unchanging knowledge must derive from unchanging
reality. So if certain knowledge exists the foundations of reality
must be eternal, outside time and change.
28 The Greek scientific community
divided at this point. Some decided that the whole of reality is
eternal and that the motion we perceive is an inferior form of
existence or an illusion. Others sought to understand how the world
of physics could be both still and moving and the world of
metaphysics both certain and uncertain.
29 The first group is called
rationalist . A rationalist prefers rational argument to
experience if the two appear to contradict. Zeno was a student of
Parmenides. Parmenides is believed to have held that the
multiplicity of existing things, their changing forms and motion, are
but an appearance of a single eternal reality ('Being'). Zeno's
paradoxes clearly exposed some of the problems involved in a rational
understanding of motion and continuity.
30 The
other group is represented here by Aristotle. Aristotle took up the
study of motion in his Physics and established, by
definition, a relationship between continuity and infinity what
is infinitely divisible is continuous (Physics Book III:1
(Aristotle)). He then went on to develop his
doctrine of potency and act (Metaphysics, Book IX:1
(Aristotle)) to explain motion, but failed
to produce a satisfactory treatment of continuity and infinity.
31 This failure meant that
rationalism, in the form of Platonism, became the dominant input into
the theoretical development of Catholicism. Parmenides' eternal being
became the foundation for the Catholic model of god and the visible
universe took on a 'second best' status.
32 The Catholic god is completely
other than the universe and not therefore observable. Instead of
observation, the Catholic theologian must rely for data on official
interpretations of the Bible and other traditions provided by the
Catholic Church (Code of Canon Law Canon 749) A scientific theology
is thus not possible in the Catholic milieu and one finds that
experience is subservient to dogma in many other traditional
religions. (Back)
Cantor
33 The
modern inquiry into continuity began with the invention of calculus
and culminated in the nineteenth century in the work of Georg Cantor.
Cantor's study of the continuum led him to the theory of sets and
transfinite symbols. (Cantor)
34 Cantor defined a set (Menge)
to be any collection into a whole M of definite and
separate objects m of our intuition or our thought (ibid,
85). He then defined two abstract representations of set,
cardinal number and ordinal type, and explored
their properties.
35 The cardinal number of M,
card(M), is the general concept which, by means of our
active faculty of thought, arises from the aggregate M when we make
abstraction of the nature of its various elements m and the order in
which they are given (86). Sets with finite cardinal
numbers are called finite sets; all others are called
transfinite sets and their cardinal numbers transfinite
cardinal numbers represented by aleph, the first letter
of the Hebrew alphabet.
36 The ordinal type of a set M,
ord(M), abstracts from the nature of its elements m but
retains their ordering. Cantor believes that
The concept of 'ordinal type' ... embraces, in
conjunction with the concept of 'cardinal number' ... , everything
capable of being numbered that is thinkable, and in this sense cannot
be further generalised (117) .
37 The smallest transfinite cardinal
aleph(0) is the least upper bound of the finite cardinal
numbers 0, 1, 2, 3 ... . Using the theory of ordinal types,
Cantor showed that given any transfinite cardinal number aleph(m)
there exists a set of higher cardinal
numberaleph(m+1) Cantor's theorem (CT) establishes
an ordered and unending abstract hierarchy of cardinal and ordinal
numbers which is called the Cantor Universe (CU).
38 I propose to use the CU
for my MoG. The rationale for this is that both god and the
CU are defined nonconstructively. CT tells us that
given the countably infinite set of natural numbers it would be
inconsistent for the other transfinite numbers not to exist. Aquinas,
the Catholic Church and millennia of mystical tradition tell us that
we can only talk about god by the via negativa, removing
inconsistencies.
39 My purpose in this article is
theological, not mathematical. Like a physicist, I treat mathematics
as a store of possible models to be exploited without too much
respect for mathematical finesse. All my logical steps must be
tested, and the model itself must be interpreted in a way which
yields testable physical predictions. (Back)
Abstraction
40 Following Cantor, we may
distinguish four levels of abstraction. The zeroth is concrete
reality, not abstract at all but included for completeness. The first
is mental or imaginative, and corresponds to the mental images we
have of concrete realities such as trees and stones. It is at this
level that we imagine sets such as a set of trees or stones. The
second and third are ordinal and cardinal numbers as defined.
41 Each level of abstraction is a
model of those before it and a realisation of
those after it. I propose the CU as a model of the universe,
or equivalently, that the universe is a realisation of the CU.
This hypothesis suggests that I may identify god and the universe,
since the CU is my MoG.
42 A model abstracts from the detail
of reality. If the model is good, this abstraction gives a true
account of reality, and manipulations of the model reflect reality.
If the CU is a good model of the universe, we should find
that the laws of cardinal and ordinal numbers established by the
theory of sets hold in reality. In fact we have never found any
practical flaws in the working of arithmetic, which underpins the
whole accounting function in commerce and all computations in
science, engineering and all other disciplines.
43 When we
talk about the properties of numbers, we also talk about properties
of the real world. This fact has fascinated scientists (and
accountants and entrepreneurs) for a long time. Einstein said The
most incomprehensible thing about the world is that it is
comprehensible (Schilpp 112). The
proposition under examination here, that we can conceive of the
observable universe as an implementation of all possible consistent
symbolic structure, provides us with some insight into the
comprehensibility of the universe. (Back)
The Limits to
Mathematics
44 Although Cantor's theory is quite
simple, it soon began to throw up paradoxes that forced a careful
reexamination of the foundations of mathematics. The result of this
reexamination, which continues today, is that there are limits to
mathematical certainty.
45 A most productive attack on the
foundation problem was led by David Hilbert. Hilbert took the
formalist point of view, which regards mathematics as a purely formal
game played with marks on paper. For a formalist, mathematics is text
about text, and so is bound by its own discoveries.
46 The only rule is to avoid
inconsistency. The assumption behind this approach was that the
paradoxes of set theory lay concealed in the seminatural language
used by the mathematicians of the day. By eliminating natural
language altogether, Hilbert hoped to eliminate the hidden paradoxes.
47 By 1928 Hilbert was able to
encapsulate his thoughts on the nature of mathematics in three
questions:
q1 Is mathematics consistent?
q2 Is mathematics complete?
q3 Is mathematics computable?
48 He
believed that the answer to all three questions would be yes, proving
that there were no limits to mathematics. He was to claim in 1930
that there is no such thing as an unsolvable problem
(Hodges, 92).
49 Goedel and Turing destroyed this
belief. Consistency in mathematics can only be bought at the expense
of incompleteness and undecidability, just as consistency in quantum
mechanics requires us to accept uncertainty. I feel that these
results are related and that the exploration of this relationship may
lead to a new understanding of motion and stillness and open the way
for a new understanding of god. (Back)
Revelation
50
Revelation is the traditional foundation of theology. Catholicism
understands revelation to be direct knowledge of god revealed to
prophets and passed on by them to nonprophetic individuals
(Abbott,111). God, although not normally
observable, becomes observable to the prophet through a special
process known as inspiration.
51 Here revelation will mean
scientific knowledge, which includes the aggregate of a priori
knowledge we call logic and mathematics. The essential content
of a priori knowledge is the boundary of consistent formal
systems defined by the aggregate of all proofs.
52 Like the mathematical models of
physics, this knowledge does not come to us directly through our
senses but through the mental processing of sense data. It is
abstract knowledge which may be represented by marks on paper or any
other medium.
53 This
revelation is my MoG, the aggregate of all consistent formal
structures. Such revelation is available to all rather than a few
chosen ones. This understanding of revelation is not inconsistent
with the ancient belief that the light of human intellect is
participation in god (Aquinas,
I:79:4). (Back)
Metamathematics
54 Practical mathematics lies in the
realm of consistency. The study of the boundaries of mathematics as
pursued by Hilbert and others is called metamathematics.
Metamathematics tells us just how far we can go with formal systems
without abandoning consistency. The establishment of consistency is
called proof, so that it is for proof to tell us how far
proof can go.
55 CT is the foundation of
symbolic operations because it establishes the existence of the
number system upon which all else is built. The numbers are an
ordered set of discrete symbols. CT holds for consistent
systems of symbols. We may think of it as a source of
symbols, here called the Cantor source, CS.
(Back)
Completeness and
Computability
56 Goedel and Turing showed that some
of the apparently pathological behaviour which Hilbert attributed to
natural language is essential to consistent formal systems.
Mathematics is complete if every mathematical statement
that obeys the formal rules can be either proved or disproved.
Mathematics is computable if there exists a definite
mechanical process, like the execution of a computer program, which
can decide whether a given proof is valid or not. The proof of
completeness is thus logically dependent on the proof of
computability.
57 Turing proved that mathematics
contains incomputable statements by devising a universal
machine that could perform all possible logical operations and
showing that there were proofs that this Turing Machine
could not complete. Turing concentrated on computable
numbers, since they allowed a simplified form of a proof which
also applies to functions, predicates and, in fact, any formal
symbolic system. Turing considered a number computable if its decimal
could be written down by a machine.
58 Using the structure of the Turing
machine as a mapping tool, Turing transformed the problem of
computability into a question about the relationship between
aleph(0), the cardinal number of the set of rational numbers
and aleph(1), the cardinal number of the set of reals, using
the diagonal argument pioneered by Cantor. I will approach the result
more simply using algorithmic information theory.
59
Algorithmic information theory defines the algorithmic information
content I(x) of x as the size of the smallest program
to calculate x. (Chaitin, 55)
It then invokes an idea similar to the law of requisite variety
in cybernetics (Ashby, 202216): if a
theorem contains more information than a given set of axioms, then it
is impossible for the theorem to be derived from the axioms.
60 In the case of the Turing Machine,
this principle shows that a system whose complexity or information
content is measured by aleph(1) cannot be computed by a system
whose complexity is measured by aleph(0).
61 We can extend this argument to get
a transfinite hierarchy of computability. This generalization of
Turing's argument says that for n > m, a system whose
complexity is measured by aleph(n) cannot be computed by a
system whose complexity is measured by aleph(m).
62 Using the CU, complexity
and computability, we can produce a rudimentary description of the
universe. Insofar as the universe is consistent, the CS is on
and emits symbols of ever increasing complexity. This looks like the
'big bang', starting simple and growing more complex. CT
tells us that any formally consistent system must create greater and
greater complexity.
63 Now assume that one system A
may know another system B only insofar as B is
computable using the resources of A. Assume further that
insofar as the complexity of B is beyond the computing
resources of A, we are justified in calling B
mysterious relative to A . Since we know from
CT that given any system X there must be a system of
greater complexity Y, we are guaranteed the existence of
mystery for any system. This, in outline, is my proof for the
existence of god.
64 As I interpret it, CT
requires that any infinite consistent formal system be a source of
complexity, that is be creative. The ancients such as
Aristotle could not see how the visible universe could create itself
and therefore postulated god as the invisible creator. Now
we can say that insofar as the universe can be modelled by a
consistent formal system it must express CT and so create
itself. We need not understand how this is so, but simply accept that
it is inconsistent for a formal system to be otherwise.
(Back)
Metaphysics
65 I take metaphysics to be primarily
the study of knowledge. All men by nature desire to know
(Aristotle, Metaphysics I:1). I am an individual component of
the universe seeking to reflect its structure in myself and express
my reflection in this text.
66 CT guarantees that a
consistent formal system gives rise to the CU. We may conceive
the CU as a static formal space existing in the world of
mathematical symbols.
67 I now wish to take a further step
toward reality by extending the model to account for the process
of modelling as I experience it and apply the extended
model to the universe as a whole.
68 I do
this with a device I call the Hilbert Oscillator (HO) which
operates between different levels of the transfinite hierarchy. The
increase in complexity is driven by CT. The decrease in
complexity, corresponding to abstraction, knowledge or modelling is
driven by a theorem of mathematical communication theory known as the
E theorem. (Khinchin, 5458)
(Back)
The Hilbert
Oscillator
69 Figure 1 is a first version of a
formal representation of the HO. It is based on Hilbert's
questions is mathematics consistent? and is
mathematics computable? The four states of the oscillator
03 simply represent the four possible
combinations of yesno answers to these two questions in their
natural binary order.
70 Let us attempt an intuitive
correlation of these states with the search for understanding. We
begin with a problem, a situation best represented by state 1.
In this state one is certain that there is inconsistency  a previous
belief has been successfully falsified.
71 The next
step is to consciously and unconsciously permute the set of known
facts or observations in an attempt to arrive at a consistent
understanding. This corresponds to state 0, a state of
darkness lacking both consistency and computability. Einstein gave us
a classic expression of this and the next phase: The years of
searching in the dark for a truth that one feels but cannot express,
the intense desire and the alternations of confidence and misgiving,
until one breaks through to clarity and understanding are known only
to him who has himself experienced them
(Pais, 257).
72 The breakthrough to clarity
corresponds to the reestablishment of consistency and the ability to
see order in an infinite number of observations from the new point of
view. This corresponds to state 2. Finally the new insight
can be reduced to writing and definite methods of computation,
corresponding to state 3
73 Experience shows that this point,
the answer to the original problem, gives rise to a large number of
further problems. The volume of scientific work awaiting completion
grows rather than diminishes with each discovery.
74 Figure 2 is an attempt to
summarise this interpretation of the Hilbert cycle in a manner
reminiscent of the temperatureentropy diagram used in
thermodynamics.
75 We now turn to establishing a more
formal relationship between the Hilbert cycle and the Cantor
Universe.
76 CT interpreted dynamically
establishes the existence of the CF, which acts to increase
the complexity of the universe. Since the source of the CF is
consistency, it acts at every point in the universe. We may
understand the CF as the foundation of the second law of
thermodynamics which tells us that the entropy of the universe never
decreases (and usually increases) in any process.
(Back)
Knowledge
77 The force driving knowledge, the
desire to know, is the inverse of the CF. Let us call this the
E Force (EF) because it corresponds to the Eproperty.
Knowledge is to some degree abstract and general.
Knowledge is simpler than the thing known because we abstract
from individual detail. As a consequence a particular item of
knowledge applies to many individuals, and hence is general. So we
have principles of accounting which apply to all enterprises because
they abstract from the details of individual businesses.
78 In the light of the theory of
evolution, we explain the natural desire to know by the competitive
advantage that knowledge confers. An organism with knowledge of its
environment is in a better position to predict future events and
adjust its behaviour accordingly. The mathematical theory outlined
here suggests that knowledge is not simply confined to living
organisms, but is a fundamental component of all symbolic systems,
and of the universe insofar as it is modelled by text.
79 The transfinite cardinal numbers
aleph are a natural measure of complexity. Their exponential
structure makes a nice mathematical fit with the logarithmic scale
usually chosen to measure entropy and complexity (Khinchin, p 2).
80 A communication system is modelled
as a source and a channel. Shannon founded
information theory on a consideration of Markov sources. Khinchin
extended the treatment to any ergodic source and any stationary
channel with finite memory (Khinchin 30). Here I speculate on
applying the theory to the CU. Space constrains me to speak
only of sources.
81
Communication theory is an application of probability theory
(Kolmogorov). The output of a source is
regarded as a random process. We take no heed of any 'meaning' which
may be encoded in the output. The probabilistic structure of this
process constitutes the mathematical definition of the given source.
82 The first part of the definition
of source S is a set A of symbols called the
alphabet of S whose elements are called letters.
The second part of the definition of S is a measure of
the probability of emission of sequences of letters, mu.
83 From this probability structure we
can compute a number H called the source entropy which is a
measure of the average amount of information carrying capacity per
letter of the source output. The source is said to be stationary if
the probability regime of its output does not change with time.
84 The only property of a source
which concerns is here is the Eproperty established by
McMillans's Theorem (Khinchin, 5458). The Eproperty
divides the set C of n letter words in the output of a
given source into two groups known as the high probability group
and the low probability group.
85 With the exception of the case
when all the letters of the alphabet are equiprobable we find that
for large n the high probability group contains only a
negligibly small share of possible n letter words from the
source. Almost all words fall into the low probability group. On the
other hand, most of the entropy of a source is carried by words from
the high probability group.
86 This fits our experience. Day to
day english speech uses only a few thousand different words out of
the millions we find in big dictionaries. Some little words used very
frequently carry most of the traffic. The Eproperty is not confined
to natural language, however, but applies to all ergodic
sources.
87 Mathematics is a timeless
formalism. A correct proof establishes relationships between symbols
which are eternally valid. A source, on the other hand, emits a
sequence of letters in time. By the time you have spoken the last
sound in a sentence the first has died away. It no longer exists, and
so cannot be talked about in the eternal present of mathematics.
88 It has become customary in physics
and mathematics to represent such time sequences by a mathematical
space. In figure 4 below, for instance, the time axis is drawn just
like the space axis, and we treat all instants of time as existing
simultaneously just like all the points in space. In general terms, a
source is ergodic if we can satisfactorily study its behaviour in
time through arguments in space.
89 Now without argument or proof, let
us assume that the Eproperty allows us to encode almost all the
information in a set of complexity aleph(n) into a set of
complexity aleph(n1)This encoding (which would be performed
by a computing machine) I use as a model of knowledge.
90 It fits our experience of
knowledge. Knowledge reduces complexity by abstracting the salient
features of a situation. These salient features correspond in our
model to the high probability sequences. Knowledge is an imperfect
representation of reality. There is some information (corresponding
to the low probability sequences) lost by confining our attention to
the high probability sequences.
91 The number of different low
probability sequences which lie outside the realm of knowledge is
much greater than the number of high probability sequences, although
the low probability sequences are much rarer and more interesting.
This corresponds to the common belief that the scope of the mystery
in the universe is much greater than what can be known.
92 We are now in a position to draw a
third picture of the HO.
(Back)
Parmenides
93 Armed with this elementary model
of the universe and a model of knowledge within it, I wish to look
again at Parmenides' position. All we have from him are fragments of
a long didactic poem. He begins with an allegory of his chariot
journey through a gate leading from night to day, where he is
welcomed by a goddess whose words form the rest of what we've got
94 She
tells him (Hammond,782):
Only three methods in philosophy ... are
conceivable, viz. (to assume) that (the reality to be studied)
necessarily is or that it necessarily is not or that it both is and
is not, comes to be and perishes, changes and moves. The last two
methods are excluded by the argument that only what is and
cannot not be can be known. The goddess therefore proceeds
to a summary deduction of the characteristics of what is; given that
it necessarily is whatever it is, it must be ungenerated and
imperishable, indivisible, selfidentical, unique, motionless,
determinate, perfect and in perfect equilibrium like a solid sphere.
It follows that many things of which mortals speak as being and not
being, etc (i.e. as having a contingent and relative being), exist
only in name. (my emphasis)
95 I think we can now add a little to
this story. From the principle that only what is and cannot not
be can be known Parmenides deduces that reality must be
motionless. If we demand that the universe be consistent, however, we
find that motion too becomes necessary. The formal path this
conclusion is through CT, which tells us that a given
transfinite number must generate its successors and that generation
is motion.
96 How do we reconcile motion and
stillness? The formal answer, I believe, is that we cannot. We are
here face to face with mystery. We have some evidence for this
mystery in physics. Heisenberg's uncertainty principle
tells us that there is a fundamental limit to the precision with
which we can measure certain quantities.
(Back)
Physics
97 Physics studies natural motion. It
seeks the invariant structure of the changing universe which we call
principles and laws. Because these laws are invariant they can be
written down and remain through time, whereas the motions they
describe cannot.
98 We know they are there because our
lives are stable as well as changing. We look for them because we are
curious, and because knowing them usually pays off handsomely.
Knowing the law is like having the street directory of a big city 
journeys from point to point become much quicker.
99 The whole of our present way of
life depends upon the discovery of fossil fuels and heat engines,
which were brought together by the eternal laws of thermodynamics.
And where would we be without our quantum mechanical understanding of
lasers and semiconductors? (Back)
Cosmological
principle
100 This essay revolves around
two cosmological principles which I will call the Cantor
cosmological principle and the ergodic cosmological
principle.
101
Cosmological principles are a scientific antidote to human
selfcentredness. (Silk, 26) When we first
became conscious, we considered ourselves to be at the centre of the
universe. Since then observational astronomers have shown us that
seems to be nothing special about the position of the earth. This
observation is enshrined in the Copernican cosmological
principle: there is nothing special about our vantage point. The
laws of physics everywhere else are the same as here, and the
universe looks essentially the same from every other point.
102 The Copernican cosmological
principle is concerned with sameness in space. The perfect
cosmological principle is that the universe looks the same from
any point in space and time. The current popularity of the
big bang theory weakens the perfect cosmological principle, since it
holds that the universe had a definite beginning and has evolved
through time, so that things looked different a long time ago.
103 Even given the big bang, however,
the perfect cosmological principle holds to the extent that the laws
of mathematics and physics have not changed. If they have changed,
out attempts to talk about the distant past on the basis of present
laws lose credibility.
104 A third cosmological principle is
the anthropic (human) cosmological principle,
which is just the opposite of the perfect principle. Our existence is
an historic fact which (possibly) could have been otherwise. When we
look around us (which is equivalent to looking into the past) we see
the special history which bore us. If things were different, we would
not be here to see them. We are therefore seeing only that part of
the universal wilderness which is congenial to our existence.
105 Let us counter the anthropic
cosmological principle with the Cantor cosmological principle.
As I have described it, the HO models an oscillation
between two levels of complexity represented by two transfinite
cardinal numbers. The CU is a static structure. We model
motion by the HO moving in the CU. The elementary
motion in the CU lies between aleph(n) and
aleph(n+1). The complexity of this motion is measured by the
subscript n. A HO modelling motion of complexity n
may be written {aleph(n), aleph(n+1)} and I will call
it HO(n).
106 Because Cantor's proof is
nonconstructive, it simply allows us to distinguish between two
possibilities, CT consistent, and CT inconsistent. Each
time it is applied it provides us with one bit of information. The
fundamental argument of the proof does not change with the complexity
n of our position in the CU. All that changes is the
input and output of the proof.
107 We may say that CT is
invariant with respect to complexity. As Cantor puts it To every
transfinite number a there is a next greater proceeding out of it
according to a unitary law ... (my emphasis).
108 As I understand it, the inverse
of this unitary law is also a unitary law, and so we may say that the
the model of knowledge represented by the HO is also invariant
with respect to complexity. Insofar as human knowledge is modelled by
the HO, it is no different from any other knowledge. From the
point of view of the HO, the life and knowledge of a human
being is indistinguishable from the life and knowledge of an
electron.
109 We should therefore detect the
same structures recurring at different levels of complexity. In
particular, the Cantor cosmological principle underlies my attempt to
take ideas from the simple world of physics and apply them in the
complex world of human politics and economy.
110 Finally we arrive at the
ergodic cosmological principle, which states that the same
information may be encoded in different ways. I noted above that an
ergodic source is one whose output in time can be represented by a
structure in space. There we were talking about physical space
represented by a line, but the term space can be
generalised to anything which endures through time. A language may be
considered a space in this general sense, and it is common knowledge
that the same ideas can be translated into different languages.
111 The ideas behind this essay exist
in me, felt but not yet expressed. I am trying to transform them into
language. I know that anything I say can be expressed in an infinity
of different ways. My task is to get the thing finished in time and
in a way that conveys my idea to you. It is very unlikely that I will
succeed completely. Every transform of information involves
uncertainty, and this fact places a boundary on the universe of each
observer. (Back)
Testing
112 Scientific method requires that
every physical model must eventually be interpreted in a way that
makes it falsifiable. A model may be declared false either if it
contains formal inconsistencies, or if it is inconsistent with
observation. To test this model, I must make it physical.
113 My theory is that I have a
MoG which fits the world I see, so that I can call the
universe divine.
114 Why do I call it a MoG?
Because I think it meets the specifications for a MoG as they
have developed over millennia. As I see it, the basic feature of god
is that it is a mystery so far beyond our ken that we can say nothing
about it. All that we can say is that it exists, and deny to it those
predicates which are inappropriate.
115 The second criterion is that god
be consistent and act consistently. This criterion is reflected in
the belief current, at least among scientists, that the world is
consistent. Every time we falsify a cherished picture the enterprise
soldiers on through the confusion toward a new level of consistency
whose existence we never doubt.
116 The model implies, I believe,
that the observable universe is constrained only by consistency. Does
this fit our experience? To sharpen the question a little, I will
write down what I see to be the correspondences between the model to
date and the reality history has delivered to us.
 It passes as a MoG, as described above;
 It provides us with a nonconstructive source of motion (the
CS) to play the role of primum movens immobile;
 It establishes a relationship between the discrete and
continuous elements of the world, thus allowing motion to coexist
with stillness.
 It explains why the arrow of time points toward
increasing complexity (entropy) and provides an inverse to this
creation in the form of knowledge or abstraction to complete the
complexity cycle modelled by the HO. (Back)
How universal is
the universe?
117 Since human beings became
conscious of themselves and began to record their thoughts in writing
we have seen our conception of the size of the universe greatly
expand. From Aristotle to Copernicus to Newton to Einstein we have
seen the universe grow to an unbounded spacetime 20 billion years
across. But has Einstein told us the whole story?
118 At first glance, the CU
looks far too big. Even the smallest transfinite number is enough (we
think) to count all the particles in the universe. What are we to
make of the transfinite hierarchy that lies beyond aleph(0)?
How can we cut the model down to size? Do we have to restrict it in
some way?
119 The old theologians argued
something like this: The universe is explained either by itself
or by something else. It is not explained by itself.
Therefore it is explained by something else. That something
else we call god.
120 The thought dawned on me ever so
slowly that the assertion it is not explained by itself may
be wrong, a deficiency of contemporary theory rather than an inherent
deficiency in the universe. What if the universe could explain
itself? Then it is god, and we are parts of god.
121 My theological mentors judged
this (correctly) to be heresy in their religious world, and I was
duly expelled from their company. It has nevertheless turned out to
be a very fruitful line of thought for me. In those days it seemed
possible to show that the universe and god are the same by equating
them both to an infinite binary string, like the tape of a Turing
Machine [see How universal is the
universe (1967)]
122 Since then I have come to feel
that the proper space for talking about the universe is the
CU. How can this abstract model come into existence, that is
become the subject of human experience? My initial idea was that the
whole universe could be modelled by a sufficiently large and fast
universal computer.
123 Suppose operations of this
computer manifest as action, a fundamental physical
quantity. Quantum mechanics teaches us that there is a smallest
quantum of action which is represented by the constant h,
discovered by Max Planck in 1900. Assuming a mass M for
the universe allows us to estimate the processing rate f of
the universal computer from the relationships hf = E = mc squared
. A plausible M puts f in the vicinity of
10**100 operations per second.
124 We represent the data and
software in such a computer by an infinite binary string. But what of
the computer? Specifically, what is the hardware? What does the
universe run on?
125 In an ordinary computer a single
logical operation involves action by millions of electrons. If each
operation of the universal computer involves many microoperations in
the hardware, the search for hardware seems to involve us in an
infinite regress, since each operation in the hardware must itself be
modelled by the universal computer.
126 So perhaps the software is
everything. How can this be? My answer I call logical
confinement. The idea is that the processes of the universe are
not confined to any hardware, but are carried on within the
boundaries of logic itself. What is consistent is observable. What is
not consistent ipso facto is not observable. This is
essentially the content of Heisenberg's uncertainty principle in the
quantum theory.
127 Since CT holds in a
logical (consistent) universe, we are provided with a a source and a
embarrassing richness of structure in the transfinite cardinal and
ordinal numbers.
128 The world, on the other hand,
appears finite. Certainly meaningful measurements are finite, and
when infinities creep into physical calculations, we are inclined to
suspect pathology.
129 If the model represents the
transfinite substance of the universe, how do we cut it down to
finite observations? The answer I propose lies in the relationship
between knower and known. We see the universe as finite because what
we see has the same complexity as ourselves. Finity and infinity are
relative. This is the idea behind the Cantor Cosmological principle.
130 The theory of relativity tells us
that no observer can see or act outside its light cone.
Quantum mechanics tells us the the precision of our observation is
limited by the quantum of action. (Back)
Locality and
Uncertainty
131 I am located in my own present
which I take to be the region of spacetime that I can experience. An
event in the present is a transition from the future to the past,
that is the realisation of a possibility.
132 Future possibilities are infinite
and not observable, although they may be predictable from the
present. The present is to some degree finite and observable. The
past is infinite and not observable, except as it exists in the
present. An event, then, is a transition from infinite future to
infinite past through the finite present.
133 Now we ask why is the present
finite for any observer? The answer is that the observer is part of
its own present, and, we will suppose, has the same cardinal number
as its own present. The fact is not limited to human beings, but
because of its mathematical generality, applies to any particle. I
believe that the mathematical notion equivalence which
Cantor placed at the foundation of his set theory places a boundary
around the present. The position of this boundary is determined by
the cardinal number of the present.
134 The idea here is analogous to
what computer people call machine infinity. No computer can store a
number bigger than its memory. From the point of view of the machine,
any number that it cannot store is infinite, and the best it can do
is an uncertain approximation. On the other hand I am part of the
system, a bigger machine than some, and they therefore look finite to
me. (Back)
Relativity
135 The core of relativity theory is
a precise definition of what we mean by present in
spacetime. Relativity is a local theory, and so contrasts with the
theory of Newton. Newton did what he could with the ideas around at
the time, but he had to make one big simplification: he assumed
instantaneous action at a distance, even though no plausible physical
mechanism could achieve this. In the Newtonian picture one can see
instantaneously the edges of the universe.
136 Einstein founded his first theory
of relativity on the principle that the velocity of light is finite
and is the same for all observers in free fall, no matter what the
relative velocities of these observers may be. Technically free fall
is called inertial motion. In the theory of relativity,
local means inertial.
137 What we see depends upon our
point of view. The inertial reference frame is the point of view that
makes motion in spacetime look simple because it yields fixed points
to work from, particularly the local constancy of the velocity of
light (c) . In a world of relative motion, we can give no fixed
meaning to the symbol at rest. c on the other hand,
points to a physical constant. Velocity, meaning distance
travelled per unit time, has a definite upper bound.
138 No thing or influence can travel
faster than light. This enables us to put limits on the causal
relationships that exist between events in spacetime, as illustrated
in figure 4.
139 An event in the past can only
influence the present if the event is close enough in space for a ray
of light to travel from it to the present event. Similarly, the
present can only influence an event in the future if it is spatially
close enough for as ray of light to travel from the present to the
future event. Outside this light cone, there can be no causal
relationships between events and the present, so it is labelled
mystery.
140 Only massless particles (eg
photons) travelling at c can move along the actual light cone, which
is a bundle of null geodesics whose spacetime length is zero. This is
not obvious from the diagram, which illustrates only one aspect of
the whole theory.
141 All points joined by null
geodesics are zero spacetime distance apart. This is why we can
observe galaxies billions of light years away. The photons entering
my eye have experienced no time since they left the star billions of
years ago. From the point of view of an observer (who always exists
in the present) null geodesics are the boundary of spacetime.
142 This suggests that we can
identify photons (particles of light) in the physical world
with symbols in the model. Like the symbols of formal
mathematical theory, they are eternal, outside spacetime. Photons can
be classified into a spectrum parametrized by frequency measured in
inverse time. The standard unit of frequency is the Hertz (Hz) ,
one cycle per second. This spectrum contains as many different
photons as there are distinct frequencies.
143 My guess is that this is a
countable number, so that we can write n(gamma) = aleph(0)
where n(gamma) is the number of categories of photon
distinguishable by frequency.
144 Photons move in a massless world,
but we massive particles see them from a vantage point where time
passes. We express frequency in terms of time, but in the timeless
world of the photon, it is a number. Outside spacetime, photons
cannot be distinguished by place or time, but they can be
distinguished by number. This transformation from information encoded
inside spacetime to information encoded on the boundary of spacetime
is an application of the ergodic cosmological principle.
145 Both the
special and general theories of relativity are classical theories.
They model a physical point using the geometric point defined by
Euclid: a point is that which has no part
(Heath, 153). Their mathematical structure
derives from classical analysis which is the branch of mathematics
that grew out of the invention of calculus. An important task of
analysis is to produce meaningful arguments about infinitesimal
quantities, that is quantities arbitrarily close to zero.
146 The resulting mathematical
techniques have arbitrarily high precision, since we can get
infinitesimally close to where we want to go. This precision led to
the idea that we could compute the entire future of the universe if
we had adequate knowledge of the present.
147 This idea has turned out to be
wrong. The present has a finite size, represented by the box in
figure 4. If we want to look into this box we have to turn to quantum
mechanics. (Back)
Quantum
mechanics
148 The whole structure of the
universe, from the point of view of myself or any other particle,
depends upon what goes on in my own present.
149 The theory of relativity allows
us to look out into the vast spaces of the universe and make sense of
them. Quantum mechanics deals with the inner space which is vastly
smaller than ourselves.
150 Observation shows that many
features of the universe are not defined to arbitrarily high
precision. They are fuzzy. The scale of the fuzziness is measured by
the quantum of action, h .
151 In relativity the velocity of
light represents an outer boundary to our spacetime. We cannot see or
be influenced by anything outside our light cone, and the same is
true for every other particle in the universe. h represents
an inner boundary.
152 We have traced a path from the
velocity of light to photons and numbers. To get a good look at the
inner boundary we need to talk about computation again.
(Back)
Computation
153 A Turing machine, like any other
computer, moves through a computation by a series of steps. Each step
is a motion which changes the state of the computer. The steps are
choreographed by the program, which may be a carefully constructed
bit of software designed to achieve a certain result, or simply a set
of random numbers. The details of the program do not matter here.
154 Each step is initiated by a clock
pulse. We model a clock as a source with an alphabet of two letters
emitted alternately. If we assume that the pulses are equiprobable,
the entropy of a clock is 1 bit.
155 We are using computation as a
model of the universe. Let us assume that the clock of the universe
is the simplest HO, that is an HO based on aleph(0)
, the first transfinite number. We will name this HO HO(0).
Let us further assume that one cycle of HO(0). is
equivalent to one quantum of action, h .
156 Now we seem to be back to the
hardware/software problem mentioned above. HO(0) cycles
between aleph(0) and aleph(1) (fig 3) and so the
computer within HO(0) may need to perform something like
aleph(1) operations per cycle of HO(0) , each of which
would be accompanied by a quantum of action.
157 The way out of this problem seems
to be through the new understanding of point introduced by Cantor's
set theory and a corresponding boundary to mystery. Euclid's point
has no parts. Cantor's point, modelled by a set, may have an infinity
of parts, but we abstract from the internal structure of the set and
simply consider it as a named entity, analogous to Euclid's point.
Whether a given entity behaves as a point or a set is a matter of
knowledge, and so a question of interpreting the HO.
158 Just as sets may be elements of
sets, HOs may be elements of HOs. Their nested
structure reflects the nested structure of the transfinite numbers.
159 In a consistent world there can
be no largest set, since CT tells us that any set generates a
bigger one. Set theory does admit the existence of a smallest set,
however, represented here in the transfinite realm by the smallest
transfinite number, so that any aggregate of sets is bounded below.
160 The existence of a least member
is a fundamental property of an ordered aggregate and
underlies many mathematical proofs, since it gives us a starting
point for establishing one to one correspondences. When we are
talking about sets, we are talking about the abstract eternal space
of mathematics.
161 When we come to talk about the
HO, however, we are talking about something like time. Let us
imagine that it operates in the opposite way to space, that there is
an upper bound to time, but no lower bound. We can make this idea
plausible using the ergodic cosmological principle. Since space
modelled by the CU has a lower bound but no upper bound, we
would expect time to be bounded at one end and unbounded at the
other.
162 Now the quantum of action appears
to us as neither space nor time but as a mixture of both. In the
model it is represented by the substitution of one symbol for another
which we call a logical operation. Such a logical operation is a step
in a proof or a step in the execution of a computer program.
163 In computing, the size of a step
is measured by the complexity of the symbols substituted. The
substitution of the symbol 2 for the symbol 1+1 is a
simpler step than the substitution of 22 for 11+11.
Practical computing also tells us that up to a point a fast computer
taking little steps takes the same time to do a particular job as a
slow computer taking big steps. There is a tradeoff between speed and
wordlength.
164 We have ordered HOs in a
spatial way, assigning the name HO(0) to the one with the
smallest wordlength aleph(0) . Let us say that HO(0)
operates at unbounded frequency, corresponding to the idea that time
has no lower bound.
165 Because space has no upper bound,
there is no largest HO in this ordering of HOs. If on
the other hand we decide to order our HOs by time rather than
space, we have no smallest HO but we do have a largest one.
From this discussion, we conclude that the size of an HO
depends upon whether we look at it from a space point of view or a
time point of view or a mixture of both. There is an uncertainty in
matching these two points of view whose unit is the quantum of
action.
166 This minimum uncertainty gives
scale to the universe. It determines the size of fundamental
particles, and therefore ultimately of ourselves and everything else.
167 Let us now try to interpret this
structure in terms of the law of conservation of energy and
the principle of least action, which I feel are the two
most general and unifying concepts in physics.
(Back)
Conservation of
Energy
168 Physics observes that energy is
conserved, that is the measure of energy in a closed system does not
change with time. Physics recognises two forms of energy,
kinetic or energy of motion K and potential
or energy of structure or relationship V.
169 An oscillator is a
system (like a pendulum) that transfers its energy from K to
V and back again. At the top of its swing it is motionless,
with minimum K and maximum V. At the bottom if its
swing its velocity and K are at a maximum and its V
at a minimum. If we average over the cycle, we find that its total
energy spends half its time potential and half kinetic.
170 Now let us translate these ideas
from the pendulum to the HO (figure 3). We associate kinetic
energy with the less complex (lower) side of the oscillator, and
potential energy with the more complex (upper) side. The HO
moves between structure in space and structure in time. Unlike
pendulums (but like quantum oscillators which have similar
properties), HOs can be nested as we have discussed above.
171 Physically, action is a product
of energy by time. The conservation of energy corresponds to a
constant flow of action. We can imagine a set of HOs , H = {HO}
with a certain total energy operating between two levels of
complexity. Structure in space feeds on structure in time through
Cantor's theorem. Structure in time feeds on structure in space
through the Eproperty. The size of H (and therefore the
size of the present for H ) is measured by its energy.
172 We can try to imagine the whole
ensemble of HOs which constitute the universe. At one end we
have the timeless formal aspect which has no maximum size. At the
other end we have the spaceless moving aspect which has no limit to
the frequency of its motion. Between these two boundaries we have the
structure of the CU inhabited by HOs which are a
varying mixture of time and space, spacetime as we experience it.
173 The real size and complexity of
the CU is beyond our imagination, but it is obvious that we
are capable of encapsulating that mysterious splendour in some way
because we are sitting here talking about it in what purports to be a
meaningful manner. To me it seems a fitting model for the wonders I
see when I look out into the universe or into myself.
(Back)
The principle of
least action
174 There
is one more weapon to forge before the arsenal is complete, the
principle of least action. Action has a classical history.
The principle of least action was first formulated by the
French mathematician Maupertuis in 1744. Whenever any action
occurs in nature, he said (in French), the quantity of
action employed in this change is the least possible.
(Yourgrau, 19)
175 Euler established it as an exact
dynamical theorem in the same year. The variational method, which
seeks understand physical systems by finding the structure
corresponding to the least action has proved very fruitful, and may
be used to derive the major theories of physics.
176 Here, where we equate the quantum
of action and the smallest observable operation of the universal
computer, the principle of least action is equivalent to the search
for the most efficient program to perform a given action. In
Darwinian terms, this most efficient program is the most likely to
survive, since it achieves its task with the least resources.
177 The principle of least action or
efficient algorithm gives us a means of selecting the observed world
out of the immense space of possibility (entropy) represented by
the CU.
178 The concept of efficient
algorithm, formulated in the general space of the HO, can
model an economy as easily as a particle, and so gives us means for
expressing the concepts of economic efficiency and economic survival
in a competitive system. (Back)
II BANKING LAW
179 We have now a small kit of rather
imperfect ideas drawn from the last century of mathematics and
physics. It remains to apply them to the law of banking.
Law
180 The old god created the universe
because he wanted to. He made it to reflect his own majesty. It is
not his child, identical in substance, but a made object, other than
and less than god. He gave it natural laws, but these laws are an
arbitrary subset of the divine law which constitutes the nature of
god.
181 We receive divine law through two
channels. First, by the study of natural law, since nature is a
reflection of god; and second, by direct revelation from god, for
example the laws of the Pentateuch and the supreme law of love given
to us by Jesus in the New Testament.
182 A third category of law is that
made by human agencies, kings, popes, parliaments and their lawyers:
human law.
183 The theological picture presented
above blurs these distinctions. By making the universe into god, we
have removed the distinction between divine and natural law. We will
call this category of law natural, and take it to be the law revealed
to us by science.
184 On the other hand, human beings
now become part of nature rather than a special creation of god, and
so human laws are in a general sense natural laws. They are rather
more complex than the laws of physics, but do not differ in
principle.
185 Wise human lawmakers act to
preserve peace. Less wise ones generally act in their own interests,
or at least try to. It is the substance of fairy stories and moral
tales that unwise self interest leads to trouble. A wise lawgiver
adheres to the perfect cosmological principle and tries to produce
laws that apply equally to all persons at all times.
(Back)
The rule of law
186
One of the chief currents
that can be observed in the history of government is the antinomy
between two conflicting forces, power and law. In their pure form,
power and law are polar opposites, the former standing for arbitrary
might, the latter for a system in which power is checked by
institutions or individual rights and channelled in such a way as to
conform with a people's values and established patterns of
expectation. Neither of these two forces, by itself, can found a
stable system of government; one because it is capricious, coercive
and unpredictable, the other because, in practice, it can become
inflexible and may adapt only with difficulty to changing conditions.
Both tend to lead to the buildup of pressures that produce their
downfall.
The rule of law (sometimes capitalised as the Rule of Law to
distinguish it from legal rules in other contexts) is a legal and
constitutional doctrine which reconciles these two antagonistic
drives (Walker, 1).
187 In the first section of this
paper I have been trying to delineate the principles of the rule of
natural law translated into human language by science. An important
feature of natural law is that it adheres to the perfect cosmological
principle. We assume that similar circumstances will produce a
similar outcome regardless of location in spacetime. In modern
economic parlance, spacetime is a level playing field .
188 I have taken consistency to be
the foundation of natural law, and tried to show (by conceiving the
universe as divine) that the laws of nature are not only consistent,
but occupy the whole space of consistency as understood by
mathematics. An indication of this fullness is the existence of
horizons in spacetime beyond which we are faced with mystery. I take
these horizons to be observational equivalents of the limits of
formal mathematics.
189 I have argued elsewhere
[A Theory of
Peace/Lecture10/Freedom] that peace in human affairs requires not
only that human law be consistent with itself and the world, but that
it must allow us to occupy the whole space of consistency .
Peace comes when the system is designed for maximum entropy. Trouble
comes when we attempt to place an arbitrary limit on human action
either locally or globally.
190 In human affairs sheer complexity
makes both consistency and the rule of law very difficult to define
(Walker, 742). It is for this reason that I have turned to the
simpler and more formal worlds of mathematics and physics for clues.
There I see that law is essentially the boundary between mystery and
knowledge. (Back)
Power
191 Power is the hands of individuals
can be dangerous because they have a tendency to use it for self
interest. The foundation of the rule of law is the distribution of
power among the people. This general idea goes under the name
democracy, people power.
192 The power of an individual may be
manifested in all sorts of ways, but most of them are not
measureable, and therefore not the subject of scientific inquiry.
What did Hitler have, or Mao, that gave them such influence over
human affairs?
193 Modern democracies devolve power
to the people through the vote, but any observer of the modern
political system can see that the individual right to vote is a
tenuous hold on power in the sea of misinformation that passes for
political debate.
194 Without meaning to demean the
vote, I will here equate power with money. Money is measurable, and
so we can measure the power a person wields by the cashflow he
controls. (Back)
Money, good and
work
195 Money is an abstract
representation of good. Good is anything abstract or concrete which a
person wants enough to work for. Work is to be understood by analogy
to computation as an ordered set of actions designed to possess good.
196 Figure 5 is an economic version
of the HO, designed to illustrate the relationship between
money and good in the model.
197 Money is considered to be a human
invention, but when we look at it closely it has many of the features
of action and energy. We have equated power with cashflow. Power is
the ability to obtain good.
198 Physically power is measured as
energy per unit time. Cashflow is ¢ per unit time, so
we can expect money to exhibit features of energy on the buying side
of the HO.
199 Money represents good which can
be produced by labour. Labour is a flow of action, that is operations
per unit time. The simplest method of valuing labour is to pay for
time worked, that is action flow is represented by cashflow. Money is
thus analogous to action on the selling side of the HO.
200 The principle of least action
favours the existence of money, since it greatly increases the ratio
of successful transactions to enquiries, so effectively reducing the
action necessary to achieve certain sales.
(Back)
Banking
201 One of earliest known uses of
writing (and hence one of the earliest sciences) was the keeping of
accounts, that is the mapping of things and events in the world onto
a set of numbers. One may use accounts for inventory control in a
barter system, or one may use a pricing system to reduce all items in
the inventory to numbers of a common unit, the ¢.
202 Money is an abstract
representation of goods, and since it is abstract, we may think of it
as knowledge of goods. In a market system, knowledge of prices comes
through the market. The movement of goods can be abstractly modelled
by the movement of money, so that sets of numbers and arithmetical
computations can model the movement of goods. Accounting calculations
become much simpler if all value is consolidated in the form of
money.
203 It is not a large step from the
notion of accounting to that of security. An account is a record of
title to some good. Transfer of the good is reflected in a record in
the account. A security not only records title to a good, but
transfer of security causes the transfer of the good.
204 A bank is a source of money.
Originally a banknote was a security entitling the bearer to payment
by the bank in gold or some other good on presentation. The banknote
issue was secured against the bank's holding of gold.
205 Modern banking has dispensed with
gold, and the underlying security of the banking system is now
usually the taxing power of the government.
(Back)
Tax
206 Tax is a form of protection
money. The taxing body contracts in one way or another to deliver
peace and possibly welfare in return for payment. The history of
taxation, like the history of banking, owes as much to blood and iron
as to the rule of law.
207 Now that the last major war is
over, however, it may be possible to unite the functions of banking,
taxation and the delivery of peace in a global system that delivers
the same results for less action and would therefore have a selective
advantage over the present system, if ever it were to come into
competition with it. (Back)
Consumer
sovereignty
208 In the absence of large scale
space travel, we can regard the planet earth as a closed system of
n natural persons A each with a name
A_{i} (i = 1 ... n). n changes with births and deaths,
but that does not affect the story.
209 Each of these people is assumed
to know what he wants, either directly or through an appropriate
agent. Economically this is known as consumer sovereignty.
Advertisers, educators and others may wish to influence these
desires, as may people with criminal intent. We presume that these
are controlled by an adequate system of accountability and criminal
law.
210 Associate with each person a
cashflow $_{i} per week. If the value of the system
becomes sufficiently obvious, people may provide this information
voluntarily. Otherwise we can learn how to do it from existing tax
collectors.
211 In the following all statistics
are computed on a weekly basis. The global cashflow, referred to
human beings G_{A} is thus Sum_{i}
$_{i} per week. We can define a corporation C as
any entity other than a natural person that has a cashflow and
construct a similar measure of global cashflow referred to
corporations, C_{A} .
212 Now we can regard the population
A as a source whose alphabet is A_{i} and
assign as the probability of each letter the normalised cashflow
p_{i} = $_{i} / G_{A} . From this we can
compute a figure which we will call the entropy of human cashflow
H($_{i} ) =  Sum_{i} p_{i} log p_{i}
which will serve as an figure of merit for the cash element of
distributive justice.
213 In the present state of the
globe, with incomes heavily skewed toward zero, we can expect this
figure to be quite low. The next measure is designed to raise it.
214 The entropy of a source increases
as the probability of its letters approaches equality. We institute a
flat rate tax T equal to a fraction x of individual
and corporate cashflow so that the total taxflow T = x(
G_{A} + G_{C} ) and (after deducting
administrative costs) distribute this money equally to all human
individuals A _{i} . It should not be hard to find
people to accept their weekly dividend. The system needs some form of
identification (eg fingerprint) to prevent too many people being paid
twice.
215 This redistribution has two
effects. First, it moves income from those with a higher cashflow to
those with lower, thus improving the distributive justice measure;
and second, it gives every person (and therefore every corporation)
current information about the health of the global economy, allowing
them to adjust their behaviour accordingly. The tax collection and
distribution data can also be processed to yield many other
interesting statistics while maintaining anonymity.
216 The divine aspect of money
referred to in the title is that in itself it is purely a number,
with no respect whatever for persons. We are all equal before the
dollar. A major input to the global economic system is solar energy
provided free by god, and it seems reasonable that some element of
this benefit should shine equally upon us all.
217 This task is to be entrusted to a
global bank (the bank of earth, BE ?) whose task, apart from
collecting and distributing T is to establish a world
currency for its operations, secured against its tax revenue and
convertible (at market rates) with all other currencies.
218 In the initial stages of this
system, the tax rate x will be very low and its collection
and distribution might be subcontracted to existing administrative
entities. All existing economic systems may remain undisturbed,
except that those governments already administering taxation and
social welfare systems may wish to adjust them in the light of the
new tax and dividend payments. (Back)
Discussion
219 I believe that the general effect
of these measures will be to unite the human population into a single
functioning economic entity operating at maximum productivity. This
hope is based on the idea of working toward the maximum consistent
flow of entropy in the global economic HO.
220 I feel that in time the
advantages of this system will lead to the removal of most economic
functions from national governments and, with increasing productivity
and environmental awareness among consumers, will be the most
efficient way to achieve a globally sustainable economy.
221 In time (if the system works)
x may grow until reaches a level where everybody can freely
purchase many of the services like education and health now delivered
to them by the public sector, if at all. The distribution of new
consumers around the globe will encourage the growth of business in
their vicinity.
222 I feel also that by removing the
political stresses caused by uneven distribution it will contribute
to peaceful government based on the return of something resembling a
tribe (a group of people known to eachother) as the fundamental
political entity. The existence of tribes as local administrators
will greatly assist the BE in its task.
223 Obviously the next job is to
build a computer model of this system and see how it works in
competition with the existing system. Unfortunately even rudimentary
calculations have been impossible within the time frame of this
competition. (Back)
Conclusion
224 The scientific method requires
that one accept oneself as a phenomenon in the universe. The old
story that we are a defective being as a result of ancestral sin in
the garden of Eden will not stand up. The scientific history of our
species is far more interesting and encouraging.
225 What is true of an individual in
a particular society is partly true of the society as a whole. The
world is blighted with ancient religious and political ideas, but the
future is uncertain and it is not easy to know what to do about them.
226 Many years ago I was asked to
leave a Catholic Monastery for saying too much. At the time I did not
know what I was talking about, but I felt strongly that I was onto
something and could not retract. Since that time I have been trying
to put my feeling into words and this is the latest attempt.
227 Niels Bohr, when asked What is
complementary to truth? Answered: Clarity (Pais 1991, p 511). As with
noncommuting quantum mechanical variables, one can have one or the
other, but it is impossible to get hold of them both at once. I hope
the obscurity of this essay suggests the truth of the idea behind it.
There is a long way to go yet before I achieve clarity. I thank the
organisers of this competition for giving me the incentive to move a
little further on the way. (Back)
Further readingBooks
Click on the "Amazon" link to see details of a book (and possibly buy it!)Abbott, Walter M, The Documents of Vatican II: in a new and definitive translation, with notes and commentaries by Catholic, Protestant and Orthodox authorities, Geoffrey Chapman 1972 Jacket: 'All 16 Documents of Vatican II are here presented in a new and readable translation. Informed comments and appraisals by Catholics and nonCatholics make this book essential reading for anyone, of whatever shade of belief, who is interested in the changing climate of today's world.' Amazon back 
Aristotle, Metaphysics IIX, Harvard University Press, William Heinemann 1980 Amazon back 
Ashby, W Ross, An Introduction to Cybernetics, Methuen 1964 'This book is intended to provide [an introduction to cybernetics]. It starts from commonplace and well understood concepts, and proceeds step by step to show how these concepts can be made exact, and how they can be developed until they lead into such subjects as feedback, stability, regulation, ultrastability, information, coding, noise and other cybernetic topics' Amazon back 
Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.' Amazon back 
Chaitin, Gregory J, Information, Randomness & Incompleteness: Papers on Algorithmic Information Theory, World Scientific 1987 Jacket: 'Algorithmic information theory is a branch of computational complexity theory concerned with the size of computer programs rather than with their running time. ... The theory combines features of probability theory, information theory, statistical mechanics and thermodynamics, and recursive function or computability theory. ... [A] major application of algorithmic information theory has been the dramatic new light it throws on Goedel's famous incompleteness theorem and on the limitations of the axiomatic method. ...' Amazon back 
de Witt, Bryce S and Neill Graham (eds) , The ManyWorlds Interpretation of Quantum Mechanics, Princeton UP 1973 Amazon back 
Goedel, Kurt, Kurt Goedel: Collected Works Volume 1 Publications 19291936, Oxford UP 1986 Jacket: 'Kurt Goedel was the most outstanding logician of the twentieth century, famous for his work on the completeness of logic, the incompleteness of number theory and the consistency of the axiom of choice and the continuum hypotheses. ... The first volume of a comprehensive edition of Goedel's works, this book makes available for the first time in a single source all his publications from 1929 to 1936, including his dissertation. ...' Amazon back 
Hammond, N G L, The Oxford Classical Dictionary, Clarendon Press 197079 Amazon back 
Heath, Thomas L, Thirteen Books of Euclid's Elements (volume 1, III), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time  the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.' Amazon back 
Hodges, Andrew, Alan Turing: The Enigma, Burnett 1983 Author's note: '... modern papers often employ the usage turing machine. Sinking without a capital letter into the collective mathematical consciousness (as with the abelian group, or the riemannian manifold) is probably the best that scoience can ofer in the way of canonisation.' (530) Amazon back 
Khinchin, A I, Mathematical Foundations of Information Theory (translated by P A Silvermann and M D Friedman), Dover 1957 Jacket: 'The first comprehensive introduction to information theory, this book places the work begun by Shannon and continued by McMillan, Feinstein and Khinchin on a rigorous mathematical basis. For the first time, mathematicians, statisticians, physicists, cyberneticists and communications engineers are offered a lucid, comprehensive introduction to this rapidly growing field.' Amazon back 
Kolmogorov, A N , Foundations of the Theory of Probability, Chelsea 1956 back 
Kramer, Edna E , The Nature and Growth of Modern Mathematics, Princeton UP 1982 Preface: '... traces the development of the most important mathematical concepts from their inception to their present formulation. ... It provides a guide to what is still important in classical mathematics, as well as an introduction to many significant recent developments. (vii) Amazon back 
McKeon, Richard, The Basic Works of Aristotle, Random 1941 Introduction: 'The influence of Aristotle, in the ... sense of initiating a tradition, has been continuous from his day to the present, for his philosophy contains the first statement, explicit or by opposition, of many of the technical distinctions, definitions, and convictions on which later science and philosophy have been based...' (xi) Amazon back 
Misner, Charles W, Gravitation, Freeman 1973 Jacket: 'Einstein's description of gravitation as curvature of spacetime led directly to that greatest of all predictions of his theory, that the universe itself is dynamic. Physics still has far to go to come to terms with this amazing fact and what it means for man and his relation to the universe. John Archibald Wheeler. ... this is a book on Einstein's theory of gravity (general relativity).' Amazon back 
Pais, Abraham, 'Subtle is the Lord...': The Science and Life of Albert Einstein, Oxford UP 1982 Jacket: In this ... major work Abraham Pais, himself an eminent physicist who worked alongside Einstein in the postwar years, traces the development of Einstein's entire ouvre. ... Running through the book is a completely nonscientific biography ... including many letters which appear in English for the first time, as well as other information not published before.' Amazon back 
Popper, Karl Raimund, The Logic of Scientific Discovery, 1992 Jacket: 'A striking picture of the logical character of scientific discovery is presented here ... Science is presented as ... the attempt to find a coherent theory of the world composed of bold conjectures and disciplines by penetrating criticism.' Amazon back 
Schilpp, Paul Arthur, Albert Einstein: PhilosopherScientist, Open Court Publishing Company 1949 Contains Einstein's autobiographical notes in German and English, 25 descriptive and critical essays on the Work of Albert Einstein, Einstein's reply to these essays, and a bibliogrphy of Einstein's writings to May 1951, Amazon back 
Silk, Joseph, The Big Bang: The Creation and Evolution of the Universe, Freeman 1988 Jacket: 'Written for the nonspecialist, The Big Bang describes the greatest contemporary puzzles and achievements in astronomy, cosmology and astrophysics, clearly recounting the history of the universe and examining current controversies from several points of view. The book concludes with a self contained appendix providing the basic mathematical framework for understanding modern cosmology."
Amazon back 
Walker, Geoffrey de Q, The Rule of Law: Foundations of Constitutional Democracy, Melbourne University Press 1988 Jacket: 'The author argues that the survival of any useful rule of law model is currently threatened by distortions in the adjudication process, by perversion of law enforcement (by fabrication of evidence and other means), by the excessive production of new legislation with its degrading effect on longterm legal certainty and on longstanding safeguards, and by legal theories that are hostile to the very concept of rule of law. In practice these trends have produced a great number of legal failures from which we must learn.' Amazon back 
Yourgrau, Wolfgang, Variational Principles in Dynamics and Quantum Theory, Dover 1979 Amazon back 
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