\documentclass[12pt]{article}
\usepackage{harvard}
\usepackage{bibtexlogo}
\usepackage{html}
\usepackage{tcr}

\newcommand{\hlnkf}[2]{\htmladdnormallinkfoot{#1}{#2}}
\newcommand{\hlnk}[2]{\htmladdnormallink{#1}{#2}}

\newcommand{\hoturl}[1]{\hlnk{\texttt{#1}}{#1}}
\newcommand{\mailto}[1]{\hlnk{\texttt{#1}}{mailto:#1}}

\begin{document}

\tcrdate{26-Nov-1996}
\tcrvolume{01}
\tcrnumber{02}

\input{tcrfront}

\begin{latexonly}
\copyrightdate{1994--96}
% This must be manually echoed in the .latex2html
% also - l2h cannot (perversely) pick it up from here.
\end{latexonly}

\title{The Metaphysics of Scarcity:\\
Popper's World~3 and the Theory of Finite Resources}

\author{\htmlref{Dr. Ray S. Percival}{authors}\\
\htmlref{\copyright \thecopyrightdate}{copyright}}

\maketitle

\begin{quote}
\textbf{Keywords:} Scarcity, Metaphysics, Invention,
Information-Content, World-3, Diminishing-Returns,
Increasing-Returns, Theories, Means-End-Scheme.
\end{quote}
\newpage
\vspace*{\fill}

\begin{abstract}
  Natural resources are infinite. This is possible
  because humans can create theories whose potential
  goes beyond the limited imaginative capacity of the
  inventor. For instance, no number of people can work
  out all the economic potential of quantum theory.
  Economic Resources are created by an interaction of
  Karl Popper's Worlds~1, 2~\&~3, the worlds of
  physics, psychology and the abstract products of the
  human mind, such as scientific theories. Knowledge
  such as scientific theories has unfathomable
  information content, is universally applicable, and
  infinitely copyable. The point can be made with
  technological knowledge such as that embodied in the
  wheel. The theory of the wheel has unbounded
  potential to be embodied in unforeseeable new
  technologies, is useful on the Moon as on Earth, and
  can be infinitely copied. Unlike a piece of land
  (using fixed factors), such knowledge shows
  increasing returns. This helps to explain Julian
  Simon's observation that \tcrquotes{natural}
  resources are now less scarce than they used to be
  and why an increasing population can increase
  resources in the long-run.  It was Simon's
  breakthrough to elaborate on the abstract character
  of \tcrquotes{natural} resources. I further explore this
  abstract character and thereby explain why natural
  resources are infinitely expandable.
  
  Economic growth and the creation of natural resources
  depends on the rate of invention. F.~Machlup's
  suggestion \cite{Machlup:SupplyOfInventorsShort} that
  the opportunity for new inventions increases
  geometrically with the number of inventions at hand
  is acknowledged for its suggestiveness, but
  criticised for its conservative position. Frank
  Tipler's fascinating argument for indefinite economic
  growth \cite{Tipler:PhysicsOfImmortality}, is
  reinforced by my argument by making a distinction
  between information in the engineer's sense and the
  infinite potential \tcrquotes{information} in our
  scientific knowledge based on Popper's notion of
  information content.
\end{abstract}

\vspace*{\fill}

\tableofcontents

\section{Introduction}

\tcrpar
How is it possible for a resource to be both scarce and
infinite?

Most philosophical treatments begin their discussions
of global problems by taking the finite-resource view
for granted.  I contend that economic raw material
resources are in an important sense infinite in the
long-run. With the invention of the rocket and nuclear
power, we may have already created the theoretical
knowledge that will make economic resources infinitely
expandable.\footnote{See
  \cite{Tipler:PhysicsOfImmortality}. Tipler would add
  computing theory to nuclear power and rocketry as the
  sufficient conditions for permanent economic growth.}
Like all theoretical knowledge, these inventions have
unfathomable economic potential and instead of showing
diminishing returns (like a piece of land), show
increasing returns. Just think how more more useful a
transistor is now than it was 50 years ago when first
invented. How much more useful are the natural numbers
now than they were when first invented simply for
counting?  But this marvelous quality of abstract
knowledge needs a philosophical explanation. A
combination of Julian Simon's analysis of the concept
of a natural resource \cite{Simon:UltimateResource}
plus Popper's Worlds~1, 2~and~3
\cite{Popper:KnowledgeAndMindBody}, helps us to
understand the nature of a resource and, as a bonus,
helps to explain how natural resources might be both
scarce and infinitely expandable.

\tcrpar
I argue, along with Simon, that the conventional notion
of a natural resource is profoundly misleading. It
implies that resources are simply portions of
matter/energy just waiting to be discovered. On the
contrary, even \tcrquotes{natural} resources are created by an
interaction between the human mind, theories and the
physical world. In a sense, all resources are created,
artificial. Seen from this angle, the issue of the
finite/infinite extent of resources solicits a very
different sort of answer. But we may consider that even
Simon's notion of a resource needs to be extended to
more accurately describe the multifaceted nature of a
resource, taking into account its physical,
psychological and logical (or abstract) character.

\tcrpar The growth of knowledge or World~3 in general
impacts on economic growth and the scarcity of
resources through the affect it has on the rate of
invention.  F. Machlup's suggestion
\cite{Machlup:SupplyOfInventorsShort} that the
opportunity for new inventions increases geometrically
with the number of inventions at hand is criticised for
its conservative position.  Frank Tipler's fascinating
argument for indefinite economic growth
\cite{Tipler:PhysicsOfImmortality}, is reinforced by my
argument by making a distinction between information in
the engineer's sense and the infinite potential
\tcrquotes{information} in our scientific knowledge.

\tcrpar
My thesis is that at any given time, raw material
resources are scarce in that humans have more uses for
them than they are able to satisfy 
\citeaffixed{Steele:MarxToMises}{for the most
in-depth criticism of the possibility of eliminating
scarcity in this traditional economic sense, see }.
However, the flow of services from raw material
resources is in principle infinitely expandable over
time without diminishing returns. This is made possible
by the infinite potential of our theoretical
understanding of a resource. A piece of copper may be
used over and over again indefinitely without
diminishing its quantity. But this does not capture its
full potential as a useful item. To capture this you
need to examine the theories we use in our use of the
copper.

\tcrpar
Popper divides all that exists into three domains:
World~1 (the world of physics, chemistry and biology),
World~2 (the world of psychological states,
dispositions and processes), and World~3 (the sum total
of the objective abstract products of the human
mind). The purpose of the theory of World~3 was to
account for the objectivity of scientific criticism,
creativity and the relationship between the mind and
the body. Both Popper and Eccles see World~3 as
containing such things as theories, numbers, and even
tools and institutions considered as abstractions
\cite{Popper:SelfAndItsBrain}. They
are our products, but once created they have autonomous
existence, properties and relations that go beyond our
expectations and intentions, indeed beyond any
psychological states. The natural numbers were created
by us but we then discovered, as an unforeseeable and
irrevocable consequence, that this sequence has odd and
even numbers. Through a sort of plastic control, world
3 is able to affect and constrain our thought and, only
through our thought, influence World~1. The invention
of numbers, for example, enabled us to develop counting
and calculating methods such as calendars, balances and
computers that in turn greatly changed the physical
world.

\tcrpar
I argue that inventions and, less obviously, natural
resources have three ontological aspects: a World~1
aspect, a World~2 aspect and a World~3 aspect.

\begin{enumerate}
\item Economic theory was first worked out as it
applies to humans, but at least some economic theorems
apply to all organisms. Economics is as old as life
itself and like other aspects of life, it has an
evolutionary history. Thus there were relatively crude
economic resources even before the advent of
consciousness. But with consciousness and especially
the emergence of human time-binding and the ability to
create and visualize new independent, distant goals and
means for reaching them, more interesting economic
actions became possible.

Previously useless physical substances become
resources, created by our valuations, intentions and
imagination with respect to their physical
properties. Copper became a new economic resource when
people placed various portions of it within new
means-end schemes. There was a time when humans had no
use for copper, but now it can be used in copper wiring
in TV sets etc.  At this level of analysis, a resource
is a World~2 object.

\item A resource can also be captured in an objective
abstract theory or cluster of theories.  The metal in a
computer becomes very useful only because of the
enormous amounts of abstract mathematics, logic and
other technological theory embodied in it. Thus a
resource is a World~3 object.

\item Along with the inventions that make natural
resources useful and expand them, their infinite
long-term serviceability without diminishing returns is
made possible partly by their World~3 characteristics:
namely, the objective logical properties of the
theories that partly constitute them.

\end{enumerate}

\tcrpar
That a natural resource such as copper is a World~1
object might seem obvious.  But every resource is not
simply a portion of matter or energy, but becomes a
resource on account of being interpreted (correctly) as
part of a theoretical means-end relationship. (This
means-end scheme may be a rather complex cluster of
technical and social hypotheses). For it is the
services we can obtain from a natural resource that we
are interested in when we speak of a natural resource.

\tcrpar A similar point has been made about other
\tcrquotes{objects} of social science: money, for example, is not
money until it is interpreted and valued as
money\footnote{Of course, certain sorts of objects will
  be objectively more suited to perform the function of
  money.}
\citeaffixed{Searle:MindsBrainsAndScience}{cf.~}. Such
thinking can, I think, be traced back to the
marginalist revolution in economics, inaugurated by
\citeasnoun{Menger:Grundsatze}, 
\citeasnoun{Walrus:PureEconomics} and
\citeasnoun{Jevons:PoliticalEconomy}.
% in the early 1870's. 
This
placed a general theoretical emphasis on the importance
of subjective factors in social science. To reinforce
the point made by Searle and others, it might be said
that these interpretive theories may only become
conscious when something goes wrong and they are
refuted. For example, during a hyperinflation money
ceases to be money because people no longer see it as
money, i.e., each person no longer entertains the
theory that most other people think that it is valuable
to most other people. My point is that just as there is
no chemical or physical analysis alone that will
determine what makes money money, so there is no
chemical or physical analysis alone that will determine
what makes a resource a resource.

\tcrpar On the other hand, I want to say that because
of the, in many ways laudable, emphasis on subjective
factors instigated by the marginalist school of
economics, there may be a tendency to overlook the part
played by the objective content of our theories and
other abstract products of our minds. This is easily
done without the distinction between Worlds~1, 2~and~3.
Theories, considered as objective knowledge, have a
special economic status that makes them quite different
to, say, a piece of land. Economic knowledge may be
divided into the unspecifiable personal knowledge of
circumstances and skills (such as the ability to ride a
bicycle or use a tool or machine) stressed by 
\citeasnoun{Hayek:UseOfKnowledge} and
\citeasnoun{Polanyi:PersonalKnowledge} and the
specifiable objective knowledge (such as scientific
theories) stressed by
\citeasnoun{Popper:ObjectiveKnowledge}\footnote{The
  most recent strong argument for the non-specifiable
  nature of personal skills and even the ability to
  discover mathematical proofs is presented by Roger
  Penrose in \emph{Shadows of the Mind}
  \cite{Penrose:Shadows}.}.
Popper argues that objective knowledge, the kind we
find represented in books, tapes, computer memory, has
an autonomous existence from the psychological or
physical states that produced it and in which it may be
represented.\footnote{One might say that the division
  into personal versus objective knowledge obscures the
  fact that some intentional states (e.g.,
  dispositional expectations) shade over into
  propositional intentional states, i.e., World~3
  status. Peoples' \tcrquotes{constitutive} attitude to money is
  a good example, for it may only be a shock like a
  hyperinflation that jolts people into making their
  unconscious expectations explicit and placed in
  linguistic form. But we do need the distinction
  between Worlds~2 and~3 to talk about the transition
  from one state to the other.} I would like to suggest
that focusing on the World~2 psychological aspects of
economic knowledge obscures the interesting
ramifications that flow from the infinite content of
the theories that interpret sections of World~1 to make
resources.  Therefore, some economic knowledge has a
special metaphysical status.  The special autonomous
properties of this domain of economic knowledge has
interesting implications and ramifications for the
nature of scarcity and the power of the inventions we
use to reduce scarcity.


\section{Respectable Metaphysics 
Versus Partial Operationalism}

\tcrpar
In dealing with the issue of the infinite versus finite
nature of resources, we are dealing with a metaphysical
issue. Expressed colloquially, metaphysics provides a
view of the world as a whole. An example of such a view
was Faraday's conception of the universe as a network
of fields of forces.

\tcrpar
Methodologically, I think it useful to interpret
metaphysical issues in terms of theories and their
logical relationships with one another. Provisionally,
we may say that a metaphysical theory is one that taken
alone is empirically untestable by confrontation with
basic statements. A basic statement is one that
describes an observable event of definite space-time
coordinates. \tcrquotes{There is red ball of at least 8 cm
diameter within one meter of the entrance to Bolton's
central library} is a basic statement. An example of a
metaphysical statement would be \tcrquotes{There is a red ball
which will give everlasting life to the person who
touches it}. To emphasise the fact that this
distinction is one of logic and not between the
everyday and the mystical, this last example can be
changed to \tcrquotes{There is an incompressible red
ball}. Another example would be \tcrquotes{There is an
inexhaustible barrel of oil}.  This definition of
metaphysical conforms with the traditional sense in
which metaphysics transcends experience, but conformity
with tradition is not crucial. More importantly, it
allows us to talk about interesting logical
relationships between statements of quite differing
character.

\tcrpar
All economic theories, even if they are scientific in
being empirically falsifiable, contain explicit or
implicit metaphysical assumptions or
presuppositions. In this they are in good company as
all the great scientific theories were not only
inspired by, but also contained, metaphysics. In some
cases the metaphysical elements are simply weak logical
implications of the empirical theory. In some cases the
metaphysical elements are more like adjuncts that can
be dispensed with without diminishing the empirical
content of the theory. In many cases, however, they
play an important role not only in augmenting empirical
content, but in guiding research.  
cf.~\cite{Watkins:ConfirmableMetaphysics}.
Also \cite{Watkins:MetaphysicsAndAdvancement}. See also 
\cite{Agassi:NatureOfScientificProblems}.
More recently \cite{Zahar:EinsteinsRevolution}.

\tcrpar I have said that metaphysical statements are
untestable by direct confrontation with basic
statements, but things are not so simple. Popper has
even shown that combining what are individually
untestable metaphysical statements can sometimes yield
a testable theory \cite[Chapter III]{Popper:Realism}.
This clearly shows
that a statement may be only relatively metaphysical.
This perhaps should not be so surprising as even
scientific statements of the law-like variety only
become fully testable in the presence of initial
conditions and other theories, though some crude
testable implications are derivable from the
law-statement alone.\footnote{For example, from the
  law-like statement \tcrquotes{At atmospheric pressure silver
  melts at 960 degrees centigrade.} one cannot derive a
  statement predicting a melting of silver at some
  definite spatio-temporal coordinates, or indeed even
  a purely existential statement to the effect \tcrquotes{that
  some silver at some place and time has melted at 960
  degrees centigrade at atmospheric pressure}. But one
  can derive negative predictions, such as the
  statement \tcrquotes{One will not observe the melting of silver
  below 960 degrees centigrade at spatio-temporal
  coordinates $w,x,y,z$ at atmospheric pressure}.} In any
case, it is clear that in the analysis of metaphysics
we have moved a long way from the cavalier attitude of
the Vienna circle, and that metaphysical elements in
science require serious and discerning treatment. In
assessing the empirical value of a statement that is
metaphysical when taken alone, one has to investigate
its role within a larger system of hypotheses, how it
contributes to the empirical consequences of the
system.  The point here is that even though the
assumption that resources are infinite may itself be
untestable, the explanatory theory in which it occurs
may be the more powerful (and more testable) for its
presence.

\tcrpar Examples of metaphysical doctrines are atomism,
determinism, the irreversibility of time, and recently
the idea of locality in physics.\footnote{The logical
  form of metaphysical doctrines may be of the all-Some
  variety. For example, determinism may be stated thus:
  for every event there is a cause. Or more
  informatively, for every event $x$ there exist a $y$ and
  a $z$ such that $y$ is a lawful relationship describable
  by some true universal law $u$, and $z$ is an event (set
  of initial conditions) preceding $x$, and $x$ is
  predictable (deducible) from $z$ in the presence of $y$
  (or of $u$) \citeaffixed[p.~196]{Popper:OpenUniverse}
  {cf.~}.
  This is
  clearly untestable by confrontation with a basic
  statement, for suppose someone presents the
  determinist with a putatively uncaused event. The
  determinist always has two defensive options. No
  matter how far you have searched for the cause of
  some unexplained event and failed, the determinist
  can say either that you failed to look hard enough
  for the initial conditions or that you have
  insufficient imagination to formulate the correct
  lawful relationship connecting the two events (the
  initial conditions and the event to be explained). He
  can say this because you cannot logically exclude the
  possibility that the very next search will identify
  the cause.}

\tcrpar
Let us become acquainted with the feel of metaphysical
statements. I was struck recently by how common
unacknowledged metaphysical assumptions are in every
day conversations. One I encountered recently might be
called failure-metaphysics: there are some people who
have fallen into circumstances of poverty and misery
from which it is impossible to escape. If any putative
confirming candidate subsequently becomes rich and
happy, then the advocate of failure-metaphysics can
always say that he misidentified the example. Of
course, the contrary (success-metaphysics), that there
is always a way of escaping from any circumstances of
poverty providing one uses the correct methods is also
untestable by itself. Such metaphysical assumptions are
more to do with self-help methods than economic theory
as such.  An example from economics might be the
following:

\begin{quote}
A person is willing to sacrifice some bit of any
desired thing if he can obtain a sufficient increase in
the amount of some other desired 
goods. \cite[p.~21]{Alchian:ExchangeAndProduction}
\end{quote}

\tcrpar
This is not empirically testable as it stands, for no
limit is placed on what might be a \tcrquotes{sufficient}
increase in other goods: does it stop short of owning
the Earth, the Solar system, the Universe? Nor does it
specify a limit on the smallest \tcrquotes{bit} of sacrifice
allowed: does it include the sacrifice of one 1/1000th
of a cup of coffee per year? However, I do suspect that
this \tcrquotes{postulate} does contribute to a theoretical
system that is empirically testable when taken as a
whole. The situation is analogous to Einstein's use of
Lobachevskian geometry in his theory. By itself,
Lobachevskian geometry cannot be empirically refuted,
but it can contribute to the refutable empirical
content of a theory.  Assuming the economic postulate
to be true, it does lead one on to interesting
speculations about values and the structure of the
world. For example, most people will not sacrifice any
amount, however small, of their moral values such as
\tcrquotes{Thou shalt not kill}.  Is this a refutation of the
postulate or could it be that the world is such that
the \tcrquotes{pay-off} just could not be arranged, perhaps
because of the constraints of physical laws
etc. Perhaps some might murder if the reward were
everlasting life for them and their family---the
Vampire option. But the world is such that everlasting
life is impossible.

\tcrpar
Although the assumption of infinite resources and its
contrary are metaphysical they play an important part
in the empirical theories to which they belong. The
assumption of infinite resources can be stated as
follows: for every increase in resource-scarcity, there
is at least one resource-augmenting invention or
discovery that will more than compensate for this
increased scarcity. The assumption of finite resources
can be stated thus: at some point in the future the
amount of resource-scarcity will increase
irretrievably. Taken alone each of these statements is
obviously metaphysical in the narrow sense of being
unfalsifiable by confrontation with basic
statements. The first places no limit on the delay
between an increase in scarcity and its correction and
it is not clear whether the compensatory correction
takes account of time-preference. The resource finitist
can sustain their gloom no matter how long humans live
in abundance and luxury.

\tcrpar The resource-finitists do try to include
testable assumptions.  The standard approach to
estimating global resource quantity is first to
estimate the presently-known physical quantity of the
resource, secondly estimate the current rate of use,
and finally predict a diminution of the first estimate
over successive periods of time until the first
estimate is exhausted. This approach, as Simon points
out, is based on a confusion of the micro and macro
contexts \cite{Simon:LinguisticConfusion}.  It is based
on Hotelling's theory of the optimum rate of use of a
spatially definite mine or well. The resource
finitist's error can be put this way. The world's
resources are seen as a large store-house which will be
emptied in a finite period even if our technology
improves the speed with which we empty it. But the
world is not a store-house with pre-defined finite
contents, since it is interpreted by our creative minds
and theories.

\tcrpar Prima facie, one might think (as Julian Simon
does) that there are two ways of challenging the
assumption of finite resources in the long run in a
macroeconomic context.  One is to point to empirical
statistics, such as trends in resource prices and
invention creation, and project from past data, a
method pioneered by
\citeasnoun{Barnet:ScarcityAndGrowth}.  For example,
price data for minerals over the past 200 years was
adduced by Barnet and Morse (and more recently by
Julian Simon) to show that the scarcity of minerals has
actually been declining over the centuries, contrary to
popular imagination. The argument is that if a resource
were becoming more scarce, its price would be bid up by
speculators, so reflecting its greater future level of
scarcity. To reinforce this, statistics on the
increasing rate of creation of inventions (some of
which increase resources) is also adduced. When one
projects this data into the future, the picture looks
quite rosy. The supposed alternative is take a more
theoretical approach.

\tcrpar
But such a division into empirical versus theoretical
approaches is unsound. One can do theory without
empirical research, for one can test a theory for
internal inconsistency, simplicity, axiomatizablity and
consistency with other background theories; but one
cannot do empirical research without some, perhaps
implicit, theory, as even singular statements
describing definite events or objects contain universal
terms that have dispositional (and counterfactual)
implications that transcend any immediate (or indeed
any finite set of) experiences or
observations \cite{Popper:LScD}. Even the seemingly trivial and
untheoretical statement \tcrquotes{There is petroleum oil in that
barrel} is highly theoretical in that its implications
go far beyond any experience. The statement implies
that the liquid in the barrel can be refined into
petrol and other useful organic substances.

\tcrpar
Simon thinks that the reason why economists have been
slow to adopt the Barnet and Morse approach is that it
is merely empirical or rule of thumb, without a
theoretical justification. The resource finitist is
interpreting the data in the light of a metaphysical
theory. The resource finitist sees the world as a
finite store-house. Humans may be getting the goods out
faster and faster with improvements in the machines
they use, but this will only empty the store-house
sooner.

\tcrpar
To provide a theoretical interpretation of the data
Simon introduces the idea of partial
operationalism. Simon stops short of a wholesale
operationalism, but insists that in order for any
economic theory of finite resources to be testable, one
has to construct an operational definition of the word
\tcrquotes{finite} in order that it may be measurable. The
attempt to make any theory more testable is
admirable. However, this aspect of Simon's approach is
unsuccessful. Testability and measurability are, though
related, quite different concepts. Strictly,
introducing measurement-orientated concepts may
increase the empirical information content of our
theories, and as such is to be valued, but a theory may
lack measurable concepts and still be
testable. Moreover, a theory may be replete with
measurable concepts and still be metaphysical, for
example: \tcrquotes{there exists a cylindrical iridium rod of
precisely 1 centimetre in diameter and 1 metre long.}
One may consistently maintain such a theory no matter
how much of space-time is surveyed. Even if some
plausible candidate were found, the testability of the
statement is confined to ranges of measurement outside
the physical limits of measurement because of the
qualification \tcrquotes{precisely}.

\subsection{Explaining versus Justifying 
Predictions of Scarcity}

\tcrpar
I think that there is another more profound problem
with the assessment of past price data. I think that
the relevance of past price data is to be understood
through a separation of the notions of justification
and explanation. I would suggest that all the really
interesting so-called \tcrquotes{justifications} of a scientific
theory (those that are obtained by trying to test the
theory) should be looked at as cases of explanation. It
is, after all, the intellectual revelation and economic
usefulness of an explanation that we want, and not its
justification. One can act no better than in the light
of the truth, so adding justification is both
intellectually and economically superfluous. This puts
past price-data in another light. It is not something
that completely determines the theory (and
projections), but something that can only test
alternative theories (along with their projections).

\tcrpar Those who wish to argue that raw material
resources are infinite should aim for the best
empirically testable theory that both explains past
data and how resources might be indefinitely expanded
in the future. One need not \tcrquotes{justify} future
predictions on the basis of past data.  On the
contrary, the question should be: how do we explain
both the past price data and explain the hypothetical
projection?  Ever since Hume undermined both simple and
probabilistic induction, we have known that an infinite
number of future projections are logically compatible
with any series of past data taken
alone.\footnote{Popper deals with the methodological
  problem of picking the best curve through some given
  graphical points in sections 32 \& 38 of \emph{Logic
    of Scientific Discovery} \cite{Popper:LScD}. Popper argues
  that given that one wants the most informative, and
  therefore most falsifiable, theory one should opt for
  the (theory) curve that has the lower degree of
  dimensionality, i.e., the one whose statement
  requires the smaller number of parameters. Theories
  with higher dimensionality require a greater number
  of basic statements to falsify them. For example, to
  refute the theory that all planetary orbits are
  circles requires only 4 singular statements, whereas
  to refute the theory that all planetary orbits are
  ellipses would require six singular statements.} 
\citeaffixed{Miller:CriticalRationalism}{For
a recent refutation of inductivist thinking see } 
In order to exclude at
least some projections one needs a universal theory.
Notice I am talking about explaining declining
scarcity, not justifying such a prediction. Ideally, in
a scientific prediction one needs a universal theory
plus initial conditions to logically derive (but note,
not demonstrate) a definite testable description of
some future state of affairs. But short of this, one
can settle for a more or less schematic explanation,
which is often the case in the social sciences. An
extreme example is our everyday schematic explanations
of human actions: even though we do not scientifically
predict in detail a piece of behaviour we can
understand it afterwards in the context of the person's
knowledge, aims and problems.\footnote{With the spread
  of psychological knowledge, this is a little unfair
  to \tcrquotes{everyday} explanations. The knowledge of visual
  illusions, for example, is quite wide spread, and
  with this knowledge one can make quite precise
  predictions about another person's subjective
  experience and possible introspective reports.} If
someone shivers on a cold day, this alone will be
insufficient to allow us to scientifically predict that
he will put his coat on; but if he does put his coat
on, we have no hesitation in explaining this as his
attempt to get warm, because we have a general theory
that sees humans as goal-directed, rational agents etc.

\tcrpar
The sort of thing that can be done is for the
resource-finitist and resource-infinitist to put some
time limit on the predictions:

\begin{enumerate}
\item Resource-finitist: measured in real prices over
20 year periods, resources will consistently become
more scarce;

\item Resource-infinitist: Any diminution of resources,
measured in real prices over 20 year periods, will be
more than compensated for within a 20 year period.  
\end{enumerate}

\tcrpar Are there any general theoretical ways of
explaining why resources and resource-augmenting
inventions do not simply dry up? I do not think it
possible to develop a theory from which one can
scientifically derive the prediction of continued
growth in economically useful inventions. I do not
think that this is possible because, as Popper has
argued, technical developments are strongly influenced
by the emergence of new ideas, but there are logical
reasons why one cannot predict (scientifically) any
radically new idea \cite{Popper:OpenUniverse}. The
argument is involved, but briefly and crudely put it is
that if one succeeded in predicting now the emergence
in the future of some new idea, this would be
paradoxical since how can an idea that is only new in
the future be predicted now. The idea would, after all,
have to be stated now as part of the prediction, but
it's newness would evaporate as soon it was stated.
This argument leaves open shrewd conjectures about
future technical development that are not strictly
derived from theory plus initial conditions, and also
the possibility of predicting types of new ideas rather
than definite descriptions of future inventions. It
also allows for carefully stated conditional
predictions. One must also bear in mind that a single
new unpredictable idea may lead to the destruction of
society as such.

\tcrpar
Nevertheless, I do believe it possible to offer a very
general argument that suggests the sort of
considerations that are necessary (though not
sufficient) to explain schematically:

\begin{enumerate}
\item The emergence of an invention or new application
of an old invention in response to a problem. This may
only be possible in retrospect.

\item Why indefinitely continued invention is
possible. One might think that resources will be finite
in the long run because one will eventually \tcrquotes{use up}
the old inventions. My argument shows that possible
inventions are not finite in the long run.

\end{enumerate}

\tcrpar
Natural resources are infinite because of the following
two fundamental facts:

\begin{enumerate} 
\item A theory (such as the theory of boolean logic
gates used in computers) can be applied an infinite
number of times and in an infinite number of different
useful projects because of its universal reference to
all space and time and because of its infinitely varied
logical and (in the case of scientific theories)
information content.

\item Any two theories of technological use can be
usefully combined not in just one way, but in an
infinite number of ways. Not necessarily alone: a
hair-dryer and a computer, for example, may not be
easily combined directly, but they may be combined in a
larger means-end scheme.

\end{enumerate}

\section{A Summary of Simon's Argument for the 
Infinite Extent of Resources}

\tcrpar
The most powerful general theoretical argument against
the claim that natural resources are finite comes from
the pen of Julian Simon. Simon places the issue of
natural resources in the context of the debate on the
impact of population growth on the world's
resources. Simon's basic point is that the concept of a
natural resource can be defined for economic theory
only relative to a service that we obtain from it. In
this approach, Simon is systematically applying Alfred
Marshall's conception of an economic good.  Simon thus
focuses on the World~2 aspect of a resource. Before a
use for oil was invented it was not a resource (in
fact, it was a nuisance to farmers). Similarly, petrol
was at first regarded as a dangerous by-product of the
extraction of paraffin. I guess there is at least ten
million tons of copper in the core of the star Alpha
Centauri, but it is not a resource (at present) because
no one can possibly use it for any service. (This is
not to say that it may not become a resource in the
future. It is an extreme example to show that the
concept of a resource includes the notion of service.)
Things become less clear in some respects and clearer
in others when one considers such things as beautiful
views of sunsets over the Amazon forest. Views, or at
least ones that can be appreciated as resources, only
come into existence with the emergence of self
conscious minds that can frame them and regard them as
a product of, but also as independent of, the self. A
beautiful view is less clearly part of a means-end
scheme than a piece of copper, but E.~Gombrich has
argued that they are constituted by a theoretical
framing of the visual world
\cite{Gombrich:ArtAndIllusion}. Moreover, if people are
willing to pay for a view then we can more easily see
that they are part of a means-end scheme.

\tcrpar
On this analysis of natural resource, the implication
of the claim that our natural resources are finite is
that the amount (or even types) of services that we can
obtain from the world are limited. You can see that we
are already moving away from the idea that a resource
is identifiable with a lump of matter/energy alone and
that we must bring in a psychological interpretation
(World~2) that links any given resource material to a
goal in such a way that the agent regards the material
as a means to the goal. As I have already hinted in the
introduction, this psychological element already
involves a theoretical component that enables the
subject to grasp the goal and the means and their
relationship to one another and to the self. Popper
argues in \emph{Objective Knowledge} that all understanding
involves the grasping of a World~3 object or
relationship \cite{Popper:ObjectiveKnowledge}.
We can extend this to the understanding
that is involved in a subjects' economic
behaviour. Understanding a means-end relationship often
involves understanding that there are alternative means
and alternative goals to choose from. It is impossible
to construct this understanding except in terms of the
subject's use of theoretical interpretations---that is,
World~3 objects.

\tcrpar
The involvement of World~3 in the question of scarcity
becomes clearer once we review the ways in which the
scarcity of a resource (i.e., its services) can be
reduced:

\begin{enumerate}
\item Discoveries of further deposits.

\item Invention of substitute products.

\item Invention of more efficient ways of surveying,
mining, processing and transporting the resource. 
\end{enumerate}

\tcrpar
All these ways of augmenting resources involve not only
psychological (intentional) states (World~2), but the
use of linguistically formulated theories (world
3). The discovery and invention of new resources and
products can be an extremely theoretical task. The
discovery of further deposits of a mineral, for
example, often involves the use of astrophysical
theories and the theoretical interpretation of satellite
and seismic data.

\tcrpar
Simon argues that as population growth puts a pressure
on the demand for services from resources, the real
cost and prices of these resources may increase. But
this very pressure creates the incentive for further
inventions which not only compensate for the increased
demand but increase production to lower costs and
prices below their pre-shortage 
values. \citeaffixed{Boserup:PopulationAndTechnology}
{This is the
best explanation for the results of extensive
historical research. cf.~}

\tcrpar
So the argument moves from whether there is a fixed
amount of copper in a mine or the Earth etc., to
whether there are diminishing returns in the long run
given (a) the growth of our total imaginative capacity
(World~2) and (b) the growth of our inventions and in
general our World~3 objective knowledge. Both of these
increase with increases in population, since the more
people there are, the more minds there are to work on
problems of scarcity and add to our objective
knowledge.

\tcrpar
We have shown that it is the volume of services we
obtain from a resource that is the most meaningful
index of the quantity of a resource. Having established
this, it becomes possible to envision the uninterrupted
physical depletion of a non-renewable resource in a
never ending process in which the total volume of
services obtainable per unit of physical resource
increases practically forever. (The only limit might be
the atomic nature of matter. But what might be more
important than physical quantity is accessibility for
use. The last remaining atoms of the resource may just
become less frequently available for use in any given
project.)

\tcrpar 
This is how
\citeasnoun[p.~55]{Simon:TheoryOfPopulation} sums up
his position with respect to his predecessors:

\begin{quote}
  It is important to notice that there need not be
  diminishing returns over time to additional people,
  because the stock of technology with which people may
  combine their creative talents grows with time.
  Kuznets makes an argument for increasing returns on
  two grounds: (a) the stimulative effect of a dense
  environment, and (b) \tcrquotes{interdependence of
    knowledge of the various parts of the world in
    which we human beings operate}
  \cite[p.~328]{Kuznets:PopulationChange}; for example,
  discoveries in physics stimulate discoveries in
  biology, and vice versa. Kuznets discounts the
  possibility of diminishing returns because
  \tcrquotes{the universe is far too vast relative to
    the size of our planet and what we know about it}
  \cite[p.~329]{Kuznets:PopulationChange}. Machlup
  suggests that every new invention furnishes a new
  idea for potential combination with vast numbers of
  existing ideas \tcrquotes{\ldots [and] the number of
    possible combinations increases geometrically with
    the number of elements at hand}
  \cite[p.~156]{Machlup:SupplyOfInventors}. It is this
  latter idea of an increasing number of possible
  permutations of the available elements of technology
  as the stock increases, when combined with the idea
  of a reduced likelihood of duplicate discoveries as
  the number of possibilities increases faster than the
  number of potential technology producers, that seems
  most compelling to me.
\end{quote}

\tcrpar
The implication is that the number (or amounts) of
resources also can increase geometrically in parallel
with the inventions. (It is important to bear in mind
that Simon's complete theory is fairly elaborate and I
can only hint at its full structure here.)  Simon's
theory does actually explain the price trends of
(non-monopolistic) mineral resources. However, it must
be granted that the form of the explanation is
schematic, for it cannot predict or explain price data
precisely (say, the price of gold 2 months from now) by
deducing them from the model plus initial conditions,
but it is, nevertheless, a valuable and testable
alternative theory. Indeed, this is typical of economic
theories and constitutes no demerit.  It shares its
schematic character with its rival, the hypothesis of
finite resources. However, unlike its rival, the
elaborate theory in to which the assumption of infinite
resources is placed has a high level of testability.
\citeaffixed{Meadows:DynamicsOfGrowth}
{To be completely fair, some resource-finitist models
have made definite predictions, but often it is not
clear how the prediction is logically derived from the
theory, assumed initial conditions and other background
knowledge. Indeed, it is not even clear what the
background knowledge is. Cf.~}

\tcrpar Let us return to the more general question. A
sceptic might ask why he should accept the idea that
the number of possible combinations of inventions grows
geometrically with the number of elements. He might
suggest that a geometric combinatorial increase might
still be finite. Inventions, he could say, are like
melodies: one can easily get the impression that the
class of possible melodies is infinite, but though
large it is far from infinite in size.\footnote{I am
  talking about melodies that can be appreciated by
  humans. Thus I am talking about discernible note
  lengths and pitches, which will obviously be finite
  in number, and melodies of finite length.} I believe
that the best way of answering him is to explore the
theoretical content of inventions, that is, their World
3 aspect. This will show how it is possible for an
invention to be applied in a literally infinite number
of different ways in production, not only with other
inventions but alone. It will become apparent that it
is misleading to say that the number of possible
combinations of inventions increases geometrically with
increases in the number of elements at hand.  Even
keeping the number of basic inventions constant, the
number of ways in which inventions might be combined is
infinite. This fact is obscured by looking at
inventions as simply physical objects that one adds
together in different groups or permutations of order.
This sees inventions as simply World~1 objects.
Popper's notion of an autonomous and causally active
world of abstract entities, World~3, is, I think, the
only way to understand how the potential of our
inventions and hence our resources is infinite.

\section{World 3 and the Unfathomable
Content of our Knowledge}

\tcrpar
To recapitulate, World~3 is the realm of abstract,
objective products of the human mind: theories, logical
relationships, numbers, symphonies. Its content
includes a diverse mixture from the humble Acheulian
axe to the magnificent constructions of modern
mathematics such as Godel's proof of the incompleteness
of arithmetic.  The point I wish to stress here is that
World~3 is a factor of economic production that cannot
be reduced to Worlds~1 or~2, and whose contribution is
quite distinct.

\tcrpar
Suppose I make the following inference on paper: If it
is cloudy, it will rain; it is cloudy, therefore it
will rain. On Popper's model there are three aspects to
this inference. There are the marks on the paper, which
constitute the physical embodiment of the logical
relationship between the premises and the conclusion;
there is the psychological process of inference that I
make (or go through) in grasping the logical
relationship together with the ability to make the
inference again; and there is the actual logical
relationship as such.

\tcrpar It is important to stress that World~3 makes a
difference to our thought and therefore economic
production. For example, the existence of logical
standards makes a difference to our thought. Even a
fallibilist can allow that we sometimes get things
right, and not only that, but also that we get things
right because they are right. You believe that you are
reading an article in English (at least partly) because
you are.  Many people will accept the validity of the
above argument (at least partly) because it is valid;
they will also reject other arguments as invalid (at
least partly) because they are invalid. You may include
training in logic etc.\ as other contributing factors in
our discrimination between valid and invalid
inferences, but this does not make validity and
invalidity causally impotent, because validity cannot
be defined in terms of training or dispositions to
discriminate.\footnote{To deny the influence of logical
  relationships would be to imply something very
  strange indeed: that the way things are never has the
  slightest influence on what we think or are prepared
  to maintain.  Correctly identifying errors in
  reasoning is on this view a purely accidental affair.
  But if the set of hypotheses we maintain in science
  is controlled even slightly by the process of trial
  and error elimination, in which the false hypotheses
  are cast from the body of science because they
  contradict true observation reports, then the
  maintainance of some hypotheses after each period
  of elimination is partly explained by their being
  true and the rejection of false ones is partly
  explained by the fact that they are false.} The
logical contradiction between say, Newton's theory and
Einstein's theory of gravitation, made a difference to
the way scientists think.

\tcrpar
World~3 has the following properties:

\begin{enumerate}
\item It is an abstract product of the human mind.

\item It is partially autonomous in four senses: 

\begin{description}

\item (a) it has a partly unfathomable content; 

\item (b) it has an unintended and irrevocable structure and
consequences; 

\item (c) it has properties and relations that we discover but 
cannot alter; 

\item (d) it is irreducible to psychological or physical 
states of affairs or laws.

\end{description}

\item It is causally active on World~1 via World~2.
\end{enumerate}

\tcrpar
One of Popper's arguments for the reality of World~3 is
that our grasp of a World~3 object (such as the natural
numbers) can affect our interaction with the physical
world and that it is common sense that only real things
can affect a physical object. (However, it is wrong to
see the affect on physical objects as \tcrquotes{constituting}
the reality of World~3 objects.) An architect's
understanding of mathematics makes a difference to the
buildings that he can construct or would even consider
constructing, but such an understanding would hardly
exist if the mathematics had not first been
invented. Hence World~3 makes a difference to economic
production. But how much of a difference can it make?
To answer this you need to look closely at its infinite
content.

\tcrpar
Part of the content of World~3 will always remain in
principle unfathomable. World~3 has been likened to a
library of human knowledge, but although this is a good
metaphor as far as it goes it is misleading. When a
theory is created it is written down in a book and some
of its implications may even be worked out and also
written down. This is the part of the theory that
becomes represented in a physical form. Think of all
the future worked-out implications of the theory. This
is still only a part of the theory's content, the rest
is the part that never gets represented in physical
form. There is always a residual because the content is
infinite. There are two ways of bringing out the
infinite content of a theory: by talking of the
information content and the logical content. The
logical content of a theory is the class of all the
(nontautological) consequences that can be logically
derived from the theory (it may be identified with
Tarski's \tcrquotes{consequence class}).

\tcrpar
The argument for the infinite logical content of a
theory $t$ can be put thus. Suppose an infinite list of
statements that are pair-wise contradictory and which
individually do not entail $t$: $a, b, c\ldots$ Then the
statement \tcrquotes{$t$ or $a$ or both} follows from $t$. The same
holds for each and every one of the statements in the
infinite list. Since the statements in the list are
pair-wise contradictory one can infer that none of the
statements \tcrquotes{$t$ or $a$ or both},
\tcrquotes{$t$ or $b$ or both} etc., is
interderivable. Thus the logical content of $t$ must be
infinite.

\tcrpar This in itself is not so important, but when
combined with the idea of information content, the two
notions produce some very interesting ramifications. In
the \emph{Logic of Scientific Discovery}
\cite{Popper:LScD}, Popper put forward the idea that a
statement says more the more it forbids.  Carnap,
accepting Popper's suggestion, defined the assertive
power of a sentence as the class of possible cases it
excludes \cite[p.~151]{Carnap:Semantics}.  Later Popper
reformulated the intuitive idea in terms of
theories, of both high and low universality
\cite[see esp.~note 15]{Popper:UnendedQuest}. The
information content is the class of all those
statements that are logically incompatible with the
given theory. Thus since Einstein's theory contradicts
Newton's theory, Einstein' theory is part of the
information content of Newton's theory. Newton could
hardly have known this, and so it could not have been
part of his psychology.

\tcrpar
Most philosophers resist this analysis because of their
adherence to psychologism and conceptual analysis. But
Popper's analysis reveals that much of interest is to
be discovered in the analysis of theories as such,
considered as objective entities, and their logical
relationships with one another.

\tcrpar
As Popper shows, when we combine this result with the
idea of logical content we obtain a parallel result,
for if $E$ is part of the information content of $N$ then
Non $E$ is part of $N$'s logical content. Thus both the
logical and information content of theories consist of
an infinite number of non-trivial consequences. As
Popper says, it follows that the task of understanding
a theory is infinite.  Furthermore, there are an
infinite number of unknown theories that form part of
the information content and logical content of Newton's
theory, and indeed of any empirical theory. Since the
concept of knowledge is often quite restrictively
defined, it might be better to speak in more general
terms of \tcrquotes{representations}, and say more generally that
Newton's mind obviously did not have a representation
of Einstein's theory or its negation, let alone most of
the other theories that are part of its information or
logical content.

\tcrpar
Of course, one can also mention the work of G\"{o}del in
this connection as pointing to the unfathomable content
of our theories, and therefore of our inventions and
resources. It can be shown that the complete system of
all true propositions in the arithmetic of numbers is
not axiomatizable and is undecidable. This means that
there will always be an infinite number of unsolved
problems in arithmetic.  I wanted to focus on the ideas
of logical and information content because these are
rarely discussed or applied to new interesting
areas. The exception has been the late Professor
Bartley, who has applied these ideas in a fascinating
way to art criticism and the Marxian idea of
alienation \cite{Bartley:UnfathomedKnowledge}.

\section{The Special Economic Status 
of Objective Knowledge: 
Resources and Inventions as World 3 Objects}

\tcrpar
The above has some interesting implications for an
economic analysis of resources and the issue of
long-run diminishing returns.

\tcrpar
Just as a resource needs to be interpreted, so an
invention cannot be reduced to its chemical and
physical properties and relations, but must be placed
in a means-end relationship. All inventions are
means/ends relationships; they are invented and adopted
for a purpose. This shows their World~2 aspect. But a
means/end relationship is partly constituted by its
theoretical interpretation, and thus all inventions are
theory impregnated. Inventions, therefore, have a world
3 aspect also.

\tcrpar
I must make clear at this point that I do not subscribe
to the popular view that every technological decision
and action (including inventions) is prescribed by one
or more scientific theories; in fact none are. This
would overlook the fact that scientific laws are
universal and therefore can only proscribe; alone, they
can tell us only what cannot happen, not what will
happen, and therefore alone cannot tell us what we
should do to achieve a given end. Building a bridge,
car, space-ship and tube of toothpaste is a matter of
engineers discovering sets of constructible initial
conditions that typically lead efficiently to the
desired result. This is a conjecture and refutation
affair. Universal theories of science help the engineer
insofar as they can be used to eliminate some of the
hopeful candidates of efficient sets of initial
conditions, namely the ones whose description
contradicts the accepted scientific theories.

\tcrpar
In talking of the theories that help to constitute and
identify a given invention or resource, I include these
low level theories. But I also want to make it clear
that even these theories plus our psychological
dispositions toward the invention or resource do not
exhaust the useful possibilities inherent in a type or
particular invention or type or particular portion of a
resource. It is sufficient for my argument that at
least part of the range of its useful possibilities is
encompassed by these low level theories.

\tcrpar
Now theories as World~3 objects have three relevant
properties:

\begin{enumerate}
\item They are universal, applying to all space and time.

\item They have infinitely varied logical/information
content.

\item They are infinitely copyable.

\end{enumerate}

\tcrpar
Now, on the basis of these facts it becomes clear how
it is possible for a) an invention to be applied in a
literally infinite number of significantly different
possible ways and b) combined in a literally infinite
number of significantly different possible ways with
other inventions. But an exactly analogous argument
applies to resources, for resources are also
interpreted as a means/end relationship.

\tcrpar
Any particular wheel or lever or computer will wear
out, but due to the universality of the theory that
helps to constitute the invention, the invention type
may be applied a potentially infinite number of times
without diminished effect. Due to the infinitely varied
logical/information content of the invention-theory,
the invention may be applied in an infinite number of
significantly varied ways. No one person, therefore,
can work out all the useful content of a theory of a
resource. If one employs more and more people on a
given piece of land, one will get diminishing returns;
after all, there is only so much room on a piece of
land. But one may not get this when employing more and
more people on a given theory, say the theory of levers
or the quantum computer. Due to its infinite content,
it has an infinitely varied terrain to work on, as it
where. This is most clearly the case with something
like the theory of arithmetic, where in the light of
the work of G\"{o}del and Tarski, there will always be an
infinite number of problems to work on.  Unlike a piece
of land, an indefinite number of copies can be made of
a theory. At any time, any number of people can be
working out useful ramifications and implications of
the theory and applying them.

\tcrpar
Let us return to Machlup's suggestion that the number
of possible combinations of inventions increases
geometrically with the number of elements at hand. From
our analysis it is clear that any two inventions will
each have a cluster of theories that explains, partly
constitutes and identifies it. It is the logical and
information content of these theories that allows us to
combine them to make further inventions. But because of
the infinite content of the theories they can be
combined in a potentially infinite number of ways. To
explain the emergence of any given combination one will
look to see what the inventor's problem situation was,
how the inventor searched through different
combinations of different portions of logical and
information content, and finally, how the two or more
invention-theories were combined. On this analysis, it
becomes clearer that a \tcrquotes{fusion of two inventions} may
consist of the following possible combinations:

\begin{description}

\item (a) Proper subsets of the theoretical contents of
the two inventions.

\item (b) A proper subset of one with the whole of the other.

\item (c) A newly discovered subset of the content of
one with a familiar subset of another.

\item (d) Whole or part contents brought together via a
bonding theory.

\end{description}

\tcrpar
In fact, since the aim of the fusion of two inventions
is a new invention, the two invention-theories will
form part of a larger action schema, and so it will
always be through some third theory that the two
inventions are combined.

\tcrpar I suggest that this logical analysis is a more
subtle and powerful way of revealing the way in which
the number of possible combinations of inventions
increases much more rapidly with the number of elements
at hand, than saying with Machlup that they increase
\tcrquotes{geometrically}.  This is reinforced when one considers
that in combining two theories one sometimes obtains
interesting implications and ramifications not
contained in the content of either theory considered
alone. Watkins has fruitfully explored this possibility
in his book \emph{Science and Scepticism}
\cite{Watkins:ScienceAndScepticism}. It is often said
that an invention that simply combines previous
inventions is not really a new invention, but a logical
analysis of inventions in terms of content allows us to
see that such invention-combinations can bring
radically new useful consequences, emergent properties,
into existence.

\section{The Possibility of Infinitely New 
Information Processing: Why Tipler Needs World 3}

\tcrpar
Before I conclude this paper, I must discuss the
relevance of Frank Tipler's fascinating argument for
indefinite economic growth to my own argument.

\tcrpar
Tipler argues that life and hence economic growth can
continue forever
\cite{Tipler:Anthropic,Tipler:PhysicsOfImmortality}.
Carbon based life-forms like ourselves are doomed
eventually in the extremely high temperatures to be
encountered at the approach to the final singularity in
a closed universe. However, our successors---highly
sophisticated, self-reproducing computers---will
colonize the whole of space and will effectively
undergo an infinite amount of economic growth and
experience before the singularity.

\tcrpar
Tipler accepts that growth rates within the present
epoch are limited by physics, but thinks that most
predictions of physical limits have ignored economics
and thus grossly underestimated the potential for
economic growth. Following Simon, Tipler assumes that
economic production is equivalent to the production of
services, but Tipler adds that each of these units of
service are in turn equivalent to the production and
transfer of amounts of information. Thus the limits of
economic growth are the limits of the growth of
knowledge: the amount of information that can be read,
stored and processed.

\tcrpar
Tipler says: \tcrquotes{Information processing is
  constrained by the first and second laws of
  thermodynamics. These laws imply that the amount of
  information that can be processed at a given
  temperature $T$ is $I = E/(kT \ln 2)$ where $E$ is
  the energy available for processing, $(\ln 2)$ is the
  natural logarithm of 2, and $k$ is Boltzmann's
  constant. Now any temperature $T$ that we can use is
  greater than the background radiation, which is 3
  degrees on the Kelvin scale, and if we limit
  ourselves to operations on the Earth, the greatest
  available energy is $E = Mc^2$ , where $M$ is the
  mass of the Earth.}
\cite[pp.~4--5]{Tipler:HumaneStudies} Thus even life
based on steady state economies is doomed, if it
remains confined to the Earth.  But life need not be
confined to the Earth, since there is the possibility
of expanding into and beyond the solar system using von
Neumann probes. In any case, Tipler points out, this
relationship between energy and information processing
allows for a one hundred-billion-fold increase in
economic wealth before we reach the physical limits of
the Earth. This estimate is based on the assumption
that we can increase economic wealth at the same rate
as we increase the speed of our computers.

\tcrpar
Tipler argues that what is important to future
self-conscious intelligent life will be subjective
time, measured in terms of the number of thoughts
experienced, not what physicists call proper time, that
measured by atomic clocks. Tipler identifies thought
with information processing of high speed computers.
Tipler argues that indefinite economic growth is
possible in a closed universe only if an infinite
amount of information can be processed before the end
of time. This is possible if the rate of information
processing increases to infinity before the big crunch.
This, Tipler argues, is possible. The required energy
for the information processing is obtained from the
differential speeds at which different regions of the
universe will collapse, a phenomenon known as sheer.

\tcrpar
However, there is an important qualification. If this
information processing is done on a finite state
machine, then eventually it will start to repeat
itself: no new thought would be possible. This
information processing must be conducted on an infinite
state machine. To supply this machine, Tipler
conjectures that life, taken as a whole, can be
regarded as an infinite state machine; but Tipler fails
to explain how.  I conjecture that the body of
scientific knowledge provides the basis for an infinite
state machine. The unfathomable information content of
theories provides the \tcrquotes{infinite tape} for the
Turing machine.  Stated more generally, the possibility
of thinking an infinite number of new thoughts is made
possible by World~3. It must be born in mind that there
are two senses of \tcrquotes{information} here. Tipler's
\tcrquotes{information} is the engineer's conception of
actually encoded information (Shannon and Weaver),
existing as realized distinct states of matter/energy,
such as on-off electrical states in a computer. The
\tcrquotes{Logical content} and \tcrquotes{information
  content} of say, scientific theories, are mostly
unencoded, existing as only \emph{potential}
information in the engineer's sense.  In some places
Tipler explicitly identifies knowledge and information.
But the above analysis of the unfathomable content of
scientific theories makes it important to distinguish
between knowledge and information. Tipler's bold and
fascinating argument is thereby reinforced.

\section{Conclusion}

\tcrpar
One might think that there is a lacuna in my
argument. The fact that an invention can be applied in
an infinite number of significantly different ways does
not show that those ways are all economic or even
useful. But this misunderstands my case. I am not
expounding an a priori argument that concludes that
resources are necessarily infinite. I am making a
conjecture that resources are infinite in the long-term
and the trying to explain how this is possible by
looking at the interaction of Worlds~1, 2~and~3.

\tcrpar
I think that my argument points to the World~3 abstract
structure of our resources and inventions that acts as
both a \tcrquotes{scaffolding of thought} for the inventor and as
a means whereby the human mind can endlessly expand the
usefulness of the world. Whether this be for good or
ill, is another question.


\nocite{Levinson:MindAtLarge}
\nocite{Simon:UltimateResource}
\nocite{Simon:ResourcefulEarth}
\nocite{Simon:StateOfHumanity}

\bibliography{v01n02}
\bibliographystyle{tcr}

\newpage
\section*{Author Contact Information\label{authors}}

\label{percival}
\begin{tabular}{ll}
    \multicolumn{2}{l}{\hlnk{\textbf{Dr.\ Ray S.~Percival}}
{http://www.eeng.dcu.ie/\~{}tkpw/people/percival.html}}\\[3mm]
    \multicolumn{2}{l}{70 Hillview Court,}\\
    \multicolumn{2}{l}{Astley Bridge,}\\
    \multicolumn{2}{l}{Bolton   BL1 8NU}\\
    \multicolumn{2}{l}{England.}\\[3mm]
    \textbf{Telephone:} & \texttt{+44-1204-593-114}\\
    \textbf{E-mail:}    & \mailto{100525.373@compuserve.com}\\
    \textbf{Web:}       & 
      \hoturl{http://www.eeng.dcu.ie/\~{}tkpw/people/percival.html}
\end{tabular}
    
\section*{Copyright\label{copyright}}

This article is copyright \copyright~\thecopyrightdate\ by
\htmlref{Dr.\ Ray S.~Percival}{authors}.

Permission is hereby granted to private individuals to
access, copy and distribute this work, for purposes of
private study only, provided that the distribution is
complete and unmodified, is accompanied by this
copyright notice, and that no charges are levied.

The work may \emph{not\/} be accessed or copied, in
whole or in part, for commercial purposes, except with
the prior written permission of the author.

All other rights reserved.

\newpage
\section*{Retrieval\label{retrieval}}

The resources comprising this article are electronically
retrievable, in various formats, via the \hlnk{World Wide Web}
{http://www.w3.org/hypertext/WWW/TheProject}. The base
(directory) URL is:

\begin{quote}
\hoturl{http://www.eeng.dcu.ie/\~{}tkpw/tcr/volume-01/number-02}
\end{quote}

This URL directly accesses the root node of the HTML format
article.  Alternatively, an additional filename may be appended
to this URL, to retrieve one of the other formats, or related
resources, as follows:

\begin{itemize}
\item The complete set of files making up the HTML format of the
article is available in the single compressed archive file
\hoturl{v01n02.html.tar.gz} (40~KByte),
which can be conveniently downloaded for local and/or offline
browsing.
\item  A printable version of the article is available in 
postscript format
as file \hoturl{v01n02.ps} (256~KByte,
32~pages), or in compressed form as 
file \hoturl{v01n02.ps.gz} (87~KByte).
\item The original \LaTeX\ source of the article is available as file
\hoturl{v01n02.tex} (70~KByte), or in compressed form as 
file \hoturl{v01n02.tex.gz} (25~KByte).
\item The \BibTeX\ source for the reference
list is available as file \hoturl{v01n02.bib} 
(10~KByte).
\item Finally, a complete set of the files just listed
is available as a single compressed 
archive file \hoturl{v01n02.tar.gz} (157~KByte). 
\end{itemize}

\input{tcrback}

\end{document}
