Parsimony The Fourth Substance
Dr. Sam Vaknin
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Occasionalism is a variation upon Cartesian
metaphysics. The latter is the most notorious case of dualism (mind
and body, for instance). The mind is a "mental substance".
The body a "material substance". What permits
the complex interactions which happen between these two disparate
"substances"? The "unextended mind" and the
"extended body" surely cannot interact without a
mediating agency, God. The appearance is that of direct
interaction but this is an illusion maintained by Him. He moves
the body when the mind is willing and places ideas in the mind
when the body comes across other bodies. Descartes postulated
that the mind is an active, unextended, thought while the body is
a passive, unthinking extension. The First Substance and the
Second Substance combine to form the Third Substance, Man. God
the Fourth, uncreated Substance facilitates the
direct interaction among the two within the third. Foucher raised
the question: how can God a mental substance
interact with a material substance, the body. The answer offered
was that God created the body (probably so that He will be able
to interact with it). Leibnitz carried this further: his Monads,
the units of reality, do not really react and interact. They just
seem to be doing so because God created them with a pre-established
harmony. The constant divine mediation was, thus, reduced to a
one-time act of creation. This was considered to be both a
logical result of occasionalism and its refutation by a reductio
ad absurdum argument.
But, was the fourth substance necessary at all?
Could not an explanation to all the known facts be provided
without it? The ratio between the number of known facts (the
outcomes of observations) and the number of theory elements and
entities employed in order to explain them is the
parsimony ratio. Every newly discovered fact either reinforces
the existing worldview or forces the introduction of a new
one, through a "crisis" or a "revolution" (a
"paradigm shift" in Kuhn's abandoned phrase). The
new worldview need not necessarily be more parsimonious. It could
be that a single new fact precipitates the introduction of a
dozen new theoretical entities, axioms and functions (curves
between data points). The very delineation of the field of study
serves to limit the number of facts, which could exercise such an
influence upon the existing worldview and still be considered
pertinent. Parsimony is achieved, therefore, also by affixing the
boundaries of the intellectual arena and / or by declaring
quantitative or qualitative limits of relevance and negligibility.
The world is thus simplified through idealization. Yet, if this
is carried too far, the whole edifice collapses. It is a fine
balance that should be maintained between the relevant and the
irrelevant, what matters and what could be neglected, the
comprehensiveness of the explanation and the partiality of the
pre-defined limitations on the field of research.
This does not address the more basic issue of
why do we prefer simplicity to complexity. This preference runs
through history: Aristotle, William of Ockham, Newton, Pascal
all praised parsimony and embraced it as a guiding
principle of work scientific. Biologically and spiritually, we
are inclined to prefer things needed to things not needed.
Moreover, we prefer things needed to admixtures of things needed
and not needed. This is so, because things needed are needed,
encourage survival and enhance its chances. Survival is also
assisted by the construction of economic theories. We all engage
in theory building as a mundane routine. A tiger beheld means
danger is one such theory. Theories which incorporated
less assumptions were quicker to process and enhanced the chances
of survival. In the aforementioned feline example, the virtue of
the theory and its efficacy lie in its simplicity (one
observation, one prediction). Had the theory been less
parsimonious, it would have entailed a longer time to process and
this would have rendered the prediction wholly unnecessary. The
tiger would have prevailed. Thus, humans are Parsimony Machines (an
Ockham Machine): they select the shortest (and, thereby, most
efficient) path to the production of true theorems, given a set
of facts (observations) and a set of theories. Another way to
describe the activity of Ockham Machines: they produce the
maximal number of true theorems in any given period of time,
given a set of facts and a set of theories. Poincare, the French
mathematician and philosopher, thought that Nature itself, this
metaphysical entity which encompasses all, is parsimonious. He
believed that mathematical simplicity must be a sign of truth. A
simple Nature would, indeed, appear this way (mathematically
simple) despite the filters of theory and language. The "sufficient
reason" (why the world exists rather than not exist) should
then be transformed to read: "because it is the simplest of
all possible worlds". That is to say: the world exists and
THIS world exists (rather than another) because it is the most
parsimonious not the best, as Leibnitz put it of
all possible worlds.
Parsimony is a necessary (though not sufficient)
condition for a theory to be labelled "scientific". But
a scientific theory is neither a necessary nor a sufficient
condition to parsimony. In other words: parsimony is possible
within and can be applied to a non-scientific framework and
parsimony cannot be guaranteed by the fact that a theory is
scientific (it could be scientific and not parsimonious).
Parsimony is an extra-theoretical tool. Theories are under-determined
by data. An infinite number of theories fits any finite number of
data. This happens because of the gap between the infinite number
of cases dealt with by the theory (the application set) and the
finiteness of the data set, which is a subset of the application
set. Parsimony is a rule of thumb. It allows us to concentrate
our efforts on those theories most likely to succeed. Ultimately,
it allows us to select THE theory that will constitute the
prevailing worldview, until it is upset by new data.
Another question arises which was not hitherto
addressed: how do we know that we are implementing some mode of
parsimony? In other words, which are the FORMAL requirements of
The following conditions must be satisfied by
any law or method of selection before it can be labelled "parsimonious":
- Exploration of a higher level of causality
the law must lead to a level of causality, which
will include the previous one and other, hitherto
apparently unrelated phenomena. It must lead to a cause,
a reason which will account for the set of data
previously accounted for by another cause or reason AND
for additional data. William of Ockham was, after all a
Franciscan monk and constantly in search for a Prima Causa.
- The law should either lead to, or be part
of, an integrative process. This means that as previous
theories or models are rigorously and correctly combined,
certain entities or theory elements should be made
redundant. Only those, which we cannot dispense with,
should be left incorporated in the new worldview.
- The outcomes of any law of parsimony
should be successfully subjected to scientific tests.
These results should correspond with observations and
with predictions yielded by the worldviews fostered by
the law of parsimony under scrutiny.
- Laws of parsimony should be semantically
correct. Their continuous application should bring about
an evolution (or a punctuated evolution) of the very
language used to convey the worldview, or at least of
important language elements. The phrasing of the
questions to be answered by the worldview should be
influenced, as well. In extreme cases, a whole new
language has to emerge, elaborated and formulated in
accordance with the law of parsimony. But, in most cases,
there is just a replacement of a weaker language with a
more powerful meta-language. Einstein's Special
Theory of Relativity and Newtonian dynamics are a prime
example of such an orderly lingual transition, which was
the direct result of the courageous application of a law
- Laws of parsimony should be totally
subjected (actually, subsumed) by the laws of Logic and
by the laws of Nature. They must not lead to, or entail,
a contradiction, for instance, or a tautology. In physics,
they must adhere to laws of causality or correlation and
refrain from teleology.
- Laws of parsimony must accommodate
paradoxes. Paradox Accommodation means that theories,
theory elements, the language, a whole worldview will
have to be adapted to avoid paradoxes. The goals of a
theory or its domain, for instance, could be minimized to
avoid paradoxes. But the mechanism of adaptation is
complemented by a mechanism of adoption. A law of
parsimony could lead to the inevitable adoption of a
paradox. Both the horns of a dilemma are, then, adopted.
This, inevitably, leads to a crisis whose resolution is
obtained through the introduction of a new worldview. New
assumptions are parsimoniously adopted and the paradox
- Paradox accommodation is an important
hallmark of a true law of parsimony in operation. Paradox
Intolerance is another. Laws of parsimony give theories
and worldviews a "licence" to ignore paradoxes,
which lie outside the domain covered by the parsimonious
set of data and rules. It is normal to have a conflict
between the non-parsimonious sets and the parsimonious
one. Paradoxes are the results of these conflicts and the
most potent weapons of the non-parsimonious sets. But the
law of parsimony, to deserve it name, should tell us
clearly and unequivocally, when to adopt a paradox and
when to exclude it. To be able to achieve this formidable
task, every law of parsimony comes equipped with a
metaphysical interpretation whose aim it is to plausibly
keep nagging paradoxes and questions at a distance. The
interpretation puts the results of the formalism in the
context of a meaningful universe and provides a sense of
direction, causality, order and even "intent".
The Copenhagen interpretation of Quantum Mechanics is an
important member of this species.
- The law of parsimony must apply both to
the theory entities AND to observable results, both part
of a coherent, internally and externally consistent,
logical (in short: scientific) theory. It is divergent-convergent:
it diverges from strict correspondence to reality while
theorizing, only to converge with it when testing the
predictions yielded by the theory. Quarks may or may not
exist but their effects do, and these effects are
- A law of parsimony has to be invariant
under all transformations and permutations of the theory
entities. It is almost tempting to say that it should
demand symmetry had this not been merely an
aesthetic requirement and often violated.
- The law of parsimony should aspire to a
minimization of the number of postulates, axioms, curves
between data points, theory entities, etc. This is the
principle of the maximization of uncertainty. The more
uncertainty introduced by NOT postulating explicitly
the more powerful and rigorous the theory /
worldview. A theory with one assumption and one
theoretical entity renders a lot of the world an
uncertain place. The uncertainty is expelled by using the
theory and its rules and applying them to observational
data or to other theoretical constructs and entities. The
Grand Unified Theories of physics want to get rid of four
disparate powers and to gain one instead.
- A sense of beauty, of aesthetic
superiority, of acceptability and of simplicity should be
the by-products of the application of a law of parsimony.
These sensations have been often been cited, by
practitioners of science, as influential factors in
weighing in favour of a particular theory.
- Laws of parsimony entail the arbitrary
selection of facts, observations and experimental results
to be related to and included in the parsimonious set.
This is the parsimonious selection process and it is
closely tied with the concepts of negligibility and with
the methodology of idealization and reduction. The
process of parsimonious selection is very much like a
strategy in a game in which both the number of players
and the rules of the game are finite. The entry of a new
player (an observation, the result of an experiment)
sometimes transforms the game and, at other times,
creates a whole new game. All the players are then moved
into the new game, positioned there and subjected to its
new rules. This, of course, can lead to an infinite
regression. To effect a parsimonious selection, a theory
must be available whose rules will dictate the selection.
But such a theory must also be subordinated to a law of
parsimony (which means that it has to parsimoniously
select its own facts, etc.). a meta-theory must,
therefore, exist, which will inform the lower-level
theory how to implement its own parsimonious selection
and so on and so forth, ad infinitum.
- A law of parsimony falsifies everything
that does not adhere to its tenets. Superfluous entities
are not only unnecessary they are, in all
likelihood, false. Theories, which were not subjected to
the tests of parsimony are, probably, not only non-rigorous
but also positively false.
- A law of parsimony must apply the
principle of redundant identity. Two facets, two aspects,
two dimensions of the same thing must be construed
as one and devoid of an autonomous standing, not as
separate and independent.
- The laws of parsimony are "back
determined" and, consequently, enforce "back
determination" on all the theories and worldviews to
which they apply. For any given data set and set of rules,
a number of parsimony sets can be postulated. To decide
between them, additional facts are needed. These will be
discovered in the future and, thus, the future "back
determines" the right parsimony set. Either there is
a finite parsimony group from which all the temporary
groups are derived or no such group exists and an
infinity of parsimony sets is possible, the results of an
infinity of data sets. This, of course, is thinly veiled
pluralism. In the former alternative, the number of facts
/ observations / experiments that are required in order
to determine the right parsimony set is finite. But,
there is a third possibility: that there is an eternal,
single parsimony set and all our current parsimony sets
are its asymptotic approximations. This is monism in
disguise. Also, there seems to be an inherent (though
solely intuitive) conflict between parsimony and infinity.
- A law of parsimony must seen to be at
conflict with the principle of multiplicity of
substitutes. This is the result of an empirical and
pragmatic observation: The removal of one theory element
or entity from a theory precipitates its
substitution by two or more theory elements or entities (if
the preservation of the theory is sought). It is this
principle that is the driving force behind scientific
crises and revolutions. Entities do multiply and Ockham's
Razor is rarely used until it is too late and the theory
has to be replaced in its entirety. This is a
psychological and social phenomenon, not an inevitable
feature of scientific progress. Worldviews collapse under
the mere weight of their substituting, multiplying
elements. Ptolmey's cosmology fell prey to the
Copernican model not because it was more efficient, but
because it contained less theory elements, axioms,
equations. A law of parsimony must warn against such
behaviour and restrain it or, finally, provide the ailing
theory with a coup de grace.
- A law of parsimony must allow for full
convertibility of the phenomenal to the nuomenal and of
the universal to the particular. Put more simply: no law
of parsimony can allow a distinction between our data and
the "real" world to be upheld. Nor can it
tolerate the postulation of Platonic "Forms"
and "Ideas" which are not entirely reflected in
- A law of parsimony implies necessity. To
assume that the world is contingent is to postulate the
existence of yet another entity upon which the world is
dependent for its existence. It is to theorize on yet
another principle of action. Contingency is the source of
entity multiplication and goes against the grain of
parsimony. Of course, causality should not be confused
with contingency. The former is deterministic the
latter the result of some kind of free will.
- The explicit, stated, parsimony, the one
formulated, formalized and analysed, is connected to an
implicit, less evident sort and to latent parsimony.
Implicit parsimony is the set of rules and assumptions
about the world that are known as formal logic. The
latent parsimony is the set of rules that allows for a (relatively)
smooth transition to be effected between theories and
worldviews in times of crisis. Those are the rules of
parsimony, which govern scientific revolutions. The rule
stated in article (a) above is a latent one: that in
order for the transition between old theories and new to
be valid, it must also be a transition between a lower
level of causality and a higher one.
Efficient, workable, parsimony is either
obstructed, or merely not achieved through the following venues
- Association the formation of
networks of ideas, which are linked by way of verbal,
intuitive, or structural association, does not lead to
more parsimonious results. Naturally, a syntactic,
grammatical, structural, or other theoretical rule can be
made evident by the results of this technique. But to
discern such a rule, the scientist must distance himself
from the associative chains, to acquire a bird's eye
view , or, on the contrary, to isolate, arbitrarily or
not, a part of the chain for closer inspection.
Association often leads to profusion and to embarrassment
of riches. The same observations apply to other forms of
chaining, flowing and networking.
- Incorporation without integration (that is,
without elimination of redundancies) leads to the
formation of hybrid theories. These cannot survive long.
Incorporation is motivated by conflict between entities,
postulates or theory elements. It is through
incorporation that the protectors of the "old truth"
hope to prevail. It is an interim stage between old and
new. The conflict blows up in the perpetrators' face
and a new theory is invented. Incorporation is the sworn
enemy of parsimony because it is politically motivated.
It keeps everyone happy by not giving up anything and
accumulating entities. This entity hoarding is poisonous
and undoes the whole hyper-structure.
- Contingency see (r) above.
- Strict monism or pluralism see (o)
- Comprehensiveness prevents parsimony. To
obtain a description of the world, which complies with a
law of parsimony, one has to ignore and neglect many
elements, facts and observations. Godel demonstrated the
paradoxality inherent in a comprehensive formal logical
system. To fully describe the world, however, one would
need an infinite amount of assumptions, axioms,
theoretical entities, elements, functions and variables.
This is anathema to parsimony.
- The previous excludes the reconcilement of
parsimony and monovalent correspondence. An isomorphic
mapping of the world to the worldview, a realistic
rendering of the universe using theoretical entities and
other language elements would hardly be expected to be
parsimonious. Sticking to facts (without the employ of
theory elements) would generate a pluralistic
multiplication of entities. Realism is like using a
machine language to run a supercomputer. The path of
convergence (with the world) convergence (with
predictions yielded by the theory) leads to a
proliferation of categories, each one populated by sparse
specimen. Species and genera abound. The worldview is
marred by too many details, crowded by too many
apparently unrelated observations.
- Finally, if the field of research is
wrongly too narrowly defined, this could be
detrimental to the positing of meaningful questions and
to the expectation of receiving meaningful replies to
them (experimental outcomes). This lands us where we
started: the psychophysical problem is, perhaps, too
narrowly defined. Dominated by Physics, questions are
biased or excluded altogether. Perhaps a Fourth Substance
IS the parsimonious answer, after all.
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