Parsimony – The Fourth Substance
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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 parsimony?
The following conditions must be satisfied by any law
or method of selection before it can be labelled “parsimonious”:
Efficient, workable, parsimony is either obstructed, or merely
not achieved through the following venues of action:
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
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 of parsimony.
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 disappears.
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
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
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 the
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.
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
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) above
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
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.