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Chapter 2
Free Will in the Probabilistic World
No idea cat1 be so rigkrly called undererrnitled,
po~vse~nantic,
open /or greatest nrisunderstandings a n d
rherefore ocr~rallysubjecr ro them as !he idea of
freedofn.
G . W . F. HEGEL
Indeed, the antinomy o f freedom and necessity has been the concern
of European thought at least since the Middle Ages. Thomas Aquinas
(1225-1274) believed consistency to be a self-evident and indubitable
principle o f thinking and existence. Freedom was always unavoidably
opposed to necessity. This followed naturally from the rigid determinism
o f the rationalistic vision of the world. The French scholastic Buridan
(1300-1358) believed the problem of free will to be logically unsolvable.
His arguments seem to have greatly affected the further evolution of this
problem. In any case, well known is the paradox (which, incidentally,
does not belong to Buridan) o f Buridan's ass which, not having logical
grounds for making a decision, starves to death between two identical
stacks o f hay. By the way, this paradox is a brilliant illustration of how
the problem o f free will merges with the concept of chance. Indeed, a
metaobserver could not tell the manifestation of free will from random
behavior if he observed, on the one hand, the fall of a coin (which can
fall down heads or tails) and, on the other hand, the real ass which does
not actually starve to death between two identical stacks of hay but in the
sequence o f similar situations chooses one-sometimes the right one and
sometimes the left one. The meraobserver would describe both cases by
the same binomial distribution function. The coin may be claimed to
possess free will to the extent that Buridan's ass does. In other words,
what is called random in the physical world in psychology seems to us an
instance of free will.
Attempts to remove the antinomy "freedom versus necessity" are still
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Language of the Probabilistic Vision
being made, in Marxism (this will be elucidated elsewhere) and in
positivism. A s a n example I would like t o quote an elegant paper by Gill
(1971) in which freedom is considered within a calculus: t o define
freedom, three postulates are introduced. and then two theorems are
formulated. Naturally. when a calculus is constructed, chance has to be
excluded:
If a given command is contingent, its contradictory opinion is
necessary-cannot be rejected without self-contradiction. The agent
cannot control himself if he commands or permits an inconsistency.
If a prohibition is contingent-not necessary-its contradictory permission is necessary @. 9).
However, we find such reasoning fairly artificial.
A probabilistic vision o f the world removes the problem o f free will
from consideration, and its dramatic quality disappears. Everything is
explained in a very simple way.
This can be illustrated by the perception o f verbal meanings. In an
earlier book (Nalimov. 19810) 1 claimed that the perception of verbal
meanings is determined by two distribution functions, namely, p h ) , the
prior distribution function o f the meaning o f the word p, and pblp),
the filter arising in the person's mind when he analyzes the meaning o f
the word p in a phrase y. During comprehension of the phrase y, the
prior distribution function of the word's meaning is reduced by the filtering function according t o Bayes's theorem
and we get the posterior distribution function' of the word p that determines its meaning in the new concrete phrase y. Distribution functions
pb) a n d p b l p ) are both individual, and in this sense they manifest a person's free will. At the same time, they are to some extent determined by
the culture in which the person lives, by his social status and upbringing.
This is the manifestation of non-freedom, or of the notorious necessity.
Manifestations o f freedom are not absolute. Freedom exists in the
possibility o f varying the distribution functions determined to some extent by the culture to which the person belongs.
In my earlier book (Nalimov, 19810) 1 introduced a scale o f languages
which determines the degree of freedom in comprehending texts. For
hard languages there is less freedom and for soft languages there is more.
But even for such a hard language as that o f physics, texts can be interpreted with a certain amount o f freedom. In quantum mechanics the
' Hcrc k
unity.
ir a constant of normalizing: thc area limiccd by the abscissa and thc cur*c&ly) murl bc
Free Will in the Probabilisric World
21
constructions a r e compared t o experimentation in a certain definite way,
but at the same time metastatements o n q u a n t u m mechanics are very different. This is well illustrated by the collection o f statements by physicists
and philosophers quoted previously (Nalimov, 1 9 8 1 ~o) n t h e meaning o f
the I+ function. compiled by the American philosopher Abel (1969). In
soft languages the freedom in comprehending texts is uncommonly great.
Capra (1975) describes the freedom in comprehending Ancient Chinese
texts a s follows:
Both the Confucian Analects and the Too Te Ching arc written in the
compact suggestive style which is typical of the Chinese way of thinking. The Chinese mind was not given to abstract logical thinking and
devcloped a language which is very different from that which evolved
in the West. Many of its words could be used as nouns, adjectives or
verbs, and their sequence was determined not so much by grammatical rules as by the emotional content of the sentence. The
classical Chinese word was very different from an abstract sign
representing a clearly delineated concept. It was rather a sound symbol which had strong suggestive powers, bringing to mind an
indeterminate complex of pictorial images and emotions. The intention of the speaker was not so much to express an intellectual idea,
but rather to affect and influence the listener. Correspondingly. the
written character was not just an abstract sign, but was an organic
pattern-a "gcstaltW-which preserved the full complex of images
and the suggestive power of the word.
Since the Chinese philosophers expressed themselves in a language
which was so well suited for their way o f thinking, their writings and
sayings could be short and inarticulate, and yet rich in suggestive images. It is clear that much of this imagery must be lost in an English
translation. A translation of a sentence from the Too Te Ching. for
example, can only render a small part of the rich complex of ideas
contained in the original, which is why different translations from
this controversial book often look like totally different texts. As
Fung Yu-Lon has said. "It needs a combination of all the translations
already made and many others not yet made. to reveal the richness of
the Loo-rzu and the Confucian Analects in their original form."
(p. 103-104)
In language, free will is expressed by a stratum of polymorphous
words forming fields of meanings which a r e sometimes fuzzy to such a
degree that they include even antonyms:
spare: (I) additional to what is usually needed or used; in reserve for
use when needed <- cash) (* parts) (- room) (* time); (2)
small in quantity c * diet).
lo bless: ( I ) ask God's favor for; consecrate: make sacred or holy (to
oneself) (to one's stars) ; (2) to praise, to call holy; (3) be fortu-
-
-
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Language of the Probabilislic Vision
nate in having; (4) (iron. or euphcm.) used for a word of opposite
meaning. "curse, damn." etc. I'm blest-expresses surprise.
lo abstract: (1) take out, separate; (2) to separate in mcntal conception; to consider apart from the material embodiment or from particular instances; (3) lo make an abstract of; to summarize; (4)
(colloq.) steal.
Antonyms express, perhaps, the maximal degree o f polymorphism, its
completeness. I would venture t o suggest that the most semantically
developed words are characterized by the presence of antonyms in their
semantic field. The conclusion seems to be as follows: antonyms are the
manifestation of free will in the language or even the property of free will
in its verbal form.
Since we have started to speak of the East, let us consider here the
problem of karma. 1 would like t o emphasize that this concept is rigidly
deterministic, which is rather non-typical of the free Oriental outlook
which we find s o attractive. In any case, this concept is in obvious discordance with the game model of the world which is so explicit in all Ancient Indian conceptions. It seems quite probable that the concept of karma came into Ancient Indian consciousness from the remotest past and is
unavoidably marked by the lack of alternatives peculiar to magic thinking. It has created a specific situation: this concept has preserved its archaic content. and it has also entered the soft system o f concepts of the
later outlook. A European thinker always tries to divide everything.
Manifestations of karma must be rigid. Otherwise, what is its meaning?
However, the probabilistic vision of the world allows us to perceive the
law o f karma as a fuzzy one which admits the existence of free will. 1
consider it relevant here to draw a parallel with the physical picture of the
microworld presented by Born (1949):
[in quantum mechanics] we have the paradoxical situation that
observablc events obey laws of chance, but that the probability lor
these events itself spreads according to laws which are in all essential
features causal laws. (p 103)l
I n quanlum lhcory causality is erprerred by ihc lamour Schradinger cqualion. Ilsrimpler! lorm is z
followr:
rn*=
a,
H~
.
Now i l v c deal with a quamum rnwmblc mhauruvely dscribcd by a set of d y n m c vxiabla q,. y,. . . ,
which arc mczurcd rimullanmurly and indcpendenlly. Ihc probabilily or finding a definilc value olrhc
wl ir dclcrmined by rhc probability denrily
htq)
IwM~~)~'
The wave function y l is~~xplicilly~ i v c nar a luncrion or ihccmrdinalcr q I! dctcrminer ihcrm,;sr;cror
any rnearurernml of ihc microryrtcm compaliblc wich amacroryrlm Mwhich dimalcr ihecondicionr lor
Free Will in the Probabilisric World
23
Perhaps the effects of karma should be described by a similarly
paradoxical system of notions in order that its principal idea be
understood by a European. Indeed, personal possibilities may be said to
be determined by a field of elementary events. Karma determines a
distribution function of probabilities on this field of potentially possible
events [the function p(p) in our terms] for each person and in each birth.
The distribution function evolves according to the karmic predetermination, but this latter proves to be fuzzy, or probabilistically weighted. In
each concrete situation a person can make his own decision. This latter
can again be represented by a fuzzy, probabilistically weighted filter [our
functionp@lp)]. Then we can apply the Bayesian theorem which explains
the way the potential possibility determined by the past interacts with the
freedom of actions. Freedom exists in the possibility of redistributing the
weights in the karmic situation. And though this state of things is much
more complicated than in the case of Buridan's ass, it is also describable
within the system of probabilistic notions. Whereas in the case of
Buridan's ass free will may be represented as a generator of random
binary numbers, people facing the karmic predetermination respond by
generating the probabilistic filter of perception, and this is, as a matter of
fact, the manifestation of their individuality. Within this model, overcoming the karmic predetermination will unavoidably lead to the
elimination of individuality. Indeed, if p, the function of the karmic
predetermination, degenerates into a n untruncated (tending to infinity)
uniform distribution, then the interval cut off on the ordinate by the
straight line determining this distribution will tend to zero, and generation of any filtering function will become a n utterly useless task. It will
turn into nothing or into everything, if this everything is void o f selective
abilities (attachments or preferences). This is exactly nirvana expressed in
the language of probabilistic notions.
In the terms of the American protestant theologian Paul Tillich (1955).
p(p) is a fuzzily determined destiny, p o p ) is an ethical constituent in
overcoming one's destiny in a situation y , and p(ply) is a reduction o f
destiny by the ethical constituent.
And now a few words on the game model of the world. A European
who has just begun his acquaintance with the East is most of all puzzled
by the concept of the world as a game: to his mind this idea is
humiliating, and arbitrarily flimsy. But matters stand differently in the
probabilistic interpretation. A game is a fuzzy, probabilistically weighted
situation which enables us to overcome the karma of individuals and en-
lhc changer in (he mlcroryrlem (Blokhinlrcv. 196). 11 ir porriblc l o rpeal o l a probabilirlically d m r mined pocenlial lor a ccrlain beha,ior under certain nmcrocondilionr wen ror a single elcccron. Polmrialily may be raid ro bc n probabilirlically dclcrminrd limir o l hccdorn glvcn lo an clcslron.
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Language of the Probabilistic Vision
tire peoples. The advent of the spiritually elevated aeon on the Earth can
only change the distribution of probabilities but in no way determines the
rigid way o f evolution. This is what happened during the advent of
Christ.
And the last thing is Theodicy,' the poignant problem of rationalistic
theology. In the probabilistic vision of the world it merely disappears, as
follows from all said above. Christ refused to take the power from the
tempting devil and thus let the world retain its spontaneity. However,
something very important in the world did change, and different people
could perceive these changes differently.
Everything said above concerns the partial, relative manifestation of
freedom. According to Indian concepts, its complete realization becomes
possible only when a person overcomes his karma and, according to
Christianity, when he learns the truth. But to leave the cycle of births is
the same as to learn the truth: this means communion with God.
Freedom proves to be the synonym for God, but God then becomes the
synonym for nothing-nirvana.
Stated another way, for a person in a state of absolute freedom, the
meaning of the concept "freedom" is lost. Is there any sense in asserting
the freedom o f choice of the filter pblp) if the function p h ) becomes a
specific case of the rectangular distribution with the ordinate tending to
zero? (Strictly speaking, if the argument tends to infinity, the concept of
rectangular distribution loses the probabilistic meaning.)
1 am quite aware of the fact that the ideas presented in this chapter
may irritate many readers. The probabilistic vision of the world does not
solve the poignant problems of European philosophy and theology; it
only withdraws them from consideration. All the problems underlying
our culture turn out to be merely pseudoproblems. Indeed, it has suddenly become obvious that European thought came to a standstill in
dealing wilh the following dichotomous concepts generated within its
culture:
subjective
mind
continuity
chance
freedom
objective
matter
discreteness
necessity
predetermination
Theodiy is i h r junilicahn ol C d .The mm, introduced by Leibnizin 1110, u n i m nvmcrousdoclrinrr which altcmpl lo coordinate ,he idea d i h c bcnrvolcnc God, r h o rationally ruler iheuorld, with
!he cxirlcnce or evil.
Free Will in fhe Probabilisfic World
dialectics
gnosticism
spontaneity
infinite
life
good
truth
conscience
mercy
God
25
logic
agnosticism
regularity
finite
death
evil
falsity
duty
punishment
atheistic mechanisticity
o f the world
If we make these concepts fuzzy, the dichotomous oppositions will
disappear: each opposition will merge into one semantic field and,
moreover, all the oppositions will turn out to be synonymous. All the
problems thus disappear. But will new, real problems arise? In other
words, will a new culture emerge?
It is noteworthy that European culture emerged while solving problems which actually were unsolvable. Did the Orient solve real problems?
And what are real problems?
To Chapter 3