Tucker S. McElroy
This is a brief examination into the subject of randomness. Namely, what do we mean by "random"; what do we mean by "chance"? These seem to be philosophically loaded terms, and one cannot employ them without invoking religious presuppositions. This essay is an attempt to delineate the various approaches taken to the concepts of chance, randomness, and determinism. We can only provide an introduction to the rich collection of concepts, else this minor endeavor would swiftly become a book. More meticulous readers, disturbed by the cursory treatment of epochal metaphysical topics, should consult the references for further information. The latter portion of this essay takes an innovative "density" approach to the concept of probability, and thereby places a non-standard interpretation upon the axiomatic definition; this is an attempt to utterly remove the ambiguous notion of "chance" from the parlance of probability theory.
The terminology is profuse, and, unfortunately, in practice the definitions are not standardized. An engineer, businessman, and theologian all mean something distinct when they use the word "random". For the purposes of clarity, the following definitions will be used throughout this paper. Chance: An abstract principle, which governs the outcome of events in our cosmos (this term will hereafter refer to the empirically knowable universe, our own space-time domain). Roughly speaking, the doctrine of chance states that when an event occurs, it could have just as easily come about some other way, and there is no particular reason or cause for things having occurred this way. Note that the principle of chance is assumed to operate through certain rules or laws (the probabilities). Chance does not attempt to explain these rules/laws, but taking them for granted, often operates through them. Ergodic: This is a precise mathematical term, which in various contexts usually means that spatial and temporal averages or computations are asymptotically identical. In probability theory, the comparison is made between a distributional average and a temporal/sampling average. This latter perspective is the one taken here—namely that an average of sampled data approximates with high probability the average over every theoretical possibility. Stochastic: An adjective, which denotes that which pertains to probability theory. It is often used synonymously with "random", but has a more precise interpretation—the word "stochastic" is the technical term for what we often call "random" in common parlance. A stochastic object does obey certain rules and patterns, but is not completely predictable. Chaos: The common usage of the term denotes disorder, unpredictability, and fluctuation. Paradoxically, it cannot be the complete absence of order (such a concept can never be defined, since "defining" is an order-imposing operation), but rather is the apparent loss or corruption of order, perhaps relative to some subjective aesthetic. In mathematics, a chaotic phenomenon is a deterministic structure (i.e., it has a functional form, with theoretical predictive capacity) which appears to be stochastic. Random: Some use this word when they discuss raw chance. Others mean a stochastic number between zero and one, generated in such a way that any outcome is equally likely (i.e., a uniform random variable on the unit interval). This is a quite limited meaning. Others refer to a sequence of numbers that have no probabilistic relationship to one another; this is redundant terminology, since the probabilistic concept of independence covers this idea. A mathematician merely uses random as a synonym for stochastic—as such it does not preclude the possibility of some outcomes being more likely than others. Deterministic: This idea says that a phenomenon has some cause, which necessarily determines the outcome (which often comes temporally). Yet, it is more subtle than fatalism, since determinism allows for the possibility of primary and secondary causes—which just means that some things are apparent (perhaps empirically), whereas others are hidden to finite understanding. From a scientist's standpoint, these secondary causes become somewhat moot, since they may never be detected or measured. As used in this paper, determinism can allow for non-predictability in the cosmos, and at the same time allow full causality once the viewpoint is extended beyond the boundaries of this world. This concept will be fleshed out more fully below. Fatalistic: Everything in the cosmos is completely determined by
forces acting within the universe—thus everything is predictable (in theory,
though it may be unfeasible) if only sufficient information can be gathered. This
view seems to bear uncomfortably against the edifice of quantum mechanics,
which preaches the inherent unpredictability of the small particles within the bed
of quantum foam. Probability: This is a mathematical theory, which is the basis of all modern studies in stochastic processes and statistics. The measure-theoretic (or axiomatic) formulation of the theory nicely lends itself to a density interpretation, which is described below. We often speak of the "probability of an event," by which we mean the chance that something happens. Depending on our notion of chance, this has various nuances.
The term "natural philosophy" refers to the belief that there are sufficient explanations of the observed phenomenon to be found within the cosmos. In fact, all supernaturalism is precluded by axiom, since the universe is assumed to be a closed topological space—no information comes in, and none goes out. Thus the natural includes the full scope of the "possible," so that the supernatural becomes synonymous with the "impossible." No discussion will be given here of why this is an attractive or adequate belief system for many scientists and intellectuals; the point here is to describe the concept of "chance" that naturally flows from this epistemological source. Certain laws and rules appear to be operating in the cosmos. Physics, chemistry, and the other sciences have attempted, over the past centuries, to trace these relationships through a partnership of reason and empiricism. The premise that empiricism leads to true knowledge is taken as a given in the current academic community. If a certain phenomenon is observed repeatedly, we notice the pattern and look for a cause. Upon such foundations modern science is avidly and faithfully pursued, and new truths established. For each observed phenomenon, some cause within the cosmos is to be sought; if no such reason can be determined, then one may either speculate or attribute the behavior to chance. Now there is considerable variation in the natural philosopher's
position on the concept of chance. Perhaps in older times (i.e., the beginning of
the twentieth century), there was a current of optimism that various laws and
rules would be worked out to such an extent that complete predictability would be
a theoretical possibility. This would imply strong fatalism, and a
mechanistic conception of man and his realm. But certain experiments in quantum
mechanics have cast serious doubts on the tenability of such doctrine.
Apparently, small particles move about randomly in the fullest sense of the word.
Without any traceable cause, a particle may move to one location or another, and
nothing in this world can account for the difference! It appears that a
probability distribution on space is attached to each particle, and after "rolling the
dice," the little particle moves to the appropriate
location. Notice the logical deduction that a consistent natural philosopher
makes at this point: since there are no forces external to the universe that can account for the motion of this particle, and nothing within the universe can be said
to cause or determine its alteration, there is no recourse but to say that
"chance" decides the path. Beyond this, no further progress can be made. We have
no hope of comprehending the impersonal "chance" any more than the
ancient pagans did (they often conceived of Fortune as a fickle woman!); so we
shut the book and declare: "It is a mystery." Of course nothing prevents us
from continuing the project of determining the exact probabilistic laws through
which raw chance operates, but we must not try to probe the nature of chance
itself. As a passing remark, we observe that similar conclusions from the premise
of natural philosophy have been used to logically deduce the theory of
evolution. To some of us, this is extremely
questionable.
Things are quite a bit different if we once admit the possibility of non-trivial "other-worldliness". If indeed there is "something" beyond and outside our own cosmos, and if interaction in some definable sense is possible, then we may have an alternate explanation for the phenomenon we observe. Indeed, it may just be possible to completely eliminate raw chance from the picture, and thus obtain a more satisfying science—one that attempts to maximize explanatory power and reduce the dominion of the unknown. First of all, perhaps we should present a brief argument as to why
this would be a desirable situation. The objective of science is to explore
and describe various aspects of our own cosmos, employing axioms laid
down centuries ago. Thus, to a scientific mind, order is preferable to raw chance. But
perhaps this is all wishful thinking—after all, experimental evidence does lead us
to the conclusion that events in the cosmos are ultimately unpredictable. So a
few words should be said to show that "raw chance" is intellectually repugnant.
If all history is, in the last analysis, the product of raw chance, we may
conclude that things could have turned out quite differently—and no meaning or
cause can be adduced to justify one history versus another. Then why are we
scuttling around so busily making propositions and theories, if our whole
civilization, our language, our brains—are the product of a meaningless
happenstance? It is fruitless to seek order in a world where that appearance of
order only came about by chance. Now in defense of the opposition, one could mention that there is
structure together with chance; perhaps the variance of the distributions is low,
so that there is a concentration of events, and "on average" things tend to
obey strict rules. This indeed seems to be the case, and it begs the next
question—how did these certain distributions or laws (it is an elegant coincidence
of probability theory that probability distributions are also called "laws")
come about? Perhaps through chance—distributions on distributions
The word "supernatural" is taken in the old sense—that which is above or beyond the natural. This is to be developed in the next section;
Christianity presents the most coherent treatment of a supernatural
system.
The ancient religion of Christianity gives a consistent outworking of these ideas, which combines supernaturalism and the apparent randomness of this world in a subtle but lovely marriage. Here I will attempt to outline the corollaries of the basic doctrines which apply to this discussion. Firstly, any concept of ultimate chance is utterly excluded from the beginning, since all events in reality are governed and determined (yea, caused) by a single intelligent entity—God. As a weak formulation, some theologians have conceived of God as only possessing foreknowledge; such a being would be too deficient in puissance to merit the epithet of "omnipotent", and thus we discard such feeble conceptions. In fact, God foreordains all events in this cosmos. Also, we must keep in mind that this entity "resides" outside and beyond our own cosmos, and thus it is utterly fallacious to apply our own limitations of space and time to One who transcends this order. And as God made the laws and rules of this cosmos, he also has power and authority to break or surpass them. Now it becomes apparent that objects in this world, on the average,
obey the laws discovered by science. Any small deviations can be attributed
to either measurement error or noise—the conglomeration of a plethora of
small effects, which it is unfeasible to compute. As for the mysterious
quantum effects, we can now assert that the particles move according to the direction
of God; and if the overall pattern is measured, it is seen to follow certain
well-studied probability distributions. Thus God gives us many instances or samples, from a divinely scrutinized theoretical
distribution. This formulation is consistent with unpredictability "within the cosmos"— since there is no possibility (for things within the natural world, obeying natural laws) of ascertaining a cause originating "outside", we may well perceive any happenstance as "uncaused." To speak mathematically, we may conceive of these supernatural causes as functions from beyond, which take values in our own history (the range space is contained within the events of our cosmos). Then, we observe only the values of the functions, but sadly have no knowledge of the function itself. In addition, this is a formulation with which theologians should be quite comfortable: God is continually managing the most minute matters of our world, operating upon matter supernaturally. This does not constitute miraculous activity, since there are no natural laws being broken; rather the supernatural economy is the foundation for natural law.
Let us now contrast the former view with fatalism. In this picture, God
(or a supernatural agent) merely makes one initial cause, which commences
the growth of the cosmos; from that point onwards, every event causes every
subsequent event in a theoretically predictable fashion. This is disagreeable
to Christianity, which preaches a God that continually upholds the universe. It
is at odds with modern science, which has noticed theoretically
unpredictable phenomena. And it is odious to the human
aesthetic. How is a theory of probability to be granted within a fatalistic framework? The concept of chaos, as defined above, gives the only tenable sense. Indeed everything is caused and determined by prior cosmic events, but this is so intricate and complicated that no computational machine could possibly make sense of the data. Thus, while being in essence cosmically deterministic, phenomena are nevertheless apparently stochastic, defying even the most diligent scrutiny. If finding the deterministic laws and functions is unfeasible empirically and mathematically, then from a practical standpoint the underlying fatalism is somewhat irrelevant—we are better off (from the perspective of predictability) modeling the cosmos stochastically, so that we may employ the full power of probability theory. Contrast this with natural philosophy—which says that randomness is not merely apparent, but is a fundamental reality—and the Christian determinism here formulated, which says that randomness is "random" as far as this cosmos is concerned, but ultimately there is a cause for everything found in supernatural realms. In natural philosophy, the probability model is an absolute reality—the pure abstraction of a mathematical theory constantly intrudes and permeates our world. In fatalism, the probability theory is a convenient tool, which is implemented due to the loss of information in the whirl of chaos. Between these extreme views, supernatural determinism preaches a probability theory, which is concrete and undecipherable from a finite perspective, and yet is completely tangible to the supernatural entity generating cosmic events. The mathematical definition of random variables and probability spaces lends itself nicely to this latter interpretation.
Some discussion should now be given on the issue of "ergodicity"
and stochastic structures in general. First, some important terminology will be
introduced. When a probabilist speaks of a We could compute an average of a random variable two different
ways: theoretically and empirically. The theoretical average would involve taking
a sum of the values weighted by their corresponding probabilities, which
are determined by the distribution. The empirical average would be obtained
by repeating a phenomenon in such a way that we generate a sequence of
identically distributed random variables. Then we simply measure each
outcome, and take the usual average of all the observations. The basic "ergodic
theorem" states It is strangely apparent that our universe is ergodic. By this, I mean that many phenomena satisfy the ergodic theorem in practice. In some cases, one may have postulated the distribution of a random variable, computed the theoretical average; then this is compared with an empirical average conducted upon data generated by the same stochastic mechanism, and behold!—the stated convergence is eerily obtained. In other cases, we have no idea what the theoretical average is, but we do see the empirical average closing in on the same number (for instance, generate a large data set and calculate the average; then repeat this whole process many times—each of the averages will often be quite close to one another!). Why should this be the case? Since the conclusions of the ergodic theorem surround us (as well as the subtler "central limit theorems"), it seems to lend validity to our probabilistic modeling of the cosmos. So it behooves us to take a closer look at the precise statement of the ergodic theorem (which, under other contexts, is called the "Law of Large Numbers"). What do we mean by an average of random variables converging to a fixed number? In its strongest formulation, the theorem can be interpreted in the following way: with probability one (i.e., all the time except for fluke instances), the empirical average tends to the distributional average as the sample size grows toward infinity. Thus, in theory it could happen that for a given experiment convergence would not occur, but in practice you would never see it happen.
Probability theory is a fairly recent development upon the scale of
human civilization. The classical formulation dealt with a class of random
variables where the number of possible outcomes was finite, and each was equally
likely (for example, the flipping of a fair coin has two equally likely outcomes).
But what if some outcomes are just more likely than others (consider an
unevenly shaped coin, which gives a bias to one outcome over another)? Clearly a
better theory was needed. Another suggestion was that of "empirical
probability"— that we define theoretical probabilities as the limits of empirical
proportions. This is essentially equivalent to assuming the ergodic theorem from the
beginning and using it as a definition. In the event that the ergodic theorem does
not hold for certain phenomena (this can and does happen, e.g., for random
variables that fluctuate "too wildly"), this definition falls flat on its face. An
axiomatic approach was developed in the twentieth century (by A.
Kolmogorov Here I'll make an aside on axiomatic mathematics: this is a great
covert whereby mathematicians may completely dodge the hydra of
epistemology. Questions like "how do we know it is true?" and "why is that type of
reasoning valid?" are banished to the philosopher's (and theologian's) circle. The
method merely consists of the declaration of certain delicate axioms, from which
the subsequent collection of Theorems, Propositions, Lemmas, and
Corollaries are carefully constructed through the operation of logic ("modus
ponens" One other paradigm, the "subjective" theory of Bayesian probability, conceives of probabilities as perceived "degrees of confidence"; the resulting mathematics is identical with the above axiomatic formulation, but a very different interpretation is placed upon the quantities of interest. I will not comment further on this intriguing theory, but concentrate on Kolmogorov's system. Modern probability theory has great explanatory power. It is from these axioms that such results as the ergodic theorem were established. This is, however, only the first item among a wealth of propositions. As a passing remark, we observe that the subject got its beginning in the various gambling problems of the Renaissance, bantered back and forth between intellectuals. It seems worthwhile to construct a coterie of examples drawn from a less nefarious context.
Below is a mathematical formulation which gives an acceptable model of the cosmos and is compatible with the above observations. It is an attempt to remove the concepts of chance, randomness, and so forth from the vocabulary of statistics. Following are some essential notations:
Let us unpack this terminology: each w contains an incredible store of cosmic information—somehow it encodes all the facts of our world from beginning to end. Suppose that absolutely every phenomenon in the cosmos was really the value of some function—the outcome C(w); whenever we speak subjunctively of what could have occurred, we are referring to C(w') for a distinct history w'. Note that the possibility of C(w) = C(w') is not excluded for some random variables C—this all depends on the particular phenomenon. Perhaps one way to picture this set of histories is to imagine a tree that is constantly bifurcating in time, according to which many possible cosmic alternatives occurs. Our own history w* has countless parallel histories w; these may be similar or even identical in many ways, but in at least one aspect they actually differ (the details of this distinguishability is embedded in the rich tapestry of s-algebras). It follows that the set of all such histories, denoted by W, is unimaginably vast. In probability theory, it is sufficient to leave W as an abstract set—our attention is focused upon the distributions of the random variables. Now the "probability" itself is a measure on the space W, which assigns to each set a number between 0 and 1; the axioms of Kolmogorov state that some monotonicity and summability properties should be satisfied. Let us see how this concept may be applied. Consider some cosmic event which we wish to model statistically. Then we generally conceive of the event as some subset of all possible histories w, such that specified outcomes occur. Observe that in our notation, this means that there is a random variable C which measures the phenomenon in which we are interested, and the event may be indicated by C taking on a particular value or values—the w's for which that value occurs constitute the event. Stated another way, we are interested in all possible worlds w for which the phenomenon occurs. The following specific example illustrates these ideas concretely. Suppose an electron is in one of two compartments (with equal dimensions) of a box—either the right (R) or left (L) side. If at a particular time we measure the location, we can model this by considering the random variable C, with possible values of R or L. The event that "it's on the left side" is equal to the set of all w's (all world histories) in which the electron really is on the left side. In general, C(w) can be either R or L. If we actually measure the value R, then we know that C(w*) = R. Let's denote the event "it's on the left side" by the letter A (so we can write A = {all w such that C(w) = L}). But this event is actually a subset of W, and we may and will apply the probability measure P to it; then P(A) gives us a number between 0 and 1, which is interpreted as the "probability that it's on the left side." This might be modeled to be the number 1/2, which means that in one out of two worlds w, the electron will be on the left side.
A probabilist will notice that there is nothing innovative in these definitions, except for the idea of "histories". But let us further imagine that we calculate the probability of an event by taking the following ratio: consider the count of all histories w in which the event occurs, and divide this by the count of all histories w in W. Thus, if we think of each w as a "particle" within the total "object" W, then the probability of an event is simply the density of the corresponding histories within the scope of cosmic possibility. Thus the name "density model". With this in mind, we can explain the apparent randomness of our
world. Through our senses, we are able to observe the values
C(w*) for various random variables C, and we are able to deduce through reason the existence
of a single state of things, namely
w*. But we are at a loss to determine exactly which
w is From the perspective of natural philosophy, the random variables come at us any which way, and there is no possible way of knowing the functions C. From the perspective of supernatural determinism, we should view the space W as grounded outside our cosmos, so that the functions C are movements or mappings between a supernatural realm into a collection of potential universes (not just our cosmos, but all subjunctive cosmoses as well!). Of course w* holds a special place in this story; to a mathematician, we might say it is an element of the dual space of random variables over W. And it is very possible that an entity from beyond could know and determine both w* and the random variables C. The ergodic theorems will guarantee that our w* does not deviate too greatly from the "center of mass" of possible realities.
San Diego, California ______________________
Billingsley, Patrick. Greene, Brian. Grene, David and Richmond Lattimore, eds.
Johnson, Phillip E. Kolmogorov, Andrei. Van Til, Cornelius. |