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Mises’ Theory of Probability and the Principles of Physical Statistics
A. Ya. Khinchin, Moscow.
In recent years the German mathematician and physicist, Professor of the University of Berlin R. Mises, has published a number of works on probability theory and physical statistics. These works had as their aim, on the one hand, to rebuild on a new scientific-philosophical foundation the entire edifice of probability theory and mathematical statistics, and, on the other, to subject to substantial revision the basic principles of the scheme according to which statistical methods have hitherto been applied in physics. In both directions the influence of Mises’ ideas on the course of scientific thought has by now become so clearly defined that acquaintance with his doctrines ought to become the common property of all those who, in one way or another, in their scientific work come into contact with statistical methods, and especially of physicists, since Mises’ reforming activity is directed chiefly at physical statistics.
It is necessary to bear in mind that Mises’ views on the role of statistical methods in physics have no direct connection with his new, highly original conception of the general notion of probability. This is important to remember because Mises’ general doctrine raises a number of methodological and mathematical doubts, which will be discussed in detail below; the special doctrine concerning the principles of statistical methods in physics, however, is to a considerable degree free from these
objections and may be accepted without hesitation even by those for whom Mises’s general probabilistic conception is, for one reason or another, unacceptable.
For the same reason, in what follows we shall attempt to set forth Mises’s general doctrine and his theory of physical statistics quite independently of one another.
I. Mises’s Doctrine on the Foundations of Probability Theory.
If we have an ordinary die and focus our attention on any one of its six faces, then considerations of symmetry compel us to suppose that the probability of the occurrence of this fixed face is equal to one sixth. The basis for this calculation is, as is well known, the decomposition of all possible outcomes of the event (throwing the die) into “equally possible” or “equally probable” cases; and the traditional definition of probability as the ratio of the number of cases “favorable” to the event to the number of all possible cases essentially presupposes that all these cases are “equally probable.” That such a definition contains a vicious circle has long been known; but more than that: as soon as the state of affairs becomes even a little more complicated, this definition becomes very difficult, or even loses its content altogether. Mises gives the example of an “irregular” die, made, for instance, of nonhomogeneous material. Here too we speak of the probability of the occurrence of a given face; but of what “equally probable” cases can there be any question here? How can the postulated probability be computed, what is its meaning, its definition? And further—when we speak of the probability that a 40-year-old man will live to 50 years, how can this probability be represented as a ratio of numbers of equiprobable cases? These difficulties, along with many others, in Mises’s opinion, show with sufficient definiteness the untenability of the classical definition of probability.
Let us return to the case of a die. If we throw it many times in succession, and if in \(n\) throws the face we have marked comes up \(m\) times, then the fraction \(\frac{m}{n}\), for large values of \(n\), will almost certainly be very close to the probability of the appearance of the marked face (on the basis of the well-known theorem of Bernoulli). This fraction is a number that we can easily find by experiment. But the probability of the appearance of the marked face is a number which, generally speaking (the case of an unfair die), is not only unknown to us, but of which we do not even possess a theoretical determination; in speaking of this probability, we are guided by vague and poorly justified analogies.
From this arises the fundamental principle of Mises’ doctrine—the empirical definition of probability. The probability of the appearance of the marked face, since it is measured ever more accurately by the fraction \(\frac{m}{n}\) as the number of throws increases, coincides with it in the limit; i.e., denoting this probability by \(p\), we must have
\[ p=\lim_{n\to\infty}\frac{m}{n}. \]
In this relation, which in the traditional understanding symbolizes one of the properties of probability, Mises proposes to see its definition. Every a priori definition, in his opinion, is doomed to failure; the empirical definition he proposes is the only one from which the possibility of predicting the course of events follows logically.
For the moment let us make only a few simple remarks concerning this definition: 1) in contrast to the traditional one, it is free of a vicious circle; 2) it applies just as well to an “unfair” die as to a fair one; 3) it requires that the limit in question exist, and thereby imposes a certain requirement on the series of experiments being performed—a requirement which, as we shall see below, is fraught with serious difficulties.
And now let us pass from this special case to the general definition of probability. We see that at the basis of such a definition there always lies a certain infinite sequence of experiments—what Mises calls a collective. In this collective each of its elements (experiment) either is, or is not, endowed with some definite attribute (in our example, such an attribute was the falling of a definite fixed face). If among the first \(n\) members of the collective there are \(m\) endowed with the given attribute, then the probability of this attribute in the collective under consideration is called the limit to which the fraction \(\frac{m}{n}\) tends as the number \(n\) increases without bound.
Thus the probability of every event (attribute) is determined exclusively within the limits of a certain given collective, and Mises constantly and insistently emphasizes this circumstance, considering that inattention to it accounts for many errors and absurdities in the calculation of probabilities.
From what has been said we already see that a collective is by no means any sequence of elements, some of which are marked (endowed) with one or another definite attribute. We see that each of the attributes that interest us must have in the collective a definite proportion (the limit of the ratio \(\frac{m}{n}\)), which is its probability. This existence of all such limits can in no way be justified a priori and is therefore a definite condition to which the sequence must be subject in order to be a collective.
But this condition is not the only one. In order that the structure of the collective should have the character necessary for the calculation of probabilities, we must require of it yet another very specific property, which Mises calls the property of irregularity (Regellosigkeit) and which, for the sake of greater clarity, we shall approach from the standpoint of a special case.
Let us return to the example with a die. Let us again imagine an infinite series of throws, but now we shall regi-
not every toss, but, for example, only the 1st, 3rd, 5th, 7th, etc., i.e. only tosses with odd numbers. Suppose we have considered \(n\) such tosses, and suppose that among them our marked face has appeared \(m\) times. We are again entitled to expect that the ratio \(\frac{m}{n}\), for large values of \(n\), will be close to the probability under investigation. Thus our collective must be constructed in such a way that, having selected from it, according to a definite law, any infinite part (for example, the elements with odd numbers), we obtain in this selected part, for the limit of the fraction \(\frac{m}{n}\), the same number as in the whole collective. This property of a collective Mises calls its irregularity.
In general, a sequence of elements is called irregular if in every partial sequence selected from it according to a definite law, each of the attributes that interest us has the very same share as in the whole original sequence.
Now we can establish the exact definition of a collective: a sequence of elements with respect to a given group of attributes is a collective if, first, each of the attributes of the given group has in it a definite share, and if, second, this sequence is irregular with respect to each of the attributes of the given group.
As has already been said, the share of an attribute in a given collective is called its probability. Knowing the probabilities of all the attributes of a given group, we know the distribution of the given collective. Mises sees the problem of computing probabilities exclusively as this: knowing the distribution of certain initial collectives, to find the distributions of new collectives obtained from the original ones by means of certain definite operations. Hence the task is to identify those operations on collectives with which the theory of probability is concerned. According to Mises’s teaching, there are four such basic operations; all the others, however complicated, are obtained by combining these four in any
number and in any sequence. Let us now consider these four fundamental operations.
1. Selection (Auswahl). This simple operation consists in the fact that, from a given collective, some part of it is singled out, for example the elements with odd numbers. Here it is important that the principle of such a selection cannot be some property of the element connected with its relation to the attributes under consideration; the decisive role must be played exclusively by the number of the element. From the definition of a collective (namely, from the property of irregularity) it follows that the distribution of the new collective coincides with the distribution of the original one.
2. Mixing (Mischung). This too is a very elementary operation, which in essence creates no new collective, but only, within the given collective, combines two or several attributes into one. Suppose, for example, that we are dealing with a collective consisting of successive throws of a die. The attributes in this collective may be, for instance, the occurrence of a two, a four, and a six; these are three mutually distinct attributes. But we may also speak of an attribute consisting in the occurrence of an even number of points. This transition from a group of attributes to one attribute uniting them is the operation of mixing, which in this way does not take us outside the bounds of the given collective. Obviously, in the classical theory this operation corresponds to the scheme usually called the theorem of addition of probabilities. It is easy to understand that the “theorem of addition” itself, in the new conception as well, is an immediate consequence of the definition of probability.
3. Partition (Teilung). In the classical treatment this operation corresponds to the so-called Bayes scheme (the scheme of “a posteriori” probabilities—a terminology against which, incidentally, Mises polemicizes sharply and justly). The operation of partition has an outward resemblance to the operation of selection in that here too the new collective is defined as a part of the original one; however, the principle by which this part is singled out is entirely different here: if there, in selecting a partial sequence,
we were forbidden to make this selection depend on the realization of the characteristics under consideration, and were instructed to carry it out purely “arithmetically,” guided only by the numbers of the elements, here, on the contrary, the operation consists in selecting from the collective the elements possessing some one of the characteristics under consideration, and in the new, selected collective then studying the distribution of the remaining characteristics.
Example: in a series of throws of a die we may select those throws in which an even number of points occurs; this gives us a new collective, within which we may, for example, raise the question of the probability of a six occurring. In ordinary language this problem is formulated as follows: assuming it already known that an even number of points has occurred, calculate the probability that this number is six. In the scheme of Mises this problem acquires the following meaning: suppose that among the first \(n\) throws an even number of points occurs \(m\) times, and among these there are \(r\) sixes; it is required to find the limit of the fraction \(\frac{r}{m}\) as \(n\) increases without bound; here, obviously, the limits of the fractions \(\frac{m}{n}\) and \(\frac{r}{n}\) are assumed known, as a consequence of which the problem obtains a very simple solution, evidently coinciding with the usual one.
- Coupling (Verbindung). This operation corresponds to the scheme of the so-called multiplication theorem and consists in forming, from two initial collectives, one new collective according to the following principle: as an element of the new collective one takes a pair consisting of one element of the first and one element of the second collective; in this new collective one studies the distribution of characteristics determined as follows: if \(A\) is some characteristic of the first collective, and \(B\) some characteristic of the second collective, then we obtain a definite characteristic \(C\) within the new collective if we require that the element taken from the first collective possess the characteristic \(A\), and its partner, taken from the second collective, the characteristic \(B\).
Example: the initial collectives are series of throws of two dice; the \(n\)-th element of the collective is a pair consisting of the \(n\)-th throw of the first and the \(n\)-th throw of the second die; we may pose the question, for instance, of the probability of the event consisting in the occurrence of a six on the first die and, simultaneously (i.e. in the throw with the same number), the occurrence of a two on the second die.
As we have already noted, according to Mises’s assertion, the task of probability theory consists in the study of distributions within collectives obtained, starting from certain given collectives, by applying the four operations described above in any number and in any sequence. The fact that these four operations do indeed exhaust the circle of the fundamental problems of probability theory contains nothing unexpected and even, in essence, nothing new: we have seen, in fact, that these operations contain in their totality all the basic principles on which, in the classical treatment, the edifice of probability theory was built (the principle of addition, the principle of multiplication, and Bayes’s principle); hence it is clear that by applying these four operations we can indeed construct any problem to which classical theory leads. Therefore it is not in this constructive part of it, where all is well, that Mises’s theory can give rise to doubts; criticism can and should be directed at the principled part of the theory, which does indeed give rise to a number of methodological, mathematical, and natural-philosophical objections.
II. Evaluation of Mises’s General Doctrine.
In the course of the forthcoming evaluation it is necessary first of all to note the brilliant merits of Mises’s reforming activity. The mere fact that this outstanding scholar, with all his authority, attacked the moss-grown prejudices and traditional absurdities of the contemporary doctrine of probabilities has indisputable scientific significance. There was and is no justification for the fact that the mathematical doctrine which in our time, perhaps, primarily before
PRINCIPLES OF PHYSICAL STATISTICS
...all the other is called upon to serve natural science and the practical sciences, precisely in its logical completeness for a century, if not more, has lagged behind most other mathematical disciplines. Even the best modern textbooks on probability theory, which are able to interest and satisfy a mathematician by the depth and seriousness of the problems posed and solved in them, leave the critically minded reader in grave perplexity with regard to the formal foundations of the doctrine set forth in them; and this perplexity is by no means one of those which the natural scientist is inclined to ascribe to the merely formal “captiousness” of the pure mathematician that does not interest him. No, here there remain under the sign of doubt questions that urgently occupy and trouble the physicist, the biologist, the economist—anyone who strives to bring to full clarity the scheme on the basis of which statistical methods of investigation are applied in his science. In the merciless criticism of these obscurities, in the unswerving tendency to replace them by impeccably clear constructions, lies the indisputable historical merit of Mises’s doctrine. And not the least place here is occupied by the systematic struggle against obsolete, plainly absurd terminology; for everyone knows what great importance terminology has in mathematical doctrines, especially when it is a question of clarifying the fundamental foundations of one or another mathematical discipline.
Suppose, says Mises, that a good tennis player has probability \(0.8\) of receiving first prize at a tournament taking place today in Berlin. Suppose further that the same player has probability \(0.7\) of receiving first prize at a tournament taking place on this very same day in New York. Since the two events in question here are obviously incompatible, then, guided by the literal traditional formulation of the addition theorem, we should have to say that the probability of our player’s receiving any one of the two first prizes is equal to the sum of the above probabilities, i.e. \(1.5\). Of course, anyone who understands the matter to any extent will avoid the crude error that leads to this absurdity. But it is enough
of the fact that the traditional formulation of the addition theorem does not contain a single word that could guard against conclusions of this kind.
Another decisive example is furnished by the multiplication theorem, which in its classical formulation, employing insufficiently clear terminology, usually conflates two problems that have no direct connection with one another and that, in the Mises scheme, correspond to the operations of separation and combination. A theorem proved for one of these two schemes is then, without any reservations, applied to the other, and the groundlessness of this transition is concealed behind vague terminology.
Many examples of this kind could be adduced.
But, while deserving universal sympathy in its destructive part, Mises’ theory in its constructive part calls forth a number of serious critical remarks, to which we now turn.
A. Criticism of principle and method
Probability is defined in Mises’ doctrine as the limit of a certain ratio, obtained, generally speaking, as the result of some series of experiments. If we have a die and ask about the probability that a six will fall when this die is thrown, then we naturally expect that the probability sought objectively expresses a certain property actually inherent in the object under study—the die. This property, really belonging to the object, may be studied by means of an experiment or a series of experiments; but it must be defined on the basis exclusively of the structure of the object itself, irrespective of whether the experiment is or is not performed.
We can measure the temperature of the air with the aid of a thermometer; but what would we say if we were invited, as the final, ultimate definition of the essence of air temperature, to accept the height of the mercury ...
of the column in a thermometer placed in this air? Would this not mean condemning oneself to a complete renunciation of understanding the essence of the phenomenon, to a principled substitution of knowledge about the object by knowledge of how this object reacts to our senses and instruments; and recognizing this knowledge as final, ultimate, admitting in principle the possibility of no other knowledge? Would this not mean condemning oneself to the crudest form of scientific relativism—a form in comparison with which even Machism has certain advantages? For we would then have to acknowledge that temperature is created by the introduction of a thermometer, that where there is no thermometer there is no temperature.
The definition of probability proposed by Mises suffers from an entirely analogous defect. If we wish to build our theory on an objective basis, if we wish the probability of throwing a six to express some property objectively inherent in our die, then we must define this probability in such a way that our definition retains its meaning even in the case when no experiments are performed with this die. If, however, probability is defined in such a way that its definition acquires meaning only in the presence of an experimenting intellect, and if, as Mises asserts, it cannot have any other definition, then such a probability cannot express any property of the studied object itself.
Such is the first, fundamental objection to Mises’s probabilistic conception.
The second objection arises in connection with that idealization of the empirical material upon which Mises builds his mathematical theory. Idealization of this kind, generally speaking, is not only permissible, but also completely inevitable when we wish to encompass some empirical domain by means of a mathematical apparatus. However, such idealization can prove fruitful only under one indispensable condition: we must be certain that the regularities which hold in the idealized object will, approximately, also hold in the real object. Thus, we often
we have the possibility, in the order of mathematical analysis, of replacing a very thin plate by a plane figure; in hydrodynamics, in the order of idealization, we replace a real liquid with its molecular structure by a continuous medium. In all these cases the success of such an idealization is guaranteed by the fact that the degree of closeness of the real object to the idealized one can be regarded as known, and that, consequently, we can estimate the limit of the error we risk obtaining by replacing the real object with the idealized one. If, in order to compute the area of a regular thousand-sided polygon, we replace it by a circle, then such an idealization is productive, because the bound of the error thereby incurred can easily be indicated.
Let us now toss a coin \(n\) times, with heads falling \(m\) times. The probability of heads is equal, according to Mises, to the limit of the fraction \(\frac{m}{n}\) as the number \(n\) increases without bound. This idealization of empiricism (the replacement of the real fraction \(\frac{m}{n}\) by its ideal, empirically unattainable limit) constitutes one of the most fundamental features of Mises’ theory. Can it be called productive? It is easy to see that it cannot, in any degree; and this circumstance is of fundamental importance for the whole theory: Mises reproaches the classical theory very vigorously for the fact that, without supplementary postulates of a natural-philosophical character, it is in no way capable of predicting the actual course of phenomena. It is therefore especially important to show that, in this respect, Mises’ theory has decidedly no advantages over it; if it is capable of predicting anything, it is only certain regularities in the course of an idealized phenomenon; but the character of the idealization adopted by Mises is such that no conclusions whatever can be drawn from it for the real phenomenon.
Indeed: let the probability of heads be equal to one half, and let us have in mind tossing a coin 1,000 times; can we, from our knowledge of the probability of heads, extract even the most modest prediction concerning ...
of how many times in the course of this thousand tosses heads will actually occur? Obviously, not the slightest; because any number of supposed occurrences of heads, from 0 to 1,000 inclusive, is compatible with the fact that the probability of heads occurring (defined as \(\lim \frac{m}{n}\) as \(n \to \infty\)) is equal to one half, and without an additional postulate we can make no choice among these possible numbers. True, we may be certain that if the experiments are continued without limit, then the fraction \(\frac{m}{n}\) will approach one half; but, unfortunately, this certainty applies only to an idealized (infinite) series of trials and gives way to complete ignorance, to an absolute impossibility of predicting anything whatever, when we are dealing with a real series of trials. True, the theory of Mises (like the classical theory) makes it possible to calculate the probability of various numbers of occurrences of heads in 1,000 tosses of a coin; but these probabilities, in their significance in Mises’ theory, cannot by themselves predict anything for us concerning the 1,000 tosses that interest us. They give us only an indication of what will happen if we undertake not one thousand tosses, but an infinite sequence of such thousands; in other words, they outline for us the picture of an idealized process and do not give the slightest indication of how the real process will proceed.
Thus Mises’ assertion that his conception of probability, in contrast to the classical one, has a direct relation to the actual course of events and permits immediate statements about this course must be recognized as unfounded and even simply incorrect. Mises’ probabilities do indeed characterize a certain process; but this process is an idealized one, and without additional postulates, as in the classical theory, there is no logically necessary transition from this idealized (infinite) process to real (finite) processes.
A. Ya. Khinchin
B. Mathematical criticism.
From the mathematical side, the objections that arise in connection with Mises’ conception relate to the very notion of a collective. The contemporary mathematician, trained by a series of bitter trials to exercise extreme caution in dealing with infinite sequences, cannot be satisfied with Mises’ indications on this matter, and cannot even reconcile them with one another. We shall now try to show that freeing the concept of a collective from internal contradictions can be achieved only by giving its definition an interpretation that deprives it of any content whatsoever. Here we are dealing, to a considerable degree, with a false idea, where beneath phrases that captivate our intuition, careful analysis reveals the absence of real significance.
Let us focus our attention on a possibly simple scheme of a collective: suppose we have an infinite sequence of numbers, each of which is zero or one, for example:
\[ 00111010110001\ldots \tag{1} \]
If we want this sequence to be a collective, then we must first of all require that the fraction \(\frac{m}{n}\), where \(m\) is the number of zeros occurring among the first \(n\) terms of our sequence, tend to a definite limit as \(n\) increases without bound. But this is not enough; following Mises, we must also require the “irregularity” of our sequence; this means that, choosing any infinite part of our sequence, we must necessarily have in this part, for the relative number of zeros, the very same limit as in the original sequence.
The first crude objection might consist in the fact that, if, for example,
\[ \lim_{n\to\infty} \frac{m}{n} = \frac{1}{2}, \]
then we can choose, as a part of our sequence, simply the aggregate of those of its terms which are equal to zero; in this part the limit of the relative number of zeros would obviously be not one half, but unity. However, this objection must be rejected, since Mises, in choosing a partial sequence, forbids us to make use of our knowledge of the positions at which the zeros and the ones stand; the choice must be made “arithmetically,” guided exclusively by the numbers of the terms of the sequence; we may, for example, take the aggregate of terms of our sequence whose numbers are even numbers, or perfect squares, or absolutely prime numbers, etc.
In general form the principle of this choice may be formulated as follows: let \(\varphi(n)\) be an arbitrary function of an integral argument \(n\), taking, for positive integral values of \(n\), likewise positive integral values, and suppose that for \(n_1<n_2\) we always have \(\varphi(n_1)<\varphi(n_2)\) [i.e., the function \(\varphi(n)\) is increasing]. Then the sequence
\[ \varphi(1),\ \varphi(2),\ldots\ \varphi(n)\ldots \tag{2} \]
is an increasing sequence of positive integers; from our basic sequence (1) we may choose the aggregate of those terms whose numbers occur in the sequence (2). This will be a choice of a partial sequence, legitimate in the sense of Mises. And it is obvious that, conversely, every such choice is necessarily realized in precisely the manner just described, by means of a suitably chosen function \(\varphi(n)\). Moreover, Mises imposes no restrictions whatsoever on the nature of this function, apart from its properties enumerated above.
Let us now return to the initial collective (1). Denote by \(f(n)\) the number of the \(n\)-th zero in this sequence. It is obvious that \(f(n)\) is a function possessing all those properties which we required of the functions \(\varphi(n)\) realizing legitimate choices. Why do we not dare, what forbids us, to put
\[ \varphi(n)=f(n)? \]
We cannot do this because, in defining the function \(\varphi(n)\), we are forbidden to make use of our knowledge of the distribution of zeros and ones in the collective (1). We may set \(\varphi(n)=2n\), \(\varphi(n)=n^2\), \(\varphi(n)=n!\), and so on, but we may not set \(\varphi(n)=f(n)\). Let this be so. But what can guarantee us against, in constructing our function \(\varphi(n)\), accidentally hitting upon a function identically coinciding with \(f(n)\)? Obviously, only some definite feature in the nature of the function \(f(n)\) could guarantee us against this; this is perfectly understood by Mises, who also clearly sees the necessity of such a guarantee, without which his entire definition collapses. Therefore Mises demands, with sufficient definiteness, that the function \(f(n)\) must be such that it cannot be “guessed,” that it must be impossible by any means, without knowing the distribution of the collective (1), to construct such a function \(\varphi(n)\) which would identically coincide with \(f(n)\). This is precisely the meaning of the formulation of the requirement of irregularity which Mises calls the “Prinzip vom ausgeschlossenen Spielsystem.”
The reader will, of course, already see the serious perplexities that arise here. What mathematical meaning is possessed by a property of a function consisting in the fact that it “cannot be guessed,” that, whatever function we may choose, it will always be “not that one”? Mathematicians know well how hopeless is the task of drawing an exact objective distinction between functions “lawfully determined,” which “can be guessed,” on the one hand, and “lawless” functions, which “cannot be guessed,” on the other. In any case, this vague distinction, in its raw form appealing to the properties of human intellect, cannot serve as the basis for the axiomatization of a mathematical discipline; yet it is precisely this role that Mises’s conception assigns to it, and we see no possibility of avoiding this difficulty along the path taken by his theory. The second property of a collective, its “irregularity,” is thus internally inconsistent: in order to free it from contradictions, we are compelled to give it a vague meaning that has no objective scientific content.
True, at first glance there seems to be one possible way out here. Modern mathematics distinguishes two types of sequences of the form (1): first, completed, finished sequences, whose specification already contains within itself a law that makes it possible to judge at which places there will be zeros and at which ones; and second, sequences that are “being created,” “becoming” (freie Wahlfolgen), where we arbitrarily or by chance put a zero or a one in the first place, then a zero or a one in the second, and so on.
A distinctive methodological feature of these sequences of the second type is that, by their very definition, we cannot think of such a sequence as completed, finished. For such a sequence, in general, only those of its properties have real content whose presence can be established after acquaintance with some number of its first terms. If, for example, we have such a sequence of positive integers which is in the process of eternal becoming, then the question whether even numbers occur among its first ten terms has a precise meaning; but the question whether this sequence contains an infinite set of even numbers has no meaning whatever, because it does not draw a distinction between two factual circumstances. The fact that these “becoming” sequences can be the object of mathematical analysis has in recent years been emphasized in a number of works by Brouwer and Weyl.
Up to now we have considered the collective as a sequence of the first type, i.e. as lawful and therefore necessarily law-governed, and have convinced ourselves that, under such an understanding, this concept is untenable. But should one not regard the collective as a sequence of the second type, i.e. as incomplete, in the process of “eternal becoming,” and therefore “lawless”?
At first glance this idea seems very natural, since the collective is a sequence generated by chance (and only subsequently idealized by the requirement
- Advances in the Physical Sciences. Vol. IX, issue 2.
...finiteness). We might, perhaps, attempt along this path even to avoid idealization and to speak not of finished, but only of “becoming” collectives. However, this possibility is an illusion, broken already at the first step. For we want there to exist limits in our sequence; this is the first, fundamental property of a collective. For a sequence “forever becoming,” which, by its very definition, we cannot think of as finished, completed, the requirement that a limit exist is an empty phrase behind which there is no real content. The disjunction of the existence and nonexistence of a limit obviously acquires meaning only for a sequence conceived as a completed series, determined to the end. The very word “limit,” and the precise concept standing behind it, do not allow this question to be posed for unfinished, forever becoming sequences.
Thus, if our first understanding of the collective led us to the impossibility of realizing “irregularity,” then the second understanding renders illusory, deprived of all reality, the basic property of a collective—the existence of limits. Mathematics knows no sequences possessing those properties with which Mises endows his collectives. And therefore we must acknowledge that Mises’ theory has, up to the present time, lacked a firm mathematical foundation.
C. A critical remark of a physical nature.
From the physical point of view, Mises’ doctrine raises one substantial objection, concerning the so-called a priori probabilities or statistical weights of molecular states. When we say, for example, that an individual molecule has equal probability of finding itself in any of the cells of its phase space (this point usually serves as the starting point of statistical thermodynamics), then already in the classical understanding of probabilities it is difficult to understand what this means; the definition of such “a priori probabilities,” in our view, has not yet been...
has proper clarity. For it is physically impossible to devise an experiment that would make it possible to verify an assertion of this kind. However, in the classical interpretation we can still reckon with the possibility (and hence entertain the hope) of giving this definition a real meaning; where the probability under study must in any case be understood as a property objectively inherent in a given molecule, we may hope, sooner or later, having sufficiently studied the nature of the molecule, to give a satisfactory theoretical foundation for the “a priori probabilities” characteristic of that molecule.
In Mises’ conception the matter stands fundamentally otherwise. Here the sought-for “a priori probability” is understood exclusively as the proportion of a characteristic in a collective. What, then, in such a case can it mean, what physical sense can an assertion have such as that “all cells are a priori equiprobable”? Obviously the meaning of this assertion can only be the following: in some initial collective the molecules are present in such a distribution that the relative number of molecules falling into a given cell is one and the same for all cells. Mises himself, incidentally, explicitly formulates the assertion of interest to us precisely in these terms1. It is now permissible to ask: what is this collective of which there is question here? Where is it realized, or where can it be realized in nature? For it is clear that in any actually realizable collection of molecules not all cells will be represented equally often, if only because the real physical world places definite limits on the energy of an individual molecule. This means that the collective to which Mises here appeals has no relation to the real world and cannot have any; it is the product of a purely theoretical construction; and if so, it remains fundamentally incomprehensible what real meaning, in Mises’ doctrine, the a priori equiprobability of all cells for a molecule of a given type can have. On the same basis one may assert that, in general,
the concept of statistical (a priori) weight in Mises’ theory can in principle have no physical significance, since it is based on the consideration of collectives which are often theoretical constructions to which, in the real world, nothing actually existing corresponds, nor can correspond.
There is one way out of this difficult situation which, incidentally, may also be recommended to the classical conception: not to ascribe any probabilistic meaning at all to “statistical weights,” recognizing such meaning only for certain combinations of them. Statistical weights would then acquire merely the role of certain coefficients having no direct physical significance; operating with them according to certain formal rules, we would obtain genuine probabilities under real physical conditions.
However, at the present time physical statistics is still far from a principled and consistent implementation of this tendency; and Mises’ theory, operating with fictitious “initial collectives,” is still farther from it. And so long as this is the case, all our objections remain in force.
III. Mises’ doctrine on the principles of physical statistics1
As we indicated in the introduction, Mises’ doctrine on the principles of physical statistics can be developed independently of his views on the general conception of probability, and this possibility is especially valuable because, as we have seen, Mises’ general doctrine calls forth various objections.
One of the chief fundamental difficulties of any statistical molecular theory consists, as is well known, in the fact that such a theory always sets before us the task,
at first sight insoluble and even meaningless: to take account of the influence of chance in a process in which, in essence, there is nothing random, but, on the contrary, everything is determined1. Statistical mechanics always represents an aggregate of molecules as a mechanical system moving according to exact laws, governed by differential equations; and yet the summary picture of this complex, but precisely lawful, motion we must determine by the methods of statistics, i.e. by being guided by the laws of chance.
It is known that the mathematical foundation of statistical mechanics is still in a very unsatisfactory state. The reason for this lies partly in considerable difficulties of a purely mathematical character, and partly—and chiefly—in the insufficient precision with which the fundamental problems are posed and in the insufficient clarity of the method itself; in particular, the place and character of the contact between theoretical considerations and the real world are outlined vaguely and unsatisfactorily.
This unfortunate state of affairs, in Mises’s opinion, has its roots in that fundamental incorrectness of the scientific approach of which we have just spoken. One cannot at the same time subject the motion of a given mass of molecules to differential equations and, together with this, require that the summary characteristics of this motion be formed according to the laws of chance. From exact data, according to exact laws, only exact consequences can be derived; there is no place here for any probabilities or statistical means. If we wish to apply statistical methods, we must abandon the idea of characterizing the motion under study by exact laws.
But would this not mean abandoning, along with it, the principle of the determinacy of natural processes, and therefore also the necessity of causal connection between them, and thus parting altogether with firm natural-philosophical ground? Mises shows (or, at least, it can be shown,
proceeding from his conception, that this is not so: it is possible rationally to assign to chance a very considerable share in physical processes; it is even possible, in other phenomena, to give it complete dominance, while at the same time in no way departing from a causal-deterministic understanding of these processes and phenomena. The scheme of this exceedingly fruitful natural-philosophical approach is as follows.
Every physical process proceeds according to absolutely exact laws that fully predetermine its course. But these laws, in their entirety, are extraordinarily complex, and therefore, in the course of scientific investigation, we are compelled to subject real processes to a certain idealization. This idealization usually consists, first of all, in the fact that we conceive the physical system we are studying as isolated, i.e., we neglect those influences exerted upon it by external bodies. An astronomer studying the motions of the solar system abstracts, in doing so, from the attractions experienced by the bodies of this system from distant stars, although he knows that the slightest motion of an atom on Sirius deflects the Earth from its regular path. In exactly the same way, a physicist constructing a molecular theory first of all imagines the mass of gas he is studying as isolated from external influences, although he knows that in the real world such isolation is unattainable.
By isolating and idealizing the given phenomenon, i.e., by disregarding certain details in its course, we usually achieve the possibility of characterizing this idealized scheme by a few simple laws; it is in this way that scientific theory is constructed. At the same time we must not forget that real processes differ from idealized ones and, above all, proceed infinitely more complexly than they do. Thus not all the elements of each given process enter into our law-governed description; the true course of phenomena differs to a certain degree from that outlined by theory; and these deviations we call random, because they are conditioned by causes that have not entered into our theory, and consequently this theory is not in a position either to explain or to predict them. We know,
that these deviations, like everything in nature, are strictly lawful; however, the laws governing them, by their great complexity, completely elude our study, as a result of which these deviations have an essentially random character.
And here we must distinguish two possible cases. Suppose, for example, that we are studying the motion of the Earth around the Sun. Besides the mutual attraction of the Sun and the Earth, we may, wishing to have an exact theory, take into account the influence of the Moon, the planets, and other bodies of the solar system. But it is clear that this is still not all; every motion of every smallest particle on the Earth, on the Sun, on any body of the solar system, and in general anywhere in the world, deflects the Earth from the orbit prescribed by our theory. These deviations are small; the regularities governing them are so complex that we have no possibility of including them in our theory and prefer not to explain or predict them, thereby ascribing them to chance.
In this example, what is important for us is the assurance that these “random” deviations (which here, as in every physical process, are necessarily present) are not capable of altering the Earth’s orbit to any appreciable extent; that, consequently, in its essential features the picture of the phenomenon that we have constructed as an idealization will correspond to the real course of the phenomenon. This phenomenon, therefore, is adequately described (with a sufficient degree of accuracy) by our theory; chance does not for a minute cease to exert its influence, but this influence is negligible—and practically we may disregard it. It is precisely in this sense that we say that “the motion of the Earth around the Sun is subject to exact laws, in which there is no place for chance.”
Let us now consider another example. Suppose we are dealing with a given mass of gas enclosed in a certain vessel and consisting of an enormous multitude of “chaotically” moving molecules. As an idealization, we consider this physical system isolated from the external world. Let us assume that the molecules collide with one another and with the walls of the vessel according to the laws of absolutely elastic bodies and have a spherical shape.
shape of a definite radius, and the walls of the vessel are absolutely smooth. In this idealization the laws governing the motion of the molecules can be formulated quite simply. The course of the idealized phenomenon is exactly determined by these laws and leaves no room whatever for chance. Let us now pass from the idealized process to the real one. Since in the real world our gas, first of all, cannot be isolated, this fact alone already creates for the real process an innumerable variety of “random” deviations from the idealized one. In passing to the real world, we, as always, open wide the doors to chance.
And chance bursts through the open doors. But, in contrast to the preceding example, this time it is not limited to introducing small, practically vanishing corrections into the course of the phenomenon; it completely destroys the theoretical picture of the process that had been constructed and replaces it with a new one, subordinated to its own laws. In fact, if at a given moment an external force, negligible in magnitude, has deflected a molecule ever so slightly from the path prescribed for it by the theory, then, taking into account that the mean free path of the molecule exceeds its diameter by a very large factor, we easily understand that the fate of this molecule will be changed in the most radical way by the insignificant change mentioned in its motion: our molecule will encounter on its way a whole series of other molecules with which it was not at all supposed to collide, and, conversely, will avoid collision with a number of molecules with which it theoretically ought to have met. And if we now take into consideration that each molecule experiences an enormous number of collisions per second and that each molecule is constantly subjected to the above-described random actions of external forces, then we immediately see that the result of these random actions, in the very near future, must be a radical change in the entire picture of motion prescribed by the theory.
Thus, in the example under consideration, we, in contrast to the preceding one, establish that the theory, how
however detailed it may be, in practice it in no way determines the basic features of the course of the real phenomenon. On the contrary, the entire determining role is assumed by “random” external influences, which in their essence are lawful and deterministic, but which have not entered into our theory and therefore, with respect to it, retain in every way the character of chance.
According to Mises, such is the situation in every molecular statistical theory. We see that, without in the least abandoning the premise of lawful determinacy of physical processes, we have a fully rational possibility of assigning to chance a very significant, and sometimes even decisive, role in these processes.
Probability theory studies the laws of chance, and therefore its role in molecular-statistical theories becomes intelligible and justified. The very scheme of applying the methods of probability theory to molecular doctrines has been set forth with great clarity by Mises in several particular examples, of which the most complete is his theory of Brownian motion.
If only in other important cases as well it proves possible to carry out this scheme satisfactorily, then in statistical mechanics we shall have made a very considerable step forward. This step will signify a fundamental transition from a mechanical-probabilistic study of molecular systems to a purely probabilistic study—a transition rationally justified by the consideration that, in a real molecular process, the power of chance manifests itself to a far greater degree than the influence of the mechanical laws governing the motion of the idealized system.
Literature
1) R. v. Mises. Marbes „Gleichförmigkeit in der Welt“ und die Wahrscheinlichkeitsrechnung. Naturwiss., 7, S. 168, 186, 205, 1919.
2) Id. Fundamentalsätze der Wahrscheinlichkeitsrechnung. Math. ZS. 4, 1, 1919.
3) Id. Grundlagen der Wahrscheinlichkeitsrechnung. Math. ZS., 5, 52, 1919.
4) Id. Ausschaltung der Ergodhypothese in der physikalischen Statistik. Phys. ZS., 21 S. 225, 256, 1920.
5) Id. Das Gesetz der grossen Zahlen und die Häufigkeitstheorie der Wahrscheinlichkeit. Naturwiss, 15, 497, 1927.
6) Id. Wahrscheinlichkeit, Statistik und Wahrheit. Wien (Springer.) 1928.