A physicist's second reaction to Mengenlehre

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A PHYSICIST’S SECOND REACTION TO MENGENLEHRE

By P. W. BRIDGMAN[The Scripta Mathematica takes pleasure in publishing this interesting article

by Professor P. W. Bridgman. As in the case of all articles, however, it is under-stood that this does not necessarily represent the views of the editors.—Editor.]

THE difficulties which physics has experienced in the last few years in adjusting its fundamental methods of description and thought

to our rapidly expanding store of physical fact has been widely adver-tised and much discussed. Under the goad of this insistently press-ing physical situation the physicist has been making serious attempts to find what was the matter with his previous methods of approach. These attempts have not been entirely fruitless, but have given him confidence that he has now in his hands the first principles of a general method of attack which ensures that errors similar to the errors of the past will not be repeated in the future.

The critical analysis which has given physics its new confidence has, up to the present, been almost exclusively confined to an examination of the nature of the physical concepts which the physicist uses. But since mathematics is coming to play an increasingly important role in the new physics, it is evident that a critical examination of the nature of the fundamental concepts of mathematics is a task of the immediate future for the physicist. It was therefore not without a certain amount of dismay that I suddenly became aware that in the mathematics of the present day there are doubts, uncertainties, and differences of opinion on fundamental questions which are at least not unlike the bewilder-ment of physics when confronted with the new phenomena of the quantum domain. My awakening to a consciousness of the situation in mathematics I owe to that extraordinarily well written little book by E. T. Bell, The Queen of the Sciences. Within a few weeks of my reading of this book A. F. Bentley’s Linguistic Analysis of Mathemat-ics appeared. This I skimmed hastily, gathering from it a most vivid impression of the chaotic state of affairs in the “fundamental” fields of mathematics, and then put it aside for later more careful study of some of its technical aspects.

1The term Mengenlehre is commonly rendered in English by Theory of Classes or of Sets or of Assemblages or of Aggregates.

Scripta Mathematica 2 (1934), pp. 101–117; 224–234.

102 A SECOND REACTION TO MENGENLEHRE

These two expositions made it evident that Mengenlehre was that branch of mathematics in which perhaps there were the most seri-ous differences of opinion and in which fundamental questions were most to the fore. This awakened recollections of an early acquaintance with the rudiments of Mengenlehre, going back to years of Univer-sity study (1900–1908) when, as usual to youth, my chief interest in mathematics lay in acquiring manipulatory facility as rapidly as pos-sible, and also awakened recollections of feelings of growing doubt and insecurity during the immediately subsequent years in the face of the whole transfinite number situation, chiefly aroused whenever I dipped into any of the writings of Bertrand Russell. In the stimulating latter years, however, these questions became lost to mind in concern with the questions presented by the fundamental developments in physics.

As the most direct way to the debatable places of mathematics, which apparently are chiefly concerned with questions of infinity, I attempted to refamiliarize myself with the elements of Mengenlehre by a reading of various popular accounts of it, as in Tobias Dantzig’s Number, the Language of Science and in Rademacher and Toeplitz’s Von Zahlen und Figuren, and then by a reading and careful study of those parts which seemed relevant of E. V. Huntington’s The Contin-uum and Other Types of Serial Order and A. Fraenkel’s Einleitung in die Mengenlehre. Some idea of the magnitude of the controversial litera-ture may be obtained from this latter book.

The reactions of this second acquaintance with Mengenlehre were as different as can well be imagined from those of my first naive con-tact, when I suppose I got the usual kick out of feeling that I was play-ing with the infinite. My second contact had behind it the implications of recent physical developments, and it is the chief reaction of this second contact which I wish to expand here, namely that the situation in mathematics is strongly analogous to that in physics, and that the difficulties of mathematics, particularly those of dealing with infinity, are to be met by the same technique which is capable of overcoming the difficulties of physics.

The technique which has come to the rescue in physics scored its first notable triumph in its first important application by Einstein to relativity theory, and the same technique is at the back of quantum mechanics; for this technique I have proposed the name “operational”. I shall attempt to exhibit what this technique is and to suggest some of its implications when extended to a field broader than that of its first physical applications. It appears that a procedure very much like the “operational” is already in extensive use by mathematicians, but it

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does not seem to have received a clean cut formulation and its applica-tions in mathematics are not consistent or systematic. After a prelimi-nary discussion of the operational method I shall make application of it to some of the elementary situations presented by Mengenlehre, and hope to show that by a consistent use of the method the paradoxes disappear.

We begin by an examination of the simple application which Ein-stein made of the method in handling two physical concepts which his analysis showed occurred in connection with all phenomena cov-ered by his special theory of relativity, the concepts, namely, of length and simultaneity. What Einstein did in effect was to demand that the meaning of these concepts, which purport to apply to concrete physi-cal situations, should be sought in the concrete physical operations involved in the physical application. More definitely, the meaning of length is to be sought in those operations by which the length of con-crete physical objects is determined, and the meaning of simultane-ity is sought in those physical operations by which it is determined whether two physical events are simultaneous or not. Contrasted with this was the earlier procedure, in which the length of an object was defined as the difference of the coordinates in absolute space of its terminal points, and the simultaneity of two events meant the equality of their absolute times. Both absolute space and absolute time were metaphysical concepts, purposely and selfconsciously divorced from physical reality, as may be seen in the explicit definitions of Newton. The mere formulation of the two methods is sufficient to establish the superiority of the new and to make understandable its success in avoiding the erroneous conclusions of the old with regard to physical fact. For if any physical situation is described only in terms of concepts which themselves are defined in terms of physical processes actually performed, the whole description reduces ultimately to a description of an actual physical experience, and as such must have the validity of all direct observation of physical fact, which for our purposes is to be accepted as an ultimate. It is especially to be noticed that our concepts, being thus framed in terms of operations actually performed in physical experience, must lead, at any stage of physical inquiry, to conclusions in which room is left for future refinements within the uncertainties and approximations of our present physical operations.

The operational procedure, as thus formulated, at first strikes one as entirely negative, or rather neutral, in character. It is, of course, obvious that this is only part of the story, and that there is still room for the genius of Einstein in discovering that when a particular and


somewhat unnatural procedure is used for measuring the length of moving objects an enormous simplification of formulation results. However, in spite of the neutrality of this procedure, it is of the utmost importance, for we have here a method by which the form in which we phrase our statements about nature establishes no prejudice whatever as to what the content of these statements shall be when extended beyond the present range of experience. In the discovery of this non-commital medium of expression it seems to me that Einstein made an even more important contribution than in the precise form which he gave to the transformation equations, which must be subject to con-tinual physical control and possible revision as the range of physical measurement increases.

When we attempt to extend the spirit of this procedure, which has been so successful in physics, to mathematics, we find necessary a little more careful examination of what is involved than sufficed for the comparatively simple purposes of physics. We can distinguish a general and a more specific application of the operational point of view. Examination will show that the operational procedure is greatly concerned with the specification of meanings. Now it is evident that “meaning” can run through the widest possible gamut from the vaguest, highly emotional, and purely personal, reactions of individual experi-ence to the precise and universally accepted terms of scientific usage, but in all these cases what I find involved in the notion of meaning is operational in character. The meaning that I ascribe to “beautiful”, for example, I find in the operations which I perform, or more simply, in what I do, in applying the term to any event or object, even if the operation is as amorphous as confronting myself, either in thought or in actuality, with the event or object, and observing whether it awakens in me that familiar and well recognized feeling with which I associ-ate the epithet “beautiful”. The operation is, of course, usually more complicated and more sophisticated, but operation of some sort there always is whenever I make the selfconscious attempt to assure myself that I know what I mean. Conversely, if there is no operation by which it may be determined whether a term or concept is applicable or not to an individual situation to which it ostensibly should apply, then the term or concept is meaningless. That is, the meaning of a term or con-cept is contained in those operations which are performed in making application of the term or concept to relevant situations.

I can only report that as a matter of personal analysis I find this operational aspect at the bottom of all meaning, but my inference is that other persons go through similar processes. This general attitude,

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or something similar to it, is fundamental in what follows.It is important to notice that in spite of the very vague and general

account which we have thus given of meaning, there may nevertheless be certain restrictions involved because of the general nature of any possible operations. The most important such restriction which occurs to me, and perhaps the only one, is the restriction with respect to time; operations are performed in time, and are thus subject to restrictions as to order.

Turning now from the general operational significance of mean-ing, it is obvious that the operations fundamental to meaning must be subject to further special restrictions when the terms are to be profit-ably employed for specific purposes. The most casual examination of what we do is sufficient to establish the existence of such additional restrictions. The precise nature of the additional restrictions has sel-dom, if ever, been made the object of systematic study, but is almost always left more or less to instinct or personal experience. Fortunately, or unfortunately, the experience of different persons is so much alike that sufficiently close agreement for ordinary social purposes is attain-able without any selfconscious or formal analysis of the matter. It is obvious enough, however, that when greater precision is needed, as in scientific matters, a more careful and articulate analysis has to be made. The restrictions which Einstein proposed in his theory of relativity for terms appropriate in describing physical situations is perhaps only the beginning of what may be done in this field. In physics, fortunately, the situation is very simple; the statement that if we describe actual situ-ations in terms of actual experience we shall not run counter to that experience needs only to be made to command assent.

But closer scrutiny discloses more than at first appears, even in the simple case of physics. Our concepts must be described in terms of actual physical operations, that is, operations which can actually be carried out. In what terms shall we describe these operations, and how shall we be sure that they can actually be carried out? I think that no logically completely satisfying answer can be given; we describe certain physical operations in terms of others which appear simpler to us in the light of our physical experience, and these may again perhaps be defined in terms of others still simpler, until presently we are con-fronted with operations which we must accept as unanalyzable and apprehendable only intuitively by personal experience. The operation of observing “coincidences” is such an unanalyzable ultimate opera-tion for Einstein. Another important point is that these intuitively ultimate operations of our analysis are considered as fixed and


identifiable in character, little frozen bits of procedure, so that the “same” operation may always be performed, whenever required, at whatever time.

We shall not try to get nearer to the bottom of the situation than I have suggested here, and in fact I believe that “bottom” is a meaning-less term in this connection. We are content with the fact of experience that it is possible to describe physical operations in such terms that when we are confronted with any concrete physical situation we know unequivocally and without question how to perform the operations demanded by the situation.

It is obvious, I think, that the remarks just made about the opera-tions of physics must apply also to the operations permissible in math-ematics, or, for that matter, to the operations of any discipline.

It is most important to stress the fact that such an unequivocal and straightforward specification of the operations as is necessary de-mands a specification of the operations in terms of other operations intuitively acceptable as simpler. A specification of the operation in terms of the properties of the operation after it has been completed is not such a straightforward and unequivocal specification. To convince oneself of this one has only to imagine oneself trying to perform an operation specified in this way. With such a definition all the questions as to the possibility of the operation at once become open, and we re-vert to the pre-Einsteinian situation. There is one particularly vicious method of specifying an operation in terms of its properties which presents itself in Mengenlehre, which we shall see more of later, but which should be mentioned here. It is possible to set up rules which determine a non-terminating series of operations, as for instance, the rules by which the sequence of the natural numbers is engendered. But it is obviously not legitimate to specify in this way a non-terminating operation, and then to treat this non-terminating complex as itself a simpler operation which may be used as an intuitive ultimate in the specification of another operation. Such a non-terminating complex can be treated in this way only when it can be proved equivalent to some other procedure specifiable in finite terms, and which can, there-fore, be actually executed. Otherwise, the non-terminating complex must be treated as the end, and no other operations be demanded after it; our ordinary experience of the order of operations as performed in time evidently requires this.

Corresponding to the restrictions necessary on operations which may be profitably employed for special purposes, we may recognize a restricted significance of the term “meaningless”. A term is “mean-

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ingless” within a given range (as of mathematics or physics) if the op-erations involved in giving meaning to the term are not subject to the restrictions that have been accepted as necessary in that range. “Mean-ingless” is used mostly in this sense in the following.

We are now ready to apply the operational procedure to mathemat-ics. Corresponding to the requirement of physics of consistency with physical facts, the ideal for mathematics, which apparently all math-ematicians accept without debate, is self-consistency or freedom from internal contradiction. The paradoxes and inconsistencies of Mengen-lehre are definitely not wanted, and it is because we do not understand how they have managed to insinuate themselves that the situation is so very disturbing. Let us first examine the mathematical and logi-cal notion of self-consistency or freedom from contradiction. What is this self-consistency and why do we so insistently demand it? The no-tion is apparently inseparably connected with those very hazy notions: “existence” and “reality”. If we try to make these notions a little more precise, we may perhaps say that self-consistency is in some way inti-mately connected with real things. The use which the mathematician makes of this feeling of the significance of self-consistency in Pos-tulate Theory is illuminating. The accepted method of proving that some system of postulates does not conceal some contradiction is to exhibit some “real”, “existing” system which satisfies the postulates. Nothing further in the way of proof or analysis is felt to be necessary; the feeling that actually existing things are not self-contradictory is so elemental as almost to constitute a definition of what we mean by self-consistent. Now when we are concerned with “things” we are evidently concerned with some form of experience, so that we may make an even broader statement and say that experience is not self-contradictory.

We may digress to inquire how it is that we can make a statement of such perfect generality about experience in view of that other principle, which is almost at the bottom of all recent thought in physics, that experience cannot be limited, but is a brute fact, only to be accepted. From this point of view, if self-contradictory experi-ences occur, that is that, and all there is to it. If one wants to make “self-contradictory” as a term applicable to facts more than a mere matter of words or pure definition, the justice of this criticism must be recognized, and all that we can assert is that no experience of the past has been self-contradictory, but that we have no absolute assurance for the future. Any mathematics must therefore be recognized to rest on at least this experimental basis, and in the last analysis to contain


at least this, if no other, experimental element. If we are made un-easy by this cloud over the possible future of mathematics and try to analyze further into the self-consistent character of all experi-ence, I believe that we can find a clue to it in the singleness of our self-consciousness. The sharper our attention becomes the more defi-nitely single and focussed becomes our consciousness. Experience is not self-contradictory because only one thing happens to us at a time; these are two equivalent ways of saying the same thing. Per-haps if we could learn to split our consciousness so that we could simultaneously experience two foci of mental activity, the principle of self-contradiction would lose its inevitableness. I may record as a matter of personal experience that I have had no luck in trying to thus multiply self-consciousness.

Returning now to the main argument, it is at once obvious that the operational technique automatically secures to mathematics the sine qua non of self-consistency, for operations actually carried out, whether physical or mental, are a special form of experience, so that any mathematical concept or argument analyzed into actual opera-tions must have the self-consistency of all experience. This statement is not to be regarded as rigorous; it contains too many vague terms. I believe, however, that it gives a qualitative suggestion of the nature of what is involved sufficiently good to lead to the anticipation of success. I doubt whether any useful purpose would be served by trying to make the procedure more precise, but we may proceed at once to applica-tions; the proper details will suggest themselves as need arises.

The reason that our emphasis on the operations may modify con-ventional mathematical procedure, which is already admittedly largely operational, is because of the requirement that the operations should be actually carried out, or capable of being actually carried out, and this in turn demands a straightforward and unequivocal specification of each detail of the operation. In particular, an operation defined in terms of its properties is not straightforwardly defined. Mathematics, however, is full of examples of operations defined in terms of their properties—every inverse function involves such operations. But it should be a matter of elementary knowledge that no inverse function can be admitted until it is shown that there is a procedure by which it may be converted into a direct function. For example, the square root of a number involves a definition in terms of properties, for we are instructed to find such a number that when multiplied by itself the given number shall be produced. But the operation correspond-ing to this definition cannot be used until it is shown that there

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is a direct procedure, with each step in the process uniquely and un-equivocally determined, for producing such a number. The elementary algorithm for extracting the square root as a decimal fraction is such a process. Once the direct process has been found, we may go back and use the definition in terms of properties if it is more convenient, and actually the definition in terms of properties is often more compact than the direct definition and is therefore often used. In fact, definition in terms of properties is so common in mathematics that the neces-sity of an underlying direct process may be lost sight of. The absolute necessity of establishing in every special case the possibility of a defi-nition in terms of properties is shown by the great number of inverse procedures which turn out to be impossible, as for example, finding the angle whose sine is greater than 1, or extracting the square root of a negative number.

It seems to me that the use of operations defined in terms of properties should be scrupulously avoided in “foundation” studies. The appearance of operations defined in this way should always be regarded with suspicion, and the definition replaced if possible by another. If such proves impossible, one is justified in summarily dis-missing the entire situation. Corresponding to Einstein’s dictum for physics that physical concepts be defined in terms of physical op-erations, I would propose for mathematics a corresponding dictum, namely that in foundation studies only operations “directly” defin-able be admitted.

What mathematics did when confronted by the impossible de-mand to extract the square root of a negative number is most in-structive for our purposes. Mathematics of course introduced the idea of complex numbers, and in particular √−1. What is √−1 ? Is it a concept defined in terms of its properties? But its properties are impossible properties. I think that √−1 must be regarded as a new symbol, standing for certain operations which present themselves in connection with the formal manipulation of algebraic equations, to be treated as intuitively given and an ultimate of experience, as unanalyzable as any other operation accepted as ultimate, such as adding 1 to a given number. The fact that a complicated experience is necessary before any one can intuitively grasp this new concept as an unanalyzable, and that a certain amount of sophistication is in-volved, is irrelevant. The fact is that mathematicians who have had experience can handle √−1 as an unanalyzable. The further fact that a geometrical procedure can be found permitting a one to one cor-respondence between actual geometrical manipulations and corre-


sponding formal manipulations of the equations I believe also to be an irrelevant, although happy, accident.

Corresponding to the situation presented by complex numbers, one can see the possibility of a more general situation correspond-ing to concepts defined in terms of properties impossible in actual experience. Self-contradiction is, par excellence, such a property. By analogy, we may call such a concept an imaginary concept, the imaginary feature consisting in its self-contradiction. It is not in-conceivable that such imaginary concepts will prove useful if one wants to play the game of putting words together, as does Bertrand Russell, in every possible combination, with complete disregard of any operationally precise background, and that from the study of them may arise a more general mathematics. I suspect that the mathematical paradise which Hilbert claims was opened by Can-tor is situated in this domain and that the condition of entry into this Paradise is willingness to admit paradox. One may, however, surmise that generalized concepts permitting of self-contradic-tion will not prove very useful unless one or a few simple types of imaginariness are discovered corresponding to the single √−1, which is adequate to meet all the situations arising from the for-mal manipulations of analysis. If any such generalization as this is attempted, the current definition of mathematics as the study of all logical (that is, self-consistent) operations, will have to be much broadened.

With these preliminaries we are now sufficiently prepared for Mengenlehre, and to make matters definite we may as well begin by following Huntington’s book and see what account we can give in purely operational terms of his procedure. At the very beginning occurs his description of “class” or “Menge”. I shall in the future use Huntington’s alternative Menge rather than the “class” because it makes direct contact with most of the literature and with the title of this paper. “A Menge is said to be determined by any con-dition or test which every entity (in the universe considered) must either satisfy or not satisfy.” This in itself is strongly reminiscent of the operational method, and is an example of the fact that many aspects of mathematics are already operational in character. But there is this important point, that in a thorough-going application of the operational method we have to actually make the tests (or imagine ourselves making them, which for our purposes is just as good) which determine the Menge, and so have to verify that the operations implied in the tests are such that they can be actually carried out.

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It is the easiest thing in the world to frame the proposed test in words of a specious appearance of legitimateness, but which a closer scrutiny discloses demand the performance of impossible operations. In the ab-sence of any systematic analysis of what constitutes possibleness in an operation, it is fortunate that in the situations which arise in practise the conditions are so simple and obvious in character that they need only be stated to be accepted, so that even a naive operational attack is able to remove the paradoxes of at least Mengenlehre.

As an example of how the actual execution of the operations discloses the seat of paradox consider a well known example, not taken from Mengenlehre, but supposed to be typical of an even more general logical situation. The example is that of the soldier, a bar-ber by profession, who is ordered by his superior officer to shave all the members of his company, and only those, who do not shave themselves. This is an innocent enough appearing order; the paradox enters with regard to the barber himself. If he shaves himself, he has thereby shaved one who shaves himself and has disobeyed the order. But if he does not shave himself, he has thereby failed to shave one who does not shave himself, and again has disobeyed the order. The innocent sounding words, therefore, conceal a contradiction—the words somehow fail to register with the actual situation. That the verbal form of the order is faulty, so that we may expect para-dox, is at once obvious if one examines what is involved operation-ally by trying to give operational meaning to the terms of the order. Imagine the barber preparing to execute the order. He divides all the men in his company into two groups—those who shave themselves and those who do not. But what does this mean, “a man who shaves himself ”. If it means a man who ever in the past has shaved himself, then the answer is easy; the company may at once be divided into two groups on the basis of questions unequivocally answerable by yes or no, and the order is exactly executed, whether the barber is or is not found in the group of those who have at some time in the past shaved themselves. But the catch comes in the fact that this is not what is tacitly understood by “one who shaves himself ”. It is obvious from the way the paradox runs that a man is to be classed as one who shaves himself if at any time, either in the past or future, he shaves himself. But the future is unpredictable, and it requires merely to be said to recognize that our operations must not involve predicting the fu-ture. It is sufficiently evident, therefore, that the order was not properly framed, because it involves an operationally indeterminate situation, and we must be prepared for paradox. That paradox will actually


result from such an operationally indeterminate situation cannot cer-tainly be stated, and in specific situations can be decided only by spe-cific examination. But unless we are prepared in each case to make the longer specific examination, we would obviously do well to avoid such operational situations.

There is another point in the paradox of the barber, perhaps even more important that the fact that the operations have been so framed as to demand a predicting of the future. For if this were the only point, we might gamble a little, and perhaps have paradox only half the time. The more important point is that the actions of the present are speci-fied in terms of future situations which are themselves affected by the actions of the present. The command to our barber effectively demand-ed that he should so act that when his actions were completed and the whole performance was contemplated as one finished and static whole, certain conditions should be satisfied, entirely ignoring the fact that the final situation must be arrived at by the step by step passing through of intermediate situations during which the conditions which by definition determine conduct may be continuously changing. This, of course, is what actually occurs in our case. When the barber has half shaved himself, is he at that moment in the class of those who shave themselves or of those who do not?

It seems to me that the difficulty in the case above is typical of a great many that occur in Mengenlehre. An operation must not be defined “self-reflexively”, and in particular, not in terms of itself contemplated from a future standpoint, which is a particular form of definition in terms of properties. The necessity for straightforward specifications of the operations, with each sub-step explicitly deter-minable, is a point that has been previously made. It is so elementary a matter that even the Patent Office has long recognized it and de-mands that an invention should be specified in terms of its construc-tion and not in terms of function.

Now consider another term in Huntington’s definition of Menge, his term “entity”. This term is evidently purposely chosen noncom-mitally, so as to allow the greatest possible latitude in the construc-tion of the Menge. It obviously includes the recognizable physical objects of our ordinary physical experience, but just as obviously is meant to include all sorts of intangibles, such as mathematical sym-bols, or abstract constructions of all sorts. It includes even complicated constructions fabricated as we go along, the execution of each succes-sive step depending on the execution of the previous one and defined by it. In particular the Menge itself is a possible entity, as shown by the

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fact that we are presently discussing Mengen which include them-selves as elements. But in spite of the great generality which ostensibly inheres in “entity”, the way in which it is used shows that it is tacitly subject to at least one restriction, namely, the rules of Aristotelian logic are supposed to apply to it. That is, the principle of self-contradiction applies, so that if given a statement, A, can be made with meaning about the entity, we can say: “either A is true of the entity or it is not true”. This in itself I believe constitutes a very much more important restriction than is commonly recognized. What sort of thing is it to which the Aristotelian principle of the excluded middle applies? Ex-amination of what we do in handling it discloses that it must have the proper amount of a quality variously to be characterized as perma-nence, or identifiability, or “staticness”. Obviously we cannot say of x “either A is true of x or A is not true of x” if x itself changes while and because we make the statement A about it. A trivial example may make the point. Let us imagine that x is a man whose ambition to be com-pletely masterful goes to such an excess that he desires to be always angry. This has as a result that x becomes angry if it is said of him that he is not angry, but on the other hand, x is human enough to become pleased (not angry) if it is said of him that he is angry. Then obviously we cannot make of this man x either one or the other of the two state-ments: x is angry; x is not angry.

Another example of the sort of situation which may arise when our terms do not have the proper amount of staticness is afforded by the example of the map of London, made on one of the flag stones of Lon-don itself. The difficulty is that the map itself, being a part of London, must be represented in the map, and this map of the map must again be represented, and so in unterminating progression. The point here is to be sought in the operational meaning of London. “London” is not a fixed and static thing, but itself changes as the action in question progresses, for the London with a map is not the same as the original London without a map. Here, therefore, by employing a non-static, fluent, object instead of the fixed object of Aristotelian convention, we find ourselves involved in an infinite process, which of course cannot be carried out actually, so that the corresponding physical object has no “existence”.

It seems to me that in allowing the most complete generality to the elements which may compose the Menge we are going too far and admitting the possibility of situations like the above. In particular, if we admit as a possible entity the Menge itself, we may easily become involved in situations of this character, as will appear more explicitly


later, and we are therefore going further than we probably wish. At any rate, we must not be surprised if such systems prove self-contradictory.

One might profitably stop for an elaborate analysis of what is in-volved in our resolution of all situations into permanent, unchanging, static, elements. We would find, I believe, not only that such analysis is at the bottom of all communication between persons, but also is necessary to all rational thought itself. Further elaboration and jus-tification of this thesis would carry us much further than we can at-tempt to go, and perhaps beyond our depth altogether. However, the broad facts of the situation are sufficiently obvious, and a very crude statement of them is all that is required here. The permanent, rec-ognizable, identifiable, recurring, elements in our experience may be of the most varied sorts, as material and mental “objects”, or simple operations, or relations between objects and operations. Such things, analyzed out of our experience, are the foundations of our thought, and it is only to such things that Aristotelian logic necessarily ap-plies. In particular, a Menge, defined as broadly as above, may be so variable that the logic is not applicable. If the object does not have this aspect of permanence, Aristotelian logic may nevertheless apply to it in special cases, but a special examination is always necessary to show that the variability is not of such a sort as to vitiate the applica-tion of the logic.

An unsophisticated examination of what we do in analyzing experience into recognizable, identifiable, fixed elements shows at once that these elements do not occur as such, naked, in our di-rect experience, but are our work, our invention. Furthermore, if the analysis is carried far enough, the fixity and permanence of the elements can be recognized to be only an approximation. The sun which rises for me this morning is not exactly the same as the sun which rose yesterday morning, if for no other reason than that today’s phenomenon is the 15347th which I have experienced, whereas yesterday’s was the 15346th. Similarly, no thought ever exactly recurs, but each thought as it occurs carries with it the con-tinually reborn and continually new fringe of all past happenings. It follows that all our processes of thought, which assume an analysis into fixed elements, must be affected with the uncertainty of this analysis, and complete rigor, even in the purely logical processes of mathematics, is unattainable.

The same sort of situation presents itself with respect to those operations which we accept as ultimate and intuitive and in terms of which more complex operations are specified. These ultimate oper-

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ations we treat as frozen, identifiable, repeatable at pleasure. This again is only an approximation.

The question at once forces itself whether the operations which we assume for our purposes to be fixed and static actually have this character in sufficient degree. To this I can conceive of no possible an-swer except that of experience itself; they are “sufficiently” static if we observe that they work that way. Mathematics thus becomes as truly an empirical or experimental a subject as does physics. From this point of view all science reduces simply to recording the observation that certain analyses of ours have in experience shown themselves capable of meeting the demands which we make of them.

Returning now to Huntington’s characterization of the class or Menge, notice that he does not attempt to say what a Menge “is”, but says only that it is “determined” by the tests. This omission is doubtless intentional, in order to avoid the difficulties of infinite Mengen, but other authors do attempt to say what a Menge “is”, and of course in-trinsic difficulties cannot be escaped by such a simple device. A defini-tion given by other authors would run something like this: “A Menge is the aggregate of all the elements which satisfy a certain condition”. Two essentially different situations may arise. The operations involved in applying the conditions may be capable of complete execution, so that the entities which satisfy the condition may be exhaustively cata-logued, as for example, the apples in Massachusetts. Here the aggre-gate of all the apples in Massachusetts is a perfectly definite thing, and the definition of the Menge as the aggregate of all its elements is natu-ral and justifiable. But on the other hand, the operations involved in applying the conditions may be such by construction as to constitute a non-terminating sequence. By this we mean that at any step in the selection of the elements there is no restriction imposed by the condi-tions themselves against taking the next step. The process by which the positive integers are engendered is the example of this sort of thing, par excellence. In such a case the aggregate of all the elements cannot be constructed, and the Menge as the “aggregate of all the elements sat-isfying the condition” is an operationally meaningless combination of words. An essential part of such situations involving non-terminating sequences of operations is to be sought in the rules and operations involved in applying the conditions. I believe that it would often lead to clarification if in such cases the word Menge should be replaced by “program”.

Both sorts of situation, whether leading to finite or to infinite Mengen could be characterized by the operations, so that a Mengen-


lehre would be possible in which the emphasis was laid on the rules, and in which the Menge was defined as a certain program of opera-tions. The historical development, however, has been in the other di-rection. A finite Menge may indifferently be characterized by the rules by which its members are determined or by the members themselves. It is perhaps natural to prefer the characterization in terms of the members for the finite Menge. But this point of view has been car-ried over to the other situation, and the infinite Menge thought of and treated in many respects as having the operational properties of a group of objects. I believe that here will be found many of the difficul-ties of Mengenlehre.

Closely connected with the treatment of an infinite Menge as a “thing” is the meaning to be assigned to the term “all” which fre-quently occurs in Mengenlehre. In the case of finite collections it has meaning operationally to talk about “all” the members of a Menge. The expression “all the apples in Massachusetts” has meaning, and it also has meaning to say that “all the apples in Massachusetts” have some property or other, such, for example, as being contained in North America. The direct operational meaning to be assigned to this statement is as follows: The aggregate of “all the apples in Massachusetts” is to be envisaged as a single collection, or a single thing, and we then state that we can verify by direct observation that this aggregate is in North America. But this same statement as to fact may be verified in another way, namely by establishing that any individual apple in Massachusetts has the property of being in North America. Corresponding to this second method of verifica-tion there is a second operational meaning to be assigned to “all”. The two meanings are equivalent for finite collections, but obviously for infinite collections the only meaning to be assigned to “all” cor-responds to the second situation, where “all” must be rendered by “any”. Often the use of the word “all” with respect to infinite aggre-gates may be quite harmless, in spite of carelessness in the mind of the author himself, for it may often occur that the theorem or proof may be rephrased, replacing the word “all” by “any”. But on the other hand, it seems to me that the distinction is sometimes lost sight of, and that “all” the members of an infinite Menge is sometimes used with operational implications which apply only to finite collections. The matter is a difficult one, and rests on inherent imperfections in language. Thus, the phrase “all the natural numbers” has no meaning, but it may make sense to say “all the natural numbers have this or that property”. If the phrase “all the natural numbers” had meaning

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by itself, then we should be able to say “all the natural numbers has this or that property”. This is a good example of the fact that lan-guage is not composed by the juxtaposition of words or blocks of words having a frozen meaning, but the meaning of a phrase changes with the context in which it finds itself.

(To be concluded.)Harvard University.

*Readers desiring to study Professor Bridgman’s article in toto may obtain a reprint of the total article by applying to the editor including 6c. in stamps for postage.

224

A PHYSICIST’S SECOND REACTION TO MENGENLEHRE

By P. W. BRIDGMAN(Concluded from page 117)

WE are now in a position to deal with what appears to me the most important part of all Mengenlehre, namely the question

of the “number” of points in a line and Cantor’s famous proof that the number of such points is of a “higher order of infinity” than the number of the positive integers. The first point is to declare that there is a one to one correspondence between all the points of a line of unit length, for example, and all the non-terminating decimal fractions less than 1, so that for the purposes of our discussion the one may be replaced by the other. This procedure we shall examine in detail later, but for the present we may accept it. The contention is that it is not possible to arrange all non-terminating decimal fractions less than 1 in sequence one after the other, or in other words, that it is not possible to set up a procedure by which any and every non-terminating decimal may be numbered. The proof involves the celebrated “diagonal Verfahren”, which finds frequent other applications in Mengenlehre. Briefly sum-marized, the proof runs as follows: “Suppose that such a pairing of all the decimal fractions against the integers is possible. Then I undertake to produce a fraction which from its method of construction can have no place in the sequence, thus proving the assumed construction is not possible. Such a fraction is easily described, namely any decimal whose first digit is any number different from the first digit of the first deci-mal in the array, whose second digit is different from the second digit of the second decimal of the array, and so on indefinitely”. When the procedure involved in the proof is examined operationally the impos-sibility of it stares one in the face. For in the first place, what are these non-terminating decimals, which are supposed to be arranged in se-quence, like apples in a row? “Non-terminating” is obviously only a po-lite way of saying “infinite”, and such are not things like apples, for no one can present me with one which I may fit into its place. Operation-ally an infinite decimal means only a program of procedure. It is not possible to actually carry out the operations involved in the diagonal Verfahren; the operations involved in producing the non-terminating decimal cannot be completed, so that it is not legitimate to postulate the performance of other operations, that is arrangement in sequence, after the impossible completion of the non-terminating decimal. A proper

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attention to the meaning of the terms would therefore demand that the proof show that all possible rules by which non-terminating deci-mals can be constructed cannot be arranged in a numbered sequence. But because of the vagueness inherent in “all possible rules”, this would obviously be a matter of great difficulty, and doubtless accounts for the turn which the proof actually takes. We return presently to this matter of classifying all possible rules. In lieu of this, the proof treats the non-terminating decimal as an ordinary object. Now there is an operational technique by which such an infinite object is replaced by a finite object; this is implicit in all work on infinite series, and so far as I know this is the only such operational technique. Namely, any non-terminating decimal is defined and determined and treatable as an ordinary finite thing if its first n digits can be written down, no matter how large the number n or by what process arrived at. If a non-termi-nating decimal is to be handled or arranged in sequence like a thing it is sufficient to know how to handle and arrange a finite decimal of n digits, the number n being subject to no restriction as to magnitude. The theorem would now demand that it is impossible to set up any scheme for arranging all possible decimal fractions of n digits in a definite order, n being subject to no restriction as to magnitude. But such a theorem is obviously false, for there are not more than 10n pos-sible decimals of n digits, so that, no matter how large the number n, I can set up a scheme by which any possible decimal of n digits will be found to correspond to some number less than 10n. What is done in the actual diagonal Verfahren when translated into this technique is this: it is shown that given a proposed array and any number n, no matter how large, it is then possible to set up a decimal the first n digits of which are different from the first n digits of any decimal to be found in the first n places of the proposed array. But this is clearly not what is required.

The ordinary diagonal Verfahren I believe to involve a patent con-fusion of the program and object aspects of the decimal fraction, which must be apparent to any one who imagines himself actually carrying out the operations demanded in the proof. In fact, I find it difficult to understand how such a situation should have been capable of persist-ing in mathematics. Doubtless the confusion is bound up with notions of existence; the decimal fractions are supposed to “exist” whether they can be actually produced and exhibited or not. But from the opera-tional point of view all such notions of “existence” must be judged to be obscured with a thick metaphysical haze, and to be absolutely meaningless from the point of view of those restricted oper-


ations which can be allowed in mathematical inquiry.This repudiation of the conventional proof by the diagonal Ver-

fahren of the non-denumerability of the non-terminating decimals will be found to be very similar in spirit, although not in detail, to the argument in Bentley’s book. It may be worth while to record that the argument above was reached by me independently of Bentley, and in fact without any clear consciousness on my part as to what his position in this matter was.

A repudiation of the proof may not affect the fact. Is it possible to approach the matter from what appears to me a rigorous point of view and show that it is not possible to correlate with the natural numbers all possible rules for producing non-terminating decimals? This I be-lieve to be certainly impossible at present. For it is in the first place ob-vious that a very elaborate preliminary study of all possible methods of formulating such rules would be necessary. There are great difficulties here, for rules which are couched in different language but give rise to the same sequence of figures must be called one rule, or otherwise the situation could be made entirely chaotic by adding irrelevant matter at the end of that part of the rule which determines the sequence of figures. Furthermore, any analysis of the rules into words or symbols would at once run into difficulties entirely unresolved at present, for such words do not have the necessary Aristotelian fixity, but change with the setting, as will be emphasized later. It seems to me a very plausible expectation, however, that the very act of classifying the rules in such a form as to allow operational meaning to be assigned to the expression “all possible rules” will contain within itself mate-rial which will make possible a denumeration. A suggestion is con-tained in the proof that all the algebraic numbers are denumerable. For to say that a number is algebraic, that is, the root of a properly characterized algebraic equation, is simply to specify a certain very broad group of rules for producing a non-terminating decimal, and when the specification of the nature of the rules is made as definite as this, the specification itself is found to contain the material for the denumeration.

One can obviously say that all the rules for writing down non-terminating decimals formulatable by the entire human race up to any epoch in the future must be denumerable, for any such rule requires a finite number of symbols and requires a finite time, entirely neglect-ing any considerations of non-uniqueness in the meaning of the sym-bols, and therefore the total number of such rules must be finite. To go further, and talk or think about an “existence” of such decimals in

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their own right seems to me absolutely meaningless. I do not know what it means to talk of numbers existing independent of the rules by which they are determined; operationally there is nothing correspond-ing to the concept. If it means anything to talk about the existence of numbers, then there must be operations for determining whether alleged numbers exist or not, and in testing the existence of a number how shall it be identified except by means of the rules?

What does it mean to say that there is a number, or that a num-ber “exists” corresponding to such and such a rule for writing down a non-terminating decimal? or in greater generality, what does it mean to say that a limit “exists” of a convergent series? One may of course make this a matter of definition, and say that by definition any deci-mal determined by such a rule is a number. But this is unsatisfactory, for it is contrary to the spirit of the question. The question evidently implies some real content; if the rule determining the non-terminating decimal determines a number, then there must be some operation by which it can be proved that something defined to be a number by some other method than the decimal method is equivalent to the decimal, equivalent being defined below. For instance, the decimal 0.333 . . . is the decimal determined by the rules for converting 1/3 into a decimal fraction, 1/3 having already been defined as a number. Similarly the se-quence 3.14159265358979323846 . . . is determined by a transcription into decimal terms of π, the ratio of circumference to diameter of a circle, by definition a number. The meaning of “equivalent” above is the ordinary calculus meaning: given a number ϵ no matter how small, then I can assign a number n such that the number given by the first n terms of the non-terminating decimal differs from the number equivalent to the whole non-terminating decimal by less than ϵ. So in the general case; it means nothing to say that any non-terminating decimal is a number unless this decimal can be shown to be equiva-lent, by an e or other proof, to something independently known to be a number. (This equivalent number is usually directly defined in terms of a finite process.) But how in the example above do I know that π is a number? The answer would seem to be simply one of con-venience; it is convenient to define as numbers, (because they handle like numbers) the ratios of all lines which can be specified in finite terms (integrations count for this purpose as finite processes because they are capable of finite definition) by any known geometrical con-struction.

This introduces at once the connection between the points on a line and the non-terminating decimals. It is said that there is a one to one


correspondence between all the points of a line of unit length and all non-terminating decimals less than unity. What does this mean? An approximate meaning is obvious enough. Given any terminat-ing decimal, of no matter how many digits, then we can find by a perfectly definite geometrical procedure a unique corresponding point on the line. If one wants to know how I am sure the point corresponding to this construction exists, I believe the only answer is, “It exists by definition”. It would be silly to say that I know the point exists because I can reach it by actual construction with ruler and compass—how far may construction with a compass “actually” be carried and what is the connection between an “actual” compass and geometry? But what about the converse; how shall we show that given any point on the line we can approximate as closely as we please to it by a variable point, itself determined by a terminating decimal of a continually increasing number of digits? The crux of the whole situation is contained in the expression “any point”. What is the operational meaning of point? Suppose someone presents me with what he purports to be a point or the line—how shall I check up to see whether he is telling me the truth or not? If I cannot find at least as much operational meaning as this in the word point, I think it will be agreed that it is not the sort of thing I want to play with, and for my purposes must be meaningless. But the mere act of saying that the alleged thing is a point involves at least some way of talking about it and therefore identifying it. If point is to mean anything, it must be identifiable, and this involves some operation or procedure for describing it. The simplest method perhaps is to specify its distance from one end of the line. This may be given in terms of terminating decimals or in terms of other things defined as numbers, such as the rationals. But there are other purely geo-metrical procedures possible. First there are Euclidean procedures expressible in terms of compass and rule; we can add to these other procedures involving the intersections of algebraic curves of any orders. All such procedures will obviously give us algebraic points. We can add other procedures, corresponding to integrations, and involving lengths of arcs or areas, thus introducing various sorts of transcendentals. But in any event, “point” has no meaning unless it is defined, and this involves the specification of some sort of proce-dure. “All the points of a line” as a purely intuitional concept apart from the rules by which the points are determined, can have no operational meaning, and accordingly is to be held for mathematics an entirely meaningless concept.

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A line is not composed of points as the forest is composed of trees, nor may a line be produced by putting together “all” the points in it. Points may be “determined” on a line, and the determi-nation involves an operation according to a rule of some sort. “All the points of a line” means no more than “All the rules for determining points on a line”, and “all” in this latter expression means no more than the “all” in “all the rules for determining non-terminating decimals”. In other words, we have no more reason to describe the points on a line as non-denumerable than the non-terminating decimals. The re-pudiation of the diagonal Verfahren for the decimals at the same time removes all reason for thinking the points on a line non-denumerable.

In fact, a consistent application of the operational criterion of meaning appears to demand the complete discard of the notion of infinities of different orders. We never have “actual” infinities, a favorite expression in Fraenkel’s book, but only rules of operation which are not self-terminating. How can there be different sorts of non-self-terminatingness? At any stage in the process the rule either permits us to go on and take the next step or it does not (our formulation of the rule in straightforward operational terms en-sures this), and that is all there is to it. This is not to deny that there may be various characteristics of different sorts of rules specifying non-terminating procedures which may profitably be the object of study, but under an operational technique I can see no place for the sort of differences that are supposed to be typified by the 2א ,1א ,0א, etc., of Cantor.

Finally, we consider what account the operational method of ap-proach gives of some of the well known paradoxes of Mengenlehre. A number of these are given in Fraenkel’s book; most of them will be found to involve the use of terms with fluent or evolving meanings. For instance, there is the Russell paradox, namely the Menge of all the Mengen which do not include themselves as elements. This is at once seen to be a self-contradictory concept, for obviously the Menge cannot contain itself as an element, without violating the fundamen-tal condition, neither can it fail to include itself as an element, for it would thereby become by definition such a Menge that it must include itself as element. The paradox is at once seen to be avoided under the restrictions which we have proposed as to permissible operations for determining the Menge, for we see that it is not possible to give a straightforward operational account of such a Menge. Imagine our-selves in the act of trying to decide whether the Menge should be included in itself as one of its own elements. At this stage of the


process the elements are not all selected, and therefore the Menge is not completed. But to know whether to include the Menge as an ele-ment or not involves the completion of the Menge. That is, it is not possible to give a straightforward operational procedure for deciding whether to include the Menge as one of its own elements or not. We have seen that paradoxical situations are to be expected whenever the operation must be specified from the standpoint of itself regarded as completed, that is, in terms of its properties.

There is another difficulty in the situation presented by the Russell paradox. What sort of a thing is the Menge which we are examining to determine whether it includes itself or not? It is established that no finite Menge can include itself as element. The Menge must therefore be infinite, which means that it is not an ordinary thing, but is merely the rule for selecting the elements. But if the Menge may be an ele-ment of itself, then the rule must be applicable to itself, which is a rule. That is, the rule must be a rule acting on a rule. But this again is an element of itself, that is, subject to the action of the rule, so that the rule must be a rule acting on a rule which acts on a rule. And so on in infinite succession. Now it is not evident that a rule defined in finite terms can be found equivalent to the rule defined in this way in terms of an infinite series of self-reflexive terms. This must be recognized to be a particularly aggravated way of defining a thing by its properties, and until this equivalence is established we have no right to a Menge presupposing an operation of this kind.

The objection may be put in another form. We have seen that the only possible Menge satisfying the Russell specification must be an infinite Menge, which means an infinite series of operations. But we have already seen that an infinite series of operations must be the last in our program; it is not allowed to consider that the infinite series can be completed and that we then go on and perform other opera-tions depending on the completion of the infinite set. But if “Menge of all Mengen which do not include themselves as elements” is to have operational meaning, there must be some operation for determining whether a Menge which purports to be one of this sort is or is not; that is, the operations by which the concept acquires meaning must be performed after the completion of a non-terminating series of opera-tions, or in other words, the situation is operationally impossible and therefore the concept is meaningless.

The next example in Fraenkel is Russell’s “Class of all thinkable classes”. This is at once shown to be a self contradictory concept by means of the theorem that the class of all the parts of a class is itself

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a class greater than the original class. Then the class of all parts of the class of all thinkable classes is itself greater than the class of all thinkable classes, but is in itself thinkable, as shown by the fact that I am thinking about it. The catch here is that the words are not used with the fixed meanings demanded in all logical formulations free from self-contradiction. The offending term is evidently “think-able”. The logical conclusion to be drawn above is that the class of all the parts of all thinkable classes is not itself a thinkable class. Such is indeed true with the original significance of “thinkable”. But the significance of “thinkable” changes in the using of it, for having once thought about something, I may now think about my thought of it, and so on with continually changing and growing operational mean-ing. But our fundamental principle requires the ruthless elimination of such terms.

Terms with fluent operational meanings should be specified with a date. A dangerous feature is that a term which is obviously fluent when directly described in terms of ultimate operations may appear static when described in terms of its properties, as “thinkable”, for example.

Analogous considerations apply to many of the paradoxes aris-ing from counting the symbols required to define a concept. For instance, the paradox of Richards, also described by Fraenkel, of all the numbers describable with the use of less than 1000 letters of the English alphabet. There are obviously a finite number of such combinations of 1000 letters, and therefore a finite number of such numbers. Write down the number larger by unity than the largest of these numbers. This is not a number included in my set, but nev-ertheless a number defined by less than 1000 letters, as one sees by counting the letters in the last sentence—hence the paradox. The term with fluent meaning here is “definable with the use of 1000 letters”, for the number “defined” by a combination of letters is not uniquely determined by that definition, but by the entire context with which the definition appears. The same combination of letters may determine an indefinitely valued set of numbers, as, for ex-ample, the number defined by the following letters: “Multiply by 10 the number I have just written down”. Strictly speaking, any term de-fined with fixed words or symbols is fluent and depends on the whole context of the definition, and in particular, on the entire past history of the individual using the symbols. That we can nevertheless by skill and practise achieve sufficient fixity in our terms for our practical pur-poses is most fortunate, but we should not allow this to obscure our


expectation of trouble when the definition is maliciously so arranged as to exploit its fluent features, as in the example above.

Another example, depending in a slightly different way on the flu-ent meaning of terms, is the famous paradox embodied in the state-ment “Epimenides, the Cretan, says that all Cretans are liars”. From the operational point of view the crux of the matter is not in the self-reflexive form of the statement, namely that a Cretan makes a state-ment about all Cretans. It rather lies in the operational meaning of the statement “all Cretans are liars”. Operationally this demands that the statements of all Cretans are to be examined, and, if found false, then the statement is true. But the examination of statements of all Cretans has to be made at some instant of time, so that all that can be attained is a statement as to Cretan veracity with a date attached. If Epimenides had said “All Cretans were liars when I began mak-ing this statement”, the paradox disappears. The paradox here arises evidently because the operations are not subject to the restriction as to straightforward description and step by step performance which we must demand as fundamental. The statement in its original form is operationally meaningless. This paradox is essentially the same as that of the barber already considered.

What is the practical mathematician going to say to all this skep-ticism and repudiation? Will he not want to be shown how it is that Mengenlehre has attained the important results that it has outside its own field? To this one may reply in the first place that the attain-ment of results is no guarantee of the correctness of the process, and it may be a difficult matter to pick out just the features which are responsible for particular successes of a complicated process, some parts of which are not above reproach. It seems to me that a simi-lar situation presents itself with respect to Einstein’s generalized, as opposed to his special, theory of relativity. Further than this I can give only a very incomplete answer to the question because I am not mathematician enough to know what these important results are, but it does seem to me that the few results mentioned in the elementary sources of my information can be readily dealt with. Thus, Fraen-kel lays emphasis on the fact that Mengenlehre gives an off-hand proof of the fact that there are other numbers than the rationale, because the rationals are denumerable, whereas all the numbers have the “Mächtigkeit” of the points on a line, which are non-denumerable, a fact so astounding and so contrary to current notions with religious implications, that it is said to have cost the Greek who first established it his life. But it does not follow that because the fact is true the proof

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is correct, and as has already appeared, it seems to me that the proof must be rejected as fallacious. I believe, however, that the mere ex-istence of non-rationals can be established with a turn of the hand, once willing to define any non-terminating decimal as a number, a privilege assumed in the Mengenlehre proof. For all rational numbers when expressed as decimals eventually become repeating. There are, therefore, non-rationals if it can be shown that rules can be set up for writing down non-terminating decimals which certainly can never become repeating. The simplest of examples is sufficient to show such possibilities, as for example: 0.1121231234 etc. To go further and show that the number determined by any specified geometrical procedure, as for example √2, is an irrational, of course requires special examination. But it is also beyond the powers of Mengenlehre to show that any specified number, deter-mined by a specified procedure, is irrational.

Mengenlehre is similarly supposed to have established the existence of transcendentals by showing that all algebraic numbers are denu-merable. This proof I would reject, holding that the mere act of assign-ing operational meaning to the transcendentals of itself ensures that they are denumerable. As a matter of fact, only a few transcendentals have been established. Mengenlehre is powerless to show whether any given number, such as e or π, is transcendental or not, and the detailed analysis necessary in any given case for establishing transcendence is not avoided by Mengenlehre. From the operational point of view a transcendental is determined by a program of procedure of some sort; Mengenlehre has nothing to add to the situation. And this, as far as my elementary reading goes, exhausts the contributions which Men-genlehre has made in other fields.

Finally, we may briefly summarize as follows: Mathematics has meaning only in so far as it is amenable to the same sort of an opera-tion control that has been found necessary in physics. If it is agreed to admit only those operations in mathematics which can be actu-ally carried out, then an operational description of any mathematical procedure becomes a description of an actual experience, and as such must have the freedom from contradiction of all experience. To en-sure that an operation can be actually carried out, it must be capable of straightforward specification in terms of other operations accept-able as simpler, the final reduction being to operations intuitively acceptable as ultimate and unanalyzable. A specification of an opera-tion in terms of its properties is not a straightforward specification; it conceals latent impossibilities of performance, which may lead to


paradox and self-contradiction. Neither is a specification straightfor-ward which is framed in terms of operations with “fluent” meanings. Aristotelian logic and the ordinary forms of thought demand the use of “static” terms. If such terms are not used, paradox is to be expected. In addition to these requirements the permissible operations are sub-ject to restrictions as to order, for “operations actually carried out” are carried out in time.

Mengenlehre is found not to involve only operations subject to the requirements just enumerated. Use only of this sort of operation leads to the rejection of the “diagonal Verfahren” for proving the non-denumerability of the points on a line, to rejection of the concept of non-denumerable infinities as opposed to denumerable infinities, and to rejection of the entire hierarchy of infinities of different orders. A few examples are considered in detail showing how paradox disappears in this operational setting.

Harvard University

A physicist's second reaction to Mengenlehre

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