Is a Universal

Is a Universal Nocturnal Expansion Falsifiable or Physically Vacuous?Author(s): Adolf GrünbaumReviewed work(s):Source: Philosophical Studies: An International Journal for Philosophy in the AnalyticTradition, Vol. 15, No. 5 (Oct., 1964), pp. 71-79Published by: SpringerStable URL: http://www.jstor.org/stable/4318481 .

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UNIVERSAL NOCTURNAL EXPANSION 71

a result of the insertion of a colored filter in the projector. In the real world such distinction between types of events does of course not exist. All events are of the representative kind; they reflect real events themselves. When the sentence "Overnight everything has increased in size" is not expressing a meaningful statement, then the term "size" in it has been illegitimately used in such a way that it does not apply to objects of the universe that are not images of something else. When, however, the term "size" is employed in a legitimate way, then the statement is verifiable.

Received June 3, 1963

NOTES 'In reviewing Passmore's "Philosophical Reasoning" (Philosophical Books, April

1962, p. 18) he argues that obviously there are statements which are logically unverifiable and cites "Everything has just doubled in size" as his example.

2 In his "Geometry, Chronometry, and Empiricism" (in Herbert Feigl and Grover Maxwell, eds., Minnesota Studies in the Philosophy of Science, Vol. III (Minneapolis: University of Minnesota Press, 1962), p. 429) he says, ". . . the utter vacuousness of the following assertion is evident at once: overnight everything has expanded (i.e., increased its length) but such that all length ratios remained unaltered."

Is a Universal Nocturnal Expansion FalsiJiable or Physically Vacuous?

by ADOLF GRftNBAUM

UNIVERSITY OF PITTSBURGH

G. SCHLESINGER has maintained that "a set of circumstances can be con- ceived under which one would have to conclude that overnight everything has doubled in size" and that "in the absence of these circumstances we are entitled to claim that it is false that overnight everything has doubled in size."' And Schlesinger went on to contend that the only way of actually rendering the hypothesis of nocturnal doubling physically vacuous is as follows: construe this hypothesis so as to make it a singular instance of the tautology that no unverifiable proposition can be verifiable, a triviality which does not qualify as an interesting ascription of physical vacuousness to the particular hypothesis of nocturnal doubling.

In order to appraise Schlesinger's contention, we must note at the outset that he takes quite insufficient cognizance of two relevant facts: (1) the hypothesis of nocturnal doubling-hereafter called the "ND-hypothesis"-



72 PHILOSOPHICAL STUDIES

can be construed in several different ways, and (2) the charge of physical meaninglessness, inherent non-falsifiability, and unverifiability has been leveled against the ND-hypothesis in the philosophical literature on a number of quite distinct grounds.

For my part, I have always construed the ND-hypothesis as being predicated on the kind of conception of spatial and temporal congruence which is set forth by Newton in the Scholium of his Principia.2 On this kind of conception, container-space (and, mutatis mutandis, container-time) exhibits congruence relations which are intrinsic in the following sense: its structure is such that the existence-as distinct from the epistemic ascertainment-of congruence relations between non-overlapping intervals does not depend at all on the transport of any kind of material congruence standard and therefore cannot depend in any way on the particular behavior of any kind of physical object under transport. This transport-independence of all congruence relations existing among intervals in principle permits the invocation of these intrinsic congruences to authenticate as a congruence standard any standard which is "operational" in the following special sense: the standard's performance of its metrical function is transport-dependent either because the standard is itself a transported physical object or because its application depends logically in some way on physical transport of a body or bodies. And, by the same token, once a unit of length is chosen, metrical equality and inequality of non-overlapping intervals could be authenticated on the basis of the intrinsic congruences. Given the context of the New- tonian assumption of the existence of a container-space having intrinsic congruences between different intervals at the same time and between a given interval and itself at different times, it was therefore possible to construe the ND-hypothesis as asserting the following: there has been a nocturnal doubling of all extended physical objects contained in space, and also a doubling of all "operational," i.e., transport-dependent, congruence standards. Because of the legacy of Newton's metrical philosophy, it was this non-trivial version of the ND-hypothesis which I appraised logically in my recent book.3 And when I denied the falsifiability of this interpretation of the hypothesis, I did not rest my denial on Schlesinger's triviality that no unverifiable proposition is verifiable; instead, my denial was based on the following weighty presumed fact: the actual failure of physical space to possess the kind of structure which is endowed with intrinsic congruences. For this (presumed) failure deprives space of the very type of metric on whose existence the ND-hypothesis depends for its physical significance and hence falsifiability. But, one might ask: is it logically possible that there be a world whose space does have a structure which exhibits intrinsic congruences?




I shall answer this question in the affirmative by giving an adumbration of such a world. First I wish to point out, however, that Newtonian mechanics and the nineteenth-century aether theory of light propagation both cut the ground from under their own thesis that their absolute space possesses intrinsic congruences, and they did so by postulating the mathematical continuity of their absolute space. For, as Riemann pointed out in his In- augural Lecture, if physical space is mathematically continuous, the very existence and not merely the epistemic ascertainment of congruences among its intervals is transport-dependent, and nothing in the continuous structure of space can vouch for the rigidity (self-congruence) of the congruence standard under transport.4 Thus, if physical space is continuous, it is devoid of intrinsic congruences, since the very existence of its congruences derives solely from the behavior of conventional standards all of which are "operational" in the special sense noted above: the performance of the metrical function by each and every standard is transport-dependent, because either the congruence standard is itself a transported body or its application depends logically on the physical transport of a body or bodies. This Rie- mannian account of the status of the metric in continuous physical space can be summarized by saying that this space itself is metrically amorphous. Given, therefore, that the continuous absolute space of Newtonian mechanics and nineteenth-century wave optics cannot be adduced to prove the logical possibility of a space endowed with intrinsic congruences, can a kind of world be imagined which is such as to establish this logical possibility?

In now confronting this question, rigorous care must be exercised lest the challenge to furnish this proof of logical possibility be turned into the following very different second challenge: to specify empirical conditions which would justify that we reinterpret our present information about the physical space of our actual world so as to attribute a structure endowed with intrinsic congruences to that space. This second challenge has to be faced not by my impending characterization of an appropriate imaginary world, but rather by contemporary speculations of space and time quantiza- tion going back to Democritos' mathematical atomism, a kind of atomism which must be distinguished from his much better known physical atomism.5 Moreover, we must guard against being victimized by the tacit acceptance of a challenger's implicit question-begging requirement-his implicit de- mand that the spatial vocabulary in which I couch my compliance with the first of these challenges be assigned meanings which are predicated on the assumption of spatial continuity which is ingredient in present-day confirmed physical theories. For the acceptance of that question-begging requirement would indeed import into the description of the sought-after imaginary world inconsistencies which could then be adduced to maintain




that a physical space endowed with intrinsic congruences is logically impossible. And this conclusion would then, in turn, enable Schlesinger to uphold his contention that my version of the ND-hypothesis cannot be falsi- fied in any logically possible world and hence is only trivially meaningless.

Having cautioned against these pitfalls, I now maintain that just as the three-dimensionality of actual physical space cannot be vouchsafed on a priori grounds, so also it cannot be guaranteed a priori that in every kind of spatial world the congruences would be transport-dependent rather than intrinsic. For imagine a world whose space has elements (space-atoms, lumps, or granules) such that (1) for any one space-atom, there are a fixed finite number of space-atoms which are next to it or contiguous with it (the magnitude of this finite number will determine the granulo-"dimensionality" of the atomic space by being two in the case of one-space, eight for two- space, etc.), (2) there are only a finite (though large) number of distinct space-atoms all told, (3) there is no physical foundation whatever for mathematically attributing proper (non-empty) parts to the space-atoms, as attested by the existence of only finitely many rest positions for any object. It is clear that the structure of this granular space is such as to endow it with a transport-independent metric: the congruences and metrical attributes of "intervals" are intrinsic, being based on the cardinal number of space-atoms, although it is, of course, trivially possible to introduce various other units each of which is some fixed multiple of one space-atom. To be an extended object in this kind of granular space is to comprise more than one space- atom. And the metric intrinsic to the space permits the factual determina- tion of the rigidity under transport of any object which is thereby to qualify as an "operational" congruence standard. By the same token, the sudden nocturnal doubling of all "operational" (i.e., transport-dependent) metric standards along with all extended physical objects could be said to obtain in such a granular space, if on some morning they each suddenly occupied or corresponded to twice as many space-atoms as before. Accordingly, the ND-hypothesis formulated above is indeed falsifiable in the context of an atomic theory of space. But it is not falsifiable within the framework of any of the confirmed modern physical theories pertaining to our actual world: their affirmation of the mathematical continuity of physical space precludes that there be an intrinsic kind of metric, which is the very kind on whose existence the ND-hypothesis depends for its physical significance and hence falsifiability.

Ellis and Schlesinger mistakenly believe they have corrected all previous treatments of the ND-hypothesis in the philosophical literature, because they have completely overlooked a fundamental logical fact which is an immediate corollary of my Riemannian justification for making alternative




choices from among incompatible spatial and temporal congruences.6 This logical fact is that Riemann's doctrine of the intrinsic metrical amorphousness of the spatial continuum permits the deduction both of the non-falsifiability of the ND-hypothesis and of the falsifiability of a certain modified version of the ND-hypothesis which I shall call the "ND-conjecture." The ND-conjecture differs from the ND-hypothesis by not attributing the doubling to all operational standards of length measurement but only to one of them while exempting the others from the doubling. In order to demonstrate the falsifiability of the ND-conjecture on the basis of Riemann's metrical doctrine, I must first clarify the logical relation of that doctrine to the view that the physical concepts of congruence and of length are each multiple- criteria concepts or open cluster concepts rather than single-criterion concepts.

Riemann's thesis that the spatial and temporal continua are both intrin- sically amorphous metrically sanctions a choice among alternative metriza- tions of these continua corresponding to incompatible congruence classes of intervals and thus to incompatible congruence relations. By thus es- pousing the conventionality both of the self-congruence of a given interval at different times and of the equality of two different intervals at the same time, Riemann denies that the existence or nonexistence of these congruence relations is a matter of factual truth or falsity. But clearly, Riemann's conception of congruence fully allows that any one set of congruence relations be specified physically by each of several different operational criteria rather than by merely one such criterion! For example, this conception countenances the specification of one and the same congruence class of intervals in inertial systems by the equality of the travel times required by light to traverse these intervals no less than by the possibility of their coincidence with an unperturbed transported solid rod. Thus, as I have pointed out else- where,7 Riemann's conventionalist conception allows that congruence (and length) is an open, multiple-criteria concept in the following sense: no one physical criterion, such as the one based on the solid rod, can exhaustively render the actual and potential physical meaning of congruence or length in physics, there being a potentially growing multiplicity of compatible physical criteria rather than only one criterion by which any one spatial congruence class can be specified to the exclusion of every other congruence class. And hence the conventionalist conception of congruence is not at all com- mitted to the crude operationist claim that some one physical criterion renders "the meaning" of spatial congruence in physical theory.

A whole cluster of physical congruence criteria can be accepted by the proponent of Riemann's conventionalist view during a given time period, provided that the members of this cluster form a compatible family during




this time period in the following twofold sense: (1) the various criteria yield concordant findings during the time period in question as to the self- congruence or rigidity of any one interval at different times, and (2) one and the same congruence class of separated spatial intervals must be specified by each member of the cluster of congruence criteria. It is a matter of empirical fact whether the concordance required for the compatibility and interchangeable use of the members of the cluster obtains during a given time period or not.

Now imagine a hypothetical empirical situation relevant to the falsifiability of the ND-conjecture: up to one fine morning, there has been concordance among the various congruence criteria in regard to the self-congruence of particular space intervals in time, but after that particular morning, that concordance gives way to the following kind of discordance: intervals which, in the metric furnished by metersticks, remain spatially con- gruent to those states of themselves that antedated the momentous morning thereafter have twice their pre-morning sizes in the metric furnished by all the other congruence (length) criteria, such as the travel times of light. In that case, the requirement of logical consistency precludes our continuing to use metersticks interchangeably with the other congruence criteria after the fateful morning to certify continuing self-congruence in time. And it is perfectly clear that in the hypothetical eventuality of a sudden discordance between previously concordant criteria of self-congruence, the thesis of the intrinsic metrical amorphousness of the spatial continuum sanctions our choosing either one of the two incompatible congruences as follows: either we discard the meterstick from the cluster of criteria for self-congruence of space intervals in time or we retain the meterstick as a criterion of self-congruence to the exclusion of all the other criteria which had been interchangeable with it prior to the fateful morning.

The two alternatives of this choice give rise to the following two physical descriptions of nature, the first of which states the ND-conjecture: Description I: Since the fateful morning, all metersticks and extended

bodies have doubled in size relatively to all the other length criteria be- longing to the original cluster, which are being retained after the fateful morning, and the laws of nature are unchanged, provided that the lengths, 1, ingredient in these laws are understood as referring to the latter retained length criteria;

or Description II: Relatively to the retention of the meterstick as a length

standard after the fateful morning and the discarding of all other previously used length criteria, there is the following unitary change in those




functional dependencies or laws of nature which involve variables ranging over lengths: all magnitudes which were correlated with and hence meas- ures of a length 1 prior to the fateful morning will be functionally correlated with the new length 1/2 after the fateful morning. Given that the lengths symbolized by "1" are referred to and measured by metersticks after the fateful morning no less than before, this mathematical change in the equations means physically that after the fateful morning, the same lengths will correspond to twice the previous values of other magnitudes.

A number of important points need to be noted regarding the two descriptions I and II.

1. Descriptions I and II are logically equivalent in the context of the physical theory in which they function. And, contrary to Schlesinger's alle- gation, they incontestably have the same explanatory and predictive import or scientific legitimacy. Descriptions I and II both enunciate an unexplained regularity in the form of a time-dependence: I asserts the unexplained new fact that all metersticks have expanded by doubling, while II enunciates an unexplained change in a certain set of equations or laws according to a unitary principle.

2. Schlesinger mistakenly believesi he has shown on methodological grounds that the doubling of the metersticks asserted by I is the uniquely correct account as against the change in the laws of nature affirmed by II. Specifically, he supposes that he can disqualify II as a valid equivalent alternative to I on the grounds that I explains all the phenomena under con- sideration by the single assumption of doubling, whereas II allegedly can claim no comparable methodological merit but is a linguistically unfortunate futile attempt to evade admitting I. And he derives this supposition from his belief that in Description II, the laws describing nature after the fateful morning can be derived from the previously valid different laws by the mathematical device of replacing the lengths I in the original laws by 1/2. But even if this belief were correct, the error of Schlesinger's inference lies in having misconstrued the mere computational stratagem of replacing 1 by 1/2 as a warrant for claiming that the only physically admissible account of the phenomena is that all metersticks have doubled. For he overlooks the fact that the alternative metrizability of the spatial continuum allows us to retain the meterstick as our congruence standard after the fateful morning to the exclusion of the other criteria in the original cluster, no less than it alternatively permits us to preserve the others while jettisoning the meterstick. Thus, Description II is based on the convention that the meterstick does not change but is itself the standard to which changes are referred,




since the lengths I are measured by the meterstick after the fateful morning just as they were before then. And, in any case, the replacement of "1" by "1/2" can yield the new equations needed in Description II only in those cases in which the dependence on "I" is linear.

In this connection, I should remark that I find Schlesinger's own compu- tations misguided. What is at issue here in the context of the assumed continuity of physical space is only the logical rather than the physical possibility of an observational falsification (refutation) of the ND-conjecture. It therefore strikes me as ill-advised on Schlesinger's part to use the actual known laws of nature (e.g., the conservation laws for angular momentum, etc.) as a basis for calculating whether the hypothetical changes postulated by the ND-conjecture would, be accompanied by other compensatory changes so as to render the detection of the hypothetical changes physically impossible. All that is at issue is whether we can specify logically possible observable occurrences relevant to the falsifiability of the ND-conjecture, not the compatibility of such occurrences with the time-invariance of the actual laws of nature.

3. The sudden discordance between previously concordant criteria of self-congruence in time is indeed falsifiable. Being a proponent of Riemann's view of congruence, I do countenance Description I no less than Description II to formulate that hypothetical discordance. And I therefore maintain that if the language of Description I is used, the falsifiability of the discordance assures the falsifiability of the ND-conjecture. By contrast, suppose that a narrowly operationist defender of the single criterion conception of length were to declare that he is wedded to the meterstick once and for all even in the event of our hypothetical discordance. The putative defender of such a position would then have to insist on Description II while rejecting I as meaningless, thereby denying the falsifiability of the ND-conjecture.

4. Schlesinger is not at all entitled to claim that II is false while I is true on the basis of the following argument: if one were to countenance II as veridically equivalent to I as a description of our hypothetical nocturnal changes, then one would have no reason to reject analogously equivalent descriptions of phenomena of temperature change and of other kinds of change. This argument will not do. For suppose that one could construct analogues of my Riemannian justification of the equivalence of Descriptions I and II and thus could show that temperature changes also lend themselves to equivalent descriptions. How could this result then serve to detract from either the truth or the interest of my contention that II is no less true than I?

As is shown by the treatment of nocturnal expansion in my book,8 I have not denied the physical meaningfulness of the kind of doubling (expansion) which results from a sudden discordance between previously concordant




criteria of self-congruence in time. And it is puzzling to me that Reichen- bach's writings and mine both failed to convey our awareness to Ellis and Schlesinger that the very considerations which serve to exhibit the physical vacuousness of the ND-hypothesis also allow the deduction of the falsifiability of the ND-conjecture. It is with regard to the ND-hypothesis but not with respect to the ND-conjecture that I have maintained and continue to maintain the following: on the assumption of the continuity of physical space, this hypothesis can have no more empirical import than a trivial change of units consisting of abandoning the Paris meter in favor of a new unit, to be called a "napoleon," which corresponds to a semi-meter.

The views which I attribute to G. Schlesinger in this paper are not found in their entirety in his own paper but were partly set forth by him in oral discussion at the meeting of Section L of the AAAS in December 1963 and also in the department colloquium of the University of Pittsburgh. The stimulus of Schlesinger's remarks was very helpful to me in articulating my views. I wish to thank Professor Wesley C. Salmon for useful comments on matters of exposition.

Received March 20, 1964 NOTES

'G. Schlesinger, "It Is False That Ovemight Everything Has Doubled in Size," Philosophical Studies, this issue. Similar views were put forward earlier by B. Ellis' "Uni- versal and Differential Forces," British Journal for the Philosophy of Science, 14:189-93 (1963).

2 See the edition by Florian Cajori (Berkeley: University of Califomia Press, 1947), pp. 6-12.

'Philosophical Problems of Space and Time (New York: Knopf, 1963), Chapter I, especially pp. 42-43.

' For details and references, see ibid., pp. 8-16, 45, and 404-5. I am indebted to Mr. Peter Woodruff for pointing out to me that Riemann's insight was anticipated by D. Hume in his Treatise of Human Nature, Part II, Section IV. It should be noted, however, that Riemann's thesis of the intrinsic metrical amorphousness of the continua of space and time does not apply to all kinds of continuous manifolds; cf. Philosophical Problems of Space and Time, pp. 16-18.

For details on this distinction, see S. Luria, "Die Infinitesimaltheorie der antiken Atomisten," Quellen und Studien zur Geschichte der Mathematik, Astronomie, und Physik (Berlin, 1933), Abteilung B: Studien, II. Luria's reading of Democritos has been contested by Gregory Vlastos in a forthcoming publication.

6 See Griinbaum, Philosophical Problems of Space and Time, Chapter 2, pp. 22-23, and Chapter 3, part B.

'See ibid., pp. 14-15. 8 See ibid., pp. 42-43.

Books Received ABBOTT, EDWIN A. Flatland. New York: Barnes and Noble, 1963. $1.00.



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Is a Universal