Abstract:            The Logical Problem with Cosmological Measurements

 

 

 

Alfred North Whitehead noted that the theory of general relativity collapsed the necessary relations of geometry into the contingent relations of physics, thus rendering the possibility of meaningful cosmological measurements dubious if not impossible. By rendering all spatial relations contingent upon the effects of matter and energy, general relativity denies us prior access to the projective relations of space, relations that are absolutely essential for the very possibility of spatial measurements. In contrast, Whitehead argued that there must be a kind of uniformity to space in order for our physical cosmology to be possible, a uniformity that requires geometry, with its logically necessary structures, be separated from the contingencies of physics. It turns out that the nature of this uniformity goes far beyond the realm of macro-physics, and reveals a logically necessary feature of any metrical inquiry.

 


 

 

 

 

 

The Logical Problem with Cosmological Measurements

 

(Traditional Paper)

 

 


 

            It is the central thesis of this paper that there is a fundamental problem at the heart of general relativity that relates to the very possibility of generating meaningful cosmological measurements; measurements, that is to say, that are fully interpretable and logically coherent within the framework of general relativity and its presupposed philosophy of nature. The above claim does not relate (directly, at least) to the empirical results that are obtained when numerical calculations based upon the general theory of relativity are compared with various observations. Rather, it relates to the underlying logical basis by which it is assumed that those numerical calculations and refined observations are, and can be, meaningfully interpreted and related the one to the other. The critical aspects of my argument were primarily developed by Alfred North Whitehead in a triptych of works published between 1919 and 1922. However, my interest here is not exegetical in character. Rather, it is with a problem that persists to this day in contemporary physical cosmology, and whose implications reach much, much further.

            This “measurement problem of cosmology” to which I am referring is situated right at the heart of orthodox, Einsteinian formulations of general relativity – or “GR” for short. The problem itself is easy enough to state and understand, and is directly related to what has commonly been presented as one of the principle strengths of GR. This, ironically, is part of our difficulty today. The measurement problem of cosmology is so straight-forwardly described and so effortless to understand, that we are instantly inclined to conclude that it could not possibly be correct. General relativity is just far too successful – or so we have been repeatedly assured – for any such difficulty to be real. But these are our assumptions speaking. We must set aside our certainties if we are to genuinely examine what it is, exactly, that we know.

            Now, in characterizing this problem as a “logical” one, I do not mean to suggest that there is a flaw in the formal, mathematical reasoning involved. Rather, the notion of “logic” that I am using here is very much akin to that of Dewey’s, where logic is characterized as the “theory of inquiry.” But this approach has also been independently argued for by Jaakko Hintikka, who views erotetic reasoning as the central purpose of logic, a thesis which Hintikka traces back at least as far as Aristotle (Hintikka 1999). Moreover, the erotetic/inquiriential issues raised by the measurement problem of cosmology go far beyond any specific facts about physics per se, and raise questions about the kinds of things that must be presupposed by any form of measurement based inquiry. We will find that a substantive degree of uniformity of subject matter must be presupposed throughout by any act of measurement in order for said inquiry to terminate in meaningful operations and logically valid conclusions. Moreover, this uniformity has a very well developed mathematical structure. I will have more to say on this at the end.

            Right now, let us take a step back from the details of GR for the moment and ask instead, “What kinds of logical relations are presupposed by spatial measurements in general?” Well, any measurement, whether it is quantitative in nature or of a merely qualitative character (i.e., more or less), requires the comparison of like to like. This means that, for spatial measurement, the comparison of one spatial segment to another is an essential aspect of this kind of operation. In other word, some established spatial segment must be held up as the “standard” to which others will be compared. Whether this standard is taken to be universal for all such comparisons, or specific to some single act of comparison, it must nevertheless be fixed and functionally uniform throughout the act of measurement. Another way of saying this is, the standard of measurement must “mean the same thing” throughout the act of measurement. Moreover, this standard must ultimately be manifested in the form of an extended physical object such as a yardstick or a tape measure so as to fully ground the spatial standard of measurement upon that of a rigid physical one. It is this physical standard that establishes the baseline of spatial extension that is the necessary basis in any spatial measurement. This standard is the possibility of “conjugacy,” of a meaningful act of comparison, within the logic of measurement. But as vital as this physical standard is, by itself it is not enough.

            It is also necessary that it be possible to bring the likeness of the standard into comparison with that which is to be measured. We must be able – in some respect or another – to bring our yardstick up to that which we intend to measure. If our unit of measure is locked in a vault in the National Institute of Standards in Washington, DC, while the bit of extended space we need to measure is in the form of a 2X4 piece of wood in a lumber yard in Los Angeles, CA, then we must either carry that standard from the one location to the other to make our measurements, or we must have a more readily transported surrogate that we can use for the job. For the purposes of measuring yards of wood on this planet, such surrogates are readily found in the forms of the vast array of acceptable measuring devices that permit us to project our standard of measurement (in the NIST vaults) to our various points of interest. This is the second, absolutely essential, relational factor in the logic of measurement. While we must have a standard of spatial comparison for conjugacy, we must also have standard(s) of spatial projection such as will allow that unit of comparison to be uniformly brought into comparison with – i.e., meaningfully projected onto – those things we mean to measure.

            It is this latter, second step which GR appears to deny us. This is because the uniform geometrical relations that make such projective comparisons possible are rendered dubious if not downright impossible by the fundamental assumptions of general relativity. These assumptions most importantly include the idea that the necessary and uniform relations of geometry are collapsed into the contingent and heterogenous physical relations of matter and gravity. In other words, the very geometry of space is dependent upon the varying physical effects of matter and energy at each point of space. This move by Einstein, of collapsing the formal and logical structures of geometry into the contingent relations of physical space, has long been viewed by the physics community as one of Einstein’s most brilliant postulates. But by so eliminating the distinction between geometry as a purely logical discipline and physical space as this is studied in empirical cosmology, Einstein also compromised the relational structures which make a coherent theory of measurement possible.

            For the question now arises, how are we to bring into comparison our chosen standards of measurement and the objective spatial extensions that are to be measured? We have no a priori basis for making such a comparison, since the geometrical structures of physical reality are now dependent upon the contingent distributions of matter and energy throughout the universe. If we had direct access to every point of physical space, we could determine by those selfsame direct means the projective relations that would logically connect those spatial regions with our standard of measurement. But we do not have such direct access. We have only barely transitted beyond the outer reaches of our own solar system with a few primitive probes. Our only access to distant space is by a purely formal and a priori process of projecting our standards of measurement. Yet, by identifying the contingent factors of physical (notably, gravitational) nature with the geometrical relations of space, Einstein eliminated the uniform and necessary relations that would make it possible to thus project our standards.

            Once again, according to Einstein, the very geometry of space is contingent and variable, its nature at any given point being dependent upon the influences of matter and energy throughout the universe upon that point. And here we have the crux of the measurement problem of cosmology. If the very structure of space is a contingent aspect of physical influences, then we must first know the nature and distribution of those physical factors before we can know the geometry of any spatial region. But in order to know this distribution of physical factors, we must be able to make accurate and reliable spatial measurements to properly place and relate those contingent, physical influences. But in order to make accurate and reliable spatial measurements, we must have a robust understanding of the geometry of the spaces in and through which we are measuring. Only with this latter can we understand the effects on our standard of measurement of the non-uniform and contingent projective relations of those spaces, and thereby establish a logically meaningful system of conjugacy with the things to be measured. Yet such a robust understanding of the geometry of space is precisely what we do not have, and cannot establish, for it is exactly what GR refuses to grant us. We must know the complete distribution of matter and energy in the universe prior to knowing its geometry. But we must have a comprehensive grasp of this geometry in order to discover this distribution. As Whitehead pointed out, with GR as our theory of space and gravity, we are saddled with a situation where we must first know everything before we can know anything.

            Again, the very simplicity of the problem can disguise its profound nature. So permit me to restate the situation from a slightly different angle. When we carry our tape measure from our house to the lumber yard, we are confident that the tape measure continues to mean the same thing at the lumber yard that it meant at the house. Measuring out X number of square feet of plywood, for instance, possesses all of the necessary functional characteristics at the store that we used at home when we first determined how much wood was required for our project. How is it possible that the tape measure should achieve such a continuity of meaning from the one locale to the other? Well, the tape measure is itself an extended segment of space, and when we carry it from one place to another so as to exploit its characteristics as a measuring instrument, we are asserting that the space itself – at the two locales and as represented within the tape measure – is appropriately comparable. We are asserting that there is a functional uniformity of projective relations that permits a workable conjugacy to take place.

            But even if the respective spaces changed in some manner, this would still not be excessively problematic, provided we could actually go to all the relevant points of space and directly determine what our tape measure means at each of these new locales. This would enable us to continue to use our measuring tape, because the rules of its application would nevertheless be knowable, although somewhat more complex than if the spaces were all uniformly of a kind. But on cosmological scales, even on scales only slightly beyond the boundary of our own solar system, we simply do not have this option. We cannot directly test such spaces to see what the projective relations are; projective relations that general relativity tells us are no longer uniformly knowable from Earth. We must come up with some reason to believe that our Earth based measurements can be legitimately projected to these distant spaces in order to have even the hope of a meaningful cosmology. But if the very geometry of space is something we cannot know until after we can confidently engage in measurement, then cosmology as a science teeters on the brink of nonsense. For while we must first measure before we can know; GR requires that we know before we can measure.

            This is the philosophical quandary in which we find ourselves. GR saddles physics with a general theory of space that renders the very possibility of measurement questionable, because the essential requirements for the possibility of spatial measurement are explicitly denied. And yet, GR appears to be successfully employed in formulating and evaluating cosmological measurements all the time. Indeed, general relativity and quantum mechanics are often held up as the premier examples of the most successful physical theories ever conceived. How are we to reconcile such practical successes with the supposed philosophical issues raised above?

            The above question almost undoes itself in the very asking, for it is a well known fact that these two theories are mathematically irreconcilable. But it is not just at the small scales of micro-physics that GR runs into problems. At the very large scales of physical cosmology, GR has also proven to be inadequate. In an effort to account for observed phenomena – phenomena of which physicists assume they have meaningful measurements – it has proven necessary to reintroduce the idea of the “cosmological constant” as well as invent such extravagant new ideas as “dark matter” and “negative energy” in an attempt to account for the evident behavior of the cosmos. And even at the most comparatively mundane levels, GR is problematic. For example, the mathematical theory of general relativity is thoroughly non-linear. This means that GR as it stands is almost useless when it comes to producing numbers which can be compared with actual measurements. Rather, relevant sections of GR must first be approximated with a linear theory, and then this linear approximation is used to calculate theoretical values. Furthermore, this failure of linearity within GR is a significant contributing factor to the irreconcilability of GR with quantum mechanics. This latter theory is linear in important aspects, and as such defies any direct resolution with a macro-physical theory that lacks such linearity. Once again, let me emphasize that there is nothing controversial in the immediately preceding statements regarding general relativity. Any contemporary text on GR will suffice to confirm these facts.1

            What GR does offer is a conceptual framework in which scientific cosmology can be engaged. However, given the above difficulties, if an alternative framework were to be proposed that avoided and/or minimized some of those difficulties, then that framework would merit consideration. And in point of fact, such a framework is in hand: it is Whitehead’s theory of relativity, first proposed in detail in 1922. But while Whitehead proposed his own applied, scientific theory, what is of greater significance is that he described an entire class of alternative theories, and it is this general class of theories which is of interest to us here.

            Now, Whitehead had no knowledge of the more contemporary troubles with GR that were noted above; his sole reason for criticizing it was the measurement problem of cosmology. It was his observation that if we do not maintain the separation between geometry and physics, then we lose the logical basis of the rules for conjugacy and projection that make spatial measurement possible. Whitehead’s solution is reminiscent of the old joke:

            Patient: “Doc! Doc! It hurts every time I do this!”

            Doctor: “Well then, stop doing that!”

General relativity collapses geometry and physics together by enveloping both within a single metrical tensor that eliminates the distinction between necessary relations and contingent ones. Whitehead’s argument was that we should stop doing that.

            So instead of Einstein’s ‘mono-metrical’ approach, Whitehead proposed a ‘bimetric’ solution. In other words, instead of shoe-horning all of the metrical relations into a single tensor (the “gμν” of standard relativity theory), thus collapsing geometry and physics together, Whitehead’s theory utilized two metrical tensors: His “J” tensor represented the contingent physical relations of gravity and other forces, while his “G” tensor stood in for the necessary spatial relations of geometry (Whitehead 1922, 81 – 6). As Whitehead claimed, and Eddington demonstrated, Whitehead’s theory was equivalent to Einstein’s at the limit of special relativity (Eddington 1924). Moreover, for almost fifty years after the publication of Whitehead’s theory, no one was able to propose an empirical test that could decide matters between Whitehead’s specific theory, and GR. Clifford Will finally suggested an effect known as “earth tides,” which gave a hope of a measurable difference between the two theories (Will 1971). But Will’s proposal was predicated upon the dubious assumptions that all the matter in the galaxy is concentrated at a point at the center, while no account was made for the rest of the matter in the universe. Yet, despite these flaws, most physicists – or, at least, most physicists who are even aware of Whitehead’s theory – assume that Will’s argument is conclusive. And this assumption comes after several generations of physicists gave no attention of any sort to Whitehead’s theory, despite the fact that there were neither formal nor empirical reasons for its neglect.

            All of this is largely moot, however. Even if Will’s purported refutation can be modified so that it is still valid, Whitehead’s specific theory was never more than an example of an entire class of theories. There are now many other members of this class than just Whitehead’s, and many of these theories are known to be viable alternatives to GR. Because each of these bimetric theories separates the geometrical relations from the contingent facts of physics, they have the necessary level of uniformity available to them to permit of the projective relations required for meaningful measurements. So it is a matter of no great philosophical relevance whether Whitehead’s specific scientific proposal still offers a viable alternative to GR or not. Whitehead himself never viewed this as anything more than an application of his general philosophical principles. What we have instead is much more interesting: a developed system of philosophical ideas that are well vindicated by an entire family of scientific exemplars that are known to be formally and empirically valid. Moreover, these theories are also known to be, in general, much more linear in their formulations. This means that testable predictions can be calculated directly from these theories, and that they are far more easily reconciled with quantum mechanics.2

            Finally, the uniformity criterion that Whitehead argued for, and which can only be achieved by separating the necessary relations of geometry from the contingent ones of physics, lends itself to a precise formulation. Indeed, this uniformity means that the geometrical relations exhibit a high degree of symmetry, such that the formal spatial relations have what is known as “constant curvature.” Euclidean space is a paradigm example of a space with constant curvature – namely, none. But it is the constancy that is the essential structural feature of Whitehead’s uniformity criterion. Given such constancy, Whitehead is quite explicit that non-Euclidean geometries are fully capable of meeting the necessary level of uniformity to avoid the measurement problem (Whitehead 1922, pg. v.) At the time that Whitehead was writing, the formal understanding of spaces of constant curvature was not yet fully developed. However, it is now the case that the relations between classes of symmetries and spaces of constant curvature – and hence, Whitehead’s uniformity criterion – is well-known. The basis of this symmetry is to be found in the mathematical theory of groups.3, 4

            This is an important discovery. To begin with, as we saw above the measurement problem is not quarantined within the limits of physical cosmology. Rather, Whitehead’s uniformity criterion is a generic requirement of any spatial measurement what-so-ever. The requirement of uniformity will be constant in any measurement operation. While the kinds of relevant uniformities will vary between different situations, the need for stable relations of comparison and conjugacy between the things measured and the standards of measurement will always be present. Even when the act of measurement is as simple as counting, the need for some uniformity is so manifest that we mark it by cautioning against comparing apples to oranges.

            Formally, such uniformities will always manifest themselves in those structures most appropriately modeled by the mathematical theory of groups. We thus find ourselves in possession of a fully developed, mathematically rigorous, universal logical requirement of all metrical inquiries. Moreover, group theoretical structures are astonishingly omnipresent. They are the beating heart of most, if not all, physical laws of nature, and are even of vital importance in understanding perceptual invariances.5 Just how far a study of the logical requirements of uniformity might take us, and just what sorts of connections it might reveal, is a very open question. But it is one worthy of pursuit.


 

Notes:

  1. See for example (Carroll 2004) for a recent technical treatment of the subject, or (Greene 2004) for a more popular development of the relevant ideas.

 

  1. For example, (Rosen 1940) and (Moffat 2003) both emphasize the linearity of their respective bimetric theories.

 

  1. For an accessible discussion of the connections between symmetry and groups, see (Weyl 1980). Standard references on the subject of spaces of constant curvature are (Helgason 1964) and (Wolf 1967). For a discussion of the role of (maximally) symmetric spaces in contemporary cosmology, see (Carroll 2004, pp. 139 – 144 and 323 – 329). Maximally symmetric spaces are, in essence, the highest degree of uniformity possible in a space.

 

  1. It is worth noting that Whitehead, from the very beginning of his mathematical career, was intimately familiar with the contemporary work on the subject, done at what was the earliest stage of the study of spaces of constant curvature. Even a casual glance at (Whitehead 1898) will find numerous citations of the principal figures and their publications on these developments. See (Epple 2002) for an excellent overview of the early history of the subject.

 

  1. For the former, see (Van Fraassen 1989). As to the latter, (Cassirer 1945) is the first place that I am aware of this relation being explicitly noted. J.J. Gibson also highlighted the connection between groups and perception (Gibson 1966). Since Gibson also mentions Cassirer in this regard, this may have been his original source of the idea.

 

 

References:

Carroll, Sean.               2004. Spacetime and Geometry: an Introduction to General Relativity, Pearson Education Inc. publishing as Addison Wesley, San Francisco.

 

Cassirer, Ernst. 1945. “Reflections on the Concept of Group and the Theory of Perception,” found in Symbol, Myth, and Culture, Yale University Press, 1979, pp. 217 – 291.

 

Eddington, Arthur S.     1924. “A Comparison of Whitehead's and Einstein's Fomulae,” Nature, (Feb. 9, 1924), pg. 192.

 

Epple, Moritz.               2002. “From Quaternions to Cosmology: Spaces of Constant Curvature, ca. 1873–1925,” Proceedings of the ICM, vol. 3, 935—946, from the International Congress of Mathematicians, Beijing, August 20 - 28, 2002.

 

Gibson, J.J.                   1966. The Senses Considered as a Perceptual System, Houghton Mifflin Company, Boston.

 

Greene, Brian. 2004. The Fabric of the Cosmos, Alfred A. Knopf, NY.

 

Helgason, Sigurdur.       1964. Differential Geometry and Symmetric Spaces, 2nd edition, Academic Press, NY.

 

Hintikka, Jaakko.         1999.  Inquiry as inquiry : a Logic of Scientific Discovery; selected papers, v. 5; Kluwer Academic Publishers, Dordrecht / Boston.

 

Moffat, J. W.                2003. "Bimetric Gravity Theory, Varying Speed of Light and the Dimming of Super-novae," International Journal of Modern Physics D Vol. 12, No. 2 (2003) 281–298.

 

Rosen, Nathan.             1940. “General Relativity and Flat Space I & II,” Physical Review, vol. 57, January 15, 1940, pp. 147 – 153.

 

Van Fraassen, Bas.       1989. Laws and Symmetry, Oxford University Press, Oxford.

 

Weyl, Hermann.            1980. Symmetry, Princeton University Press, Princeton.

 

Whitehead, Alfred North.         1922. The Principle of Relativity with applications to Physical Science, Cambridge University Press, Cambridge.

 

                        ---        1898. A Treatise on Universal Algebra, Cambridge University Press, Cambridge.

 

Will, Clifford.                1971. “Relativistic Gravity in the Solar System II. Anisotropy in the Newtonian Gravitational Constant,” The Astrophysical Journal, 169: 141 – 155.

 

Wolf, Joseph A.            1967. Spaces of Constant Curvature, McGraw-Hill Book Company, NY.