**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.

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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 “g_{μν}”
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:*

- 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.

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

- 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.

- 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.

- 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.

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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
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Rosen, Nathan. 1940. “General Relativity and Flat Space I & II,” *
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Van Fraassen, Bas. 1989. *Laws and Symmetry*, Oxford University
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Weyl, Hermann. 1980. *Symmetry*, Princeton University Press,
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Whitehead, Alfred North. 1922. *The Principle of Relativity with
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--- 1898. *A Treatise on Universal Algebra*,
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Will, Clifford. 1971. “Relativistic Gravity in the Solar System
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Wolf, Joseph A. 1967. *Spaces of Constant Curvature*,
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