In my previous blog in this series, I talked about measurement, and whether it is permissible to express various quantities in terms of other quantities which we normally think of as quite different, such as mass and length. I cited Sir Arthur Eddington’s derivation of the mass of the sun in kilometres, and pointed out that some of the reasoning he used in order to make that derivation was, at the very least, questionable.

In this episode I will go through another argument that Eddington uses on the same subject. The mathematical derivation might, after all, look a bit like a conjuring trick even if you accept all the slightly questionable treatment of units. But he also makes another, more philosophical claim: that the mass and length of a body are “merely measure-numbers, in different codes, of the same intrinsic quality”.

By way of explanation of this statement, he cites examples of other pairs of quantities that he thinks are the same. One is mass and energy, of which I spoke at length in another recent blog piece, hopefully managing to convince the reader that the equivalence of these two is by no means proven; the other is parallax and distance.

Parallax is the apparent angular motion of a nearby star, relative to more distant stars, over a 6 month period, as the Earth travels from one side of the Sun to the other. As the diagram shows, the parallax (*θ*) is certainly *related* to the distance *d* of the star, albeit by a *reciprocal *relationship (the greater the distance, the smaller the parallax); but even if we were to grant that two quantities connected by a reciprocal relationship were in some sense *the same*, we might worry about what would happen if the radius (*r*) of the earth’s orbit changed; the parallax would change, but the distance of the star wouldn’t; so can we really say that there is any meaningful sense in which these two quantities are the same? In fact, is this argument any more than just another unconvincing mathematical trick?

When I gave a talk on this topic at a recent conference, it was suggested to me that Eddington “knew” from his researches into general relativity that mass and distance “really were” the same under the extreme circumstances (large masses, near-light speeds) in which general relativity comes into its own, despite not having taken the opportunity in his book to explain this. My knowledge of GR is not nearly comprehensive enough to question that conclusion, but I think it is still legitimate to ask, “even though, in such extreme conditions, two quantities may look the same, does that mean that we should regard them as the same *in everyday life*?”

Despite not being qualified to comment on how mass and length might be regarded as the same, I do have an inkling about similar claims that are often made with regard to distance and *time*. One of the big results of special relativity is that two observers, travelling at a constant relative speed, will not agree on time intervals or on distance intervals measured along the line of motion, but they *will* agree on the “invariant interval”:

(*x*^{2} + *y*^{2} +* z ^{2} *–

*c*)

^{2}t^{2}^{½}

(where *c* is the speed of light). So while both would agree on the *size* of this quantity, they would not agree on the sizes of the contributions to it from *x,y,z* and* t. *Hence it appears that length and time get “mixed up”, and we cannot uniquely specify what is length and what is time; so shouldn’t we regard length and time as being “the same thing”?

Well, once again, as is the case with parallax and distance, the two quantities are not* identical* but merely *related.* There is the small matter of *c*, and there is also the minus sign to contend with. And we can define length and time unambiguously if we insist that they are measured in a reference frame in which the thing being measured is at rest. This situation is to some extent analogous to that of mass and energy (which were also on Eddington’s list of quantities which are “measure numbers in different codes of the same intrinsic quality”); as mentioned above, there is little persuasive evidence of this.

What motivates the desire to declare two or more quantities “the same”? Well, presumably it is one of the main driving factors behind science, which is to offer *simple* explanations of the universe. If we have two competing explanations and one is shorter or simpler in some way, then we tend to choose that one. This principle is known as *Occam’s Razor*. But can reducing the number of distinct quantities with which we describe the universe count as “simplifying”, especially when such reduction is so counter-intuitive, at least at an everyday level?

Worse still – simplification often works directly against another fundamental goal, that of ever-greater *generality*. Mixing up distance and time, or mass and distance, only works (if at all) in a very limited domain of physics. Likewise, as I’ve already explained, reducing the number of dimensions, or apparently reducing it, by working in a particular system of units, limits us to a very small (if fundamental) part of theoretical physics (and in any case, if at any stage we want to calculate a real physical quantity, as of coure we will, we have to translate back into more conventional everyday units).

There are unfortunately few texts on the philosophy of measurement, and each of those that have been written laments the paucity of other available sources. Percy Bridgman, in *Dimensional Analysis*, has this to say about the symbols we use to represent measurements:

[a quantity *x*] “might stand for the number which is the measure of a velocity …. By a sort of shorthand method of statement we may abbreviate this long-winded description into saying that [x] is a velocity, but of course it really is not, but is only a number which measures velocity”.

We can see that the language Bridgman is using here (“number which measures …”) is reminiscent of Eddington’s “measure-numbers”. But even if we accept that the algebraic quantities we work with *are *just numbers, and hence are of a kind, we must not forget that they form only part of the compound entities that Bridgman and Campbell call *magnitudes*, and there is no reason why we should think that the magnitudes themselves are interchangeable, as Eddington seems to want.

It is true that D.C. Ipsen, in his book *Units, Dimensions & Dimensionless Numbers*, offers two alternative representations of a physical quantity (or a physical variable, as he calls it), one of which is the aforementioned “magnitude” format, where a magnitude is a compound entity consisting of a number and a unit, whereas the other includes the unit in the definition of the variable; an example of this version would be “length in metres”, and in this case an algebraic expression such as *x*, which stands for such a variable, is indeed just a pure number, and not only in the “shorthand” sense referred to by Bridgman. And it does seem that Eddington may have been thinking of this latter usage, since his exact words were “the gravitational radius in centimetres, the inertia in grams, and the energy in ergs, are merely measure-numbers in different codes of the *same* intrinsic quantity of the particle”. But this version is not in widespread use, and even if it were, it would not permit Eddington to claim that mass and length were in any sense the same, since the unit has simply shifted from being part of the magnitude to being embedded in its definition. And as Ipsen points out, the disadvantage of using this second format would be that we would actually have to treat measurements of a particular quantity in different units (e.g. length in metres, length in millimetres, length in feet etc) as *distinct* *quantities*.

In summary, I cannot see how Eddington can fit the square peg of mass into the round hole of length; they are simply not the same thing, whichever way I try to view them. So I cannot go along with his project of mixing different types of quantity up, and for the same reason I have little time for those who seek to reduce the number of dimensions by introducing obscure units or claiming that it all looks the same at the scale of the atom or the galaxy. In fact, far from wanting to *reduce* the number of dimensions, or the number of physical constants, as some want to do, in my next piece in this series I will argue for an *increase* in this number.

*To be continued …*

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