What do you mean, “it’s a silly question”? Sir Arthur Eddington wouldn’t agree with you. He argued in 1924 that the mass of the Sun could be expressed in kilometres (1.5 km actually) and that the mass of the Earth was about 5 millimetres (on which scale, the mass of a person would be a length considerably smaller than an atomic nucleus).
But how can a mass, or a weight (which of course is not quite the same thing as a mass, but let’s not be too picky at this stage) be a length? Well, if you think about it, although the measurements we make each have their own units (mass has kilograms, grams, pounds, ounces etc; length has metres, kilometres, feet etc etc) which we tend to think of as qualitatively distinct from one another, we do sometimes express one measurement in terms of a completely different one. Holiday brochures, for instance, might say that a holiday cottage is “10 minutes from nearest shops”. Of course there are underlying assumptions in such statements which it is assumed we are all aware of, so they don’t need to be explicitly stated. In the case of the distance to the shops, what’s given is the time taken when travelling by car.1 This is sometimes explicitly stated – e.g. we might be told the shops are “10 minutes drive” or “an hour’s walk” away – but even then, if it really is the distance we are talking about, we need to know the actual speed at which the car or the walking person moves. (In actual fact, we may be more interested in the time in this case, if weighing up whether we’ve got time to nip to the shop before dinner, for instance – but if we were driving a car with very little petrol we might be primarily interested in the distance, in order to work out whether we’ve got enough fuel). 2
In science, we need slightly better precision than just knowing roughly how long it will take to get to the shops. So if we are going to specify a time instead of a distance, we will need to say the exact speed by which the time and distance are related. The speed of light is a useful standard for such purposes; in fact there are actually units of distance such as the light-year and light-second which are in common use. Astronomers who quote distances in light-years are really primarily interested in distance, and just find the light-year a handy-sized unit; but terrestrial physicists may quote cable lengths in light-nanoseconds, and here it is actually a time they are interested in rather than a distance. Electrical signals travel at the speed of light, so if you are doing an experiment that involves very precise timing, you need to know how much of a delay each piece of cable will introduce. Some cables are actually labelled in light-nanoseconds, and sometimes the “light” gets left off because it’s assumed we will understand what it means. (A light-nanosecond is almost the same as a metric foot (0.3m) which introduces a rather bizarre link between cutting-edge physics and the timber trade, since the metric foot is the unit in which timber is sold). The nub of this is that we know that distance is speed multiplied by time, but sometimes we don’t bother to explicitly mention the speed.
This practice of expressing one quantity in terms of another to which it is related, is fairly widespread in science. A good example of this is the way we describe pressure. Pressure is, of course, force per unit area; many of us, at least those of a “certain age”, will know that atmospheric pressure at the earth’s surface is about 14 pounds 3 per square inch, while others will be more familiar with the figure of 100,000 pascals (the SI unit, equivalent to newtons per square metre). But pressure is also measured in millimetres of mercury (otherwise known as torr, commemorating the inventor of the barometer, Evangelista Torricelli). So here we are identifying a pressure, which is force per unit area, with a length – the length of a column of mercury in a barometer. Isn’t that the same sort of thing as Eddington identifying the mass of the sun with a length?
The exact relationship between atmospheric pressure and the height of the column is that the pressure equals the product of the height with the density of mercury and the acceleration due to gravity (g). The latter two are constants as long as the barometer stays where it is and is always read at the same temperature; so it is tempting just to forget them and use the height as a measure of pressure. (Actually we don’t quite forget the density, since it is implicit in the name of the unit, millimetre of mercury). If we put in the numerical values of all three quantities, we will have the height in metres (where I have switched to the base unit, for reasons that will soon become apparent) multiplied by the density in kilograms per cubic metre, multiplied by g in metres per second per second. Hence the pressure comes out as a number of (metres x kilograms per cubic metre x metres per second per second), or kilograms per metre per second per second – aka pascals. I don’t think we would say the pressure is a number of metres or millimetres in quite the same sense as Eddington meant when he said the mass of the sun was a length; the name of the unit, “millimetre of mercury”, effectively indicates that we are not just talking about straight millimetres here. 4
Analysing Eddington’s derivation in the same way, we note that the formula he actually comes up with gives the mass of the sun as a length (the distance between the sun and an orbiting planet, let’s say the Earth) times the square of the orbital speed, and divided by the gravitational constant. But what Eddington does is to express the orbital speed as a fraction of the speed of light (and that fraction is just a pure number) so now we have the sun-Earth distance times the square of the fraction, times the square of the speed of light, divided by the gravitational constant. He then announces that he is going to work in a new system of units in which the two constants (the speed of light and the gravitational constant) have the value 1, so that when he inserts these 1s, together with the numerical values of the speed fraction and the sun-earth distance, the calculation does appear to leave us with a length.
We can see that this is the equivalent of us expressing air pressure in a system of units in which the acceleration due to gravity and the density of mercury both have the value 1. But that wouldn’t allow us to simply equate the pressure to the height of the column of mercury, because the two constants, while they may have the value 1 and hence be “invisible” in the calculation, still have units. In fact the acceleration due to gravity at the earth’s surface is often referred to as simply g, particularly by astronauts, as though it were a unit, to enable comparisons with the artificial accelerations produced by jet engines (as in “we hit 6g on that flight”). 5 So we’d have to state the pressure as “X millimetres mercury g” (where X is the height of the column) and not just “X millimetres”. Re-doing Eddington’s calculation with the units included would involve multiplying the Sun-Earth distance in metres by the square of the orbital speed in (metres per second) squared, and dividing by the gravitational constant, which has the units of cubic metres per kilogram per second per second. All of which combines together to give us … well, just kilograms actually, since the metres and seconds cancel.
To be precise though, Eddington’s calculation of the mass of the Sun also includes an implicit change of basis. We find that derivations of purely mechanical quantities tend to require just three types of unit – the kilogram, metre and second – which correspond to three types of quantity – mass, length and time – which are often referred to, though rather confusingly, as dimensions (“confusingly” because they are not the same thing as the familiar spatial dimensions). But while it seems we do have to have three distinct quantities, it doesn’t have to be precisely those three quantities. We could, for instance, think in terms of force rather than mass, defending this choice on the basis that we are more familiar with the concept of force than with mass; so we could have force, length and time as our three quantities, remembering that, from Newton’s Second Law, mass is force divided by acceleration. But we could equally replace mass with density, and work with density, length and time. You can see that we could use pretty well any three combinations of M, L and T as our three basic quantities, subject to certain restrictions (and if you have a maths background and this reminds you of vector spaces, that is no coincidence). 6 In the pressure example, we could be said to be using length, density and acceleration. Eddington preferred to use length, speed and gravitation as his three basic quantities; in particle physics, on the other hand, energy, speed and action are chosen, where action is energy multiplied by time; it has no analogue in everyday experience, although it has the same dimensions as angular momentum. So when he introduces his system of units in which the speed of light and the gravitational constant both have the value 1, he is not talking about 1 metre per second, or even 1 light-second per second; what we need here is a base unit of speed (i.e. one that does not have to be defined in terms of other units), but we don’t have such a unit, so it is necessary to invent one; we could call it the einstein, so that the speed of light is 1 einstein. Similarly we need a base unit of gravitation, which we also have not got, but let’s call it the eddington. Thus the derivation, properly done so that the names of the units are included, will give the mass of the sun as 1.5 kilometre einstein2 per eddington – not just 1.5 kilometres.
Incidentally, if you don’t agree that physical quantities like the speed of light cannot be just numbers, try this simple test. Take your car (if you have one; if not, borrow one) to the garage and, assuming it’s one of those old-fashioned ones with an attendant (this is a thought experiment after all) ask for some petrol. When he or she asks how much, just say “1”. “But 1 what?” comes the tetchy reply from the bored attendant standing there holding the nozzle in the freezing cold (I forgot to mention that it’s winter in this thought experiment). “One litre or one gallon?” In response to this, stand your ground: “Just 1”. Cite Eddington if you have to. See how long it takes before the attendant stomps off in despair and/or sprays you with petrol.
Let’s indulge in a little fanciful speculation. Suppose the company that makes the cables for physics experiments starts making hi-fi cables, and calibrates these in light-nanoseconds too, claiming that it’s essential that the cables connecting your amplifier to your speakers must all be the same length to the nearest light-nanosecond. (This is utter tosh of course: the time it takes for the loudspeaker cone to complete one cycle, even at the highest audible frequencies (say 20kHz) is of the order of 50 microseconds, which is 50,000 nanoseconds; and besides, the human ear cannot distinguish sounds which arrive less than 60 milliseconds (that’s 60 million nanoseconds) apart. But never underestimate the power of marketing people!)
So the company gets very big and successful because everyone wants to have a perfect hi-fi, and in fact it branches out and starts making oil pipelines as well, which are also calibrated in similar units – obviously somewhat bigger units though, say light-microseconds. Now, the oil industry is exteremely powerful, especially in the USA, where they kinda like having things measured in metric feet, as they haven’t yet really severed themselves from the imperial system. Pretty soon, the various companies involved lobby governments to officially adopt this new system of measuring, and this is done, sold as a “simplification”, since distance is henceforth to be defined in terms of the distance travelled by light in a given time, and the same units (seconds, milliseconds, microseconds, nanoseconds) will do for both time and distance – thus getting rid of a whole set of unnecessary units (those of length). The “speed of light” is now measured in nanoseconds per nanosecond, so that it is simply a dimensionless number, with the value 1.
Well, if such an unlikely scenario were to come about, not only would some people find it awkward measuring everything in nanoseconds (which, after a respectable interval, could perhaps be renamed “feet”) but there would always be a few who would maintain that, however much you define away, distance and time simply aren’t the same sort of quantity, and hence should not be measured in the same units. I think I would be among that number.
But we don’t have to hypothesise about unlikely scenarios, because a similar system of measurement is in existence today, in the world of particle physics. The Standard Model of particle physics makes the same stipulation that the speed of light is 1, and also makes a similar stipulation regarding another physical constant, Planck’s constant. As a result, all physical quantities – if we leave aside for now the sticky subject of electrical quantities – are measured in terms of the same unit – usually the GeV or gigaelectronvolt. This practice appears to originate from Richard Feynman’s work on quantum electrodynamics (QED) in the 1940s. (He might, or might not, have learnt the trick from Eddington – he doesn’t tell us.)
Not that particle physicists have actually redefined distance or time or anything else – any more than Eddington did. Instead they defend the formulae they derive on the basis that they are working in a particular system of units – the so-called “natural units”.7
I count myself as particularly fortunate in having learnt maths and physics in the UK in the 1960s, when two distinct, consistent systems of units existed side by side: the c.g.s. (centimetre, gram, second) system and the f.p.s. (foot, pound, second) system. So we learnt from the very beginning that the physical laws we used, in their algebraic format – again leaving aside electrical ones – were true in either system, or in any other consistent system. 8 They were, in other words, units-invariant – something that would have pleased Joseph Fourier and William Thomson, who both expressed the desire to make our physical laws general enough to make sense in any system of units. But it is clear from reading textbooks and papers that there are physicists out there who are not used to such generality, and tend to think of physical laws as being true only in certain units. This approach made it easier for Eddington, with his idiosyncratic system, or Feynman, in the case of the nascent QED, to calmly announce that “we will now work in a system of units in which c = 1 etc”. They saw no problem with what they saw as merely changing units, and certainly no loss of generality, which is something we might worry about today.
But why is this important, I hear you ask? Well, measurement underpins the whole of quantitative science, which means all of physics, most of chemistry and a fair bit of biology; and it is making inroads into the social sciences as well. But measurement is also an everyday activity which everyone is deemed to know how to do. It would seem to be an insult to the intelligence to suggest that people need to be taught how to measure. But, as with many other things, what most of us understand about measurement is just enough to be able to carry out our normal day-to-day activities – to measure wood for a shelf, weigh flour to make bread, time something cooking in the oven. But the philosopher’s job is to go further – further even than the scientist, if the scientist isn’t interested or lacks the resources to do it him or herself.
In my next blog in this series I will discuss a parallel philosophical argument of Eddington’s in which he regards length and mass as in some sense “the same”.
To be continued …..
1 It’s assumed we are all aware of it, although sometimes some of us aren’t. Those of us without cars can sometimes get caught out by this.
2 In which case perhaps the answer we are actually seeking might be “it’s about half a gallon away” …
3 Actually pounds weight or pounds force; a pound is a unit of mass, not force.
4 Just to close the section on pressure, I should note that it probably has more distinct units than any other quantity. As well as the units already mentioned, there is also of course the bar or atmosphere, set at 100,000 pascals.
5 However, in this context g usually appears as G, capitalised but not italicised.
6 The restrictions are that the basis vectors are linearly independent, and span the space (i.e. any vector can be expressed uniquely in terms of the basis vectors).
7 Not to be confused with the perfectly respectable practice of defining units in terms of the values of physical constants.
8 A consistent system of units is a closed set consisting of a number of base (fundamental) units and a number of secondary (derived) units, such that if the units we put into a calculation all belong to the set, so will the units of the result. Thus if a calculation required an energy and a density, say, a combination of joules and kg m-3 would work, but not joules and g cm-3.