# When is a Kilogram Not Equal to One Kilogram?

When is a kilogram not equal to one kilogram? One obvious answer is when we are talking about weight: what we usually refer to as a kilogram in weight is more correctly a kilogram-force, representing the force exerted by a body of a certain mass in kilogram in standard Earth gravity. Then last week, the *Guardian*’s science section ran a story by Gavin Haynes about the International Committee for Weights and Measures working towards redefining the kilogram as part of a much wider redefinition of units scheduled for 2018. The platinum-iridium cylinder that has been the standard definition for a kilogram for over a century, the International Prototype of the Kilogram (IPK) lovingly nicknamed ‘Le Grand K’, will be replaced by a new system that will express mass in terms of Planck’s constant. So this kilogram prototype would soon cease to be a kilogram in mass if one finds a more accurate and stable definition of weight from which this object may diverge. But there is another answer to this question, rather another way in which this question can be put, which yields an interesting philosophical problem.

One can arrive at the same problem when considering a different quantity and asking ‘when is a metre not equal to one metre?’ For Ludwig Wittgenstein, the answer was the metre rod in Paris that, until the adoption of the Krypton standard for the definition of length in 1960, served as the prototype for the definition of the SI metre. This rod, he argues, is the only object ‘of which one can state neither that it is 1 metre long, nor that it is not 1 metre long’, because the prototype is itself ‘not something which is represented, but is a means of representation’ (§50). In other words, it is meaningless to ask whether the metre rod is a metre long because the rod itself is now a part of the various rules that govern our language, and so the question does not apply to it. One could perhaps say the same about the IPK: one can neither state that it is one kilogram in mass, nor that it is not, so it is a kilogram that is not a kilogram.

But the picture is more complicated and more fascinating. Saul Kripke, in *Naming and Necessity*, disagrees with Wittgenstein that one cannot say of these prototypes that they are one metre in length (54) or, in this case, one kilogram in mass. While these objects are incorporated into the various rules of the language game, they still are physical objects (unlike, say, words like ‘green’ which are abstract), and one can therefore attribute length and mass to them (even though one cannot attribute ‘greenness’ to the word ‘green’). But Kripke poses a more interesting question that follows from this: is it necessary that the metre rod will be a metre long? And similarly, one can ask whether it is necessary that the kilogram prototype will have a mass of one kilogram? Can there not be conditions where the metre rod is longer or shorter than a metre, or the IPK has more or less mass than a kilogram?

The answer to this question, according to Kripke, complicates a long-standing view regarding three categories of modality, particularly ‘necessary’, ‘a priori’ and ‘analytic’ truths. A necessary truth is something that is true in all states of affairs, something which is impossible to be false (the opposite being ‘contingent’). Something is true ‘a priori’ when it is knowable prior to any empirical experience of the facts of the world, something which can be known in the mind alone (the opposite of which is ‘a posterior’). Something is an ‘analytic’ truth if its truth is evident from an analysis of the meaning of the terms alone (the opposite being ‘synthetic’). Because of some prominent overlaps between them, they have been used synonymously. If something is an analytic truth like ‘all bachelors are unmarried’, then it is knowable a priori because one need not observe every bachelor to verify this but merely understand the language, and it is necessarily true because an unmarried bachelor would be a contradiction. Similarly, one would assume if something is knowable a priori, then it is true regardless of verification with the world, and is therefore necessarily true. But Kripke’s example of the metre rod demonstrates that this is not the case.

If we consider the question of whether the IPK has a mass of one kilogram, then this statement is knowable a priori: one need only know the definition of a kilogram as a mass equal to that of the IPK to know that the IPK has a mass of one kilogram. However, this statement is not necessarily true: one can imagine another possible world in which the IPK would have, instead of say 2.20 lb of mass that it actually has, more or less than that1. It is possible, then, that the IPK would have a mass that differs from one kilogram in the actual world, and therefore it is not necessarily true. This statement is an example of a contingent a priori statement, analogous in structure to Kripke’s own example of the metre prototype.

The difference between necessity and a prioricity is that they are both different orders of modality: necessity is a matter of ontology, or the possible ways in which certain things can exist, whereas a prioricity is a matter of epistemology, or how a certain thing can be known. So the two are far from being synonymous, even though there are overlaps between truths that are necessary and a priori. Analytic truths, however, belong to a stronger category: they are truths that are both a priori and necessary. The difference between these modalities are made clear when considering the ways in which the kilogram prototype might not have a mass of one kilogram, or the metre prototype might not be one metre in length. The metre rod has, of course, been retired. So as far as examples go, it is somewhat dated. The IPK, however, is now the only prototype that is still based on a physical object, and so is still relevant, until 2018 by when Le Grand K will join the metre rod.

There is a pressing need to do this: as Elizabeth Gibney reports in *Nature*, physical objects are unstable as they can gain or lose atoms and hence their mass can vary. Defining mass using a constant will, in theory, result in a measurement that will not just be stable, but also be available to anybody who does not have access to the prototype to compare it with. This is why teams of metrologists had been trying to find a stable and convergent value to fix Planck’s constant by the start of July this year. Then using both the Planck-Einstein Relation *E = hν* (where *E* is the energy of the photon, *ν* the frequency of radiation and *h* Planck’s constant) and Mass-Energy Equivalence *E* = *mc*^{2} (where *m* is mass and *c* the constant velocity of light) one can relate mass to Planck’s constant. This will form the basis of the new definition of the kilogram.

Once this happens, we will be in a rather peculiar position: as it stands, one can say of the kilogram prototype that it has a kilogram of mass. (One cannot make a similar analogy for the current definition of a metre.) Once the definition of a kilogram changes, this will not be the case. Moreover, we would lose the last remaining physical object which could be used as a current and valid example for Kripke’s argument (rather than referring back to a historical case). However the good news is that the definition of a kilogram will always be traceable backwards to this Le Grand K: the value of Planck’s constant that the International Committee for Weights and Measures will use for this definition will be derived experimentally with reference to the current prototype, and then this value of Planck’s Constant will then be used for future definitions of mass. So there is some solace in knowing that some aspect of this history will be preserved in the history of the definition of the kilogram.

**Note**:

1 One might be tempted to assume that in this world, the definition of a kilogram would correspond to this new mass of the IPK and therefore its mass would still be one kilogram. However, this is not the case because of the way in which we use language. Without wishing to complicate this too greatly and venture into more specific two-dimensional semantics, it would suffice to say this much: the definition of one kilogram (as, say, equivalent to 2.2 lb) has a trans-world identity that is constant because its meaning is fixed with reference to the actual world. What we consider the definition of ‘one kilogram’ is not what the IPK is in any possible world, but what it *actually* is, i.e. what its mass is in this (the actual) world. So even in a world where the IPK is 3 lb, one kilogram would still be 2.2 lb. The only way the definition of a kilogram will vary is if some other world were considered the actual world, but that is complicating the model for semantics beyond what is relevant here. See, for further on this subject, Davies and Humberstone and Soames.

**Works Cited**:

Davies, Martin and Lloyd Humberstone. ‘Two Notions of Necessity.’ *Philosophical Studies* 38.1 (1980): 1-30. Web. 28 Jul. 2017. http://www.mkdavies.net/Martin\_Davies/Language\_files/TwoNotions.pdf

Gibney, Elizabeth. ‘Kilogram Conflict Resolved at Last.’ *Nature* 526 (15 Oct. 2015): 305-6. Web. 28 Jul. 2017. doi:10.1038/526305a

Haynes, Gavin. ‘Au Revoir to the Original Kilogram: Le Grand K Joins the List of Relegated Metrics.’ *The Guardian* 26 Jul. 2017. Web. 28 Jul. 2017.

Kripke, Saul. *Naming and Necessity*. Oxford: Blackwell, 1980. Print.

Soames, Scott. ‘The Early Two-Dimensionalist Semantics of Davies and Humberstone.’ *Reference and Description*. Princeton: Princeton UP, 2009. Print. 106-29.

Wittgenstein, Ludwig. *Philosophical Investigations*. Trans. G.E.M. Anscombe, P.M.S. Hacker and Joachim Schulte. Oxford: Blackwell, 2009. Print.