(Metrological-philosophical reflections not submitted to any journal)
Ingvar Johansson
Professor emeritus in philosophy, Umeå University, Sweden
(Address: Virvelvindsvägen 4P, SE-222 27 Lund, Sweden) ingvar.johansson@philos.umu.se
Abstract
The architects of the “new SI” claim to have achieved a theoretical-metrological unification of the SI system by basing all re-definitions of base units on fundamental constants of nature. This achievement is critically examined upon its merits and weaknesses. Logical and metrological arguments are put forward commensurate with the nature of the Avogadro constant and the Planck constant. As a result, competing alternative proposals for a re-definition of the mole and the kilogram are supported.
1. Introduction
The development of physics and chemistry can be seen as an ongoing interplay between finding empirical (but theory-laden) diversities and unifying them by means of some explicit theory, then on this new basis finding new empirical diversities, which, in turn, become unified, and so on. From this perspective, the so-called “new
SI” [1] is a remarkable attempt to reduce metrological diversities in the present SI system [2] by means of a unifying notion that is alternately called “fundamental constant” and “constant of nature”; the architects of the new SI lately renamed them appropriately “reference constants” [3]. All three notions refer to what in present-day physics and chemistry is regarded as invariant quantities in nature. In this wide sense, constants (or constancies) in nature do not only include constants such as c , h , and k that appear in fundamental equations, but also entities that are regarded as being exactly the same, independently of place and time, such as the elementary charge e and the number of periods per unit of a specified radiation in certain atoms.
In the proposed new SI, the seven base units are each defined by one or more such reference constants. Thereby, the very concept of a material prototype of unit (the kilogram) is abandoned. Another very important characteristic of the present SI is extended: it is considered allowable that definitions of base units (e.g. the metre) rely on derived quantities (e.g. velocity). For sure, the intent is good, but the proposal will not reach its unifying goal if the reference constants used differ too much in character.
I will show that the Avogadro constant (usual symbol N
A
), used to define the mole, is not a true constant of nature. And I will point out that another of the reference constants, the Planck constant ( h ), used in the proposed re-definition of the kilogram, although indeed being a constant of nature, is very special compared with other such constants. Therefore, the architects of the new SI should reduce the stress put on theoretical unification, and also should modify some of their proposals (see
Section 5). My paper takes its departure from earlier discussions [4-9].
1

2. The Avogadro constant/number and the mole
In history the ideal gas “law,” pV = nRT , was first formulated without any connection to statistical thermodynamics, but a similar law involving the Boltzmann constant, pV
= NkT , has been derived within statistical thermodynamics. Today, these two quantity equations cannot but be regarded as being two equivalent formulations of one and the same “law.” They are based on two different models of our knowledge about gases, but relate the same macroscopic quantities of pressure, volume, and temperature.
Therefore, if one of them contains a constant of nature, so does the other. In the first formulation R (the gas constant) represents such a constant, and in the second k (the
Boltzmann constant) does.
From the stated equivalence of these two equations, it follows that nR = Nk . Since n is a variable for a quantity that has the mole as its SI unit, and N is a variable for number of molecules, the gas constant R must be regarded as being a constant (of something) per mole of an ideal gas , and the Boltzmann constant k as being a constant (of something) per molecule of an ideal gas . The so-called Avogadro constant appears to be the Avogadro number as it enters the scene as a conversion factor from the gas constant R to the Boltzmann constant k :
R = N
A k .
In this equation, the Avogadro constant cannot possibly be regarded as a constant of nature on a par with the constancy of the velocity of light in vacuum ( c , used in the definition of the metre), the constancy of the number of periods of a specified radiation of the
133
Cs atom (used in the definition of the second), and the Boltzmann constant (used in the definition of the kelvin). This is, however, how it is regarded in the new SI [1].
If the two equations pV = nRT and pV = NkT are considered not only equivalent, but also valid for all values of their variables, then the names ‘gas constant’ and
‘Boltzmann constant’ would be two different names of a true constant of nature. But in such a case the Avogadro constant must be an Avogadro number (symbol e.g. N avo
) as the equation R = N
A k shows. A conversion factor between two representations of one fundamental constant cannot be but a number.
This makes me align with some chemists concerned with metrology; I am a philosopher. Ever since the SI system in 1971 introduced the mole as a base unit for the kind-of-quantity (entry 1.2 in [10]; or dimension , section 1.3 in [2]) amount-ofsubstance , many chemists have complained and claimed that this seemingly continuous kind-of-quantity forces stoichiometry – our way of thinking about number of atoms in molecules where all entities are discrete – into an alien metrological form.
They want to do away with the notion of a continuous “amount-of-substance” as well as with its presumed conventional measurement unit, the mole. Instead, they think that the Avogadro number can, and should, be regarded as a scaling factor (such as the dozen) for non-conventional elementary entities. The view that the quantity equations pV = nRT , and pV = NkT are equivalent in the practice of measurement, underpins the criticism more than it supports the architects of the new SI. Let me explain.
( N
The distinction between the Avogadro constant ( N
A
) and the Avogadro number avo
) with N
A
= N avo
mol
-1 is of utmost importance in what follows. Traditionally, i.e. since 1971, it is claimed in the SI that
1) since nR = Nk implies R = ( N / n ) k , and
2) since R is a constant per mole of a gas,
2

we must regard ( N / n ) as being the Avogadro constant, N
A
, i.e. as being the Avogadro number per unit mole of a gas ( N
A
= N avo mol
-1
). However, there is another and simpler interpretation possible.
Using the Avogadro number, one can claim that nR = Nk implies R = N avo k (not:
R = N
A k ). But then, in the name of consistency, since k is a constant per molecule of a gas, and N avo
a pure number, the mole cannot in this equation possibly be understood as being a measurement unit. In terms of the SI brochure’s notion of “dimension,” the left-hand and the right-hand sides of the equation R = N avo k must have the same dimension, but they have not if the right hand side needs no measurement unit at all, but the mole on the left hand side is an ordinary measurement unit. However, if the mole is regarded as only a pure scaling factor (i.e. a pure number) for discrete elementary entities such as atoms, molecules, ions, and electrons, then this dimensional requirement is met ! If one frees oneself from the false view that the mole must be an ordinary measurement unit for logical reasons, then all arguments go in favor of regarding it only as a scaling factor.
As far as I can see, all metrologically-interested scientists, especially chemists, ought to take a step back and, without any preconception, ask themselves: is the mole an ordinary conventional measurement unit, or is the so-called Avogadro constant in fact an Avogadro number, i.e. nothing but a scaling factor in relation to naturally given discrete elementary entities?
According to the view I have made mine, the mole can be regarded as being defined by the equality below, where E represents an arbitrary discrete kind of entity
(compare: x dozen E  x 12 E ):
 x mol (entities E )  x N avo
(entities E ).
As all pure numbers, N avo can take on many functions. In the equation x g = x N avo
Da, it functions as a conversion factor for two different conventional units of mass, namely gram and dalton; it functions here exactly the way 1.09 functions in x m
= x 1.09 yd (yard), a conversion of units of length.
When put down explicitly, it becomes obvious: one and the same number can function both as a scaling factor for a given natural unit, and as the number in a unit conversion formula for two conventional units (of the same kind-of-quantity). In a sense, a conversion formula is a special kind of scaling operation for a certain kindof-quantity.
One can choose whether to make the gram or the dalton (or something else) the unit for the continuous quantity mass, and then use the unit conversion formula above, but one cannot choose whether the notion ‘mole entities E ’ or the notion ‘entities E ’ should be the primary notion. It must be the latter. Why? Because E designates a discrete and in itself (i.e. without the help of any conventional measurement unit) countable sort of entity.
The view I have defended implies a rehabilitation of the use of the expression
‘number of moles’. On this view, it is as acceptable to use it as it is to use the phrase
‘number of dozens’. On the present SI conception of the mole, however, ‘number of moles’ is quite a misnomer.
(I have only discussed the Avogadro constant in relation to the gas constant and the
Boltzmann constant, but it appears also in relation to the Loschmidt number/constant and the Faraday constant. I am prepared to argue, however, that even in the latter two cases only a scaling factor is needed.)
3

3. The Planck constant and the kilogram
At present, the kilogram is defined as being the mass of the concrete platinum-iridium
International Prototype Kilogram (IPK) kept at BIPM in Paris. The new SI wants this definition exchanged for a theoretical definition where both the constancy of the velocity of light in vacuum ( c ) and the Planck constant ( h ) play very central roles.
Taking for granted that energy of relativity theory is the same as the energy of quantum mechanics, the equations E = mc
2
and E = hv can be combined into m =
( hv )/ c
2
; thereby creating the possibility of defining what is 1 kg mass by means of stipulating that 1 kg mass corresponds to specific values for h , c , and some radiation frequency v . The new kilogram definition is not directly related to the rest mass of any particle, but to the energy of photons ( hv ). The constant h is used only in the definition of the kilogram, but c is re-used; it figures in the definition of the metre, too. Also, the constant radiation frequency of the
133
Cs atoms is presupposed, since the definition presupposes the definition of the second.
As far as I can see, the definition put forward can, taken some of the presuppositions mentioned for granted, be phrased like this (compare: One metre is the distance travelled by light in vacuum when its speed is given the value
299 792 458 m s
-1
):
 One kilogram is the (relativistic) mass of a photon whose frequency in hertz is such that the Planck constant is equal to 6.626 06X  10
23
J s (X should be decided on later).
All measurement units are units in relation to a kind-of-quantity, and the same is true of constants of nature. A constant of nature is in itself not tied to any specific measurement unit, but it is tied to a specific kind-of-quantity. The constant c , for instance, can be stated in metre/second, kilometre/hour, yard/minute, and many other such units, but all of these units have to be units for the kind-of-quantity velocity. The
Planck constant is related to the kind-of-quantity action , i.e. energy times time (in SI units: joule times second, i.e. J s).
Ordinary physical and chemical kinds-of-quantity can – when regarded as referring to something real and not as being only simplifying mathematical tools – be regarded as in a commonsensical way being simple or complex properties of particles, waves, statistical ensembles, movements, or processes. That is, as something that directly – at a certain point of time – is an aspect of at least one of the kinds of entity mentioned.
By definition, even though action is a property in the wide sense of VIM [10], it cannot be a property in the ordinary sense just distinguished. Let us explain.
Action is defined as being energy multiplied by time ; which means that it cannot but be thought of as extended in time. It is as impossible to think of it as existing at a point of time, as it is impossible to think of a volume as existing in a plane, a surface existing in a line, or a line in a point. Speed is different; it can be defined as distance multiplied by reciprocal time (or divided by time). Therefore, the constancy of nature that is reflected in the notion of ‘the Planck constant’ cannot possibly be regarded as the time- and place-independent constancy of an ordinary property. And this is true both before and after the constant became part of quantum mechanics. Instead, the
Planck constant seems to represent (among other things) the “property” of some entities and/or processes to occur only in discrete quanta.
This fact – not discussed in the new SI – ought to make metrologists cautious before they make the Planck constant central to any base unit in the SI system. A
4

move from the IPK and its problems to a theoretical definition that relies on a constancy of nature can be done in other ways. The kilogram can very well be defined by the mass of some atom that is known to have a constant mass. According to one such proposal (see Section 5), the kilogram should be defined as the mass of a defined number of
12
C atoms (in their nuclear ground state). This definition has a structure that is completely analogous to the definition of the second in the present SI, a definition whose essential features are retained in the new SI. As said earlier, the second is defined by means of the constant radiation frequency of
133
Cs atoms.
A further oddity of the new SI definition of the kilogram might also be noted. If one calculates the “definition frequency” ( v d
) for 1 kg – using 1 = ( hv d
)/ c
2
– one obtains a frequency (rounded number: v d
= 1.4 x 10
50
hertz) that is much higher than any frequency that physicists today thinks any photon has. That is, the new kilogram definition is not only not directly related to any particle with rest mass, it is not related to any presently known kind of existing particle at all.
4. Metrological unifying and the constancy of constants of nature
In the introduction I mentioned the importance in physics and chemistry of unifying efforts, but added that what looks like a unifying feat may not always be one. Here, I would like to add some characteristics of a goal that is specific to metrology, and in whose light I think all unifying proposals in metrology should be judged.
The basic metrological unit problem is to find entities that do not change but are constant in time, be these individual property instances in concrete things or theoretically defined quantities . With respect to concrete prototypes such as the IPK, the problem implies that the prototype has to be constructed in a certain way, be stored in a certain way, be handled in a certain prescribed way, and now and then checked against copies similarly constructed and treated. As the history of the IPK shows, changes can nonetheless occur.
The corresponding problem with theoretically defined units based on constancies of nature is that all such constancies are part of some empirical theory, and that no such theory is immune to revision as a matter of principle. As the history of physics and chemistry shows, now and then a seemingly well established theory has to be refined, or even replaced by a more sophisticated one. Therefore, even seemingly well established constants of nature may not forever be regarded as such. For any kind of definitions, epistemological uncertainty is part and parcel of empirical science.
The upshot of these remarks is the following. If metrology faces a conflict between defining the base units
(a) by means of definitions that seem to secure unit constancy over time , but are theoretically heterogeneous, and
(b) by means of some theoretically homogeneous notions , but with less secured time constancy, then alternative (a) should be chosen. In metrology, demonstrated unit constancy should take precedence over theoretical unit homogeneity .
Therefore, there is nothing wrong with the heterogeneous mix in the present SI of prototype definitions and theoretical definitions; nor is there anything wrong with the mix in the new SI proposal between constants of nature taken from fundamental equations and supposed constancies of nature. There is no mandatory or strict metrological need saying: the more base units based on constants from basic quantity equations, the better.
5

As far as I can see, there is nothing in today’s physics and chemistry that makes it
12 reasonable to think that it is less certain that each C atom (in its nuclear ground state) will forever be regarded as having the same mass everywhere, than it is that c and h will forever be regarded as being “fundamental” constants of nature. Let me add two comments, which show that – surely – this is not just a fancy of mine.
In July 2012, the physicist John Webb and collaborators were awarded the
Australian Museum Eureka Prize in Scientific Research [11]. They received it for work that may show that the so-called fine structure constant (  ) varies across the universe. This constant is in physics regarded as a fundamental constant, and one of the equations in which it appears is  = e
2 c 
0
/2 h . Here 
0
represents the magnetic permeability of free space, which so far has been regarded as a constant, too. Now, however, if the prize winners’ measurements contain nothing wrong, then one of the assumed constants e , c , h , and 
0
must in the future be regarded as a variable. If the new SI is accepted, then e , c , h should be regarded as constants, and 
0
must be regarded as what co-varies with  . I find it odd that a decision among metrologists should have such an immediate substantive implication for physics.
Moreover, the same researchers claim that a variation of  “would show the existence of a preferred cosmological frame, which would demonstrate the incompleteness of the Einstein Equivalence Principle” [12]. If this conclusion is correct, it directly undermines the kilogram definition of the new SI. As already said in Section 3, this definition has as one of its assumptions that the equations E = mc
2
(the equivalence principle) and E = hv can be combined into m = ( hv )/ c
2
. Therefore, the new SI definition presupposes that the equivalence principle is complete, but in the paper referred to it is said to probably be incomplete.
In September 2011, physicists within the so-called OPERA collaboration released results that seemed to show that neutrinos can move faster than light, but already in
February 2012 they themselves contested it. Now, if nonetheless their initial claim would be repeatedly confirmed, something would have to be changed in relativity theory; qualifications of some sort would have to be introduced. However, whether or not such qualification proposals would affect the view that c is constant is a completely open question. They may and they may not. I am mentioning this
“neutrino case” only because it illustrates the possibility that the symbol c may after all not necessarily name a real property constancy of nature. And the same possibility exists with respect to h . The move from the IPK to a definition of the kilogram by means of c and h is not a move from a precarious temporal constancy to a certain constancy .
Certain, however, is that ‘the Avogadro constant/number’ is not a name for a constant of nature. But it might, of course, nonetheless be given a place in an SI brochure as a scaling factor, as is done in the proposal below. For historical reasons, if no other, it would be curious now to delete it altogether.
5. Possible new definitions of the mole and the kilogram
In the light of the reflections made above, and in order to prevent oddities to be built into the new SI from its inception and, of necessity, surfacing later, I connect to two quite independently of me proposed changes to the new SI. Also, by the way, these definitions are much more transparent and easier to teach [13].
There is a problem with the quantity concept of which the mole is the unit that needs to be resolved before the unit can be tackled; the concept ‘amount-of-
6

substance’ should be replaced by ‘number of entities’ (entry 1.4 Note 3 in [10]).
Why? Because the former term gives the impression that the concept refers to a quantity that takes continuous values, whereas in today’s physics and chemistry it refers to naturally discrete entities. That is, it refers only to entities that , without the
help of any measurement unit can be counted in principle, but which often for the sake of easy “communication” are in need of a scaling factor. In my opinion, VIM:s definition of quantity (entry 1.1 in [2]) should be interpreted in such a way that collections of entities that are countable without a measurement unit can be called quantities. Collections of elementary entities must be reckoned quantities.
In the new SI it is stated that an elementary entity “may be an atom, molecule, ion, electron, any other particle or a specified group of such particles.” This long and open-ended expression can just as well be exchanged for one that simply says that the entities in question must be identical and specified.
The number used in the definition below could, in principle, be chosen at will.
However, in order to ensure continuity of the worthiness of data measured in the past,
I guess the same numerical value ought to be taken as has been obtained in the international projects on the so-called Avogadro constant (obtained in the Watt balance and Si projects); metrological compatibility of results has now been achieved in these projects. My proposed new re-definition of the mole looks like this:
 The mole, symbol mol, is the unit for measurements of a large number of discrete elementary entities; the entities must be identical and specified. 1 mole consists of 6.022 141 79  10
23
entities.
Except for the phrase ‘a large number of’, several chemists are in favor of this definition; I have added the phrase in order not to stipulate that the mole has to be used when it comes to talk about one or a few elementary entities. Probably, some of the chemists I think of have argued for their view inside CCQM (Consultative
Committee for Amount of Substance). The same goes for this definition of the kg:
 The kilogram, symbol kg, is the unit of mass. It is the mass of 6.022 141 79 x
10
23
atoms of
12
C atoms in their nuclear ground state, multiplied by 1000/12.
A logical consequence of the definition is that the kilogram becomes an exact number of what to my knowledge many chemists regard as a natural unit of mass:
1/12 of the mass of a
12
C atom.
Another good consequence is this. In the present SI, the Dalton (Da) and the unified atomic mass unit (u) cannot be defined within the system, since their value in kg has to be determined experimentally; both units are nonetheless accepted for use with the SI. Now, if the kilogram definition is, as above, made
12
C-based, then both
Da and u (1 Da = 1 u) can be defined within the SI system and be regarded as integral parts of it.
According to present-day physics and chemistry, the proposed magnitude for the unit of mass is always and everywhere the same. Thus an important requirement on units – being traceable to a phenomenon, stable and the same anywhere anytime – is fulfilled.
7

References
1. BIPM (2010) Draft Chapter 2 for SI Brochure, following redefinitions of the base units .
http://www.bipm.org/utils/common/pdf/si_brochure_draft_ch2.pdf
. Accessed 17 October 2012
2. BIPM (2006) The International System of Units (SI), 8th edn.
http://www.bipm.org/en/si/si_brochure/ . Accessed 17 October 2012
3. Taylor B (2011) The Current SI Seen From the Perspective of the Proposed New SI. J Res Natl
Inst Stand Technol 116:797–807
4. Johansson I (2011) The Mole is Not an Ordinary Measurement Unit. Accred Qual Assur 16:467–
470
5. De Bièvre P (2007) Numerosity versus mass. Accred Qual Assur 12:221–222
6. Johansson I (2010) Metrological thinking needs the notions of parametric quantities, units and dimensions. Metrologia 47:219–230
7. De Bièvre P (2011) Integer numbers and their ratios are key concepts in describing the interactions of atoms and molecules. Accred Qual Assur 16:117–120
8. Price G, De Bièvre P (2009) Simple principles for metrology in chemistry: identifying and counting. Accred Qual Assur 14:295–305
9. Cooper G, Humphry SM (2012) The ontological distinction between units and entities. Synthese
187:393–401
10. International Vocabulary of Metrology – Basic and General Concepts and Associated Terms, VIM
3rd edition JCGM 200:2012. http://www.bipm.org/vim . Accessed 17 October 2012
11. http://eureka.australianmuseum.net.au/eureka-prize/scientific-research7
12. King J, Webb J, et al. (2010) Possible Cosmological Spatial Variation in the Fine-structure
Constant. Journal & Proceedings of the Royal Society of New South Wales 143: 35–39
13. Hill T (2012), The Kilogram Cabal. The Chronicle of Higher Education, July 2 2012. http://chronicle.com/article/The-Kilogram-Cabal/132617/
8