A DEGLI STUDI DI PAVIA
Dipartimento di Studi Umanistici
Laurea in Filosoﬁa
The Deﬁnition of Units in the New International
System: Historical and Epistemological Issues
Tesi di laurea di
Relatore: Prof. Lorenzo Magnani
Correlatore: Dott. Tommaso Bertolotti
Anno Accademico 2015-2016
and whom she is carrying
For in pursuing the truth one must start
from the things that are always in the same state
and suﬀer no change.
The Bureau International des Poids et Mesures (BIPM) is scheduled to pub-
lish in 2018 a new edition of the Brochure, with the deﬁnition of the seven
base units of the International System (SI). The new deﬁnition, now circu-
lating in draft, is the most substantial change in the SI since its foundation
in 1875, at the signature of the Metre Convention: in fact, for the ﬁrst time
all base units (and thereafter also all derived units) will no longer be deﬁned
based on a real physical object (as the kilogram prototype), or on a known
physical phenomenon of a known reference material (as the triple point of
water), or on a thought experiment (as the force of attraction between two
parallel wires, placed at known distance and carrying an electric current),
but based on “constants of nature”, whose numerical values will be ﬁxed.
The epistemological value of this extraordinary event cannot be ignored:
afterwards, all measurement instruments will be calibrated based on these
constants, ideally ﬁxed once and forever.
In this thesis, the author introduces ﬁrst the approach to the concept of
measurement with some examples from the history of philosophy: a path is
traced, which starting from ancient Greek philosophy and through modern
science, brings to the current deﬁnition of measurement in the International
Vocabulary of Metrology, used nowadays as the isomorphism between mea-
surement quantities (the physical magnitude: length, time, mass or others)
and the real numbers. Then the ontological approaches to the process of
measuring are shown: from subjective relativism, to operationism, to rep-
resentationalism. Switching to measurement units, the importance in their
standardization and in the creation of the SI is stressed, also as “moral media-
tors”. The new deﬁnitions of the second, the metre and the kilogram are then
analysed (with the controversial ﬁxation of the hyperﬁne splitting frequency
of caesium atom, of the speed of light in vacuum and of Planck constant).
Some critical arguments against the New SI are then presented: whether or
not the Metre Convention is fulﬁlled; the diﬀerence between “constants of na-
ture” and “technical constants”, with the meaning and consequences of their
ﬁxation; whether constants of nature are “true constants”, or they are “as-
sumed to be constant”. Eventually, the problem of “true” constants of nature
is critically addressed, according to either a realist or an idealist ontological
point of view, showing where the New SI proposed by BIPM demonstrates
to follow one or the other approach, referring back to the examples from the
history of philosophy presented in the introduction.
1 Introduction to measurement and the philosophy of measure-
1.1 Examples from the history of philosophy . . . . . . . . . . . . 1
1.2 Deﬁnition and theory of measurement . . . . . . . . . . . . . . 5
1.3 Epistemological value of measurement . . . . . . . . . . . . . . 8
2 Fixing constants of nature 11
2.1 Measurement units and knowledge value of standardization . . 11
2.2 Anthropometric, SI and natural units . . . . . . . . . . . . . . 13
2.3 TowardstheNewSI ....................... 18
2.3.1 The second and the metre . . . . . . . . . . . . . . . . 18
2.3.2 The kilogram: ﬁxing Planck constant . . . . . . . . . . 20
3 Further epistemological considerations 23
3.1 Some arguments against the New SI . . . . . . . . . . . . . . . 23
3.2 Physical constants as immutable entities . . . . . . . . . . . . 25
3.3 Bridges between real and ideal entities . . . . . . . . . . . . . 28
4 Conclusions 31
A Tables 33
Introduction to measurement
and the philosophy of
1.1 Examples from the history of philosophy
“If he wasn’t one of the greatest philosophers of the ancient Greece, he would
have probably been the greatest poet”, so or something similar said Nigel
Warburton on Plato, during the presentation of his book A Little History of
Philosophy at Waterstone’s Piccadilly, London [War11]. An essay of Plato’s
skills as a dramatist can be found in the Theaetetus [Pla04, Kah13]. There,
Euclid and Terpsion of Megara report a dialogue occurred about 30 years be-
fore between Socrates and the two mathematicians Theodorus and Theaete-
tus, Theodorus’ disciple. A dialogue in a dialogue, then: the latter taking
place presumably in 369 BC, the day after a battle of Corinth, in which
Thaetetus was deathly injured, as is reported by Euclid at the beginning of
the play; the former, while Socrates was going to the King’s Porch to face
the indictment of Meletus in the famous trial (399 BC).
There are some issues making the Theaetetus an interesting starting point.
The title character and mathematician Theaetetus (ca. 414 - 369 BC) is sup-
posed to have demonstrated that there are ﬁve and only ﬁve regular polyhe-
drons [Boy68], the stable “elements” in Plato’s cosmology of the Timaeus;
and he was also the ﬁrst to classify irrational numbers (Theaet., V, 147e-
148b). His friend and master Theodorus of Cyrene demonstrated the irra-
tionality of the square root of some odd integers from √3 to √17 (Theaet.,
V, 147d). The irrational numbers are fundamental to set an isomorphism be-
tween physical and mathematical quantities in the theory of measurement.
2CHAPTER 1. INTRODUCTION
These and other discoveries were probably sources of reference for the Book
X of the Elements by Euclid, not to mention that Plato himself (428/427 -
348/347 BC) was a distinguished mathematician who inspired Greek mathe-
maticians and Euclid himself. And Euclidean geometry has been almost the
unique source of reference for the whole theory of measurement until at least
the XIX century.
The Theaetetus is considered as Plato’s work on epistemology, challenging
the concept of knowledge from what is essentially an empiricist perspective,
raising the question: is knowledge possible at all from sensation? In fact, an-
other co-protagonist of the dialogue is the “ghost”-character Protagoras (484
or 481 - end of V century BC): the sophist philosopher, who is not present
in the dialogue, but whose philosophy is central in it. Most importantly his
subjective relativism, according to which
man is the measure of all things - of the things that are, that they
are; of the things that are not, that they are not (Theaet., VIII,
Protagoras’ argument is illustrated by Socrates; then it is also confuted
by him, arguing that subjective relativism cannot be (according to Socrates-
Plato) the basis of knowledge (epist¯em¯e). Then, what is the target of
epist¯em¯e? In the Theaetetus, Socrates explicitly refuses to comment on the
Eleatic claim (for example in the Parmenides) that knowledge requires a
stable being as its object: the dialogue limits itself to a refutation of the ﬂux
theory and of the reliability of a sensation-based knowledge. According to
Kahn [Kah13], this negative conclusion implies “a positive conclusion that
some elements of unchanging Being is required as an object for knowledge
[...]. Hence the positive sequel to the negative outcome of our dialogue will
be assigned not to Socrates, Plato’s spokesman for the classical theory, but
to a sympathetic visitor from Elea” in the sequel dialogue of the day after,
Disciple of Plato, Aristotle (384-322 BC) stresses in its Metaphysics [Ari14]
(see Met. I, 6) the problem of the distance in Plato between the ideal Forms
and the changing things that make nature and are object of sensation and
observation (and of measurement, I add). Less a mathematician then his
master, he was quite far from the technicalities and abstract mathematics of
the other Plato’s disciples in the Academy1: indeed, he suggested (Met. I,
9, 991b) the possibility to set the numbers-Forms (though, strictly speaking,
1M`edeis ageˆometr`etos eisitˆo mou t`en steg`en (“let no one ignorant of geometry come
under my roof”) was engraved at the door of Plato’s Academy, according to the tradition
1.1. EXAMPLES FROM THE HISTORY OF PHILOSOPHY 3
numbers were not Forms, in Plato) as the cause of the numerical ratios that
we see in natural things (as for example, in measures). Measurement, and
measurement units, are therefore objects of metaphysics:
Measure is that by which quantity is known, and quantity qua
quantity is known either by unity or by number, and all number
is known by unity. Therefore all quantity qua quantity is known
by unity, and that by which quantities are primarily known is
absolute unity. Thus unity is the starting point of number qua
number. Hence in other cases too “measure” means that by which
each thing is primarily known, and the measure of each thing is
a unit - in length, breadth, depth, weight and speed. [...]
In all these cases, then, the measure and starting-point is some in-
divisible unit (since even in the case of lines we treat the “one-foot
line” as indivisible). For everywhere we require as our measure
an indivisible unit; i.e., that which is simple either in quality or
in quantity (Met., X, 1, 1052b, 20-35 [Ari14]).
Here some concept of unit of measurement is introduced: a concept that
according to Aristotle is used in physics as in all other sciences, but it is
studied and analysed itself in the “ﬁrst philosophy” as a source of knowledge:
The essence of what is one is to be some kind of beginning of
number; for the ﬁrst measure is the beginning, since that by which
we ﬁrst know each class is the ﬁrst measure of the class; the one,
then, is the beginning of the knowable regarding each class (Met.,
V, 6, 1016b, 15-20 [Ari14]).
Thereafter, while Plato in the Theaetetus just criticizes the subjective
relativism in measurements, Aristotle goes even further, in what can be seen
a ﬁrst rationalist approach to measurement. Few decades later, this will ﬁnd
a ﬁrst attempt of mathematical rigour in the Euclidean geometry. In fact, in
Euclid’s Elements, Book V, the deﬁnition of Aristotle’s Metaphysics, Book
X above, is formalized [Mic04]: the measure of a magnitude Mrelative to a
unit uis deﬁned as the positive number Qgiven by
and the equation holds even when the quantities Mand uare incommen-
surable, as a heritage of Theaetetus’ theory of irrational numbers: in mod-
ern terms [Rus20, RW13] this is to say that Mand Qare real numbers
(M, Q ∈R), when u= 1.
4CHAPTER 1. INTRODUCTION
Quantitative magnitudes as lengths can be added and their measurements
add as well, but this is not the case of certain physical magnitudes as temper-
ature: in this case adding volumes does not result in adding temperatures.
It was John Duns Scotus in the Middle Age to account of those magnitudes,
which are now called intensive: thanks to him, medieval scientists were able
to distinguish between the concept of quantitative magnitudes and extensive
or intensive addition. In other words, even properties of physical bodies like
temperature that were thought to be “qualitative” (therefore not quantiﬁ-
able, not measurable) were indeed quantitative. Following J. Michell [Mic04]
such breakthroughs ﬂourished in the XIV century and ﬁnally led to Galileo’s
philosophy [of nature] is written in this grand book - I mean the
universe - [...] in the language of mathematics, and its charac-
ters are triangles, circles, and other geometrical ﬁgures, without
which it is humanly impossible to understand a single word of
it; without these, one is wandering around in a dark labyrinth
(Galileo Galilei, The Assayer, [Dra57]).
The possibility of measurement as source of knowledge of the physical
objects and the role of geometry in that are discussed by Immanuel Kant
(1724-1804) in the Critique of Pure Reason, aiming to solve Hume’s concerns
on the knowability of the objects themselves. A description of Kant’s theory
of geometry and therefore measurements can be found in Magnani [Mag90],
referring to the Axioms of Intuition ([Kan67], p. 206-209). It is well-known
that, according to Kant, space and time are known a priori, which means
independently on the experience. In his transcendental exposition, Kant
shows that it is only as an act of synthesis that space becomes measurable,
as all physical objects as intuitions in space and time. Asking the reader to
take the proposition
two straight lines can neither contain any space nor, consequently,
form a ﬁgure (Immanuel Kant, Critique of Pure Reason, A47-
and then to try to derive it analytically from the concepts of a straight line
and the number two, Kant concludes that it is simply impossible. Thus,
since this information cannot be obtained from analytic reasoning, it must
be obtained through synthetic reasoning, i.e., a synthesis of concepts (in this
case “two” and “straightness”) with the pure (a priori) intuition of space.
According to Brittan:
1.2. DEFINITION AND THEORY OF MEASUREMENT 5
The unity of consciousness requires that in an important respect
we “construct” the objects of our experience. From this stand-
point the proof of the Axioms [of intuitions] turns on the fact that
our “construction” are extensive magnitudes. We are reminded
also that quantities are not “given”; they are “constructed” by
some of the same acts of synthesis that “constructs” the objects
that have them. [...] It is simply that objects are objects of expe-
rience, “really possible” objects, on the condition that we concep-
tualize them in certain ways. In particular, objects are objects
of experience insofar as they are determinate, measurable. This
is a fundamental condition of objectivity. But the measurability
of objects, and the possibility of taking them up into empirical
consciousness, depends ultimately on certain conceptual abilities
of us (G. G. Brittan, Kant’s Theory of Science, p.112 [BJ78]).
In the contemporary philosophy of science, according to Thomas Kuhn
[Kuh61] the fact that a measurement is quantitative and therefore produces
numbers has a fundamental role in the conﬁrmation of scientiﬁc theories:
but, according to Kuhn, in “normal science” measurement is almost always
targeted to explicitly demonstrating the success of the existing theory. Most
importantly “it is through abnormal states of scientiﬁc research that mea-
surement comes occasionally to play a major role in discovery”: this “abnor-
mal situation” leads to the “crisis state”. Quantitative measurements then
provide the authority that may trigger, through crisis, the development of
After having sketched the progress in the concept of measurement from
Plato to the contemporary philosophy of science (some more in-depth analy-
sis on modern philosophical discussions about measurement is performed in
section 1.3), in the next section I will give the current standard deﬁnition of
measurement, with its mathematical formalism, and show how this matches
much of the ancient Aristotelian/Euclidean deﬁnition (1.1).
1.2 Deﬁnition and theory of measurement
According to the International Vocabulary of Metrology (VIM, [BIP12]), of
the Bureau International des Poids et Mesures (BIPM), a physical quantity
property of a phenomenon, body, or substance, where the prop-
erty has a magnitude that can be expressed as a number and a
reference [clause 1.1],
6CHAPTER 1. INTRODUCTION
where the number is referred to as the quantity value and the reference is
typically a measurement unit.
Ameasurement is then the
process of experimentally obtaining one or more quantity values
that can reasonably be attributed to a quantity [clause 2.1].
Calling Mthe magnitude, Qthe quantity value and uthe measurement
unit, than the VIM deﬁnition of measurement equals the Aristotle-Euclidean,
300 BC deﬁnition of eq. (1.1).
Measurement is then a map between the set of magnitudes MQof a phys-
ical quantity and the set of real numbers R. In the mathematical jargon
[BV03], such a relation is an order isomorphism2, because it preserves a re-
lation of order. If we indicate with (MQ,) and (R,≤) the two sets with
their order relation (an empirical one, , and a numerical one, ≤), then, with
reference to the quantities in eq. (1.1), the order isomorphism is the map
µ: (MQ,)→(R,≤), so that q→µ(q) = M
and, given q1, q2∈MQ, then
if (and only if)
The order relations and ≤have the following properties:
1. reﬂexivity: for every µ∈Rand q∈MQ,µ≤µand qq;
2. antisymmetry3: for every µ1, µ2∈Rand q1, q2∈MQ, if µ1≤µ2and
µ2≤µ1, then µ1=µ2; and if q1q2and q2q1, then q1=q2;
2Diﬀerently from Boniolo and Vidali [BV03], I will consider the map between MQand
Ran isomorphism and not an homomorphism because MQis here the set of magnitudes
of a physical quantity, not the set of physical objects with that particular quantity, as in
[BV03]. In this case, the map is bijective (and therefore the homomorphism is by deﬁnition
an isomorphism): in fact, from µ(q1) = µ(q2) it does follow that q1=q2(e.g. if two rods
1 and 2 have µ(q1) = µ(q2)=1 m, then the two rods have the same length, that is q1=q2,
even if they are indeed diﬀerent physical objects). The reason of this choice is based on
epistemological considerations, as will be clear in the next section.
3Again, Boniolo and Vidali [BV03] refers to a “weak” order relation, in which the
antisymmetric property does not apply to the set of physical objects; it does apply here
to the set MQof magnitudes.
1.2. DEFINITION AND THEORY OF MEASUREMENT 7
3. transitivity: for every µ1, µ2, µ3∈Rand q1, q2, q3∈MQ, if µ1≤µ2and
µ2≤µ3, then µ1≤µ3; and if q1q2and q2q3, then q1q3.
Is such an isomorphism possible at all, and is it uniquely deﬁned? The
applicability of this deﬁnition relies on the deﬁnition of the unit of mea-
surement uin eq. (1.1-1.2): based on the choice of u, then, it is possible
to assign to a magnitude q∈MQof a physical quantity a quantity value
Q∈Rand therefore to perform a measurement, getting the measurement
value M=Q×u∈R. Then a measurement can be performed (i.e. the
isomorphism can be established) when the unit of measurement uis deﬁned,
which again leads us to the Aristotelian deﬁnition: “measure means that by
which each thing is primarily known, and the measure of each thing is a
Diﬀerent units of measurement can be chosen and this challenges the
uniqueness of measurement: the deﬁnition of the units is driven by historical,
geopolitical, technological and theoretical constraints, as shown in chapter
2 and I will discuss in chapters 3 some epistemological consequences in this
Before concluding this chapter with a tentative overview of the various
possible epistemological approaches to the concept of measurement in the
next section, a ﬁnal comment on the standard deﬁnition of measurement.
According to the standard ISO Guide of the expression of uncertainty in
measurement (GUM, [BIP08]) the expression of measurement result requires
the indication of its uncertainty. Quoting from the VIM [BIP12]:
The Uncertainty Approach is to recognize that, owing to the in-
herently incomplete amount of detail in the deﬁnition of a
quantity, there is not a single true quantity value but rather a set
of true quantity values consistent with the deﬁnition. However,
this set of values is, in principle and in practice, unknowable
[clause 2.11, Note 1, my emphasis].
This appendix to the deﬁnition of measurement seems to redirect it away
from the quasi-Aristotelian approach above. The shift from measurement
error to measurement uncertainty, in fact, is far more than purely termi-
nological, as stated for example by Luca Mari in [Mar03]. As I will show
in the next section (and further comment in Chapter 3), this may look like
more a Platonic-idealistic approach to the philosophy of measurement, than
a Euclidean-realistic one. In the words of Mari, “numbers are symbols (i.e.
information entities) for [...] empirical entities: numbers do not belong to
8CHAPTER 1. INTRODUCTION
Figure 1.1: Concept of measurement and epistemological strands (in bold);
the corresponding ontological approaches are shown in italic.
1.3 Epistemological value of measurement
In this last introductory section I will try to answer the question “what is
the epistemological value of measurement?” Possible approaches in contem-
porary philosophy are highlighted in bold in the scheme of Fig. 1.1: in italic,
the corresponding ontological point of views are also addressed.
To answer the question, let us start with the sceptic attitude of those who
think that there is no value at all in measurement: for the sceptic as a result
of a measurement a quantity value can be assigned to the measurand, but no
information to the object itself (if any) can be inferred, which could even lead
to solipsism as the extreme consequence. This approach can be considered
ﬁctionalism in measurement. I claim that also a certain extreme version
of Protagoras’ subjective relativism of the Theaetetus can be ascribed to
this strand in the epistemology of measurement.
Following the example by J. Michell in [Mic94], consider a measurement
of a rod whose result is
L= (1.25 ±0.01) metres (1.5)
(where the given uncertainty has been calculated assuming a k= 2 coverage
factor, corresponding to a 95% level of conﬁdence, according to the require-
ments of the GUM [BIP08]). Then the ﬁctionalist argues that, independently
on the existence of the rod, on whether or not it-itself has any property called
“length” and on whether we are measuring that property or anything else, it
is useful and convenient to think “as if” the rod exists, it has a quantitative
1.3. EPISTEMOLOGICAL VALUE OF MEASUREMENT 9
property called “length” and its measurement value is approximately 1.25
metres. A subjective relativist like the Theaetetus ’s Protagoras would say
that he-himself sees a rod of length approximately 1.25 metres, but cannot
say anything on what other observers may see (in the extreme version, they
may even see nothing at all!).
Consider now those strands that have no doubts on the epistemological
value of measurement. First I distinguish between those who consider that
the results of a measurement depend on the operator, rather than on the
measurand. This is the position of the operationist, according to whom
the outcome of measurement (1.5) is what the operator’s own attitude and
procedure deliver when applied to some external measurand called “rod”.
The operator than is 95% conﬁdent that the rod has a length between L=
1.24 and L= 1.26 metres, thus leaving a 5% probability that Lis totally
wrong. I argue that from an ontological point of view this may be Kant’s
position, and that of idealism as an extreme consequence.
Consider now the realist account. In this case there is no doubt that the
object of the measurement is the measurand, more or less independent on the
action and inﬂuences of the operator; but then there are three possibilities
depending on by which means the operator gets in touch with the measurand:
1. A “Baconian” empiricist is an objective realist, believing in his senses
and believing that his senses give the most direct and appropriate view
of reality. In the example (1.5), the objective realist will try to improve
the measurement environment and instrumentation to reduce to zero
the uncertainty (which is seen as a possible “error” in measurement)
and ﬁnally state that the “true value” is L= 1.25 metres.
2. A traditional realist is an empiricist like the previous one, but he
acknowledges that he interacts with the measurand by means of one
or more instruments, that can more or less inﬂuence the result of mea-
surement. In eq. (1.5) he asserts the isomorphism between the length
of the rod and the real number 1.25, as a ratio between the length of
the rod and a reference length (the standard metre). He has measured
the length of the rod, technically not the rod itself (even if of course he
is sure that the rod itself exists): that is why I think is more correct to
consider it an isomorphism (and not a homomorphism as in [BV03]), as
I noticed in the previous section. The traditional realist also estimates
that environmental conditions and calibration of the measurement in-
strument (with traceability to the standard metre) may give rise to an
“error” of maximum ±0.01 metres4.
4With respect to the (either Kantian or idealist) approach of the operationist, I argue
10 CHAPTER 1. INTRODUCTION
3. Between the traditional realist and the operationist stands the ratio-
nalist position of the representationalist [Mic94, Mar03, VF08], for
whom the theory of numbers in measurement represents empirical real-
ity and, as an extreme consequence, measurement can be characterized
in pure mathematical terms, in an asbract measurement theory5. For
the representationalist, in eq. (1.5) the real number 1.25 represents
an empirical reality (the length of the rod): other numbers could do
the same job, then it is important to state the unit of measurement,
which is chosen by a standardized convention (but, diﬀerently from my
view of he pure conventionalist and in agreement with Michell [Mic94],
“behind that convention stands a fact and eq. (1.5) represents that
fact numerically). The “uncertainty” value ±0.01 metres is calculated
in perfect agreement with the measurement theory.
The example in eq. (1.5) of the length measurement of a rod may look
quite trivial: but it is not, in particular with reference to the current def-
inition of the metre unit, as discussed in the next chapters. But of course
the analysis would look less trivial and more troublesome if it was about the
diameter of the Milky Way, for example, or that of an electron or a neutrino,
just to keep the analysis on measurements of physical length.
The concept of measurement mostly diﬀused among scientists and philoso-
phers of science from the beginning of the XX century until nowadays is
somewhere between the realistic representationalism and the idealistic oper-
ationism. In the next chapters I will ﬁrst put the attention on measurement
units and the importance of standardization on the choice of the units (chap-
ter 2); then in chapter 3 I will comment on the possible epistemological im-
plications, within the strands just introduced in this section, that lay behind
the choices described in the draft Brochure of the new International System
of measurement units.
here that the traditional realist (the same as the objective realist above) considers any
deviation from the true value a “measurement error”, rather than a “measurement un-
certainty”. For the realist, the true value can be known, by virtue or by chance; for the
operationist the true value is in principle unknownable.
5Some may argue that this position is also not far from the Kant’s approach of a priori
synthesis brieﬂy introduced in section 1.1.
Fixing constants of nature
2.1 Measurement units and knowledge value
John D. Barrow in his The Constants of Nature [Bar02] reminds us on the
importance of the agreement in measurement units referring to a well-known
scientiﬁc disaster (worthy of 125 million dollar lost) due to unit misunder-
standing: the Mars Climate Orbiter mission, launched by NASA on Decem-
ber 11, 1998, to study the climate on planet Mars. The spacecraft disin-
tegrated when entering Mars atmosphere 100 km below the target orbit of
around 150 km from the ground: the error was admitted by the incredulous
head of the scientiﬁc committee, due to input data transmitted from the
ground-base software in “imperial” units1(the pound-second, lbf s) to the
spacecraft software, running in the metric units of the International System
(Syst`eme international d’unit´es, SI): the newton-second, N s.
Unfortunately this is just the most famous event of the like, not the only
one. In 1983 Air Canada switched to metric SI units for the measurement of
fuel on their air-crafts, but in the ﬁrst ﬂight afterwards the crew reﬁlled using
imperial units, thus running out of fuel in the middle of the ﬂight (the emer-
gency landing was safe). In 1999 a patient was reported by the American
Institute for Safe Medication Practices for having received 0.5 grams of seda-
tive instead of 0.5 grains (approximately 0.03 grams). And even Columbus
had a relevant unit-conversion problem: he miscalculated the circumference
of the Earth (Roman miles instead of nautical), assuming he was in India
while he reached the coasts of the Bahamas [Con10].
1This is the common name of the practical units in use in the United Kingdom, in
the most of the countries of the former British Empire and, with little variations, in the
12 CHAPTER 2. FIXING CONSTANTS OF NATURE
The examples above highlight the role of standardization of measurement
units as a form of “moral mediator” in the technological society, as Lorenzo
Magnani illustrates in his Morality in a Technological World [Mag07]. In
fact, standardization leads to a more or less general agreement in measure-
ment that, other than avoiding possible disaster of various entity and nature,
presents several more advantages for people in the contemporary world. The
advantages of standardization came into evidence ﬁrst in the ﬁeld of trans-
port, commerce and communication [MP14, Gal03]. With the development
of railways in Europe and North America in the XIX century it was soon
realized that local time (based on the position of the Sun in each town) was
highly impractical, for example to ﬁx reliable timetables. On December 1st
1847 in Great Britain the railway companies started adopting a common
standard time based on the Greenwich Mean Time (GMT, the local time
at the Greenwich Royal Observatory, used since 1675 by British mariners
to determine longitude). Similar decisions were taken in North America, in
the British colonies, in France and worldwide. The synchronization of time
between diﬀerent towns ﬁrst, then between diﬀerent countries, was ﬁnally
possible due to electriﬁcation of the telegraphic network, allowing accuracy
within few seconds all over distant countries.
I will brieﬂy describe in section 2.2 how standardisation was linked to a
rationalisation of measurement units during the Age of Enlightenment. I will
show how measurement unit were originally anthropometric, dating back to
the ancient Roman units or before. With no standard reference, they were
subject to the will of the dominant authority: often the reference length of
“one inch”, for example, was the thumb of the king, and the king had the
authority to change the reference, according to his desiderata and conve-
nience. Thus standardization of measurement units acted also as a form of
“democratization” of measurements, defending people interests (merchants,
artisans, peasants, travellers,...) against the governing authority.
As stated by Accredia (the Italian body providing accreditation to cer-
tiﬁed laboratories and research centres), around 10% of the Gross Domestic
Product of the more economically developed countries is made by testing,
analysis, measurements and the like, with constantly growing trend. The
quality of measurements is based on the conﬁdence in measurement results:
this is related to standard test methods and standard rules to calculate the
Standardization of units allows traceability of measurements to reference
standards, and measurement traceability is a strong moral mediator. In
fact, there is a “conﬂict of interests” in measurements and trading [UKA12].
Assume I go to the market to buy 1 kilogram of potatoes: if the weighting
machine reads high, then the trader gains (and I loose); if it reads low, then
2.2. ANTHROPOMETRIC, SI AND NATURAL UNITS 13
I gain (and the trader looses). We (myself and the trader) need an agreed,
common measurement: this means that traceability is fair to both the seller
and the buyer; furthermore, it means that I can buy from diﬀerent sellers and
get the same amount of good, therefore traceability enhance and “regulate”
the free market.
Standardization is typically international and it was so from the Metre
Convention onwards: it does not permit, at least in principle, monopoly in
metrology by a single nation and its possible consequences in international
trade, because all nations cooperating in the standards are allowed to build
their own replicate of any primary standard unit. This is an additional reason
why standardization can be seen as a moral mediator.
Accurate measurement underpins trade, commerce and manufacturing
all around the world. In a global economy, the wheel bearings for my car
could be made in China, for example [UKA12], but the car itself in Eastern
Europe: they have to ﬁt very well to avoid accidents and therefore a common
measurement unit is essential.
Finally, measurement provides conﬁrmation (or refutation) of theories.
I do not want to go through details of the debate between veriﬁcationism
and falsiﬁcationism. Here I just want to stress a last cognitive aspect that
could arise from standardization as a negative aspect: the conﬁrmation bias
[Mag11]. Of course standardization is a strong tool against the possible
bad-faith of the experimenter to embubble himself (voluntarily, but often
even involuntarily) defending his scientiﬁc credo. But on the other hand
standardization (especially that of measurement procedures) can also act in
limitation of the scientist’s freedom of investigation and originality. In the
next sections I will introduce the development in the choice and deﬁnition of
reference measurements: in all possible deﬁnitions, the risk of conﬁrmation
bias is present, with its possible negative consequences (polarization of the
scientiﬁc opinion, illusory association between events,...).
2.2 Anthropometric, SI and natural units
The BBC few years ago funnily reported about old English ladies showing
few carrots when asked how much 100 kg of vegetables could be2. SI units
2“Although a great deal has happened in the UK over the past 40 years, there is still
a lot of ignorance of the current situation. Some people think it has already gone too far,
others do not seem to be aware of what has actually happened. One school of thought is
that SI (metric) is OK for some ﬁelds, e.g. scientiﬁc and engineering endeavours (indeed,
some people think SI was designed only for such purposes, rather than general use), but
it is not suitable for others. For example, as Britain is an island, and the design of our
14 CHAPTER 2. FIXING CONSTANTS OF NATURE
are not popular in most parts of the civilized world. Then when and how the
SI units were deﬁned and introduced?
Originally measurement units were anthropometric, very practically de-
ﬁned as parts of human body (e.g. the foot or the inch) or based on speciﬁc
measurement procedures (e.g. the knot, to measure the speed of a vessel,
based on the knot counting through sailor’s ﬁngers using a chip log). Most
of them were dating back at least to the Roman Empire (e.g. the mile, from
the latin mille passus). Many of those practical units were roughly based on
sexagesimal numerical system (base-60, probably introduced by the Babylo-
nians), allowing to deal with many simple fractions that could be more helpful
in the practice without modern calculators and before the introduction of the
base-10 arabic numerical system.
The path towards SI units dates back to the French Revolution: according
to Barrow [Bar02], the enthusiasm around French Revolution boosted the
introduction of new units (poids et mesures), disregarding the social obstacles
that old practical conventions inevitably cause. The idea of egalitarian and
universal units was as a stronger argument, with respect to some adaptation
diﬃculties, the same assessed even nowadays by the English ladies of the
Other diﬃculties were geopolitical. When the English and the Americans
(soon followed by others [Gal03]) had agreed in using the same time reference
in the GMT, while refusing certain French attempts (the French Republican
Calendar; the reference time in Paris), the scorned French started forcing
the path towards the metrication of the unit of length and of weight: both
should have been proposed worldwide following the rationalist ideas of the
Age of Enlightenment, opposed to the old-fashioned and conservative units
of the British Empire.
In fact, while the GMT was introduced in 1847, the two platinum stan-
dards of 1 metre and 1 kilogram had already been deposited in the Archives
de la R´epublique in Paris on the 22 June 1799: this can be seen as the ﬁrst
step in the development of the present SI units [BIP06]. The metre was origi-
nally deﬁned as one ten-millionth ( 1
10,000,000 ) of the distance from the Equator
to the North Pole3; the kilogram was the mass of 1 dm3of water. We can
see how these deﬁnitions were natural : far from being idealized beings, they
refer to real natural quantities (water and the Earth meridian), rather than
part of human body like the anthropometric units.
road signs does not aﬀect how we do trade with the rest of the world, (so it is argued) it
is not necessary for road signs to convert to metric” [Ass13].
3The metre had been previously deﬁned in the XVII century as the length of a pendulum
with a half-period of 1 second.
2.2. ANTHROPOMETRIC, SI AND NATURAL UNITS 15
With the Convention du M`etre (Metre Convention) on 20 May 1875,
which created the BIPM4, work began on the construction of new inter-
national prototypes of the metre and kilogram. Together with the astro-
nomical second as the standard unit of time, these units constituted the
three-dimensional mechanical unit system called MKS system (metre, kilo-
gram, and second). In 1901, Giovanni Giorgi (1871-1950) proposed that the
MKS system should be extended with a fourth unit to be chosen from the
units of electromagnetism. Giorgi’s proposal opened the path to a number of
new developments. Finally, in 1971 the current version of the international
standard system of units (named SI since 1960) was introduced, bringing to
the seven base units listed in Table A.1 in the Appendix, together with the
current deﬁnition and year of adoption.
Those units are the “base” units: other units can be “derived” by them, as
for example the unit of the force, the newton (symbol: N), equal to 1 kilogram
times 1 metre divided by 1 square second (in symbols: 1 N = 1 kg m s−2).
But at a ﬁrst glance, it should be evident that also base units are kind
of interdependent: for example the metre depends on the deﬁnition of the
second and similarly the mole on the deﬁnition of the kilogram, and the
candela from that of the watt and the hertz (thus the metre, the kilogram
and the second). The reasons behind the choice of these base units are many,
some of which could be grouped as follows [HH12]:
historical: as shown, base units were introduced all over one century,
where physics has developed so that some quantities that originally
were thought to be independent, later on were shown to be interdepen-
dent (e.g. length and time, via the constancy of the speed of light);
practical: certain units are practically more useful as base units,
rather than derived from other units (e.g. in chemistry is more straight-
forward to deal with the mole, than with kilograms, via the Avogadro
pedagogical: physics textbooks often follow an historical approach
starting, for example, from the Newton equations, where it is impor-
tant to diﬀerentiate between length and time, to derive other mechan-
ical units such as speed, acceleration,...; furthermore, the dimensional
4Together with BIPM, two other institutions were founded: the General Conference
for Weights and Measures (Conf´erence g´en´erale des poids et mesures, CGPM), a meeting
between member states, typically every 4-6 years (during these meetings standard units
can be redeﬁned); and the International Committee for Weights and Measures (Comit´e
international des poids et mesures, CIPM), an advisory body.
16 CHAPTER 2. FIXING CONSTANTS OF NATURE
analysis that comes from the choice of the base units is a useful tool to
check the correctness of exercises;
technical: from a metrological point of view, this is the most relevant
reason: it is often questioned why the ampere (current intensity) is a
base unit, rather than the coulomb (quantity of charge): the reason
is a better accuracy and reproducibility with which up to now can be
measured 1 A of current, rather than any accurately and reproducibly
deﬁned quantity of charge.
Furthermore, it is clear how some of the seven SI base units “suﬀer” from
the maturity of physics at the time they were thought and introduced. The
most obvious example is the kilogram, whose very classical deﬁnition dates
back more than a century ago and still refers to the mass of the physical
artefact in Paris.
In terms of the reference concepts on which the seven base units are
deﬁned, I will group them as follows:
1. Units referring to a physical object: the kilogram;
2. Units referring to a property of a reference material: the second (refer-
ring to a property of caesium 133) and the kelvin (to a property of
3. Units referring to an experiment: the ampere and the candela;
4. Units referring to constants: the metre (to the speed of light in vac-
uum c= 299,792,458 m/s) and the mole (indirectly, to the Avogadro
constant NA= 6.02214129 ×1023 mol−1.)
Before introducing the incoming development of the SI units into the
“New SI” in the next section, I will comment on how the approach of the
last of the groups above leads to the deﬁnition of a “natural” system of
units, that is largely (and variously) in use among theoretical physicists and
is described for example in the work by L. Hsu and J. P. Hsu [HH12] or in
various works by Barrow5(see for example [Bar02, Bar07]).
5Max Planck (1858-1947) proposed a natural system of units (then referred to as
“Planck system of units”), based on combinations of his constant h,c,k= 1.38064852 ×
10−23 m2kg s−2K−1(the Boltzmann constant) and G= 6.67408 ×10−11 m3kg−1s−2
(the gravitational constant): see [Bar02, HH12] for details.
2.2. ANTHROPOMETRIC, SI AND NATURAL UNITS 17
The approach behind “natural unit systems” is indeed the idea of referring
all units to constants of nature, close to some of what already happens in the
SI deﬁnition of the metre (and, with much carefulness6, of the mole).
While the epistemological consequences of such approach are discussed in
the next chapter, some rationales are quite straightforward: an immediate
consequence is that the constants of nature become physically dimensionless,
as pure numbers πor e. Then, a deﬁnition of a unit referring to a physical
object (such as the kilogram) suﬀers the possible deterioration of the object
itself with time, thus leading to a continuous and costly control of the primary
reference, together with periodic control and alignment of the secondary units
all around the world. This is a practical reason. There is also a “political”
reason: reference to a physical object contrasts one of the original ideas
behind the standardization philosophy of the Metre Convention, which is to
avoid a country to be in a leading role (while the possession of the primary
reference poses itself a country in a leading role). Furthermore, referring to a
physical object is still a sort of anthropometric, non-rational approach: the
reference mass of 1 kg (the mass of “that mass” of 1 kg) is clearly man-centred
and man-related, and it does not seem to be referred to a man-independent,
immutable natural entity.
On the other hand, if constants of nature do exist, then they provide the
most reliable references for measurement units, since they are “constants”:
therefore temporally stable (i.e. immutable entities) and not “man-related”
(they would exist even if men did not exist). One may think that a being from
a diﬀerent part of the universe (or living at diﬀerent temperature, length and
mass ranges) may not agree that the Paris 1 kg prototype is a convenient and
useful reference unit, but will probably agree that cis a constant of nature.
These motivations seem to support the deﬁnition of measurement units
referring to constants of nature: let us then see here how a natural system
of units can be deﬁned. Following the approach of [HH12], a natural system
can be directly deﬁned starting from the SI deﬁnition of the metre of Table
A.1. This deﬁnition could be re-read as an equivalence (an isomorphism)
between length and time, saying that one metre “is” 1/299,792,458 second,
stating a unit conversion between metre and seconds, thus leading c= 1 to
be a dimensionless unit.
The same can be done with the mass, by means of a combination of Ein-
stein’s relation E=mc2(being now cdimensionless, it follows that the mass
6In this contest one should bear in mind that, while cis a constant of nature, NAis not:
it is a scaling factor between macroscopic and microscopic (atomic scale) observations of
nature. There are also concerns on the mole deﬁnition and on the use of the mole as a
base unit at all. I will not go through details on that, which can be found in for example
in [Mei11, Joh11, Fel11].
18 CHAPTER 2. FIXING CONSTANTS OF NATURE
mis homogeneous in natural units to the energy E) and Planck’s equation
E=hν, where h= 6.6260693 ×10−34 J s is the Planck’s constant, acting
as a conversion coeﬃcient between E(and thus mass) and the frequency ν
(and thus time). Playing with physical units,
h= 6.6260693 ×10−34 kg ·m2/s = 6.6260693 ×10−34
the kilogram could be redeﬁned as 1 kg = (299,792,458)2
66,260,693 ×1041 s−1, giving h= 1
Similar redeﬁnitions can be performed on all other base SI units, as shown
in Table A.2 in the Appendix, and so forth on all derived physical units,
referring then all physical quantities in terms of only one base unit (instead
of seven as in the SI): the second. They may look quite impractical: in
fact they are impractical for common people, only theoretical physicists use
2.3 Towards the New SI
That of 2018 will be the most signiﬁcant revision of the SI in its history: all
base units will be redeﬁned with reference to constants, apparently just in
line with the current deﬁnition of the metre and, to some extent, of the mole.
According to the circulating draft of the New SI Brochure [BIP15], the seven
constants are listed in Table A.3 in the Appendix; the New SI units are listed
and deﬁned in Table A.4.
Some disapproving arguments against the choices of the New SI deﬁni-
tions will be presented in Chapter 3. In the next subsections I will comment
only on the relevant variations if the deﬁnitions of the ﬁrst three units: the
second, the metre and the kilogram.
2.3.1 The second and the metre
Let us ﬁrst consider the old and new deﬁnitions compared:
7A reason for theoretical physicists to use the natural units of Table A.2, which was
not listed above, is that being c= 1 and h= 1 they are allowed simply to throw these
symbols away in most of their equations, which is very practical indeed for them.
2.3. TOWARDS THE NEW SI 19
SI New SI
second The duration of 9,192,631,770 Deﬁned by taking the ﬁxed numerical
periods of the radiation value of the caesium frequency ∆νCs,
corresponding to the the unperturbed ground-state hyperﬁne
transition between the two splitting frequency of the caesium 133
hyperﬁne levels of the ground atom, to be 9,192,631,770 when expressed
state of the caesium 133 atom in the unit Hz, which is equal to s−1
for periodic phenomena
metre The length of the path travelled Deﬁned by taking the ﬁxed numerical
by light in vacuum during a time value of the speed of light in vacuum c
interval of 1
299,792,458 of a second to be 299,792,458 when expressed in the
unit m s−1, where the second is deﬁned
in terms of the caesium frequency ∆νCs
The new deﬁnitions of the second and the metre (and the new order is
done by purpose) are signiﬁcant because the natural phenomena to which
they refer has not changed: the hyperﬁne splitting frequency of caesium 133
∆νCs and the speed of light in vacuum c, respectively. (Indeed, the same
happens to the candela and the mole, but I will limit the discussion here to
metre and second).
The numerical values of cand of ∆νCs had already been ﬁxed in the
previous revisions of the metre and the second deﬁnitions: as a matter of
fact, nowadays the speed of light in vacuum is not “circa” 299,792,458 m
s−1, but “exactly” that value (no decimals); similarly for the unperturbed
ground-state hyperﬁne splitting frequency of Cs133. Therefore this is not a
But the new deﬁnitions hide a fundamental change, which is full of epis-
temological consequences: while in the current SI the object of the deﬁnition
is a physical magnitude (“the duration” and “the length”), which is the
category “quantity” in the ontological jargon, in the New SI the subject
is in both cases a “ﬁxed numerical value” (∆νCs = 9,192,631,770 Hz and
c= 299,792,458 m s−1), from which the categories “period” and “length”
are deducted. Take the metre: in the old deﬁnition the logical order of the
quantity (length) −→ number 1
299,792,458−→ unit (metre); (2.2)
in the New SI the order has changed to:
299,792,458−→ quantity (length) −→ unit (metre).(2.3)
20 CHAPTER 2. FIXING CONSTANTS OF NATURE
In other words: reading the old deﬁnition one has no doubt that the
quantity “length” is in a sense pre-existent to the particular numerical value
of the “speed” c(another physical quantity which is incidentally constant),
that has that value when length is measured in metres, and thus gives the
deﬁnition of that unit. The New SI starts from a statement of the “the ﬁxed
numerical value” of c, which being deﬁned in unit m s−1, then leads to the
introduction of the quantity “length” which is measured in metres.
The New SI lists the second at the top of the list. This is not incidental,
as mentioned above, because its deﬁnition cascades down into all other deﬁ-
nitions. This looks very familiar with what theoretical physicists do with the
“natural units” illustrated in the previous section: in the New SI, all units
(apart from the mole) depend on the deﬁnition of the second.
But while almost no XXI century physicist doubts in considering ca
constant of nature, the same cannot be said of ∆νCs in the deﬁnition of the
second. This is an integer number representing the number of cycles of an
electromagnetic wave from a transition of the caesium stable isotope Cs133:
it is “constant” as are constant “1”, “the year I was born”, “the number
37,847” or any other integer number: it is a “technical constant”. Since
almost all units are dependent from the second and the second is not based
on a “true” constant of nature, in Chapter 3 I will show how this is related
to one of the major concerns against the New SI by many scholars.
Furthermore, as Eran Tal suggests in [Tal16], the new deﬁnition “stipu-
lates that a particular frequency associated with the caesium atom is uniform,
that is, that its periods are equal to one another. However, the frequency
in question is a highly idealized construct. As far as the deﬁnition is con-
cerned, the caesium atom in question is at rest at zero degrees Kelvin with
no background ﬁelds inﬂuencing the energy associated with the transition.
It is only under these ideal conditions that a caesium atom would constitute
a perfectly stable clock”, and “paradoxically, under these ideal conditions
it would be impossible to probe the caesium atom so as to induce the rel-
evant transition. Hence the duration of the second is deﬁned in a doubly
This introduces a topic that will be discussed in the particular example
of the third unit in exam in the following section, the kilogram, i.e. the
challenges linked to the practical realizations of the New SI units, or, in the
BIPM jaergon, the mise en pratique.
2.3.2 The kilogram: ﬁxing Planck constant
Resolution 1 of the 24th CGPM, held in 2011, set the path towards the
redeﬁnition of the kilogram, being the last unit to be deﬁned in terms of a
2.3. TOWARDS THE NEW SI 21
material artefact: the kilogram is in fact the eldest unit, dating back to 1889,
when the 1st CGPM deﬁned it as the mass of the international prototype.
In the current edition of SI, from the deﬁnition of the kilogram depend the
deﬁnitions of the ampere, the mole and the candela: thereafter, possible
deviations of the prototype mass aﬀect the deﬁnition of those units as well.
Such deviations have been estimated to the order of 50 µg every 100 years.
The reason behind such a longevity is the diﬃculties encountered when
trying to reproduce accurately any particular experiment in which a quantity
can be measured, related to a mass of 1 kilogram via a known physical
constant. This is the important phase that should be ﬁnally described in the
mise en pratique document of BIPM, where hints are given to metrologists
on how to build up experimental equipments to apply the New SI deﬁnition
to practical masses to be calibrated towards the standard deﬁnition.
As the CGPM resolution states, “many advances have been made in re-
cent years in relating the mass of the international prototype to the Planck
constant h, by methods which include Watt balances and measurements of
the mass of a silicon atom”. In fact nowadays the draft mise en pratique
for the deﬁnition of the kilogram describes the two approaches. In the ﬁrst
one (the Watt balance) the calibrating weight m×gof the artefact (being g
the gravitational acceleration) is balanced by the electromagnetic force pro-
duced on a circular coil immersed in a radial magnetic ﬁeld when a current
ﬂows through the coil: it can be shown that the calibrating mass mis then
proportional to a frequency (to be measured) and to Planck constant h.
The second method comes from the classical idea where the mass of a
pure substance can be related to the number of elementary entities in the
substance of known atomic mass. Certain pure crystals (e.g. crystalline
silicon) are nearly perfect, with atoms displaced in a tetrahedral shape where
the distance between nearest neighbours is called lattice parameter and can
be measured with great precision and again is ﬁnally related to the mass of
the crystal via Planck constant h.
In both methods described in the mise en pratique of the New SI the
Planck constant has the ﬁxed value speciﬁed in Table A.3, which is the
novelty of the new deﬁnition and for which similar considerations hold as for
the metre and the second. But Planck constant is a “true” constant of nature
for physicists: we have already found it in the deﬁnition of the natural units
in section 2.2.
22 CHAPTER 2. FIXING CONSTANTS OF NATURE
3.1 Some arguments against the New SI
The proposed changes of the deﬁnitions in the Bruchure of the New SI have
necessarily raised a debate between BIPM and a number of opponents among
metrologists, scientists and philosophers of science.
Ulrich Feller in [Fel11] stresses the need of a new foundation of the whole
SI, based the formal language of mathematics, to distinguish between purely
abstract concepts (physical quantities and real numbers), observable things
(units) and elements of ordinary language (the unit names). He argues that,
both in the current and the New SI, these elements are mixed up, generating
confusion. Speciﬁcally on the New SI, Feller criticizes ﬁrst the lack of clarity
in the deﬁnitions themselves: the “measurable things representing the unit
value of the physical quantity are not deﬁned and the scientist is left alone
to ﬁnd out how experiments have to be designed for realizing the real units”.
He acknowledges that the practical realization of each unit is described in the
mise en pratique, but then the deﬁnition itself “serves nobody and should be
discarded”. He also comments on the “circularity” of deﬁnitions, for which
some “base units are deﬁned in terms of derived units that are said to be
derived from the base units” (for example in the kilogram the derived unit
joule is introduced). Finally, as already disclosed in the previous chapter, a
third concern is about the constants given in the mole and in the candela
deﬁnitions: “there is nowhere a thing in nature corresponding to these con-
stants [and] to call the Avogadro constant a true invariant throughout time
and space is about the same as one would claim that the dozen is a true
invariant throughout time and space”.
24 CHAPTER 3. FURTHER EPISTEMOLOGICAL CONSIDERATIONS
Gary Price in [Pri11] agrees that the only “genuine” constants of nature
are four (c,h,eand k, see Table A.3 in the Appendix): “the spectral charac-
teristics of a speciﬁc atom and the luminous eﬃcacy of a speciﬁc frequency
are not fundamental” and “the Avogadro ‘constant’ is a human artefact,
not a universal fundamental physical constant”. However Price goes further,
challenging “natural units” and asking if it is really necessary to refer to con-
stants of nature at all; he brings the example of the second as a proper use
of a “non-fundamental physical invariant [referring] to an invariant natural
example of the quantity time interval”; while the new kilogram deﬁned in
terms of h“does not exist in our world, it is a virtual thing, conjured into
existence, disconnected from the everyday notion of mass”. According to
Price, “deﬁning constants as constant and making them the anchor of our
measurements” puts constants themselves beyond any empirical test, that
will be aﬀected by circularity. He brings the example of the ﬁne-structure
hc , a combination of the four constants of nature above
characterizing the strength of the electromagnetic interaction: “the possibil-
ity that αmay change has been the subject of serious theoretical discussion
for three quarters of a century”, but the New SI is now going to state that
αis constant and has always been constant.
Luca Mari in [Mar15a, Mar15b] highlights that the deﬁnitions in the New
SI follow the logical sequence similar to the one I sketched in eq. (2.3): (i) a
constant numerical value c= 299,792,458 for the speed of light in vacuum is
stipulated; and (ii) the metre is deﬁned in order to be c= 299,792,458 m s−1.
But the risk according to Mari lies again (apart from the complex deﬁnition
that looks circular at the ﬁrst glance) on what happens if the constant in
the future appears not to be constant: any possible variation in the “true” c
value, for example, may not be seen as a variation of cor of the metre (which
is linked to c), but any measurement of length will be inevitably aﬀected
by the drift of the calibration of the primary and so forth all instruments
connected with the metre, which is technically, “conceptually and socially”
problematic. The solution proposed by Mari is to redeﬁne (at the expense
of adding more complexity) the logical sequence to: (i) a constant numerical
value c=nis stipulated for the speed of light in vacuum; (ii) the metre is
deﬁned in order to be c=nwhen measured in m s−1; (iii) as to the current
knowledge is n= 299,792,458.
Franco Pavese in [Pav14] warns also “that only a few countries in the
world will (or can aﬀord to) perform direct measurements of the constants
relevant to the New SI proposed deﬁnitions of the base units”. This repre-
sents a risk in metrological traceability chain, a contradiction to the principles
of the Metre Convention and, I add, a limitation for the New SI as “moral
mediator” as speciﬁed in section 2.1.
3.2. PHYSICAL CONSTANTS AS IMMUTABLE ENTITIES 25
To some of these concerns, the draft Brochure of the New SI provides
some explanations. As to Feller’s concern on complex (and useless) deﬁni-
tions, “the use of a constant to deﬁne a unit disconnects its deﬁnition and
realization. This oﬀers the possibility that completely diﬀerent or new supe-
rior practical realizations can be developed, as technologies evolve”.
As to the general concern about “true” or “technical” constants, the
Brochure acknowledges the distinction. In particular the hyperﬁne splitting
frequency ∆νCs “has the character of an atomic parameter, which may be
aﬀected by the environment”: however, “this transition is well understood,
stable and is also a good choice as a reference transition under practical con-
siderations”. Indeed, this looks like a diﬀerent approach than for other units:
as to the second, the New SI does not look for the universality of a constant of
nature (in line with the “natural units” of theoretical physicists and leaving
the practicality to the mise en pratique ), but is driven by “practical consid-
erations”. And at the same time it keeps a shadow of old anthropometric
units and conventionalism, for example in the reason behind the choice of
the luminous eﬃcacy Kcd, which is only “related to a conventional spectral
response of the human eye”.
As to the question about “true” constants and their possible hypothetical
change with time and space, the Brochure simply claims that “the experi-
mental limits of the maximum possible variation[s] are so low, however, that
any eﬀect on foreseeable measurements can be excluded”.
3.2 Physical constants as immutable entities
The importance of referring to immutable entities as a source of knowledge
can be dated back to the Ancient Greek philosophy. Relevant examples
were discussed in chapter 1: Theaetetus’ regular polyhedrons used by Plato
to shape the elements constituting the physical universe of the Timaeus.
Aristotle also put immutable objects as the starting point in pursuing the
truth, as in the quotation of the foreword of this thesis.
Ancient astronomers could easily notice the repeatability of planets mo-
tions and the constancy of the star positions in the sky and study the motion
of the planets with great precision. As the Nobel laureate physicist Steven
Weinberg notices in his To Explain the World [Wei16] “the sky must have
been commonly used as a compass, a clock, and a calendar”. Modern physi-
cists still look with similar enthusiasm to immutable concepts as the constants
of nature or combinations of them, as the ﬁne-structure constant α.
According to John D. Barrow [Bar02], constants of nature for physicists
of the XX century represented continuing the progress of physics from the
26 CHAPTER 3. FURTHER EPISTEMOLOGICAL CONSIDERATIONS
The impact over the following centuries of Copernicus’ leap away
from the prejudices of anthropocentrism was felt across the whole
spectrum of human investigation. [...] The march towards estab-
lished constants of Nature that were not explicitly anthropocen-
tric, but based upon the discovery and deﬁnition of universal
attributes of Nature, can be seen as a second Copernican step.
The fabric of the Universe and the pivotal structure of her uni-
versal laws were now seen to ﬂow from standards and invariants
that were truly superhuman and exraterrestrial. The fundamen-
tal standard of time in Nature bore no simple relation to the
age of man and woman, no link to the period of days, months
and years that deﬁned our calendars, and was too short to al-
low any possibility of direct measurement (John D. Barrow, The
Constants of Nature [Bar02]).
Since units of measurements need to be stable and immutable over time,
the New SI has put similar eﬀorts in trying to anchor them to the most stable
and immutable concepts available nowadays for physicists: the constants of
nature c,h,eand k(plus other three controversial “technical” constants).
Anthropometric units such as the “foot” or tangible physical object as the
kilogram prototype are deﬁnitely more practical and “easy” than the complex
deﬁnition proposed: but they both lack of universality and suﬀer inevitable
changes with time, which the New SI wants to avoid.
The naive realist may agree with Gary Price that those units simply “do
not exist in our world”. A mature realist (if not an idealist) will argue instead
that the speed of light in vacuum cis as a real physical “object” as the old
metre prototype, the platinum bar standard: but cis more stable in time
than the platinum bar. In fact there can be no diﬀerence, if one abstracts
from the tangibility of the unit as BIPM seems to propose in the New SI.
More importantly, highly idealized concepts as the speed of light “in vacuum”
cor the “quantum of action” hsuﬀer no change by deﬁnition.
But the hyperﬁne splitting frequency ∆νCs, which is so important in the
deﬁnition of all units, is not a constant of nature, as BIPM readily admits.
It is constant based on induction from many past observations. More severe
concerns regard Avogadro “constant” and the luminous eﬃcacy Kcd. One
may argue that probably BIPM does not fully share the metaphysical ap-
proach to constants of nature of some contemporary physicists: maybe we
should look at those numbers in the New SI deﬁnition in a weaker sense,
simply as physical (real or idealized) “objects” that are conventionally pre-
3.2. PHYSICAL CONSTANTS AS IMMUTABLE ENTITIES 27
Figure 3.1: Evolution of the units of time, mass and length towards an idealist
approach in the New SI.
sumed to be constant as it was for the length of the platinum metre bar or
the mass of the kilogram prototype more than one century ago.
Impoverished from the role of “true constants” down to the weaker role
of “conventional constants” (or “assumed to be constants”), one may also
accept the fact that in the future their value may change, in the slightly
modiﬁed approach proposed by Mari. This may prevent a certain embub-
blement of physics against any possible future research aiming to show, for
example, that the ﬁne structure constant αis actually not constant.
But on the other hand, “true” constants of nature guarantee that the
New SI units based on them are no longer anthropometric: but is it true
that they are not anthropocentric, as in the intentions of the XX century
physicists in Barrow’s quotation? Are they really “superhuman and extrater-
restrial”? Anthropocentrism is usually deﬁned as “the belief that considers
human beings to be the most signiﬁcant entity of the universe” but also “it
interprets or regards the world in terms of human values and experiences”
(online Merriam-Webster dictionary). In science, it is typically associated
with the Ptolemaic system of the universe, with the Earth at the centre and
the Sun, the Moon and the other planets orbiting around it. Provided that
the New SI does not pretend to put the man back to the centre of the uni-
verse, can we say that it does not interpret nature and physics in terms of
human values and entities? Probably not so deﬁnitely, as all SI units are
obviously deﬁned to be of practical use for mankind: a metre is a man-sized
dimension of length and the same can be said for all other units. Therefore
the risk of anthropocentrism is inevitable when man deﬁnes standard units
Furthermore, the claim that the speed of light in vacuum is less a human
artefact than a platinum bar of given length can be thought as somewhat
naive, because “speed” itself is a human concept, belonging to a certain
physical theory that is certainly not superhuman; the “vacuum” is also an
idealized concept that helps us to ﬁx our deﬁnition. And ﬁnally the constancy
28 CHAPTER 3. FURTHER EPISTEMOLOGICAL CONSIDERATIONS
of cis part of “our” current knowledge and view of nature: could anyone
really be sure that an extraterrestrial being agrees that cis a constant of
nature, as pointed out in section 2.2? To answer yes is somewhat to agree
with Salviati in Galileo’s Dialogue:
I say that the human intellect does understand some of [the propo-
sitions of Nature] perfectly, and thus in these it has as much
absolute certainty as Nature itself has (Galileo Galilei, Dialogue
Concerning the Two Chief World Systems [Gal53]).
3.3 Bridges between real and ideal entities
Mari and Petri in [MP14] describe measurement as a bridge over a “soft
ground” between two islands: the “empirical island” of physical phenomena
and the “information island” of mathematical formal structure (numbers).
In between the ground is “soft”. “Measurements and their results cannot
be yes-no sharp: their goal is to be objective and inter-subjective; however,
they cannot be completely. [Hence] the conclusion that the two pillars of
objectivity and inter-subjectivity that construct the measurement science
bridge lean on soft ground”. Also “measurement scientists are usually, and
wisely, at least moderately realist and recognize as unproblematic that their
instruments actually interact with parts of the territory, i.e., objects and
their quantities, whose existence is independent” of mathematical models.
On the other hand the results of such interactions are numbers, therefore
mathematical objects and not empirical ones.
In this picture, I claim that measurement units are the pillars of the
bridge connecting the two islands. They are objective as far as they are
inter-subjective: objective, from the realist point of view; inter-subjective
for the conventionalist one. The New SI deﬁnitions then try to hold up
the link between the two sides of the bridge: a realist side in the empirical
island and an idealist one in the information island. The realist side is rather
implicit in the new deﬁnition (e.g. there is no reference to mass in the new
kilogram deﬁnition), while the idealist information side is more evident (e.g.
the kilogram is deﬁned ﬁxing the numerical value of h).
Seen within the development of measurement units in the recent history
(see Figure 3.1) the New SI seems therefore to represent a step towards a more
rational and idealistic approach in measurement, corresponding to somewhat
between the representationalist and the operationist epistemological strands
of Figure 1.1. The route towards the reference to ﬁxed and immutable con-
cepts (a goal stated early in Plato’s and Aristotle’s works) is now heading
3.3. BRIDGES BETWEEN REAL AND IDEAL ENTITIES 29
to the ﬁxation of constants of nature, that can be seen in ontological conti-
nuity with both the immutable forms of idealism (Plato); the objects of an
immutable ethereal universe (Aristotle), where constants of nature “suﬀer
no change” and are “true” constants, as agreed by many theoretical physi-
cists since the early XX century; or, in the Kantian jargon, objects from an
act of synthesis of a priori forms of space and time, as in the contemporary
operationist approach of many philosophers of science.
In the representationalist approach of Bas C. Van Fraassen [VF08] (a re-
alist position), measurement is a process between three diﬀerent domains,
that the author also refers to as the Appearance from Reality Criterion: (a)
the observable phenomena (macro objects, motion, tangible and visible bod-
ies); (b) the theoretically postulated reality (micro structures, forces, ﬁelds,
constants of nature); (c) the appearances (measurement outcomes). The New
SI units of measurements link the three domains, while showing appearances
(c) of observable phenomena (a) via reference units that are deﬁned in a
theoretically postulated reality (b).
The representationalist approach is challenged by quantum physics and
the concept of measurement in it:
Can we describe the process in which observable Ais measured
on a system in quantum state ψby a suitable apparatus, starting
in its ‘ready state’ ϕand interacting with that system through a
certain interval of time, so as to show how the outcome is pro-
duced? (Van Fraassen, Scientiﬁc Representation [VF08])
According to Van Fraassen, the process described above does not apply
to quantum mechanics, since “the theoretical description of this interaction
[between the observable phenomena (a) and the theoretically postulated re-
ality (b)] in quantum theoretical terms just does not seem to provide a place
for the speciﬁc outcome [(c)] in question”. The Appearance from Reality Cri-
terion is also challenged by the supervenience of mind, and therefore the idea
of a certain autonomy of psychology and cognitive science with respect to
Both the quantum mechanical and the supervienence of mind challenges
are beyond the scopes of this work, but represent an interesting chance for
further investigations on the impact of the New SI to the philosophy of sci-
30 CHAPTER 3. FURTHER EPISTEMOLOGICAL CONSIDERATIONS
There are many reasons to consider standardization in general and standard-
ization of measurement units as a strong moral mediator in the contemporary
society. It is “democratic”, establishing ﬁx references and their traceabil-
ity that guarantees both the interests of merchants against the government
authority and of customers against merchants. It fosters internationalisa-
tion and helps to establish treaties of global trade. Standard measurements
are important in supporting new scientiﬁc theories or conﬁrming established
ones; among possible negative counterparts, it may strengthen conﬁrmation
bias, thus embubbling the scientist against possible new experimental evi-
The New SI deﬁnitions of base units, which are scheduled to be oﬃcially
launched in 2018, represent the most important change made by the BIPM
since its foundation. The seven traditional base units are going to be re-
ferred to certain “constants of nature” that are supposed to be a more stable
and immutable reference than the ones in the current deﬁnition. It aims to
complete the route that switched from the pre-rational anthropometric units
to the rational approach of the Metre Convention, following the example of
the current deﬁnition of the metre. This change is most evident with the
kilogram, whose deﬁnition has never changed since its introduction. Deﬁn-
ing units with reference to “constants of nature” resembles the approach of
“natural units”, commonly used by theoretical physicists, even though the
practicality of such deﬁnitions is questionable.
A diﬀerent philosophical perspective has been highlighted when the cur-
rent and new deﬁnitions of the metre and the second were compared. In fact,
current deﬁnitions already refer to the same “constants” that are used in the
New SI: but here the focus is more on the constants themselves, than in the
physical quantities to be deﬁned. One may then have the impression of a cer-
tain ontological predominance of the constants (ideal entities) with respect
32 CHAPTER 4. CONCLUSIONS
to the physical quantities (attributes of real entities) that these constants
The choices made in the circulating draft of the New SI opened a de-
bate on the soundness of these “constants”. While some of them (c,h,e
and k) are commonly accepted by theoretical and experimental physicists
as “true” constants of nature, others (the Avogadro constant, the hyperﬁne
splitting frequency of caesium and the luminous eﬃcacy of a given radiation)
are widely recognized only as “technical” constants. This raises questions
on the rationale behind the choice: for the theoretical physicist, “natural
units” (and the New SI thereafter) should be deﬁned in terms of “true”
constants of nature because in principle they are totally non-anthropometric
and non-anthropocentric (they are “superhuman and extraterrestrial”, hence
they “suﬀer no change”); but from the naive realist, technical constants may
be more acceptable in being part of the “real” world and not based on ide-
alizations that may be falsiﬁed by future better physical theories.
Considering the process of measurement from an epistemological perspec-
tive, the New SI approach of referring to “constants of nature” echoes relevant
examples from the philosophy of science where sources of (scientiﬁc) knowl-
edge was based on immutable concepts, either real or ideal objects. Exam-
ples from Plato, Aristotle, Galileo and Kant have been proposed. This again
raises questions on the ontological soundness of the new deﬁnitions: metrol-
ogists may avoid the philosophical conundrum with a weaker conventionalist
approach for which these constants are only “assumed to be constants” and
subject to possible periodic check of their numerical value in the future.
Philosophers of science picture measurement as a link between real (the
empirical world) and ideal entities (physical theories and measurement out-
comes as real numbers). The New SI deﬁnitions make explicit reference to
the ideal entities (constants of nature), thus supporting an epistemological
position that swing between operationist idealism and representationalist re-
Further investigations regard the possible challenges to the representa-
tionalist interpretation of the New SI represented for example by the problem
of measurement in quantum mechanics and the supervenience of mind over
fundamental physics claimed by certain strands of psychology and cognitive
34 APPENDIX A. TABLES
Quantity Name Symbol Deﬁnition Year
length metre m the length of the path travelled by 1983
light in vacuum during a time interval
299,792,458 of a second
mass kilogram kg the mass of the international prototype 1889
time second s the duration of 9,192,631,770 periods
of the radiation corresponding to the
transition between the two hyperﬁne
levels of the ground state of the
caesium 133 atom
electric ampere A that constant current which, if maintained 1946
current in two straight parallel conductors of
inﬁnite length, of negligible circular
cross-section, and placed 1 metre apart
in vacuum, would produce between these
conductors a force equal to 2 ×10−7
newton per metre of length
temperature kelvin K the fraction 1
273.16 of the thermodynamic 1954
temperature of the triple point of water
amount of mole mol the amount of substance of a system 1971
substance which contains as many elementary
entities as there are atoms in 0.012 kg
of carbon 12. When the mole is used,
the elementary entities must be speciﬁed
and may be atoms, molecules, ions,
electrons, other particles, or speciﬁed
groups of such particles
luminous candela cd the luminous intensity, in a given direction, 1979
intensity of a source that emits monochromatic
radiation of frequency 540 ×1012 hertz
and that has a radiant intensity in that
direction of 1
683 watt per steradian
Table A.1: SI units of physical quantities: name, symbol, deﬁnition and year
of adoption of this deﬁnition [BIP06].
Quantity SI Natural Deﬁnition
length m s the same
mass kg s−1the mass of a body whose equivalent
energy is equal to that of a number of
photons with frequency (299,792,458)2
66,260,693 ×1041 s−1
time s s the same
electric A s−1that constant current equivalent to
66,260,693 ×1034 s−1
temperature K s−1the change in the thermodynamic temperature
of a system whose energy has increased by an
amount equal to the energy of a collection
of photons whose angular frequencies sum to
66,260,693 ×1011 s−1
amount of mol s−1the same, referring to the new deﬁnition of kg
luminous cd s−2the luminous intensity, in a given direction,
intensity of a source that emits monochromatic radiation
of frequency 540 ×1012 hertz and that has
a radiant intensity in that direction of
66,260,693 ×1041 s−2per steradian
Table A.2: Convertion of SI units to natural ones [HH12].
Constant Symbol Value Unit
hyperﬁne splitting of Cs ∆νCs 9,192,631,770 s−1
speed of light in vacuum c299,792,458 m s−1
Planck’s constant h6.626070040 ×10−34 kg m2s−1
elementary charge e1.6021766208 ×10−19 A s
Boltzmann’s constant k1.38064852 ×10−23 kg m2s−2K−1
Avogadro’s constant NA6.022140857 ×1023 mol−1
luminous eﬃcacy Kcd 683 cd sr kg−1m−2s3
Table A.3: The seven ﬁxed constants on which are based the New SI units
36 APPENDIX A. TABLES
Quantity Name Symbol Deﬁnition
time second s deﬁned by taking the ﬁxed numerical value of the
caesium frequency ∆νCs, the unperturbed ground-
state hyperﬁne splitting frequency of the caesium
133 atom, to be 9,192,631,770 when expressed in the
unit Hz, which is equal to s−1for periodic
length metre m deﬁned by taking the ﬁxed numerical value of the
speed of light in vacuum cto be 299,792,458 when
expressed in the unit m s−1, where the second is
deﬁned in terms of the caesium frequency ∆νCs
mass kilogram kg deﬁned by taking the ﬁxed numerical value of the
Planck constant hto be 6.626070040 ×10−34
when expressed in the unit J s, which is equal to
kg m2s−1, where the metre and the second are
deﬁned in terms of cand ∆νCs
electric ampere A deﬁned by taking the ﬁxed numerical value of the
current elementary charge eto be 1.6021766208 ×10−19
when expressed in the unit C, which is equal to
A s, where the second is deﬁned in terms of ∆νCs
temperature kelvin K deﬁned by taking the ﬁxed numerical value of the
Boltzmann constant kto be 1.38064852 ×10−23
when expressed in the unit J K−1, which is equal
to kg m2s−2K−1, where the kilogram, metre and
second are deﬁned in terms of h,cand ∆νCs
amount of mole mol the amount of substance of a speciﬁed elementary
substance entity, which may be an atom, molecule, ion,
electron, any other particle or a speciﬁed group
of such particles. It is deﬁned by taking the ﬁxed
numerical value of the Avogadro constant NAto
be 6.022140857 ×1023 when expressed in the unit
luminous candela cd deﬁned by taking the ﬁxed numerical value of the
intensity luminous eﬃcacy of monochromatic radiation of
frequency 540 ×1012 Hz, Kcd , to be 683 when
expressed in the unit lm W−1, which is equal to
cd sr W−1, or kg−1m−2s3cd sr, where the
kilogram, metre and second are deﬁned in terms
of h,cand ∆νCs
Table A.4: The New SI (2018): name, symbol and deﬁnition [BIP15].
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