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UNIVERSIT `

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

Mauro Pravettoni

matr. 412526

Relatore: Prof. Lorenzo Magnani

Correlatore: Dott. Tommaso Bertolotti

Anno Accademico 2015-2016

to Elisa

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.

Aristotle

Abstract

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-

v

vi

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.

Contents

1 Introduction to measurement and the philosophy of measure-

ment 1

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

Bibliography 33

vii

Chapter 1

Introduction to measurement

and the philosophy of

measurement

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.

1

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,

152a [Pla04]).

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,

the Sophist.

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

[BLV14].

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

M=Q×u(1.1)

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

belief that

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-

48/B65 [Kan67]),

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

scientiﬁc research.

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

is a

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

u=Q∈R(1.2)

and, given q1, q2∈MQ, then

q1q2(1.3)

if (and only if)

µ(q1)≤µ(q2).(1.4)

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

unit”.

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

choice.

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

physical world”.

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.

Chapter 2

Fixing constants of nature

2.1 Measurement units and knowledge value

of standardization

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

United States.

11

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

measurement uncertainty.

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

BBC program.

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

principle);

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

water);

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

(299,792,458)2kg ·s,(2.1)

the kilogram could be redeﬁned as 1 kg = (299,792,458)2

66,260,693 ×1041 s−1, giving h= 1

dimensionless.

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

them7.

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

novelty.

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

terms is:

quantity (length) −→ number 1

299,792,458−→ unit (metre); (2.2)

in the New SI the order has changed to:

number 1

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

counterfactual manner”.

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

Chapter 3

Further epistemological

considerations

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”.

23

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

constant α=2πke2

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

Copernican revolution:

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

of measurements.

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

fundamental physics.

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-

ence.

30 CHAPTER 3. FURTHER EPISTEMOLOGICAL CONSIDERATIONS

Chapter 4

Conclusions

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-

dences.

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

31

32 CHAPTER 4. CONCLUSIONS

to the physical quantities (attributes of real entities) that these constants

deﬁne.

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-

alism.

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

science.

Appendix A

Tables

33

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

of 1

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].

35

Quantity SI Natural Deﬁnition

unit unit

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

current q4π×299,792,458

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

2π×13,806,505

66,260,693 ×1011 s−1

amount of mol s−1the same, referring to the new deﬁnition of kg

substance

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

1

683 ×2

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

[BIP15].

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

phenomena

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

mol−1

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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