Open Journal of Philosophy
2013. Vol.3, No.4A, 6-9
Published Online November 2013 in SciRes (
Open Access
The Origin of Quantification
Edward MacKinnon
California State University East Bay, Hayward, USA
Received May 14th, 2013; revised June 14th, 2013; accepted June 21st, 2013
Copyright © 2013 Edward MacKinnon. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited.
Neither the Greek nor the Alexandrian nor the early Arabic philosopher/scientists ever developed a
mathematical representation of qualities, a prerequisite for a mathematical physics. By the early seven-
teenth century the quantification of qualities was a common practice. This article traces the way this prac-
tice developed. It originated with a medievally theological problem and was developed by philosophical
logicians who did not have mathematical physics as a goal. The verbal algebra they developed was given
a mathematical formulation in the late fifteenth century. This was subsequently assimilated into a
neo-Platonic revival that stressed mathematical forms. The quantification of qualities developed in phys-
ics supplied the paradigm for quantification in other fields.
Keywords: Quantification; Early History of Physics
Greek and Alexandrian scientists developed geometry and
arithmetic. Yet, neither they nor the Arab successors ever deve-
loped a quantitative science of qualities. At the start of the scien-
tific revolution in the early 17th century, a quantitative treatment
of qualities was a common practice. We will consider what
prepared the way for this practice (MacKinnon, 2011: chap. 2). It
helps to begin by considering why the Greeks and Alexandrians
did not have this practice.
Historical Development
Two basic reasons might explain this deficiency. The first
stems from the Greek understanding of the relation of mathe-
matics to reality. For Plato mathematical forms exist in a sepa-
rate realm of ideas. For Aristotle mathematics is derived from
physical reality by a process of concept abstraction. Since his
position supplies the background for the medieval philosophers
we will consider, it requires a more detailed treatment.
In his de Interpretation (chap. 1) Aristotle declared that
though different languages have different word to symbolize
mental experiences, all men have the same mental experiences.
The alleged reason is that the things determine our concepts of
them. Aristotle’s ordered list of categories: substance, quantity,
quality, relation, place, time, situation, state, action, and passion,
provide the most basic concepts. The first three categories have a
conceptual ordering that determines the way quantities are
treated. Consider the predicate “red”. This cannot be applied in a
literal sense to an immaterial being, such as an idea, or an
unextended being, such as a point. A quality, such as color or
taste, pre-supposes extension which, in turn, presupposes a
substance that is extended and colored. Any discussion of the
quantity of a quality perverts the proper conceptual ordering.
In addition to speculative difficulties, there were practical
difficulties. The Greeks and Romans represented integers by
arbitrarily chosen letters. This blocked any mathematical repre-
sentation of continuously varying qualities. It also made it ex-
tremely difficult to develop any sort of isomorphism between
numbers as a system and the quantity of qualities. Numbers were
generally treated in a way that obscured the logical structures
needed. In the Pythagorean tradition numbers were classified as
even and odd, and then into evenly even (powers of two), evenly
odd, and oddly even, with a further distinction into prime, com-
posite and perfect numbers. This supported the popular trend
seeking properties of particular numbers that were associated
with particular qualities, rather than the properties of numbers as
a system.
Aristotle’s doctrine of categories were well known to me-
dieval scholars even before the systematic translation of Aris-
totle’s works in the late twelfth and early thirteenth century.
Porphyry, a disciple of Plotinus, wrote a commentary on the
Categories, the Isagoge. This was translated into Latin, Syrian,
Arabic, and Armenian and served as a staple text for the early
arguments between Nominalists and Realists. The basic cate-
gories were accepted as something determined by the nature of
reality. This put analysis in ontological terms. Since the basic
categories as well as concepts of particular objects are deter-
mined by the reality known, one can analyze objects by analyz-
ing concepts of objects. This linguistic analysis was carried on in
a theological perspective.
This theological perspective seemed to require a quanti-
fication of qualities. One’s rank in heaven, according to accepted
teaching, depends upon the degree of grace, or charity, that one
has at the moment of death. Dante’s Divine Comedy vividly
illustrates the different places assigned in hell, purgatory, and
heaven. Since grace is a quality, albeit a supernatural one, com-
paring degrees of grace is comparing quantities of qualities.
Accordingly, a way had to be found to discuss the quantity of a
quality. St. Thomas Aquinas, the most Aristotelian of the me-
dieval philosophers, seems to have been the first to give a
coherent account of the way in which quantitative determi-
nations can be given to qualities. Aquinas distinguished between
quantity per se, or bulk quantity, and quantity per accidens, or
virtual quantity. (Summa Theologiae, I, q. 42, a.1, ad 1) Virtual
quantity, or the quantity of a quality, can have magnitude by
reason of the subject in which it inheres, as a bigger wall has
more whiteness than a smaller one, or it can have magnitude by
reason of the effect of its form. The first effect of a form is a way
of existing, e.g., as human. The secondary effect of a form is
shown through its action on objects. A comparison of relative
effects serves as a measure of virtual quantity. Thus, one with
greater strength can lift heavier rocks.
Measurement did not rest on the modern idea of a unit of
measure or even a practice of measuring things. In medieval
thought the Platonic idea that the perfect form is the measure for
any being that participates in that form was reinforced by the
scriptural statement that in creating the world God disposed all
things in number, weight, and measure. Even when measure-
ment is separated from a doctrine of participation and treated in
terms of numbers, as in Aristotle’s treatment of time, the
constraint is that everything must be measured “by some one
thing homogeneous with it, units by a unit, horses by a horse,
and similarly times by some definite time” (Physics, 223b14).
Applying quantities to qualities broke this Aristotelian constraint.
This new idea of the quantity of a quality was the pivot leading
from the Aristotelian philosophy of nature to a mathematical
The historical development of medieval natural philosophy
has been treated in detail elsewhere (Lindberg, 1992: chap. 12;
Clagett, 1959). In summaries, this is often done by presenting the
aspects leading to Newtonian physics in a manner intelligible to
a modern audience. Here I wish to do the opposite, to bring out
the complexities and confusion involved in developing an
account of properties of matter and motion that admitted of a
mathematical representation. The evolution of the quantification
of qualities followed the general evolutionary pattern of advan-
cing by coping with particular problems and without any goal of
developing a mathematical physics.
The idea of the quantity of a quality matured into a doctrine of
the intensification and remission of qualities. This led to a nest of
conceptual problems. Does the quality itself change, the degree
of participation in a quality, or does one quality replace another?
If the quality changes by addition, rather than replacement, how
is the addition of qualities to be understood?
The nominalism, spearheaded by William of Ockham, led to a
de-emphasis on the ontological aspects of this discussion. Ins-
tead of asking how intensification and remission of a quality
takes place he sought a criterion allowing one to predicate
“strong” or “weak” of the qualities a thing has.
The mathematization of this came chiefly from the “Cal-
culators” of the Merton school in fourteenth century Oxford and
later from the Parisian school. What mathematics did they have
(Mahoney, 1987)? In the twelfth century, three new elements
were introduced and gradually assimilated: Hindu-Arabic
arithmetic with its superior notation, Euclidean geometry, and
the algebra in the first (of three) parts of al-Khwarizmi’s treatise.
This part ended with the rule of three, or how to infer a fourth
number on the basis of three. Thus, if eight cost five, how much
do eleven cost? It was a verbal algebra. No symbols were used
even for numbers. Jordanus Nemorarius in the thirteenth century
first introduced these, in a very limited way. This treatment of
proportions was gradually fused with Euclidean geometry.
Euclid, more an organizer than an originator, had two distinct
theories of ratios and proportions. The one in Book VII,
stemming from Pythagoras, was limited to integers. The one in
Book V, stemming from Eudoxos, treated continuous mag-
nitudes. The mediaevals who adapted this lacked Eudoxos’s
concern with incommensurables and existence theorems. From
these elements, they fashioned conceptual tools that could treat
intensification of quantities through a kind of verbal algebra.
To grasp the conceptual problems it is important to change
perspectives. Today we would express the rule of three in a ratio
which we would symbolize as A/B = C/D. The Merton cal-
culators would symbolize this rule as (A,B) = (C,D). What is
significant is not the change in format, but the interpretation
given to it. The terms we treat as numerators and denominators
were understood as parts of a system of classification which
admitted of groups and sub-groups. If A is a multiple of B then
(A,B) is a multiple ratio. If A contains B once with a remainder
of 1 then (A,B) is a superparticular ratio. This admits of
different kinds. (3,2) is a sesquialterate ratio; (4,3) is a ses-
quitertian ratio. If A contains B once with a remainder greater
than one, then (A,B) is a superpartient ratio, which also admits
of sub-groups. If A contains B more than once, then one has the
general categories of multiple superparticular ratios and mul-
tiple superpartient ratios. In this way, Euclid’s system of pro-
portions gradually became assimilated to the arithmetic of
fractions. This was extended to ratios of ratios, but still using
terms and categories rather than mathematical symbols for
numbers. Thus a limited verbal algebra was developed for ex-
pressing proportions for quantities of qualities. It could not yet
treat a continuous variation in the quantity of a quality.
This systematization was given a kind of state-space repre-
sentation in the late fifteenth century by Nicole Oresme at the
University of Paris. Since this represents the transition from a
verbal algebraic treatment of quantification to the beginning of a
mathematical treatment is deserves more consideration. In
interpreting this it is important to realize the very abstract level at
which the physical-mathematical correspondence is found. I will
present Oresme’s explanation of this and then comment on it.
Every measurable thing except numbers is to be imagined in
the manner of continuous quantity. Therefore for the men-
suration of such a thing, it is necessary that points, lines, and
surfaces, or their properties be imagined. For in them (i.e., the
geometrical entities), as the philosopher has it, measure or ratio
is initially found, while in other things it is recognized by
similarity as they are being referred by the intellect to them (i.e.,
to geometrical entities). Although indivisible points, or lines, are
non-existent, still it is necessary to feign them mathematically
for the measures of things and for the understanding of their
ratios. Therefore, every intensity which can be acquired succe-
ssively ought to be imagined by a straight line perpendicularly
erected on some point of the space or subject of the intensible
thing, e.g., a quality. For whatever ratio is found to exist be-
tween similar ratio is found to exist between line and line, and
vice versa (Clagett, 1966: pp. 165-167).
Consider a body and a variable quality, such as motion or heat.
Represent the quality by a base line (eventually an x-coordinate)
and the intensity by a perpendicular (or y-coordinate). “Measure-
ment”, as Oresme uses this term, does not presuppose any unit or
method of measurement. The length of the lines representing
intensities has no absolute significance. What counts are the
ratios. If the intensity doubles then the length of the perpen-
dicular line should double. The initial assignments are always
Open Access 7
arbitrary, e.g. “Let the intensity of a quality have a value of four”.
As the intensity of a quality changes, the ratio of line lengths
representing the ratio of the changing intensities also change.
The mathematics of proportions handled ratios.
F. This seems to involve vanishingly small distances and times.
The development of this path led to Newton’s theory of fluxions,
or differential calculus. The notion of quantity of motion was
still obscure. However, it was represented by the area under a
line. For uniformly difform motion the areas involved triangles
and rectangles, something these medieval logicians could handle.
However, for difformly difform motion the quantity of motion is
the area under an irregular curve. Archimedes had treated some
such areas by his method of exhaustion. His works were not yet
available. The development of this path led to the integral
calculus of Leibnitz and his account of functions. For Leibnitz a
function expressed the relation of a dependent variable to an
independent variable. The ultimate independent variables are
space and time. Functions, so defined, express the relations
represented in Oresme’s diagrams.
The men who developed these systems were, by our occu-
pational categories, logicians, not physicists. They developed
logical systems that admitted of mathematical representation and
which might, incidentally, admit of physical examples. The term
“motion” still meant change in a quality with local motion
gradually emerging as the most significant type and “velocity” a
term for the intensity of local motion. Wallace (1981) has
clarified the basic systematizations used. I will use his notation.
A uniform motion (U) is one with a constant velocity (v), while a
difform motion (D) has changing velocity. Motion may be
uniform or difform in two different ways: with respect to space
(U(x) or D(x)) depending on whether all the parts of a body
move with the same velocity; and with respect to time, U(t) or
D(t). Difform motion is of various kinds. Motion that is difform
with respect to the parts of the object moved may be uniformly
difform, U(x), in the sense that there is a uniform spatial
variation in the velocity of the various parts of the body, or
difformly difform, D(x), if there is no such uniformity, or it may
be uniformly difform, U(t), or difformly difform, D(t), with
respect to time. This allows of various schema for treating
velocity. I will outline the one that Oresme used (Figure 1).
In the sixteenth century, Italy became the center for the deve-
lopment of mechanics, and of a mechanics that became more
clearly related to dynamics. There were three somewhat separate
traditions. The first, an academic tradition, was an extension of
the work of the Calculators. Paul of Venice studied at Oxford
and then, on his return, taught Mertonian ideas (Wallace, 1981:
Part II). Here the treatment of motion was caught up in the
traditional clash between Nominalists and Realists. This concern
influenced the dynamic tradition that Galileo redeveloped. The
Nominalists focused on the new mathematical treatment and
slighted the realistic significance of their formalism. The other
two traditions, developed apart from the universities, were
strongly influenced by the publication, in 1453 and later, of
Moerbeke’s translation of some of Archimedes’ works. The
Northern group, Tartaglia, Cardano, and Benedetti, was very
concerned with applications of mechanics to ballistics. The
group in central Italy, Commandind, Ubaldo, and Baldi, were
more mathematically oriented. (Drake & Drabkin, 1969).
The standard example of (3) is a rotating wheel, or, for
Oresme, the heavens, as circling around the earth or the sun (a
hypothesis he considered). The different parts move at different
velocities, but do not change in time. An example of (2) is a
falling stone. All the parts move at the same velocity, but the
overall velocity increases. This is not as good an example, since
it is not really uniformly difform. The increase in velocity
gradually stops, especially for light bodies. Either (2) or (3) may
be represented by a diagram which illustrates the Merton
theorem on uniformly difform motion. AC is the time axis, while
After the fall of Constantinople (1453) some Greek scholars
moved to Italy and taught Greek literature and philosophy in the
Academies sponsored by some noble families. This led to a type
of Neo-Platonism opposed to the Aristotelianism taught in the
universities. The Neo-Platonists especially studied Plato’s
Timaeus, where the astronomer, Timaeus, presents an extended
explanation (29e - 92c) of the order of the universe, beginning
with the Demiurge fashioning preexistent matter in accord with
mathematical forms and proportions and terminating with an
explanation of illness and disease resulting from a lack of
proportion of the four elements. A fusion of this idea of the
universe as an embodied mathematical system with the Biblical
account of creation supported the idea that God created the world
in accord with ideal mathematical forms. The further fusion of
these rather nebulous ideas with the mathematical treatment of
quantities of qualities led to the conclusion that in coming to
know things through their proper mathematical forms human
knowledge matched divine knowledge.
BF and CD represent intensities of uniformly difform motion.
The quantity of motion is represented by the area under the line
representing motion. Flipping the triangle, FGD, over so that D
coincides with C produces the rectangle, ACGE. This de-
monstrates that the total motion of a uniformly difform motion is
equal to the total motion of a uniform motion with half the
terminal velocity. It also follows that the total motion in the
second interval is three times the motion in the first interval. This
is the aspect Galileo adapted in treating acceleration.
This abstract approach raised questions that could not be
handled in the received Aristotelian terminology. Thus, half the
terminal velocity meant the instantaneous velocity at the point,
(2) U(x) D(t)
(3) D(x) U(t)
(4) D(x) D(t)
(1) U(x) U(t)
In the Preface to his Mysterium Cosmographicum, Kepler
wrote: “The ideas of quantities have been in the mind of God
from eternity, they are God himself; they are therefore also
present as archetypes in all minds created in God’s likeness”
(Cited from Koestler, 1960: p. 65). In a letter to a friend he said:
“For what is there in the human mind besides figures and
magnitudes. It is only these which we can apprehend in the right
way, and if piety allows us to say so, our understanding is in this
respect of the same kind as the divine, at least as far as we are
able to grasp something of it in our mortal life. Only fools fear
that we make man godlike in doing so; for the divine counsels
Figure 1.
Oresme’s representation of the Merton theorem of uni-
formly difform motion.
Open Access
Open Access 9
are impenetrable, but not his material creation” (Citation from
Baumgardt: p. 50).
Galileo did not share Kepler’s mathematical mysticism. Yet,
he assumed a similar relation between quantitative forms as
archetypes in the mind of God and men: “I say that human
wisdom understands some propositions as perfectly and is as
absolutely certain thereof, as Nature herself; and such are the
pure mathematical sciences, to wit, Geometry and Arithmetic. In
these Divine Wisdom knows infinitely more propositions, be-
cause it knows them all, but I believe that the knowledge of these
few comprehended by human understanding equals the Divine”
(Galileo: p. 114).
As these citations indicate, physical explanations were still
functioning in a theological context that had been stretched to
accommodate the mathematical representation of qualities. This
was regarded as objective, representing reality as it exists
independent of our knowledge of it. There was, however, the
beginning of a shift from a theological perspective to an obser-
ver-centered perspective. Early in the 15th century, the Floren-
tine architect and engineer, Filippo Brunelleschi, developed the
basic laws of linear perspective, reportedly by painting a copy of
part of the cathedral on top of its mirror image. Massaccio, della
Francesca, and others transformed painting by making
perspective basic. Leon Batitista Alberti codified the rules of
perspective in his book, Della pittura, (1436) with a vanishing
point and a horizon, both determined by the position of the
observer. This is a geometrical representation of space. In linear
perspective, the two dimensional representation of a three-di-
mensional space is thought of as a projection on a two dimen-
sional surface of light rays travelling from the source to the eye
of the observer, rather than the flat space of medieval painters.
The space represented is a Euclidean homogeneous space
organized from the standpoint of an outside viewer. In spite of
strenuous opposition this new way of organizing representations
of reality from the perspective of an outside observer rapidly
spread to other fields. (Frey, 1981; Chevalley, 1993).
Classical French drama respects the “Aristotelian” dramatic
unities of an integrated story completed in one day at one locale.
Aristotle had only insisted on unity of action. The “classical
Aristotelian doctrine” was articulated by sixteenth century Ita-
lian critics influenced by perspective. The dramatic action
should be presented from the perspective of an observer. Pers-
pective spread to physics when Kepler, influenced by Dȕrer’s
perspectival methods as well as Galileo’s account of his teles-
cope, showed, in his Dioptrice, how a correct geometrical
analysis of light rays explained vision. The theory it replaced,
Aristotle’s doctrine of transmitted images received as impressed
sensible species, was never able to account for the fact that
distant objects look smaller. Descartes’ La Dioptrique extended
Kepler’s work by giving a correct law of refraction. He
explained different colors in terms of light producing different
pressures on the eyeball. In his earliest experiments Newton
refuted this by inserting a stick behind his own eyeball and
exerting pressure.
Perspective entered mathematics with Descartes’ analytic
geometry and the representation of bodies through coordinates in
Euclidean space. Most analyses of this focus on the fact that the
geometry is Euclidean. What was also novel was the portrayal of
space from the perspective of an outside observer. The idea of
the detached observer regarding physical reality from an external
viewpoint culminates in Descartes’ Discourse on Method and
Meditations. This detached-observer view of reality was gra-
dually transformed into the notion of classical objectivity that
Husserl sharply criticized.
Regardless of whether gunnery practice or perspective was the
primary factor relating the observer to the described motion, the
final result is clear. In place of the abstract ratios of the Cal-
culators, the new methods begin with a three dimensional space,
which supplies a framework for the measurement of motion. The
bodily presence of the subject anchors this framework. Galileo
extended this through his development and use of the telescope,
describing in precise detail the positions of the Medicean stars
(Jupiter’s moons) as he saw them on January evenings in 1610.
Galileo’s work is not simply an extension of the preceding
developments. Galileo played a pivotal role in developing em-
pirical science, shaping and judging mathematical formulas by
the way they fit controlled observations. In spite of his early
exposure to Aristotelian natural philosopher, he took Archi-
medes, the prototypical mathematical scientist, as his ideal.
However, these advances were only possible because of the
developments we have been surveying. The mathematical treat-
ment of motion and forces emerged from three centuries of
muddling through quantities of qualities, verbal algebra based on
proportions, and a gradual switch from a theological inter-
pretative perspective to an observer-centered viewpoint. This
quantification of qualities gradually spreads to other fields,
something treated by other authors in this issue.
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