THOMISTS AND THOMAS AQUINAS ON THE
FOUNDATION OF MATHEMATICS
MAURER
ARMAND
I
modern
Oome
nas,
to follow
lead of Thomas
the
Aqui
in the
known
the objects
types of mathematics
Eu
as
and
numbers
of
whole
the
arithmetic
such
century,
claiming
of the
that
hold
thirteenth
clidean
thomists,
geometry,
of reason
beings
his once-popular
are
real
rationis)
(entia
scholastic
manual,
terms
In scholastic
entities.
they
real beings
(entia
Elementa
Philosophiae
but
Gredt maintains
that,
ico-Thomisticae,
Joseph
the object of mathematics
and Thomas
Aquinas,
or continuous
in arithmetic
discrete
quantity
are not
In
realia).
Aristotel
to Aristotle
according
either
is real quantity,
quantity
in geometry.
in abstraction
essence
of quantity
"When quantity
in bodily substance.
"it is not a being of reason
he writes,
in this way,"
is considered
(ens
so
it
is
a
Nevertheless
but
real
abstractly
being
(ens reale).
rationis)
ex
both real and conceptual
that it leaves out of account
considered
considers
The mathematician
istence."
ulation
the
to real existence
its relation
from
Recent
mathematicians,
to fictitious
quantity, which
Gredt
and belonging
mathematics,"
1
tatem
esse
to Gredt
According
their
spec
real being;
positively
this is a
from
distinct
essentially
to it only by reduction.1
mathematics
transcendental
special,
but not
conceptual
essence
its
which
by
has
the fourth dimension,
example,
a relation
to real existence.
excludes
for
extend
continues,
"real
est quantitas realis ita tarnen secundum
"Obiectum Matheseos
quiddi
suam abstr?ete
ut non dicat ordinem ad
et inadaequate
considerata,
reale
in substantia
corp?rea
seu
in ente
mobili.
.
. .
Quantitas
ita
con
ens rationis,
sed ens reale, tarnen ita abstr?ete
Recentes
etiam ab esse reali et esse rationis.
consideratur,
fietam ex
mathematicam
mathematici
usque ad quantitatem
speculationem
tendunt, quae non est ens reale, sed rationis tantum, ut est quarta dimensio,
suam positive exeludit ordinem ad esse reale.
Ita
essentiam
quae secundum
transcenden
constituitur Mathesis
quaedam specialis, quae vocatur Mathesis
talis et quae a Mathesi
reali essentialiter
neque ad earn pertinet
distinguitur
nisi reductive"; Joseph Gredt, Elementa
Aristotelico-Thomis
Philosophiae
ticae, 5th ed. (Freiburg: Herder,
1929) 1194.
siderata
Review
Metaphysics
non
est quidem
ut abstrahat
of Metaphysics
47
(September
1993):
43-61.
Copyright
?
1993
by
the Review
of
ARMAND MAURER
44
read the works
of Gredt,
his Ele
including
in his magistral
he
with
agrees
Degrees
of Knowledge
that at least the objects
of Euclidean
and the arith
geometry
Maritain
Jacques
and
menta,
Gredt
metic
of whole
are
numbers
entia
in distinction
realia
to the objects
rationis.
The
of modern
which
he calls entia
types of mathematics,
of
the
former
of
Maritain
says, are
objects
types
mathematics,
sense
the philosophical
that they can exist outside
the mind
whereas
the
world,
so exist.
cannot
physical
matics
geometries.2
more
Delving
than Gredt
into the nature
of mathematics,
deeply
that when
the mathematician
conceives
his
stresses
Maritain
jects they acquire an ideal purity in his mind which
real
existence.
By
world
the
mode
of being
intellect
but
their
purification
tion or reconstruction.
"Quantity
corporeal
structed
thus
[in mathematics]
but
substance,
or
constructed
it remains
idealized
itself witness
same
of
the mind
it has
built
and
now
the common
by
the
with
also
the conditions
describes
the
as a construc
to Metaphysics,
as a real accident
material
reason.
sensible
only their
are no
There
is affected.
studied
the
that not
Maritain
as
recon
of entities
Nevertheless
of
even
when
to bear
in
continues
corporeal,
it is derived."4
He makes
the
"In . . . mathematical
of Knowledge:
it has drawn
entities
from sensible
data
something
whence
in the Degrees
point
knowledge,
or which
the matter
from
in mathematics
of quantity
in his Preface
He writes
is not
ob
they lack in their
a way
in the real world
abstraction.3
or idealization
entities
in such
definition
very
numbers
lines,
to mathematical
proper
these
abstracting
idealizes
them
or whole
points,
in the
of mathe
objects
types
a line, and a whole
are real
A point,
number
or the constructions
irrational
numbers
of non-Eu
not
but
beings,
clidean
the newer
of
in
real
grasps
on them.
constructs
It grasps
them
or reconstructs
through
them
their
on
consti
the
same
tutive
elements,
level.
These things in the real [world] (when they are entia realia)
are
accidents
or properties
of bodies,
but
the mind
treats
them
as
2
to Unite; or, The Degrees
of Knowledge,
Jacques Maritain, Distinguish
of Gerald B. Phelan
trans, under the supervision
Bles,
(London: Geoffrey
on p. 460.
1959), 3 145-6. Maritain refers to Gredt's Elementa
166.
4 Ibid.,
toMetaphysics
(London: Sheed and Ward,
Jacques Maritain, A Preface
82.
1943),
45
AQUINAS ON THE FOUNDATION OF MATHEMATICS
and as
subsistent
they were
beings
of them were
free of any experimental
though
makes
In The Philosophy
sence of the scientific
becoming
Edward
of Mathematics
of mathematical
habit
and
"conformable
as quantified."6
solely
are not
mathematics
He
us
that
in nature,
discovered
completely
the properties
of the elements,
but neither
of the mind.
The mind,
he says, discovers
discovers
products
in nature
stance
and
mathematical
"inducts"
entities
verse
of
invention
Does
is one
the mathematician
with
the materials
this mean
that
least
the point
not
tionis
imaginative
he calls
these
in claiming
tain, however,
also real entities.8
Taking
distinguished
somewhere
of
substance
the
of
objects
as the scientist
they
pure
sub
quantified
with
the help of
in extramental
Thus
construction
the uni
and
of experience.7
are purely fictitious?
Not at
objects
is not an innate or sheerly
Mathematics
self
would
have
it.
at
furnishes
Experience
For this reason Maziarz
of departure
for mathematics.
a
as a logical
mathematical
object
regard
to Maritain's
but as an ens naturae.
Referring
Knowledge,
es
the
these
to Maziarz.
all according
creative
affair, as the idealist
does
of
sees
are
It is only in the imagination
and not
imagination.
as such exist.
of mathematics
reality that the objects
the
it
notion
as the mind's
to the nature
to assure
the
origin."5
Maziarz
abstraction
identifiable
hastens
though
entities
entia
that the objects
realia.
He
of modern
entity or ens ra
The Degrees
of
goes beyond Mari
mathematics
are
a suggestion
a
of John of St. Thomas,
Yves
Simon,
of Maritain,
the objects
of mathematics
pupil
places
real beings
of reason.
between
and beings
On the one
up
that mathematics
is not an ontology
of real quantity.
hand, he asserts
On the other hand, he is equally
insistent
that mathematical
entities
are not beings
of reason,
for this would
can have
imply that none
On
the
root
of
the
sixteen
square
counterparts.
physical
contrary
5
32.
In The Range of Reason, Ma
Maritain, The Degrees
of Knowledge,
ritain writes of science using as its instruments
"explanatory
symbols which
are ideal entities (entia rationis) founded
on reality, above all mathematical
entities built on the observations
and measurements
collected by the senses";
Jacques Maritain, The Range
(London: Geoffery Bles, 1953), 32.
of Reason,
See also p. 36.
6
Edward Maziarz, The Philosophy
of Mathematics
(New York: Philo
the mathematical
sophical Library, 1950), 194. On pp. 208-209, however,
nature is said to be "treated as if itwere a substantial
thing."
7
240.
8 Ibid.,
See also p. 227 n. 125.
Ibid., 208.
ARMAND MAURER
46
does
have
does
not.
a counterpart,
All mathematical
such
the
though
root
square
one
of minus
an
of
objects,
however,
which
Simon calls a "condition
of reason";
for they abstract
unreality,
from sensible
and
which
makes
it impossi
qualities
primary matter,
to exist outside
ble for them, as mathematical,
the mind.
They are
said
enigmatically
Simon
quotes
to exist
John
have
"beside
element
the world
of reality."9
statement
that mathematical
of St. Thomas's
true being;
it does not exclude
the
indeed,
In this respect
itself.
it differs
from purely
reality of quantity
imagi
a
reason.
which
is
of
"Mathematical
nary quantity,
being
quantity,"
admits
quantity
of real and
a being
of St. Thomas,
"is not determinately
a real being, but it is indifferent
is it determinately
dition and admits
of either of them."10
John
says
of reason;
to either
ther
nei
con
as teaching
that the objects
of
interpret Aquinas
not only have counterparts
in the real world
mathematics
but are in
fact the real quantities
of bodies
abstracted
from them.
An
Thomas
Some
derson
Thomists
writes
that the mathematician
as real.
precisely
Like
studies
the philosopher
though not
of physical
so in abstraction
real quantities,
he treats
of nature
of nature he does
but unlike the philosopher
quantities,
from the sensible
and
motions
of natural
qualities
Furthermore,
things.
as such,
since quantity
is an accident
of substance
and is only intelligible
enter into
quantity must
son states that mathematics
of St. Thomas
are sort
reason
of
that
"mathematical
intermediate
nor
real beings,
That mathematics?more
real world
the object
of mathematics.11
treats of real quantities,
is the theme
beings,
but share
of Vincent
Ander
Though
he grants with John
as Thomas
entities,
i.e.
they
are neither
features
of both."12
exactly
geometry?is
Smith's Aquinas
describes
purely
a science
Lecture,
them,
of
beings
of
the
St. Thomas
9
Yves Simon, "Nature and the Process
of Mathematical
Abstraction,"
on Maritain's
The Thomist
29 (1965): 135 n. 23. Commenting
of
philosophy
Simon describes Maritain's
of a mathematical
mathematics,
conception
object
as "a preter-real
and which
entity always affected by some conditio rationis
in the real"; Yves
often turns out to be a mere ens rationis with a foundation
in The Philosophy
of the Sciences,"
Simon, "Maritain's Philosophy
of Physics,
36.
ed. Vincent E. Smith (Jamaica: St. John's University
Press,
1961),
10
135-6 n. 23.
See also John of St.
Simon, "Nature and the Process,"
Cursus
Thomas,
theologicus,
disp. 6, a. 2 (Paris: Descl?e,
1931) 1:534 n. 20.
11
Matter
and
the
of Mathe
Thomas C. Anderson,
Objects
"Intelligible
in Aquinas," The New Scholasticism
43 (1969): 555-76.
matics
12
Ibid, 559 n. 13.
47
AQUINAS ON THE FOUNDATION OF MATHEMATICS
can it be called a science,
he asks,
How
the Object
of Geometry.
In fact, geometry
studies
sub
if it is not a study of the real world?13
of
is
stance
its
and
this
with
dimensions
quantity,
something
along
"exist in the sensible
world
real.
Its objects,
he writes,
they
although
on
as bearing
He
this existence."14
of mathematically
thought
the statement,
is the
concludes
his lecture with
"Euclidean
geometry
not
but
science
of what
is real but not physical,
sensible,
imaginable
In his Science
and mobile."15
and Phi
but not natural
truly essential
are not
losophy he presents
to Bertrand
Russell's
a similar
view
realistic
of mathematics.
Contrary
as an affair of logic
of mathematics
conception
as a linguistic
or symbolic
and David Hubert's
view of mathematics
is quantified
he insists that the subject of mathematics
system,
being?
as so
in
matter
not
it
is
of
that
exists
though
something
thought
existing.16
From
this brief
survey
of some
leading Thomistic
it is readily
of mathematics
of Aquinas's
conception
that there are significant
differences
tations
view
agree that in Aquinas's
or
constructs
mental
purely
beings
tinuous or discrete,
such as circles,
erally
abstracts
sus
that
these
these
Most
from
are
objects
this by adding
of mathematics:
qualify
in the object
they do not exist
to a "condition
forms
in reality
of reason."
sensible
of the
interpre
apparent
The Thomists
gen
them.
among
the objects
of mathematics
but
forms
triangles,
matter.
of quantity,
and numbers.
are not
either
The mind
There
is general
in the external
real, for they exist
that there is also an element
they are
as mathematical
and unreality
of these objects
Some Thomists
contend
It is not
can be
reconstructed
objects,
clear, however,
reconciled.
con
consen
world.
of unreality
or
in the mind,
or they are subject
how
the
reality
the object
is
of mathematics
that, since
on which
real quantity,
it cannot
be abstracted
from substance,
it de
see no place
for its being
and intelligibility.
Others
for sub
pends
in
in
the
the
for
this
would
sense,
stance,
object,
categorical
imply that
13
Vincent
E. Smith, St. Thomas on the Object of Geometry
(Milwaukee:
University
Press, 1954), 65.
Marquette
14
66.
15Ibid,
84.
Ibid,
16
and Philosophy
Vincent E. Smith, Science
(Milwaukee: Bruce,
1965),
219.
See also Vincent E. Smith, Philosophical
Physics
(New York: Harper
and Row, 1950), 11-12.
ARMAND MAURER
48
is a subsistent
the
object
holds
that
subsistent
the mathematician
entity.17
treats
It will
be
recalled
of quantity
as
that Maritain
though
it were
a
being.18
II
Both
ematics
the Thomists
and
unreality,
who
on the reality of the object
of math
as containing
an element
that object
of
insist
who
regard
as a reconstruction
those
such
by
or a "condition
the mind
of
in the works
find support
for their interpretations
of Aquinas.
reason,"
on the De Trinitate
In his commentary
of Boethius
his ear
(1258-59;
treatment
liest and most
extended
of the subject),
he describes
pure
as
mathematics
the
science
of abstract
and its properties,
quantity
considers
such as the circle and triangle.19
and
Geometry
magnitude
as
num
Mathematical
arithmetic
such
lines
and
number.20
objects,
on sensible matter
for their existence
but not
bers, are said to depend
this sort of matter
for our knowledge
of them, for they do not include
in their definitions.21
be
Boethius,
Aquinas
Following
distinguishes
tween
of physics,
the theoretical
sciences
and
divine
mathematics,
by the kind of form which
the form of quantity,
and
of form
abstraction
(abstractio
science
(namely, metaphysics)
is said to concern
Mathematics
abstraction
Quantity
stance.
is called
is understood
Since
the
as
the
depends
quantity
the mathematician
intelligibility,
only from the sensible
qualities
first accidental
on
cannot
substance
abstract
and movements
form
for
they study.
its mode
of
formae).22
in sub
inhering
its existence
and
it from
substance
but
of bodies.23
17
see Scriptum
For Aquinas's
notion of substance
super libros Senten
tiarum 2, d. 3, q. 1, a. 6, ed. P. Mandonnet
(Paris: Lethielleux,
1929), 2:103.
in Si. Thomas Aquinas
See also Etienne Gilson, "Quasi Definitio Substantiae,"
Commemorative
ed. Armand Maurer (Toronto: Pontifi
1274-1974:
Studies,
cal Institute of Mediaeval
1:111-29.
Studies,
1974),
18
See note 5 above.
19
super librum Boethii De Trinitate
Expositio
(hereafter EBT), q. 5, a.
E.
Bruno
Decker
J.
ed.
and
3,
(Leiden:
Brill, 1955), 184.20-22; The Division
trans. Armand Maurer, 4th ed. (Toronto: Pontifical
Methods
of the Sciences,
Institute of Mediaeval
Studies,
1986), 38-9.
20
a.
44.
ad
and Methods,
q. 5,
3,
6, 188.25; Division
21EBT,
and Methods,
14.
Division
q. 5, a. 1, 165.21-24;
EBT,
22
and Methods,
Division
41.
q. 5, a. 3, 186.16-18;
23EBT,
a.
and
Division
37-40.
q.
5,
3,
184.2-186.12;
Methods,
EBT,
AQUINAS ON THE FOUNDATION OF MATHEMATICS
49
in the object
of mathematics
of substance
is also
presence
as
matter
of
As
the
Aristotle
quantity.
required
maintains,
intelligible
there is form instantiated
in sev
there must be some matter wherever
The
eral
Since
individuals.24
be noticed,
however,
and De anima
it is not
forms.27
to be
that
is said
This
conforms
the generic
calls
principle
but known
by the intel
The notion
to Aquinas,
is substance.25
matter
of
also appears
in Aqui
quantity
intelligible
on Aristotle's
and his commentary
theologiae
Physics.26
Summa
physics
or body
to the senses
as the
of substance
It should
trian
circles,
of individuation
mathematical
be a material
there must
gles, and numbers,
not perceptible
in mathematics,
lect.
This principle,
according
nas's
are many
there
notion
in his
that
on
commentaries
substance
but
the Meta
the continuum,
surface,
of mathematical
matter
the
intelligible
use of the term;
to Aristotle's
he
of plane
the
figure
for example,
matter
of
intelligible
the
circle.28
in terms of the form
This description of the object of mathematics
of quantity
and
interpretation
stance with
mathematician
ing from
bodies.
stantial
the
The
entity
its underlying
of mathematics.
substance
As
its sensible
takes
sensible
qualities
for his domain
object
characterized
easily
so
actions,
substance
active
would
itself
treats
the physicist
and
and
qualities
of mathematics
lends
with
to a realist
of material
sub
it would
seem
quantity,
abstract
and
properties
passive
to be a real
appear
the
of
sub
only by the form of quantity.
on Aristotle's
and Physics
Metaphysics
a
to
in mathematics.
realist point of view
In the
commentaries
also
Aquinas's
lend themselves
former
the forms of figures,
commentary
or body
are said to exist in the continuum
or triangles,
as the form of human nature
such
as circles
24
7.11.1037a4-5.
For Aristotle's
Aristotle, Metaphysics
7.10.1036a9-12,
see Hippocrates
notion of intelligible matter
G. Apostle, Aristotle's
Philoso
of Chicago Press, 1952), 50-2, 106;
phy ofMathematics
(Chicago: University
and Joseph Owens, The Doctrine
in the Aristotelian
of Being
Metaphysics,
3d ed. (Toronto: Pontifical
Institute of Mediaeval
Studies,
1978), 342-3.
25
and Meth
EBT, q. 5, a. 3, 184.16-20; q. 5, a. 3, ad 2, 187.2-13; Division
42.
ods, 38,
26
Summa
super
Theologiae
(hereafter ST) I, q. 85, a. 1, ad 2; Sentencia
n.
lect.
2,
3,
332, ed. Angeli M. Pirotta (Naples: M. d'Auria Pontifi
Physicam
cus, 1953).
27
Sententia
super Metaphysicam
7, lect. 11, n. 1508, ed. M.-R. Cathala
and R. M. Spiazzi (Turin: Marietti,
See also lect. 10, n. 1496; and Sen
1950).
tentia super De anima 3, lect. 8, n. 714, ed. Pirotta (Turin: Marietti,
1938).
28
8.6.1045a33-35.
Metaphysics
ARMAND MAURER
50
in the organic
body.29
are said to have
mathematics
In the
exists
the same
lines, and surfaces?but
he says, studies
them
Physics,
ple,
points,
ways.
whereas
and
mathematics
Statements
and
commentary
physics
exam
of
objects
inquiry?for
they consider
as the termini
in abstraction
them
studies
their motions.30
latter
such
them
in different
of natural
from
as these would
bodies,
bodies
these
lead
to the con
of mathematics,
that for Aquinas
the objects
like those of phys
in short, that they are entia realia.
in the real world;
found
we find others
these statements
in the same works
of
Along with
clusion
ics, are
that
the objects
Aquinas
implying
such a way
that,
as Maritain
mode
but
of existing
matical
line has
it properties
only
different
those
not
The
fact
of a real
without
or natural
in
abstracted
are affected
says, they
in their very definition.31
one dimension?length
from
are
of mathematics
only in their
that a mathe
breadth?gives
line.
Aquinas
lines and circles
exist in the real world,
that, although
they
recognizes
are not of the same sort as those studied
He
mathematics.32
by
points
was aware of this, for he realized
a
out that Aristotle
that in geometry
straight
circles
touches
and
a circle
to real
that do not belong
properties
a straight
in Euclid's
For example,
and lines.
line
geometry
a circle at only one point, but this is not true of a real circle
line and
straight
have
line.33
between
Aquinas
clearly differentiates
properties
he explains
and those of mathematical
objects when
room while
the doors
have
entered
his disciples'
seems
matical
having
through
to be
contrary
straight
its own
the closed
lines
door,
how
Christ
could
were
shut.
This
to the principles
coincide
cannot
starting
of the real world
for two mathe
of geometry,
differ
in place,
but must
each
end point.
When
Christ
passed
and
point
at that moment
two bodies
occupied
the same
29
In VIIMetaph.,
lect. 10, n. 1496.
lect 3, n. 329.
30In IIPhys.,
31
Anderson
lists texts of Aquinas and works on Aqui
See note 3 above.
nas that "say in effect that mathematicals
exist with their specific character
Anderson
istics only in the mind of the mathematician.";
"Intelligible Matter
in Aquinas,"
558 n. 10.
and the Objects of Mathematics
32
is in
The statement
In IIIMetaph.,
lect. 7, n. 416; 11, lect. 1, n. 2161.
an objection,
but Aquinas does not deny it. He only denies Plato's position
that there are separately
existing objects of mathematics.
33
78. See Aristotle,
and Methods,
EBT, q. 6, a. 2, 216.20-26; Division
The tangency of a
3.2.997b35-998a4.
De anima
1.1.403al2-16; Metaphysics
a
one
is
to
at
line
demonstrated
circle
by
Euclid, Elements
only
point
straight
lect. 7, n. 416.
3, prop. 15, 16, as Aquinas points out; In IIIMetaph.,
AQUINAS ON THE FOUNDATION OF MATHEMATICS
51
at only two points,
terminated
and
place, and two lines of these bodies
Since this contradicts
each line at the same two points.
the notion
of
a mathematical
have done what
is mathe
line, God would
straight
matically
impossible.
In replying, Aquinas
between
and nat
mathematical
distinguishes
so
in
two
be distinct
ural lines.
The former must
that
lines of
place,
be thought
of as coinciding.
Two natural
lines, on
are
in the bodies
in which
if
distinct
the contrary,
they exist.
Now,
as in the miracle
we assume
in the same place,
that two bodies
exist
sort
this
cannot
of Christ's
the room with
entering
the doors
coincide.
and two surfaces,
lines, two points,
a
but the miracle
did not violate
miracle,
by
This
matics.34
it follows
that
two
could
only happen
the principles
of mathe
conceives
the
of
that Aquinas
clearly
objects
from those of nature
quite different
properties
shows
as having
mathematics
shut,
This
or reality.
is true not
This
between
distinguishes
the number
but also in arithmetic.
only in geometry
Aquinas
or numerable
a multitude
that is numbered
and
we number
or
it.
In a sense
by which
things numbered
can be called a number,
as we speak of ten men or horses.
is that by which
This
Number
they are numbered.
itself, however,
or counting
numeration
is an act of the human mind.
The existence
numerable
of
to
the multitude
is due
to the divine
mind;
its numeration
is owing
ours.35
Numbers
Aquinas,
what
today
an act of our mind.
For
originate
through
one is a number;
one is the starting point of
natural numbers.36
Each natural number
is an
themselves
neither
zero nor
are called
by adding one to its immediate
aggregate
predeces
produced
one to three.
four is produced
In other
sor; for example,
by adding
one
each
number
is
caused
several
Number
times.37
words,
by taking
of ones,
is also
said
seems
to be number
tude.38
to be
Number
of the continuum,
by the division
or numerable
in the sense of a numbered
caused
itself, while
presupposing
this multitude,
but
this
multi
depends
on
34
De potentia,
q. 1, a. 3, sed contra 8 and reply, ed. Paul M. Pession
See also In IV Sent., q. 2, a. 2, sol. 3, ad 2 (Paris:
1953).
(Turin: Marietti,
Vives, 351874), 11:325.
In rVPhys.,
lect. 23, n. 1209; lect. 17, n. 1113; and In VIIMetaph.,
lect.
1722.
3, n. 36
ST I, q. 11, a 1, adl.
37
In
lect. 17, n. 1020.
38 VMetaph.,
De potentia,
q. 1, a. 16, ad 3.
ARMAND MAURER
52
an act
of the mind.
Incidentally,
that God made
necker's
saying
are the work
of humans.
From
it seems
this
are
of mathematics
owe
their
existence
Aquinas
the whole
In fact we make
would
not
numbers
agree with Kro
but all the others
all of them.
clear
the objects
that, in Aquinas's
opinion,
not simply
from the real world
abstracted
but
or reconstructive
to a constructive
of
activity
the mind.
It is not
how this view of mathemat
immediately
clear, however,
ics can be reconciled
with
the statements
of Aquinas
cited
above,
seem
which
to imply a realist
of mathematics.
If the
interpretation
of mathematics
have properties
different
from those of nature
objects
or reality,
even
and if they differ from
it
in their definitions,
how can
or be abstracted
to exist
in the real world
they be said in any sense
from it?
In a neglected
of mathematics
quaestio
in a new
disputata
Aquinas
approaches
and original
offers
way which
the object
a possible
to these
solution
at Rome
between
1265 and
problems.
Composed
as
so
the quaestio
that he inserted
1267, Aquinas
regarded
important
on the Sentences,
it into his commentary
a de
which
he had written
cade
earlier
years,
when
Thus
the quaestio
from his mature
dates
(1252-1256).39
a master
he was
in theology,
unlike
the rest of the com
as a bachelor
which
he wrote
of the Sentences.40
It cannot
mentary,
be dismissed,
therefore,
as an early
expression
of Aquinas's
teaching
on mathematics.
In the quaestio
into the distinction
between
di
Aquinas
inquires
vine attributes,
like goodness
and wisdom,
and their possible
foun
in God.
dation
We are not concerned
here with his reconciliation
of
the plurality
39
of
these
attributes
with
the
absolute
oneness
of God.
It is so important, Aquinas
in book 1 of
says, that practically
nothing
can be understood
the Sentences
without
it. See In I Sent., d. 2, q. 1, a. 3,
1:66.
sol, ed.
40 Mandonnet,
On the origin and au
1:63-72.
Ibid, d. 2, q. 1, a. 3; ed. Mandonnet
see
of
this
Antoine
"Saint
Thomas et la dispute
thenticity
quaestio
Dondaine,
des attributs divins (I Sent. d. 2, a. 3): authenticit?
et origine," Archivum
Fratrum Praedicatorum
8 (1938): 253-62.
For an analysis of the notion of
see my article, "A Neglected
mathematics
in this quaestio
Thomistic
Text on
the Foundation
of Mathematics,"
21 (1959): 185-92; re
Mediaeval
Studies
in Thomas Aquinas
Studies
printed in Armand Maurer, Being and Knowing:
and Later Medieval
Pontifical
Institute
of Mediaeval
Philosophers
(Toronto:
The quaestio disputata
has been a part of Aquinas's
Studies,
1990), 33-41.
on the Sentences
since his own day, but for the sake of conven
commentary
ience I shall refer to it simply as the quaestio.
53
AQUINAS ON THE FOUNDATION OF MATHEMATICS
for our subject
importance
are related
the ways
concepts
of a reality
may be a likeness
What
is of
tween
concept
"man."
A concept
of this sort has
is his
threefold
be
distinction
to extramental
reality.
(1) A
outside
the mind,
for example
an immediate
in extra
foundation
the truth of the concept
reality: the reality causes
through
term
of
mind
and
the
and
the
reality,
conformity
concept
signifying
mental
properly predicated of the reality.
of an extramental
reality,
of our way
but
the
is
(2) A concept may not be a likeness
the mind
may
devise
it as a
(adinvenit)
A concept
reality.
its immediate
basis
of knowing
extramental
consequence
in reality;
of this sort has only a remote
foundation
An example
is the concept
is an activity
of the mind
itself.
of genus.
to
There
is nothing
outside
the mind
this
but
concept,
corresponding
from
the
attribute
fact
that we
to animal
we
there are many
of animals
species
of genus.
The proximate
foundation
of
act of the mind; but the concept
is a constructive
know
the notion
a concept
of this sort
has a remote
basis
in the extramental
in forming
taken
so the mind
is not mis
world,
it. Another
of this type of concept
sug
example
or the
is the abstraction
of the mathematicians,
by Aquinas
gested
of mathematics
abstraction
to the mathematician's
(abstractio
mathematicorum).
act of abstracting
but
He
is not
to the mathe
referring
or intentio
matical
he forms by means
of this act.
But more
concept
a
nor a
A
about
this later.
may have neither
concept
proximate
(3)
remote
in reality,
of a chimera.
like the concept
is
foundation
This
nor do we form it as a conse
not a likeness
of anything
in the world,
of our way
quence
false concept.41
41
of knowing
the world.
Hence
Aquinas
calls
it a
"Unde sciendum,
intellectus
quod ipsa conceptio
tripliciter se habet
con
enim hoc quod intellectus
ad rem quae est extra animam.
Aliquando
extra animam, sicut hoc quod concipitur de
cipit, est similitudo rei existentis
hoc nomine
in re
intellectus
habet fundamentum
'homo'; et talis conceptio
ad intellectum,
facit quod
immediate,
inquantum res ipsa, ex sua conformitate
sit verus, et quod nomen significans
intellectus
illum intellectum proprie de
re dicatur.
autem hoc quod significat nomen non est similitudo
Aliquando
ex modo
rei existentis
extra animam, sed est aliquid quod consequitur
intel
rem
est
et
sunt
extra
intentiones
quae
quas intel
animam;
hujusmodi
ligendi
lectus noster adinvenit; sicut significatum
hujus nominis
'genus' non est sim
ilitudo alicujus rei extra animam
sed ex hoc quod intellectus
existentis;
attribuit ei intentionem
intelligit animal ut in pluribus speciebus,
generis[; ] et
non sit in re, sed in in
intentionis
licet proximum
fundamentum
hujusmodi
non
est res ipsa. Unde
intellectus
tellectu, tarnen remotum fundamentum
est falsus, qui has intentiones
Et simile est de omnibus aliis qui
adinvenit.
ex modo
sicut est abstractio mathematicorum
et
consequuntur
intelligendi,
ARMAND MAURER
54
on the Sentences
a decade
first commented
before
Aquinas
we have been following,
a
the quaestio
he made
similar dis
between
three orders of "things signified
names":
by
(1) those
When
he wrote
tinction
in extramental
like man and stone;
being
reality,
having a complete
(2)
those having no being at all in reality outside
the mind,
like dreams
and
a foundation
the image of a chimera;
in extramental
(3) those having
are
an
but
whose
notions
act
of
the
mind.
reality
by
completed
Aquinas
as an example
gives
species.
it does
manity
of the latter
a universal,
type of concept
is something
real, but outside
such
as a
for instance,
the mind
Humanity,
no
common
not have the nature of a universal,
for there is
hu
in the external world.
When
the mind
forms the notion
of hu
it acts upon it and adds to it the meaning
of
(intentio)
same is true of time, continues
a
It
has
basis
Aquinas.
as time, namely
in the before and after of motion,
but its formal character
the numbering
of the before
and after,
is completed
by an act of the
a
Truth also has
mind.
in reality, primarily
foundation
the being (esse)
manity,
however,
a species.
The
of things,
it knows
but
its notion
is completed
(ratio)
as
they are.42
things such
here
between
Aquinas
distinguishes
concepts
that he
described
in the quaestio
by an act of the mind,
the
on
same
three
the divine
when
orders
of
attributes.
vero id quod significatur per nomen, non habet fun
hujusmodi.
Aliquando
in re, neque proximum,
damentum
chimerae:
neque remotum, sicut conceptio
ex
alicujus rei extra animam, neque consequitur
quia neque est similitudo
rem
vere:
et
est
I
modo
ideo
ista
In
aliquam
conceptio
falsa";
intelligendi
1:67. For an analysis of this text from the
Sent, d. 2, q. 1, a. 3; ed. Mandonnet
of logic see Robert W. Schmidt, The Domain
of Logic according
viewpoint
to Saint Thomas Aquinas
(The Hague: Martinus Nijhoff,
1966), 85-9.
42
in
dicendum,
nominibus,
quod eorum quae significantur
"Respondeo
esse totum
enim sunt quae secundum
venitur triplex diversitas.
Quaedam
sunt extra animam; et hujusmodi
sunt entia completa,
sicut homo
completum
et lapis. Quaedam autem sunt quae nihil habent extra animam, sicut somnia
autem sunt quae habent fundamentum
in
et imaginatio chimerae.
Quaedam
re extra animam, sed complementum
rationis eorum quantum ad id quod est
enim
Humanitas
formale, est per operationem
animae, ut patet in universali.
cum non sit extra
est aliquid in re, non tarnen ibi habet rationem universalis,
in
sed secundum
animam aliqua humanitas multis communis;
quod accipitur
intellectus
intentio, secundum quam
intellectu, adjungitur ei per operationem
inmotu,
dicitur species: et similiter est de tempore, quod habet fundamentum
sed quantum ad id quod est formale
scilicet prius et posterius
ipsius motus;
nu
in tempore,
intellectus
scilicet numeratio,
per operationem
completur
in re, sed ratio
merantis.
Similiter dico de veritate, quod habet fundamentum
eo
per actionem
intellectus,
quando scilicet apprehenditur
ejus completur
1:486.
See also
In I Sent, d. 19, q. 5, a. 1; ed. Mandonnet,
modo quo est";
De potentia Dei q. 7, a. 6; Schmidt, The Domain
of Logic, 82-5.
AQUINAS ON THE FOUNDATION OF MATHEMATICS
55
not mention
this passage
does
mathematical
quaestio,
a
as an example
with
of notions
in reality but
foundation
an
act
this
of
the
mind.
fit
into
They readily
by
category,
as does
or genus.
of species
the logical notion
of the two passages
is somewhat
different.
The no
language
the
Unlike
concepts
completed
however,
The
are here said to be "completed"
in this category
by an act of the
an act
in order to achieve
their formal character.
In the quaestio
mind
on
a
real foundation.
Itwould
of the mind
them
"devises"
(adinvenit)
tions
seem,
however,
completing.
nas regularly
not discover
to be
the same mental
act that does
the devising
and
ens
ens
in an
This act results
not an
reale.
rationis,
Aqui
an ens rationis
as a being
describes
that the mind
does
a
as
in reality but devises
of
consequence
(adinvenit)
He uses this term of logical notions
and others,
for
reality.43
to make.44
the craftsman's
idea of what
he intends
example,
Among
as mental
re
them he places
the objects
of mathematics,
elaborations
knowing
based
motely
tive activity.
on real quantity
on the mind's
but proximately
If this be true, for Aquinas
the mode
of abstraction
to the modes
in the other
is only analogous
employed
in general,
it is a way of knowing
in which
abstraction
one
siders
aspect
of a thing,
construc
in mathematics
sciences.
Like
con
the mind
out of consideration
other
aspects
leaving
it considers
the quantity
of bodies,
Specifically,
thing.45
and motions.
from their sensible
But the abstrac
qualities
abstracting
or
as
as
tion is constructive
well
selective.
The mind must
completive
of the same
add
to the real foundation
formal character,
None
and truth.
in extramental
comes
character
mathematical
play
43
the notions
of these, Aquinas
reality.
They have
from
the mind.
notion
and complete
of species,
universal,
told us, enjoys a complete
a foundation
there, but their
has
The
same would
seem
to be
its
time,
being
formal
true of
notions.
In this new
an
of the mathematical
as it does with
intrinsic
notion
role
of mathematical
as the
intelligible
objects
matter
real substance
of quantity.
does
not
Intelligible
"Ens autem rationis dicitur proprie de illis intentionibus,
quas ratio
in rebus consideratis,
sicut intentio generis,
et
speciei
similium,
in rerum natura, sed considerationem
rationis
quae quidem non inveniuntur
In TVMetaph.,
lect. 4, n. 574.
It should be noted that here
consequuntur";
For
Aquinas asserts that beings of reason are properly the subject of logic.
see Schmidt, The Domain
the meaning
of ens rationis
75-93.
of
Logic,
44
q. 8, a. 1, ed. R. Spiazzi (Turin: Marietti,
1949), 162.
45Quodlibet,
and Methods,
37.
EBT, q. 5, a. 3, 183.26-184.3; Division
adinvenit
ARMAND MAURER
56
matter
and
is within
sometimes
The mathematical
for sensible
it is placed by Aristotle
of quantity
itself, where
on his works.46
himself when
by Aquinas
commenting
to
still
related
is
essentially
object
substance,
however,
the order
is the remote
reality from which
of Aquinas's
consequences
substance
are
There
important
in the
of mathematics
tions
instead
second
of
the first.
order
second
Unlike
concepts
exist
do not properly
speaking
itself.
of existence
is the mind
subject
in the external
world.
Hence
on
of his
the
They
the no
placing
quaestio
first
level,
the mind.
outside
disputata
on the
those
Their
proper
are not
terms
mathematical
it is abstracted.
signs of anything
cannot properly
be
in the external
of anything
real: there is no referent
world
predicated
no
for a mathematical
line, circle, or number.
Finally, mathematical
are they said to be true, in that they
tions are not false; but neither
to anything
the mind.
does not suggest
outside
that
conform
Aquinas
be true
they might
The originality
It seems
noticed.
in some
other
sense.
not go un
should
if
in
the Middle
any, precedents
few,
none
to the best of my knowledge
that placed
logic and
to the real world.47
in relation
It has
in the same order
and
Ages,
mathematics
of
this notion
to have
of mathematics
had
to reconcile
of enabling
the Thomist
the real
important
advantage
in
remote
of
the
it
is
real
its
unreal
mathematical
features
and
object:
in the construction
in the sensible
but it is unreal
foundation
world,
the
or completion
the minds
adds
to it through
its act of abstraction.
Ill
Not
long after
the quaestio
disputata
the death
were
of Aquinas
challenged
his views
by Giles
on mathematics
of Rome
in
(1247-1316),
46
See notes 27 and 28 above.
47
Andrew Molland shows that Albert the Great, Aquinas's
teacher, places
inclines towards a con
in the real order, but that he strongly
mathematics
in the
G. Molland,
"Mathematics
Andrew
view of mathematics;
ceptualist
and the Sciences:
Commem
Thought of Albertus Magnus," Albertus Magnus
Institute of
orative Essays
(Toronto: Pontifical
1980, ed. James A. Weisheipl
on pp. 469-70:
Molland
Mediaeval
quotes Albert,
Studies,
1980): 466-7.
"Many of the geometers'
figures are in no way found in natural bodies"; Albert,
c.
A. Borget, Opera omnia
tr.
ed.
2,
3,
17;
(Paris: Vives, 1890). Au
Physica
between
thin lines, which are like a
gustine, in a Platonic vein, differentiates
the eye does not see
which
spider's web, and the lines of pure mathematics,
also between
the numbers by which
and which are known within ourselves;
we count and mathematical
numbers
10.12.19).
(See Augustine,
Confessions
to Edward Synan.
to Augustine
I owe the reference
57
AQUINAS ON THE FOUNDATION OF MATHEMATICS
a member
he had
Giles
studied
under Aqui
of St. Augustine.
the
latter
had
left
after
in 1269-1272,
Rome, where
shortly
on
attributes.
Giles
is best
the divine
the quaestio
disputed
known
for
nas
of the Hermits
in Paris
as Aquinas
In his
Aquinas's
verbatim
orders
teaching
that
taught, two principles
on
own commentary
and heartily
quaestio
disagrees
in which
the passage
Aquinas
an
those with
concepts:
a
remote
foundation,
only
notes
that Aquinas
placed
of
He
and
and the divine attributes
with
it.
things
takes
He
those with
the conceptions
no
(res).48
of
notice
quotes
between
distinguished
immediate
foundation
and
are not,
existence
of a being, but two
the Sentences
Giles
those with
all.
essence
in creatures
almost
three
in reality,
at
foundation
of natural
things
in the first order, logical and mathematical
to Giles
in the second.
But according
this misconstrues
conceptions
both the divine attributes
and the objects
of mathematics.
Perfections
in God and in a higher way
he says, but
than in creatures,
truly exist
fall short of their object.
when we affirm them of God our affirmations
I understand
When
immediate
foundation
derstand
the divine
wisdom,
the divine
for my knowledge
ad rem intellectam),
essence
furnishes
an
as regards
the reality
I un
but not for my way of know
(quantum
it
ad intelligendi
for wisdom
does not exist
modum),
(quantum
ing
as
we
in God
of God are not like those
it. Our conceptions
understand
A
like
of mathematics.
of natural
but
rather
conceptions
things
in a
has an immediate
foundation
continues,
line, Giles
the thing that is understood,
for a mathematical
line as regards
in the natural world.
it does
line exists
But my way of understanding
not have an immediate
in reality,
for I know
it without
the
foundation
mathematical
natural
matter with which
in that world.
and qualitative
it exists
My
a
it
remote
in
of
foundation
like
all
has
way
reality,
only
knowing
intentions."
The remote
foundation
for my knowing
"second
mathe
sensible
is the fact
matics
and
the
that quantity
lends itself
from the sensible.49
imaginable
to be abstracted
from quality
48
Giles of Rome, Theoremata
ed. E. Hocedez
de esse et essentia,
(Lou
vain: Museum Lessianum,
See
Etienne Gilson, The Christian
Philos
1930).
trans. L. K. Shook (New York: Random House,
ophy of St. Thomas Aquinas,
1956), 49369.
"Cum intelligo lineam mathematicam,
isti conceptioni
intellectus quan
tum ad rem respondet
etiam ut fundamentum
linea naturalis
immediatum,
esse habet in re naturali.
Sed quantum ad modum
quia linea mathematica
in re, quia
ei aliquid immediatum
fundamentum
intelligendi non respondet
licet linea mathematica
sit in re naturali et illam intelligam, tarnen non habet
esse sine materia quali vel sensibili, quam tarnen sine materia quali et sensibili
ARMAND MAURER
58
to the
three classes
of concepts
by
distinguished
a fourth,
an
in between
with
immediate
concepts
a remote
in reality and those with
foundation.
Concepts
which
divine
fourth
include
attributes
and
mathe
class,
Consequently,
Giles adds
Aquinas
foundation
in Giles's
and a remote
have, as noted above, both an immediate
notions,
an
in reality.
immediate
have
foundation
in the ex
foundation
They
as regards
a
the reality
but they have
ternal world
they designate,
our way of understanding
remote
that reality.
In the
basis as regards
matical
latter
respect
With
the
they are like "second
introduction
of Giles's
a radically
poses
nas's
different
notion
intentions."50
fourth
of concepts
from that
class
of mathematics
Unlike
does
not
and
circles
he pro
of Aqui
that mathe
grant
Giles,
Aquinas
in reality;
have a proximate
basis
their proximate
concepts
are natural
act of the mind.
There
foundation
is a constructive
lines
quaestio.
matical
and
circles
There
are
not
are
but
also
are
these
the numbers
based
not
the
lines
or numerable
numbered
of mathematics.
remotely
though
are products
of a mental
on
The
the sensible
activity
multitudes
which
world
of mathematics.
in reality but they
of mathematics,
objects
and abstracted
alters
their
definitions
from
it,
and
res immediate
non respondeat,
Et licet isti modo
intelligendi
intelligo.
re
in
est
sic
immediate
quod
quia
ipsa
intelligi possit, cum quantum
respondet
Giles of Rome, In libros
possit separari a quali et imaginabile a sensibili";
Sententiarum
1, d. 2, q. 3, ql. 2 (Venice, 1521), f. 18rb. Giles does not refer
Antoine
to Aquinas by name but, as was the custom of the time, as quidam.
is to Aquinas.
that the reference
See his
has shown conclusively
Dondaine
et
des
256.
For
Giles'
attributs
Thomas
la
"Saint
divines,"
article,
dispute
see Robert J. McLaughlin,
Abstraction
abstraction
doctrine of mathematical
toAristotle
as Constitutive
and Saint Thomas Aquinas
of Science according
33-9.
of
University
Toronto,
1965),
diss.
(Ph.D.50
divinam quantum ad rem intellectam ut im
"Cum intelligo sapientiam
in qua verissime
talis
fundamentum
mediatum
respondet
ipsa divina essentia
non
modum
divina
Sed
ad
existit.
respondet
quantum
intelligendi
perfectio
in Deo
essentia ut immediatum
fundamentum,
quia non est eo modo sapientia
rerum
sed
Unde non assimilatur
ut nos intelligimus.
naturalium,
conceptioni
nee
immediatum
quae
mathematicorum,
penitus
conceptio
conceptioni
magis
Sed quantum ad rem habet
habet in re, nee penitus mediatum.
fundamentum
et convenit cum naturalibus,
immediatum
quantum ad modum
[intelligendi]
Giles of Rome, In
et convenit cum secundis
habet mediatum,
intentionibus.";
f. 18rb-va.
I Sent,
A
is immediately
the concept of an external
A "first intention"
thing.
of something
is a concept
intention"
"second
upon that first
consequent
and
It is formed by reflection on the primary act of knowledge
knowledge.
See St. Thomas, Depotentia,
its concept.
q. 1, a. 1, ad 10; q. 7, a. 9. For the
117-29.
of these intentions see Schmidt, The Domain
of Logic,
meaning
59
AQUINAS ON THE FOUNDATION OF MATHEMATICS
in the external world.
like them exists
Aquinas was
Nothing
mathemat
aware
of
his
from
of
this
contemporary
knowledge
clearly
have seen
he would
ics. Had he known modern
types of mathematics,
natures.
As
range of the mind's mathematical
behalf we
in Thomas's
and defender
unlimited
the almost
a witness
inventiveness.
have
Robert
of
A
polemic
against
di
threefold
his confrere's
Orford
Dominican
and Thomist,
supported
fourfold
Giles's
in his quaestio
vision
of concepts
disputata
against
in the first class
the divine attributes
he places
Like Aquinas,
division.
Giles
Orford's
of Rome,
logical concepts
tion mathematics.51
about
but unfortunately
in the second,
and
written
1288-1292.
he does
not men
IV
did not bring
Aquinas
matics.
They are scattered
the De
on mathe
in
chiefly
writings,
while
often
expressed
his many
and
abstraction
In his
to a
find an approach
on Aristotle
commentaries
do we
of Boethius
Trinitate
of mathematics.
philosophy
he stays close to the Greek
of mathematical
his views
place
Only in the highly original questions of his
treating of other topics.
on
in one
throughout
and Aristotle
on Boethius
commentaries
treatise
together
literal
philosopher's
as a mental
his doctrine
echoing
thought,
exclu
act that concentrates
to
of physical
bodies,
quantities
motions.
and
qualities
and geometry
arithmetic
In these commentaries
presents
Aquinas
Euclid's
Elements
and
Arithmetic
from
he
knew
Boethius's
of
(which
as
arts
sciences.
Their ob
liberal
and
realistic
frankly
Geometry)52
sively on the extensive
of their
the disregard
jects
are not
they free
in sensible
separately
constructions
bodies,
51
Robert
and numerical
sensible
existing
of the
though
forms,
as Plato
imagination
considered
apart
are
taught, but neither
intellect.
exist
They
from them.
Logic alone?
and
dictorum
of Orford, Reprobationes
inprimum
afratre Egidio
q. 1, d. 2, q. 9, ed. A. P. Vella (Paris: J. Vrin, 1968), 57-9.
Sententiarum,
52
ed. G. Friedlein
arithmetica,
Boethius, De Institutione
(Leipzig: Mi
ed. I. L. Heiberg, Opera omnia,
nerva, 1867). Euclid, Elementa
Geometriae,
vols. 1-5 (Leipzig: Teubner,
1883-1888).
ARMAND MAURER
60
not so much
regarded
of mental
constructs
which
Aquinas
science?treats
The
realistic
view
as a science
as an instrument
(entia rationis).
seems
to have
of mathematics
It was
in the thirteenth
held
been commonly
Albert
the Great,53
by
of arts at Paris.55
We
century.
accepted
Robert
and
in
the masters
general
Kilwardby,54
seen
have
indications
in the writings
of Aquinas,
this generally
held view of mathematics
qualified
a constructive
ality, suggesting
in the elaboration
Albert
him
preceded
writes
Andrew
Molland,
egy,"
link between
Aristotle
the
While
world."56
have
hardly
that science.57
role
of mathematical
the Great
failed
on
commenting
to be struck
for
the
and
intellect
claimed
that
"Albert's
general
tenuous
than
more
mind
geometer's
Euclid's
and
Geometry,
constructive
the mind's
by
that he
however,
as a science
of re
imagination
It has been
objects.
in this regard.
"was to make
the
of
the
strat
had
outside
Albert
could
in
function
There
he
a quasi-factive
attributes
role to the mind
in
explicitly
we
in the quaestio
been
have
disputata
considering.
as
the concepts
of mathematics
in the same order
places
those
of
logic,
Aquinas
mathematics
own
as devised
by
on
the mind
on
and
the proximate
basis
He uses
the same
activity
only remotely
reality.
has
adinvenit
the
double
of discovering
(which
meaning
to
the
denote
of
both
kinds
of
concepts.
genesis
ing)
it seems
inevitable
to conclude
not
intentions,"
though
on acts
relations
consequent
ond
clude
These
bodies.
relations
exist
that both
the
of
the mind.
but primarily
such
as natural
in reality
The mathematician
same
of
items
and
visualizes
are entia
its
word
invent
Consequently
rationis
and "sec
kind.
Logical
Mathematical
as lines, circles,
visual
quantitative
them
and
of
in the
notions
are
in
notions
and numbers.
features
imagination
of
and
53
See
Albert, Physica
1, tr. 1, c. 1, ed. A Borgnet, Opera omnia, 3:2a.
in the Thought
also Molland,
"Mathematics
of Albertus Magnus,"
466, and
note 47 above.
54
Robert Kilwardby, De Ortu Scientiarum,
ed. Albert G. Judy (Toronto:
Institute of Mediaeval
Pontifical
Studies,
1976), 13-14, 29-31, 36-41, 53-81.
This work is dated c. 1250.
55
au XlIIe si?cle, ed. Claude
See Quatre introductions
? la philosophie
of four treatises by mas
This is a compilation
Lafleur (Paris: J. Vrin, 1988).
ters of arts at Paris before 1250.
56
in the Thought of Albertus Magnus," 470.
"Mathematics
57Molland,
See Paul M. J. E. Tummers,
"The Commentary
of Albert on Euclid's
of Geometry,"
in Albert the Great and the Sciences,
479-99.
Elements
61
AQUINAS ON THE FOUNDATION OF MATHEMATICS
reconstruct
them from their natural
abstracts
mentally
setting, while
are
an
of
in
The
them
ideal
way.
objects
geometry
proximately
ing
on such a reconstructive
are directly
act and numbers
founded
based
on the acts of adding
and counting.
Giles of Rome criticized his former teacher for deviating from the
commonly
his usual
notions
held
doctrine
Aristotelian
on
of mathematics
from
Giles
that mathematical
subject.
granted
sense
our
in
the
that
of knowing
way
intentions,
in abstraction
from sensible
and qualitative
mat
entities
teaching
are second
mathematical
the
and not on reality.
But he rejected Aquinas's
is known
what
is
that in mathematics
(the res intellecta)
a
remote
in
with
the
foundation
the
sensible
only
mind,
by
and that as such it is an ens rationis.
ter depends
contention
devised
world,
on the mind
If this be indeed the case, the ambiguities
on
even
and
the
objects
can be
subject
of mathematics
Are
removed.
at
all?or
or are
real beings,
between
real
in Thomistic
least
literature
some?of
the
or do
entia
they
rationis,
and entia
rationis?
The
beings
are
to
the
conclusion
constructs
leads
that
all
of
present
study
they
the mind,
but they have a real remote
in the sensible world.
foundation
as the intelligible
or is
matter
Is substance
needed
of mathematicals,
lie somewhere
they
substance only thought of as fulfilling this function?
is only the remote
foundation
of mathematical
does not enter into a mathematical
notion.
not
a substantial
ten categories
cals is within
entity;
of Aristotle.
as an ens
can be
resolved
itself
objects,
A mathematical
it does
not
is
object
fall within
the
matter
of the mathemati
intelligible
an
ens
like
rationis
order,
itself; for ex
of line as divisible
into straight
and curved,
The
the conceptual
the generic
concept
ample,
and the generic
of number
concept
These
difficulties
and ambiguities
matics
rationis
If real quantity
substance
in the
as divisible
in Thomistic
light of Thomas's
into whole
and fraction.
accounts
neglected
of mathe
text
on
the
subject.58
Pontifical
58
See note
40 above.
Institute
ofMediaeval
Studies