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Aristotle's Philosophy of MathematicsAuthor(s): Jonathan LearSource: The Philosophical Review, Vol. 91, No. 2 (Apr., 1982), pp. 161-192Published by: Duke University Press on behalf of Philosophical ReviewStable URL: http://www.jstor.org/stable/2184625 .

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The PhilosophicalReview,XCI, No. 2 (April 1982)

ARISTOTLE'S

PHILOSOPHY

OF MATHEMATICS

Jonathan

Lear

r

he fundamentalroblem

n thephilosophyfmathematics,

which

has

persisted

rom

lato's

day

until

urs,

s to

provide

an accountof mathematical ruth

hat s

harmonious

with our

understandingfhowwe cometoknowmathematical ruths.1n

Physics 2 and Metaphysics

3 Aristotle rovided he seeds

of

a

unified hilosophy

f

mathematics.

his has

not

been

generally

appreciated

or

wo reasons. irst, t

is

commonly

ssumed

that

Aristotle hought hat the

objects n the natural world do

not

perfectlynstantiatemathematical

roperties: physical

phere

is not

ruly pherical; straightdge

s

not

trulytraight.2

n

con-

sequence, hough ommentatorseeAristotles railing gainst

Platonic ntology fgeometrical

nd arithmeticalbjects, hey ee

him

as unable to offer

genuinely lternative pistemology.

Mathematicians,

ccording

o one nfluential

nterpretation

f

Aristotle, treat objects

which are different

rom

ll sensible

things, erfectly

ulfill

iven

conditions nd are

apprehensible

by pure thought. 3

his interpretation ust view

Aristotle s

caught

n

the middle

of

a

conjuring

rick:

rying

o offer n

ap-

parentlyPlatonic account of mathematicalknowledgewhile

refusing

o allow

theobjects

hattheknowledge s knowledge

f.

Second,

Aristotle's

hilosophy

f

mathematics

s

often abeled

'Meno 73e-87c,Republic

507e-527d. cf. Paul

Benacerraf,

Mathematical

Truth,

ournal

fPhilosophy, 0

(1973); Kurt Gddel, What is

Cantor's

Con-

tinuum

Problem?

in

Philosophyof Mathematics:

Selected

Readings,

ed.

Paul

Benacerraf nd Hilary Putnam (Englewood Cliffs: PrenticeHall, 1964);

Charles

Parsons, Mathematical

ntuition, roceedingsf

the

Aristotelian

ociety,

80

(1979-80); PenelopeMaddy, Perception

nd

Mathematical

Intuition,

PhilosophicalReview,89 (1980).

2Ian Mueller, Aristotle n Geometrical

Objects, Archiv ir

Geschichte er

Philosophie, 52 (1970);

Julia Annas,Aristotle'sMetaphysics

M

and

N

(Oxford:

Clarendon

Press,1976),

p.

29.

The

literature

n

which

this

ssumption

s

ex-

plicit

or implicit s enormous. mention

wo excellentworks

which, think,

provide

he

bestdefense f n interpretationithwhich

wish o

disagree.

3Mueller, p. cit., p. 157.

161








JONATHAN

LEAR

abstractionist -mathematical

objects

are formed

by

abstract-

ing

from he

sensible

properties f

objects-and it is

thoughtthat

he fallsvictimto Frege's attacks on abstractionist philosophies

of

mathematics.4

However,

Aristotle's

abstractionism

s

of

enduring

philosoph-

ical

interest.For

not

only

does it

differ

undamentally

from

the

psychologistic

heories

that

Frege

scorned;

it

represents

serious

attempt

to

explain

both how

mathematics can be true and

how

one

can have

knowledge

of

mathematical truths.

I

In

Physics 2

Aristotle

begins

to define

mathematical

activity

by

contrasting t with

the

study of

nature:

The nextpointto consider showthemathematician iffersrom

the

physicist.

bviously

physical

odies

contain

urfaces,

olumes,

lines,

nd

points, nd these

re the

subjectmatter

f

mathemat-

ics ... Now

the

mathematician,

hough

e too

treats fthese

hings

(viz., urfaces,

olumes,

engths,

nd

points),

oes not

treat hem s

the

imits f

physical

ody;

nordoes he

consider he

ttributes

n-

dicatedas the

attributes

f

uch bodies. That

is

why

he

separates

them,

or

n

thought

hey

re

separable

rom

motion

kinisis),

nd it

makesnodifferenceordoesanyfalsityesult f hey reseparated.

Those

who believe he

heory

fthe

forms o

the

ame,

hough

hey

are

not aware

of

t;

for

hey eparate he

objects f

physics,which

are less

eparable

han

those f

mathematics. his is

evident

f

one

tries o

state

n

each of

he wo

asesthe

definitionsfthe

hings nd

of their

ttributes.

Odd,

even,

traight,

urved,

nd

likewise

number,

ine,

andfigure

o

not nvolve

hange;

not

ofleshnd bone

nd

man-these

are

defined

ike nub

ose,

ot

ike urved.

imilar

vidence

s

supplied

bythemorephysical ranches fmathematics,uchas optics,har-

monics,

stronomy.

hese are in

a way

the

converse f

geometry.

While

geometry

nvestigateshysical

engths,

ut notas

physical,

optics

nvestigates

mathematical

engths,

ut as

physical,not

as

mathematical.

Physics

2,

193b23-194al2; my

emphasis]

4Annas,

p. cit.,

pp.

31-33.

162








ARISTOTLE

ONMA

THEMA

TICS

Several

fundamental

features

f

Aristotle's

philosophy

of

mathe-

matics

emerge quite

clearly

from

this

passage.

(1)

Physical

bodies

do

actually contain the

surfaces,

engths,

and

points that are the

subject

matter

of

mathematics

(193b23-25).

(2)

The mathematician

does

study

the

surfaces,

volumes,

lengths,

nd points

of

physical

bodies,

but

he does not

con-

sider them as the

surfaces,

tc.,

ofphysical

bodies

(1

92b3

1-

33). Geometrydoes investigatephysical lengths,but not

as

physical

(194a9-1

1).

(3)

The

mathematician

is able to

study

surfaces,

volumes,

lengths,

nd points

in

isolation

from

heir

physical

instan-

tiations

because

(in

some

way

which

needs

to

be

explained)

he is

able

to

separate the

two

in

thought

(193b33

ff.).

(4) Having been separated by thought,mathematical objects

are freefrom

the

changes

which

physical

objects

undergo

(193b34).

(5) (For

some

reason

which

needs

to be

explained)

no falsity

results

from this

separation

(193b34-35).

Indeed, it

seems that

it is

onlythose

things

which

the

mathema-

tician separateswhich can be legitimately eparated. Platonists,

in

their

so-called

discoveryof the

Forms,make at

least

two

mis-

takes.

First,

they

do not

realize

that,

at base,

they are

doing no

more

than

engaging

in

this

process of

separation in

thought

(193b35).

Second, they

choose

the

wrong

thingsto

separate.

The

reason-and

thismust

provide a

significant

lue

to how

Aristotle

thought

egitimate

acts of

separation

could

occur-seems

to be

that Platonists tried to separate frommatterthingsthat could

not be

conceived

of

except as

enmattered.

His

paradigm

con-

trast

s

that of

snub

with

curved:n

account

of

snubness

cannot

merely

pecify

shape; snub

must

be a shape

embodied

in a nose.5

Because it is

necessarily

nmattered,

snub

thing

cannot be

con-

ceived of

as

independent of

physical

change as,

for

example, a

5Cf.,

e.g.,

Metaphysics

025b31;

1030b29

ff.;

1035a26;

1064a23;

1030b17;

1035a5; 1064a25.

163








JONATHAN LEAR

curve can

be.

It

is

thus clear that Aristotle allows that there is

some egitimate

type

of

separation, differing

rom

the Platonist

separation of the Forms, such that if we understand how this

separation occurs

and why

it is

legitimate,

we will

understand

how mathematics is possible.

The heart

of

Aristotle'sphilosophy

of

mathematics

s

presented

in Metaphysics

M3:

Just as universal

propositions

n

mathematics are not about

sepa-

rate

objects over

and above

magnitudes

and

numbers,

but are

about these, nlynot as having magnitude orbeing divisible, clearly

it

is also possible for there to

be

statements

nd

proofs

about

(perz)

perceptible magnitudes, but not

as

perceptible but

as

being

of

a

certain kind. For

just

as

there are

many

statements about

things

merely

s

moving apart

fromthe nature of each

such

thing

and

its

incidental properties

and

this does

not mean that there

has

to be

either some moving object separate from the

perceptible objects,

or

some such

entity

marked off

n

them),

so in

the

case of

moving

things there will be statementsand branches of knowledge about

them, not as

moving but merely as bodies,

and again merely as

planes and

merely s lengths, s divisible and as indivisible but with

position

and

merely s indivisible. So since it is

true to say without

qualification

not only that separable things exist but also that

nonseparable

things exist (e.g., that moving things exist), it is also

true to

say

without qualification that

mathematical objects exist

and are as

theyare said to be. It is true to say of

other branches of

knowledge,without qualification, that they are of this or that-

not

what is incidental (e.g., not the white,

even if the branch of

knowledge

deals with the healthyand the healthy

s white) but what

each

branch

of

knowledge

is

of,

the

healthy

if

it

studies ts

subject)

as

healthy,

man if

(it

studies

it)

as

man. And likewise with

geome-

try:

the

mathematical branches of

knowledge

will

not be about

perceptible objects ust

because their objects

happen

to be

percep-

tible, though

not

(studied)

as

perceptible;

but neither

will

they

be

about

other

separate

objects

over

and

above these.

Many properties

hold true of

things

in

their

own

right

as

being

each of

them of

a

certain

type-for

instance,

there re attributes

peculiar

to

animals as

being

male or as

being

female

(yet

there is no

female or male

separate

from

animals).

So there are

properties holding

true of

things merely

as

lengths

or as

planes.

The

more

what

is known is

prior

in

definition,

nd the

simpler,

the greaterthe accuracy (i.e., the simplicity)obtained. So there is

164








ARISTOTLE

ON MA THEMA

TICS

more ccuracywhere

here s

no

magnitude

hanwhere here

s,

nd

most

of

all where

here

s

no

movement; hough

f there

s

move-

mentaccuracy s greatestf it is primarymovement, hisbeing

the simplest, nd uniform

movement he simplest orm f that.

The same account

applies to harmonics

nd optics; neither

studies tsobjects

s

seeing

ras

utterance,

ut s lines nd numbers

(these being properattributes

f the

former);

nd mechanics

likewise.

So

ifone

posits

bjects eparated

romwhat s incidental

o them

and studies hem

s

such,

one will not because

of

this

peak

falsely

anymore han fone draws foot n theground nd calls t a foot

long when t is

not a foot ong; the falsehood

s not partof the

premises.

The bestway

of studying ach thingwould be this: to separate

and posit

what

s

not

separate,

s

the arithmetician oes and

the

geometer.

man

is

one and

indivisible

s

a

man,

and

thearithme-

tician

posits

himas

one

indivisible,

hen tudies

what

s

incidental

to the manas indivisible; hegeometer,

n the other

hand,

studies

him neither

s

a man noras indivisible, ut

as

a solid object.For

clearlyproperties

e would have had even

if he

had

not been

in-

divisible an

belong

to

him

irrespective

f

his

being

indivisible

or a man [aneutoutin].That is whythe geometers peak correctly:

they

alk about

existing hings

nd theyreallydo exist-for

what

exists oes so

in one of twosenses,

n

actuality r materially.Met.

M3,

1077bl8-1078a31; my emphasis]

The point of the argument is to show that, pace Plato, one can

allow that the mathematical

sciences

are

true without

having

to

admit the existence

of

ideal

objects.

This

chapter

follows a sus-

tained

critique

of mathematical

Platonism,

and it should be read

as

possessing

ts own dialectical

strategy,

irected

if not

against

Plato,

then

against

Academic Platonists.6

Thus

Aristotle

begins

with

cases

which

he thinks

either

are

or should be

most embar-

rassingto thePlatonist. By the universalpropositions n mathe-

matics (ta katholou n tois

mathimasin),

e is probably referring

to

the

general propositions

expressed by

Eudoxus'

theory

of

pro-

portion.

Aristotle

eports

hat the fact that

proportions

lternate

6Cf.

Metaphysics2.

7Cf.Metaphysics

077b17;

Euclid,ElementaLeipzig: B.G.

Teubner,

1969-73),

II, 280;and W. D.

Ross,

Aristotle'setaphysicsOxford:

Clarendon

Press,

975),

II, 415.

165








JONA

THAN LEAR

-that

if

a:b::c:d, then a:c::b:d-used to be proved

separately

fornumbers, ines,

solids,

and times

An. Pst. A5, 74a19-25).

Now,

he says, t is proved universally 74a24). I suspect that the proof

with which Aristotle was familiar

differed lightlyfrom

Euclid

V-16. For

Euclid V presents a

generalized theory of

propor-

tional

magnitudes,ut in

MetaphysicsM2, Aristotleexplains his

objection to Platonism on the basis

of the universal

propositions

of

mathematics as follows:

There

are some universal

ropositions rovedby mathematicians

whose pplication xtends eyond hese bjects sc.,Platonicnum-

bers nd

geometricalbjects].

o therewill

be

another

ype

f

object

here,between nd

separate

fromboth Forms and intermediate

objects,

neither

umber

nor

point

nor

magnitude

or time.

f

this

is mpossible,learlyt s also

mpossible

or heformer

sc.,numbers

and

geometrical

bjects]

to exist

n

separation

from

perceptible

objects. Met.M2, 1077a9-14;

my emphasis]

Thus the proofwithwhich Aristotlewas familiarprobably had

a

slightly

more

algebraic

character than Euclid

V-16. The ex-

ample

of

universal

propositions

in

mathematics

is

supposed

to

cast doubt on the

Platonists'

move from

the fact that there

are

sciences of

arithmetic and

geometry

which issue arithmetical

and

geometrical

truths,

to the existence of

Platonic

numbers,

planes, and

solids.8 Aristotle's

objection

is ad

hominem.

f

one be-

lieves

that

the

mathematical sciences guarantee the existence

of

Platonic numbers and

geometrical objects, then

one

should

also believe

in

the

existence

of

deal

objects

which the

generalized

science of

proportion

s

about. But neitherare there

any

obvious

8At

Metaphysics

079a8-1I

Aristotle

ays according

o the

rgument

rom

he

sciences here

will

be forms f verythingf

which here s a science. The

exact

structurefthe

rgument

rom

he ciences

s

a

complex nd difficultroblem,

which ies

beyond he cope ofthispaper.

For thosewho wishtoreconstructt,

cf.Metaphysics

90b8-17,

987a32-b18,Pen

Idean,

fragments and 4 in W.D.

Ross,AristotelisragmentaelectaOxford:

ClarendonPress,1974). See

also M.

Hayduck's ext

fAlexander's ommentary

n thenotedpassage nMetaphysics

A9, In Aristotelis

etaphysica ommentaria,

Berlin, 1891). And cf. Republic479a-

480a, Timaeus

1d-52a. For recent ommentaryee, e.g., W.D. Ross,

op. cit.,

I, pp. 191-93;

H.

Cherniss, Aristotle's riticism

f

Plato

and theAcademyNew

York:

Russell

nd

Russell,1964),pp. 226-60,272-318,488-512.An even more

diffi-

cult

uestion,which lso liesbeyond he copeof his

aper,

s:

towhat

xtent,

f

any,

did

Aristotle imself ccept a, perhaps

estricted,

ersion f

the

argument

from

he

sciences? ee note 27 below.

166








ARISTOTLE

ONMA THEMA TICS

candidates,

as squares, circles,

and numbers

are obvious candi-

dates for geometry

and arithmetic,nor would

the Platonist

be

happy to discover a new ideal object. For thetheory f propor-

tion

is

applicable

to numbers, lines, solids, and times,

though

it is not about

any particular objects over

and above these.

Immediatelybefore discussing

the various

distinct proofs

that

proportions

lternate, Aristotle

ays that occasionally we cannot

grasp the full universalityof

a truth because

there is no name

that

encompasses

all

the different

orts of

objects

to

which

the

truth pplies (An.Pst.A5, 74a8). In thesepassages,Aristotleuses

the word megethos

o

refer

strictly

to

spatial magnitudes;

whereas Euclid's use

of

megethos

n

ElementsBook

V

is best

understood

s

being

the

general

termAristotle

oughtafter,

ppli-

cable to spatial magnitudes,

numbers,

and times. For

this broad

Euclidean sense of megethos

shall use the

term magnitude ;

and

I

shall reserve spatial magnitude

for he Aristotelian

mege-

thos .) So Aristotle's point

at the beginning

of Metaphysics

M3

can now be

summed up

as follows: the generalized theoryof

proportion

need not

commit us to the existence

of

any special

objects-magnitudes-over

and above numbers and spatial

magnitudes.

The theory s about spatial

magnitudes and

num-

bers,only

not

as

spatial magnitude

or

number,

but rather

s

mag-

nitude: that is, they

exhibit a common

property,

nd they

are

being

considered

solely

in

respect

of this.

The second difficulty or the Academic Platonist to which

Aristotle alludes is presented

by astronomy.9

Aristotle clearly

thought

that

the Platonist should

feel

embarrassed

by having

to

postulate

a heaven

separate

fromthe

perceptible

heaven and

also

by

having

to admit that ideal objects

move.10 The actual

planets

themselveswere,

Aristotle hought, ternal and unchang-

ing (in the relevant respects)

and so could function

s the objects

that the unchanging truthsof astronomywere about. It seems

that

Aristotle

thought

he

could get his Platonist

opponent-or

perhaps

a

philosopher

who is

already retreating

romPlatonism,

9Cf.Metaphysics 077b23

ff.;

077al-5; 997a84-99al9. See

also the

references

to astronomyn

Physics

193b24-194al2,

quoted in part above.

10

owever, t s far rom lear hatPlato was embarrassed ythis.Cf.Republic

528a-b, 529c-d. Perhaps hemovement f dealobjectsbecame an

embarrass-

ment o the Academic Platonists.

167








JONATHAN

LEAR

but

not yetsure

how far

to retreat-to

admit

that

astronomy s

about the

heavenly

planets, but

considered

solely in

terms of

theirpropertiesas moving bodies.

But

having

come

this

far,

the

retreating

Platonist

cannot

stop

here.

For if

one admits

that

there are

physical

objects

that can

be treated

solely

as

moving

bodies,

in

isolation

from

all their

other

particular

properties, hen

one

can treat

these

objects

solely

as

bodies,

solely

as planes,

lengths,

and so on

(1077b23-30).

Reality,Aristotle

seems

to be

saying,can be

considered under

various aspects. Given the paradigm Aristotelian substances-

individual men,

horses,

tables, planets-we

are

able

to

consider

certain

featuresof these

substances

in

isolation.

Generalizing,

one

might

say

that Aristotle

is

introducing

an

as-operator,

r

qua-operator,

which

works as follows. Let b be an

Aristotelian

ubstance

and let b

qua

F

signify

hat

b is

being

considered

as an F.

Then

a

property

s

said

to

be true of

b

qua

F

if

and

only

f

b is

an

F

and

its

havingthatproperty ollowsof neces-

sity

from

ts

being an

F:

11

G(b qua F)

*-

F(b)

&

(F(x)

-

G(x)).

Thus to use

the

qua-operator s

to

place

ourselvesbehind a veil

of

gnorance:

we allow

ourselvesto know

only

that

b

s

F

and then

determineon the

basis of

that

knowledge

alone what other

prop-

ertiesmusthold of it. If,forexample, b is a bronze isosceles tri-

angle-

Br(b) &

Is(b) &

Tr(b)-

then

to

consider b

as a

triangle-b

qua

Tr-is to

apply

a

predicate

filter:

t filters ut

the predicates

like

Br

and

Is

that

happen

to

be true of b, but are irrelevant to our current concern.

The

filter

nables

Aristotleto make a

different

se of

the dis-

tinction

between

incidental (kata

sumbebikos)

nd

essential

(kathl'

I

usethe

turnstile ere o

signifyhe

relation

follows f

necessity,

hich

I

argue

lsewhere

ristotleook

s

primitive.X

H

P

shouldbe

read

P

follows

of

necessity from X.

Cf.

Prior

Analytics

5b28-31 and

my Aristotle nd

Logical

Theory

Cambridge

nd

NewYork:

Cambridge

University

ress,

1980),

Chap-

ter

1.

168








ARISTOTLE

ON

MA

THEMA

TICS

hauto) predication

from

that

which

is often attributed

to

him.12

On

the standard

interpretation,

ne

is

given

an

Aristotelian

ub-

stance-forexample, an individual man-and the essential pred-

icates

are

those

that

must hold

of theobject

ifthat

object

is

to

be

the

substance

that t

is. They

are what

it

is

for he substance

to

be

the

substance

that

it is. An incidental

predicate

is

one

that

hap-

pens

to

hold of

the substance,

but

is such

that

if t failed

to

do

so

the

substance

would

continue

to exist.

For Aristotle,

rational

would

be

an

essential

predicate

of Socrates;

snub-nosed

and

pale would be incidental.

In Metaphysics

M3,

however,

the

distinction

between

essential

and

incidental

predicates

can

be

made

only

for

an

object

under

a

certain escription.

f

we

are

considering

b as being

n

F,

then

every

predicate

that

is

not

essential

to its

being

F is considered

inci-

dental,

even

though

t

may

be essential

to b'sbeing

the

substance

that

it is.13

That is why

in the definition

of G(b

qua

F)

one

shouldhave (F(x) H G(x) ) on therighthand side oftheequiv-

alence

rather than

F(b)

H

G(b).

For

one

might

have

F(b)

H

G(b)

in

virtue

of

what

b s

instead

of

n virtue

ofwhat

F

and

G

are.

For example,

Aristotle

hought

that

the

heavenly

bodies

must

be

composed

of

a

special

stuff

ifferent

rom and

more

divine

than

earth,

air,

fire,

r

water; he

also thought

that

the

heavenly

bodies

must be

indestructible

De

Caelo A2, 10).

However,

if

one

takes

an arbitrary

tar

and

applies

the

predicate

filter

o that

one

considers

t solely

as a

sphere,

ll the properties

hatdo not follow

from

ts

being

a sphere

(its

being

composed

of special

stuff,

ts

indestructibility,

tc.)

arefrom

his erspective

ncidental.

By

apply-

ing a predicate

filter

o an

object

instantiating

he

relevant

geo-

metrical

property,

we

will

filter

ut

all predicates

which

concern

the

material composition

of

the

object.14

Thus

the geometer

is

able

to

study

perceptible

material

objects-this

is indeed

all

that

he studies-but he does not studythem as perceptible or as ma-

terial.

'2Cf. Metaphysics

1077b34-1078a9;1078a25.

I

am not confident

whether

Aristotle's sage here

s a deviation rom is

usual usage or whethert

provides

evidence

hat the standard nterpretations

in need of revision.

13

Cf., e.g., Annas,

op. cit., pp. 148-49; J.J. Cleary, Aristotle's

octrine

of Abstraction, npublished

manuscript.

'4Metaphysics

077b20-22, 078a28-31;

hysics

193b31-33, 194a9-11.

169








JONA

HAN

LEAR

A

difficulty

or

his

nterpretation

might

be thought

to be pre-

sentedby

Metaphysics

M3, 1077b31-34:

So

since t

is

true

o

say

without ualification

ot only that

sepa-

rablethings xist

but also that

nonseparable

hingsxist e.g.,

that

moving

bodies

exist),

t

is

also true to say

without

ualification

that mathematical

bjects xist

nd are

as they

re said to

be.

It

would be

a mistake

to interpret

his

sentence

as assertingthe

existence of

ideal objects. First,

the

expression

without quali-

fication

(haplks)

certainlymodifies

to say (legein

and eipein

respectively)and not to be (einai or estin). For Metaphysics

M2

closes with

a

categorical

statement

that mathematical

ob-

jects

do not

exist

without qualification

(ouk haplis

estin,

1077b16).

Rather, mathematical

objects are supposed

to exist

in some

qualified

fashion.

Indeed

the task of

Metaphysics

M3 is

to

explicate

the

way

in

which

mathematical

objects

can be said

to exist.

Second,

that

the sentence begins

with so since

.

(host'epei)howsthat the claim thatseparables and mathematical

objects

exist

is

the conclusion

of

the

argument

that preceded

it.

But that

argument,

s we

have

seen,

denies

that there re separate

Platonic objects

and

affirms hat geometry

s about

physical,

perceptible

magnitudes,

though

not considered

as

physical

or

as

perceptible.

Thus,

for

Aristotle,

one can

say

truly

that

sepa-

rable

objects

and

mathematical

objects

exist,but

all this tatement

amounts to-when properly analyzed-is that mathematical

properties

are

truly instantiated

in

physical

objects

and, by

applying

a

predicatefilter,

we

can consider

these

objects

as

solely

instantiating

the appropriate properties.

This

interpretation

s

confirmed

by

the

analogy

Aristotle

proceeds

to make with the

science

of health

(1077b34-1078a2).

We can

say

without

quali-

fication that

the

science

of health

is

about

health;

and

we can

disregardanythingthat is incidental to being healthy,

for ex-

ample,

being pale.

This does

not mean

that

health

exists

in-

dependently

of

healthy

men and

animals;

it means

only

that

in

studying

health we

can

ignore

everything

hat

is irrelevant to

that

study.

So far

Aristotle

has

argued

that

in

studyinggeometry

ne need

study only

physical

objects,

not

Platonic

objects,

though

con-

sidered

independently

of

their

particular

physical

instantiation.

Cf.

Ross,

op.

cit.,

I,

pp.

416-17.

170








ARISTOTLE

ON

MA

THEMATICS

The second

major

step

inAristotle's

rgument

begins

at

1078a1

7

wherehe

says

that

if

someone

should postulate

and

investigate

objects that are separated from ncidental properties,he would

not because

of

this

be

led to speak

falsely.

Why

is this?

Suppose,

forexample,

we assume

that

there

is an

object

c such

that

(VG)

(G(c)

<-+

G(c

qua

Tr) .

That is,

we are

assuming

the

existence

of

an object

whose

only

properties re those that are logical consequences of its being a

triangle.

This is the

assumption

of

a

geometrical

object,

a

tri-

angle,

separated

from ny

material instantiation.

Now

suppose

we should

prove,

via theproof

ofEuclid

1-32,

that

c has

interior

angles

equal

to

two

rightangles

(2R(c) ).

Since by

hypothesis

we are guaranteed

that the

only

properties

of c are

logical

con-

sequences

of

its being

a triangle,

we can

argue

from

2R(c)

to

(Vx)

(Tr(x)

-*

2R(x)).

By universal

instantiation

we can

infer

2R(b)

for

any

triangle

b.

The

reason

one

will not be led

to

speak

falsely as

a

result

of

the

fiction hat

there re

separated

objects,

according

to

Aristotle,

is that

the

falsity

s not

in

the premises

(1078a20-21).

The

analogy offered s with the case in which one draws a line (on a

blackboard or

in

sand)

and

says for

example,

Let

the

line AB

be

one foot ong.

Aristotle

correctly

ees that

the

line is drawn

forheuristicpurposes

and

is

not

a

part

of

the

proof.

How

is this

analogous?

In the above

proof

one

has

assumed

that

thereis

a separated

geometrical

object

c,

but

in fact

from the

point

of

view of

the

proof, here s no difference etweenc and any actual triangular

171








JONA

THAN LEAR

object b considered

as a triangle-b

qua Tr.

For theproperties

one

can

prove to hold

of c

are precisely

those

one can

prove to

hold

ofc qua Tr,and it is dasy to show from he definitionof the qua-

operator

that

these are

the same as

the properties

one can prove

to

hold

of

b

qua Tr,regardless

f the choice

of

b,provided

only

that

it

is

a

triangle.

For heuristic

purposes

we

may indulge

in the fic-

tion

that c is a

separated

triangle,

not

merely

some b considered

as

a

triangle;

this s a

harmless fictionbecause

the proof

s indif-

ferent

s to whether

c is really separated

triangle

or ust b

con-

sideredas one. It is in thissensethat thefalsity oes not enter nto

the

premises.

The falsity

would

enterthe

premises

fwe were

to concentrate

noton properties

hat

c

must

have, but on

properties

hat

c cannot

be

proved

to have.

For

example,

since

Triangle

(x)

Y-

Isosceles

(x)

Triangle (x)

Y- Scalene

(x)

Triangle (x) Y- Equilateral (x),

we

have

ilsosceles

(c

qua

Tr) & -iScalene

(c

qua Tr) &

nEquilteral

(c qua Tr),

and thus

ilsosceles

(c)

& -iScalene

(c)

&

nEquilateral

(c).

However,

Triangle (x)

H

Isosceles

(x) v Scalene

(x)

v Equilateral

(x),

and so it follows

that

Isosceles

c qua

Tr)v Scalene

(c

qua Tr)

v Equilateral

(c qua

Tr),

and thus

Isosceles (c) v Scalene (c) v Equilateral (c).

So

it

seems

we can

prove

both

that

c

is either

scalene

or

equilat-

eral or

isosceles,

and that

it is neither

scalene nor

equilateral

nor

isosceles.

The contradiction

has

arisen because the

false

assump-

tion that

c is a

pure geometrical

object

has entered

the

premises:

for t

is

essential

to

the

proof

tself hat the

only properties

c has

are the

properties

hat follow

from ts

being

a

triangle.

The

rea-

son,

for

Aristotle,

that one does

not confront

uch

problems

in

172








ARISTOTLE ON

MA THEMA TICS

geometry s that

one is really only considering

an actual physical

object

in

abstraction

from ts particular

physical instantiation.

Though one cannot prove of b qua Tr that it is equilateral, one

cannot correctly

nfer hat b is not equilateral,

only that it

is not

the case that b considered

s a triangles equilateral. The fiction

hat

cisa separated geometricalobject, a triangle

tout ourt,s harmless

only

because we are concerned

in

geometry

with the propertiesc

does have, not with

the propertiesc

does not have. It is

only if,

ignorant of the

foundations of one's

mathematical practice, one

takes the fictionof geometrical objects too seriously and begins

to enquire philorophically into their

nature that one runs

into

trouble. The analogy would

be with

someone who drew

a line

in

the sand

saying,

Let

AB

be one foot

ong, and then wenton to

say,

But

AB

is not a foot ong; therefore t both is and

is

not

the

case that

AB

is a foot ong. Similarly,

one mightobject: If some-

thing

s a

triangle,

t has a definite hape;

and if t has a shape, it

must be colored;

and yet a separated mathematical triangle

is

neither

white,

nor

green,

nor red

....

Aristotle would

respond

that one is hererunningthe riskof turning harmless fiction

nto

a

dangerously

misleading

falsehood by importing the falsehood

into the premises cf. 1078a18-2 1).

The fictionworks,

n

part,

due

to the fact that as

geometers

we

are

concerned only

with

geo-

metrical properties

that follow from

being a triangle. To think

that the

pure

mathematical

triangle

c

must be

colored,

on the

groundthat t has a definite hape, is,forAristotle, o be confused

about what

one is

doing

when one

talks

about

separated

mathe-

matical

objects.

For,

as we have

seen,

there s

one sense

in

which

the

separated

mathematical

triangle

c does not

have a definite

shape:

it is neither scalene nor isosceles nor

equilateral.

So, too,

one

must

accept

that t does not have

a

definite olor.

For

in

creat-

ing the harmless fiction

of a

separated

mathematical

triangle,

one has abstractedfrom ll considerationsofany triangle'scolor.

That the

postulation

of

separated

objects is of heuristic

value

Aristotle s certain. The lesson of

1078a9-13 seems to be that the

more

predicates

we can

filter

ut,

the more precise and simple

our

knowledge

will be.

6

The

paradigm

of a

precise

science was

the

articulated

deductive

system

of

geometry;

as one moves

toward

a

science

of nature, one's

knowledge

becomes

ever less

'6Cf.Annas,op cit.,p. 150; Ross,op. cit., I, p. 417.

173








JONA

THAN

LEAR

precise.

Astronomy

could

provide precise

deductions

of the

planets'

movements,

but

there

were anomalies

between

the

de-

ductions and observed phenomena; and a general theory of

dynamics

was

inmuch

worse

shape

than

astronomy

cf.

1078al2-

13). Our knowledge

s simpler

because

we

have

been

able to

filter

out extraneous

information.

For

if we have proved

(1)

2R(c),

thenwe

know

thatwe can

infer hat

any particular

triangle

what-

ever has interior ngles equal to two rightangles. If by contrast

we had proved

only

(2)

Triangle

(b)

&

F1(b)

&

. .. &

Fn(b)

F- 2R(b),

then of course

we

could

infer

ny

instantiation

of

(3)

Triangle

(x)

& F1(x)

&

... &

Fn(x)

H

2R(x);

but

given

any

other triangle

d

such

that

(4) Triangle (d) & G1(d) &

...

&

Gk(d),

we

would

notbe able

to

prove,

on the

basis

of (2),

that

(5)

2R(d).

For it

isnot clear

on

what

properties

f bthe

proof

of 2)

depends.

However,

(5)

is

an obvious consequence

of 1)

and

(4):

for

t fol-

lows

from 1)

that

(6)

2R(d).

The

postulation

of

separated

geometrical

objects

enables

us to

attain knowledge

that

is more general.

And it is

through

this

general

knowledge

that

one

can discover

the explanation

(aitia)

of

why

something

s the case.

For by

abstracting

one

can see that

the full

explanation

of a

triangle's

having

the

2R

property

s that

it s a triangle nd not,say, that t isbronzeorisosceles cf.An.Pst.

A5).

In

a limitedsense,

though,

the abstract

proof

s unnecessary.

For ofany particular

physical

triangle

d we can prove

that it

has

interior

ngles equal

to

two right

angles

without

firstproving

this

for

c:

we could

prove

that

d has the

property

directly.

The

proof

that

physical

object possesses

a geometrical

property

via

a

proof

hat a

pure

geometrical

object

has the property

s a useful,

but

unnecessary,

detour.

However,

if

we want

to know

why

he

174








ARISTOTLE

ON

MA THEMA

TICS

object

possesses

the

property,

he

abstract

proof

s of crucial

im-

portance.

Thus it is that thebestway ofstudyinggeometry s to separate

the

geometrical

properties

of

objects

and

to

posit

objects

that

satisfy hese

properties lone

(1078a2

1

ff.).

hough

this s

a

fiction,

it is a

helpful

fictionrather

than

a

harmful

one:

for,

t

bottom,

geometers

re

talking

about

existing

things

and

properties

they

reallyhave

(1078a28

ff.).

II

This

interpretation

f

Aristotle's

philosophy

of

geometry

rests

on

the

assumption

that

Aristotle

thought

that

physical

objects

really

do

instantiate

geometrical

properties.

One

might

wonder

both

whether

his

assumption

is

true

and to

what

extent t

needs

to be

true.

There

is,

as

we

have

seen,

strong

vidence

in

favor

of

theassumption:thepassages fromPhysics 2 and Metaphysics 3

considered

above

repeatedly

emphasize

that

the

geometer

studies

physical

objects,

but

not as

physical

objects.

There

is

no

mention

that

the

physical

objects

do

not

possess

geometrical

properties,

yet

one

would

certainly

expect

Aristotle

to

mention

this

f

he

believed it.

Further,

hroughout

he

Aristotelian

corpus

there

re

scattered

references

o

bronze

spheres

and

bronze

isosce-

lestriangles: here sno suggestion hattheseobjectsare notreally

spherical or

really

triangular.17

So

in

view

of this

prima

acie

evidence,

the

burden of

proof

shifts

o

those

who

believe

that,

for

Aristotle,

physical

objects

do

not

truly

nstantiate

geometrical

properties.

Two

passages

are

cited.

8

I

shall

argue

that

neither

need

be

construed

as

supporting

the

thesis

that

physical

objects

fall

short of

truly

possessing

geo-

metricalproperties.First,consider the passage fromMetaphysics

B2:

But on

the

other

hand

astronomy

annot

be

about

perceptible

magnitudes

or

bout

this

heaven

bove

us.

For

neither

re

perceptible

lines

uch

ines as

the

geometer

peaks

of (for

no

perceptible

hing

s

thus

17

I

discuss

his

n

detail below.

Metaphysics

97b35-998a6;

1059blO-12.

Cf.

Mueller,

op.

cit.,

p.

158;

Annas,op. cit.,p. 29.

175








JONA

THAN

LEAR

straightrround:for

Kphysical)

ircle

K

.g., hoop touches straightdge

not t a point,but s

Protagoras

sed to

say

it did in his

refutation

f

the

geometers)or are themovementsnd spiralorbits f theheavens

like

hosewhich

stronomytudies

orhave

points he

ame

nature

as stars. [Met. B2,

997b34-998a6; my emphasis]

One should not consider this

passage

in

isolation from the con-

text

in

which it

occurs. Metaphysics 2

is a

catalogue

of

philo-

sophical problems

aporiai) presented

fromvarious

points

of view.

None of it should be

thought of as

a

presentation of Aristotle's

considered view on the subject. It is rather a list ofproblems in

response to which he will form his philosophical position. Im-

mediately beforethequoted passage Aristotle s putting forward

the

problem for

the

Platonists that the belief

in

Form-like nter-

mediates involves

many difficulties 997bl2-34). The quoted

passage

can

thus

be read as an

imagined

Platonist's

response:

Yes, the belief

n

intermediates s problematic, but,on the other

hand, giving them up involves

difficulties,

oo. Here it is an

imagined Platonist

speaking,

and not

Aristotle. o Aristotle

s not

endorsing Protagoras'

view;

he is

presenting

t as one horn

of a

dilemma that must be resolved. We have

already

seen Aristotle's

proposed resolution;

and

it is one

that involves

asserting

that

some

physical objects

perfectlypossess geometrical properties.

One

mightobject:

Does this mean that

Aristotle

s

committed

to

saying

that the

hoop qua

circle

touches the

straightedge

at a

point? The straightforwardnswer to this s, Yes it does, but

this s not as

odd as

it

may initially ppear. Protagoras' objection

looks

plausible

because the

hoops

we tend to see are

not perfectly

circular,

nd so

theyobviously

would not touch a

straight dge-

let alone the surfaceson which

they actually rest,

which are not

perfectly traight-at

a

point.

But Aristotle s not committed to

saying

that

there are

any perfectly

ircular

hoops existing

n

the

world. All Aristotlemustsay is: i) insofar s a hoop is a circle it

will touch a

straight

dge

at a

point; ii)

there are

some

physical

substances

that are circular.

(Such

circular substances need not

be

hoops.)

Claim

(i)

is

true:

inasmuch as a

hoop

fails to touch a

straight dge

at a

point,thus much

does it fail to be a circle. And

there s

certainly

evidence that Aristotle believed claim

(ii).

Of

course,

Aristotle

thought

that

the

stars

were

spheres

and that

theymoved in circular orbits De CaeloB 11,8). But thereis also

176








ARISTOTLE ON MA

THEMATICS

evidence that

he

thought

that even in the

sublunary

world

physi-

cal

objects could

perfectly

nstantiate

geometrical

properties.

In Metaphysics 8 he says:

. .

. ust

as

we do not make the

substratum,

he

brass,

o we do

not

make

thesphere, xcept

ncidentally,ecause he

bronze

pheres a

spherend

we

make hat. or

to

make

particular

s

to

make

particu-

lar out

of

an

underlying

ubstrate

enerally.I

mean

that

to

make

the

rass ounds

notto

make

the

round

r

the

phere,

ut

something

else,

.e.,

toproduce

his orm

n

something

ifferent

romtself

or

if

we

make

the formwe must

make t

out ofsomethinglse; for hiswas

assumed.

or

example,

we

make bronze

phere;

nd

hat n he ense

hat

out f

his,

whichs

brass,

e

make

he ther

hich

s

a

sphere.)f,

then,

we also

make he

ubstratetself,

learly

we shall

make t

n

the ame

way, nd the

process

f

making

will

regress o

infinity.bviously,

then, he

form

lso,

or

whateverwe

ought o

call

the

hapepresent

in

the ensible

hing,

s

not

produced;

for his s

thatwhich s

made

to be

in

something

lseeither

y

art

or

bynature

r

by

some

faculty.

Butthatheres a bronzephere,hiswe make. orwe make t out fbrass

andthe

phere;

e

bring

he

orm

nto his

articular

atter

nd

the esult

is

a

bronze

phere.

1

33a28-b

1

;

my

emphasis]'9

Later,

n

Z10,

he

continues:

Those

thingshat re the

form nd the

matter

aken

ogether,

.g.,

the snub or the ronze

ircle, ass

away

into these

materials nd

the

matter

s

a

part

ofthem,

ut those hings hich o

not nvolvemat-

terbut arewithoutmatter,fwhich he ogoireoftheformsnly,

do not

pass

away,

ither

ot t

all

or at leastnot n this

way.

There-

fore

hesematerials re

principles

nd

parts

f the

concrete

hings,

whileof

the form

hey re

neither

arts

nor

principles. nd

there-

fore he

lay

statue

s

resolved nto

lay

and the

phere

nto

ronzend

Callias

into

flesh

nd

bones,

nd

again

he irclento

ts

egments;or

therere

ircles hichre

ombinedithmatter.or circle s

said

homony-

mously

oth

f

he

nqualified

ircle

nd

f

he ndividualircle ecausehere

is no pecial ameor he ndividuals.1035a25-b3; myemphasis]20

The

individual

circle is a

physicalobject;

one in

which the form

of

a circle

is

imposed on

some matter.There

is no

suggestionthat

19

f.

Metaphysics

033b

9-29

and

1036a2-6.

20

Cf.

also

Metaphysics

1035a9-14:

The

logosof a

circle

does not

contain

the

ogos f

the

segments .

. the

segments n

which

the

form

upervenes;

et

they are

nearer the form

than

the

bronze

when

roundness

s

produced n

bronze

(myemphasis).

177








JONA

HAN

LEAR

it is not

really

circular.

This

theme

is developed

further

n

the

following

hapter,

Z 11:

In thecase of things

which

are

found

o

occur

n

specifically

if-

ferentmaterials,

as

a circlemay xist

n bronze

r stone r

wood, t

seems

plain

that hese,

he

bronze

r the tone, reno part

ofthe

ssence

f

a circle,

ince t

s

found

part

fromhem.

Of

all

things

which

re

not

seen o exist part,

here

s no reason

why

he

ame maynot

be true,

just as

fall

circles

hat ad ever

een

een

were f

bronze;or

nonetheless

the

bronzewould

be no part

of

the form;

ut it

is hard to

eliminate t

in thought.1 36a31 b2; my emphasis]

Now

at times t

may seem

as though

Aristotle

s saying

that

circles

are not sensible

objects.

For example,

he criticizes

Socrates'

analogy

between

an animal

and

a circle because

it

misleadingly

encourages

the thought

that

man can

exist without

his

parts

as

the circle can

exist

without

the

bronze (1036b24-28).

However,

in

such

passages,

Aristotle s

not claiming

thatthere

re

nosensible

circles;he is claiming only that it is possible fora circle to exist

independently

of

a

particular

physical

instantiation.

These

are

the

intelligible

mathematical

circles

which

Aristotle contrasts

with

the perceptible

circles of bronze

and

wood

(1036a3-5).

It

is the

task of Metaphysics

3-4

to explain

the way

in which

such

nonsensible

circles

exist.

The

second

passage

cited

to

support

the thesis hat

Aristotledid

not believe that physicalobjectsperfectlynstantiategeometrical

properties

s two lines

from

Metaphysics

1:

...

withwhat

ort f

thing

s

the

mathematician

upposed

o

deal?

Certainly

ot with

the

things

n

this

world,

for

none of

these

s

the ort

f

thing

which hemathematical

ciences

nvestigate.Met.

Ki,

1059blO-12]

But again

we must consider

the

context

n which this

statement

occurs, n particularthe largerpassage of which it is a fragment:

. .

.these

thinkers

sc.,

Academic

Platonists] lace

the

objects

of

mathematics

etween he

Forms

nd perceptible

hings

s a kind

of third

etof

thingspart

from he

Forms nd

from

he

things

n

this

world;

but there

s

not a

third

man or

horsebesides

he deal

and

the

ndividuals.f,

n

theother and,

t

s

not

s

they

ay,

with

what sort

of

thing

must

the mathematician

e

supposed

to

deal?

Certainly ot with hethingsnthisworld;fornoneof

these

s

the

178








ARISTOTLE ON

MA THEMA

TICS

sort fthing

which

hemathematical

ciences nvestigate.Met.

K

1,

1059b6-12]

Now it should be clear that we are onlybeingfacedwith thevery

same

dilemma already presented

to

us

in

Metaphysics

2. Indeed,

Metaphysics

I is onlya recapitulation

of

the aporiai f

Metaphysics

B2,

3. Again,

we should

read Metaphysics

059blO-12

as the

Aca-

demic

Platonist's

response

to the

objection

that the

postulation

of

intermediates

nvolves

many

difficulties.

urther,

he

claim that

thethings

n thisworldare not the sort

of thing

which the

mathe-

matical sciences nvestigateneed not even be read as a claim that,

for

example,

a

bronze

sphere

could not

be

perfectly

pherical.

Rather,

it could be

the claim

that

when a geometer

considers

a

sphere

he

does not

consider

t as

made

up

ofbronze

or

any

other

matter.

Aristotle

could

then

see

himselfas

providing

a solution

superior

to the

Platonist's:

for he can explain

this horn

of the

dilemma

without

having

to

resort to

postulating problematic

intermediates.He could thus thinkof himself as dissolvinghe

dilemma,

rather han

accepting

eitherhorn. The

great

advantage

of

being

able to dismiss

these two

passages is

that it

becomes

immediately

videnthow

Aristotle

hought

geometry ould

apply

to the

physical

world.

There does,

however,

remain

room for

kepticism.

Even if one

grants

hat

there

are, for xample,

perfect

pheres

n thephysical

world,

must there be perfectphysical

instantiations

of every

figure

the

geometer

constructs?

Surely,

the skeptic

may object,

Aristotle hould not

commit himself o there

being,

for

example,

perfectly

riangular

bronze figures.

And, the skeptic

may con-

tinue,

even

if

there were

perfectly

riangular

physical objects,

there are no

physical

instantiationsof the more

complex

figures

which a

geometer

constructswhen

he

is proving a theorem.

I

think t is

clear

how Aristotle

would

respond.

At the end

of Meta-

physics 9 he says:

It

is

by

an

activity

lso that

geometrical

onstructions

ta

diagram-

mata)

re

discovered;

orwe

findthem

bydividing.

f

they

had

already

been

divided,

he constructions

ouldhave been

obvious;

but

as

it

s

they

re

present

nly

potentially.

Why

are

the

angles

of

a

triangle

qual

to tworight ngles?

Because

the

anglesabout

one

point

re

equal

to tworight ngles. f,

hen,

he

ine

parallelto

the

sidehad beenalreadydrawnupwards,t wouldhave been evident

179








JONA

HAN LEAR

whyKthe riangle

as suchangles)

to anyone

s soon as he saw

the

figure.... Obviously,

herefore,he

potentiallyxisting

hings

re

discovered y beingbrought o actuality; he explanation s that

thinking

s an actuality.... [1051a21-31]

The geometer,

Aristotle ays,

s able to carry

out geometrical

con-

structions

n

thought:

the thinking

which is an activity

makes

actual

the construction which existed

only potentially

before

the thinking

ccurred.21

Thus we can

make sense of

Aristotle's

claim:

That is why hegeometerspeakcorrectly:hey alk boutexisting

things

nd they eally o

exist-for

what xists oes

so

in

one oftwo

senses,

n

actuality

rmaterially.Met.M3,

1078a28-3

]

The

word materially

(hulikos)connotes

both the

matter

from

which the construction

s made

and also the potentiality,

associated

with matter,

of the geometrical

figure

before the

activity

of thought.22

Has, then,Aristotle evered thetie betweenpure mathematics

and the

physical

world?

Does the geometer

contemplate pure

mathematical

objects

that

are

not

in

any

way

abstractions

from

the

physical

world? don't think

o. For to retain

the

ink between

geometry

nd

the

physical

world,

Aristotle

need

only

maintain

that the elements of a geometrical

construction re abstractions

from

the physical

world. Not every

possible

geometrical con-

struction eed be physically nstantiated. n Euclidean geometry,

constructions

re

made

from

traight ines,

circles,

and

spheres.

We have

already

seen the

evidence that

Aristotle

thought

that

there were

perfectly

ircular

physical

objects. Evidence

that he

thought

there were also physical objects

with perfectly

traight

edges

is

in

the

De Anima:

If,

hen, here

s any of the

functionsr affections

f the

oul which

ispeculiar o it, t will be possiblefor tto be separatedfrom he

body.

But

if

there

snothing eculiar

o

it,

t

will not be

separable,

but t

willbe like

he traight,

owhich,

s

straight,

anyproperties

belong, e.g.,

it will touch bronze

phere

t a

point,

lthough

the

straight

21

I further iscuss the role of mental activity

n

Aristotle's hilosophy f

mathematics in Aristotelian Infinity, Proceedings f

the

Aristotelian ociety,

80 (1979/80).

22

Cf. also, e.g., Metaphysics 048a30-35.

180








ARISTOTLE ON MA THEMA TICS

if separated

will not so

touch;

for t is

inseparable,

f t is

always

found

n

some body. [403alO-16; my emphasis]

What Aristotletakes to be touching a bronze sphere at a point

is a physical traight dge, forwhen the straight ine has been ab-

stracted

t

cannot touch a physical object at all.23 Since a geo-

metrical triangle can be constructed in thought from straight

lines,Aristotle oes not have to say that a particular bronze figure

d is

perfectly riangular.

f

it is-and

given

that he

thought

there

could be a

physical straight dge,

there is no reason for

him

to

denythat therecould be a physical triangle-then one can apply

the

qua-operator o it and proceed to prove theorems bout it qua

triangle.

f

it is not,then theproperties hat have been proved to

hold of triangles will hold of it more or less depending on how

closely

it

approximates being

a

perfect riangle.

We could then

relax our claim that d qua F is

G,

and say only that d insofar s it is

Fis G.

The important point is that direct links between geometrical

practice and the physical world are maintained. Even

in

the case

where the

geometer

constructs a

figure

n

thought,

one

which

perhaps has never

been

physically instantiated, that figure

is

constructed rom

lements which are direct abstractionsfrom he

physical world. Otherwise it will remain a mysteryhow,

for

Aristotle,geometry

s

supposed

to be

applicable

to the

physical

world. Certainly it is a mistake to treat Aristotle's few remarks

about

intelligible

matter as

providing

either the basis of

a Pla-

tonic

epistemology

r

a

prototype

f the form f outer ntuition. 4

He

specifically ays

that mathematical

objects

have

intelligible

matter,

5

and

in

these contexts t is used to do ustice to the nature

of mathematical

thinking:that is,

when one carries

out

a

geo-

metrical

proof

t seems as

though

one has

a

particular object

in

mind.

To

prove a general theorem about triangles, t seems as

though

one

chooses an

arbitrary articular triangle

on

which one

23Cf. lso De Anima

431b15-17; 432a3-6. Note Sextus

Empiricus' report:

Aristotle, owever, eclared

hat he engthwithout readth ftheGeometers

is not nconceivable For

in factwe apprehend he ength f a wall without

having perception

f he

wall's breadth )

. .

(Adversus

athematicos,x,412).

24

Cf.

Metaphysics

1036a2-12; 1036b32-1037a5;

1045a33-35.

See also

1059b14-16; 1061a26-30.

25Metaphysics

1036a9-12; 1037a2-5; cf. 1059b14-16;1061a28-35.

181








JONATHAN LEAR

performs construction.

6

Thus it does not seem

that one merely

has the formof, for example, triangularity n mind: intelligible

matter s invoked to account for the fact that we are thinking

about a particular object.

Aristotle's philosophy of mind, however, falls significantly

shortof being Platonic. For Aristotledoes not

have to postulate

the existenceof

an

object that does not exist

n

the physical world

to be the

object

of

thought;

he

merely

has to explain

how

we

think

about objects that do

exist n

the world.

7

The problem forAris-

totle is the nature of thinking n general,not the existence of a

special type of mathematical object to think bout. It is true that

we can both perceive a sphere and think bout

it. In fact,we can

think

bout it

in

abstraction

from

he fact that

t

is

composed,

for

example,

of

bronze. We

can even

perform

mental constructions

and form figure hat we have perhaps never perceived.But even

in this case we are doing no more than constructing figure

n

thought fromelements that are direct abstractions from the

physical world.

One

might

wonder

whether,

on this

interpretation,

t follows

that

perceptible objects

have

intelligible

matter.

The answer is

that they have intelligible matter nsofar s

they

can be

objects

of

thoughtrather than perception: that is,

it

is

the

object

one

is

thinking bout that has intelligible matter.

The evidence for his

26

Cf. Euclid I-32; and the nterpolationfter uclid

III-3. See

also

Posterior

Analytics 4a24-35; Metaphysics

1051a26; T.L. Heath, Mathematics

n Aristotle

(Oxford:Clarendon Press,1970),pp. 31-39, 71-74, 216-17.

27

A critic fAristotlemight aythatAristotle ad notprovided significant

advanceon Plato, sinceAristotle

ften ays, .g.,thatknowledge s

of the uni-

versal cf., .g.,Metaphysics003a13-15,1060b19-21,

086b5-6,32-37;

DeAnima

417b22-23; Nicomacheanthics 1140b31-32, 1180b15-16. See also Posterior

Analytics7b28-88al 7.) and that

proper cientificroofwillbe of heuniversal

(cf.Posteriornalytics5b15 ff.).

However, he claimthat knowledges of the

universal dmitsof widelydifferentnterpretations,epending n whatone

takesAristotle's

heory

f

the universal

o

be.

I

am

inclinedto interpret

is

theory

f

universalsnd

of

knowledge

f heuniversal

n

a strongly on-Platonic

fashion; ut

I

cannot

defend

hat nterpretationere.

n

this paper

I am not

trying

o

explicate ully ristotle's

heory fknowledge, or m

I

claiming

hat

it is completely nproblematical.

wish only to make the weaker laim

that

mathematical nowledge s no more problematical orAristotle han

knowl-

edge of anything

lse. So

if

we

grant

Aristotle hat we can have

knowledge

f

the

properties

f a

physical bject,

t follows hat

we

can have mathematical

knowledge.

182








ARISTOTLE

ONMA THEMA

TICS

is

Aristotle's laim

that

intelligible matter s the

matter which

exists

n

perceptible

objects

but

not

as

perceptible,

for

example,

mathematical objects (1036all-12).

III

Aristotle's

philosophy

of

arithmeticdiffers

ignificantly rom

his

philosophy of

geometry,though

Aristotle

is not

especially

sensitive to this

difference.

The

predominant

mathematics of

ancient Greecewas geometry,nd thus t s notsurprising hathis

philosophy

of

mathematics

should be

predominantly a

philos-

ophy

of

geometry.

The main

obstacle

preventing

Aristotle

rom

iving

successful

account

of

arithmetic

s

that

number is

not a

property f

an

ob-

ject.28 Thus

one

cannot

legitimately

hink

of a

number

as

one

of

the

various

properties of an

object

that

can be

separated

from

it in thought.One can, however, ee Aristotlegrapplingwiththe

problem:

The

bestway of

tudying

ach

thingwould

be this:

o

separate

nd

positwhat s

not

separate s the

rithmetician

oes,

nd

thegeom-

eter.

A

man

is

one and

indivisible s a

man,and the

arithmetician

posits

him

as

one

indivisible;

he

geometer,

n

the

other

hand,

studies

im

neither s

a

man

nor

s

indivisible,

ut as a

solid

object.

For

clearly

roperties

e

would have

had even

f

he

had

not

been

indivisible an belong to himirrespectivef his being ndivisible

or a

man

[aneu

touton].Met.

M3,

1078a22-28]

Aristotle's

osition s,

think, s

follows.

ubstances,

that s, mem-

bersof

natural

kinds,

arry,

o

to

speak,

a

first-level

redicate

with

them

as

theirmost

natural form

f

designation.

A

man

is

first

nd

foremost

man.

Thus when

one

considers a man as a

man,

one

is

not

abstracting one

of

the

many properties

a man

may possess

from heothers; one is rather electinga unit ofenumeration. In

Fregean

terms,

ne is

bringing

bjects under a

first-level

oncept.

Elsewhere

Aristotle llows

that the

number

of

sheep,

of

men, and

of

dogs may

be the

same even

though

men, sheep, and

dogs

differ

from ach

other.

9

The

reason is

that

we

are

given

the

individual

28

Cf. G.

Frege,

The Foundations

f

Arithmetic,Oxford:

Blackwell, 1968).

29Cf., .g.,

Physics

20b8-12;

223bl-12;

224a2-15.

183







JONA

THAN LEAR

man

(or sheep or dog) as a

unit, and the enumeration of each

group yields the same result.

The arithmetician posits a

man as

an indivisible because he positshim as a unit and as a unit he is

treated as

the

least number

(of

men).30

Thus Aristotle uses the as-locution for two distinct purposes.

In

geometry t is used to

specify

which

property

f

a

physical

ob-

ject

is to

be

abstracted from thers nd from he matter.

n

arith-

metic

t is used

to

specify

he

unit

of

enumeration.

t

is

easy

to see

how in a pre-Fregean era

these two uses could be run together.

Both uses could be looselyexpressedas considering an x in re-

spect

of its

being

an F.

In

one case

we

can

consider a bronze

sphere

n

respect

of

ts

being

a

sphere;

n

the other

we

can consider

Socrates in

respect

of his

being

a

man.

Indeed,

in

both cases one

can be

said

to

be

abstracting :

in the

former ne abstracts

from

the

fact

that the

sphere

s

bronze;

in

the latterone abstracts

from

the fact hat one man is

many-limbed, nub-nosed,

tc.

But

to

run

thesetwo uses

togethermay

be

misleading.

For

in

the former ase

we are picking out one of theobject's many properties nd sepa-

rating it in thought.

In

the

latter

case we

are picking

out the

object itself,

nder

its

most

natural

description,

nd

specifying

t

as

a unit

for

ounting.

f

one

thinks, hat, trictlypeaking,

bstrac-

tion

is the

separation

of

a

property

f

the

object,

then there is

in

the case of arithmetic

no

abstraction

at

all.

Aristotlewould have

had a

more difficult ime conflatingthese uses

if

he had not

in-

stinctively witched to natural kind termswhen discussingarith-

metic. For

suppose

Aristotle

had

asked

us to consider a

sphere

as

a

spherehi sphairahhi phaira): therewould

be

no way of know-

ing

on

the basis

of the locution

whether we

were

to treat the

sphere

as a unit

in

counting spheres

or

to consider the spherical

aspect

of a

sphere

in

abstraction from

ts other

properties.

IV

Since

Frege, philosophies

of mathematics that can be labelled

abstractionist have been

in bad repute. t is, however, mistake

to tar

Aristotlewith Frege's

brush. Frege was attacking psycholo-

gism, the attempt to

reduce logic and mathematics to laws of

30

Cf. Physics

220a27-32;

Metaphysics 092bl9.

184







ARISTOTLE ON

MA

THEMA

TICS

empirical

psychology.He

was

especially

critical of the

idea that

number

was

anything

subjective or

dependent for

ts

existence

on the existence of some inner mental process.

Frege

treats

abstraction as a

deliberate

lack of

attention:

We

attend ess

to a

property nd it

disappears.

By makingone

characteristicfter

another

disappear,we get

more and

more

abstract

concepts

.

. .

.

Inattention is

a most

efficacious logical

faculty;

presumably

his

accounts for

the

absentmindedness f

professors.

uppose

there re a

blackand a

white at

sitting ide

by

side

before s. We

stop

attending

o their

olor

and theybecome

colorless, utthey restill

ittingide

by side.We

stop

ttending

o

their

osture nd

they re no

longer

itting

though

hey

have

not

assumed nother

osture) ut each one

is still

n

its

place.

We

stop

attending

o

position;

they

cease to

have

place but still remain

different.

n

this

way,

perhaps,

we

obtain

from ach one

of them

general

oncept

fCat.

By

continued

pplication

f this

procedure,

we obtain

from

ach object a

more

nd more

bloodless

phantom.

Finallywe thus btainfromachobject

something

holly eprived

of

content;

ut the

omething

btainedfrom ne

object

s different

from he

omethingbtainedfrom

nother

bject-though

t is

not

easy to

say

how.32

Frege's point,

made

repeatedly with

varying

degrees of

humor

and

sarcasm,

is that if

we

abstract from ll

the

differences e-

tween

objects,

then t will

be

impossible

to count them

as different

objects.

One must

distinguish wo strandsof

thought

that

run

together

through Frege's criticisms.

First,

there

are the

attacks on

psy-

chologism,

on

treating

abstraction

as a deliberate

lack of atten-

tion.

Second,

there

is his

criticismof

treatments f number

as

a

first-level

oncept, of

number as a

property

of an

object.

Since

number is

not a

property

of an

object,

it is not

there to be ab-

stracted

ven

if

bstraction

were

a

legitimate

process.

As

one reads

Frege'scritique ofhis predecessors n the FoundationsfArithmetic,

one can

see that their

problem lies

mainly

in

their

assumption

that

number s a

first-level

oncept,

not

in

whetheror not

they

re

abstractionists.

3

Cf.

Frege,

op.

cit.,

pp.

33-38.

32

From

Frege's

review

of

Husserl's

Philosophie

der

Arithmetik,

n

Translations

from

he

hilosophical

Writings

f

Gottlob

rege,

d. P.

Geach

and

M.

Black

(Oxford:

Blackwell,

970),pp. 84-85.

185







JONA

THAN

LEAR

If,

however, one

considers

genuine

properties

of

objects, in

particular,

geometrical

properties

ctually

possessed

by

physical

objects, then it becomes clear that not all formsof abstraction

need

be

treated

as

psychologistic

r,

indeed, as

forms f

inatten-

tion.

For

Aristotle, abstraction

amounts to no

more

than

the

separation

of one

predicate

that

belongs

to

an

object and

the

postulation of

an

object that

satisfies

hat

predicate

alone.

This

separation

may

occur

in

thought

(Phys.

193b34),

but this

is no

more

damning

than

admitting

that

one

carries out a

geometrical

proof

or

arithmetical

calculation

in

thought.

One is

carrying

out a

determinate

procedure and there

is no

irremediably sub-

jective

element.

V

Of

course

Aristotle's

philosophy of

mathematics

does

have its

limitations,but what is remarkableis that,even from contem-

porary

perspective,

t

retains certain

strong

virtues. The

limita-

tions re

obvious

enough.

In

arithmetic

we are

given

only

a

means

of

selecting

a

unit

for

enumeration. In

geometry

Aristotle's

method

depends

on

there

being actual

physical

objects

which

possess

all the

relevant

propertieswith

which we

reason

geomet-

rically. There

are

two

features of

mathematical

experience

to

whichAristotle's heorydoes not seem to do justice. First, much

of

mathematics,

for

xample

set

theory,

annot

easily be

thought

of

as an

abstraction

from

any

aspect of

physical

experience.

Second,

there

s the

plausible belief

that

mathematical

theorems

are

true

rrespective f

whether

here s

any

physical

instantiation

of them.

Under

scrutiny,

owever,

this belief

turns

out to be

less

categorical

than

one

might have

expected.

For

our belief in

the

truth f a mathematical theoremmay not depend on the actual

existenceof

a

bronze

triangle,

but it is

inextricably

inked to

our

belief in

the

applicability

of

mathematics to the

world.

Euclid

1-32, once

thought

to be

an a

priori

truth, s no

longer even

thought

to

be

true. The

reason

is that it

is now

believed that

physical

triangles,

f

there

were

any,

would not

have

interior

angles

equal

to

two

right

angles.

Euclid

1-32

may

be

a conse-

quence

of the

Euclidean

axioms-and

thus we can

make

the

186







ARISTOTLE ON MA

THEMA

TICS

limited claim

that the theorem is true of triangles n

Euclidean

space-but one of the axioms which enters essentially into the

proof s thought to be false of the physical world. So while one

may believe

a

theorem true while remaining agnostic on the

question, say,

of whether

there are any physical triangles, one

must believe that a triangle s a physical possibility nd that the

theorem truly describes

a

property t would

have.

The virtues of Aristotle'sphilosophy of mathematics are most

clearlyseenby comparingit with thephilosophyof mathematics

advocated by Hartry Field in Science WithoutNumbers.33ield

argues that to explain the applicability

of

mathematics one

need

not assume it to be true. One need only assume that one's mathe-

matical

theory M)

is

a

conservative

xtensionf one's

physical theory

(P). Suppose S is a sentence using solely termsof physical theory.

Then to say that M is a conservativeextension of P is to say that

if

M,

P

F-

S,

then

P F-

S.

That is, any sentenceof physical theorywhich can be proved with

the

aid of mathematics

can

be proved without

t. The

invocation

of

mathematics makes the proof simpler, shorter, nd more per-

spicuous, but

it

is

in

principle eliminable.

And

for mathematics

to be a conservative extension of science, it need not be true; it

need only be consistent. 4

Aristotle reated

geometry

s

though

it

were

a

conservativeex-

tensionof physical science. If,as we have seen,one wantstoprove

of

particular

bronze isosceles

triangle

b

that

t has

interior

ngles

equal to

two

right

angles,

one

may

cross to the realm of

pure

geometricalobjects

and

prove

the theoremof

a

triangle

c. The

2R

property

as been

proved

to

hold

of c in such a

way that

it is evi-

dent

that

it

will

hold

of

any triangle,

o one can then

return

to

Hartry

ield,

Science

Without

umbers

Oxford:

Blackwell,

1980).

It isnotgenerallyrue hat heconsistencyf theorywillguarantee hat

it

will

apply

conservatively

o

the

theory

o

which

t is

adjoined.

But

Field

makes

ngenious

se

of

the

fact

hat

mathematics

an be

modeled n

set

theory

to

show

that

n the

special

case of

mathematics,

onsistency

oes

suffice

or

conservativeness.ee

Field,

op.cit.,

Chapter

1

and

Appendix.

To

see that

con-

sistency

oes

not in

general

mply

conservativeness,

onsider

wo

consistent

theories

, which

has P

as a

theorem,

nd T2,

which

has

P D

Q

as

a

theorem.

Even f

he

unionof

T,

and

T2

s

consistent,

,

+

T1

is

not

conservative

xten-

sion

of

either

T,

or

T2,

ince n

T,

+

T?,

Q

s

a

theorem,

lthough

t

is

not

a

theorem f

eitherT, or T,

187







JONA

HAN

LEAR

the

physical world and conclude that b

has

the

2R

property.

The

crossing

was not,

however, strictly

peaking necessary:

one

could have proveddirectlyof the bronze isoscelestriangleb that

it had interior ngles

equal to two

right ngles.

The

reason

why

the

crossing is

valuable, though, is that

one

thereby proves

a

general theorem

applicable

to all

triangles

rather than

simply

proving that a

certain property holds of

a

particular

triangle.

Thus

if

we let AP

stand

for

Aristotle's

physical

theory,

AG for

Aristotelian

geometry,and S for an

arbitrary

sentence

in

the

language of AP, Aristotle would have allowed that

if AP,

AG

F-

S,

then

AP

K

S.

Geometry,

or

Aristotle,

was

a

conservative

extension of

physical

theory.

The great

virtue of Aristotle's

account

is that

Aristotle also

takesgreat

pains to

explain

how

mathematics

can be

true.

A

con-

servative extension of

physical

theory

need

not

merely

be con-

sistent; t can also be true. Aristotletries to

show how

geometry

and

arithmetic an be

thought

of

as true,

even

hough

he existence

of

separated

mathematical

objects, triangles and

numbers,

is

a

harmless

fiction. 5 hat is, he

tries to show how

mathematical

statements

an be true

in

a

way which does not

depend on

the

singular

terms

having

any

referenceor the

quantifiers

ranging

over

any

separated

mathematical

objects.

The

key

to

explaining

the truthof a mathematical statement ies in explaining how it

can be

useful.

Aristotleconsidered the truths

of

geometry

to be

useful because there are clear

paths

which

lead

one from the

physical world to the world of

geometrical

objects.

There

may

be

no

purely

geometrical

objects, but they are a

useful iction, be-

cause

they

are an

obvious

abstraction

from

featuresof

the

physi-

cal

world.

Merely

to

say

that an

arbitrary

heory

T

is a conserva-

tive extensionof our theoryof the physical world (P) will not

3 Let

us use

the word Platonist o

describe he

position n the

philosophy

of

mathematics eld by

Plato and his followersn the

Academy. Let

us use

platonist o describe

nyone who

believes thatmathematical

tatements

are

true

n

virtue

f the existence

f

abstract bjects

which xistoutside pace

and

time.

Kurt Godel is

an

example of a

platonist.) inally, et us say

that a

mathematical

ealist s

someonewho

believes hatmathematical

tatements

are

determinatelyrue rfalse

ndependentlyf our

knowledge fthem.Then

one can

say

that

Aristotle efends

form

f

mathematical ealismwhiledeny-

ingbothPlatonism nd Platonism.

188







ARISTOTLE ON

MA THEMA

TICS

explain the

usefulnessor

applicability

of T. It

would

be easy to

formulate

consistent

heory

T,

prove that

P

+

T

is a

conserva-

tiveextension ofP, and show that T is of no use whatsoever in

deriving consequences about

the physical world. It is

precisely

because

mathematics s so

richly pplicable

to the

physical world

that we

are inclined

to believe that it is

not merely one

more

consistent

theory

that

behaves

conservatively with

respect

to

science,

but that it is

true.

What

is

needed is

a bridge between the

physical

world and

the world ofmathematical objects, similarto the bridge Aristotle

provided

between bronze

triangles and geometrical

triangles,

that

will

enable us to see how we can

cross to the

world

of mathe-

matical

objects

and return

to the

physical

world.

This

bridge

would

explain

both the sense in which

mathematics s an

abstrac-

tion of

the

physical

world and

why

it

is

applicable.

This bridge,Field

suggests, s

supplied by

Hilbert's representation

theorem

orEuclidean geometry.

6

The proof

of the representation

theoremshows that given any model of Hilbert's geometrical

axioms, there will be

functions

rom

points

in

space into the real

numbers

which

satisfy onditions for

distance

structure.Given

that,

one

can show that the

standard Euclidean

theorems are

equivalent

to

theorems

about relations between real

numbers.

So

if

one

thinksof

models

of

space as

being abstract, there is a

two-stage

process

of

moving

from the

physical

world to

the

mathematical. The first tage is Aristotle's,where one moves

from

he

physical

world to a

Euclidean model of

space;

the second

is where one moves from

the Euclidean

model to

a

model

of

Euclidean

space

in

the real

numbers.

The

homomorphic

func-

tions

would then

provide

the

second

span

of the

bridge

and

Aristotelian

abstraction would

provide

the first.

Or

one could

just

take

physical

space

as the model

for the axioms

(assuming

that the axioms are true ofphysical space), and then one needs

only

the

homomorphic

functions s a

bridge.

This

example

is

specific

to

geometry,

but it

contains

the key

to a

general

theory

of

the

applicability

of

mathematics.

For

36Field, p.

cit., Chapter 3.

Cf.

David

Hilbert,

Foundations

f Geometry

(Lasalles: Open

Court,

1971). Of course,

ince

Aristotle, e have

learned

that

to

formalize

eometry

uccessfully

e

need more

axioms than

geometers

f

Aristotle's ay

were

ware of,

particularly bout

the

relation f

betweenness,

and thatgeometry as undergone thoroughrithmetization.

189







JONATHAN LEAR

mathematics to be applicable to the world

it

must reproduce

structural eatures hat are found (at

least to some approximate

degree) in the physical world. Moreover, there must be a bridge

by which we can cross fromthe structural

features

of the world

to the mathematical

analogue,

and then return to

the

physical

world. Consider a simple example

from et theory. To say that

arithmetic an be modelled

in

set theory s to say that there

exist

at least one function

f

that

maps

the natural numbers one-one

into sets and functionsf1 ndf2 which map pairs of sets into sets

such that

x

+

y

=

z

*-*f

(fO

x),

fo

(y))

=o (Z)

and

X y

=

Z

-*f2 (o (X), fO (Y)

=o

(Z)-

The

functionsf,

ndf2

impose a structure

n sets relevantly nal-

ogous

to

that

of the natural numbers,

andfo

provides

a

bridge

between thenatural numbersand the universeof sets.There may,

of

course,

be

many triples

f<,

f,

A)

which

satisfy

hese

con-

straints.37What remains

to be

supplied

is a

bridge

between

natural numbers and

the

physical

world. This bridge was success-

fully specified by

the

logicists

in

their otherwise unsuccessful

attempt

to reduce mathematics to

logic.

A

number

n is related to

othernumbers

n

ways

which are

intimately

inked

to

the

manner

in

which n-membered ets of durable

physical objects are related

to disjoint

sets

of various cardinality.

The union

of two

disjoint

two-membered

ets of

durable

physical

objects usually

is a four-

membered

set,

and the arithmetical truth

2

+

2

=

4

reflects

this fact.

Of

course, physical objects

may perish

or coalesce

with

others-thus

the

usually -and one

of the virtues of arith-

metic s that

n

crossing

o the realm of

numbers

one can abstract

from

this

possibility.

Thus, to explain the usefulnessand applicability of mathe-

matics we have had to follow

Aristotle and

appeal

far more

strongly

o the existence of a

bridge

between

the

physical

world

and the

world

of

mathematical

objects

than

we have to

the fact

that mathematics s a conservative

xtension of science.

The con-

servativeness f mathematics

was invoked

by

Field to

explain why

37

Cf. Paul

Benacerraf,What Numbers

ould Not

Be,

Philosophical

eview,

74 (1965).

190







ARISTOTLE ON MA THEMA

TICS

we need

not

thinkof

mathematics as true,but

only

as

consistent.

But to explain why this

particular consistent

theoryrather

than

others suseful, e have had torelyratherheavilyon theexistence

of

bridges nd thus, think, o reimporthe

notion

of

truth.

For,

in

an Aristotelian

spirit, one

can

allow that

2

+

2

=

4

is

true

without having to admit

that there exist

numbers

in

a Platonic

realm

outside of space

and time. That

there must exist bridges

between the physical

world and those

portions of mathematics

which are

applicable to

it implies that the mathematics

must

reproduce (to a certain degree of accuracy) certain structural

features

f

the physical world.

It

is

in

virtueof this

accurate struc-

tural

representation

f the

physical

world

that

applicable

mathe-

matics

can

fairly

be said

to

be

true.

The crucial contrast between

Aristotle nd

Hartry

Field

is

as

follows.For

Aristotle,

mathematics is

true,

not

in

virtue of

the

existence of

separated

mathematical

objects

to which its terms

refer, ut

because

it

accurately describes the

structural

properties

and relations which

actual

physical

objects

do

have.

Talk of

nonphysical

mathematical

objects

is a

fiction,

one that

may

be

convenient and

should be harmless

if one

correctly

understands

mathematical

practice.

Field

agrees

with

Aristotlethat there are

no

separated mathematical objects, but

thinks that for

that

reasonalone

mathematics

is

not true. From an

Aristotelian

per-

spective,

Field looks

overly

committed to

the

assumptions

of

referential emantics: in particular,to the assumption that the

way

to

explain

mathematical truth s

via

the

existence

of mathe-

matical

objects.

One can

understand how mathematics can be

true,

Aristotle

thinks,

by understanding

how it

is

applicable.

Of

course,

s we now

realize

in

contrast

o

Aristotle, ot all mathe-

matics

need be

applicable.

And

where

portions

of

mathematics

are

not

applicable,

there is no

compelling

reason to think

them

true.

8

38

For

example,

he

ssumption

hat

there xists

measurable

ardinal

snot

thought

o

enhance

the

applicability

of

set

theory

o the

physical

world.

(Cf.

F.

Drake,

Set

Theoy:

An

Introduction

o

Large

Cardinals

Amsterdam:

North

Holland,

1974),

Chapter

6.)

Here all

that s

importants

that one

can consis-

tently

ugment

he

standard

xioms

of

set

theory

with n

axiom

asserting he

existence f

a

measurable

ardinal.

From

the

point

of

view

of

applicability

one

could as

well

have

added

an axiom

asserting

he

nonexistencef

measur-

able

cardinal.

Thus,

under

the

assumption

hat

set

theory

there

exists

191







JONA

THAN LEAR

Where

is

one to draw the line between

the truthsof

mathe-

matics and the

parts

of

mathematics that are

consistent with

them,but neithertrue nor false?Nowhere: for here s no demar-

cation

between applicable

and

nonapplicable mathematics

that can

be made with

any

certainty.One cannot

determine a

priorithat a portion of

mathematics is not

applicable. It is con-

ceivable,

though unlikely,

hat we should

discover that the

world

is sufficientlyarge and

dense that we need a

large

cardinal axiom

to describe t.

9

But this does

not mean that we should

treat all of

mathematics s true or all of t as

merely

onsistent.That we

can-

not draw a distinction

precisely does not

mean that there is no

distinction to be

made:

it

means

only

that

any

suggested

boundary

will

remain

conjectural

and

subject

to

revision.

Thus,

the question of how much of

mathematics is true can

only

be

answered a posterlori.

Though

not Aristotle's, this

philosophy of

mathematics is

Aristotelian. One of

its virtues

is

that

it

does

provide

a har-

monious account of truth and knowledge. Of those portionsof

mathematics that are

not

true,

the

question

of

knowledge

does

not arise.

Of

those portions that are

true,

there exist

bridges of a

fairly

irect

sortbetween the

physical

realm

and the mathemati-

cal. And it

is

in

virtue of

our

understanding

of how

these

bridges

link

the mathematical

and

the

physical that we can

be said to

know

mathematical truths.

Mathematics,

according to

Aristotle,

studies the physical world, but not as physical: I should like to

think

that this

approach

to the

philosophy

of

mathematics

provides

an

explication of that

insight.40

Clare

College,Cambridge

measurable

ardinal

has

no

physical

onsequences hat

set

theory

oes not

have on itsown,there s noreasontosaythatthemeasurable ardinalaxiom

is true

or

false.

3

For

example,

f

the

cardinality

f

the

physical

ontinuum

was

found

to

equal not

?t 1,

but the

first

measurable

cardinal

ft.

40

I

would

ike

to

thank

M. F.

Burnyeat, .

E. R.

Lloyd,

M.

Schofield, . J.

Smiley,

nd the

editors f

the

Philosophicaleview

or

riticisms f an

earlier

draft.

Aristotle's philosophy of mathematics.pdf

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