Against Against Intuitionism

8/11/2019 Against Against Intuitionism

1/19

Against against IntuitionismAuthor(s): Dirk SchlimmSource: Synthese, Vol. 147, No. 1, Reflections on Frege and Hilbert (Oct., 2005), pp. 171-188Published by: SpringerStable URL: http://www.jstor.org/stable/20118651.

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2/19

DIRK SCHLIMM

AGAINST

AGAINST

INTUITIONISM

ABSTRACT.

The main ideas behind

Brouwer's

philosophy

of Intuitionism

are

presented.

Then

some

critical

remarks

against

Intuitionism made

by

William

Tait

in

Against

Intuitionism

[Journal

of Philosophical

Logic,

12,

173-195]

are

answered.

1.

INTRODUCTION

In

the

following

I

shall

present

what

I

take

to

be the

core

of

Brouwer's

philosophy

of Intuitionism and defend

it

against

critical

remarks that

have

been

put

forward

by

William Tait

in

Against

Intuitionism

(1983).

To

understand Brouwer's

philosophy

of Intuitionism

it

is

helpful

to

first

bring

to

mind the

questions

that

it

was

intended

to

address. The

famous

quotation

from Kant's

Critique

of

Pure Reason

can

help

us

illustrating

the

fundamental difference

between

Brouwer's and other

approaches:

Thus all human

knowledge begins

with

intuitions, pro

ceeds

to

concepts,

and

ends

with

ideas

(A702/B730).

According

to

Howard

Stein,

Hilbert

chose

this

sentence

as

the

epigraph

to

his

Grund

lagen

der

Geometrie,

because

he

wanted

to

get

rid

of

the

Kantian

intu

itions and

proceed

to

the

concepts

of

mathematics,

following

Dirichlet's

call

to

a

maximum

of clear

seeing thoughts

(Stein

1988,

241).

Brou

wer's

direction

was

opposite,

he wanted

to trace

mathematics

back

to

its

origins,

which

he

considered

to

be

rooted

in

the

human intellect.

To

take

the natural

numbers

for

granted,

as

suggested,

for

example,

by

Kro

necker and

Poincar?,

was

not

enough

for Brouwer. He

wanted

to

know

where the natural numbers

came

from,

to

descend

to

the

ground,

to

find

the ultimate

explanation

for the

possibility

of

practicing

mathematics.

Mathematics, then,

was

to

be

built

up

on

these

grounds,

according

to

the

principles

that resulted

from

this

investigation.

Van

Stigt

calls

Brou

wer's

method

of

philosophical

exploration

genetic:

it searches for the

ultimate

nature

of

things

and human

activity

in

their

origins,

the

pro

cesses

that

brought

them into

being

(van

Stigt

1996,

382).

Synthese (2005)

147: 171-188

DOI

10.1007/sll229-004-6299-y

?

Springer

2005

http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp


3/19

172

DIRK

SCHLIMM

Given these

motivations

Brouwer

developed

a

very

broad

philos

ophy,

which

incorporated

epistemological,

psychological,

as

well

as

moral

aspects.

Traces of

this

philosophy

can

be found

in

almost

all

of

his

writings.

Clearly,

this

goes

well

beyond

what

are

tradition

ally

considered

to

be the

topics

of

philosophy

of

mathematics.

In

the

end Brouwer's

philosophy

of

Intuitionism

has

not

found

many

followers,

and

in

particular

most

mathematicians

(then

and

now)

have

not

regarded

it

as

necessary

for

motivating

investigations

of

intuitionistic

mathematics.

However,

given

the

importance

of Intu

itionism for

the debate about the foundations of

mathematics

in

the

early

20th

century (Mancosu 1998),

which extends

also

to

contem

porary

discussions

(Detlefsen

1990),

I

regard

a

clear

understand

ing

of

Brouwer's basic

writings

as

indispensable

for the

historically

minded

philosopher

of

mathematics.

2. BROUWER'S PHILOSOPHY OF

INTUITIONISM

2.1.

What

Intuition

Is

Let us begin this exposition of Brouwer's philosophy of mathemat

ics

by taking

a

closer look

at

the

meaning

of

'intuition',

the

central

concept

of

Intuitionism.

In

the

ordinary

use

of

language

'intuition'

means

the

ability

of

direct

apprehension,

to

grasp

something

without

the

process

of

rea

soning,

to

have

an

immediate

understanding.

Brouwer

begins

his

dissertation

of 1907

with

an

example

of what he

regards

an

intuitive

act:

counting.

His

former

student

Arend

Heyting explains

that

even

children

know what the natural numbers

are

and

how the

sequence

of the natural numbers can be constructed (Heyting 1971, 7). These

uses

of 'intuition'

are

in

accordance with the

ordinary meaning.

However,

Brouwer

introduces

a

second,

somewhat different

mean

ing

of

intuition.1

He

considers

the basic intuition

of mathematics

(and

of

every

intellectual

activity)

as

the

substratum,

divested of all

quality,

of

any

perception

of

change,

a

unity

of

continuity

and

discreteness,

a

possibility

of

thinking together

several entities.

(Brouwer

1907,

8)

It

is

in

this

sense

that

'intuition' is used

as

the

cornerstone

for

Brouwer's

philosophy.

For

the sake of

clarity

I

shall

use

the

term

'Intuition'

with

a

capital

letter

to

refer

to

Brouwer's

notion and

'intuition' when

it

is

meant in

the usual

sense.



4/19


5/19

174

DIRK

SCHLIMM

This

intuition

of

two-oneness,

the

basal

intuition

of

mathematics,

creates not

only

the numbers

one

and

two,

but

also all

finite

ordinal

numbers,

inasmuch

as one

of

the elements of the two-oneness

may

be

thought

of as a new

two-oneness,

which

process may

be

repeated

indefinitely.

(Brouwer

1912,

85-66)

On the

basis

of

Intuition

we can

connect two

things

to

form

a

new

totality,

and then

this

totality

can

be taken

together

with

another

thing

to

form

a

totality again.

This

process

allows

us

to

gather

many

particulars

and

thereby stepwise

to

build

a

unity.

Thus,

Brou

wer

also refers

to

Intuition

as

the intuition

of the

many-one

ness

(Brouwer

1907,

98)

or

unity

in

multitude

(Brouwer

1907,

179).

Not

only

can

the natural numbers be created

on

the basis

of

Intuition,

but also the

continuum

( intuition

of

the

continuum ;

Brouwer

1908,

569),

and

the entire

body

of

mathematics:

Math

ematics

(...)

develops

from

a

single aprioristic

basic intuition

(Brouwer

1907,

179).

If

mathematics

is

to

be

developed

from

Intuition,

then

Intuition

has

to

provide

means

to create

all mathematical

objects.

Here,

Brou

wer

distinguishes

two

phases

in

the

development

of

Intuitionism.

In

the first

act

of

Intuitionism ,

in

which mathematics

is

separated

from

language

and the

importance

of

Intuition

is

recognized,

new

entities

are

formed from

objects

that

have been obtained

previously.

The

second

act

of Intuitionism

recognizes

also

infinitely proceed

ing

sequences

and

mathematical

species

as

forms

of entities

that

can

be

generated

on

the basis

of

Intuition

(Brouwer

1952,

140-142).

Since

all

(intuitionistic)

mathematics

can

be tracked back

to

the

basal Intuition

and

its

self-unfolding

in

the

mind,

it

follows

that mathematics

itself is

a

construction

of the mind:

Intuition

istic

mathematics

is

a

mental

construction,

essentially independent

of

language.

It

comes

into

being by self-unfolding

of the

basic

intuition

of

mathematics,

which

consists

in

the

abstraction of

two

ity

(Brouwer

1947).

As

this

statement

shows,

Brouwer

sharply

dis

tinguishes

between

mathematics and

the

language

of mathematics.

Mathematics

is

done

in

the

mind,

not

in

an

externalized

way

using

language

or

written

signs:

The words of

your

mathematical dem

onstration

merely

accompany

a

mathematical

construction

that

is

effected

without words

(Brouwer

1907,

127).

The

presence

of such

a

construction is

in

fact

Brouwer's criterion

of existence

in

mathe

matics:

to

exist

in mathematics

means:

to

be constructed

by

intu

ition

(Brouwer

1907,

177).

A

mathematical

statement

is

true

only

when

a

corresponding

construction has been made.

Brouwer

writes:



6/19

AGAINST AGAINST

INTUITIONISM

175

truth

is

only

in

reality

i.e.

in the

present

and

past

experiences

of

consciousness

(Brouwer

1948,

1243) (see

also Detlefsen

1990).

Brouwer's

view that

all

mathematics

is

essentially

a

language-less

activity

has

led

to

the harsh

confrontation

with

views

that

are

based

on

the

possibility

of

representing

mathematics

in

a

formal

language,

as

advocated,

for

example,

by

his

contemporaries

Russell and

Hil

bert.

We

shall

return to

this

later,

when

addressing

Tait's criticisms.

2.3.

The

Origins of

Intuition

So

far the

nature

of Intuition and its role

in

mathematics have been

presented.

In this section we shall see where Intuition comes

from,

and

why

human

beings

have

come

to

be able

to

make

use

of it.

In

Brouwer's

dissertation

a

whole

chapter

is

dedicated

to

the

rela

tion

between

mathematics

and

experience.

Here

he

introduces

the

notion

of

taking

the mathematical

view

and

discusses

it

in

rela

tion

to

Intuition:

Proper

to

man

is

a

faculty

which

accompanies

all

his

interactions

with

nature,

namely

the

faculty

of

taking

a

mathematical

view

of his

life,

of

observing

in the world

repe

titions of

sequences

of

events,

i.e.

of

causal

systems

in

time.

The

basic

phenomenon

therein is the simple intuition of time, inwhich repetition is possible in the form:

'thing

in time and

again thing',

as

a

consequence

of

which

moments

of life break

up

into

sequences

of

things

which

differ

qualitatively.

(Brouwer

1907,

81)

Brouwer further

distinguishes

between

two

distinct

phases

involved

in

taking

the

mathematical view:

In

the

first,

a

temporal

succession

of

things

or

events

is

perceived,

and

in

the

second,

some

of these

are

identified

as

being

causally

related. These

two

phases

are

also

called

the

temporal

view

and the causal view

(Brouwer

1927,

153).

Underlying

and

making possible

the

mathematical view is

the

intuition of

time ,

which bears some

affinity

to Kant. In

fact,

Brouwer

names

the

purpose

of his

dissertation

to

be

to

rectify

Kant's

point

of view

on

apriority

in the

experience

and

bring

it

up

to

date

(Brouwer

1907,

113).

As is

well

known,

Kant

rejects

the

possibility

of

having

an

unstructured

experience

of

some

kind

of

raw

stuff,

but claims

that all

experience

is

determined

by

the

forms of

intuition,

space

and

time.

After the

development

of

non

Euclidean

geometries

in

the

19th

century

it

was

no

longer

tenable

to

regard

three-dimensional Euclidean

space

as

the

only

possible

con

ception

of

space

and

therefore Kant's

apriority

of

space

had

to

be

abandoned.

Brouwer

places

himself

exactly

in

this

tradition:

the

position

of

intuitionism

(...)

has

recovered

by

abandoning

Kant's



7/19

176

DIRK

SCHLIMM

apriority

of

space

but

adhering

the

more

resolutely

to

the

apriority

of

time

(Brouwer

1912,

69).

In

contrast to

the

Kantian intuition

of

time, however,

Brouwer

rejects

the view that all

experience

is

neces

sarily organized

in

advance

by

this intuition:

Mathematical attention is

not

a

necessity

but

a

phenomenon

of life

subject

to

the

free

will, everyone

can

find this

out for

himself

by

internal

experience:

every

human

being

can

at

will either

dream-away

time-awareness

and the

separation

between the Self and

the

World-of-perception

or

by

his

own

powers

bring

about

this

separation

,and

call

into

being

the

world-of-perception

the

condensation

of

separate

things.

(Brouwer

1933,

418^419)

To

take the causal view consists

in

identifying

objects

in

the

tem

poral

sequences,

and relations between

them,

causal relations.

In

this

way

patterns

are

created which

can

be observed

in the

world.

This

'seeing',

however,

is

a

human

act

of externalization:

there is

no

real

existence

of

objective

natural

phenomena

as can

be ascribed

to nature

itself:

the

seeing originates

in

man,

is

an

expression

of

man's will

alone,

independent

of

nature

which itself exists

indepen

dent of man's

will

(van

Stigt

1979,

394).2

The

ability

to

take the

mathematical view

has

contributed

to

the

survival

of

mankind,

because

of

its

great

utility

for human self

preservation.

To

be able

to

see

causal

sequences

in the

world

by

taking

the mathematical

view

allows

us

to

jump

from

the

end

to

the

means

(Brouwer

1907,

81).

If

a

sequence

of

events

is

recog

nized,

it

becomes

possible

to

estimate the

consequences

of

one's

actions. The

human

tactics

of

'acting

purposively'

then

consists

in

replacing

the

end

by

the

means

(a

later

occurrence

in

the intellec

tually

observed

sequence

by

an

earlier

occurrence)

when

the human

instinct

feels

that

chance favours the

means

(van Stigt

1979,

395).

A

simple

example

may

illustrate this

point. People

who

like

strawberries

are

likely

to

go

into the woods

in

summer

to

look

for

them.

Doing

this

requires only

the

knowledge

that the

probability

of

finding

strawberries is

higher

in

summer

than

during

the

rest

of

the

year.

Imagine

now,

that

somebody

discovered

that

strawberries

are

bigger

and

tastier

when

they

grow

where it has

rained in

spring.

This

very

simple

causal

sequence

'water

in

spring,

tastier

strawber

ries

in

summer' leads

our

person

not

only

to

look for

strawberries

in

summer,

but

also

to

take

care

that the

plants get enough

water

in

spring,

and

to water

them

if

necessary.

The

act

of

watering

the

plants

does

not

have

an

immediate

goal,

but

an

indirect

one,

namely

to

have tastier

strawberries.

The

advantage

of this

tactic is

that

it



8/19

AGAINST AGAINST INTUITIONISM

177

is easier

to

water

the

plants

than

to

look

for

strawberries

that

hap

pened by

chance

to

grow

in

areas

where

it rained

in

spring.

Exploiting

causal

sequences

that

are

projected

into the world

is

not

an

absolutely

reliable

process,

because

it

can

always

happen

that

a

pre

sumed

pattern

does

not

lead

to

success.

But,

despite

this

possibility

in

general

the

consideration

of

sequences

and

consequent

going

back

from

the end

to

the

means,

where intervention

appears easier,

show

themselves

very

efficient tactics from which mankind derives its

power

(Brouwer

1907,

82).

The

ability

to

take

the

mathematical

stance

is

not

only

a

contingent

human

ability,

but the

most

important

human

faculty,

which

secures

survival:

Indeed,

if

this

faculty

did

not

achieve its

end

it would

not

exist,

as

lion's

paws

would

not

exist if

they

failed of their

purpose

(van

Stigt

1979,

395).

This

faculty

of

the

human

intellect,

developed though

evolution,

is

present

in

every

human

being, just

as

every

lion has

paws.3

And

since

Intuition is the basis

of

the mathematical view and is

also the

origin

of

mathematics,

it follows that all human

beings

develop

simi

lar

mathematics. This is

not

necessarily

so

for

the

language

in which

mathematics

is

expressed:

it is easily conceivable,

given

the same

organizations

of the human intellect

and

consequently

the

same

mathematics,

a

different

language

would

have

been

formed,

into which the

language

of

logical reasoning,

well known

to

us,

would

not

fit.

Probably

there

are

still

peoples,

living

isolated from

our

culture,

for which this is

actually

the

case.

(Brouwer

1907,

129)

Here

Brouwer

tries

to

answer

the

objection

that

Intuitionism does

not

account

for the

public

character

of

mathematics,

which is

raised,

for

example, by

Tait.

2.4.

The

Value

of

Intuition

We have

seen

that

Intuition

forms

the basis

of

our

ability

to

perceive

sequences

of

events,

which

in

turn

allows

us

to

shift

from

actions

with

direct

goals

to

actions that

serve as means

to

some

future

end.

Herein

lies the

source

of

human

power

(van

Stigt

1979,

395).

Brouwer

does

not

give

a

value

judgment

about this

ability

in

this

published

work,

but he does

so

in

those

parts

of the

thesis

that he

was

urged

to

leave

out

by

his

advisor.

Korteweg

was

of the

opin

ion

that these

parts

represented

Brouwer's

pessimistic

view

of life

and

that

this had

nothing

to

do

with foundations

of

mathematics.



9/19

178

DIRK

SCHLIMM

However,

for

Brouwer

these

parts

contained the basic ideas

which

held

together

his whole thesis.

They

deal

with the

way

mathematics

is

rooted

in

life,

what

therefore

should be the

starting-point of

mathematical

theories;

all

particular topics

in

my

dissertation

only

make

sense

when

related

to

this fundamental thesis.

(Letter

from Brouwer

to

Korteweg,

November

5, 1906;

quoted

from

(van Stigt

1990,

491)

I

discuss Brouwer's

views here

to

give

the full

picture

of

his

philoso

phy

of

mathematics,

which

not

only

is

concerned

with mathematical

objects,

but also

with

the

way

they

are

connected

to

life.

In

Leven,

Kunst

en

Mystiek (Life,

Art,

and

Mysticism) 4

Brou

wer relates a

myth

that is the

key

to many of his ideas:

Originally

man

lived

in

isolation;

with the

support

of

nature

every

individual

tried

to

maintain

his

equilibrium

between

sinful

temptations.

This filled

the

whole

of his

life,

there

was no room

for

interest

in

others,

nor

for

worry

about

the

future;

as a

result

labour did

not

exist,

nor

did

sorrow,

hate, fear,

or

lust.

But

man was

not

content,

he

began

to

search for

power

over

others

and for

cer

tainty

about the

future.

In

this

way

the

balance

was

broken,

labour become

more

and

more

painful

to

those

oppressed

and

the

conspiracy

of

those

in

power

gradually

more

and

more

diabolical.

In

the

end

everyone

wielded

power

and

suffered

suppression

at

the

same

time.

The old instinct of

separation

and isolation

has survived

only

in the form of

pale

envy

and

jealousy.

(Brouwer

1905,

7)

This

mythological

time

is

lost for

Brouwer,

the human

race

discov

ered its will

to

power

over

nature

and

over

other human

beings.

In

contrast to

the often

told

success-story

of

science,

Brouwer's

ver

sion is

a

negative

one,

a

story

of

decay.

The

breaking

off from

the

state

of

equilibrium

was

made

possible by

the mathematical

view,

which

itself

originates

in

Intuition:

In

this

life of lust and

desire

the

Intellect renders

man

diabolical service

of

connecting

two

images

of

the imagination

as means

and end. Once

in

the

grip

of

desire for

one

thing

he

is made

by

the

Intellect

to

strive after another

as a

means

to

obtain

the former

(Brouwer

1905,

19).

Taking

the math

ematical view allows

us

to

objectify

the

world,

to

perceive

causal

sequences

and

to

communicate

with each other.

But,

the main

pur

pose

of

communication

is

to enforce man's

will

over

others

out

of

fear

or

desire

(van

Stigt

1979,

397).

Why

then,

if

these

were

his

views,

did

Brouwer, nevertheless,

become

a

mathematician?

He

answers:

But mathematics

practised

for its own sake can achieve all the harmony (i.e., an overwhelming

multiplicity

of different

visible,

simple

structures

within

one

and

the

same

all-embracing

edifice)

such

as can

be found

in

architecture and



10/19

AGAINST

AGAINST

INTUITIONISM

179

music,

and

also

yield

all the

illicit

pleasures

which

ensue

from the

free

and

full

development

of one's

force

(van

Stigt

1979,

399).

He

later

also

talks

of

the constructional

beauty,

the

introspective beauty

of

mathematics,

when the Intuition is

left

to

free

unfolding

without

the restrictions

imposed

by

the

exterior

world

(Brouwer

1948,

1239).

Even

in

logic

this

beauty

can

be

found:

in

itself,

as an

edifice of

thought,

it

[logic]

is

a

thing

of

exceptional harmony

and

beauty

(Brouwer

1955,

1).

Only through

self-reflection,

free

from

fear and

desire,

free from

the influences

of

the

world,

one can

experience

the transcendental

truth,

according

to

Brouwer.

And those

imprisoned

in

life

call

this

mysticism,

they

think it

obscure,

but

truly,

it is the

light

that is

only

darkness

to

those who

are

in

darkness themselves

(Brouwer

1905,

74).

We

turn

now

to

the discussion

of

some

critical

remarks

against

Brouwer's

philosophy.

3. AGAINST INTUITIONISM

On the first four pages of

Against

Intuitionism , William Tait puts

forward

a

number of observations and

arguments

in

order

to cast

doubt

on

the

plausibility

of Brouwer's views of

mathematics.

The

remainder

of

Tait's

article is

dedicated

to

suggesting

an

account

of

the

meanings

of

mathematical

propositions

that

is

adequate

for both

constructive and

classical mathematics.

What

distinguishes

these

two

are

then

only

the

principles

admitted for

constructing

mathemat

ical

objects

and the fact that

some

terms

are

used with different

meanings

(e.g.,

'function').

The

upshot

is

that

constructive

mathe

matics can be subsumed under classical mathematics. My concern

here,

however,

is

only

with the first

part

of Tait's

essay.

Tait

begins

his

discussion

by quoting

the

following

passage

from

Brouwer's

(1952)

Historical

background,

principles

and

methods

of

intuitionism ,

which

by

now

should sound

quite

familiar

to

the

reader. Here Brouwer

states

that

Intuitionistic

mathematics

is

an

essentially languageless

activity

of the

mind

having

its

origin

in

the

percep

tion

of

a

move

of

time,

i.e.

of

the

falling

apart

of

a

life

moment

into

two

dis

tinct

things,

one

of which

gives

way

to

the

other,

but

is

retained

by

memory.

If

the two-ity thus born is divested of all quality, there remains the empty form of

the

common

substratum

of

all

two-ities. It

is

this

common

substratum,

this

empty

form,

which is the basic intuition

of

mathematics.

(Brouwer

1952,

141)



11/19

180

DIRK

SCHLIMM

For

each

of the

following

12

criticisms

of

Intuitionism

that Tait

pro

duces,

I

shall first

very

briefly

state

his

claim

(in italics)

and

then

reply

to

it

from

a

Brouwerian

point

of view.

1. Tait

begins

with

the

claim

that Brouwer's

insistence

on

mathe

matics

being essentially

a

language-less

activity

was no

doubt

partly

motivated

by

his

polemic

against

those he called

formalists,

in

partic

ular

against

Hilbert.

We

have

seen

that

the idea

that

mathematics is

independent

of

language

is

expressed

in

Brouwer's earliest

writings

and

represents

one

of

the

core

views of

his

philosophy.

This

motivated his

polem

ics

against

Hilbert's

'formalism',

which started

five

years

after Brou

wer's

dissertation with

Intuitionism and formalism

(1912),

but

not

the

other

way

around. Brouwer

indeed

discusses

Dedekind,

Cantor,

Peano,

Hilbert,

and

Russell,

a

group

he

later

referred

to

as

the

old

formalist

school

(Brouwer

1981,

2),

in

his

1907 dissertation

and

criticizes them for

placing

too

much

emphasis

on

the

language

of

mathematics and

for

denying

the role of

intuition.

How

far

these

discrepancies

influenced

the

development

of

Brouwer's

philosophy

or

resulted from it

has

not

yet

been

determined

and

possibly

never

will be. It

should

be

kept

in

mind, however,

that Brouwer's critical

attitude towards

language

as an

adequate

carrier of

thought

is

in

an

important

characteristic

of

his

views

expressed

as

early

as

1905

(Brouwer

1905).

2.

Referring

to

the

above

quotation,

Tait

infers

that in

one

life

moment

we

perceive

infinitely

many

falling aparts,f

(p.

174),

which

he

regards

as

paradoxical.

Here

my

reply hinges

on

the

correct

understanding

of

a

life

moment .

Brouwer

characterizes

the

falling

apart

of

a

life

moment ,

which is rendered

possible by Intuition,

as a

move

of time.

This

indi

cates

that he

regards

this

as

the

perception

of

an

interval,

rather

than of

a

single point

in

time.5

Tait himself

later

interprets

Brou

wer's claim

as

being

about time

intervals

(p. 176).

Tait

argues

from the

perception

of

a

two-ity

and

the fact

that this

can

be

repeated

to

the

perception

of

an

infinity,

which

strains

the

notion of

perception

(p.

174).

But for Brouwer

the

repeated

appli

cation of

this

process,

the

unfolding

of

Intuition,

is carried

out

in

thought

and

thus

it is

not

a

perception

in the

sense

of

a

sensory

experience.

In

fact,

even

the notion of

an

intuitive continuum

is

one

that

Brouwer

describes

as

being

based

on

Intuition:



12/19

AGAINST

AGAINST INTUITIONISM

181

In the Primordial Intuition of

two-oneness

the

intuitions of continuous and dis

crete

meet:

'first'

and 'second'

are

held

together,

and

in this

holding-together

consists the intuition of the continuum

(continere

= hold

together). (Brouwer

1908,

569;

quoted

from

van

Stigt

1990, 155;

see

also the first

quote

in this

text.)

Brouwer

regarded

this

view

as

necessary

in

explaining

how

we

could

possibly

make

sense

of the

continuum.

By

1918-1919

Brouwer

had

ceased

to

mention the intuitive continuum after

he

had

developed

the

notions

of choice

sequence

and

spread

for

talking

about

the

mathematical continuum.

3.

Continuing

his

analysis of

the

1952

passage,

Tait

wants to

inter

pret

retained

in

memory

quite

literally,

as

if

the

past

were

a

sub

stance

in

a

box

that

I

could

take

out

and examine'

(p.

174).

This,

he

says,

does

not

make

sense

to

him.

To

understand

Brouwer

such

a

literal

interpretation

is

not

called

for. If

I

see a

leaf

falling

from

a

tree

as a

downward

motion of

an

object,

I

must

be

able

to

retain

at

least

some

impressions

in

my

memory. Otherwise,

I

could

not

speak

of

a

motion,

but

only

of

the

perception

of the leaf

at

various

positions

between

the branch and

the

ground.

Thus,

I

do

not

see

the need

of

keeping

the

past

as

it

were

in

a

box

for

examination,

to

be

expressed by

Brouwer's

writ

ings.

In

fact,

what Tait

calls the

ordinary

way

of

understanding

this

phrase,

in

the

sense

of

remembering

past

events

and

experiences,

is

all

that Brouwer needs for

his

account.

4. Tait

introduces

an

example for

being

conscious

of

time:

to

hear

two

successive

ticks

of

a

clock ,

which he

thinks

to

be the

likeliest

candidate

for

what

Brouwer

has

in

mind

(p.

175).

Here,

however,

he

sees no

falling

apart

of

a

life

moment.

In

none

of

his

writings

does

Brouwer

ever

talk

about

auditory

experiences

to

illustrate

the

origin

of

Intuition.

What

he talks

about

instead is

seeing

a

sequence,

objectifying

the

world

(see

above).

Nevertheless,

if

Brouwer's

analysis

is

complete,

we

should be

able

to

make

sense

of

Tait's

example.

When

we

hear the second tick

of

the

clock,

we

are aware

of

it

as one

single

tick. But because

of

the

near

past

that

is

still

retained in

our

memory,

we can

think of it

as

being

related

to

the first

tick,

and therefore

as

being

part

of

a

sequence

of

ticks.

Our

consciousness of time arises

because

we

realize

the second

tick

as

being

something

different

from the first

one,

and

at

the

same

time

recognizing

it

as

falling

under the

same

concept,

namely

'tick'.

The

second

tick

divides

time

into

two

distinct

phases: (1)

the last

tick,

and

(2)

the

rest

of the

sequence

of

ticks,

which is still

present

in



13/19

182

DIRK

SCHLIMM

memory.

These

two

phases

are

what Brouwer calls the two

distinct

things,

of which

one

gives

way

to

the

other,

but is retained in

mem

ory

(Brouwer

1952,

141).

Thus,

the

falling

apart

does take

place,

even

if

Tait refuses

to

see

it.

5.

The

above

example

is

used

by

Tait

to

infer

that

the number

thirty

can

be created

in

the

experience

of thirty

successive

ticks

of

a

clock

(p

.

175).

He then claims

to

never

have had

such

an

experi

ence,

though possibly

he has heard

thirty

consecutive ticks.

And

even

when

he had counted

up

to

thirty,

the basis

on

which he could

verify

his

count

was

objective

evidence,

e.g.,

saying

'thirty',

rather than

an

introspective

one.

Even

if Tait

never

actually experienced

the

creation of

the

num

ber

thirty,

it is the

knowledge

that

he

could

come

to

say

'thirty'

after

counting

a

certain

number of ticks of

a

clock that

constitutes

what

he

means

by saying 'thirty'.

That the

character

of the

evidence

used

to

verify

the result

of

an

act

of

counting

is

objective

comes

from the

fact

that

we

usually

count

exterior

objects,

not

internal

ones,

and

that

we can

repeat

the

process

of

counting

in

case

we

feel

uncertain about the

result.

But

the

act

of

counting

itself,

the

abil

ity

to

discern different

objects is,

for

Brouwer,

the

application

of

a

sequence

obtained

from

Intuition

to

the world.

And

if

I

hear

the

clock

striking

three

times,

and

my

friend

afterwards

tells

me

that

it

must

have

struck

four

times,

because it

is four

o'clock,

isn't

it

reliance

on

my

introspection

if

I

answer

There

must

be

something

wrong

with

the

clock,

because

I

heard

it

only

three times ?

6. Tait

argues

that

on

the basis

of

Brouwer's

account

we

cannot

justify

the

principle

that

every

number has

a

successor,

since

we can

not

possibly

have

an

experience

of

a

series

of

1010

elements.

The

underlying problem

is

that the

concept of my experience of

succes

sion has

no

precise

extension.

For

Brouwer,

we

do

not

have

to

actually experience

that

every

single

number has

a

successor,

because

numbers

are

what

are

gen

erated

by

putting

two

units

together,

then another

one

and

so

on.

Since

this is

a

conceptual point,

there

is

no reason

whatsoever

to

assume

that

the

application

of the basal

Intuition of mathematics

cannot

be continued

after

a

certain

point.

Indeed,

for

understanding

an

unlimited

iteration of

applications

of Intuition

no

corresponding

actual experiences

are

necessary.

7.

As

an

aside

and

without

discussing

it

further,

Tait

remarks

that

Brouwer's

view does

not

give

an

account

of

the

public

character

of

mathematics.



14/19

AGAINST AGAINST

INTUITIONISM

183

This

kind

of

criticism culminates

in

accusing

Brouwer's

philoso

phy

of

being

a

form

of

solipsism.

As

noted

above,

it

seems

indeed

compatible

with

Brouwer's

theory

of Intuition that

all

human

beings

possess

the

faculty

of

taking

the

mathematical

view,

and

this

leads

to

everybody

creating

similar

(intuitionistic)

mathematics.

Furthermore,

because

language

is

needed

to

practice

mathematics

as a

public

activity,

Brouwer

is

able

to

point

to

the

origin

of

some

problems

that

arise

in

mathematical

practice.

Just

recently

some

authors have

argued

that the

informal

language

of

mathematics

is

not

adequately captured

by

formal

systems

(e.g.,

Rav

1999),

oth

ers

that mathematical

proofs

are

better

understood within

a

social

context

(e.g.,

Heintz

2000).

Brouwer

would

clearly

agree

with

the

former

claim,

and

would

regard

the latter

as

arising

from

conflat

ing

proofs

as

mental

objects

with

their

linguistic

representations,

which

do

depend

on

the

social

context.

The

fact

that

proofs

can

be

accepted

or

refuted

(Grabiner

1974)

also

indicates

a

certain

amount

of

ambiguity

in

the

language

of mathematics.

These

observations

directly

follow

from

Brouwer's

account

of mathematics:

In

a

human mind

empowered

with unlimited

memory

therefore

pure

mathematics,

practised in solitude and without the use of linguistic symbols would be exact. However,

this

exactness

would

again

be

lost

in

mathematical communication between

individu

als,

even

between

those

empowered

with unlimited

memory

since

they

have

to

rely

on

language

as a

means

of communication.

(Brouwer

1934,

58)

Thus,

instead

of

this

being

a

serious criticism

of

Intuitionism,

it

points

at

a

phenomenon

in

mathematical

practice

that

can

be

accounted for

in

the framework

of Brouwer's

philosophy,

but which

can

be

explained

only

with

difficulty

by

other views

of

mathematics

that

are

less

critical

towards the

use

of

language.

8. Brouwer's affinity with Kant's argument for the a priori charac

ter

of

time

is

acknowledged by

Tait,

but

he

regards

it

as

no more

via

ble than Kanfs

analogous

view

concerning

space,

which

is

rejected

by

Brouwer.

Tait

claims

that 0

=

Sn

may very

well

be

compatible

with

our

experience for

some

n

(p.

176),

when this

statement is

regarded

as

being

about

time,

e.g.,

about

a

clock

Brouwer

regards

Intuition

as

being

a

priori

in

particular

with

regard

to

scientific

experience,

and

explicitly

stresses

the

indepen

dence

of

mathematics

and

experience.

When Tait

takes

a

statement

about a clock to be a statement about time, he is

talking

about time

in

a

scientific,

measurable

sense.

This is

a

conception

of

time which

Brouwer

regards

as

being already

infected

by

the

mathematical

view,



15/19

184

DIRK

SCHLIMM

as

he

makes

explicit

in

a

footnote

to

his

claim that time is

a

pri

ori: Of

course

we

mean

here

intuitive time which

must

be

distin

guished

from

scientific

time.

By

means

of

experience

and

very

much

a

posteriori

it

appears

that scientific time

can

suitably

be

introduced

for the

cataloguing

of

phenomena,

as

a

one-dimensional coordinate

having

a

one-parameter

group

(Brouwer

1907,

99).

True

experi

ence

of time

is

only possible

after

we remove

the

scientific

attitude:

For

example,

in

the

case

of

the

word

time

the

awareness

of

solitary

weakness,

of

roaming,

deserted after

rejection

of

guidance,

may

only

break

through

when

it is

no

longer possible

to

include the

indepen

dent,

variable coordinate

of mechanics

(van Stigt

1979,

398).

But

then

no

equation

like the

one

suggested

by

Tait

is

applicable

any

more.

9. That

counting

is

a

temporal

process

is

rejected

by

Tait

as

an

answer

to

the

question

of

why

we

understand

temporal

succession

any

better

than other

kinds

of

succession,

at

least,

once we

give

up

Brou

wer's

idea

that the

counting experience

is

itself

an

object

with

a

well

defined

structure

from

which

we can

abstract

(p.

176).

The

point

here is that

for Brouwer

the basic intuition of

math

ematics

is

the

same as

the intuition

of time

as

he understands it

(see

above).

Whenever

we

count

or

perceive

some

change

this

pro

cess

takes

place

in

time

and

it therefore

cannot

be

separated

from

time

itself. The

underlying

substratum of

any

such

process

is the

same,

namely

the basic intuition

of

mathematics

(or

of

any

intel

lectual

activity)

(Brouwer

1907,

8).

What

we

do

in

counting,

for

Brouwer,

is

to

apply

the abstract

structure

of the

ordinal numbers

obtained

by

the

unfolding

of Intuition

to

the

objects

of

experience.

10. Tait

challenges

the

argument

that without consciousness

of

temporal passage

we

would

not

understand

succession

because

it

is

very

hard

to

understand the

antecedent

of

the

counterfactual

(p.

176).

Here

I

can

only

refer back

to

Brouwer's

view

as

expressed

in

the

quotation

above

from

(Brouwer

1933,

418-419): According

to Brou

wer we can

make

sense

of the antecedent

of this

counterfactual

by

dreaming-away

time-awareness.

11. Tait

accuses

Brouwer

of

applying

a

vicious

circle

(to

use

the

explanandum

in the

explanans)

in

his

argument

that

the

concept

of

number

is

generated by

successive

applications of

the

Intuition

of

two

oneness.

To

explain

the

concept

of

number

as

iterations

of

succes

sion,

implies

that

we

already

understand

the

notion

of

number,

because



16/19

AGAINST AGAINST

INTUITIONISM

185

iteration

means

iterating

a

finite

number

of times

(p.

176;

emphasis

in

original).

The

generation

of

the

representation

of

a

number

consists of

constructing

the

successor

of

an

already

existing

entity.

Thus,

this

process

requires

only

an

initial

element

and the

application

of the

successor

function,

or

in

Brouwer's

words,

another element

that

can

be

put

together

with

the first

one

under

a new

concept.

If

we

rep

resent

numbers

by

a

series

of

strokes

on

paper,

we

need

one

stroke

to start

with

and

then,

whenever

we

have

a

series of strokes

to

rep

resent

the

number

n,

add

a new one

to

obtain

n

+

1. To

say

that

'III'

represents

3,

we

do

not

have

to

understand the

meaning

of

'3'

in

advance,

because

we

define its

meaning

to

be

'|||'.

Furthermore,

to

see

whether

a

series of strokes is

'|||',

we

do

not

need

to

know

what

'3'

means

other

than

'|||'.

We

can

compare

a

different

num

ber constructed

by

the

same

process

without

the

concept

of

number

already

present:

we

successively

take

away

one

stoke

from

both

rep

resentations,

until

one

of

them

is

empty.

If

the

other

one

is

empty,

too,

then

they represented

the

same

number,

if it is

not

empty,

the

numbers

were

not

equal.

12. The last remark

of

Tait

I

want

to

discuss here

is

the

claim that

mathematics

is

a

linguistic activity

of

a

community.

He

arrives

at

this

conclusion

by

asking

in

what

sense

is construction

according

to

a

rule

not

linguistic?

and

answering

that a rule

is

a

symbol

(p.

176).

Brouwer

vehemently

disagrees

with the

premiss

that the rule

has

to

be

a

symbol

and

is

therefore

linguistic.

The

separation

between

mathematics

and the

language

of

mathematics is

one

of

the

crucial

points

of

Brouwer's

philosophy:

The words

of

your

mathematical

demonstration

merely

accompany

a

mathematical construction

that

is effected without words

(Brouwer 1907, 127).

That such

a con

struction

is

according

to

a

rule,

Brouwer

would

respond,

does

not

mean

that

this rule has

to

be

presented

linguistically.

4. CONCLUSION

In

this

paper

Brouwer's

understanding

of

mathematics

was

pre

sented.

Questions

such

as

What

is

the

origin

of

mathematics? ,

How does

mathematics

come

into

being? ,

and

Why

did

math

ematics

come

into

being?

were

answered

according

to

his

philos

ophy

of

Intuitionism. Then

a

series of

criticisms

against

this view

were

presented

and

replied

to

from

a

Brouwerian

perspective.

These



17/19

186

DIRK

SCHLIMM

replies

have

shown,

I

hope,

that the last

word

on

Intuitionism

has

not

yet

been

spoken.

ACKNOWLEDGEMENTS

The author wishes

to

thank

Jeremy Avigad,

Michael

Hallett,

Colin

McLarty,

Wilfried

Sieg,

Bill

Tait,

and

Dirk

van

Dalen

for

their

comments

on

earlier versions of this

paper,

parts

of

which

were

pre

sented

at

the

1998

Midwest

Conference

on

the

History of

Mathemat

ics

at

Iowa State

University,

Ames,

IA.

Special

thanks

are

also due

to two

anonymous referees of this journal, whose insightful

com

ments

have been

extremely

helpful.

NOTES

1

Brouwer's

writings

are

often not

as

clear

as

one

might

wish

them to

be.

In

order

to leave it

to

the reader

to

verify (or

to

call

in

question)

my

interpreta

tion,

a

number of

passages

are

quoted

from

his

writings.

2

Quotations

from

(van

Stigt

1979)

are

from

passages

that

Brouwer

originally

wrote

for

his

dissertation,

but that

were

omitted

in

the final version.

3

Evolution

is

not

explicitly

mentioned

by Brouwer,

but it

helps

in

understanding

the

place

of

mathematics

in

the human intellect.

4

Even

though

this article

was

written

in

1905,

when

Brouwer

was

24,

he

tried

to

republish

it

in

1927 and

thought

of

translating

it

into

English

even

in

1964,

two

years

before

his

death.

3

Compare

this view

to

the

following

remarks

by

Kronecker. He

introduces the

ordinal numbers

as

a

stock

of

signs

which

we can

adjoin

to

a

collection of dis

tinct

objects

that

we

are

able

to

tell

apart

(Kronecker

1887,

949)

and tells

us

in

a

footnote what

kind of

objects

he

has

in mind: The

objects

can

in

a

certain

sense

be

similar

to

one

another,

and

only

spatially, temporally,

or

mentally

dis

tinguishable

-

for

example,

two

equal

lengths,

or

two

equal

temporal

intervals

(Kronecker 1887, 949).

REFERENCES

Benacerraf,

P.

and

H.

Putnam,

(ed.):

1964,

Philosophy

of

Mathematics

-

Selected

readings,

Prentice

Hall,

Englewood

Cliffs,

NJ.

Brouwer,

L.E.J.:

1905,

Leven,

Kunst

en

Mystiek,

Waltman,

Delft.

English

translation

(only excerpts):

Life,

Arts and

Mysticism,

in

(Brouwer 1975),

pp.

1-10.

Full

trans

lation

in

(Brouwer 1996).

Brouwer,

L.E.J.:

1907,

Over de

Grondslagen

der Wiskunde.

Maas

&

Van

Suche

len,

Amsterdam.

English

translation:

On

the Foundations

of

Mathematics,

in

(Brouwer 1975),

pp.

11-101.



18/19

AGAINST AGAINST INTUITIONISM

187

Brouwer,

L.E.J.:

1908,

'Die

m?glichen M?chtigkeiten',

Atti

IV

Congr.

Int. Mat.

Roma

III,

pp.

569-71.

Reprinted

in

(Brouwer

1975),

pp.

102-104.

Brouwer,

L.E.J.:

1909,

Het wezen der

meetkunde,

Amsterdam,

1909.

English

transla

tion:

The

Nature of

Geometry,

in

(Brouwer

1975),

pp.

112-120.

Brouwer,

L.E.J.:

1912,

Tntuitionism and

Formalism',

Bulletin

of

the

American

Math

ematical

Society

20

(1913),

81-96.

Reprinted

in

(Brouwer

1975),

pp.

123-138

and

in

(Benacerraf

and

Putnam

1964),

pp.

66-77.

Brouwer,

L.E.J.:

1927,

Berliner

Gastvorlesungen,

in

(Brouwer 1992).

Brouwer,

L.E.J.:

1929, 'Mathematik',

Wissenschaft und

Sprache',

Monatshefte

der

Mathematik

36,

153-164.

Reprinted

in

(Brouwer

1975),

pp.

417-428.

Brouwer,

L.E.J.:

1934,

Changes

in

the

Relation between Classical

Logic

and Mathe

matics. In

(van

Stigt

1990),

pp.

453?458.

Handwritten

manuscript,

German

ver

sion

presumably

from

1930-1934.

Brouwer,

L.E.J.:

1947,

'Richtlijnen

der intuitionistische

wiskunde',

KNAW Proceed

ing,

Vol.

51,

p.

339.

English

translation: Guidelines

of

Intuitionistic

Mathematics,

in

(Brouwer 1975),

p.

477.

Brouwer,

L.E.J.:

1948, 'Consciousness,

Philosophy

and

Mathematics',

Proceedings

of

the 10th International

Congress

of Philosophy,

Amsterdam 1948

III,

pp.

1235-1249.

Reprinted

in

pp.

480?196.

Excerpts

reprinted

in

(Benacerraf

and

Putnam

1964),

pp.

78-84.

Brouwer,

L.E.J.:

1952,

'Historical

Background, Principles

and Methods

of

Intuition

ism',

South

African

Journal

of

Science

49,

139-146.

Reprinted

in

(Brouwer

1975),

pp.

508-515.

Brouwer, L.E.J.: 1955, 'The Effect of Intuitionism on Classical Algebra of

Logic',

Proceedings of

the

Royal

Irish

Academy

Section A

57,

113-116.

Reprinted

in

(Brouwer

1975),

pp.

551-554.

Brouwer,

L.E.J.:

1975,

Collected

Works,

Vol.

1.

North-Holland,

Amsterdam.

Edited

by

Arend

Heyting.

Brouwer,

L.E.J..1981,

Cambridge

Lectures

on

Intuitionism

Cambridge.

Manuscript

of

lectures held from 1946-1951. Edited

by

Dirk

van

Dalen.

Brouwer,

L.E.J.:

1992,

Intuitionismus,

B.I.

Wissenschaftsverlag,

Mannheim.

Edited

by

Dirk

van

Dalen.

Brouwer,

L.E.J.:

1993,

in

'Willen,

Weten,

Spreken',

Euclides

9,

177-193.

Also in

De

uitdrukkingwijze

der

wetenschap,

kennistheoretische voordrachten

gehouden

aan

de Universiteit

von

Amsterdam

(1932-1933),

pp.

43-63.

English

translation:

Will,

Knowledge

and

Speech,

in

(van

Stigt

1990),

pp.

418?431.

Excerpts

in

(Brouwer

1975),

pp.

443-146.

Brouwer,

L.E.J.:

1996,

'Life, Art,

and

Mysticism',

Notre Dame Journal

of

Formal

Logic,

37(3),

389-429.

Translated

by

Walter

P.

van

Stigt.

Detlefsen,

M.:

1990,

'Brouwerian

Intuitionism',

Mind

99(396),

501-534.

Ewald,

W:

1996,

From Kant

to

Hilbert: A

Source

Book

in

Mathematics',

Clarendon

Press,

Oxford.

Grabiner,

V:

1974,

'Is

Mathematical Truth

Time-Dependent?'

American

Mathemati

cal

Monthly

81(4),

354-365.

Heintz, B.: 2000, Die Innenwelt der Mathematik. Zur Kultur und Praxis einer bewei

senden

Disziplin, Springer Verlag,

Berlin,

Heidelberg,

New-York.

Heyting,

A.:

1911,

Intuitionism,

an

Introduction,

3rd

edn.

North-Holland,

Amsterdam.



19/19

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Against Against Intuitionism