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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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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
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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.
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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:
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AGAINST AGAINST
INTUITIONISM
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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
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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
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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.
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DIRK
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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
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AGAINST
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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)
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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:
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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
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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.
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AGAINST AGAINST
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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,
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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
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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
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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.
This content downloaded from 138.26.16.5 on Sun, 14 Sep 2014 11:44:58 AMAll use subject to JSTOR Terms and Conditions
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8/11/2019 Against Against Intuitionism
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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.
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