80
The
Ontology
of
Mathematical
Practice
Oswaldo
Chateaubriand
Pontifícia
Universidade
Católica
do
Rio
de
Janeiro
(PUC-‐Rio)
Conselho
Nacional
de
Desenvolvimento
Científico
e
Tecnológico
(CNPq)
ochateaubriand@gmail.com
Abstract:
As
Bernays
maintained
in
his
famous
1935
article
“On
Platonism
in
Mathematics”,
in
mathematical
practice
objects,
functions,
relations,
properties,
structures,
etc.
are
treated
as
entities
that
exist
independently
of
our
discourse
and
of
our
constructions.
Bernays
maintains
that
this
form
of
platonism
is
essentially
a
manner
of
speaking,
which
does
not
involve
a
commitment
to
a
strict
form
of
platonism.
In
this
note
I
defend
Bernays’
position
and
propose
a
more
systematic
formulation
for
this
form
of
platonism,
combining
it
with
ideas
of
Frege
and
of
Gödel.
Key-‐words:
platonism;
Paul
Bernays;
mathematical
practice.
In
the
XIII
Colóquio
Conesul
I
gave
a
talk
defending
the
idea
that
the
ontology
of
mathematics
can
be
conceived
as
an
ontology
of
properties,
with
everything
else
characterized
in
terms
of
properties.
I
gave
a
similar
talk
at
the
IHPST
(Institut
d’Histoire
et
Philosophie
des
Sciences
et
des
Techniques)
when
I
was
in
Paris
in
2010,
which
led
to
a
big
discussion
about
mathematical
objects,
because
objects
are
seen
as
a
clear
aspect
of
the
ontology
of
mathematics
and
it
is
not
clear
how
to
deal
with
them
in
terms
of
properties.
Today
I
was
invited
to
speak
about
some
topics
related
to
mathematical
practice,
so
I
thought
I
would
speak
again
about
platonism,
but
from
a
somewhat
different
perspective.
Notae
Philosophicae
Scientiae
Formalis,
vol.
1,
n.
1,
p.
80
-‐
88,
abril
2012.
81
In
his
1935
paper
“On
Platonism
in
Mathematics”,
Bernays
discusses
a
tendency
of
mathematical
practice
“which
…
consists
in
viewing
the
objects
as
cut
off
from
all
links
with
the
reflecting
subject”
(p.
275).
After
outlining
this
tendency
he
goes
on
to
say
that
he
has
“set
forth
only
a
restricted
platonism
which
does
not
claim
to
be
more
than,
so
to
speak,
an
ideal
projection
of
a
domain
of
thought”
(p.
277).
And,
he
claims,
this
in
1935,
“platonism
reigns
today
in
mathematics”
(p.
276).
Although
I
am
not
a
practicing
mathematician,
or
a
historian
of
mathematics,
I
think
we
can
characterize
present
day
mathematical
practice
as
a
form
of
platonism
uncommitted
to
a
philosophical
analysis
as
to
what
one
is
talking
about.
It
is
sometimes
said
that
mathematicians
are
more
formalist
than
platonist,
but
this
does
not
seem
to
be
true.
I
think
the
basic
intuitions
are
the
platonistic
intuitions,
although,
in
fact,
mathematicians
do
not
want
to
dwell
on
the
philosophical
questions.
They
do
not
want
to
ask
what
an
object
is,
or
what
a
property
is,
or
what
other
mathematical
entities
are
from
an
ontological
viewpoint.
They
only
want
to
ask
these
questions
with
respect
to
mathematics
itself,
not
in
terms
of
some
prior
philosophical
conception.
So,
what
I
will
try
to
do
in
my
talk
is
to
reformulate
some
of
my
ideas
in
these
terms.
The
reason
I
defend
the
notion
of
property,
independently
of
asking
questions
as
to
the
nature
of
properties,
is
that
it
has
a
very
clear
basis
both
in
practice
and
in
our
way
of
thinking.
Properties
are
expressed
by
predicates,
and
it
does
not
matter
what
they
are.
What
we
do
when
we
characterize
a
predicate,
i.e.,
when
we
give
the
semantic
content
of
a
predicate—be
it
a
singular
predicate
or
a
relational
predicate—is
to
give
conditions
of
applicability
for
that
predicate.
In
mathematics
these
characterizations
are
very
precise—as
opposed
to
ordinary
predicates,
where,
in
many
cases,
the
characterizations
are
rather
imprecise.
I
think
every
mathematician
would
agree
that
these
predicates
express
mathematical
properties.
If
one
tries
to
prod
them,
however,
and
ask
“What
do
you
mean
by
a
property?”,
one
easily
sees
that
it
is
not
the
kind
of
question
they
want
to
get
into.
They
do
not
want
to
discuss
what
they
mean
by
“property”,
but
Notae
Philosophicae
Scientiae
Formalis,
vol.
1,
n.
1,
p.
80
-‐
88,
abril
2012.
82
they
probably
will
say
that
a
property
is
what
is
expressed
by
a
predicate:
what
I
define,
what
I
formulate,
and
so
forth;
these
are
properties.
The
problem
I
saw
in
my
platonistic
talk
is
that
mathematicians
also
talk
about
objects,
structures,
spaces,
etc.,
and
one
of
the
questions
that
can
be
raised
is
what
is
meant
by
“object”.
One
wants
to
talk
about
objects
in
such
a
way
that
one
is
not
committed
to
giving
a
philosophical
analysis
of
what
an
object
is.
But
we
want
to
say
that
numbers
are
objects;
many
people
want
to
say
that
sets
are
objects;
they
want
to
say
that
spaces
are
objects,
and
so
on.
These
are
all
kinds
of
objects
one
talks
about
in
mathematics,
and
how
should
we
conceptualize
that?
Well,
there
is
a
sense
in
which
it
is
rather
clear,
and
that
is
when
we
talk
about
a
domain
of
discourse.
Even
functions
can
be
treated
as
objects
if
they
are
our
domain
of
discourse.
This
is
an
idea
Frege
elaborated
in
some
ways,
even
before
the
discussion
he
has
in
the
early
90’s.
In
The
Foundations
of
Arithmetic
(p.
77
n.
2)
he
has
a
characterization
of
objects
and
concepts
according
to
which
an
object
is
that
which
can
be
subject
of
a
singular
proposition,
and
a
concept
is
that
which
can
be
a
predicate
of
such
a
proposition.
So,
he
characterizes
objects
as
subjects
of
discourse.
Something
that
can
be
a
subject
is
what
he
is
going
to
call
an
“object”.
That
is
one
of
the
reasons
he
wants
to
say
that
numbers
are
objects,
because
we
refer
to
numbers
as
subjects.
We
say
“the
number
1
is
the
smallest
positive
integer”,
“the
number
9
is
odd”,
and
so
on.
And,
of
course,
we
are
using
those
expressions
in
subject
position.
So,
in
that
context,
at
any
rate,
Frege
treats
the
notion
of
object
as
the
notion
of
what
can
be
talked
about
as
the
subject
of
an
assertion,
or
the
subject
of
a
predication.
(Of
course,
it
could
also
be
a
relational
proposition.)
When
he
gets
to
his
later
papers
“On
Concept
and
Object”
and
“Function
and
Concept”
he
seems
to
want
to
introduce
a
more
ontological
notion
of
object
and
concept,
and
then
he
comes
up
with
things
like
“objects
are
saturated
and
functions
are
unsaturated”.
And
then,
of
course,
he
gets
into
a
puzzle
(for
which
he
was
so
thoroughly
criticized)
because
he
says
that
in
“the
concept
horse
is
easily
attainable”
one
is
not
talking
about
a
concept
but
about
an
object,
Notae
Philosophicae
Scientiae
Formalis,
vol.
1,
n.
1,
p.
80
-‐
88,
abril
2012.
83
because
“the
concept
horse”
is
subject
of
the
singular
proposition.
People
have
criticized
him
for
that,
but
perhaps
his
mistake
was
to
ontologize
the
distinction.
In
fact,
Frege
has
no
ontological
analysis
of
objects—or
of
functions,
for
that
matter.
When
he
characterizes
functions
he
does
so
in
terms
of
an
expression
where
one
can
substitute
certain
names;
and,
he
says,
if
I
substitute
these
names
by
other
names,
then
I
have
the
same
function.
And
what
does
he
say
about
objects?
He
says
an
object
is
anything
that
is
not
a
function.
And
that
is
the
only
characterization
he
gives.
So,
perhaps
Frege
made
a
mistake
in
trying
to
ontologize
the
distinction
of
object
and
function.
Maybe
he
should
have
stayed
with
the
original
distinction
in
terms
of
subject
and
predicate.
He
would
have
avoided
a
lot
of
problems.
Nobody
can
object
to
“the
concept
horse”
being
the
subject
of
the
predication
“is
easily
attainable”,
but
if
one
says
that
it
is
an
object,
in
some
sense
ontological,
then
it
seems
problematic.
So,
I
want
to
say
that
in
the
practice
of
mathematics
people
talk
about
objects
essentially
in
that
Fregean
sense.
Whatever
is
a
subject
we
can
say
it
is
an
object.
Whatever
is
part
of
the
domain
of
discourse
we
are
talking
about
we
can
say
it
is
an
object.
Of
course,
we
can
talk
about
functions,
about
spaces,
about
all
kinds
of
things,
but
as
they
appear
in
the
subject
position,
we
can
say:
well,
I
am
treating
those
as
part
of
the
objects
I
want
to
talk
about.
This
may
seem
odd,
but
we
do
it
in
practice.
In
fact,
it
is
very
interesting
that
when
we
discuss
structures
for
first-‐order
logic
(or
for
logic
in
general),
the
elements
of
the
structure
can
be
anything.
Although
there
is
an
old
idea
that
the
elements
of
a
structure
are
individuals,
we
can
have,
for
instance,
a
first-‐order
theory
of
properties,
where
the
properties
are
the
elements
of
the
structure
and
are
the
objects
of
the
structure.
My
point
is
that
in
practice
being
an
object
is
just
being
a
subject
of
some
kind
of
discourse.
Of
course,
something
can
occur
both
as
subject
and
as
predicate,
even
numerical
terms.
As
Frege
himself
pointed
out,
we
assert
of
some
property
that
it
applies
to
three
things,
or
that
it
applies
to
five
things,
and
these
are
second-‐
order
predications
about
first-‐order
properties.
We
can
also
assert
that
a
Notae
Philosophicae
Scientiae
Formalis,
vol.
1,
n.
1,
p.
80
-‐
88,
abril
2012.
84
property
applies
to
finitely
many
things,
or
to
infinitely
many
things,
and
these
are
also
numerical
predications—or
cardinality
predications—about
properties.
Thus,
numbers
can
also
appear
as
predicates
and
not
only
as
subjects.
My
idea
for
getting
out
of
having
objects
and
having
only
properties
was
essentially
this:
characterize
the
numbers
in
terms
of
properties
and
say
this
is
what
numbers
really
are.
But
I
think
that
from
the
point
of
view
of
mathematical
practice,
as
far
as
I
understand
it,
one
does
not
get
into
these
questions.
We
say
that
sets
are
objects,
but
if
we
start
asking
ourselves
why
sets
are
objects,
there
does
not
seem
to
be
a
clear
answer.
Why
are
sets
objects?
Why
aren’t
they
something
else?
Sets
are
treated
sometimes
as
characteristic
functions;
so
why
is
a
set
an
object
and
not
a
function?
Also,
a
set,
in
an
arbitrary
sense,
is
a
selection
of
things,
and
why
isn’t
that
selection
a
function
rather
than
an
object?
It
seems
that
sets
are
objects
because
in
doing
set
theory
we
treat
them
in
subject
position.
And,
of
course,
we
have
relations
among
sets,
and
operations
on
them,
and
so
on.
In
the
case
of
structures
there
is
a
tendency
of
viewing
structures
as
the
fundamental
notion
of
mathematics.
This
has
been
a
recent
trend:
all
mathematics
deals
with
structures.
I
agree
this
is
a
fundamental
notion
of
mathematics,
but
I
do
not
think
one
can
characterize
everything
in
terms
of
structures.
Structures
appear
often
as
objects,
but
they
also
appear
in
a
very
general
way.
We
may
talk
about
a
particular
structure—say,
a
specific
group—as
an
ordered
pair
(say)
of
a
set
and
an
operation,
but
we
may
also
talk
about
the
group
structure
more
generally.
In
fact,
it
seems
perfectly
acceptable
to
talk
about
the
group
structure.
But
what
does
it
mean?
What
is
the
group
structure?
The
group
structure
is,
in
a
sense,
a
group
function,
a
group
operation.
That
is
what
all
groups
have
in
common;
an
operation
that
behaves
in
a
certain
way.
It
seems,
therefore,
that
all
these
notions
appear
in
mathematical
practice
in
a
way
that
is
not
ontologized.
They
are
discussed,
they
are
used,
but
they
are
not
treated
ontologically
in
a
philosophical
sense.
This
is
related
to
what
I
was
arguing
in
the
XIII
Conesul,
but
in
that
talk
I
was
trying
to
make
a
reduction,
Notae
Philosophicae
Scientiae
Formalis,
vol.
1,
n.
1,
p.
80
-‐
88,
abril
2012.
85
which
I
am
not
trying
to
do
now.
The
question
that
comes
up
now
is
about
the
kind
of
epistemology
that
goes
together
with
this
practice
of
talking
about
objects
and
properties
in
a
platonistic
but
uncommitted
sense.
When
Bernays
wrote
his
paper
and
characterized
the
restricted
form
of
platonism,
he
also
said
that
there
is
a
robust,
full-‐fledged,
platonism
which
is
what
has
been
shown
to
be
untenable
by
the
paradoxes.
Eventually,
however,
he
goes
back
on
that
when
he
reviews
Gödel’s
paper
“Russell’s
Mathematical
Logic”
in
The
Journal
of
Symbolic
Logic.
There
he
says
(p.
75)
that
full-‐fledged
platonism
leads
to
paradox
only
if
it
is
combined
with
an
aprioristic
epistemology.
If
the
epistemology
is
not
aprioristic,
as
Gödel’s
is
not,
then
the
situation
is
different.
Gödel
is
a
very
famous
full-‐fledged
platonist,
but
he
also
considers
the
possibility
of
separating
a
realistic
platonism
from
a
more
restricted
form.
In
fact,
there
is
a
very
explicit
passage
in
the
appendix
he
wrote
for
the
paper
on
the
continuum
problem,
where
he
says
(p.
272):
The
question
of
the
objective
existence
of
the
objects
of
mathematics
(which,
incidentally,
is
an
exact
replica
of
the
question
of
the
objective
existence
of
the
outer
world)
is
not
decisive
for
the
problem
under
discussion
here.
The
mere
psychological
fact
of
the
existence
of
an
intuition
which
is
sufficiently
clear
to
produce
the
axioms
of
set
theory
and
an
open
series
of
extensions
of
them
suffices
to
give
meaning
to
the
question
of
the
truth
or
falsity
of
propositions
like
Cantor’s
continuum
hypothesis.
Moreover,
one
of
the
aspects
of
his
epistemology,
emphasized
by
Bernays,
is
to
suggest
that
we
can
evaluate
mathematical
propositions
by
looking
at
their
consequences—especially
consequences
in
number
theory,
which
he
compares
with
the
most
elementary
domain
of
clear
intuition.
So,
if
one
has
principles
that
have
new
consequences
in
number
theory,
this
brings
evidence
for
those
principles
even
if
they
do
not
have
intrinsic
evidence.
And
that
is
the
reason
he
compares
the
mathematical
situation
with
the
physical
situation.
He
says
explicitly
(p.
265)
that
the
reason
one
accepts
a
mathematical
principle
can
be
the
same
reason
one
accepts
a
physical
law;
because
there
are
so
many
verifiable
consequences
of
the
principle—verifiable
in
mathematics,
not
somewhere
else.
Gödel
is
thinking
of
something
like
number
theory
as
Notae
Philosophicae
Scientiae
Formalis,
vol.
1,
n.
1,
p.
80
-‐
88,
abril
2012.
86
corresponding
to
the
domain
of
perception,
but
the
epistemological
claim
is
much
more
general
than
that,
and
I
think
the
example
of
the
axiom
of
choice
is
an
excellent
example.
As
long
as
one
is
talking
about
classical
mathematics,
there
is
very
little
doubt
that
people
accept
the
axiom
of
choice
today
in
their
everyday
work.
The
reason
for
this
is
that
the
axiom
of
choice
has
many
consequences
in
every
domain
of
mathematics
and
that
these
consequences
have
been
verified
in
those
domains.
Even
the
results
that
were
thought
to
be
odd,
such
as
the
Tarski-‐
Banach
paradox,
were
diagnosed
as
such
because
the
decompositions
effected
by
the
axiom
of
choice
are
not
at
all
like
ordinary
geometric
decompositions.
The
conclusion
to
which
all
the
verifiable
consequences
and
explanations
led
was
that
there
is
clear
indirect
evidence
of
this
kind
for
accepting
the
axiom
of
choice,
and
that
is
exactly
what
happened
in
practice.
Of
course,
a
constructivist
can
have
reasonable
objections
to
the
axiom
of
choice,
but
now
we
are
getting
into
a
specific
conception
of
mathematics,
which
I
do
not
think
is
the
normal
practice
of
mathematics.
Gödel
also
thinks
that
the
axiom
of
choice
has
an
intrinsic
justification.
As
far
as
I
know,
Gödel
was
the
first
person
to
make
the
distinction
between
the
logical
conception
of
set,
which
he
attributed
to
Frege,
and
the
mathematical
conception
of
set,
which
he
attributed
to
Cantor.
The
logical
conception
is
the
conception
of
dividing
the
totality
of
things
into
all
those
that
have
a
certain
property
and
those
that
do
not.
This
is
Frege’s
conception
of
extensions,
which
led
to
paradox.
The
other
conception,
which
Gödel
conceptualizes
in
terms
of
the
operation
“set
of”,
consists
in
starting
with
a
well-‐defined
totality
of
things
and
“generating”
(in
a
manner
of
speaking)
sets
of
these
things,
and
sets
of
the
things
thus
generated,
and
so
on.
This
mathematical
conception,
in
terms
of
selections,
does
not
give
rise
to
any
paradox,
unless
one
makes
the
mistake
of
supposing
that
the
process
has
to
come
to
an
end
in
a
set
of
all
sets.
Moreover,
Gödel
sees
the
axiom
of
choice
as
being
part
of
the
very
idea
of
these
selections.
Notae
Philosophicae
Scientiae
Formalis,
vol.
1,
n.
1,
p.
80
-‐
88,
abril
2012.
87
He
says,
in
fact,
that
the
axiom
of
choice
and
the
axiom
of
definable
subsets
are
the
axioms
that
characterize
the
iterative
conception.
I
think
that
in
practice
this
epistemological
conception
is
the
main
conception
used
by
mathematicians,
though
I
do
not
think
it
is
conceptualized
in
exactly
the
same
way
it
is
conceptualized
by
Gödel.
One
thing
I
have
found
out
by
talking
to
my
colleagues
in
the
mathematics
department
is
that
the
idea
many
logicians
have
that
mathematicians
work
in
terms
of
axiomatic
systems
is
just
not
true.
Mathematicians
do
not
work
axiomatically.
In
fact,
in
most
cases
where
one
has
axioms,
the
axioms
are
really
definitions:
groups,
rings,
fields,
topological
spaces,
etc.
And
in
other
cases
mathematicians
do
not
work
from
axioms;
in
fact,
they
often
do
not
even
know
what
the
axioms
are—I
suspect
that
a
lot
of
mathematicians
do
not
know
the
Peano
axioms,
for
instance.
They
work
from
what?
They
work
from
practice,
from
a
common
tradition
of
working
in
analysis,
or
in
number
theory,
or
mixing
things
up,
or
doing
many
different
things.
So,
just
as
there
is
not
an
ontology
separate
from
mathematical
practice,
I
think
there
is
not
an
epistemology
separate
from
mathematical
practice.
And
again,
if
one
tries
to
talk
epistemology
to
mathematicians,
they
tend
to
blank
out—although
they
may
be
polite
about
it.
Talking
about
epistemological
issues
or
ontological
issues
is
not
the
kind
of
question
they
want
to
discuss;
and
they
do
not
think
it
is
relevant
to
discuss
it
for
their
work.
My
impression,
then,
is
that
Bernays’
idea
that
his
restricted
form
of
Platonism
reigns
in
mathematics
is
still
true
today.
Following
a
suggestion
of
José
Ferreirós
at
the
end
of
my
talk
we
may
say
that
what
reigns
in
mathematics
is
a
methodological
platonism.
Notae
Philosophicae
Scientiae
Formalis,
vol.
1,
n.
1,
p.
80
-‐
88,
abril
2012.
88
References
BENACERRAF,
P.
and
PUTNAM,
H.
(Eds.)
(1964)
Philosophy
of
Mathematics:
Selected
Readings.
Prentice-‐Hall,
New
Jersey.
BERNAYS,
P.
(1935)
“On
Platonism
in
Mathematics”.
In
Benacerraf
&
Putnam,
274-‐286.
_____.
(1946)
Review
of
K.
Gödel
“Russell’s
Mathematical
Logic”.
The
Journal
of
Symbolic
Logic,
v.
11,
75-‐79.
FREGE,
G.
(1884)
The
Foundations
of
Arithmetic:
A
Logico-‐Mathematical
Enquiry
into
the
Concept
of
Number.
Blackwell,
Oxford,
1950.
GÖDEL,
K.
(1944)
“Russell’s
Mathematical
Logic”.
In
Benacerraf
&
Putnam,
211-‐
232.
_____.
(1947/1964)
“What
is
Cantor’s
Continuum
Problem?”
In
Benacerraf
&
Putnam,
258-‐273.
Notae
Philosophicae
Scientiae
Formalis,
vol.
1,
n.
1,
p.
80
-‐
88,
abril
2012.