
Fictionalism in the Philosophy of Mathematics [pdf] - monort
http://www.colyvan.com/papers/fictionalism.pdf
======
__mbm__
It is interesting to interpret the body of mathematics as you would a
collection of fictional tales. However, the philosophy begins to unravel (to
me) when it asserts that "8 is larger than 5" is false while "Sydney is larger
than San Francisco" is true because the latter statement "has referents".

What is it that makes Sydney and San Francisco real objects with meaningful
sizes while 8 and 5 are not real and do not have meaningful sizes? Sydney and
San Francisco are defined by political and legal "stories" in the same way
that 8 and 5 are defined in mathematical "stories". The theory only seems to
be consistent if all out-of-context falsifiable statements are taken to be
false.

This theory placates me, since it leaves the truth value of mathematical
statements (in the context of the mathematical story) to mathematicians.
However, it renders any conclusions meaningless to mathematics, even if it is
meaningful for a philosophy dealing with human stories.

~~~
Retra
I've always considered true and false to be properties of models, which are
(necessarily) approximations of some underlying reality. So I'd say "8 is
larger than 5 _in some sense_ " and "Sydney is larger than San Francisco _in
some sense_ ", where we might admit that the senses differ, and maybe 8 > 5 is
'truer' because the senses in which it are true are more general and require
fewer experiences to verify.

But at the end of the day, you don't want to just say "everything can be true
if you stretch far enough", you want to say that things are true only when
we've demonstrated some utility in saying that they're true. So we just defer
the issue: if you want to say something is true, you always need to know what
difference it makes.

~~~
__mbm__
> if you want to say something is true, you always need to know what
> difference it makes.

Absolutely. It seems that fictionalism avoids both the "in some sense"
qualifications and worrying about what difference it makes by asserting that
statements out of context are simply false. While consistent, I'm similarly
not convinced that its useful.

------
KKKKkkkk1
I'm having trouble understanding the significance of Hartry Field's work. It
sounds like he replaced the axiomatic system of what we call mathematics with
an axiomatic system that he developed specifically for Newtonian gravity, and
based on this he was able to claim that mathematics is dispensable. As an
undergrad in logic class, I got the impression that mathematics is the
discipline that studies axiomatic systems, so if you build an axiomatic
system, you're doing mathematics. If that's true, then isn't all he did is
just redefining the word mathematics in a very narrow sense and then
dispensing with that narrowly defined notion?

~~~
hyperpape
The question concerns the quantifiers that you see in math ("there exists an
even prime number", "all numbers have a unique prime factorization"), etc.

Field shows that for the theories necessary for Newtonian mathematics, we can
treat those as quantifying over points in space-time, as opposed to anything
distinctively mathematical. So we avoid Platonism.

It sounds like you, like most people who haven't studied philosophy of math,
assume some sort of pseudo-formalist account, and therefore don't find the
question very compelling. Please don't take that disparagingly (most
mathematicians are in the same boat, and I'm not sure I have much of an
opinion at this point, save to note that people who haven't studied philosophy
tend to think the issue is obvious in a way that a lot of philosophers don't).

~~~
samth
I think most mathematicians are more likely to be Platonists than formalists.
Formalism is also a real position in philosophy of math, but a minority one
for I think two reasons. One is that the math we've developed seems
significant enough and effective enough to deserve more explanation than just
"it's one of many possible formal systems". Second, if you take a
straightforward formalist approach, then there aren't really any interesting
questions in philosophy of math, so people who think that are less likely to
become philosophers of math.

~~~
cokernel
It's possible that a mathematician with philosophical commitments is more
likely to be a Platonist than a formalist, but I suspect that most
mathematicians are neither ("Philosophy is for the philosophers"). The so-
called "naive" view that most mathematicians may be described as having ("I'm
discovering something real when I prove a theorem", "Mathematics is
objective", and so on) certainly sounds similar to the language used by
Platonists, but I have yet to run into a mathematician who is comfortable with
the logical conclusions of a strict Platonism. In particular, there are
epistemological problems with Platonism (how can we know anything about non-
physical entities?) that cause problems in the justification of mathematics
and would be nice to be able to ignore.

~~~
samth
I think we mostly agree on the empirical issues -- mathematicians usually
don't care about the philosophical issues, and espouse naive Platonism which
they haven't thought through.

However, I don't think it's right to say that they're formalist just because
they wouldn't agree, upon reflection, with Platonism's conclusions. I (like a
lot of computer scientists) am a formalist, but it's not fair to say that
other people secretly agree with me because I think Platonism has major flaws.

------
jules
Does this represent the state of philosophy of mathematics, because I find
this view quite naive. Mathematics is in the business of making a model of the
real world and then making falsifiable predictions with it, just like science.
Take Fermat's last theorem: for all positive a,b,c and n>2, the value a^n +
b^n - c^n never comes out 0. This is an empirically falsifiable statement. You
can even make it a statement about real world objects if you wish: represent a
number by a jar of coins, and do addition C=A+B by filling jar C with the
coins in jar A and jar B together, multiplication AB by taking a whole jar B
for each coin in A, etc. Rules of mathematical deduction are just devices for
making predictions. So mathematical objects are neither "real" as in
platonism, nor meaningless as in fictionalism.

~~~
roywiggins
Embodied mathematics is what you are describing I think. The idea is that
mathematics arises out of ordinary experience extended by metaphor. It charts
a pretty satisfying middle ground between Platonism and Fictionalism.
Mathematics is a story, but it is "at the bottom" tied into our experience of
everyday reality. Do real numbers exist? Probably not, but the the continuum
is like a long stick you can make marks on.

Can you have a trillion apples, so you know that a trillion + a trillion is
two trillion? No, but you can reason about "trillion" as existing
metaphorically.

~~~
hyperpape
That actually just sounds like good old fashioned nominalism or fictionalism--
you'd have to sketch out a more detailed story, but I don't think it's
actually distinct.

~~~
roywiggins
I'm (probably badly) summarizing Lakoff & Nunez's book:

[https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From](https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From)

Neither are capital-P Philosophers, and I don't remember if they actually make
a claim that their ideas are philosophically distinct from fictionalism-
they're more worried about platonists!

The other approach I'm fond of is saying mathematical objects exist but only
inside the heads of mathematicians. As long as mathematicians share a
particular idea and can talk about it, it's math. So when I point to the real
numbers, I'm pointing at a real thing: the shared idea of them. There's no
Platonic realm, but they exist in the mind. It parallels how mathematics is
really done (a proof is only a proof if it convinces mathematicians) so I find
it compelling on that front.

------
wyager
There's a bit of a litmus test you can apply to belief systems to find out if
they're definitely objective.

An intelligent alien species that's never met humans would almost certainly
invent math. The syntax and organization would probably be different, but the
rules would be the same.

On the other hand, aliens would almost certainly not write The Hobbit.

~~~
calibraxis
I don't know, it may be the current intellectual culture's speciesism that we
often think that. For instance, they may have many modes of perception better
than math. Or even if they have a "math" sense that's oddly like ours, they
may be unaware of it because it's unconscious... in their underlying
circuitry... just like we're not built to directly introspect many of our own
capacities.

We may even be in constant contact with "aliens" now, but unable to perceive
them meaningfully. Like most lifeforms maybe couldn't tell us apart from a
rock or rubber.

------
Houshalter
This is absurd. Clearly mathematical statements are highly predictive of the
real world. They aren't false.

These people seem confused over the meaning of the word "exist". Regardless of
whether or not numbers "exist", we can show that objects in the real world can
obey the same laws as abstract numbers. If I have 2 apples, and take 2 more
apples, I will never have 5 apples. The properties of math are real and apply
to the real world.

If you insist on modelling philosophy on the language we happen to use, then
just treat numbers as adjectives. As if 5 is a property an object can have,
rather than a physical object itself. You don't need to worry about 2
"existing" any more than worrying about "tallness" exists, when talking about
objects that are taller than other objects.

~~~
abrezas
You think that they are confusing the term exists but the reality is that what
"exists" means is very hard to define.

Where is the line between abstract objects and the real world you mentioned?
Can you give a rule that separates the two?

Do black holes exist? Electrons? Magnetic fields? Is one of these a model but
doesn't _really_ exist?

~~~
Houshalter
Arguing about the definition of a word is a huge red flag that a discussion is
going nowhere. Sadly a lot of philosophy seems to be basically arguing about
words.

>Do black holes exist? Electrons? Magnetic fields?

Yes. I mean we might not know what they are exactly or how they work, but they
are clearly real phenomena that we can observe. In the same sense, "twoness"
is a real property that a set of objects can have.

------
MaysonL
Whatever one chooses to say about the truth or falsity of mathematical
statements (& imo, calling them all false is ludicrous), it is hard to refute
the argument that they are quite _effective_. See Hamming and Wigner[0,1]

[0][http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.htm...](http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html)

[1][https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...](https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html)

------
catnaroek
> Fictionalism in the philosophy of mathematics is the view that mathematical
> statements, such as ‘8+5=13’ and ‘π is irrational’, are to be interpreted at
> face value and, thus interpreted, are false.

Arrrgh! So annoying! How much time will it pass until philosophers of
mathematics finally understand that mathematical truth has nothing to do with
philosophical truth?

> Fictionalists are typically driven to reject the truth of such mathematical
> statements because these statements imply the existence of mathematical
> entities, and according to fictionalists there are no such entities.

Crash course in logic: If mathematical objects don't exist, then statements
about them aren't “false” - they're _meaningless_.

~~~
cokernel
A disclaimer: I am not currently a fictionalist, but I find it remarkably
interesting to consider.

Fictionalists are not making claims about mathematical truth; they are making
claims about philosophical truth. When Colyvan (who I believe is not a
fictionalist) describes fictionalism as an error theory of mathematical
discourse, he does not mean that mathematicians incorrectly assign the label
of mathematical truth to statements that are mathematically false, but rather
that it is an error to conflate mathematical truth ("true in the story of
mathematics") with philosophical truth. In other words, you and the
fictionalists are in agreement on your first point.

I don't quite understand your second point, or its justification. For example,
would you consider the statement "All even primes greater than 2 are divisible
by 1001" to be true, false, or meaningless? I think most people will agree
that even primes greater than 2 do not exist. Fictionalists would assert that
this statement is false; with my background in formal logic, I tend to believe
that this statement is vacuously true. I would be interested in seeing an
argument for the claim that this statement is meaningless.

~~~
techbio
Even while Captain Obvious is not wrong, eyes will rightly glaze over.

Double negative--I'm not a mathematician, I paint, but Mom taught me right and
wrong, two of the latter do not make a right.

"All even primes greater than 2 are divisible by 1001." => "everything else in
this analysis will be ignored"

(because it's not math)

~~~
cokernel
Could you clarify who "Captain Obvious" is?

~~~
techbio
The Wikipedia link is the reference I intended. "Captain Obvious" refers to
the one in a group or conversation who states what everyone else knows, and as
such impedes the group's progression toward their shared goals.

Thanks, HN user "douche", you nailed it.

Strange to me how folk wisdom is so unwelcome in a philosophy discussion.
While not explicitly stated, the subtext appears to be putting my comment as
the "Captain Obvious." I'll avoid commenting in this context in the future.

------
force_reboot
Fictionalism, and other formalist theories, will have to confront that problem
that when we speak of mathematics as abstract rules governing strings of
symbols, these rules themselves are mathematical. So it only replaces "numbers
are real" with "abstract symbols are real". There are axiomatic systems that
are strong enough to express manipulations of abstract symbols, but weaker
than the usual systems that mathematicians deal with (e.g. see the work of
Edward Nelson on so called predicative arithmetic). But to my knowledge these
have not been explored much.

