What are Numbers? Philosophy of Mathematics


Hello and welcome to Elucidations, a philosophy
podcast recorded at the University of Chicago, I’m Matt Teichman & I’m Jamie Edwards.
With us today is Daniel Sutherland, Professor of Philosophy at the
University of Illinois at Chicago. And he’s here to talk to us about
the philosophy of mathematics. Daniel Sutherland, welcome.
Thank you. Thanks for having me. One of the perennial questions in
the philosophy of mathematics is: “What is a number?”
Maybe we can just start there. Well, that’s a very good question.
Numbers appear to be abstract objects. And by ‘abstract object’ philosophers
mean objects that don’t seem to occupy a spatial or a temporal location; and
also a kind of thing that doesn’t enter in causal relations with other things. And
in fact, numbers are one of the prime candidates for abstract objects, should
there be any. Now, various people have been quite puzzled about the
role of abstract objects in general: Whether there are any–there are people
who have been opposed to admitting that there are abstract objects and have tried to
explain away cases of apparent abstract objects, and that’s the case in the
philosophy of mathematics as well. One of the troubling things about allowing
for abstract objects is that it’s hard to see how they fit into our conception of
the world as we understand it today, that is to say, through science, where
we try to explain things through physical laws and causal interactions. And that
has given a lot of people motivation to resist the idea of abstract objects
altogether. Another more pressing problem within philosophy is to explain how it
is that we could have knowledge of abstract objects and their properties
if they really are abstract. If they don’t occupy a spatial & temporal location,
if they don’t enter in causal relations, then how do we have knowledge of them? This issue has actually been
tremendously formative to philosophy itself because it looks like,
typically, the way you know something is to stand in some sort of causal relation
to it. So the paradigm case here would be perceiving the things in the
room that you’re in right now. Light strikes an object, it reflects
off of it, it impinges on your eye, it causes neurons to fire. That’s a more
detailed scientific picture that we have now than was available at an earlier
point of time, but still, the idea is that you enter in some sort of causal
relation to an object, and that’s what justifies your knowledge that, say, for
example, there’s a cup on the table right now. Now if we’re talking about an abstract
object, it is, by definition, something that doesn’t enter into causal relations.
So, how can you have knowledge of it? Plato and other Rationalists thought that
you have some sort of perception of things that are abstract, and described in
perceptual metaphors our knowledge of the Forms — which you may’ve heard
about — and also numbers. These are ideal, abstract objects that we don’t enter
into causal relations with, or at least no direct causal relations with. So
that raises a puzzle: Are you attributing to human beings a distinct mode of
perception? And if so how do you explain it? And that has been a big problem
for those who believe that you have knowledge of abstract objects, is
giving an account of the epistemology. That is to say, the theory of knowledge, giving a
theory of knowledge of these abstract objects. How is it that you can know things about
them? This problem has perplexed people so much that some have looked for
alternative views about what numbers are. They have tried to locate numbers in the
mind or perhaps in the physical world. One way to do this, and this is a
simplified way of thinking about it. You might think that numbers are in the
world in the sense that the number two is really summarizing or somehow an indirect way of thinking about two
things, two things that you experience in the world. That’s to give numbers some
sort of a concrete realization and then what you know is something in the world.
There are much more sophisticated versions of making numbers a part of
the world that I won’t get into. Another idea is to say that numbers are
constructed by us. They’re in some sense just in our minds. And there are
different approaches in the philosophy mathematics corresponding to the
different attitudes about what a number is and theories about how we know
numbers. So, if I step back a bit to what I was saying at the beginning. Some philosophers think that numbers are
abstract objects, and those philosophers have a way of thinking about numbers
that seems very natural. We think of THE number 3, we don’t think of 3
things here, 3 things there. We think that there is THE number 3. And
it also happens to be a way that most mathematicians seem to think about
numbers, that they are objects. And it also fits very neatly with our theories
of truth. I won’t go into the details there. But there’s a way in which, if you
just take numbers as abstract objects, it seems to follow from the way we talk
about numbers that they’re abstract objects, it fits with a very natural conception of
them, and it fits with our theory of truth. But if you take that option, then you
have to explain how we KNOW these abstract objects. So, your work
then is gonna be cut out in the epistemological side. Those who claim
that numbers are really manifested in the world have an easier time
explaining how you know numbers. Oh, you know numbers by seeing
collections of 2 and then maybe you abstract out an idea of 2, and
when you’re talking about 2 + 3=5, that’s shorthand for groups of
2 put together with groups of 3 give you groups of 5.
Something like that. But it makes the epistemology more
tractable. The problem is is giving an account of numbers that really fits the
way we think about numbers, the way we talk about numbers, the way we
theorize about numbers in a way that gives us a comprehensive theory of what
numbers are. And that requires a lot of work. Plus you have to, even in the
epistemology, explain how you get from seeing 2 things to the number 2. And
that turns out not to be as easy as you might think. It looks like it’s a quick
step, but there’s a big difference between having a collection of 2 and
talking about the number 2, and some complicated relations between them.
Would you say just a little more about this? This seems like we would have the
same issue with colors for instance, abstracting the color blue away from
blue things. It seems like this would be a similar step we would have to make.
But people don’t seem particularly worried about making that step with colors, why would we be worried about
making that step with numbers? That’s a very good question. So an answer
to this actually pulls us into questions about the status of what it is that you abstract
out and what you’re talking about then. So some of the same issues may arise
here. There’s two issues here, one is abstraction and then there
is being an abstract object. So you can abstract — there’s the
abstraction process that might get you the idea of the number 2, an abstraction
that gets you the idea of the color blue. Then there’s a further question,
what those correspond to. Is the one representing an abstract object, the number
two? And then, what about the color blue? What is the status of the color blue
whose representation you’ve abstracted from particular instances of blue?
Is that something that exists independently of the things that it’s
instantiated in? So what we’re seeing here is that the issues in philosophy
of mathematics are closely tied to perennial issues in philosophy about
how to think about universals and how to think about the process of abstraction and what the content of our thoughts are
after we’ve engaged in a process of abstraction. So some of the same issues are
on the table. I think in the case of mathematics it’s particularly problematic
because the idea is that we seem to be talking about an abstract object, and
that gives the problem more bite than it might otherwise have. But there are
similar issues in both cases that would need to be addressed. And I’m glad
you raised this because the problem of universals — this classic problem in
philosophy — you have the same kinds of ways of dealing with it that you find in the
philosophy of mathematics. That is to say, some people will say there IS the color
blue. It exists independently of us and abstraction allows us to get an
idea that will allow us to refer to that. That’s somebody who takes the existence of
blue apart from its instantiations quite seriously. Other philosophers might say that
you have an abstraction that gets you a mental representation that doesn’t
correspond to anything in the world, It’s just an idea. That corresponds to one
of the ways of thinking about numbers. And other people will say all you
really have are the instantiations of blue, and the word ‘blue’ or the
idea ‘blue’ is just a way of talking about those instantiations. So there’s an exact
parallel to handling the problem in the case of color to the ways you handle it
in the philosophy of mathematics. So that’s a bit of background about one way
to think about philosophy of mathematics. You get in a dilemma between giving the
objects of mathematics the ontological status they seem to demand and answering
the epistemological question of how we could have knowledge of them. So there’s
a dilemma here. And this dilemma was famously articulated by Paul Benacerraf
in an article. And one of the interesting things about that article is that with
this understanding of the dilemma, the pull on the one hand, toward giving a
metaphysically robust interpretation of what a number is and
the problem of explaining how we can have knowledge of it
gave him a way of reflecting on different schools of philosophy
of mathematics that had arisen. And it looked like the different
schools fell into these different camps: those who take numbers to be objectively
real and independent abstract objects, those who take them to be creations of our mind,
and those who think that they’re in some way manifested in the world. So we’ve
been contemplating three possible responses to the question
“What is a number?” The first answer says that numbers are
abstract objects. They’re not located in space and time, and they don’t have the
sort of causal influence on things that we perceive that things located in
space and time seem to. The second answer to the question “What is
a number?” that we’ve looked at says something like: Numbers are mental
constructs or they’re things in our minds. And then the third answer we’ve been
thinking about says that numbers are somehow in the concrete spatio-temporal
world, maybe they’re physical things even, but they’re things that we directly experience
the same way we experience chairs and tables. What would that answer look like?
I mean, would it be the number 2 is the set of all things of which there are 2?
Or how does that answer work exactly? Like so much in philosophy,
the devil’s in the details, right? It’s one thing to think you just get numbers
from the world by seeing pairs of things in the world and then
you’re home free. The difficult part is in coming up with a theory
that can really make sense of how that is to work. One thing to notice here is that those who
want to make the epistemological problem easier by locating numbers in the world
in some way make knowledge of numbers very much like my knowledge
that there’s a cup on the table here. Whereas, from the get-go, from Plato
and before Plato, there was this sense that mathematical knowledge was quite
special, that we actually do know it in a different way. I said earlier that
somebody who is committed to the idea that numbers are abstract objects is
stuck with having to give an account of how we have knowledge of them, and
that Plato and others following Plato, have resorted to talk of perception.
But of course, perception is like my perception of the cup, so it’s just a
metaphor, and they’ve had trouble spelling out the metaphor. Do we have a special faculty of perception
that allows us to know things? And Plato’s answer was yes. And that was
the answer of other people as well. One of the nice features of
biting that bullet is that it makes mathematical knowledge special in
a way that it does seem to be special. So, mathematical knowledge does not seem to
be dependent on the way the world happens to be. Mathematical knowledge also comes
with what seems to be a necessity that it be true. It isn’t just that
3 + 5=8 happens to be true, in the way it can be that it
happens to be the case that there’s 2 cups next to me
on the table right now. But it seems like it MUST be the case.
There’s a force of necessity there. And it’s also the case that mathematical knowledge, if you think
through things carefully, has a level of certainty that comes with it that has
made a deep impression on people. In fact it made such a deep impression that it’s
shaped all of philosophy. Mathematics provides a paradigm of the best kind of
knowledge we seem to be capable of. It makes us want to know everything or as
much as possible with that same level of certainty. So, this is a very interesting feature
of mathematical knowledge and it was again, I want to emphasize, deeply
influential in the history of philosophy. There’s a special kind of knowledge.
Now, go back to where we were. If you’re going to explain mathematical
knowledge as being based on the things that we experience in the world, you’re
making mathematical knowledge very much like my knowledge of the cup on the
table; and it’s no longer clear how you could explain, for example, that
my knowledge that 2 + 3=5 is actually a knowledge of a
necessary truth. How would you get that from experience by seeing apples
when you were a child or something? There seems to be something very special here.
The justification of mathematical knowledge does not seem to be dependent
on experience, it seems that the propositions of mathematics are necessary, and it seems that we can be more
certain of mathematical claims than a lot or most or all claims that we get
through our senses about the world. So that’s already an issue that’s on
the table once you take this route. John Stuart Mill is one of the most famous
proponents of the idea that you don’t need to have any kind of special faculty
of knowledge to have mathematical knowledge, you get it as a child by
seeing that 2 apples together with 3 apples equals 5 apples. John Stuart
Mill then has the problem of explaining why mathematical knowledge seems to be
knowledge of necessity. And his claim is that you’re not having insight into a
necessary truth, it’s just that you experienced things coming together in
groups in that way with such incredible consistency that it feels necessary
to you. But really it’s of the status of a claim like “there are two cups on the
table right now”. But that’s actually a challenge for the whole approach.
You’re gonna have to explain the apparent necessity of mathematical claims
in a way that seems convincing. Another challenge is really to get an
explanation of how we get from seeing 2 cups on the table to the number 2
in a way that doesn’t already presuppose the number 2, right? It’s very easy to say
that you could see 2 cups on the table and then think of the “two-ness” or
something like that. It’s not so clear that you could really do that unless you
already had the concept of 2 to bring to the table, both figuratively and
literally in this case. So you have to be very careful of how you describe how
if you’re being a strict empiricist and thinking that all your knowledge comes
from the world, how this account is supposed to work. Now, one thing you
might say is, well, there’s this two-ness of the cups here and there’s a two-ness of
the glasses on the table–the sunglasses and the reading glasses–and then there’s
what those two groups have in common. And now you are abstracting out a
representation. A common way of thinking about that is that I’m seeing the set of
2 here and the set of 2 here. But now we have to be very careful because,
remember, that the whole point of this exercise was that you were in
perceptual contact with 2 things here and 2 things there. But
we’ve already made a shift to talking about the SET of 2 here and the set of 2
here. Why is that problematic? On most accounts, a set is an abstract object,
every bit as abstract as the numbers that we were trying to explain in
the first place. So the lesson of this is once you start digging into it, giving an
account of how we know mathematical truths based on experience doesn’t seem
to be as straightforward as you might’ve at first thought. You have to account
for the necessity, the apparent independence of justification on particular
sense experiences. And you can’t, so to speak, cheat and import abstract concepts when
you’re giving an account of how we get the number concepts themselves.
That makes the project interesting, difficult, but interesting. It isn’t fatal to it.
And in fact, there are proponents of all three positions that I described earlier.
I should say though about this view that numbers are somehow manifested or
physical in the world and exist in the world; I’ve given the example of collections
of things in the world and talked about John Stuart Mill, but in fairness,
I should say that there’s a much more sophisticated version of this view
that developed at the beginning of the last century called “Formalism”.
And Formalism attempted to account for mathematical knowledge by interpreting
it as rules for the manipulation of physical symbols on a piece of paper.
That gives Formalism more resources than the view we were talking about earlier
with respect to John Stuart Mill. But it is important to see that some of the
motivations and the, so to speak, location in the classification of possible
responses to the dilemma of mathematical knowledge and the status of
mathematical objects is in that camp still. Another influential approach to the
question “How do we come to know facts about numbers and how do we come to
have knowledge of arithmetic?” is to say that we have knowledge of numbers in whatever
way it is that we have knowledge of logic; of what implies what,
of what follows from what. And this school of thought has been
given the name Logicism. So how does the Logicist view work exactly? Well that’s a very good question, an
important historical development that did impact, deeply impact, our views on the
nature of mathematics, both the nature of mathematical objects and our knowledge
of them. So, as you’re pointing out, Logicism was a project that began at the end
of the 19th century to show that mathematics is really nothing more
than logic. Now, one could have various motivations for doing this, but I could
just say quickly that if you were worried about abstract objects and it turned out
that mathematical knowledge was just logical knowledge of a special form,
you might feel better about the status of mathematics because logic does not
appear to assert the existence of objects. Also, if you were concerned about
the nature of mathematical knowledge insofar as you’re interested in how it
can be justified, and it turned out that mathematics really was based on logic,
then the justification of mathematical truths would just rest on logic. Now, in a
certain way, that just pushes everything back to logic, but it would tell you
something very interesting about mathematics and will tell you something
very interesting about logic, namely you can get mathematics out of logic and
mathematics is nothing more than logic, and so on. This raises a host of issues
even if Logicism were successful. However, Logicism was only partially
successful in the sense that if the project was to show that mathematics
is nothing but logic, that was not a successful project. What was shown was
that mathematics is something that you can obtain from logic plus set theory.
Now recall that a few moments ago I was pointing out that sets are abstract
objects. So you might have all the same worries and concerns that you had before
Logicism was on the table. And in fact, the introduction of set theory really
throws a twist into the whole Logicist project for more than one reason. First of all, set theory itself has
developed into its own field, but set theory appears to be committed to
an awful lot of abstract objects because you can build sets out of sets and so on
and so forth; or anywhere there is a set you might think there’s a set of those
sets. And very quickly you get beyond uncountably many sets and way beyond
that as well. So suddenly, what looked like a very neat reduction of
mathematics to logic seems to depend on something that is immensely rich,
powerful, and clogs the universe with infinite numbers of sets that a lot of
people aren’t too comfortable with. So what looked like a nice streamlined
reduction of mathematics to logic turns out to involve set theory which comes
with its own commitments, its own theory and a tremendous number of
abstract objects all on its own. One thing that is important to
understand is that Logicism arose at the end of the 19th century after a
period in which mathematics was a arthimetized. What do we mean by that? Prior to the 19th century, there was the
study of geometry and then the study of mathematics more generally, and there
wasn’t a unified account of mathematics. Also, very importantly, mathematical
proofs and mathematical methods appeared to rely on intuition taken in a very
broad sense, that is to say, geometrical proofs seem to require that you do
constructions and that you have some sort of insight — again I’m using
intuition in a very broad sense here. Mathematics hadn’t achieved all of the
rigor that it has today, yet there was a push for rigor, in particular, in the
foundations of calculus and that kept pushing mathematicians to think more
and more rigorously about mathematics and try to distinguish good proofs from bad
proofs, and that motivated a lot of mathematicians to try to get rid of any
kind of apparent appeal to intuition and have as strict as possible proofs. All of
that push toward what was called the arithmetization of mathematics in which
you have a theory of numbers that begins with the natural numbers and then
you explain how to, so to speak, build rational numbers or account for rational
numbers in terms of the natural numbers, then you extend also to
negative numbers and so on. Once you do that, you have now made it
possible to try to think of geometry as simply an application of mathematics.
And that gives a priority to numbers and the natural numbers in particular over
all other branches of mathematics, geometry included. That is what I mean
by the arithmetization of mathematics. There was a big drive, as I said, to get
rid of any appeal to intuition and this broad sense and to reduce everything
to logically rigorous proofs. A lot of that was not formally articulated until
the end of the 19th century when a philosopher named Gottlob Frege
develop symbolic logic of a form that we know it today. It was subsequent to the
arithmetization of mathematics that certain philosophers and mathematicians
such as Frege started thinking about what lay at the foundation of the
natural numbers themselves, and led them to the Logicist thesis. The point is that
Logicism could be seen as part of this long path of getting rid of any reference to
intuition at all in mathematics. And if you think that logic is simply a matter of
conceptual knowledge, you have given an account of mathematical knowledge that
does away with any kind of intuition or anything that could give you false
results. That was the ideal that was appealed to here. Given that Logicism
wasn’t successful, given that it depends explicitly on set theory, that gave a lot
of philosophers of mathematics pause about the Logicist project and how we
know mathematical truths after all. Is our mathematical knowledge really just
logical knowledge based on concepts or does it go beyond it in ways that depend
on more than conceptual representation? Now, even independently of Logicism,
but in particular, because of some of the difficulties that it faced, there were
philosophers of mathematics who thought that there must be some role for
intuition in mathematics; again, intuition understood in a broad sense. And Henri
Poincaré was a famous example of this. He thought that mathematics required some
sort of mathematical intuition. Those who are inclined to agree with Poincaré, in one
form or another, often look back to Kant because Kant had insisted that our
mathematical knowledge was not simply conceptual, but required also intuition. Now what’s important to understand here
is that when Kant said ‘intuition’ he wasn’t just talking about some quasi-
perceptual insight into facts or something like that, he had a very
precise notion of intuition that contrasted with concepts. Both
concepts and intuitions are kinds of representation that we have. Conceptual
representation is distinguished from intuitive representation in that the
latter has a, what I will call, quasi- perceptual format. That is to say,
it’s not the kind of thing that you think paradigmatically fits into the form of a sentence where you
break things down into subject and predicate and make a simple assertion
like “the cat is on the mat”. To give an example of what Kant had in mind,
an intuition is like a spatial representation in which you have the
representation of parts next to each other located in space, and that’s a very
different kind of representation than a concept. Kant claimed that all knowledge
requires both concepts and intuitions to work together. Those who recognize
the limitations of Logicism or were in fact opposed to Logicism look back to
Kant to understand why he thought intuition was required in mathematical
cognition and mathematical knowledge. Now I said that Kant says that intuitions
are quasi-perceptual, the reason I say quasi-perceptual is that he doesn’t
think that they are perceptual in the way that most perception is empirical,
the way things [like] my perception of the cup on the table is empirical. Rather,
Kant wanted to allow that some kinds of quasi-perceptual representation are
pure and a priori. Notice that this would make mathematical knowledge very
special and different from my knowledge of the cup on the table, but at the same
time, it has something in common with it because it has some of the features
of perception, the way spatial representations are perceptually
represented for example. The big question then was for those who are looking
back to Kant and are interested in getting at the bottom of Kant’s views is “How
does intuition play into our mathematical knowledge in the sense of intuition that
Kant is interested in?”. So there is still a very live question today, 200 years
after Kant, about what is the nature of mathematical cognition, where the form
of the question is what sorts of concepts or quasi-perceptual representations are
presupposed by mathematical knowledge. And that means that Kant’s views are still
of very much of interest today because even though mathematics is developed
tremendously since Kant’s time, we still do not have a clear theory about
how we have mathematical knowledge. Daniel Sutherland, thank you very much for
joining us. Thank you, thanks for having me.