Episode Transcript
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Speaker 1 (00:08):
We know a lot of things about the universe. We
know that everything around us is made of tiny little
particles that obey strange quantum rules. We know that our
planet moves through space curred by the mass of the Sun.
We know that the earth is four and a half
billion years old and that the universe is almost fourteen
billion years old, and all of that knowledge has something
(00:31):
in common. It's all expressed in terms of mathematics. Our
quantum theories, our ideas about gravity, our understanding of the
age of the earth and the universe all depend deeply
on math. And if we're going to dig deep into
the foundations of reality and see if we understand what's there,
shouldn't we do the same thing and ask ourselves some
(00:53):
hard questions about the mathematics, asking what it is, why
it works and whether it's even necessary? If math is
a language of physics, then how certain are we that
it reflects something true about the universe rather than something
about our minds? Hi, I'm Daniel. I'm a particle physicist
(01:28):
and a professor at UC Irvine, and I love speaking
the language of math. Sometimes, when I'm confused about how
something works, it's the math that leads me through and
shows me the answer. There's something wonderfully crisp about mathematics.
I love how the Patterns Click together with an exactness
and reliability. I love that it doesn't zag or break
(01:50):
or wilt, that six times nine is the same today
as it is tomorrow and will be forever and welcome
to the podcast Daniel and Jorge Explode in the universe,
where we dig deep into the universe around us and
try to find some answers. We have an unquenchable thirst
to understand and an insatiable appetite for asking questions. My
(02:12):
friend and Co host, Jorge is on vacation and so
today we are going to ask some of the deepest
of questions. Regular listeners know that we mostly talk about
the physics of the universe, but that sometimes we dig
a bit deeper and ask about the philosophy of it.
We don't just want to know what the fundamental particles are,
but we want to know why those particles, what does
(02:35):
it mean that it's these particles, and also what is
a particle and why is the universe made out of
them instead of something else? That's the philosophical side of physics.
Today we're gonna follow our noses all the way down
the philosophical rabbit hole and ask questions about what lies
(02:55):
underneath all of that. If you dig far enough into physics,
you always end up face to face with math. Our
equations are written in Math, our predictions and calculations are mathematical.
So math provides the bricks for building our castle of physics.
But that should intrigue us, that should inspire our curiosity.
(03:16):
What are these bricks, the numbers and shapes and functions
and sets that we use to build up our physics?
Where do these mathematical bricks come from? Where do they live?
Can we smash them together to learn about them? How
do we know what rules they follow? Is this the
only way to build physics, or could we have done
it without math? Would aliens use math in their theories?
(03:40):
And if you're a regular listener, you'll hear me saying
this all the time. It's amazing that math does describe
the universe, that it works so well, that we can
devise these beautiful and simple mathematical stories about the universe
and all different scenarios. Tiny particles seem to follow group theory,
rushing rivers obeyed differential equations and massive galaxies are bound
(04:03):
together by geometry. What does it mean that it works
so well? Is it something about how our mind works,
or is it something deep and true about the universe itself?
So today on the podcast will be answering the question.
Is Math the language of the universe? And to help
(04:25):
me soart through some of the slippery issues at the
heart of this deep question, is our guest, professor, Mark Coldy.
Van Mark is a professor of philosophy at the University
of Sydney in Beautiful Australia, where he thinks deeply about
these questions all day long. He's also an accomplished writer,
publishing the indispensability of mathematics and an introduction to the
(04:46):
philosophy of math, which I read recently cover to cover
and found to be very compelling and accessible. The title
makes it sound a little bit like a textbook from
an introductory philosophy course, but it's very conversation national and
very easy to read. I learned a lot and it
inspired me to invite mark to join us on the
podcast to chat about some of the questions at the
(05:08):
heart of mathematics. So it's my pleasure, then, to welcome
Professor Mark Colevin to the podcast. Mark, thanks very much
for joining us, thanks for having me, and I understand
while I have never been to your part of the world,
you have actually spent some time here in Irvine. Is
that right? That's right. Yeah, I had a visiting fellowship
in Irvine back in two thousand and one. So you
(05:31):
can compare for us then, the glorious weather of Orange
County with the weather of your local Sydney. Nothing compares
with the weather of Orange County. Is it's faculous every day?
Correct answer. Correct answer. All right, now that we have
your qualifications sorted out. Hell me, you're a philosopher of
math and I've never spoken to a philosopher of math before.
So tell me. What does a philosopher of math do
(05:52):
all day? I mean, is it reading and writing and
coffee and emails? What got you excited about philosophy of math? Well,
I started out in mathematics. The usual story for a
philosopher of mathematics. You start out in mathematics, you start
getting interested in certain questions in mathematics. That leads you
to more philosophical pondering and at some stage then you
(06:16):
defect to the dark side and become a philosopher, which
is what happened with me mathematics, that undergraduate an honors
level and then that PhD, switched to philosophy, mainly because
I was interesting questions about what counts as the right
logic for mathematics and what is a proof in mathematics,
and these are questions that mathematicians have a good handle on,
(06:40):
but mainly by doing them, I mean you're trained in
mathematics to do proofs, by just doing proofs, and the
question of what, why is this a proof and that
not a proof, is mostly given to you by way
of example. Right, there's a flaw in this proof, there's
a gap in this proof, or this one is a
good proof, and so on and so for but as
a philosopher of mathematics you're much more interested in a
(07:02):
systematic answer to such such questions. What is the correct logic?
Is that classical logic? Is it something other alternative logic
mathematicians are using, and so on and so forth. So
these are sorts of questions that I was interested in,
or became interested in by studying mathematics and found that
the answers really weren't in the mathematics department so I
(07:23):
straight over to the philosophy department occasionally and they didn't
have the answers either, but at least they recognized that
these were interesting questions. So that was my particular path
into the philosophy of mathematics at least. And so why
do you think it is that mathematicians aren't that interested
in like why proofs work or whether proofs should work?
(07:44):
You know. Well, why is it that it's the philosophy
department to ask those kinds of questions? I mean, are
there folks in the mathematics side of it that do
that and just don't call it philosophy? Yeah, I think so.
I wouldn't say that mathematicians not interested in this. I
mean one of the things that I think is interesting
about philosophy of x, whatever X is. For me it's
(08:04):
philosophy of mathematics primarily. But if you're doing philosophy of
something rather, then you need to engage with there's something
or other. So if you're doing philosophy of quantum mechanics,
you've need to talk to folks doing quantum mechanics. If
you're doing philosophy of biology, you need to speak to
folks doing biology and you learn a great deal about
the other discipline as well. So for me philosophy of
(08:26):
mathematics was an excuse to kind of do a bit
more mathematics, talk to mathematicians. Mathematicians, some are interested in
such questions, some are not. That's as you would expect.
Some are interested in Topology, some are not. So it's
just a particular bunch of questions that some mathematicians are
interested in and as a philosophy of mathematics it's good
to talk to them about these things. You know, I'm
(08:48):
interested in mathematical intuitions about such things, not just sitting
back in the philosophical armchair, as it were, and coming
up with my own theories of these things. Right. Do
you feel like there's something of an asymmetry there, that
maybe philosophers of mathematics are more interested in what mathematicians
are thinking about than mathematicians are interested in what philosophers
are thinking about? In the case of the physics department,
(09:10):
for example, we have a lot of people over here
who are doing physics and a few of us are
interested in what philosophers of physics are saying about what
we're doing, but a lot of people seem to subscribe
to you know, fine, men's approach. Philosophy of science is
about as useful to scientists as ornithology as the birds right.
Do you have that same reaction from mathematicians? They're like look,
proofs work. Why do we care why they work? It
(09:31):
depends again on the mathematicians, as you rightly point out.
Amongst physicists you've got things like trying to sort out
the interpretation of quantum mechanics. That's a deeply philosophical question
that a lot of physicists are engaged with. You know'
you've got to just dismissed that. As you know, that's philosophy,
that's not physics. It's crucial to quantum mechanics to have
(09:52):
an appropriate interpretation of what's going on there. So there's
a place where some physicists, not all physicists, are interested
in the reputation of quantum mechanics, but those who are
recognized that as a philosophical problem and interested in well
informed opinions from suitable philosophers. Not every philosopher has an
opinion on that either, and so in mathematics I would
(10:15):
say most mathematicians are not particularly interested in the philosophy
of mathematics. But there are some. Well, in physics it
seems sort of natural to ask these questions. We discover
the universe is this way and then we can ask
like what does that mean, or why is it this way,
in that some other way? In the case of mathematics,
what are the sort of foundational questions here? What are
the questions? The philosophy of math is answering basic questions
(10:39):
about what the subject matter is. I think that one
of the interesting things about philosophy of mathematics is the
problems start really early on. So if you say you
know someone who gives you a scientific discipline, biology, what
is biology? It's not always easy to answer such questions.
But you can say something helpful like it's the study
of living organisms or it's the study of evolution, and
(11:02):
say what that is. What's physics? Well, it's the study
of the fundamental particles and large scale structures, theories of
spacetime and so forth. Mathematics is the study of dot
dot dot, filling the dots right. It's not easy to
answer that question, attempting to say that it's about the
study of numbers, functions, sets and the like, but that
(11:25):
immediately raises a question of what are they, then these
are not the sorts of things that one can gain
access to. It's not like fundamental particles. You can build
accelerators and you can smash things into one another and
you can find traces of fundamental particles, but numbers are
not the sorts of things that even leave traces. So,
(11:47):
you know, if the mathematics is the study of numbers,
then how is it that mathematicians gain access to this
mathematical realm? And there's some of the you know, the
questions you get right from the get I think this
is one of the most fascinating questions. Is this question
of like, what our numbers? Are they things in our
minds or are they things in the universe? You know,
(12:10):
what are the rules by which they operate? Are they
rules that we invented the way we like invented the
rules of checkers, or are they rules that we've discovered
that are true and deep in the universe? But it's
amazing to me that we can do math without knowing
these things right, that we can calculate one plus one
equals two and one plus two equals three and all
sorts of complicated into rules in multiple dimensions without knowing
(12:32):
the answers about what it is we are doing. How
do we reconcile that? How do we understand why it's
possible to do it without knowing what it is we
are doing? Right? That's the really, I think the heart
of philosophy of mathematics is to understand mathematical progress in
light of these difficult questions. So it's not like you
know as a philosopher mathematics. So I'm going to the
(12:53):
mathematics department and telling them to hang on, stop, stop,
until we sort these things out. Right, mathematics is business
as usually. With a few exceptions, there have been some
interesting cases in the history of mathematics. So early in
the twentieth century there was a movement called intuitionism or
constructive ism, and that came from within mathematics. So one
of the Great Mathematicians of the Twentieth Century, Alie J Brauer.
(13:18):
Many theorems named after Brauer, became concerned that if mathematics
is a kind of construct, a mental construct, then you
can't just assume that every mathematical proposition is either true
or false until it's constructed. So if you think about
the fiction, for instance, in this way, what's true in
a fiction is all that's said to be true in
(13:40):
the fiction, plus a bunch of natural implications. Right. So,
for example, she collins lives that twenty two and a
half Baker Street or whatever. That's true in that story.
That's right. And implied by that that he lived near
other streets that are nearby London. The geography of London
is supposed to be held fixed. So without even saying that,
(14:00):
that's a natural implication of that. So all of that
can be taken be true. But Sherlock Holmes walked down
Good Street exactly fourteen times in his life. Neither true
nor false. Surely right? There's not stated in any of
the books. It's not stated that he didn't. So it
seems like that's neither true nor false. That's very different
(14:20):
to the actual world, the actual real world. Even if
you don't know whether something is true or false, it's
true or false nonetheless, or at least that's a natural position.
I don't know how many pairs of gray socks Napoleon owned,
but there is a fact of the matter about that
and we'll probably never know that. Right. So brow became
concerned that if mathematics was a kind of construct like this,
(14:44):
then you can't use certain proof methods, proof methods that
require that the proposition in question is either true or false,
in particular reductive of proof methods which proceed by assuming
the negation of the thing that you want to prove
and then derive a contradiction from that and therefore include
(15:04):
that the proposition unnegated, is true. But if it's neither
true nor false to start with, that neither negation nor
the proposition itself are true or false. So brow became
concerned about particular proof methods and wanted to restrict mathematics
too purely constructive methods, and this movement continues. There are
(15:24):
still mathematicians who stand by this very, very strongly and
think that classical mathematics, which uses these kinds of non
constructive methods, problematic. That's saying that you want to reconstruct
some of the important theorems of mathematics and provide alternative
proofs that are constructive. There may be that all of
(15:46):
mathematics is just a big castle built on sand and
then the end none of it is real. Is that
the idea? Yeah, so the idea is if it is
some sort of construction, mental construction. That doesn't mean that
it's nothing, it's still kind of you know, Shakespeare is
a meant. You know, shakes, works of Shakespeare Mental Constructions.
That great, great things, but you just need to be
(16:06):
careful about the logic that you're using. So, according to
this line of thought, then some proofs are in fact
not proofs at all. I think this touches on a
really interesting question here, which is like, why does it
matter whether or not these things are real or whether
not these things are just constructed? And I think it
goes to the heart of sort of what we're trying
(16:26):
to do, at least in physics, which is learned about
the universe. Right, I am interested in physics because I
want to know what's out there and what's real, not
because I want to build a complicated mathematical construct that
I can use to play around with in my head
and with my friends. Right, I want to know what's
actually out there. So, in the question of mathematics, that
brings up this issue like our numbers real, are these
(16:47):
proofs real, or are they just a game that we
have invented? And to make it more concrete, I like
to think about it in terms of like aliens. You know,
if alien scientists showed up here. Could we talk to
them about out the things in physics that we've discovered?
If the things we've discovered are real and part of
the universe, then yes, if they're just like in our minds,
(17:07):
then no, they would have different ways of thinking about it.
So can we take that same sort of question and
ask it about mathematics and ask like what would aliens
have developed mathematics if it's part of the universe, or
where they have some other way of like putting together
structured thought to figure out the universe? If, in fact,
mathematics is just part of our minds and the way
we think, is that a reasonable way to think about
(17:28):
the questions of the philosophy of math? Yeah, I think
that's a very good way of putting it. You might
think so. For instance, some things in mathematics are just
artifacts of the way we are. So you might think
using based ten, for instance, that's you know, to do
with numbers of fingers and so on and so forth.
But something very natural about base two. And I don't
believe you would know more about this than me, but
(17:50):
I believe that it's thought that if you were going
to contact extraterrestrials, then the initial segments of the international
expansion of Pie in base two would be something that
an intelligent life form would recognize. That's assuming that sometimes
there's something really objective about Pie. It's hard to imagine
that that's just the kind of construct of ours. You know,
(18:12):
you think that surely pie turns up in the most
unexpected places. It's not just about the ratio of the
circumference to the diameter of a circle. That's the initial definition,
but you know, as you know, it turns up in
it's just about everything right everywhere, in complex analysis, in geometry.
So yeah, so that the thought that that's something that
(18:33):
would be recognized by another intelligent life form seems reasonable.
But that pushes you to this sort of objective point
about mathematics, that it seems to be something that's objectively true,
not just mental construction. You wouldn't expect another an alien
life form too recognize facts about Sherlock Holmes, for instance.
(18:55):
You know, the detective novel could be a universal kind
throughout that could be exists and all intelligent beings everywhere.
You know, I imagine sitting across the table from alien
mathematicians and introducing them to ours, and I can imagine
that it might be that the kind of elaborate constructs
that we've built calculus and geometry, they might have very
(19:15):
different ways of doing these kind of things. I mean,
even in the history of our mathematics, our path to
these sorts of things have been varied and could have
gone differently. But it feels like maybe at the heart
of it there could be something in common that if
you drill down to the core of mathematics, of fundamental
ideas on which everything else is built, maybe we could
compare those with alien mathematicians. How well have we done
(19:36):
in terms of like examining the foundations of our own mathematics,
of understanding what our castle is built on? You know,
what are the basic rules of mathematics? Great Question. I mean,
when we think about the foundations of mathematics, a lot
of it is this program of trying to construct other
bits of mathematics from some other mathematics. So set theory
(19:59):
or category, if the theory for prefer, but set theoryt'st
stick for that for the moment. There's beautiful constructions in
set theory where you can construct the natural numbers out
of sets and then you can construct ordered pairs of
natural numbers out of sets and then you can get
functions and so on and so forth. So you can
get a great deal of mathematics built just out of
(20:22):
set theory. Can you explain that to me? Like, how
do you get natural numbers out of sets? What does
that mean? So you just have a series of sets.
So you start with the empty set, right. So the
empty set is the set that has nothing in it.
You identify that with zero. Just call that zero. It's
not zero. That's an empty set, but let's let's just
humor me call that zero. Then you have the set
(20:44):
that contains the empty set that has one member, you know.
So it's the set that has the empty set inside it.
So it has one member. Let's call that one. This
is just arbitrary. We're just making this up as we go.
And then you collect all of the sets from the
previous stages and collect them together. So the next stage
you take the empty set plus the set that contains
(21:06):
the empty set that has two members in it. Suggestive
name for that one too. This is a construction due
to the mathematicians John Von Neuman, and they're called the
von Neuemen ordinals. So you can construct the natural numbers.
In this way, you can then define, you know, I
won't go into details, but you can define edition and
so forth in this set theoretic way. And what does
(21:28):
that accomplished for you? Now, instead of having zero, one
and two, you have these weird sets. Why is that
better or more foundational, or what have you learned from
doing that? That's the kind of really interesting question here.
In one sense, it's now no longer sort of transparent, right,
we're familiar with the natural numbers and sort of building
it out of these these sets. They get really is
(21:49):
you can imagine. They get really ugly really quickly. So
once you start talking about numbers like seventeen, it's hard
to write it down on the one page what that is.
But no one suggests thing that you need to use
these things instead of natural numbers. But it's an interesting
exercise that you can construct the natural numbers out of sets.
(22:10):
So you might think in a way sets are all
you need. Sets really like the fundamental particles of physics. Right,
no one says there are no tables and chairs and
there are no people. Well, some people say such things,
but just because we can show that people are made
out of fundamental particles no one says stop doing biology
(22:31):
or sociology or psychology handed all over particle physicists, because
it's all particles. It's an interesting discovery that we're all
made up out of these fundamental particles. So in that
sort of vein you might think it's interesting that you
can reconstruct almost all of mathematics out of set theory
like this. Not suggesting you do it that way, but
(22:52):
it's an interesting construction. And that may be that the
fundamental mathematical particles, as it were, our sets, I see.
So it's like reduct Shinism to say what bits are
fundamental and what bits emerge. If we can figure out
which bits are fundamental and then we can ask questions
just about those and try to get some insight into,
like the actual nature of mathematics. So does that work?
(23:15):
I mean, can you say I'm going to start with
sets and from that build everything geometry and into roles
and differential equations? Can you base all of mathematics on
these weird sets? Yes, I mean with certain caveats. Are are
are a couple of little areas of mathematics that don't
succumb to this primarily category theory. But but set that
(23:36):
aside all of the mathematics that most of us know
and love. You can build out of sets in this
kind of way, and that's, you know, again, that's just
an interesting fact about mathematics. It demonstrates, firstly, the power
of set theory, but really it's such a versatile tool
set theory. Secondly, you know, it does lend support to
(23:58):
this idea that sets are the equivalent of the fundamental
particles and mathematics. And again, business is usual for topology
and all the other areas of mathematics. Not suggesting they
quit and go and do set theory instead, but rather
it's an interesting fact that their area can be reduced
in this, you know, admittedly cumbersome way, just as reducing
(24:19):
a table or a chair to fundamental particles. Try and
do that in particle physics, give the full description of
what a table is in particles. Right, if it's possible
at all, it's going to be incredibly cumbersome and not
terribly useful to you furniture removalists and other people working
with furniture. All right, well, I have a lot more
questions about the foundations of mathematics, but first let's take
(24:42):
a quick break. Okay, we are back and we're talking
to professor Mark Collivan about the fundamental part of goals
of mathematics and he is suggesting that if aliens arrive
(25:05):
and we are sitting across the table from their mathematicians,
that we might be able to talk to them about
the foundations of mathematics, which may be built out of sets.
We understand now that these sets follow some of the
rules that we identify with, for example basic arithmetic, and
that from that you can build everything else. So then,
what does it mean that the fundamental units of math
(25:26):
are sets? Does that mean that sets are real in
some way, or does it just mean that if sets
are real, then everything else is real? Or if those
rules about sets are real, they're from the universe, then
we can rely on everything else being true? Is that
the situation? Yes, so I think there are realists in
mathematics and those people will say maybe there's just sets,
(25:49):
but the sets are at least real. You've got to
sort of think that the fundamental furniture of the universe includes,
you know, all of the things that your particle for
physicists tell us about, plus sets. And the anti realists
about mathematics say no, the sets are some sort of
construction and their their role is to build this edifice
(26:11):
of mathematics upon sets. But that doesn't tell us anything
about the nature of sets, whether they're real or not.
The action, I think, in modern debates in philosophy mathematics
turns to applications of mathematics pretty quickly. Then you can
think about it in parallel with physics. Why is it
that we believe in certain bits of physics and not others?
(26:33):
So take bits of physics that are more speculative at
the moment. I take string theory to be such an area.
Some people believe in strings, some people don't, and it's
yet to be settled. Happy to take pure advice on this,
but that's my understanding of the current state of play there.
And why is it that people are not concerned about
(26:54):
other particles like electrons? Well, because electrons do a lot
of work in of theories and it's hard to imagine
any of our current physical theories functioning with our electrons
or something very much like them, whereas there are alternatives.
So some of the more speculative parts of physics now,
the speculty parts, very often get settled down the track somewhere,
(27:17):
but that's how we go about deciding whether things are real.
In physics, it seems right. Is it indispensable to sort
of greater physical theory? So people have turned to mathematics
with that same kind of view and thought, okay, for mathematics.
What would it take for mathematics to be real? Well,
perhaps if it's indispensable to our best science, that's a
(27:38):
clue that is real. It's not just some constructor of
the Human Mind, for instance. So this line of thought,
often called it indispensability argument. If mathematics is indispensable to science,
the bits of mathematics that are in fact indispensable should
have the same status as the science itself. It's certainly
(27:59):
a common feeling among physicists that mathematics is somehow the
language of the universe itself, because we can so effectively
describe these rules of physics in terms of this language.
You know, Stephen Weinberg said it's positively spooky how the
physicist finds a mathematicians has been there before him or her.
(28:19):
And you know there are often these cases, especially in
particle physics, where we struggle to understand something then we
discover this some bit of mathematics like group theory invented
just because of the curiosity of mathematicians playing games basically
in their minds, turns out to be applicable to the
world in a gorgeous way that clicks into place and
suddenly gives us insight. And those moments, I mean that
they're not religious moments or spiritual moments, but there are
(28:43):
moments where you feel like you've gained some deep insight
into the way the universe works, and it doesn't feel
like here's a useful description of the universe, it feels
like you're revealing the inner mechanisms of the universe itself.
But how can we know the difference right? How can
we tell whether these things are real, not just the
physical particles we're talking about, but the mathematics that describes them?
(29:06):
How do we distinguish between whether real or not before
we meet alien scientists? So I'm very interested in this
argument you suggest about the indispensability of mathematics, and I
read a book recently called Science Without Numbers, much, which
I'm sure you're familiar with, by hartree field, because he
suggests that math is very, very useful, you know, like
(29:26):
the way making it to do list is a good
way to organize your day, but you could probably get
through your day without it. But it's not actually necessary,
and he says in this book, and I'll quote him
because I find this outrageous, I am denying that numbers
or any similar entities exist. What a statement to make.
Can you help us wrap our minds around this opposite argument,
the one that suggests that we don't actually need math,
(29:48):
that it's useful but not indispensable. How do we make
sense of that? Yeah, I mean, let me say from
the get go I just think that's a fantastic book.
It's I disagree with heartree field on these issues, but
it is one of the absolute gems in philosophy of mathematics.
That book. It's, you know, outrageous, audacious, incredible project. I
(30:09):
mean so in response to this indispensability argument that mathematics,
you know, you can think of this argument of the
following form. We ought to believe in all of them,
only the entities that are indispensable to our best science.
So just should we believe in electrons? Well, just go
and see. Are they indispensable? Could you do science without electrons? No,
(30:29):
you can't. Could you do science without point masses? Well,
it would be difficult, but you could recast all talk
of point masses a little bit more carefully. Right. Could
you do science without coffee? No, for a coffee is three.
I think it was the old joke about the mathematician
being a machine that turns caffeine into theorems. You've solved
(30:54):
the question of the philosophy of math. What is mathematics exactly?
It's turning coffee into papers. We're telling us about the
argument about indispensability. Right, the indispensability argument says, you know
is something indispensable. That's really all you need to do.
So you look to your science. You don't need to
philosophize too much about this. In a way, just philosophizing
is done by recognizing that things that are indispensable to
(31:16):
your best scientific theories. That's what you ought to be
committed to. All right, but let's explore that a little
bit more deeply actually, before we get back to hartshred
field this indispensability argument, because there's some wrinkles there that
I don't really understand. I read like Putnam and quine
arguing that because our best theories are mathematical and those
theories are confirmed that's sort of like also confirms the
(31:37):
mathematics as you go along with it. Like if you
have electrons in your theory and your theory works, then
you believe in electrons. Well, if your theory also has
math as part of it, you're adding numbers, then you
that sort of like comes along with the proof. But
you know, that makes me wonder about things like infinity.
You know, we can do experiments in the universe to
explore particles. But as far as we know there's a
(31:58):
certain number of particles in the univer like tender to
eat or something, depending on how many you count. Does
that mean that only numbers up to tend to the
EAD are real and indispensable and numbers bigger than that,
like infinity, are not real or just parts of our mind?
You need to be careful there. I mean that's not
the only application of infinity. Right. So if you think
(32:18):
space time is continuous, as it is treated in general relativity,
at least quantum mechanics, it's argument that it's treated discreetly there,
but at least in general relativity spacetime is treated continuously.
So how many space time points are there right continuum? Many,
you know, not just the basic infinity there. That's, in
(32:39):
mathematical terms, to do the able of zero. That's the
infinity of the continuum. So it's not just a number
of particles. But are you going to need infinity in
other places? Try and do probability theory without continuous distributions, right.
So infinity crops up in all sorts of places in science,
(32:59):
not just think things. So that argument allows us to
believe in numbers and also believe in numbers like up
to infinity. They are real, but that's only if we
actually need them in our science. If we could do
the science without the numbers, then we wouldn't be able
to necessarily argue that the numbers are also real. So
help us understand Hartley field's argument that we don't need
numbers to do science right. So that's the starting point.
(33:21):
Is this this argument that you know science, you've committed
to everything that's indispensable to your best science. Taken for
granted that mathematics is indispensable for science. So the action
was really kind of on the first premise. Do you
really want to believe in everything in your best science?
We've got fictionalist planes. What about inertial restframes and so on?
And so forth. But heartrey field came along and said
(33:44):
maybe you could do science without numbers, maybe you could
just be realist about space time itself instead of having
the basic idea is, instead of treating space time in
this mathematical way, you just deal with space time itself.
So you're realist about the space time as an entity,
as it were, rather than treating space time as this
(34:07):
mathematical structure that has metrics and coordinate systems and so
on and so forth. Okay, so Hartley field hasn't smoked
so many bin in appeals that he doesn't believed in
the universe. He says space is real. Time is real,
but a mathematical description of that is not necessarily real.
Is that where we are? That's right, but in order
to say that you can't just I mean, you can
(34:27):
just say it, but for anyone to believe what you're
saying you've got to deliver the goods. You've got to
show how you can do something like newtonian mechanics. Is
that his case? Study show how you can do at
least the differential fragment of newtonian mechanics without talking about
anything mathematical. So just looking at relational properties of space
(34:49):
time points. And that's the basic trick and it's surprisingly
how far you can go with that. How is that
possible at all? I mean, if I think about Newtonian mechanics,
the first thing I comes to my mind is f
equals g MM over our squared. It's about relative distances,
it's about masses, it's about forces. If you throw that out,
what do you have left? Well, the thing is you
(35:10):
don't throw it out, you reconstruct it in a much
more direct or indirect, depending on how you're looking at it, way.
So instead of thinking about, for instance, a point in
space having a gravitational potential and the gravitational potential function,
then is this map from the space time to real numbers. Right,
(35:32):
that's the standard presentation. Newton would have talked in those terms,
of course, but now we think of it as a spacetime,
clearly in manifold, and you map from that to real
numbers and that's just your gravitational potential function. Archery field
shows that you can do that directly. Just think about
compare space time points with respect to their gravitational potential,
(35:55):
not having a gravitational potential function. That sits on top
of that and what shows is by doing it this
way you can recover the standard presentation. So you can
actually prove these results that show that you get everything
back that you would have had in the standard presentation.
So again, the take home message from field is not
(36:16):
you should be doing it this way rather than the
way everyone has done it. It's just big like the
story with sets. Right. The fact that you can do
it gives you evidence that the mathematics is not indispensable.
It's just a nice, quick and a much more elegant
way of doing it, but it's not indispensable and you
can recover everything. You can prove that you can recover
(36:38):
everything that you get in standard new Tony Mechanics this way.
I understand that it's a useful way to answer the
question do we need math? By proving that you could
do without it if you had to, doesn't mean that
you should do without it. Right, it's like asking a
question of could you live without jelly beans? You could
go without them for a year and you could prove
that you don't need to eat them. It doesn't mean
that nobody should eat jelly beans. But I'm still not convinced.
(37:01):
I mean your description here of his formulation of gravity
includes things like comparing potentials, and to me potentials are
numbers and comparing is a relationship. Are you seeing? Those
things are not mathematical or they're just not numbers. Yeah,
you've got to be realist about the points themselves, and
that's one of the criticisms of field is that you've
(37:23):
got to be realist about space time points and that
those things have properties. They have primitive properties like the
gravitational potential, electromagnetic potential and so on. So they have
those properties. So rather than those being a mathematical function
that lives on top of that, it's just these primitive
properties of the space time points. And so a lot
of people who are concerned about believing in mathematical objects,
(37:46):
because after all, that's kind of spooky, believing in the
spacetime points is also rather spooky. Right. It's not enough
for field to just believe in the manifold has actually
got to believe that individual points have these properties. But
whichever way you go on this, and as I said,
I disagree with him about the upshot of all this,
but the exercise itself is just incredible. You know, before
(38:09):
he did this, no one would have thought did you
even get started there. So you've got a lot of criticism.
People are saying, oh well, what about Hamiltonian formulations of
classical theories? What about chroantum mechanics where the underlying space
is infinite dimensional, Hilbert spaces, and so on and so forth,
and these are all fair and interesting criticisms. But before
he started no one thought you could do the differential
(38:30):
fragment of Newtonian mechanics either, you know. So for a
great deal of time, a great deal of the the
debate was about how far can you go with this
field style program because if you can go take it further,
then that's going to suggest that mathematics isn't indispensable after all.
So we have, on one hand, folks arguing that mathematics
(38:51):
is beautiful and elegant and unreasonably effective and, beyond that,
actually indispensable to understand in the universe and hurtlee field,
and some folks suggesting that maybe it's just useful but
not actually necessary. So where do you come down on that?
You're a philosopher of mathematics. You've thought deeply about these things.
Do you think that numbers are real? Are they just
(39:12):
part of our minds or are they something we found
in the universe? I'm a realist about mathematics, so I
come down on the former. So I think that mathematics
is in fact indispensable to our best scientific theories, and
that's why I take mathematical into at least some mathematics,
that I think they can still be speculative parts of
mathematics that we don't have reason to believe yet. And
(39:34):
so what convinces you? Just as there are, respectively, parts
of physics right that we don't believe yet. So you know,
you might think some of the higher reaches of set theory,
they're hard to go into the details here now, but
there are bizarre higher reaches of set theory that don't
look like they have any direct applications anywhere yet. You know,
maybe if they do, then then you'd be realist about those.
(39:55):
But it's not a blanket argument that, because mathematics is indispensable,
believe in all of it right. It's got to be
the bits that get applied, and that's one of the
criticisms of this line of thought. You don't get realism
about mathematics. You get realism about Calculus, you get realism
about Algebraic Topology, realism about differential geometry. You know you're
(40:16):
going to get realists about the bits that get used
and not all of it Um but that's for me.
That's as it should be. Just because physics is in
the business of describing the universe doesn't mean we should
believe all of physics. We should believe the bits of
physics that is really indispensable to our understanding of the universe.
And as you know, there are going to be speculative
parts of physics, not just things like string theory, but
(40:39):
nonphysical models. So, for instance, massless universes. People study things
like a universe with no mass. Can you have curvature
in the universe with no mass, for instance? Not that
we live in such a universe. We know we don't
mean one of those universes, but the question is that
will give us some understanding about our own universe if
we can study these non existent universes. So same with mathematics.
(41:01):
They're gonna be speculative parts of mathematics like that that
even someone like me is not going to be a
realist about. But I am not convinced at the end
of the day by heart tree fields project, despite the
fact that I think it's a fascinating and, you know,
beautiful technical exercise. And so I understand that being convinced
about the realism of sets doesn't mean that all of
(41:21):
mathematics is real. But what is it that convinces you
about the realism of sets? So what is the argument
that persuades you? Is it the usefulness of mathematics? Is that,
you know, seeing something beautiful in nature and seeing, you know,
mathematics in it, the FIBONACCI sequence or, you know, the
golden ratio? What is it that convinces you that mathematics
is real and not just a human construct? Well, it
(41:42):
comes back to this indispensability that we just I can't
see how we could do science without mathematics. And, moreover,
it's not just that it's this language of science, as
you often hear. I'm not quite sure what that really
even means. I mean it's not like, you know, once
upon a time all academic work had to be carried
out into Latin and Latin was the language of science.
(42:04):
It's not that that people are talking about it like
there's something much deeper about mathematics than Latin. No one
for a moment really thought that you couldn't do science
in English or whatever. It had to be in Latin.
That when people say mathematics is the language of science,
they mean something much deeper than that, I take it.
And one of the ways I think that mathematics is
indispensable is in offering up explanations. So this is very controversial.
(42:29):
Great deal of debate in philosophy mathematics about this at
the moment, about whether you can get explanations in physics.
Say That mathematically in character. So the mathematics is not
merely just his language, that is providing explanations for what's
going on in the physical world and, as I said,
(42:51):
very controversial. I do believe that. I do think that
mathematics is offering explanations and that's a really important way
in which you can be in to spend sable. If
something's it play an explanatory role in your best scientific theory,
then it really does look like you should be a
realist about it. If you'RE gonna be realist about anything,
and I mean there are anti realists about a great
(43:12):
deal of science as well, but I'm setting that aside.
If you're going to be realist about science, then it's
very odd to say that the reason for such and
such an event occurring was because of some entity, but
there's no such entity. You Haven't explained anything if you
say that. So if mathematics can played this kind of
(43:32):
explanatory role, then that looks like really good grounds for
thinking that it's indispensable to scientific explanation. It is really
compelling to me that mathematics can describe not just the
fundamental bits of the universe and the fundamental elements of mathematics,
but also that it seems like we can find these
(43:53):
fairly simple mathematical stories to describe emergent things. You can
imagine living in a universe where the basic thing are
strings and there's math about strings. And that doesn't mean that,
even if you understood how strings work, that you could
use them to predict the path of a hurricane. Right.
It's incredibly complicated to go from fundamental bits, even from
water drops, to a hurricane, not to mention from strings. Right.
(44:14):
So strings don't really like provide any explanation of the
path of a hurricane, even if you knew what all
the strings were doing. Right. But you can zoom out
and find some higher level laws. You know, you find
fluid mechanics and you find gravitational rotational theorems about how
galaxies move. We can seem to do physics at these
higher levels even without understanding the little bits underneath. And
(44:35):
the same way it seems like we can find mathematics
that describes the universe, even if it's not necessarily connected
to those fundamental little bits of the universe. Why do
you think that is? Why do you think it is
that mathematical descriptions of the universe emerge at all these
different levels, even when they're not necessarily connected to each
other or easily built from one to the other? Again,
(44:56):
one of the bright puzzles and philosophy of mathematics. It's
this often col the unreasonable effectiveness problem. How is it
mathematics just turns up in all of these places, not
just fundamental physics, but in Chemistry and biology and psychology. No,
at which level you know? If you think of science
as these levels, from the fundamental to the more complex,
(45:16):
at every level you've got relevant mathematics that appears there
and again. I wish I had a good answer to that.
One suggestion is that when you know the old, the
old sort of addage that if hammer is your only tool,
then the whole world looks like a nail. Right. So
we've got differential equations, damn it, we're going to use
them everywhere, right. But that just doesn't wash with me.
(45:39):
It's not just that we're forcing everything to be thought
of in this framework of a particular bit of mathematics
like differential equations. And to be fair, there were times,
I think, where physics was a bit like that. Everything
had to be well behaved. Differential Equations Linear First Order,
and you're try and do as much as you can
(45:59):
with is because they were well understood. But I just
don't think that's how we work now. I mean there's
so many different branches of mathematics that are turning up
in all different places. As you mentioned earlier, group theory.
A number of places where you need group theory grows
by the day. It doesn't seem to be simply we
have these tools and we're going to use it, damn it.
It's more like these are the very tools that we
(46:22):
would need to do any such science. And again, what
does that tell us about the mathematics? Well, like it's
intritically connected to the physical world in this kind of
way that you just can't understand the physical world without, yeah,
having the relevant mathematics under control. You know, as I said,
(46:42):
that's controversial. Is just try and flag the things that
are more controversial. But since we're talking philosophy here, we
can just have a general disclaimer. All of this is controversy. Well,
we like to get into the weeds on this show,
and so I have a lot more deep questions about
philosophy of math but when to take another quick break?
(47:14):
All right, so we're back and we're having a lot
of fun talking to Professor Mark Coldevan about whether mathematics
is inherent in the universe and is it real. And
I was wondering when you were talking earlier, since philosophy
of mathematics asks like what is mathematics and is it real,
is there a branch of philosophy called philosophy of philosophy
that asks like whether philosophy is real and what our
(47:36):
philosophers doing? Anyway, there is, there is. Very recently people
have been working on philosophy of philosophy. I think you know,
white philosophers have been doing this for a long time.
I just didn't come up with the phrase, but not
so much the idea of where the philosophy is real,
because you're not interested where the physics is real. You're
interested in whether the things that physics posits are real. Right,
(47:58):
and so, in such far as philosophy is positing entities,
are those things real? You could ask that sort of question.
But my understanding at least, the philosophy of philosophy is
more a kind of a systematic study of methodology, right,
which is kind of how I think of philosophy of
science is in many ways looking at science and trying
(48:20):
to discern useful things to say about methodology. And so
philosophy of philosophy is much more about methodological questions about
philosophy questions like does philosophy make progress? So philosophy cops
criticism because the questions were interested in the questions we're
talking about here now. Are Numbers Real? That goes back
(48:42):
to at least back to Plato, right. And have we
made much progress since then? Well, you know, I'd like
to think we've made some, but certainly if you look
at progress physics has made since such times to now,
physics has not much better firm a ground then. Right,
that's true. Maybe you guys just need more coffee, although
you could also say that physics is just an outgrowth
(49:04):
of philosophy. When a question becomes experimental, it decrimes its
own science and philosophy sort of loses control of it.
But speaking of concrete questions, I want to come back
to the framing we had earlier about aliens. Do you
think that, if aliens arrived, that we could use mathematics
as a sort of basis for building a mental connection
(49:25):
with them, of understanding whether or not we're thinking in
a similar way as them? Would you send of mathematicians
or philosophers of math out to meet the aliens? First thing,
I do think that there would be a good place
to start. Would be bits of mathematics do you think
likely to be universal? I must say I haven't given
a lot of thought to who I would send first
(49:45):
to meet the aliens. You don't realize that you're near
the top of the list. That should concern me for
all sorts of reasons. But but yeah, I do think
the look looking for bits of mathematics that you think
would be calm. Again, you wouldn't want decimal expansion of
pie based ten, but expansion of pie based two. That's
(50:08):
something that you might think would be recognizable fundamental theorems
suitably couched, because you know that the notation you use
is perhaps arbitrary in various ways. But the fundamental theorem,
fundamental theorem of Calculus, for instance. You think that any
reasonably advanced life forms who are capable of traveling to
(50:31):
Earth from great distance would have come across the fundamental
theorems of Calculus. So how do you express those in
a way that's not merely notational dependent too much on
the notation? You can't express it in English, obviously, but
the standard notation using integral signs and so forth. I
think that's kind of accidental that. How do you get
(50:52):
that idea across? That does seem like a good place
to start. You'd think that an intelligent, advanced race would
know the fundamental theorem of Calculus, but how would they
write it and how how should you convey it to them? Well,
we had Noam Chomsky on the podcast a few weeks
ago and we asked him this question and he said,
I'll quote, there's a good chance that arithmetic is universal.
(51:14):
It's a fair guess that at least the arithmetic would
be close enough to be absolute, so that anything we
might call intelligence that we would recognize this intelligence would
at least sit on that. And I suppose that he's
making the argument that you're making that mathematics is probably fundamental.
And in addition, he's drilling down and he's saying, let's
not start with something complicated, let's go down to the basics,
like you were saying earlier, set theory, let's find the
(51:36):
fundamental elements and see if we can begin from that.
Do you think that program is likely to be successful
if aliens arrived? Yeah, yeah, I think that's it's a
very good suggestion. You know, again back to the something
like pie not just any old numbers, because you might
think that nothing special about one to three, four in particular,
but it's really crucial to number theory. Be a concept
(52:01):
of prime number, for instance. You might think certain numbers
jump out at you like prime numbers, Pie, e. Some
of these numbers in particular, and so if you can
get a way of expressing those numbers, but fundamental parts
of arithmetic. But the notion of primness again, how do
you imagine an intelligent, advanced race not having that concept? Again,
(52:24):
just how do you convey it? But I do like
the suggestion. Yeah, well, I'd love to examine the sort
of counter idea. Like to think about what an intelligent
race might have to have in their minds in order
to not arrive at Arithmetic. You know, I can imagine
sitting across from their mathematicians and drawing a symbol for
one and pointing at one thing and an apple, and
bringing another apple and then writing the symbol for two
(52:47):
or something, and you know, then you have one plus
one equals to this kind of stuff. And you must
have thought about this more deeply than I have. What
assumptions are there inherent in that? You know, are we
assuming that the concept of abstraction to say, are these
two apples? They have the similar property. They're both apples.
Obviously they're not the same apple. They're different ones, darker,
one brighter or whatever. What if the aliens are like,
(53:08):
you know, that's one apple and that's one apple. We
don't know what you mean with this whole two business.
That's nonsense. Aren't there fundamental assumptions we're making there, even
with one plus one equals too? Yeah, I think so.
I think it's exactly right. It's got to be counting
the right sorts of things right. So one cloud and
another cloud is one big cloud right. So you don't
(53:29):
think one cloud plus one cloud equals one. That's a
falsification of basic arithmetic. You think, no, you're counting the
wrong kinds of things. They're one plus one is too
given that you're counting the right kinds of things, discreet
things that have a certain kind of property, you can
still count an apple and an orange and get to
but then you've got to have this overarching concept of
(53:51):
pieces of fruit or things before me or something. But
there's got to be some overarching concept there. So you're
absolutely right. There are some conditions for even understanding basic arithmetic.
I was reading about how Japanese people count and discovering
that Japanese counting words are actually quite different from English
counting words, that if you put a set of things
(54:12):
in front of them, they tend to group them by shape.
So those are not just two things, these are two
flat things and there's two tall things over there. It
strikes me that if, even across human cultures we count
and abstract things in different ways, it might be that aliens,
I have no idea what we're talking about when we're
demonstrating our basic arithmetic to them. There are cases in
English as well where you're not interested in the number
(54:34):
of things for various reasons. She's just interested in existence.
So we say it's raining, meaning that there are drops
of rain falling, and that's all we care about. Just
want to know whether I need my umbrellas today or not.
So you can imagine someone putting apples in front of them.
They're saying it's appling. Right, I have the concept of
(54:54):
multiple apples, but I don't need to three or four.
Who Cares how many apples? There's zero apples and there's
appling again, and we do that. There are lots of
instances of that, like raining is the obvious example, but
there are other cases as well where you're interested in
zero or existence. You know of something and you don't
(55:15):
particularly worry about counting, even though you could. There are
rain drops are discreet. You could worry about how many
rain drops there are, and it's not purely because it's
a difficult exercise to count them. It's more just no,
I'm not really interested in how many there are. I'm
just interested is it raining or not. So again, you know,
you could have this idea of zero things or many.
(55:40):
You know, zero, one and many. You can imagine an
intelligent civilization getting by with a very different kind of
sense of what many is as many? You know, more
than a thousand or you know, is it just a few?
I find that people have a different sense like effective infinity.
You know what is a lot and what needs to
be counted individually. Anyway, these are really fast and any
questions and I really thank you for answering them and
(56:02):
for exploring these questions with me. I think my last
question to you is do we expect any sort of
breakthroughs in philosophy of mathematics? I mean, you said we've
been struggling with a question of our numbers and circles
real since Plato. Do you think we're going to figure
that out, that we ever have a point where we're
like yeah, we proved that, now we can move on
to something else, or is philosophy of math basically going
to go forever until we meet the Aliens? I would
(56:24):
like to think that it would be solved, but I
don't think that's necessary for it to be worthwhile exercise. Um.
So why I think that is we learn a lot
along the way. Sometimes asking the right questions is more
interesting than finding answers to them, and I don't think
there's any special about philosophy. You know, I think you
(56:44):
would find that in physics as well. Not being able
to answer some questions, but those questions giving rise to
other questions and new areas is what motivates us and
what keeps the disciplines rolling. It sounds to an outside
of that might sound rather close shop. As you know,
you're only interested in these little questions. We're interested in
getting the exercise rolling. But I do think that we've
(57:07):
learned a lot about the relationship between mathematics and the
physical realm, a lot of understanding about the foundations of mathematics.
We don't have the answer to what the foundations of
mathematics are, but you have some interesting insights from set
theory and the like. So are we like to to
solve the problems in my lifetime? I don't think so.
(57:28):
I hope not. I'd be out of a job, you know,
but I don't think it's going to happen. But I
don't think that that means that it's all a waste
of time. I think we get in many interesting insights.
In particular, the insights that motivated me early on in
my career were this connection that why is that? The
mathematics is applicable. I read this paper, famous paper by
Eugene Vigna called the unreasonable effectiveness mathematics and natural sciences,
(57:53):
and I read that as an undergraduate and it just
captivated by that paper. And did he answer the questions? No.
Has Anyone answered those questions? No, but fascinating stuff to
think about and I think we have a much better
understanding of the relationship between Applied Mathematics and physics now
as a result of asking me sort of other questions
(58:14):
about his mathematics. Real absolutely no. I think it's very
useful and also, in a way, maybe you haven't even
anticipated that. You have spent your life preparing, I think,
to meet the aliens and when they ask me who
we should send, you know, in our first contingent to
chat with our alien technological friends, I'm going to nominate you. Good,
as long as they're friendly, you know. Well, we'll find
(58:36):
out right just will send the military. You know, make
sure they're friendly first. All right, sounds good. Well, thank
you very much for joining us and for talking to
us about these crazy ideas. I hope that one day
we do figure out our numbers real why math works.
And if, in fact, math is just a game we
invented in our minds or the fundamental code of the
universe itself. Thanks very much for joining us today. My pleasure,
(59:00):
thanks for having me, thanks for listening, and remember that,
Daniel and Jorge explained, the universe is a production of
I heart radio. For more podcast from my heart radio,
visit the I heart radio APP, apple podcasts or wherever
(59:20):
you listen to your favorite shows.