Kind: captions
Language: en
The following is a conversation with
Joel David Hamkins, a mathematician and
philosopher specializing in set theory,
the foundation of mathematics, and the
nature of infinity. He is the number one
highest rated user on math overflow,
which I think is a legendary
accomplishment. Math overflow, by the
way, is like Stack Overflow, but for
research mathematicians.
He is also the author of several books
including proof and the art of
mathematics and lectures on the
philosophy of mathematics and he has a
great blog infinitely more.xyz.
This is a super technical and super fun
conversation about the foundation of
modern mathematics and some mindbending
ideas about infinity, nature of reality,
truth, and the mathematical paradoxes
that challenged some of the greatest
minds of the 20th century.
I have been hiding from the world a bit,
reading, thinking, writing,
soulsearching, as we all do every once
in a while, but mostly just deeply
focused on work and preparing mentally
for some challenging travel I plan to uh
take on in the new year. Through all of
it, a recurring thought comes to me. how
damn lucky I am to be alive and to get
to experience so much love from folks
across the world.
I want to take this moment to say thank
you from the bottom of my heart for
everything, for your support, for the
many amazing conversations I've had with
people across the world. I got uh a
little bit of hate and a whole lot of
love, and I wouldn't have it uh any
other way. I'm grateful for all of it.
This is the Lex Freedman podcast. To
support it, please check out our
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can also find ways to contact me, ask
questions,
give feedback, and so on. And now, dear
friends, here's Joe David Hamkins.
Some infinities are bigger than others.
This idea from Caner at the end of the
19th century I think it's fair to say
broke mathematics before rebuilding it
and uh I also read that this was a
devastating and transformative discovery
for several reasons. So one it created a
theological crisis because infinity is
associated with God. How could there be
multiple infinities and also Caner was
deeply religious himself. Second there
was a kind of mathematical civil war.
The leading German mathematician
chronicer called Caner a corruptor of
youth and uh tried to block his career.
Third, many fascinating paradoxes
emerged from this uh like Russell's
paradox about the set of all sets that
don't contain themselves and uh those
threatened to make all of mathematics
inconsistent. And finally on the
psychological side, on the personal
side, Caner's own breakdown. He
literally went mad spending his final
years in and out of uh sanatoriums
obsessed with proving the continuum
hypothesis. So laying that all out on
the table uh can you explain the idea of
infinity that uh some infinities are
larger than others and why was this so
transformative to mathematics?
>> Well, that's a really great question.
I would want to start talking about
infinity and telling the story much
earlier than caner actually because I
mean you can go all the way back to
ancient Greek times when Aristotle
emphasized the potential aspect of
infinity as opposed to the impossibility
according to him of achieving an actual
infinity and Archimedes method of
exhaustion where he is trying to
understand the the area of a region by
carving it into more and more triangles
say and sort of exhausting the area and
thereby understanding the total area in
terms of the sum of the areas of the
pieces that he put into it and it
proceeded on this kind of potential
under this potentialist understanding of
infinity for for hundreds of years
thousands of years. Uh almost all
mathematicians were potentialists only
and thought that it was incoherent to
speak of an actual infinity at all. Um,
Galileo is an extremely
prominent exception to this though he
argued against this sort of potentialist
orthodoxy in the dialogue of tunu
sciences. Really lovely account there
that he gave. Um, and that the in many
ways Galileo was anticipating Kandra's
developments except he couldn't quite
push it all the way through and uh ended
up throwing up his hands in confusion
in in a sense. I mean, the Galileo
paradox is the idea or the observation
that if you think about the natural
numbers, I would start with zero, but I
think maybe he would start with one. The
numbers 1 2 3 4 and so on. And you think
about which of those numbers are perfect
squares. So 0 squar is zero and one
square is one and two squar is 4, 3
squar is 9, 16, 25 and so on. And
Galileo observed that the the perfect
squares can be put into a onetoone
correspondence with all of the numbers.
I mean we just did it. I associated
every number with its square. And so it
seems like on the basis of this one to
one correspondence that there should be
exactly the same number of squares,
perfect squares as there are numbers.
And yet there's all the gaps in between
the perfect squares, right? And and this
suggests that uh you know there should
be fewer perfect squares, more numbers
than squares because the numbers include
all the squares plus a lot more in
between them, right? And Galileo was
quite troubled by this observation
because he took it to cause a kind of
incoherence in the comparison of
infinite quantities. Right? And another
example is if you take two line segments
of different lengths and you can imagine
drawing a kind of foliation a fan of
lines that connect them. So the end
points are matched from the shorter to
the longer segment and the midpoints are
matched and so on. So spreading out the
lines as you go and so every point on
the shorter line would be associated
with a a unique distinct point on the
longer line in a onetoone way. And so it
seems like the two line segments have
the same number of points on them
because of that even though the longer
one is longer. And so it makes again a
kind of confusion of our ideas about
infinity. And also with two circles, if
you just place them concentrically and
draw the rays from the center, then
every point on the smaller circle is
associated with a corresponding point on
the larger circle, you know, in a one to
one way. and and again that seems to
show that the smaller circle has the
same number of points on it as the
larger one precisely because they can be
put into this one to one correspondence.
Now of course the contemporary attitude
about this situation is that those two
infinities are are exactly the same and
that Galileo was right in those
observations about the equinosity and
the way we would talk about it now is
appeal to what uh what I call the caner
Hume principle or some people just call
it Hume's principle which is the idea
that if you have two collections whether
they're finite or infinite then we want
to say that those two collections have
the same size they're equinumerous
if and only if there's a onetoone
correspondence between those
collections. And so Galileo was
observing that line segments of
different lengths are equinumerous and
the perfect squares are equinumerous
with the whole all of the natural
numbers and and two any two circles are
equinumerous and so on and that the
tension between the kander Hume
principle and what could be called
Uklid's principle which is that the
whole is always greater than the part
which is a principle that Uklid appealed
to in in the elements. I mean many times
when he's calculating area and so on he
wants it's a kind of basic idea that if
something is just a part of another
thing then the the whole is greater than
the part and so what Galileo was
troubled by was this tension between
what we call the canerhume principle and
Uklid's principle
and it really wasn't fully resolved I
think until Caner he's the one who
really explained so clearly about these
different sizes of infinity and so on in
a way that was so compelling and so he
exhibited two different infinite sets
and proved that they're not equinumerous
they can't be put into onetoone
correspondence
and it's traditional to talk about the
uncountability of the real numbers so
Kanto's big result was that the set of
all real numbers is an uncountable set
so maybe if we're going to talk about
countable sets then I would suggest that
We talk about Hilbert's hotel which
really makes that idea perfectly clear.
>> Yeah, let's talk about the Hilbert's
hotel.
>> Hilbert's hotel is a hotel with
infinitely many rooms. You know, each
room is a full floor suite. So there's
floor zero. I always start with zero
because for me the natural numbers start
with zero. Although that's maybe a point
of contention for some mathematicians.
The the other mathematicians are wrong.
>> Like a bunch of geo programmers. So
starting at zero is a wonderful place to
start.
>> Exactly. So there's floor 0, floor 1,
floor two or room 0 1 2 3 and so on just
like the natural numbers. So Hilbert's
hotel has a room for every natural
number
and it's completely full. There's a
person occupying room N for every N. But
meanwhile, a new guest comes up to the
desk and wants a room. Can I have a
room, please? And the manager says,
"Hang on a second. Just give me a
moment." And you see, when the other
guests had checked in, they had to sign
an agreement with uh with the hotel that
maybe there would be some changing of
the rooms, you know, during the stay.
And so the manager sent a message up to
all the current occupants and told every
person, "Hey, can you move up one room,
please?" So the person in room five
would move to room six and the person in
room six would move to room seven and so
on. And everyone moved at the same time.
And of course, we never want to be
placing two different guests in the same
room and we want everyone to have their
own private room and but when you move
everyone up one room then the bottom
room room zero becomes available of
course and so he can put the new guest
in that room. So even when you have
infinitely many things then the new
guest can be accommodated and that's a
way of showing how the particular
infinity of the occupants of Hilbert's
hotel it violates Uklid's principle. I
mean, it exactly illustrates this idea
because adding one more element to a set
didn't make it larger because we can
still have a 1:1 correspondence between
the total new guests and the old guests
by the room number. Right?
>> So, to just uh say one more time, the
hotel is full. the hotel is full
>> and then you could still squeeze in one
more and that breaks the uh traditional
notion of mathematics and and breaks
people's brains about when they try to
uh think about infinity. I suppose this
is a property of infinity.
>> It's a property of infinity that
sometimes when you add an element to a
set it doesn't get larger. That's what
this example shows.
But one can go on with Hilbert for
example. Uh, I mean maybe the next day,
you know, 20 people show up all at once,
but we can easily do the same trick
again. Just move everybody up 20 rooms
and then we would have 20 empty rooms at
the bottom and those new 20 guests could
go in. But on the following weekend, a
giant bus pulled up Hilbert's bus. And
Hilbert's bus has, of course, infinitely
many seats. There's seat zero, seat one,
seat two, seat three, and so on. And so
one wants to uh you know all the people
on the bus want to check into the hotel
but the hotel is completely full. And so
what is the manager going to do? And
when I talk about Hbert's hotel in when
I teach Hilbert's hotel in in class I
always demand that the students provide
you know the explanation of of how to do
it.
>> So maybe I'll ask you can you tell me
yeah what is your idea about how to fit
them all in the hotel everyone on the
bus and also the current occupants. uh
you uh you separate the hotel into even
and odd rooms and you squeeze in the new
Hilbert bus people into the odd rooms
and uh the [clears throat] previous
occupants go into the even rooms.
>> That's exactly right. So I mean that's
the a very uh easy way to do it if you
just tell all the current guests to
double their room number. So in room n
you move to room 2 * n. So they're all
going to get their own private room, the
new room, and it will always be an even
number because 2 * n is always an even
number. And so all the odd rooms become
empty that way. And now we can put the
bus occupants into the oddnumbered
rooms.
>> And by doing so, you have now shoved in
an infinity into another infinity.
>> That's right. So what it really shows I
mean another way of thinking about it is
that well we can define that a set is
countable if it is equinumerous with a
set of natural numbers and and a kind of
easy way to understand what that's
saying in terms of Hilbert's hotel is
that a set is countable if it fits into
Hbert's hotel because Hilbert's hotel
basically is the set of natural numbers
in terms of the room numbers. So to be
equinumerous with a set of natural
numbers is just the same thing as to fit
into Hilbert's hotel. And so what we've
shown is that
if you have two countably infinite sets,
then their union is also countably
infinite. If you put them together and
form a new set with all of the elements
of either of them, then that union set
is still only countably infinite. It
didn't get bigger. And that's a
remarkable property for uh a notion of
infinity to have, I suppose. But if you
thought that there was only one kind of
infinity, then it wouldn't be surprising
at all because if you take two infinite
sets and put them together, then it's
still infinite. And so if there were
only one kind of infinity, then it
shouldn't be surprising that the union
of two countable sets is countable.
>> So there's another way to push this a
bit harder. And that is when uh when
Hilbert's train arrives and Hilbert's
train has infinitely many train cars.
>> Mhm. And each train car has infinitely
many seats.
>> And so we have an infinity of infinities
of the train passengers together with
the current occupants of the hotel. And
everybody on the train wants to check
into Hilbert's hotel.
So the manager can again of course send
a message up to all the rooms uh telling
every person to double their room number
again. And so that will occupy all the
even-umbered rooms again and but free up
again the oddnumbered rooms. So somehow
we want to put the train passengers into
the oddnumbered rooms. And so well every
train passenger is on some car let's say
car C and seat S. So somehow we have to
take these two coordinates you know C
comma S the car number and the seat
number and produce from it an odd number
in a one to one way you know and that's
that's actually not very difficult. In
fact uh one can just use say an easy way
to do it is to just use the number 3 to
the C * 5 to the S
3 to the C 3 to the car number. So 3 * 3
* 3 you know the number of the car you
multiply three by itself the number of
the train car and then you multiply five
by itself the seat number times and then
you multiply those two numbers together
so 3 to the c * 5 to the s
that's always an odd number because the
prime factorization has only threes and
fives in it there's no two there so
therefore it's definitely an odd number
and it's always different because of the
uniqueness a prime factoriization. So
every number can be factored uniquely
into prime. So if you have a number of
that form, then you can just factor it
and that tells you the exponent on three
and the exponent on five. And so you
know exactly which person it was, which
car they came from and which seat they
came from.
>> And prime factorization is every single
number can be uh uh decomposed into the
atoms of mathematics which is the prime
numbers. You can multiply them together
to achieve that number. And that's prime
factorization. You're showing three and
five are both prime numbers odd. So
through this magical formula, you can uh
deal with this
train infinite number of cars with each
car having infinite number of seats.
>> Exactly. Right. We've proved that if you
have countably many countable sets, then
the union of those sets, putting all
those sets together into one giant set
is still countable. Yeah, cuz the the
train cars are each countable. Plus, the
current hotel, it's sort of like another
train car if you want to think about it
that way. The current occupants of the
hotel could, you know, have the same
number as as any of the train cars. So
putting countably many countable sets
together to make one big union set is
still countable. It's quite remarkable I
think. Um I mean when I first learned
this many many years ago I was
completely shocked by it and transfixed
by it. It was quite amazing to me that
this notion of countable infinity could
be closed under this process of
infinitely many infinities adding up
still to the very same infinity which is
a strong instance a strong violation of
Uklid's principle once again right so
the new set that we built is has many
more elements than the old set in the
sense that there's additional elements
but it doesn't have many more elements
in terms of its size because it's still
just to countable infinity and it fits
into Hbert's hotel.
>> Uh, have you been able to sort of
internalize a good intuition about
countable infinity? Cuz that is a pretty
weird thing that you can have a
countably infinite set of countably
infinite sets. You can shove it all in
and it still is countable infinite set.
>> Yeah, that's that's exactly right.
Right. I mean, I guess
of course when you when you work with
these notions that the the argument of
of Hilbert Sortell becomes kind of
clear. There's many many other ways to
talk about it too. For example,
let's think about say the the integer
lattice the grid of points that you get
by taking pairs of natural numbers say
so the the upper right quadrant of the
integer lattice. Yeah. So there's the,
you know, row 0, row one, row two, and
so on, column 0, column 1, column 2, and
so on. And each each row and column has
an countable infinity of points on it,
right?
So those dots, if you think about them
as dots, are really the same as the
train cars. If you think about each
column of in the in that integer
lattice, it's a countable infinity. It's
like one train car and then there's the
next train car next to it and then the
next column next to that the next train
car and so but if we think about it in
this grid manner then I can imagine a
kind of winding path winding through
these grid points like up and down the
diagonals winding back and forth. So I
start at the corner point and then I go
down up and to the left and then down
and to the right up and to the left down
and to the right and so on in such a way
that I'm going to hit every grid point
in on this path. So this gives me a way
of assigning room numbers to the points
because every every grid point is going
to be the nth point on that path for
some n
>> and that that gives a correspondence
between the grid points and the natural
numbers themselves. So it's a kind of
different picture. I mean before we used
this 3 to the C 5 * 5 to the S which is
a kind of you know overly arithmetic way
to think about it but there's a kind of
direct you know way to understand that
it's still a countable infinity when you
have countably many countable sets
because you can just start putting them
on this list and as long as you give
each of the infinite collections a
chance to add one more person to the
list then you're going to accommodate
everyone in any of the sets in one list.
Yeah, it's a really nice visual way to
think about it. You just zigzag your way
across the grid to make sure everybody's
included that gives you kind of an
algorithm for including everybody. So,
can you speak to the uncountable
infinities? So, what are the integers
and the real numbers and what is the
line that Caner was able to find?
>> So, maybe there's there's one more step
I want to insert before doing that which
is
>> the rational numbers. So, we did we did
pairs of natural numbers,
>> right? that that's the train car
basically. But maybe it's a little bit
informative to think about the rational
the fractions the set of fractions or
rational numbers because a lot of people
maybe have an expectation that maybe
this is a bigger infinity because the
rational numbers are are densely ordered
between any two fractions you can find
another fraction. Right? The average of
two fractions is another fraction.
[gasps]
And so so sometimes people it seems to
be a different character than um than
the integers which are discreetly
ordered right from every any integer
there's a next one and a previous one
and so on. But that's not true in the
rational numbers
and yet the rational numbers are also
still only accountable infinity. And the
the way to see that is actually it's
just exactly the same as Hilbert's train
again because every fraction consists of
two integers the numerator and the
denominator and so if I tell you two
natural numbers then you know what
fraction I'm talking about I mean plus
the sign issue I mean if it's positive
or negative but if you just think about
the positive fractions then you know you
have the numbers of the form p over q
where q is not zero. So you can still do
3 to the P * 5 to the Q. The same idea
works with the rational numbers. So this
is still a countable set. And you might
think, well, every every set is going to
be countable because there's only one
infinity. I mean, if that's a kind of
perspective maybe that you're um uh
adopting, but it's not true. And that's
the profound achievement that Caner made
is proving that the set of real numbers
is not a countable infinity. It's a
strictly larger infinity and therefore
there are uh there's more than one
concept of infinity more than one size
of infinity. So let's talk about the
real numbers. What are the real numbers?
Why do they break infinity? The
countable infinity
>> right?
>> Looking it up on perplexity.
Uh real numbers include all the numbers
that can be represented on the number
line encompassing both rational and
irrational numbers. We've spoken about
the rational numbers and the rational
numbers by the way are by definition the
numbers that can be represented as a
fraction of two integers. That's right.
So with the real numbers we have the
algebraic numbers. We have of course all
the rational numbers. The integers and
the rationals are all part of the real
number system. But then also we have the
algebraic numbers like the square root
of two or the cube root of five and so
on. Numbers that solve an algebraic
equation over the integers. Those are
known as algebraic numbers.
It was an open question for a long time
whether that was all of the um real
numbers or whether there would exist
numbers that are the transcendental
numbers. The transcendental numbers are
real numbers that are not algebraic
>> and we won't even go to the surreal
numbers about which you have a wonderful
blog post. We'll talk about that a
little bit later.
>> Oh great. So it was Louisville who first
proved that uh there are transcendental
numbers and he exhibited a very specific
number that's now known as the
Louisville constant which is a
transcendental number. Caner also
famously proved that there are many many
transcendental numbers. In fact it
follows from his argument on the
uncountability of the real numbers that
there are uncountably many
transcendental numbers. So most real
numbers are transcendental
>> and again going to perplexity
transcendental numbers are real or
complex numbers that are not the root of
any nonzero polomial with integer or
rational coefficients. This means they
cannot be expressed as solutions to
algebraic equations with integer
coefficients setting them apart from
algebraic numbers. That's right. So some
of the famous transcendental numbers
would include the number pi you know the
the uh 3.14159265
and so on. Uh so that's a transcendental
number also oilers's constant the e like
e to the x the exponential function.
>> So you could say that some of the
sexiest numbers in mathematics are all
transcendental numbers.
>> Absolutely. That's true. [laughter]
Yeah. Although you know I don't know
square of two is pretty
>> square. All right. So it depends. Let's
not beauty can be found in in all the
different kinds of sets. But yeah,
>> and if you have a kind of simplicity
attitude, then you know zero and one are
looking pretty good, too. So, and
they're definitely not.
>> Sorry to take that tangent, but what is
your favorite number? Do you have one?
>> Oh, gosh. You know,
>> is it zero?
>> Did you know there's a proof that every
number is interesting? [laughter]
You can prove it because
>> Yeah. What's that proof look like? How
do you even begin? I'm going to prove to
you that every natural number is
interesting.
>> Okay?
>> Yeah. I mean zero is interesting because
you know it's the additive identity,
right? That's pretty interesting. And
one is the multiplicative identity. So
when you multiply it by any other
number, you just get that number back,
right? And two is, you know, the the
first prime number that's super
interesting, right? And
>> okay so one can go on this way and and
give specific reasons but I want to
prove as a general principle that every
number is interesting and and this is
the proof.
>> Um suppose toward contradiction that
there were some boring numbers.
>> Okay.
>> Uh
>> but if if there was an uninteresting
number
>> Yes.
>> then there would have to be a smallest
uninteresting number.
>> Mhm. Yes. But that's a contradiction
because the smallest uninteresting
number is a super interesting graphology
to have. [laughter]
>> So therefore there cannot be good there
cannot be any boring numbers. I'm going
to have to try to find a hole in that
proof [laughter]
>> cuz there's a lot of big tin in the word
interesting. But yeah that's a be that's
beautiful. That doesn't say anything
about the transcendental numbers about
the real numbers. You just prove from
just for natural numbers.
>> Okay. Should we get back to Caner's
argument or
>> Sure. You you've masterfully avoided the
question. You basically said I love all
numbers.
>> Yeah, basically. Okay, that's what I my
>> back to Caner's argument. Let's go.
>> Okay. So, Caner wants to prove that the
infinity of the real numbers is
different and strictly larger than the
infinity of the natural numbers. So, the
natural numbers are the numbers that
start with zero and and add one
successively. So, 0 1 2 3 and so on. And
the real numbers as we said are the the
numbers that come from the number line
including all the integers and the
rationals and the algebraic numbers and
the transcendental numbers and all of
those numbers altogether. Now obviously
since the natural numbers are included
in the real numbers we know that the
real numbers are at least as large as
the natural numbers and so the claim
that we want to prove is that it's
strictly larger. So suppose that it
wasn't strictly larger. So that then
they would have the same size. But to
have the same size remember means by
definition that there's a 1 to1
correspondence between them. So we
suppose that the real numbers can be put
into one toone correspondence with the
natural numbers. So therefore for every
natural number n we have a real number.
Let's call it r subn.
R subn is the nth real number on the
list. Basically, our assumption allows
us to think of the real numbers as
having been placed on a list. R1, R2,
and so on. Okay. And now, now I'm going
to define the number Z and it's going to
be the integer part is going to be a
zero. And then I'm going to have put a
decimal place. And then I'm going to
start specifying the digits of this
number Z. [snorts] D1, D2, D3, and so.
And what I'm going to make sure is that
the nth digit after the decimal point of
Z is different from the nth digit of the
nth number on the list.
>> Okay. So, so to specify the nth digit of
Z, I go to the nth number on the list, R
subn, and I look at its nth digit after
the decimal point. And whatever that
digit is, I make sure that my digit is
different from it. Okay?
And then I want to do something a little
bit more and that is uh I'm going to
make it different in a way that I I'm
never using the digits zero or nine. I'm
just always using the the other digits
and not zero. There's a certain
technical reason to do that.
[sighs and gasps]
But the main thing is that I make the
digits of Z different in the nth place
from the nth digit of the nth number.
If you had ma if you had drawn out the
numbers uh on the original list R1, R2,
R3 and so on and you made it you know
and they were each filling a whole row
and you thought about the nth digit of
the nth number it would form a kind of
diagonal going down and to the right and
that for that reason this argument is
called the diagonal argument because
we're looking at the nth digit of the
nth number and those exist on a kind of
diagonal going down and we've made our
number D so that the nth digit of Z is
different from the nth digit of the nth
number.
But now it follows that Z is not on the
list because Z is different from R1
because
well the the first digit after the
decimal point of Z is different from the
first digit of R1 after the decimal
point. That's exactly how we built it.
And the second digit of Z is different
from the second digit of R2 and so on.
The nth digit of Z is different from the
nth digit of R subn for every N. So
therefore Z is not equal to any of these
numbers R subn and but that's a
contradiction because we had assumed
that we had every real number on the
list but yet here is a real number Z
that's not on the list. Okay. And so
that's the main contradiction.
>> And so it's a kind of proof by
construction. Exactly. So given a list
of numbers Caner is proving it's
interesting that you say that actually
because there's a kind of philosophical
controversy that occurs as a uh uh in
connection with this observation about
whether Caner's construction is
constructive or not. Um given a list of
numbers, Caner gives us a specific means
of constructing a real number that's not
on the list is a way of thinking about
it.
There's this one aspect which I alluded
to earlier but some real numbers have
more than one decimal representation and
it causes this slight problem in the
argument. Um for example the number one
you can write it as 1.0000
forever but you can also write it as
0.999
forever. Those two those are two
different decimal representations of
exactly the same number. And then you
beautifully got rid of the zeros and the
nines. Therefore, we don't need to even
consider that. And the proof still
works.
>> Exactly. Because the only kind of case
where that phenomenon occurs is when the
number is eventually zero or eventually
nine. And so since our number Z never
had any zeros or nines in it, it wasn't
one of those numbers. And so actually in
those cases, we didn't need to do
anything special to diagonalize. Just
the mere fact that our number has a
unique representation already means that
it's not equal to those numbers. So
maybe it was controversial in Caner's
day more than 100 years ago, but I think
it's most commonly looked at today as
you know one of the initial main results
in set theory and it's profound and
amazing and insightful and the beginning
point of so many later arguments. And
this diagonalization idea has proved to
be an extremely fruitful proof method.
And almost every major result in
mathematical logic is using in an
abstract way the idea of
diagonalization. It was really um the
start of so many other observations that
uh were made including Russell's paradox
and the halting problem and the
recursion uh theorem and so many other
principles are using diagonalization uh
at their core. So
>> can we uh can we just step back a little
bit? This infinity crisis led to a kind
of uh rebuilding of mathematics. So uh
it would be nice if you lay out the
things it resulted in. So one is set
theory became the foundation of
mathematics. All mathematics could now
be built from sets giving math its first
truly rigorous foundation. The
exiomatization
of mathematics. The paradoxes forced
mathematicians to develop ZFC and other
aimatic systems and uh mathematical
logic emerged. Uh Geredo Touring and
others created entire new fields. So can
you uh explain what set theory is and uh
how does it serve as a foundation of
modern mathematics and maybe even the
foundation of truth?
>> That's a great question. Set theory
really has two roles that it's serving.
this kind of two ways that set theory
emerges. On the one hand, set theory is
the is its own subject of mathematics
which with its own problems and
questions and answers and proof methods.
And so really from this point of view,
set theory is about the transfinite
recursive constructions or wellfounded
definitions and constructions.
And those ideas have been enormously
fruitful and set theorists have looked
into them and developed uh so many ideas
coming out of that. But set theory has
also happened to serve in this other
foundational role. It's very common to
hear things said about set theory that
really aren't taking account of this
distinction between the two roles that
it's serving. It's its own subject, but
it's also serving as a foundation of
mathematics. So in its foundational
role, set theory provides a way to think
of a collection of things as one thing.
That's the the central idea of set
theory. A set is a collection of things,
but you think of the set itself as one
abstract thing. So when you form the set
of real numbers, then that is a set.
It's one thing. It's a set and it has
elements inside of it. So it's sort of
like a bag of objects. A set is kind of
like a bag of objects. And so we have a
lot of different axioms that uh describe
the nature of this idea of thinking of a
collection of things as as one thing
itself, one abstract thing. And axioms
are, I guess, facts that we assume are
true based on which we then build the
ideas of mathematics. So there's a bunch
of facts, axioms about sets that we can
put together and if they're sufficiently
powerful, we can then build on top of
that a lot of really interesting
mathematics.
>> Yeah, I think that's right. So I mean
the history of how of the current set
theory aims known as the Zera Franco
axioms came out in the early 20th
century with with Zermelo's idea. I mean
the history is quite fascinating because
um Zerlo in 1904 offered a proof that
the what's called the axim of choice
implies the well-order principle. So he
described his proof and that was
extremely controversial at the time and
there was no theory there weren't any
axioms there. Caner was not working in
an aimatic framework. He didn't have a
list of axioms in the way that we have
for set theory now and Zerlo didn't
either. Um and his ideas were challenged
so much with regard to the well-order
theorem
>> uh that he was pressed to produce the
theory that in which his argument could
be formalized and that was the origin of
what's known as their melo set theory.
Uh and going to perplexity the axiom of
choice is a fundamental principle in set
theory which states that for any
collection of non-mpty sets it is
possible to select exactly one element
from each set even if no explicit rule
to make the choices given. This axiom
allows the construction of a new set
containing one element from each
original set even in cases where the
collection is infinite or where there is
no natural way to specify a selection
rule. So this was controversial and uh
this was described before there was even
a language for exiomatic systems.
>> That's right. So on the one hand I mean
the exo choice principle is
completely obvious that we want this to
be true that it is true. I mean a lot of
people take it as a law of logic. If you
have a bunch of sets then there's a way
of picking an element from each of them.
There's a function. And if I have a
bunch of sets, then there's a function
that uh when you apply it to any one of
those sets gives you an element of that
set. It's it's a completely natural
principle. I mean, it's called the
eximma choice, which is a way of sort of
anthropomorphizing the mathematical
idea. It's not like the function is
choosing something. I mean, it's just
that if you were to make such choices,
there would be a function that consisted
of the choices that you made. And the
difficulty is that when you when you
can't specify a rule or a procedure by
which you're making choices,
then it's difficult to say what the
function is that you're asserting
exists. You know, you want to have the
view that well there is a way of
choosing. I don't have an easy way to
say what the function is, but there
definitely is one. Yeah, this is the way
of thinking about the XMA choice. So
we're going to say the the the three
letters of ZFC maybe a lot in this
conversation. You already mentioned Zela
Frankle set theory. That's the Z and the
F. And the C in that is this comes from
this axum of choice.
>> That's right.
>> So ZFC sounds like a super technical
thing but it is the set of axioms that's
the foundation of modern mathematics.
>> Yeah. Absolutely. So one should be aware
also that there's huge parts of
mathematics that don't that pay
attention to whether the exo choice is
being used and they don't want to use
the eximma choice or they work out the
consequences that's that are possible
without the exma choice or with weakened
forms of of smela franle set theory and
so on and that's quite a there's quite a
vibrant amount of work in that area I
mean but going back to the exo choice
for a bit it's maybe uh interesting to
to give Russell's
description of how to think about the
axim of choice. So Russell describes um
this uh rich person who has a an
infinite closet and in that closet he
has infinitely many pairs of shoes
and he tells his butler uh to uh please
go and give me one shoe from each pair.
And and the butler can do this easily
because he can for any pair of shoes he
can just always pick the left shoe. I
mean there's a way of picking that we
can describe
we always take the left one or always
take the right one or take the left one
if it's a red shoe and the right one if
it's a brown shoe or you know we can
invent rules that would result in these
kind of choice functions. We can
describe explicit choice functions. And
for those cases, you don't need the axim
of choice to know that there's a choice
function. When you can describe a
specific way of choosing, then you don't
need to appeal to the axiom to know that
there's a choice function.
But the problematic case occurs
when you think about the infinite
collection of socks that the person has
in their closet. And if we assume that
socks are sort of indistinguishable
within each pair, you know, they match
each other, but they're sort of, you
know, incisible,
then the the butler wouldn't have any
kind of rule for which sock in each pair
to pick. And so it's not so clear that
he has a way of of producing one sock
from each pair because right so that's
what's at stake is the question of
whether you can specify a rule by which
the choice function you know a rule that
it obeys uh that defines the choice
function or whether there's sort of this
arbitrary choosing aspect to it that's
when you need the axim of choice to know
that there is such a function but of Of
course, as a matter of mathematical
ontology, we might find attractive the
idea that well, look, I mean, I don't I
don't not every way of choosing this
socks has to be defined by a rule. Why
should everything that exists in
mathematical reality follow a rule or a
procedure of that sort? If I have the
idea that my mathematical ontology is
rich with objects, then I think that
that there are all kinds of functions
and ways of choosing. Those are all part
of the mathematical reality that I want
to be talking about. And so I don't have
any problem asserting the actual choice.
Yes, there is a way of choosing um but I
can't tell necessarily tell you what it
is. But in a mathematical argument, I
can assume that I fix the choice
function because I know that there is
one. So it's a the philosophical
difference between working when you have
the exim of choice and when you don't is
the question of this constructive nature
of the argument. So if you make an
argument and you appeal to the axim
choice, then maybe you're admitting that
the objects that you're producing in the
proof are not going to be constructive.
you're not going to be able to
necessarily say specific things about
them. But if you're just claiming to
make an existence claim, that's totally
fine. Whereas, if you have a
constructive attitude about the nature
of mathematics and you think that
mathematical claims maybe are only
warranted when you can provide an
explicit procedure for producing the
mathematical objects that you're dealing
with, uh, then you're probably going to
want to deny the axim choice and maybe
much more.
Can we maybe speak to the axioms that uh
underly CSC? So going to perplexity ZFC
or is a Mela frankl set theory with the
axiom of choice as we mentioned is the
standard foundation from most modern
mathematics. It consists of the
following main axioms. Axiom of
extensionality, axiom of empty set,
axiom of pairing, axiom of union, axom
of power set, axiom of infinity, axiom
of separation, aim of replacement, axum
of regularity, and axiom of choice. Uh,
some of these are quite basic. Um but it
would be nice to kind of give people a
sense
>> of what it means to be an axiom like
what kind of basic facts we can lay on
the table on which we can build some
beautiful mathematics.
>> Yeah. So the history of it is really
quite fascinating. So Zerlo introduced
most of these axioms I mean as part of
what's now called Zurlo set theory to
formalize his proof from the axim choice
to the well-order principle which was an
extremely controversial result. So in
1904 he gave the proof without the
theory and then it was challenged to
provide the theory and so in 1908 he
produced the Zumelo set theory and gave
the proof that in that theory you can
prove that every set admits a well
orderering. Um and so the acts on the
list these things like extensionality
express the the most fundamental
principles of the understanding of sets
that he wanted to be talking about. So
for example, extensionality says if two
sets have the same members, then they're
equal. So it's this idea that the sets
consist of the collection of their
members and that's it. There's nothing
else that's going on in the set. So it's
just if if two sets have the same
members, then they are the same set. So
it's maybe the most primitive uh axiom
in some respect. Well, there's also just
to give a flavor,
uh there exists a set with no elements
called the empty set. Uh for any two
sets, there's a set that contains
exactly those two sets as elements. For
any set, there's a set that contains
exactly the elements of the elements of
that set. So the union set and then
there's the power set. For any set,
there's a set whose elements are exactly
the subsets of the original set, the
power set. And the axum of infinity,
there exists an infinite set. typically
a set that contains the empty set and is
closed under the operation of adding one
more element. Back to our hotel example.
>> That's right.
>> And there's there's more. But this it's
kind of fascinating. Um I used to put
yourself in the mindset of people at the
beginning of this of trying to formalize
set theory.
It's it's fascinating that humans can do
that. I read some historical accounts by
historians about that time period,
specifically about Zerla's axioms and
his proof of the well-order theorem. And
the historians were saying, um, never
before in the history of mathematics has
a mathematical theorem been argued about
so publicly and so viciferously
uh as that theorem of Smela's. Um and
it's fascinating also because the axim
of choice was widely regarded as a kind
of you know basic principle at first but
then when but people were very
suspicious of the well order theorem
because no one could imagine a well
orderering say of the real numbers and
so this was a case when Zumelo seemed to
be from principles that seemed quite
reasonable proving this obvious untruth
and so people were mathematicians were
objecting um but then Zermelo and others
actually looked into the mathematical
papers and so on of some of the people
who had been objecting so viciferously
and found in many cases that they were
implicitly using the axim choice in
their own arguments even though they
would argue publicly against it because
it's so natural to use it because it's
such an obvious principle in a way I
mean it's easy to just use it by
accident with if you're not critical
enough and you don't even realize that
you're using the axim choice. That
that's true now even people like to pay
attention to when the axim choice is
used or not used in mathematical
arguments. I mean up until this day it
used to be more important in the early
20th century it was very important
because people didn't know if it was a
consistent theory or not and there were
these antitomies arising and so there
was a worry about consistency of the
axioms. Um but then of course eventually
with the result of of Gerel and Cohen
and so on this consistency question
specifically about the eximma choice
sort of falls away. We know that it the
aim of choice itself will never be the
source of inconsistency and set theory.
If there's inconsistency with the axim
choice then it's it's already
inconsistent without the axim of choice.
So it's not the cause of inconsistency.
And so in that from that point of view
the need to pay attention to whether
you're using it or not from a
consistency point of view is somehow
less important but still there's this uh
reason to pay attention to it on the
grounds of these uh constructivist ideas
that I had mentioned earlier
>> and we should say in set theory
consistency means that it is impossible
to derive a contradiction from the
axioms of the theory.
means that there's no contradictions.
That's that's a a consistent aimatic
system that there's no contradictions.
>> A consistent theory is one for which you
cannot prove a contradiction from that
theory.
>> Maybe a quick pause, quick break, quick
bathroom break.
You mentioned to me offline uh we were
talking about Russell's paradox and that
there's a nice another kind of
anthropomorphizable
proof of uncountability. Uh I was
wondering if you can lay that out. Oh
yeah, sure. Absolutely.
>> Both Russell's paradox and the proof,
>> right? So let's So we we talked about
Cander's proof that the real numbers,
the set of real numbers is an
uncountable infinity. It's a strictly
larger infinity than the natural
numbers. But Kander actually proved a a
much more general fact, namely that for
any set whatsoever,
the power set of that set is a strictly
larger set. So the power set is the set
containing all the subsets of the
original set. So if you have a set and
you look at the collection of all of its
subsets, then Caner proves that this is
this is a bigger set. They're not
equinumerous. Of course, there's always
at least as many subsets as elements
because for any element, you can make
the the singleton subset that has only
that guy as a member, right? So there's
always at least as many subsets as
elements. But the question is whether
they whether it's strictly more or not.
And so Caner reasoned like this. It's
very simple. It's a kind of distilling
the abstract diagonalization idea
without encumbered by the complexity of
the real numbers. So we have a set X and
we're looking at all of its subsets.
That's the power set of X. Suppose that
X and the power set of X have the same
size. Suppose there's contradiction.
they have the same size. So that means
we can associate to every individual of
X a subset.
And so now let me define a new set I
mean another set. I'm going to define
it. Let's call it D. And D is the subset
of X that contains all the individuals
that are not in their set.
Every individual
was associated with a subset of X. And
I'm looking at the individuals that are
not in that their set. Maybe nobody's
like that. Maybe there's no element of X
that's like that. Or maybe they're all
like that. Or maybe some of them are and
some of them aren't. I don't it doesn't
really matter for the argument. I
defined a subset D consisting of the
individuals that are not in the set
that's attached to them. But that's a
perfectly good subset. And so because of
the equinosity it would have to be
attached to a to a particular individual
you know and but that let let's call
that person uh it should be a name
starting with D so Diana
>> and now we ask is Diana an element of D
or not but if Diana is an element of D
then she is in her set so she shouldn't
be because the set D was the the set of
individuals that are not in their set.
>> So if Diana is in D, then she shouldn't
be. But if she isn't in D, then she
wouldn't be in her set and so she should
be in D.
>> That [clears throat] that's a
contradiction. So therefore, the number
of subsets is always greater than the
number of elements for any set.
And the anthropomorphizing idea is the
following. I like to talk about it this
way. uh for any collection of people you
can you can form more committees from
them than there are people
even if you have infinitely many people.
>> Mhm.
>> Suppose you have an infinite set of
people. And what's a committee? Well, a
committee is just a list of who's on the
committee basically the members of the
committee. So there's there's all the
twoperson committees and there's all the
one person committees and there's the
universal the worst committee the one
that everyone is on. Okay. The best
committee is the empty committee with no
members and never meets and so on. Yeah.
Or is the empty committee meeting all
the time? I'm not sure.
>> Yeah. Wow. That's a profound question.
And does a committee with just one
member meet also? Yes.
>> Yeah. Maybe it's always in session. I
don't know.
>> Yeah.
>> So So the the the claim is that there's
more committees than people.
>> Mhm.
>> Okay. Suppose not. Well, then we could
make an association between the people
and the committee. So, we would have a
kind of every committee could be named
after a person in a onetoone way. And
I'm not saying that the person is on the
committee that's named after them or not
on it, whatever. Maybe sometimes that
happens, sometimes it doesn't. I don't
know. It doesn't matter. But let's form
what I call committee D, which consists
of all the people that are not on the
committee that's named after them.
Okay, maybe that's everyone. Maybe it's
no one. Maybe it's half the people. It
doesn't matter. That's a committee. It's
a set of people.
And so it has to be named after someone,
let's call that person Daniela.
So now we ask, is Daniela on the
committee that's named after her? Well,
if she is,
then she shouldn't be because it was the
committee of people who aren't on their
on their own committee. uh and if she
isn't then she should be. So again it's
a contradiction. So when I was teaching
in Oxford one of my students uh came up
with the following different
anthropomorphization of canvas argument.
Let's consider all possible fruit
salads. We have a given collection of
fruits
>> you know apples and oranges and grapes
whatever and a fruit salad consists of
some collection of those fruits. So
there's the banana, pear, grape salad,
and so on. There's a lot of different
kinds of salad. Every set of fruits
makes a salad, a fruit salad. Okay? And
we want to prove that for any collection
of fruits, even if there are infinitely
many different kinds of fruit,
for any collection of fruits, there are
more possible fruit salads than there
are fruits.
So if not then you can put a onetoone
correspondence between the fruits and
the fruit salads. So you could name
every fruit salad after a fruit might
not be that fruit might not be in that
salad. Doesn't matter. We're just it's a
naming a one to one correspondence. And
then of course we form
the diagonal salad which consists of all
the fruits that are not in the salad
that's named after them. Mhm.
>> And and that that's a perfectly good
salad. It might be the kind of diet
salad if it was the empty salad or it
might be the universal salad which had
all fruits in it if all the fruits were
in it.
>> Or it might have just some and not all.
So that diagonal salad would have to be
named after some fruit. So let's suppose
it's named after durian, meaning that it
was associated with durian in the 1:1
correspondence. And then we ask, well,
is durian
in the salad that it's named after? And
if it is, then it shouldn't be, and if
it isn't, then it should be. And so it's
again the same contradiction. So, so all
of those arguments are just the same as
Caner's proof that the power set of any
set is bigger than the set. And this is
exactly the same logic that comes up in
Russell's paradox because Russell is
arguing that um the class of all sets
can't be a set
uh because if it were then we could form
the the set of all sets that are not
elements of themselves.
So basically he's what Russell is
proving is that there are more
collections of sets than than elements
because we can form the diagonal class.
Yeah. The class of all sets that are not
elements of themselves. If that were a
set
then it would be an element of itself if
and only if it was not an element of
itself. It's exactly the same logic in
all four of those arguments. Yeah. So
there can't be a class of all sets
because if there were then there would
have to be a class of all sets that
aren't elements of themselves. But but
that set would be an element of itself
if and only if it's not an element of
itself which is a contradiction. So this
is the essence of the Russell paradox.
Um I don't call it the Russell paradox.
Actually when I teach it I call it
Russell's theorem. There's no universal
set. Uh and uh it's not really confusing
anymore. At the time it was very
confusing but um uh now we've absorbed
this nature of set theory into our
fundamental understanding of of how sets
are and it's not confusing anymore. I
mean the history is fascinating though
of the Russell paradox because
um before that time Frager was working
on his monumental work undertaking
implementing the philosophy of logicism
which is the attempt to reduce all of
mathematics to logic. So Frager wanted
to give an account of all of mathematics
in terms of logical notions
and he was writing this monumental work
and had formulated his basic principles
and those principles happened to imply
that for any property whatsoever you
could form the set of objects with that
property.
This is known as the general
comprehension principle.
and and he was appealing to the
principles that support that axiom uh
throughout his work. I mean it was
really it wasn't just an incidental
thing. he was really using this
principle and Russell wrote him a letter
when he observed the work in progress
that there was this problem because if
you accept the principle that for any
property whatsoever you can make the set
of objects with that property then you
could form the set of all sets that are
not members of themselves. That's just
an instance of the general comprehension
principle. And [snorts]
but
the set of all sets that aren't elements
of themselves can't be a set because if
it were then it would be an element of
itself if and only if it's not a member
of itself. And that's a contradiction.
And so Russell wrote this letter to
Frager and it was just at the moment
when Frager was finishing his work. It
was already at the publishers and you
know in press basically but it's
completely devastating. I mean, it must
have been such a a horrible situation
for Frager to be placed in because he's
finished this monumental work, you know,
years of his life dedicated to this and
Russell finds this basically
oneline proof of a contradiction in the
fundamental principles of the of the
thesis that completely destroys the
whole system. And
Frager had put in the appendix of his
work a response to Russell's letter in
which he explained what happened and he
wrote very gracefully, "Hardly anything
more unwelcome can befall a scientific
writer than to have one of the
foundations of his edifice shaken after
the work is finished. This is the
position into which I was put by a
letter from Mr. Burgeon Russell as the
printing of this volume was nearing
completion." And then he goes on to
explain the matter concerns his basic
law five and so on. And
>> it's heartbreaking. I mean there's
nothing more traumatic to a person who
dreams of constructing mathematics all
from logic to get a very clean simple
contradiction.
I mean that's just uh
>> you devote your life to this work and
then it's shown to be contradictory and
that must have been heartbreaking.
>> What do you think about the the Frager
project? The philosophy of logic, the
dream of the power of logic to construct
the mathematical universe. Of course the
the project of logicism did not die with
rea and it was uh continued and you know
there's a whole movement the neolicists
and so on in contemporary times even but
my view of the matter is that really we
should view the main goals of logicism
are basically completely fulfilled in
the rise of set theoretic
foundationalism. I mean uh when you view
ZFC as the foundation of mathematics and
in my view the principles of ZFC are
fundamentally logical in character
including the axim of choice as I
mentioned as a principle of logic. This
is a highly disputed point of view
though because a lot of people take even
the axim of infinity as not as
mathematical inherently mathematical and
not logical and so on. But I think if
you adopt the view that the principles
of CFC have to do with the principles of
abstract you know set formation which is
fundamentally logical in character then
it's complete success for loicism. So
the fact that set theory is able to
serve as a foundation means that
mathematics can be founded on logic.
>> I think this is a good moment to talk
about uh Girdle's incompleteness
theorems. So can you explain them and uh
what did they teach us about uh the
nature of mathematical truth?
>> Absolutely. It's one of the most
profound developments in mathematical
logic. I mean the incompleteness
theorems is when mathematical logic in
my view first became sophisticated.
It's a kind of birth of the subject of
mathematical logic. But to understand
the theorems, you really have to start a
little bit earlier with Hilbert's
program because at the time, you know,
with the Russell paradox and so on,
there were these various contradictions
popping up in various parts of set
theory and the Barali 40 paradox and so
on and and Hilbert was famously
supportive of set theory. I mean,
there's this quote of him um saying, "No
one shall cast us from the paradise that
can created for us. And what I take him
to mean by that is he was so captured by
the idea of using set theory as a
foundation of mathematics. And it was so
powerful and convenient and unifying in
a way that was extremely important. And
he wasn't he didn't want to give that up
despite the danger of these paradoxes,
these contradictions basically is how
some people viewed them. And so
>> this minefield of paradoxes. Yeah.
>> Right. A mindfield. That's a really good
way of describing the situation. And so
Hilbert said, well, look, we have to fix
this problem. You know, we want to use
the set theory foundations, but we want
to do it in a way that is trustworthy
and reliable. We can't allow that the
foundations of mathematics are in
question. You know, this is a kind of
attitude I think that underlies Hilbert
and the Hilbert program. And so he
proposed, look,
we're going to have this strong theory,
this set theory that we want to be
proving our theorems in. Um, but uh, I
mean, on the one hand, we we want it to
be as strong as possible. We would like
it to answer all the questions that
there's another famous quote of Hilbert
in his retirement address where um, he
proclaims uh, Vermusen Viss.
So we we must know we we will know in
which he's very optimistic about the
ability of mathematics to answer all of
the questions of mathematics that we
have posed. We have all these problems
we want to solve and he is saying we're
going to do it. We're going to solve all
these problems. So we want to propose
this strong theory and one has the sense
that he had in mind set theory um in
which all the questions are going to be
answered. Okay. But secondly, we want to
combine that with
in a very weak arithmetic purely
fitistic theory. We want to prove that
the reasoning process of the strong
theory is safe. Okay. So in order to
make sense of that point of view, you
you basically have to invent the
philosophy of formalism.
uh where we can look at what is a proof
what is it the nature of mathematical
reasoning and on Hilbert's way of
thinking about this a proof is basically
a itself aistic
kind of object it's a sequence of if you
think about the nature of what a proof
is it's it's a sequence of assertions
which can be viewed as sort of sequences
of symbols that conform with certain
rules of logical reasoning and and this
is a a formalist way of understanding
the nature of proof. So we think about a
proof in a kind of syntactic formal way
even though the contents of those
statements might be referring to
infinite uncountable objects. The
statements themselves are not infinite
uncountable objects. The statements
themselves are just finite sequences of
symbols. So we kind of think of proof as
maybe it's fair to say almost like
outside of math. It's like tools
operating on math. And then for Hilbert,
he thought proof is inside the aimatic
system. Something like this.
>> Yeah, that's helpful.
>> That's wild.
>> The main thing about formalism is that
you you think of the process of doing
mathematics, you divorce it from the
meaning of the mathematical assertions,
right? So the meaning of the
mathematical assertions that you make in
this infinitary theory has to do with
these huge uncountable infinities and so
on possibly and and and and that's a
very sort of uncertain realm maybe and
the source of the paradoxes and so on in
some people's minds. And so um but the
reasoning process itself is consists of
writing down sequences of symbols on
your on your page and you know
undertaking a a an argument with them
which is following these finitary rules.
And so so so if we divorce the meaning
of the symbols from just the process of
manipulating the symbols it's a way of
looking at the nature of mathematics as
a kind of formal game in which the
meaning may be totally absent. I don't
think it's necessarily part of the
formalist view that there is no meaning
behind but rather it's emphasizing that
we can divorce the meaning of the
sentences from the process of
manipulating those sentences and then
Hilbert wanted to prove in this purely
theory that if we follow the rules of
that game we're never going to get a
contradiction.
>> Mhm. So those were the two aims of the
Hilbert program is to found the strong
infinitary theory probably set theory
which is going to answer all the
questions and then secondly prove in the
finitary theory that the strong theory
is safe in other words consistent. Yeah.
>> What does the word finitary infinitary
theory mean?
>> Yeah. Well, this is of course
philosophically contentious and people
have different ideas about what exactly
it should mean and so there's hundreds
of papers on exactly that uh question,
but I I like to take it just kind of
informally. I mean it means that we're
talking about finite sequences of
symbols and we're going to have a a
theory you know finite strings of
symbols and we a finitary theory would
be one whose subject matter is about
those kinds of things so that we can
conceivably argue about the nature of
these finite strings. So a a proof is
just a finite sequence of statements so
that every statement is either one of
the axioms or follows by the laws of
logic from the earlier statements in
some specified manner like using modus
ponent or some other law of logic like
that um and such that the last line on
the list is you know the theorem that
you're proving. So that's what a proof
is in this kind of way of thinking. to
take a specific example. I mean I always
conceive of the perhaps the most natural
finitary theory that one would be called
upon to exhibit would be piano
arithmetic the theory of piano
arithmetic which which is a first order
theory of the nature of arithmetic. Um
but okay so some people say well piano
arithmetic has these strong first order
induction axioms and there's much much
weaker versions of arithmetic like I
sigma KN or I sigma 1 and so on which
are um even more finitary than piano
arithmetic. So different philosophical
positions take different attitudes about
what is what does it take to be finitary
how finitary do you have to be to be
truly finitary. So according to
perplexity, piano arithmetic is a
foundational system for formalizing the
properties and operations of natural
numbers using a set of axioms called the
piano axioms. Piano arithmetic provides
a formal language and axioms for
arithmetic operations such as addition
and multiplication over the natural
numbers. The axioms define the existence
of a first natural number usually 0 one.
The concept of successor function which
generates the next natural number. rules
for addition and multiplication builds
from these concepts. The principle of
induction allowing proofs around all
natural numbers and it goes on. So it's
a very particular kind of arithmetic
that is affinitary.
>> In my I view it asary but this a
contentious view. I mean not everyone
agrees with that. That's what I stress
trying to hint at. All right.
>> Um, piano arithmetic is one of the
hugely re hugely successful theories of
the natural numbers and and elementary
number theory. Essentially all of
classical number theory. So whatever
kind of theorems you want to be proving
about the prime numbers or factorization
or uh any kind of finitary reasoning
about finite combinatoral objects all of
it can be formalized in piano
arithmetic. I mean that's the the the
basic situation. Of course, one has to
qualify those statements in light of the
girdle incompleteness theorem, but uh
for the most part, the classical
um number theoretic analysis of the of
the finite numbers is uh almost entirely
developable inside piano arithmetic.
So if we if we go back to the Hilbert
program, so Hilbert has these two goals.
Produce the strong theory which is going
to answer all the questions and then
prove by purely means that that theory
will never lead into contradiction. And
one can think about well the
incompleteness theorem should be viewed
as a decisive reputation of the Hilbert
program. It it defeats both of those
goals decisively completely.
But but before explaining that maybe one
should think about you know what if
Hilbert had been right what would be the
nature of mathematics in the world that
Hilbert is telling us to search for
>> and if I may going to perplexity
definition of Hilbert's program it was
David Hilbert's early 20th century
project to give all of classical
mathematics a completely secure finitary
foundation in essence the goal was to
formalize all of mathematics and precise
exiatic systems
and then prove using only very
elementary reasoning about symbols that
these systems are free of contradiction.
>> Right. Exactly right. Let's imagine what
it would be like if he were if he were
had been right. So we would have this
finitary theory
and it would prove that the strong
theory was free of contradiction. So we
could start enumerating proofs from the
strong theory. We can I mean right now
we can write a computer program that
would systematically generate all
possible proofs
from a given theory.
And
so we could have like this theorem
enumeration machine that just spit out
theorems all day long in such a manner
that every single theorem would
eventually be produced by this device.
And so if you had a mathematical
question
of any kind,
you could answer it by just waiting for
either the answer to come out yes from
the machine or the answer to come out
no. So the the nature of mathematical
investigation in Hilbert's world is one
of just turning the crank of the theorem
or enumeration machine devoid of
creative thinking or imagination. it's
just getting the answer from the from
this by wrote procedure. So so Hilbert
in effect is telling us I mean in with
his program that the fundamental nature
of mathematics is wrote computation. I
mean the way I think about the Hbert
program seems extremely attractive in
the historical context of being worried
about the antitomies the the
inconsistencies
and so how can we kind of block them and
so it seems natural
>> first of all to have a strong theory
that's going to answer all the questions
because the idea of logical independence
and pervasiveness that we now know
exists um just wasn't uh you know there
was no known they didn't know anything
like that happening ever and so um it's
natural to think that it wouldn't happen
and also that they would be able to
guard against this inconsistency. So it
seems like the goals of the Hbert
program are quite natural in that
historical context. But you know when
you think a little more about what the
nature of it would be like it shows you
this kind of wrote procedure and now
you're saying well that doesn't seem so
unlikely maybe I mean in the light of
the increasing computer power and so on
it's actually maybe turning into our
everyday experience where the machines
are calculating more and more for us and
um in a way that uh could be alarming.
Okay. But okay, so to talk about the
alternative to the Hilbert point of
view, I mean, if he's wrong, then what
is the nature of mathematical reality?
Well, it would mean that we couldn't
ever maybe for the first goal, we
couldn't ever write down a theory that
answered all the questions.
So we would always be in a situation
where our best theory even the
infinitary theories
uh
would have questions that they stumble
with and are unable to answer
independence uh would occur. But then
also because of the failure of the
second goal, we would also have to be
constantly worrying about whether our
theories were consistent or not. and we
wouldn't have any truly convincing means
of saying that they were free from
contradiction.
And the fact of Girdle of Girdle's
incompleteness theorem shows that that
is exactly the nature of mathematical
reality actually those are the two
incompleteness theorems. So the first
incompleteness theorem says you cannot
write down a computably exatizable
theory that answers all the questions.
every such theory will be incomplete
assuming it includes a certain amount of
arithmetic and secondly no such theory
can ever prove its own consistency. So
not only is it the case that theitary
theory can't prove the consistency of
the strong infinitary theory but even
the infinitary theory can't prove its
own consistency. Right? That's the
second incompleteness theorem. And so
it's in that sense a decisive takedown
of the Hilbert program. Um which is
really quite remarkable the extent to
which his theorem just really answered
that whole puzzle. Um it's quite
amazing. I mean there's another aspect
kind of easy to think about. I mean, if
you're if you're wondering about
theories that prove their own
consistency, then I mean, would you
trust a theory that proves of itself
that it's consistent? I mean, that's
like it's like the used car salesman
telling you, "Oh, I'm trustworthy." I
mean, it's not a reason to trust the
used car salesman, is it? Just because
he says that. So similarly if you have a
theory that proves its own consistency
well I mean even an inconsistent theory
would prove its own consistency and so
it doesn't seem to be a logical reason
to believe in the consistency if you
have a theory that proves itself
consistent. Uh just for clarification
you use the word theory is it in this
context synonymous with axiomatic system
>> right so in mathematical logic theory is
a technical term and it means any set of
sentences in a formal language and so if
you say aiometic system it's basically
synonymous to my usage with theory. So a
theory means you know the consequences
of a set of axioms or people are
sometimes unclear on whether they just
mean the axioms or the consequences of
the axioms.
>> So theory includes both the axioms and
the consequences of the axioms and you
use it interchangeably and the context
is supposed to help you figure out which
of the two you're talking about the
axioms or the consequences. Or maybe to
you they're basically the same.
>> Yeah. But they're so closely connected
although you know all the features
aren't the same. So uh if you have a
computable list of axioms for a theory
then you can start enumerating the the
consequences of the axioms but you won't
be able to computably decide whether a
given statement is a consequence or not.
You can enumerate the consequences. So
you can you can semiide the consequences
but you won't be able to decide yes or
no whether a given statement is a
consequence or not.
So it's the distinction between a
problem being computably decidable and a
problem being computably innumerable
which um was made clear following the
work of Turing and others that came from
from that. I mean it so that's one
difference between the list of axioms of
the theory and the theory itself. The
axioms could be you can decide maybe
computably whether something is an axiom
or not but that doesn't mean that you
can decide computably whether or not
something is a theorem or not. Usually
you only get to decide the positive
instances. If something is a theorem,
you will eventually come to recognize
that. But if something isn't a theorem,
maybe at no point will you be able to
say no that's not a theorem.
>> And that's of course connected to the
halting problem and all of these all of
these contradictions and paradoxes are
all nicely beautifully interconnected.
>> That's right. Yeah. Absolutely. Yeah.
>> So can we just linger on Gears of
completeness theorem? You mentioned the
two components there. You know, there's
so many questions to ask like what what
is the difference between provability
and uh truth? What is true and what is
provable? Maybe that's a good line. This
is a really core distinction that
it's fascinating to me to go back and
read even the early 20th century people
before Girdle and Tarski
and they were totally sloppy about this
distinction between truth and proof. It
wasn't clear at all until Girdle
basically although even as late as
Borbachi has a kind of confusion in this
foundational work. So this standard
graduate level textbooks used in France
in the presentation of logic they are
conflating truth and proof. To be true
for them means to be provable. So in the
early days, maybe it wasn't clear enough
that the concept of truth needed a
mathematical investigation or analysis.
Maybe it was already taken to be fully
clear. But because of the incompleteness
theorem, we realize that actually
there's quite subtle things happening,
right? And so why don't we talk about
this distinction a bit? To me, it's
absolutely core and fundamental to our
understanding of mathematical logic.
this distinction between truth and
proof.
So
truth is on the semantic side of the
syntax semantics dichotomy. Truth has to
do with the nature of of reality. I mean
okay when I talk about reality I'm not
talking about physical reality. I'm
talking about mathematical reality. So
so we have a concept of something being
true in a structure a statement being
true in a mathematical structure. Like
maybe you have the real field or
something and you want to know does it
satisfy this statement or that statement
or you have a group of some kind or
maybe you have a graph this is a
particular kind of mathematical
structure that has a bunch of vertices
and edges and you want to know um you
know does does this graph satisfy that
statement and Tarsuski gave this
absolutely wonderful account of the
nature of truth in what's now known as
the disquotational theory of truth and
What Tarski says is the sentence quote
snow is white unquote is true if and
only if snow is white
and what he means by that is look to say
truth is a property of an assertion. So
we can think of the assertion as a
syntactically.
So the this the sentence is true if and
only if the content of the sentence is
the case. Yeah. So the sentence snow is
white, you know, in quotations
uh is true, that just means that snow is
white. And that that's why it's called
the disquotational theory because we we
remove the quotation marks from the
assertion, right? and and you can use
this idea of disquotation to give a
formal definition of truth in a
mathematical structure of a statement in
a formal language. So for example, if I
have a formal language that allows me to
make atomic statements about the objects
and relations of the structure and I can
build up a formal language with you know
with with the logical connectives of and
and or implies and not and so on and
maybe I have quantifiers right then for
example to say that the structure
satisfies f and c
that that single statement f and c I'm
thinking of that as one statement just
means means that it satisfies fe and it
satisfies C. And if you notice what
happened there, I at first the and was
part of the sentence inside the
sentence, but then in the second part I
was using the word and to refer to the
conjunction of the two conditions. So
>> yeah, it has the disquotation.
>> Yeah, it has the disquotation. And so
this idea can be done for all the
logical connectors and quantifiers and
everything. you're you're applying
Tarosk's idea of discretation and it
allows you to define by induction the
the truth of any assertion in a formal
language inside any mathematical
structure and so to say that a sentence
is true first of all it's ambiguous
unless you tell me which structure
you're talking about it being true in
um and so maybe we have in mind the
standard model of arithmetic or
something with the natural numbers and
the arithmetic structure and I want to
know is a given statement true in that
structure
then then we have a formal definition of
what that means according to the Tarski
recursive definition of truth.
Okay, that's truth. Proof on the other
hand is you know in this Hilbert way of
thinking we can develop proof theory.
What is a proof for a mathematician for
a mathematical logician? A proof is a
certain sequence or or arrangement of
sentences in the formal language that
accord with the logical rules of a proof
system. So there's certain modes of
reasoning that are allowed. So if you
know A and you know A implies B in the
proof, then at a later step you're
allowed to um uh to write B as a
consequence. So if you know A and you
know A implies B those are both two
statements that are known then you can
deduce B as a consequence according to
the rule of modus ponents this is the
rule modus ponent and you know there's a
lot of other rules some people would
call this implication elimination
there's different kinds of proof systems
there's a lot of different formal proof
systems that exist that are studied by
the proof theorists and all of them have
the property that they're sound which
means that if the premises of the
argument are all true in a structure and
you have a proof to get a conclusion,
then the conclusion is also true in that
structure. So that's what it means to be
sound that proof proofs preserve truth.
They're truth preserving arguments.
Okay. But also um
the proof systems are also generally
complete. They're both sound and
complete. And complete means
um that whenever a statement is a
consequence a logical consequence of
some other statements which means that
whenever the assumptions are true then
the then the consequence is also true in
in the structure. So whenever you have a
logical consequence then there is a
proof of it. Okay. And the proof systems
generally have both of those properties.
They're sound and complete. There's a
third property a lot of logicians talk
about sound and complete. sound and
complete this, sound and complete that.
But actually there's a hidden third
adjective that they should always be
talking about in any such case which is
that um you should be able to recognize
whether or not something is a proof or
not. So there's a computable aspect uh
to to the proof systems. We want to be
able to recognize whether something is a
proof. It should be computably decidable
whether a given sequence of statements
is a proof or not. So we don't want a
proof system in which someone claims to
have a proof but we can't check that
fact that whether it's a proof or not.
>> We want to be able to you know to
correctly adjudicate all claims to
having a proof.
>> Yeah. A mathematician comes to mind that
said he has a proof but the margins are
too small to contain.
>> Exactly.
>> So
>> so that doesn't count as proof. So
generally all the classical proof
systems that are used are sound and
complete and also computably decidable
in the sense that we can decide whether
something is a proof or not. So what is
again the tension between truth and
proof? Which is more powerful and how do
the two interplay with the
contradictions that we've been
discussing. So the incompleteness
theorem is the question whether we could
say write down a theory for arithmetic.
Say for the standard model of arithmetic
where we have the natural numbers and
plus and times and 01 and less than and
so on. In that formal language we can
express an enormous number of statements
about the nature not only of arithmetic
but actually by various coding methods.
We can express essentially all of finite
mathematics in that structure. So the
question would be can we write down a
computable list of axioms that will
answer all those questions by proof. In
other words, we want to have a complete
theory, a theory of arithmetic that
proves all and only the true statements.
That would be the goal. Hilbert would
love that. I mean that would be
supportive of Hilbert's program to have
such a complete theory of arithmetic.
And Girdle proved that this is
impossible. You cannot write down a
computable list of axioms that is
complete in that sense. There will
always be statements
if the theory is consistent. There will
always be statements that you cannot
prove and you cannot refute. So they are
independent of that theory.
>> How traumatic is that that there are
statements that are independent from the
theory.
>> I mean my view is that yeah this isn't
traumatic at all. This is rather
completely eyeopening
in terms of our understanding of the
nature of mathematical reality. I mean,
we're not
we we understand this profound fact
about our situation with regard to
mathematical truth. The incompleteness
theorem tells us look we just can't
write down a list of axioms that is
going to be consistent and is going to
answer all the questions. It's
impossible. And so I don't think of it
as trauma. I just think look this is the
nature of of mathematical reality and
it's good that we know it. And so now we
need to move on from that and you know
do what we can in light of that. Is it
fair to say that that in general it
means if I give you a statement you
can't know if your axiomatic system
would be able to prove it.
>> That's right. In general you cannot the
the provability problem we can formulate
it as a decision problem. Given a theory
and given a statement is that statement
a consequence of that theory? Yeah. This
is one of the most famous decision
problems. in fact the very first one
because it's the it's equivalent to the
Hilbert Aurman um and Chiden's problem
which is also appearing in the title of
Turing's 1936 paper that was so
important for computability theory so
it's it's a formulation of the inchings
problem does a given theory
have a given statement as a logical
consequence which because of Girdle's
completeness theorem not his
incompleteness theorem but his earlier
completeness theorem
Girdle had proved that the proof systems
that they studied did have this
completeness property that I mentioned.
So provability is the same as logical
consequence. So and this is an
undecidable decision problem. Turing
proved and uh and we now know it's
equivalent to the halting problem.
>> Can you describe the halting problem
because it's a it's a thing that shows
up in a very useful and again traumatic
way through a lot of computer science
through a lot of mathematics. Yeah, the
halting problem is expressing a
fundamental property of computational
processes. So given a program or maybe
we think of it as a program together
with its input, but let me just call it
a program. So given a program, we could
run that program, but I want to pose it
as a decision problem. Will this program
ever complete its task? Will it ever
halt? And the halting problem is the
question given a program, will it halt?
Yes or no?
And
of course
for any one instance the answer is
either yes or no. The qu that's not what
we're talking about. We're ask we're
talking about whether there's a
computable procedure to answer all
instances of this question. So it's a
decision problem is given as a scheme of
instances for all possible programs that
you could ask about. What I want to know
is is there a computable procedure that
will answer those questions. Right? And
and it turns out the answer is no. The
halting problem is computably
undecidable. There is no computable
procedure that will correctly answer all
instances of whether a given program
will halt. And of course we can get half
the answers in the sense that
you give me a program and you say will
this halt and I could take that program
and I could run it.
>> Mhm.
>> And I could keep running it and maybe in
a week it would halt and at that time I
could say yes it halted. So I can get
the yes answers correctly for halting
all the yes answers.
But the problem is
if it didn't halt yet like maybe I
waited you know a thousand years and it
still hasn't halted. I don't seem
entitled to say no it's not going to
hold yet cuz maybe in a thousand1 years
it'll halt. And so at no point can I
seem to say no. In order to say no it
won't ever hold.
It seems like I would have to really
understand how the program worked and
what it was doing.
So giving the yes answers was sort of
trivial. You didn't have to understand
it. You just needed to run it which is a
kind of wrote task. But to give the no
answers, you need to have a kind of deep
insight into the nature of the program
and what it's doing in such a way that
you would understand it and be able to
see oh no I can see this program is
never going to halt because you know
it's a much more difficult task to say
no it won't halt than it is to say yes
it halted cuz I ran it and it halted. Um
and it turns out to be impossible to
have a computer procedure that gives the
no answers. Yeah. And the argument is
not very difficult. Should we do it?
>> Yes, let's let's do it.
>> Okay. Suppose toward contradiction. I
mean all these views are by
contradiction and this argument is going
to be a diagonal argument in the same
style as the Russell argument and the
Cander argument and Girdle's argument
that we haven't talked about yet. So
many diagonal arguments come in. So
suppose towards contradiction that we
had a procedure for determining whether
a given program halted on a given input.
Now let me describe I'm going to use
that procedure as a subruine in the
following process and my process let's
call it Q process Q and it takes as
input a program P okay and the first
thing it does is it asks that subruine
hey would P halt if I ran it on P itself
okay that's the diagonal part because
we're applying P to P
Okay, so I'm describing program Q and
program Q takes as input P which is
itself a program and the first thing it
does is it asks the halting subretine
program would P halt on P
and if the answer comes back from the
subretine yeah that would hold then what
I do in program Q is I immediately jump
into an infinite loop so I don't hold if
P holds on P I don't hold
>> but if the answer came back No, P is
never going to halt on P. Then I halt
immediately.
Okay. So the and that's it. I've
described what Q does.
>> Mhm. And the thing about Q is that Q Q's
behavior on P was the opposite of P's
behavior on P. I mean that's how we
designed Q specifically
so that Q on P had the opposite behavior
as P on P. Okay. So now of course what
do we do? Well the same thing that
Russell did and so forth and counter we
we ask well what would Q do on Q?
And because of this opposite behavior, Q
would halt on Q if and only if Q does
not halt on Q, which is a contradiction
because Q has to have the opposite
behavior on Q than Q does. But that's
just contradictory.
>> What a beautiful proof.
>> It's absolutely beautiful. Yeah, I
agree. And it's following the same logic
of Russell and Caner. I mean going back
to Caner basically because Russell is
also quoting Caner in his letter to
Frag. So therefore the conclusion is
that the halting problem is not
computably decidable.
And now we can immediately prove
Girdle's theorem using this. Actually
it's an immediate consequence. So why
don't we just do that? I view this as
the simplest proof of Girdle's theorem.
You don't need the Girdle sentence to
prove Girdle's theorem. You can do it
with the halting problem.
So suppose
that we could write down a computable
aimization of all of the true facts of
elementary mathematics
meaning arithmetic and finite
combinatorial things such as turning
machine computations and so on. So in
fact all those finite combinatorial
processes are formalizable inside
arithmetic with the standard
arithmetization coding process. But let
me just be a little bit informal and say
suppose we could write down a complete
theory of elementary finite mathematics.
So we have a an aximization of that
theory.
Then we could produce all possible
theorems from those axioms in the way
that I was describing earlier with
Hbert's program. I mean if we had a
complete theory of elementary
mathematics, we could construct a
theorem enumeration machine that
produced all the theorems and only the
theorems from that theory.
So now I have this theorem enumeration
device on my desk and I announce that
I'm open for business to solve the
halting problem. So you give me a
program and input that you want to run
that program on and I'm going to answer
the halting problem and the way I'm
going to do it is I'm just going to wait
for the statement coming out of the
theorem enumeration device that asserts
either that P does halt on that input or
I wait for the statement that P does not
halt on that input. But one of them is
going to happen because it was a
complete theory that was enumerating all
the true statements of elementary
mathematics. So therefore, if I had such
a system, I could solve the halting
problem. But we already proved that you
cannot solve the holding problem. So
therefore, you cannot have such a
complete theory of arithmetic. So that
proves Girdle's theorem. Maybe to take a
little bit of a tangent. Uh can you
speak you you've written a wonderful
book about proofs and the art of
mathematics. So what can you say about
proving stuff in mathematics? What is
the process of proof? What are the
tools? What is the art? What is the
science of proving things in
mathematics?
>> So this is something I find so wonderful
to teach young mathematicians who are
learning how to become mathematicians
and learning about proof. And I wrote
that book when I was teaching such a
proof writing class in New York.
>> So many universities have such a course
the proof writing course which is
usually taken by students who have
learned some mathematics. Usually
they've completed maybe the calculus
sequence and are making the kind of
transition to higher mathematics which
tends to involve much more proof and
it's a kind of challenging step for
them. So the many math departments have
this kind of course on proof writing
where the students would get exposed to
how to write proofs and I wasn't happy
with most of the other books that exist
for those kind of courses and the reason
was that they were so often so dull
because they would concentrate on like
this the totally uninteresting parts of
what it's like to write a proof. these
kind of mechanistic
procedures about how to write a proof.
You know, if you're going to prove an
implication, then you assume the
hypothesis and argue for the conclusion
and so on. And all of that is true and
fine and that's good to know except if
that's all that you're saying about the
nature of proof, then I don't think
you're really learning very much. So I
felt that it was possible to have a much
better kind of book, one that was much
more interesting and that had
interesting theorems in it that still
admitted of elementary proofs. So I
wrote this book um and tried to fill it
with all of the compelling mathematical
statements with very elementary proofs
that exhibited lots of different proof
styles in it. And so I found that the
students appreciated it a lot. We should
say we dedicate the book to my students.
May all their theorems be true, proved
by elegant arguments that flow
effortlessly from hypothesis to
conclusion while revealing fantastical
mathematical beauty.
Is there some interesting proofs that
maybe illustrate
for people outside of mathematics or for
people who just take math classes in
high school and so on?
>> Yeah, let's do a proof. There's one in
the book. We can talk about it. But I
think it's a nice problem. It's in the
discrete math. Yeah, the 5.1. That one
more pointed at than pointing. Okay. So,
this is the following problem. Suppose
you're gathered with some friends, you
know, in a circle. And you can point at
each other however you want or yourself,
whatever. It doesn't matter. And you can
point at more than one person, you know,
use all your fingers or your feet or
whatever you want. So maybe you point at
three of your friends or something and
they point at two or three of their
friends or whatever and one person is
pointing at 10 people and somebody isn't
pointing at anybody maybe or um and
various people are pointed at also right
so the question is
could we arrange a pattern of pointing
so that everyone was more pointed at
than they are pointing at others. So, in
other words, maybe there's seven people
pointing at me, but I'm only pointing at
five people. And maybe there's, you
know, 20 people pointing at you, but
you're only pointing at 15 people or
something like that. Right? So, I want
to know there's a similar question on
Twitter.
For for a group of people on Twitter,
could you arrange that everyone has more
followers than following? Yeah. It's the
same question. Mathematically, it's
identical. Yeah.
>> Although I don't know. It's not
identical because I said you could point
at yourself and I think that's not Can
you follow yourself?
>> No, I don't think so.
>> I don't think you can. Okay. So, can you
arrange it so that everyone is more
pointed at than pointy? And in my book,
I give a couple of different proofs of
this. I think I give an induction proof
and then there's another proof. I think
there's three different proofs in there.
But why don't we just talk about the my
favorite proof. Suppose it were possible
to arrange that we're all more pointed
at than pointing. Okay. Now, what we're
going to do, we're going to agree.
We're going to give a dollar to everyone
that we're pointing at.
>> Yeah.
>> Okay. And so, what happens? Everybody
made money
because I was pointed at by more people
than I'm pointing. So, I got $10, but I
only paid out $7. And similarly, you got
paid $20, but you only paid out $15. So
if everyone is more pointed at than
pointing, then everyone makes money. But
it it's obviously impossible for us to
make money as a group by just trading
money with ourselves.
>> And therefore, it can't be possible that
we're all more pointed at than pointing.
And this proof illustrates something.
It's one of my habits that I suggest in
the book to anthropomorphize your
mathematical ideas. So
>> you should imagine that the mathematical
objects that are playing a role in your
question are people or active somehow
animals or something that maybe have a
will and a goal and so on. This is this
process of anthropomorphizing and and it
often makes the problems easier to
understand because we all are familiar
with the fact that it's difficult to
make money and the proof is totally
convincing because of our knowledge that
we can't make money as a group by
trading dollars between us, you know,
without any new money coming in to the
group.
>> And but that by itself is actually a
difficult mathematical
claim. I mean, if if someone had to
prove that that you can't make money by
trading in within a group, you know, it
can't be that everyone in the group
makes money just by shifting money
around in the group. Maybe you think
that's obvious and it is obvious if you
think about money. But if you had asked
the question, you know, about
mathematical functions of a certain kind
and so on, then maybe it wouldn't be as
clear as it is when you're talking about
this money thing because of
we we can build on our human experience
about the difficulty of getting money
and you know or other resource. It
doesn't have to be money. It could be
candy, whatever. You know, we just know
that you can't easily get more things in
that kind just by trading within a
group. And we should say that sometimes
the power of proof is such that the
non-obvious can be shown and then over
time that becomes obvious. So in the
context of money or social systems
there's a bunch of uh things that are
nonobvious and the whole point is that
proof can guide us to the to the truth
to the accurate description of reality.
We just proved a property of money.
[laughter]
It's interesting to think about well
what if there were infinitely many
people in your in your group then it's
not true anymore the the the theorem
fails in fact you can arrange that
everyone is strictly more pointed at
than pointing and also you can if
everyone has uh has even just one dollar
bill [snorts]
>> then you can arrange that afterwards
everyone has infinitely many dollar
bills because in terms of cardality
that's the same it's just say countable
infinity in each case if you had
countably many friends and everyone has
$1 bill then you can arrange a pattern
of passing those dollar bills amongst
each other so that afterwards everyone
has infinitely many dollar bills what
you need is for each person to be
attached to you know one of the train
cars or something so think of everyone
as coming from Hilbert's train but also
think of them as fitting into Hilbert's
hotel so just have everyone on the nth
car give all their money to the person
who ends up in the nth room. So they
each give $1 to that person. So
afterwards that person has infinitely
many dollars, but everyone only paid out
$1. So it's a way of making it happen.
>> To what degree, sticking on the topic of
infinity?
Should we think of infinity as something
real?
>> That's an excellent question. I mean a
huge part of the philosophy of
mathematics is about this kind of
question that what is the nature of the
existence of mathematical objects
including infinity but I think asking
about infinity specifically is isn't
that different than asking about the
number five what is
>> what does it mean for the number five to
exist what what are the numbers really
right this this is the maybe one of the
fundamental questions of mathematical
ontology I mean there's many different
positions to take on the question of the
nature of the existence of mathematical
objects or abstract objects in in
general. And there's a certain kind of
conversation that sometimes happens when
you do that. Um, and it goes something
like this.
Sometimes people find it problematic to
talk about the existence of abstract
objects such as numbers. And there seems
to be a kind of wish that we could give
an account of the existence of numbers
or other mathematical objects or
abstract objects that was more like you
know the existence of of tables and
chairs and rocks and so on. Um and so
this seems to be this desire to reduce
mathematical existence to something you
know that we can experience uh
physically in in the real world. Um but
my attitude about this attempt
is that it's very backward I think
because
I don't think we have such a clear
existence of the nature of physical
objects actually I mean we all have
experience about existing in the
physical world as we must because we do
exist in the physical world but I don't
know of any
satisfactory account of what it means to
exist physically. I mean if I ask you
say
imagine
uh you know a certain kind of steam
locomotive you know and I describe the
engineering of it and the weight of it
and the nature of the gear linkages and
and you know and I show you schematic
drawings of the whole design and so on
and you know we talk in detail about
every single detailed aspect of this
steam locomotive but then suppose after
all that conversation
I say, "Okay, now I would like you to
tell me what would it mean for it to
exist physically?" I mean, as opposed to
just being an imaginary steam
locomotive, then what what what could
you possibly say about it? I mean,
except by saying, "Oh, I just mean that
it exists in the physical world." But
what does that mean? That's the
question, right? It's not an answer to
the question. That is the question. Uh,
so I don't think that there's anything
sensible that we can say about the
nature of physical existence. It is a
profound mystery. In fact, it becomes
more and more mysterious the more
physics we we know. I mean back in say
Newtonian physics then one had a picture
of the nature of physical objects as you
know little billiard balls or something
or maybe they're infinitely divisible or
something like that. Okay. Then this
picture is upset with the atomic theory
of matter. And but then that picture is
upset when we realize that the atoms
actually can be split and consists of
electrons and protons and neutrons and
so on. But then that picture is upset
when we realize that those things
themselves are built out of quarks and
lepttons and so on and who knows what's
coming. And furthermore, all of those
things the nature of their existence is
actually as wave functions in in uh you
know some cloud of probability and so
on. And so it just becomes more and more
and more mysterious the more we learn
and not at all clarifying. And so the
the nature of what it means to say that
you know there's an apple on my desk and
to give an account of what that physical
existence really is at bottom I think is
totally absent. Whereas we do seem to
have a good a much more satisfactory
account of the nature of abstract
existence. I mean I can talk about the
nature of the empty set. You know this
is the predicate which is never true or
something like that. I can I can talk
about those kind of logical properties
or the singleton of the empty set and so
on. I mean of course it's very difficult
if you go very far with it. But the
point is that it doesn't get more and
more mysterious the more that you say it
becomes only more and more clear.
And so, so it seems to me that
um we don't really have any
understanding of what the physical world
is as opposed to the abstract world. And
it's the abstract world where existence
is much more clear. [gasps]
It is very true that we don't know
anything about the soda bottle or the
steam locomotive just cuz we can poke at
it. Again, we anthropomorphize and
that's actually get gets us into trouble
sometimes because I'm not feeling the
quantum mechanics when I'm touching
this. [laughter]
>> That's right.
>> And therefore, it's easy to forget and
feel like this is real and
mathematical objects are not. But you're
you're making the opposite argument when
you draw a distinction between numerals
and numbers, which numerals are the
representation of the the number on the
page and so on. But could you say that a
number is real? Do they do numbers
exist?
>> I I happen to think so. I mean, I'm on
the side of realism in mathematics and I
think that that these abstract objects
do have a real existence in a way that
we can give an account of in a way I
just tried to describe.
>> So like you would describe it as um as a
size of a set.
>> Well, there's different ways to
understand the nature of four. I mean
actually this gets into the question of
of structuralism which is maybe a good
place to talk about it.
>> What is structuralism?
>> Structuralism is a philosophical
position in in mathematics or the
philosophy of mathematics by which one
emphasizes that what's important about
mathematical objects is not what they're
made out of or what their substance or
essence is but rather how they function
in a mathematical structure. And so what
I call the structuralist attitude in
mathematics is that we should only care
about our mathematical structures up to
isomorphism. If I have a mathematical
structure of a certain kind and I make
an exact copy of it using different
individuals as to form the elements of
that structure then the isomeorphic copy
is just as good mathematically and
there's no important mathematical
difference that would ever arise from
from working with this isomeorphic copy
instead of the original structure. And
so therefore that's another way of
saying that the substance of individuals
you know in a mathematical structure is
irrelevant
with regard to any mathematical property
of that structure.
And so,
so to ask a question like what is the
number four really is an
anti-structuralist thing because what
you know if you have a if you have a
structure say the natural numbers you
know with all the numbers in it 0 1 2 3
4 and so on. Then I could replace the
number four with something else like you
know this bottle of water could play the
role of the number four in that
structure and it would be isomeorphic
and it wouldn't matter at all for any
mathematical purpose to use this
alternative mathematical system. You
know that's to say that we don't care
what the number four is really. That is
irrelevant. What the only thing that
matters is what are the properties of
the number four in a given mathematical
system you know and recognizing that
there are other isomeorphic copies of
that system and the properties of the of
that other systems number four are going
to be identical to the property of the
properties of this systems number four
with regard to any question that's
important about the number four but
those questions won't be about essence
so so in a sense structuralism a kind of
anti-essentialism in mathematics
So is it fair to think of numbers as a
kind of pointer to a deep underlying
structure?
>> Yeah, I think so because I guess part of
the point of structuralism is that it
doesn't make sense to consider
mathematical objects or individuals in
isolation. What's interesting and
important about mathematical objects is
how they interact with each other and
how they behave in a system. And so
maybe one wants to think about the
structural role that the objects play
you know in in a larger system a larger
structure. There's a famous question
that Fraga had asked actually when he
was looking into the nature of numbers
because in his loses program right he
was trying to reduce all mathematics to
logic and in that process he was
referring to the Caner Hume principle
that you know whenever two sets are
equinumerous then they have the same
number of elements. I mean if and only
if and he founded his theory of number
on on this principle. But he recognized
that well there was something that
dissatisfied him about that situation
which is that the can or Hume principle
does not seem to give you a criteria for
which things are numbers. It only tells
you a kind of identity criteria for when
are two numbers equal to each other.
Well, two numbers are equal just in case
the sets of those sizes are
equinumerous. So that's the criteria for
number identity, but it's not a criteria
for what is a number. And so this
problem has become known as the Julius
Caesar problem because Frag has said we
don't seem to have any way of telling
from the Hume principle whether Julius
Caesar is a number or not. So he's
asking about the essence of number and
whether
>> of course one has the sense that he he
picked maybe what he was trying to
present as a ridiculous example because
maybe you have the idea that well
obviously Julius Caesar is not a number
and there's a lot of philosophical
writing that seems to take that line
also that obviously the answer is that
Julius Caesar is not a number but the
structuralists disagree with that
position. The structuralist attitude is
look, you give me a number system. If
Julius Caesar isn't a number, then I can
just let's take the number 17 out of
that system and plug in Julius Caesar
for that role. And now I've got a new
number system and now Julius Caesar
happens to be the number 17.
>> And that's totally fine. Yeah. So the
point of structuralism is um is that the
question of whether Julius Caesar a
number or not is irrelevant to
mathematics. It is irrelevant because it
is not about structure. It's about this
essence of the mathematical objects.
>> Mhm. So um so that's the structuralist
criticism of Fragus point. You've kind
of made the case that uh you can say
more concrete things about the existence
of objects in mathematics than you can
in our physical reality about which to
us human and brains
things are obvious and not. So how
what's more real the reality we see with
our eyes or the reality we can express
uh in mathematical theorems?
>> I'm not quite sure. I mean,
I live entirely in the Platonic realm
and I don't really understand the
physical universe at all. So, I don't
have strong views.
>> Let's talk about the Platonic realm. Is
it like because you live there, is it
real or
>> Oh, yeah. totally. Yeah. This is the
realist position in mathematics is that
abstract objects have a real existence.
And okay, what's meant by that is that
there's some sense of existence in which
those objects can be regarded as real.
>> How should we think about that? How
should we try to visualize that? What
does it mean to live
amongst abstract objects,
>> right?
>> Cuz life is finite. We're all afraid of
death. We fall in love with other
physical manifestations of objects.
>> [laughter]
>> And you're telling me that maybe reality
actually exists elsewhere and this is
all just a projection.
>> Well, I mean
>> from the abstract realm.
>> Do abstract objects exist in a place and
out of time? That's very debatable, I
think. And what does place and time?
Yeah. So,
>> it's like what's more real, physics or
uh the the mathematical platonic space?
Well, the mathematical platonic realm is
I'm I'm not sure I [clears throat] would
say it's more real, but I'm saying we
understand the reality of it in a much
deeper and more yeah
>> a more convincing way. I don't think we
understand the nature of physical
reality very well at all. And I think
most people aren't even scratching the
surface of the question as I intend to
be asking it. So, you know, obviously we
understand physical reality. I mean, I
knock on the table and so on and we know
all about what it's like to, you know,
have a birthday party or to drink a
martini or whatever. And and so we we we
have a deep understanding of existing in
the physical world,
>> but maybe understanding is the wrong
word. We have an experience of living in
the world and riding bicycles and all
those things.
>> But I don't think we actually have an
understanding at all. I mean, very very
little of the nature of physical
existence. I think it's a profound
mystery. Whereas I think that we do have
something a little better of an
understanding of the nature of
mathematical existence and abstract
existence. So that's the how I would
describe the point. Somehow it feels
like we're approaching some deep truth
from different directions
and we just haven't traveled as far in
the physics world as we have in the
mathematical world.
>> Maybe I could hope that someone will
give you know the convincing account but
it seems to be a profound mystery to me.
I I can't even imagine what it would be
like to give an account of physical
existence.
>> Yeah. I wonder like a thousand years
from now as physics progresses what this
same conversation would look like,
>> right? That would be quite interesting.
>> Do you think there's breakthroughs a
thousand years from now on the
mathematics side? Cuz we've just
discussed and we'll return to a lot of
turmoil a century ago,
>> right?
>> Uh do you think there's more turmoil to
be had? It's interesting to me because I
have my feet in two worlds of
mathematics and philosophy and to
compare the differences between these
subjects and one of the one of the big
there's many cultural differences but
one of the big cultural differences is
towards the idea of progress in the
subject because mathematics has huge
progress. We simply understand the
mathematical ideas much much better. You
know, continually improving our
understanding and there's growth in
knowledge. We understand the nature of
infinity now better than they did 100
years ago. I mean, definitely better.
And they understood it better 100 years
ago than they did, you know, for the
previous thousands of years and so on.
So, in almost every part of mathematics,
there's improved understanding of the
core issues. So much so that um you know
the the questions at hand become totally
different and the field sort of moves on
to more difficult uh interesting
questions. Whereas in philosophy this
that's a little bit true that there's
progress but meanwhile it's also true
that there are these eternal questions
that have been with us for thousands of
years. And in fact, so much so that you
can find a lot of philosophers arguing
that the important contribution of
philosophy is in asking the questions
rather than answering them because it's
hopeless to [laughter] answer them. I
mean, the nature of these deep
philosophical questions is so difficult.
Less of a sense of progress is what I'm
trying to say. Um I don't see any reason
to think that the progress in
mathematics in the growth in our
mathematical understanding and knowledge
won't simply continue. And so a thousand
years from now maybe the mathematics
that they will be doing at that time uh
would probably be completely
unrecognizable to me. I maybe wouldn't
even begin to understand what they're
talking about even without sort of
witnessing you know the intervening uh
developments. Um, so if you bring
someone from ancient times to today,
they maybe wouldn't even understand what
we're talking about with some of the
questions. But I feel that, you know, if
Archimedes came and we were able to
communicate, I I think I would be able
to tell him, you know, about some of the
things that are going on uh in
mathematics now and and maybe, you know,
>> or or anyone from that time, I mean, so
uh so I think it is possible to have
this kind of progress even when the
subject kind of shifts away from the
earlier concerns. as a result of the
progress. Basically
>> to take a tangent on a tangent uh since
you mentioned philosophy maybe
potentially more about the questions and
maybe mathematics is about the answers.
I have to say you are a legend on math
overflow which is like Stack Overflow
but for math. You're ranked number one
all time on there with currently over
246,000 reputation points. How do you
approach answering difficult questions
on there? Well, Math Overflow has really
been one of the great pleasures of my of
my life. I've really enjoyed it. I mean,
and I've learned so much from
interacting on Math Overflow is um I've
been on there since [snorts] 2009, which
was shortly after it started. I mean, it
wasn't exactly at the start, but a
little bit later. Um and uh and
I think I it gives you the stats for how
many characters I typed and I don't know
how many million it is but uh uh there's
enormous amount of uh um time that I've
spent thinking about those questions and
it has really just been amazing to me.
>> How do you find the questions that grab
you and how do you go about answering
them? So I'm interested in any question
that I find interesting. So and it's not
all questions. So sometimes certain
kinds of questions just don't appeal to
me that much.
>> So you go outside of set theory as well.
>> So I think when I first joined Math
Overflow I was I was basically one of
the only one of the few people in logic
who was answering. I mean there were
other people who know some logic
particularly from category theory and
other parts of mathematics um that
aren't in the most traditional parts of
logic but they were answering some of
the logic questions. So I really found
myself able to make a contribution in
those very early days by engaging with
the logic related questions. But there
weren't many logic people asking
questions either. But what I found was
that there was an enormous amount of
interest in topics that were logic
adjacent. So a question would arise you
know in group theory but it had a logic
aspect or an analysis or whatever and
there would be some logic angle on it.
And what I found was that I was often
able to figure out an answer um by
learning enough about that other subject
matter. This is what was so rewarding
for me is because basically I I had to
learn enough my expertise my main
expertise was logic but someone would
ask a question you know that that was
about say the axom of choice in this
other subject matter or the continuum
hypothesis or something like that in in
the other subject matter
>> and I would have to learn enough about
that other subject and the context of
the question in order to answer and I
was often able to do that and so I was
quite happy to do that and and also I
learned a lot by doing that because I
had to learn about these other problem
areas and so it really allowed me to
grow enormously as a mathematician. To
give some examples of uh questions
you've answered, what are some
reasonable sounding statements that are
independent of ZFC? What are the most
misleading alternate definitions in
taught mathematics? Is the analysis as
taught in universities in fact the
analysis of definable numbers solutions
to the continuum hypothesis? Most
unintuitive application of the axim of
choice non-trivial theorems with trivial
proofs [laughter]
reductio at absurdum or the
contraositive.
What is a chess piece mathematically? We
should say you've worked quite a bit on
infinite chess which we should
definitely talk about. It's awesome.
You've worked on so many fascinating
things. Has philosophy ever clarified
mathematics? Why do we have two theorems
when one implies the other? And of
course, just as an example, you've given
a really a great almost historical
answer on the topic of the continuum
hypothesis. And maybe that's a good
place to go. We've touched it a little
bit, but it would be nice to lay out
what is the continuum uh hypothesis that
Caner struggled with. And I would love
to also speak to the psychology of his
his own life story, his own struggle
with it. It's the human side of
mathematics is also fascinating.
>> Yes.
>> So what is the continuum hypothesis? So
the continuum hypothesis is the question
that arises so naturally whenever you
prove that there's more than one size of
infinity. So Cander proved that the
infinity of the real numbers is strictly
larger than the infinity of the natural
numbers.
But immediately when you prove that
one wants to know well is there anything
in between. I mean what could be a more
natural question to ask immediately
after that? And so Caner did ask it and
he spent his whole life thinking about
this question. Um and so the continuum
hypothesis is the assertion that there
is no infinity in between the natural
numbers and the real numbers. And of
course Caner knew many sets of real
numbers. everything in between I mean
everything that's in that interval would
be equinumerous with some set of real
numbers but we know lots of sets of real
numbers I mean there's all these various
closed sets can set so on this Vitali
set we have all kinds of sets of real
numbers and so you might think well if
the condition hypothesis is false then
we've probably seen the the set already
we just have to prove you know that it's
strictly in between but it turned out
that for all the sets that anyone ever
could define or pick out or observe for
all the sets of real numbers, it was
always the case either that they were
countable in which case they're
equinumerous with the natural numbers or
else finite um or they were fully
equinumerous with the whole real line
and so they were never strictly in
between. I mean, you're in the situation
and you have hundreds, thousands of sets
that are candidates to be in between,
but in every single case, you can prove
it's on one side or the other and not
strictly in between. And so, in every
situation where you're able to figure
out whether it's in between or not, it's
always never strictly in between.
>> Now, Caner was obsessed with this.
>> I think he was. Yeah. I'm not a
historian, so I don't know the exact
history. everything I've seen it seems
to be the question that broke him. Um I
mean just struggling with different
opinions on the hypothesis within
himself and desperately chasing uh
trying to prove it.
>> So he had a program for proving it which
has been affirmed in a certain respect.
Of course the the condition hypothesis
holds for for open sets. That's easy to
see. If you have an open interval then
this is fully equinumerous with the
whole real line. Any interval is
equinumerous with the whole line because
all you would need is a function you
know like the arc tangent function or
something that maps the whole real line
into into an interval and that's a one
function. So we know the open sets have
the property that their non-trivial open
sets are all fully equinumerous with the
whole real line. So never strictly in
between. But remarkably Caner proved it
also for the closed sets and that is
using what's called the Caner Bendixen
theorem. So it's quite a remarkable
result. It's definitely not obvious and
in this theorem actually was the origin
of the ordinals. Caner had to invent the
ordinals in order to make sense of his
Canderbend Dixon process.
>> Can you define the open and the closed
set in this context?
>> Oh yeah, sure. So a set of reals is open
if every point that it contains is
surrounded by a little interval of
points the whole tiny little interval.
But that tiny little interval is already
just by itself equinous for the whole
line. So that's why the question is sort
of easy for open sets. A closed set is a
complement of an open set. And there's a
lot of closed sets that are really
complicated of varying sizes. So of
course any closed interval is a closed
set. But it's not only those. There's
also things like the caner set which you
get by omitting middle thirds. Maybe
some people have seen this construction
or you can imagine sort of randomly
taking a lot of little tiny open
intervals you know all over the line and
so on. So that altogether would be an
open set and the complement of it would
be a closed set. So you can imagine just
kind of tossing down these open
intervals and what's left over is the
closed set. Those sets can be quite
complicated and they can have isolated
points. For example, if the two open
intervals were were just kissing and
leaving only the one point between them.
But also you could have you could have
sequences that are converging to a point
that would also be a closed set or a
convergent sequences of convergent
sequences and so on. That would be a
closed set. Also
>> the caner set is constructed by
iteratively removing open intervals
middle thirds like you mentioned from
the interval and trying to see can we do
a thing that that goes in between.
>> Right. So the question would be can you
produce a set that has an intermediate
size an intermediate cardality right
>> and caner proved with a closed set no
it's impossible every closed set is
either countable or equinumerous with
the whole real on
>> and what what the caner program for
solving the continuum hypothesis was was
of sort of working up so he did it for
open sets and for closed sets and you
sort of work up maybe he wants to go
into what are called the burell sets
which are sort of combinations of open
and closed sets. Um and there's a vast
hierarchy of of burell complexity and
and it turns out that the continuum
hypothesis has been proved also for the
burell sets in this hierarchy but then
one wants to go beyond what about more
complicated sets. So there's this
hierarchy of complexity for sets of real
numbers. And Caner's idea was to sort of
work your way up the hierarchy by
proving that the continues
was more and more true for those more
and more complicated sets based on our
understanding of the earlier cases and
and that has been carried out to an a
remarkable degree. It turns out that one
needs one begins to need large cardinal
assumptions though in order to get to
the higher realms even at the level of
projective hierarchy which are sets that
you can define by using quantifiers over
the real numbers themselves. So you get
this hierarchy on top of the burell
hierarchy the hierarchy of projectively
definable sets. Um and it turns out that
if you have enough large cardinals then
the projective sets also are
always either countable or equinumerous
with the whole real line and and then
one can try to go beyond this and so on.
So, so I view all of those results which
came you know in the past 50 years um
the later ones as fulfilling this cander
idea that goes back you know 120 years
to his idea that we would prove the
conditional wellness by establishing
more and more instances for greater and
greater complexity of sets. But of
course, even with what we know now, it's
hasn't fully succeeded and it can't
because the hierarchy of complexity
doesn't include all sets of real
numbers. Some of them are sort of
transcending this hierarchy completely
in a way. And so, so that the program
can't ever fully be successful,
especially in light of the independence
results.
>> Yeah. Well, spoiler alert, can you go to
the independence results? Sure. So, what
does that mean? Um uh so continual
hypothesis were shown to be independent
from the ZFC axioms of mathematics.
>> Right? So the ZFC axioms were the axioms
that were put forth first by Zerlo in
1908 in regard to his proof of the well
order theorem using the axim of choice
that that wasn't fully ZFC at that time.
It was just zerlo theory because he sort
of there was a kind of missing axiom.
the replacement axiom and the foundation
axiom were added later and that's what
makes the zerlo franle axumization which
became sort of standard actually there's
another aspect which is zerlo's original
theory allowed for the existence of
elements or these atoms mathematical
objects that are not sets but out of
which we build the set theoretic
universe whereas
set theorists today generally don't use
elements at Paul, I
I argue that um it's really the
philosophy of structuralism that leads
them to omit the earl elements because
it turns out that if you adopt ZFC
axioms with earl elements, ZFCU it's
called or ZFA
then any structure that exists any
mathematical structure that exists in
that sethic universe with the atoms is
isomorphic to a structure that doesn't
use the atoms at all.
And therefore, you don't need the atoms
if you're a structuralist because you
only care about the structures up to
isomeorphism anyways. And the and the
theory is simply more elegant and
clearer without the atoms. They're just
not needed. And so that's why today when
we talk about set theory, generally
we're talking about the atomree version
and CFC has no elements. Okay. So we
formulate the CFC axioms of set theory.
These are expressing the main principle
ideas that we have about the nature of
sets and set existence.
And
Caner had asked about the Gand
hypothesis
in the late 19th century
and it remained open totally open
until 1938. And we should mention, I
apologize, that it was the number one
problem in the Hilbert 23 set of
problems formulated at the beginning of
the century.
>> That's right.
>> Maybe you can comment on why did he put
that as number one.
>> So, right. So, Hilbert had introduced at
his famous address at the turn of the
century this list of problems that he
thought could guide or were important to
consider in the coming century of
mathematics. I mean, that's how people
talk about it now, although I'm not sure
at all. Of course, I I can't really
speak for Hbert at all, but if you were
a very prominent mathematician,
I find it a little hard to believe that
Hilbert would have conceived of his list
in the same way that we now take his
list. I mean, having observed the
century unfold, we know that that list
of 23 problems did in fact guide whole
research programs and it was extremely
important and influential. But at the
time, Hilbert would have no reason to
think that that would be true. And he
was just giving a lecture and uh had a
list of problems that he thought were
very important. And so I tend to I would
find it more reasonable to think that he
was just making a list of problems that
he thought were extremely interesting
and important and fundamental in a way
without the kind of um heavy burden of
guiding this 20th century research.
Although it turns out that in fact
that's exactly what they did. And we
already discussed how Hilbert Hilbert's
views on the nature of set theory and
the fundamental character that quote
where he said no one will cast us from
the paradox that Caner has created for
us. So, so I think Hilbert was convinced
by Caner on the importance and the
fundamental nature of the continuum
hypothesis for the foundations of
mathematics which was a critically
important development for the unity of
mathematics. I mean before set theory
emerged as a foundation of mathematics
there were you know there there's
different subjects in mathematics
there's algebra and there's analysis
real analysis and topology and geometry
and so there's all these disperate
subjects
with their own axioms separate axioms
right and but sometimes it happens like
when you're proving say the fundamental
theorem of algebra you know that the
complex numbers are an algebraically
closed field that you can solve any
polomial equation in but the proof
methods for that theorem come from other
parts of mathematics you know this
topological proofs and so on and so what
is that how does that work I mean if you
have totally different axiom systems but
you're using results from one subject in
another subject it's somehow incoherent
unless there's one underlying
subject so the unity of mathematics was
provided by the existence of a
mathematical foundation like set theory
and at the time it was set theory. And
so it's critically important to be able
to have a single theory in which one
views all of mathematics is taking place
to resolve that kind of transfer and
borrowing phenomenon that was definitely
happening.
So that must have been part of Hilbert's
thinking about why it's so important to
have a uniform foundation and set theory
was playing that role at the time. Now
of course we have other possible
foundations from coming from category
theory or type theory um and there's
univalent foundations now so there's
sort of competing foundations now
there's no need to just use one
foundation one set theoretic foundation
although set theory continues to in my
view have an extremely successful
metathematical analysis as a foundation
I think is much more successful in set
theory for any of those other
foundations but um it's much less
amendable though to things like computer
proof and so on which is part of the
motivation to find these alternative
foundations. So yeah. Okay. So to talk
about Hilbert though I think he was
motivated by the need for a unifying
foundation of mathematics and set theory
was playing that role and the continuum
hypothesis is such a core fundamental
question to ask. So it's seems quite
natural that he would put it on the
list. There were other logic related
questions though like Hilbert's 10th
problem is also related to logic. This
is the question about dapontine
equations and he asked to provide an
algorithm to decide whether a given
dioontine equation has solution in the
integers. So a dontine equation is just
I mean it's a maybe a fancy way of
talking about something that's uh easy
to understand a polinomial equation
except it's not just one variable many
variables. So so you have polomials in
several variables over the integers and
you want to know can you solve it? So
the problem is as stated by Hilbert
provide an algorithm for answering the
question whether a given polomial
equation has a solution in the integers.
So he's sort of presuming that there is
an algorithm but he wants to know what
is it? What is the algorithm? But the
the problem was solved by proving that
there is no algorithm. It's an
undecidable problem like the halting
problem. there's no computable procedure
that will correctly decide whether a
given polinomial equation has a solution
in the integers. So that's quite a
remarkable development I think. So there
were also there's a few other logic
related questions on the list
>> and so eventually continuum hypothesis
were shown to be independent from ZFC
axioms as we've mentioned. So what how
does that make you feel
>> and what what is independence when you
tell the story the historical story is
really quite dramatic I think because
Kander poses the question you know late
19th century
>> and then it's totally open Hilbert asks
about it you know at the turn of the
20th century nobody has any clue there's
no answer coming until 1938 this is four
decades later right so a long time
And Girdle, Kurt Girdle proved
half of it. What he proved is that
if the axims of set theory are
consistent, then there is a set
theoretic world
where both the axim of choice and the
continuum hypothesis are true.
So what he's doing is showing this is
called the constructible universe
girdles L.
So, so he solved this. This is the same
result where he answers the safety
question of the axim of choice but also
for the continuum hypothesis. They're
true in the same center universe we get.
So if if ZF without the action of choice
is consistent, then so is ZFC plus the
continuum hypothesis is the result 1938.
It's really a such a beautiful argument.
It's just incredible, I think, because
he's he's building an alternative
mathematical reality. That's the
structure of the proof is that okay, if
if there's any mathematical reality, if
there's any set theoretic world, then
we're going to build another one, a
separate one, a different one, maybe
different. Maybe it's the same as the
original one. It could be if we started
already in the one that he built, then
it would be the same. But there's no
reason to assume it was the same. So he
he has this kind of model construction
method to build this alternative set
theoretic [clears throat] reality the
constructible universe and then he
proves that the aximo choice is true
there and also the continuum hypothesis
is true there and it's just amazing
really beautiful argument okay so then
for the other part of the independence
that's only half of it because girdle
shows basically that you can't refute
the continuum of witnesses but that's
not the same thing as proving that it's
true. He showed that if set theory is
consistent
without the continuum hypothesis, then
it's consistent with the continu
hypothesis. So that's not the same thing
as proving that it's true. Yeah. And
then it didn't come until 1963
when Paul Cohen invented the method of
forcing
uh and proved that if there's a model of
set theory then there's a model of set
theory in which the continuum hypothesis
is false.
So so Cohen also is giving us this
extremely powerful tool for building
alternative mathematical realities is
how I think about it. He's explained to
us how to take any set theoretic world
and build another different one in which
the conditional hypothesis is false. The
forcing extension just is such a
fascinating technique tool of forcing
maybe I'm anthropomorphizing it but it
uh seems like a way to escape one
mathematical universe into another or to
expand it or to alter it. So travel
between mathematical universes. Can you
explain the technique?
>> Exactly. It's all those things. It's so
wonderful. I mean, that's exactly how I
think about it. I mean,
>> and we should mention maybe this is a
good place to even give a bigger picture
of it. One of your more controversial
ideas than mathematics as laid out in
the paper, the set theoretic multiverse.
You describe that uh there may not be
one true mathematics, but rather
multiple mathematical universes, and
forcing is one of the techniques that
gets you from one to the other. So can
you explain the whole shebang?
>> Yeah, sure. Let's let's get into it. So
the the lesson of Cohen's result and
Girdle's result and so on these
producing these alternative set
theoretic universes.
We've observed that the continuum
hypothesis is independent and the axim
of choice is independent of the other
axioms. But but it's not just those two.
We have thousands of independence
results. practically every non-trivial
statement of infinite cominatorics is
independent of CFC. I mean this is the
fact. Um it's not universally true.
There are some extremely difficult
prominent results where people proved
things in CFC but for the most part if
you ask a non-trivial question about
infinite cardalities then it's very
likely to be independent of CFC. and and
we have these thousands of arguments
these forcing arguments that uh are used
to establish that and so how should we
take that I mean on the one hand
if you have a theory and it doesn't
answer any of the questions that you're
interested in okay so what does that
mean if you're following what I call the
universe view or the monist view
you might naturally say well look ZFC is
a weak theory and there's there's the
true
set theoretic reality out there and we
need a better theory because the current
theory isn't answering the questions.
Everything's independent. And so that
seems like a quite reasonable thing to
take. If you think that there is that
every set theoretic question has a
definite answer and there's a unique set
theoretic truth or a unique fact of the
matter. Right? This is this is the
universe view. And by the way to
reiterate
uh independent means it cannot be proved
or disproved within this aimatic system
within this theory.
>> Right. Exactly. So to be independent
means you can't prove it and also you
can't prove that it's false. You can't
it
>> that's why the statement is so traumatic
or sad that most of the interesting
stuff as you said has been shown to be
independent
>> right
>> of ZFC. But that's an interesting way to
put it, I think, because it reminds me
of this uh when I was a graduate student
in Berkeley, there was another graduate
student who's working with a non-logic
professor in um sea star algebra or
something like this. So, it's a part of
analysis or functional analysis [gasps]
and they were looking at a question and
it turned out to be independent of CFC,
right? And the attitude of this other
professor was that oh I guess I asked
the wrong question.
But my attitude and the attitude of all
the set theorists was when you ask a
question that turns out to be
independent then you asked exactly the
right question because this is the one
you know it's it's carving nature at its
joints. You're you're adjudicating the
nature of Sethic reality by finding
these two realms. You find one of these
dichotoies. You know, there's the worlds
where it's true and the worlds where
it's false. And so when you ask that
question, that's to be celebrated. It
means you asked exactly the right
interesting, fascinating question. So
it's not a kind of bleak thing that you
you can't prove it and you can't refute
it and that's such a disaster. Rather,
it means that you found this this this
cleavage in reality in mathematical
reality and it's good to know about
those when they happen.
>> Carving nature at its joint. So what can
you do about the things
uh that are shown to be independent from
ZFC? So what what are the techniques?
>> So one thing is that because of the
incompleteness theorem we know that
there's going to be for any theory that
we can write down there's going to be
things that we can't prove true things
we can't prove in it. So this those
things are going to be independent. And
so so we're already aware of the fact
that there will always be these
independent phenomenon for any theory
that we write. And furthermore, some of
those theories we won't even be able to
prove that they're consistent. Yeah.
Like the consistency of the own theory.
So that's called the consistency
strength hierarchy. So it's a direct
consequence of Girdle's second
incompleteness theorem that for any
theory we can write down then towering
over it is this incredibly tall tower of
consistency strength where the strength
and theories aren't just adding another
axiom but they're adding another axiom
even whose consistency was not provable
in the previous uh layers of the
hierarchy.
And so how lucky we are to find the
large cardinal axioms that instantiate
exactly this feature of increasing
consistency strength. this unending and
extremely tall
hierarchy of consistency strength of
axioms and it exactly fulfills the
prediction that that Girdle's theorem
makes about that kind of thing except
the axioms in the large cararchy aren't
you know metalological self-referential
statements of the form that sometimes
arise in the girdle analysis but rather
they're professing existence of big
infinities, these large cardinal axioms.
And so it's such a welcome development.
And yet it's also known that the
continuum hypothesis is independent of
all of the known large cardinal axioms.
So none of the large cardinal axioms we
can prove none of them can settle the
continuum hypothesis. So, so the
independence phenomenon is still there
for things like the like the caner
hypothesis and the cardinal um uh
cardinal combinatorics that
>> you're building this incredible
hierarchy
of axiomatic systems that are more
powerful than the ZFC
>> more more powerful than ZFC and then
more powerful than that more powerful
than that and so on. keeps going forever
and it will never be finished
>> and still to this day the continuum
hypothesis is not
>> it's not settled by any of the large
cardinal axims.
>> Wow.
Wow.
[laughter] What does that how does that
make you feel? Is that will it ever be
settled?
>> Yeah. Well, it's part of my multiverse
view I guess. So uh which we started by
I was describing the universe view which
is the view that look there are facts of
the matter about all of these questions
and that it will turn out if you're a
universe view person which I'm not but
if you are then you will hold that there
is a right answer to the continuum
hypothesis question and there's a right
answer um to the large cardinal
questions and so on and that what we
should be aiming to do is figure out
this one true set theory. Okay. In
contrast,
uh I take the developments of set theory
over the past half century or more as
evidence that there isn't such a unique
set theoretic reality. Rather, what
we've been doing for decades now is
producing more and more alternative set
theoretic universes in which the
fundamental truths are differing from
one to the other. and and that that that
is the answer to the continuum
hypothesis question. The fact that given
any model of set theory, there's a
forcing extension where the continuum
hypothesis is true and another one where
it's false. You can sort of turn it on
and off like a light switch. And that's
the fundamental nature of the continuum
hypothesis is that you can have it or
you can have the negation as you like
within a very closely related set
theoretic world. Wherever you happen to
be living, there's a closely related one
where ch is true, where the conditional
hypothesis is true and one where it's
false. And that itself is a kind of
answer. It's not a singularist answer, a
universe view answer. It's a pluralist
answer. And and this led me to my views
on the multiverse view of set theory and
pluralist truth. Namely, the the
fundamental nature of set theoretic
truth as this plural character in that
there isn't a singular meaning to the
fundamental terms, but rather there's
this choice of alternative set theoretic
universes that have different truths.
>> So, what does the multiverse view of
mathematics enable you to do? What does
it empower you to do and what are the
limitations? What are the things that
breaks about mathematics as a field as a
space of knowledge and uh what does it
enable?
>> First of all, I guess one should say
that these different philosophical
positions that you might take in the
philosophy of set theory like the
multiverse view or the universe view, we
don't ever disagree about the
mathematics. We're all agreeing on what
the theorems are. It's a question of
philosophical perspective on the
underlying meaning or the context or
really What is a philosophy of
mathematics for? Right? And um I mean if
you look back in history for example
like to the time of calculus with Newton
and linenets right they famously
developed the ideas of calculus uh using
their concepts of infantessimals and and
those foundations were roundly mocked by
Bishop Barkley and so on who talked
about you know what are these same
eancent increments and shall we not call
them the ghosts of departed quantities
and and But the foundations really were
kind of completely suspect I think at
the time. Uh and and that foundations of
infant decimal calculus really only
became rigorous in 1950s or so with the
development of non-centered analysis and
Robinson's work. Okay. So the point I'm
trying to make is that do you need a
robust rigorous foundation of
mathematics to make enduring insights
in mathematics?
And the answer regrettably is apparently
not because in calculus
even with that lousy creaky foundation
of infant decimals not even well
understood that Newton and Linus had
they proved all the fundamental theorems
of calculus and you know they had all
the main insights in those early days
with that extremely bad foundation. And
so that shows you something about the
relevance of the kind of foundational
views on mathematics and and how
important they are for mathematical
developments and progress and insight. I
mean because I view those early
mathematical developments in calculus as
genuinely mathematical and extremely
important and insightful even though the
foundations weren't any good um from you
know by contemporary perspectives. Okay.
So, so rather so when it comes to the
philosophy of set theory and the dispute
between the the universe view and the
pluralism
um my view is that the choice of the
philosophical perspective doesn't
actually have to do with the
mathematical developments directly at
all rather it tells us where should set
theory go what kind of set theory should
we be looking at what kind of questions
should we be asking so If if you have a
universe mentality, the universe view,
then you're going to be pushed to try to
find and articulate the nature of the
one true set theoretic universe.
>> And I think that remark is really well
borne out by the developments with Hugh
Wooden who's one of the most prominent
um mathematicians and philosophers with
the universe view and his theory of
ultimate L and so on. And he's really
>> was also your adviser. He was also my
supervisor. Yeah, my graduate supervisor
>> which is a a personal story as well.
>> This this fundamental dispute on this
question. But he is um has a very strong
and successful research program sort of
trying to give legs to finding the
nature of the one true set theoretic
universe and it's driving the questions
that he's asking and the mathematical
programs that he's pursuing. Whereas if
you have a pluralist view as I do, then
you're going to be led and attracted to
questions that have to do with the
interaction of different set theoretic
universes or maybe you want to
understand the nature of how are the
models of set theory related to their
forcing extensions and so on. And so
this led to things um uh that I call say
set theoretic potentialism where you
think about a set theoretic universe uh
in a potentialist way. Not in the sense
of potential infinity directly because
all of these universes have infinite
sets inside them already. But they're
potentialist in the sense that we could
have more sets. The universe could be
wider and taller and so on, you know, by
forcing or by extending upward. Um and
so we want to understand the nature of
this uh realm of set theoretic universes
and and that's quite some exciting work
and so with Benedict Lova and I we
proved some theorems on the mortal logic
of forcing and set theoretic
potentialism under end extension. And
I've done a bunch of work on this topic
and and also um I mounted uh together
with Gunter Fuks and Jonas Reitz who was
one of my own PhD students the topic of
settic geology which is studying it's
taking the metaphor of forcing I mean in
forcing you have the ground model and
the forcing extension and when I was
first working with Jonas uh
he said I want to undo forcing I want to
go backwards and I at first said, "But
Jonas, it doesn't work that way. You
start in the model, in the ground model,
and you go out. You go to the bigger
one." You know, that's how forcing
works. And he said, "No, no, I want to
go backwards." And and so he was quite
persistent actually. And um um and so
finally I said, "Okay, let's let's do
it. Let's take it seriously." And so we
sat down and and and started thinking,
you know, more precisely and carefully
and deeply about the nature of taking a
synthetic universe and seeing where did
it come from by forcing which was a new
way of thinking about forcing at the
time
>> like reverse engineering the forcing.
>> Yeah, something like that. We forcing is
a way of producing a new universe and so
you could start somewhere and go to that
new universe or you could look where you
are and say well look I got here by
doing that already in the past.
>> Mhm. So we define models of the um
bedrock model and ground you know sort
of undoing the forcing and and really it
was quite fruitful and I view this as
part of the sort of pluralist
perspective except the difference is
that stheretic geology is amenable to
the universe view. So even though the
work was inspired by this philosophical
view on the multiverse view,
nevertheless the central ideas of
geology have now been picked up by the
people with the research program in the
universe view. Um because it turns out
that synthetic geology is helping them
or us to discover the nature of the one
true universe relates to its mantle. is
this concept of the settric mantle that
I had introduced in a way that is
extremely interesting and so it's
historically quite funny I think because
this research program that grew entirely
out of the pluralist point of view uh
ended up being picked up by the universe
point of view uh research program in a
in a way that is quite important. Can
you prove something
in the world that you arrived at through
forcing and then take some of that back
to the ground model?
>> Yeah, absolutely. And that's a really
powerful argument method actually.
People often want to do that. You
suppose you're in some set theoretic
context. You know, you could think about
this living in a set theoretic universe
and you want to prove something in that
universe only. But maybe one way to do
it is to first construct this forcing
extension and then use the features
about this forcing extension to realize
that certain things must have already
been true in the ground model and then
you throw the forcing extensions away
and you yeah so this can happen to pick
a more elementary example if you think
about the early days of people reasoning
with the complex numbers before they
really understood them. So they would
have these algebraic equations that
they're trying to solve
ex you know and they would have the
tools and methods of doing it but then
in the course of you know so they would
have to do things to the polinomial and
change the factors and and and so on and
produce other polinomials and solve them
and so on and sometimes they could
produce solutions
in the middle of their construction.
They were led to like the square root of
minus5 or something, you know, in the
construction and they didn't have any
meaning for that, but they would just do
it symbolically,
you know, and and eventually it would
turn in, you know, because of the
methods that they had, they would
combine and they would cancel and so on
and all the complex parts would cancel
out and they'd end up with this, you
know, actual answer, you know, 3 + 17 or
whatever. And and they could check it
and it worked. it was a solution of the
original equation. And so it must have
been bewildering to them because they
would start with this question purely in
the real numbers, an algebraic question,
and they would march on their method and
proceed through the land of nonsense,
you know, with these square roots of
negative numbers, and then end up with
an answer that was real again that they
could verify was correct.
And and so I view this kind of forcing
argument that I was just describing in a
similar way. You start in in set theory
and you you go to this land of nonsense
in the forcing extension this imaginary
world and you argue and you come back I
mean you make a consequence in the
ground model and it's such a beautiful
way of arguing. So speaking of the land
of nonsense I have to ask you about
surreal numbers but first I need another
bathroom break.
All right, we're back and there's this
aforementioned
wonderful blog post on the serial
numbers and uh
that uh there's quite a simple serial
number generation process that can
basically construct all numbers. So
maybe this is a good spot to uh ask what
are serial numbers and what is the way
we can generate all numbers.
So this real number system is an amazing
an amazingly beautiful mathematical
system that was um introduced by John
Conway.
>> Rest in peace one of the great
mathematicians ever.
>> Yes absolutely and I really admire his
style of mathematical thinking and
working in mathematics and the serial
number system is a good instance of
this. So, so the way I think about the
surreal numbers is um what it's doing is
providing us a number system that
unifies all the other number systems.
So, it extends the real numbers. Well,
not only it extends the integers, the
natural numbers and the integers and the
rational numbers and the real numbers um
but also the ordinals and the infant
decimals. So they're all sitting there
inside the surreal numbers and it's this
colossal system of numbers. It's not a
set even. It's a proper class it turns
out because it contains all the ordinal
numbers. Um but it's generated from
nothing by a single rule. And the rule
is so we're going to generate the
numbers in stages in transfinite
sequence of stages.
And at every stage we take the numbers
that we have so far and in all possible
ways we divide them into two sets. A
lower set and an upper set or a left set
and a right set.
So we divide them into these two sets so
that everything in the left set is less
than everything in the right set. And
then at that moment we create a new
number that fits in the gap between L
and R. Okay, that's it. That's all we
do. So let me say it again.
The rule is we proceed in stages and at
any stage then in all possible ways we
divide the numbers we have into two
collections
the left set and the right set so that
everything in the left set is less than
everything in the right set and we
create a new number a new serial number
that will fit in that gap.
Okay. So for example, we could start
well at the beginning we don't have any
numbers. We haven't created anything
yet. And so well we could take nothing
and we could divide it into two sets.
The empty lower set and the empty upper
set. I mean the two empty sets. Um and
everything in the empty set is less than
everything in the empty set because
that's a vacuous statement. So so we
we're we're we satisfy the conditions
and we apply the number generation rule
which says we should create a new
number. And this is what I call the big
bang of numbers. The the surreal genesis
when the number zero is born. Zero is
the firstborn number that is bigger than
everything in the empty set and less
than everything in the empty set. Okay.
But now we have this number zero. And so
therefore we now can define new gaps
because if we put zero into the left set
and have an empty right set then we
should create a new number that's bigger
than zero and less than everything in
the empty set and that number is called
the number one. And similarly at that
same stage the we could we could have
put zero into the right set and so that
would be the firstborn number that's
less than zero which just called minus
one. So now we have three numbers minus
one, 0 and one. And they have four gaps
because there could be a number below
minus one or between minus1 and 0 or
between 0 and one or above one. And so
we create those four new numbers. The
first number above one is called two.
The first number between 0 and 1 is
called 1/2. And then on the negative
side we have minus a half and minus two.
And so on. So now we have what is that?
Seven. Seven numbers. So there's eight
gaps between them. So at the next birth
day they call them the next stage will
be born all the numbers between those
gaps and then between those and between
those and so on and as the days progress
we get more and more numbers but those
are just the finite birthdays because as
I said it's a transfinite process. So at
day omega that's the first infinite day
we're going to create a lot of new
surreal numbers. So every real number
will be born at that stage because every
real number fills a gap in the
previously born rational numbers that we
had just talked about. It's not all the
rationals because actually the rational
numbers that are born at the finite
stages are just the rationals whose
denominator is a power of two. It turns
out those are called the diotic
rationals. So the real numbers are all
born on day omega. But also some other
numbers are born on day omega. namely
the ordinal omega itself is the
firstborn number that's bigger than all
those finite numbers and minus omega is
the firstborn number that's less than
all those finite numbers but also we
have the number epsilon which is the
firstborn number that's strictly bigger
than zero and strictly less than all the
positive rational numbers. So that's
going to be an infant decimal number in
that gap and so on. On day omega + one
we get more numbers and then omega plus
2 and so on and the numbers just keep
coming forever. So this is how you build
the surreal number system. Then it turns
out you can define the arithmetic
operations of addition and
multiplication in a natural way that is
engaging with this recursive definition.
So we have sort of recursive definitions
of plus and times for the serial
numbers. And it turns out you can prove
that they make the surreal numbers into
what's called uh an ordered field.
Uh so they satisfy the field axioms
which means that you have distributivity
and commutivity of of addition and
multiplication and also you can you have
reciprocals for every nonzero number.
You can divide by the number. So you can
add and multiply and divide and
subtract. Um and furthermore you can
take square roots. Um and furthermore
every odd degree polomial has a root
which is true in the real numbers
because if you think about say a cubic
or a fifth degree polomial then you know
it's going to cross the axis because it
has opposite behaviors on the two
infinities because it's an odd degree
polinomial. So on the positive side it's
going to the positive infinity on the
negative side it would be going to minus
infinity. So it has to cross. So we know
in the real numbers every odd degree
polomial has a root and that's also true
in the surreal numbers. So that makes it
what's called a real close field. Um
which is a very nice mathematical
theory. Um so it's really quite
interesting uh how we can find copies of
all these other number systems inside
the serial numbers.
>> But the serial numbers are fundamentally
discontinuous as you write about. What
are the consequences of this?
>> Right. So the serial numbers have a
property that they form a non-standard
model of the real field which means that
they provide a notion of infant
decimality that one can use to develop
calculus on the grounds of Robinson's
non-standard theory that I had mentioned
earlier but they don't have the least
upper bound property for subolctions. So
there's no set of serial numbers, no
non-trivial set of serial numbers has a
least upper bound and there are no
[snorts] convergent sequences in the
serial numbers. And so for the sort of
ordinary use in calculus based on limits
and convergence
that method does not work in the serial
numbers at all. So that's what I mean
when I say the serial numbers are
fundamentally discontinuous. They they
have a fundamental discontinuity going
on. But you can still do calculus with
them because you have infantessimals if
you use these non-standard uh methods
the infantessimal based methods to
calculus and people uh and people do
that. I once organized a conference in
New York and we had John Conway as a
speaker at that conference and there was
a question session and someone asked him
I mean it's a bit rude question I think
um but they asked it and the question
was what is your greatest disappointment
in life I mean I would never ask a
question like that at a conference in a
very public setting but Conway was
extremely graceful and he answered by
saying that the surreal numbers Not the
numbers themselves but the reception of
the serial numbers because he had
ambition that the serial numbers would
become a fundamental number system used
throughout mathematics and science
because it was able to do nons analysis.
It was able to do calculus. It unified
the ordinals and so on. And it's such a
unifying amazing structure beautiful
structure with elegant proofs and
sophisticated ideas all around it.
and he was disappointed that it it it
never really achieved that unifying
status that he had the ambition for and
this he mentioned as his greatest
disappointment.
>> Yeah, Donald Kuth tried to celebrate it
uh but never quite took hold.
>> So I don't want to give the impression
though that that the serial numbers are
not widely studied because there's
thousands of people who are studying it.
In fact, Philip Erlick, who is one of
the world experts on this real numbers,
mentioned to me once that Conway was his
own worst enemy with regard to that very
issue because in the Conway style,
everything is a game. And he treated the
Surreal numbers as a kind of play thing,
a toy.
And maybe that makes people not take it
seriously, although my view is that it
is extremely serious and useful and
profound. [clears throat]
And I'm I've been writing a whole series
of essays on the surreal numbers for my
substack at infinitely more. And I just
find the whole subject so fascinating
and and beautiful. I mean it's true. I'm
not applying it in engineering which
maybe was uh part of this Conway
ambition. Uh and I just wanted to before
I forget mention
the Conway turning everything into a
game. It is a fascinating point that I
didn't quite think about which I think
uh the game of life is just an example
of exploration of cellular automa. I
think cellular is one of the most
incredible complicated fascinating. It
feels like an open door into a world we
have not quite yet explored. And it's
such a beautiful illustration of that
world, the game of life. But calling it
a game, maybe life balances it cuz such
a powerful word, but uh it's not quite a
game. It's a fascinating invitation to
an incredibly complicated and
fascinating mathematical world. I think
uh every time I see cellular automa and
the fact that we don't quite have tools,
mathematical tools to make sense of that
world, it uh fills me with awe. Speaking
of a thousand years from now, it feels
like that is a world we might uh make
some progress on.
>> The game of life is a sort of playground
for computably undecidable questions
because in fact you can prove that the
question of whether a given cell will
ever become alive [snorts]
is computably undecidable. In other
words, given a configuration and you ask
will this particular cell ever, you
know, be alive
>> in the evolution and you can prove that
that question is equivalent to the
halting problem. It's it's computably
undecidable. It's semi-decidable in the
sense that if it will become alive, then
you will know it at a finite stage
because you could just run the game of
life algorithm and let it run and if it
ever did come alive, you could say,
"Yeah, it was alive." But if you've run
it for a thousand years and it hasn't
come alive yet, then you don't
necessarily seem to have any basis for
saying no, it won't ever come alive. If
the behavior was very complicated, maybe
if you have a complete understanding of
the evolution of the behavior, then you
can say no. But you can prove you won't
always have that understanding precisely
because the problem is equivalent to the
holding problem. And nevertheless, when
you sit back and look and visualize the
thing, some little mini cellular
civilizations are born and die quickly
and some are very predictable and
boring, but some have this rich
incredible complexity. And maybe that
speaks to a thing I wanted to ask on the
halting problem and uh decidability.
You've mentioned this thing where if you
understand the program deeply, you might
be able to say something. So can we say
something interesting about maybe one
statistically how many programs
we know something about in terms of
whether they halt or not or what does it
mean to understand a program deeply
enough to be able to make a prediction.
The main lesson of computability theory
in my view
is that
it's never the case that you can have a
thorough understanding of the behavior
of a program by looking at the program
and that the content of what you learn
from a program I mean in the most
general case is always obtained just by
running it and looking at the behavior
and and the proof of that is there's a
theorem called ris's this theorem which
makes that idea completely robust.
But I want to just take a little detour
towards another question riffing on
something that you just said namely one
can ask the question uh what is the
behavior of a random program? So you
have some formal computing language.
Yeah. And you want to, you know, look at
the collection of all programs of a
certain size. Maybe there's only
finitely many. And can you say something
about the behavior of a randomly chosen
one? Like with a certain likelihood it
would have a certain behavior. And the
answer turns out to be extremely
interesting. Once years ago, Alexi
Maznikov asked me a question. He had
this concept of a decision problem with
a black hole. And what that means is
it's a decision problem which is
possibly difficult in the worst case but
the difficulty was concentrated in a
very tiny region called the black hole
and outside of that black hole it was
very easy. Mhm.
>> And so, for example, this kind of
problem is a terrible problem to use if
you're basing your encryption scheme,
you know, you don't want to use a black
hole problem because if someone can rob
the bank 95% of the time, you know, then
that's not what you want or even any
non-trivial percent of the time is too
dangerous. So, so you don't want to use
problems that are, you know, almost
every case as easily solved as the basis
of your encryption. And the question
Alexi asked me was does the halting
problem have a black hole? Yeah. And
[clears throat] so if we take say the
standard model of touring machines, it's
one way infinite tape with zeros and
ones on the tape and so on the head
moving back and forth and you know it
stops when it gets into the halt state.
Then it turns out we proved that there
is a black hole. And what that means is
there's a computer procedure that
decides correctly almost every instance
of the holding problem. Even though the
holding problem is not decidable, we can
decide almost every instance. So more
precisely, there's a collection of
touring machine programs
such that um we can easily decide
whether a program is in that collection
or not. And for the programs in the
collection um we can decide the halting
problem for those programs easily. And
furthermore almost every program is in
the collection in the sense that as the
number of states goes to you know
becomes large the proportion of programs
in the collection goes to 100%.
So the as asmtoic density of the
programs is one. Mhm.
>> And the proof was quite fascinating
because it's one of these situations
where the theorem sounds really
surprising I think to many people when I
first tell it I mean to computability
experts then it's sort of intriguing to
think that you can solve almost every
instance of a holding problem. Um but
then when they hear the proof it's
completely a letdown. Unfortunately
nobody likes the theorem after the
proof. And so the proof is so simple
though if you know what a touring
machine how a touring machine operates
there's this infinite paper tape on
which the machine writes zeros and ones
and the head moves back and forth
according to rigid instructions and the
instructions are all of the form if the
machine is in such and such a state and
it's reading such and such symbol on the
tape then it should write this symbol on
the tape and it should change to this
new state specified and it should either
move left or write as specified. So a
program consists of instructions like
that.
If you look at look at a program, you
know, one of the states is the halt
state and that's when the program halts.
But you can calculate how many programs
don't have any instruction that
transitions to the halt state. You can
easily calculate the proportion and in
the limit it goes to 1 over e^2 13 and a
half%. If you calculate the limit
the proportion of programs with end
states that don't ever halt because they
don't have any instruction saying halt.
>> Mhm.
>> Those programs obviously never halt
because they can't halt. They don't have
any instruction that says halt. So 13%
of programs
>> 13% you can say they don't halt because
you just look at them and you can
understand there's no they never they
never change to the whole state so they
can't halt.
>> I mean that nevertheless is beautiful to
to know to to show. So that's a kind of
trivial reason for non-halting, you
know, and when I first made that
observation, I thought, okay, this is
the proof strategy because we wanted I
wanted to say at first the goal was
>> look, that's a stupid reason for a
program not to halt. And I just want to
pile up as many stupid reasons as I can
think of
>> until it gets more than 50%. And then I
can say most, you know. [laughter]
>> Yeah.
>> Yeah, that was my goal.
>> I love this.
>> Yeah. So we thought more about it though
and we hit the jackpot because we found
one gigantic stupid reason that
converged to 100%. I mean in the limit
and so and the the the
stupid reason for a program not to halt
is that well if you think about the
behavior say the head is sitting there
it's on the leftmost cell of the tape at
the very beginning it's in the start
state and the head is following an
instruction and the instruction says
when you're in the start state which it
is and you're reading something on the
tape then you should write something and
you should change to a new state and you
should either move left and right left
or right, but half of them move left.
But if you move left and you are already
at the end, then the head falls off and
so the computation stops because the
head fell off the tape.
That's a pretty stupid reason. Okay, but
that's half of them already just like
that. Okay, and then some of them went
right and they changed to a new state.
And amongst those, you know, the new
state, half of those ones are going left
and half are going right from that
place. And then most of those are
changing to a new state. When there's a
lot of states, it's very likely that the
next state that you transition to is
new.
And so you get this random walk
behavior, if you know what that means,
where half go left and half go right at
each step. And there's a theorem due to
polio, which is called the polio
recurrence theorem, which says when you
have a random walk, a one-dimensional
random walk, then it's very likely to
come back to where you started. And when
that happens for us, then half of them
from that place fall off on the next
step.
And so you can show using this kind of
uh analysis that the probability one
behavior of a random touring machine is
that the head falls off the tape before
it repeats a state. And that is the
stupid proof that shows how to solve the
halting problem because when that
happens we can answer the halting
problem saying no the computation
stopped because the machine crashed not
because it halted. So therefore it
doesn't count as halting on some
accounts or you know if you want to
define that as halting crashing as
halting then but in any case however it
is that you set up your formalism you're
going to be able to answer the question
for the behavior of the machine
>> um when the head falls off.
>> So statistically in the limit you solve
the halting problem.
>> Yes. Exactly. Computably solve it. Yeah.
>> What do we take from that? Because you
didn't solve the halting problem. No,
it's impossible to fully solve the
halting problem correctly in all cases.
>> That's pretty cool. That's kind of I
mean I don't know. That's
>> it's a probabilistic way. I mean, it's
probabilistic in the sense that we're
solving almost all instances. Yeah.
Computably. There's versions of this
that are maybe more interesting from the
point of view of complexity theory and
actually useful. I mean, there's the
whole PNP problem and so on. And there's
this genre of NPMPlete problems which
are problems that are in feasible. They
would take exponential time to solve
them in the ordinary way and they're not
known to be polomial time solvable
although in these cases it's an open
question whether there is a polinomial
time algorithm a feasible algorithm and
for most uh for most of the nplete
problems you can prove um that there's a
polomial time approximation that solves
almost all instances in a feasible
amount of time so like the knapsack
problem you packing problems and so on
other kinds of problems satisfaction
problem when depending on how you set up
the formalism you can prove and and I've
proven many instances of this um but
also I think it's widespread
uh for almost all the npmplete problems
the difficult problems and these are
important problems for industrial
application what these are problems that
we actually want to solve
um we can have feasible algorithms that
solve almost every instances of them
>> the amount of fields and topics you
worked on is truly incredible. I have to
ask about P versus NP. This is one of
the big open problems in complexity
theory. So for people don't know, it's
about the relationship between
computation time and problem complexity.
Do you think it will ever be solved? And
uh is there any chance the weird
counterintuitive
thing might be true that P equals NP?
>> Yeah, that's an interesting question.
Sometimes people ask about whether it
could be independent which I think is an
interesting question for logicians.
Um [clears throat]
and of course well one has to say if
you're entertaining the idea of
independence uh you know over which
theory because every statement is going
to be independent over an extremely weak
theory. So that's you know it doesn't
make sense to say it's independent all
by itself. You're only independent
relative to a theory right? So the way I
think about PNP is that I mean of course
it's a theoretical question about the
asmtoic behavior of these problems. I
mean for a problem to be in P means that
there you know there is a a computable
decision procedure that that runs in
time bounded by some polomial but the
coefficients on that polinomial could be
enormous and the degree could be
incredibly high and um and so for small
values of inputs then it doesn't make
sense to talk about this polomial time
feasibility with respect to say the the
range of problem inputs that we will
give it in our lifetime or in the span
of human civilization or whatever. I
mean because it's an asmtoic property
it's really in the limit as the size of
the inputs goes to infinity that's the
only time that polomial or np becomes
relevant and [snorts] so maybe it's
important to keep that in mind when
sometimes you find kind of overblown
remarks about made about you know if p
equal np then this will be incredibly
important for human civilization because
it means that we'll have feasible
algorithms for solving these incredibly
important problems in np
you know that um you know it would cause
immense wealth for human societies and
so on because we would be able to solve
these otherwise intractable problems and
that would be the basis of new
technology and industry and so forth. I
mean people make these kind of remarks
but of course you have to temper those
remarks by the realization that P and P
equal NP or P not equal NP are not about
these practical things at all because of
the asmtoic nature of the question
itself. Okay, that's on the one hand,
but on the second hand, we already have
the algorithm, so we could use it
already. Except it's a terrible
algorithm because it it it involves all
this incredible amount of coding. And so
>> and uh on the third hand, like you said,
we already have approximation algorithms
that uh yes that from a pragmatic
perspective solve all the actual real
engineering problems of of of
humanization. like the SAT solvers work
amazingly well you know in lots and lots
of cases even though we can prove that
we don't expect if P is not equal to NP
then there won't be a polinomial time
SAT solver but actually the SAT solver
approximations you know are really quite
amazing
>> sorry to ask the ridiculous question but
who is the greatest mathematician of all
time [laughter]
who are the possible candidates Oiler
Gaus Newton Ramanogen Hilbert we
mentioned uh Girdle
touring if you throw them into the
bucket.
>> So this is I think an incredibly
difficult question to answer. I mean I
personally I don't really think this way
about sort of ranking the mathematicians
by greatness. Um
>> so you don't have like you know some
people have like a Taylor Swift poster
in their [laughter] room. You don't have
it. I mean, if you forced me to pick
someone, it would probably be Archimedes
because he is such incredible
achievements in such an early
era which totally transcended the work
of the other people in his era. So, but
I also have the view that I want to
learn mathematics and gain mathematical
insight from whoever can provide it and
wherever I can find it. And and this
isn't always just coming from the greats
and sometimes the greats are doing
things that are just first and not you
know somebody else could have easily
been first and so there's a kind of luck
aspect to it when you go back and look
at you know the achievements and because
of this progress issue in mathematics
that we talked about earlier namely we
really do understand things much better
now than they used to and when you look
back at the achievement that had been
made then maybe you can imagine you know
thinking Well, you know, somebody else
could have could have had that insight
also. Um, and and maybe they would have
it's already a known phenomenon that
disperate mathematicians end up proving
essentially similar results at
approximately the same time, but okay,
the person who did it first is getting
the credit and so on.
>> What do you make of that? Because I see
that sometimes when mathematicians, this
also applies in physics and science
where completely separately
discoveries are made at a very similar
time. What does that mean?
>> It's relatively common. I mean, I think
it's like certain ideas are in the air
and being thought about but not fully
articulated. And so this is the nature
of growth in knowledge.
>> Do you understand where ideas come from?
>> Not really. I mean what what's your own
process when you're thinking through a
problem?
>> Yeah, that's another difficult question.
I suppose it has to do with I mean my
mathematical style my style as a
mathematician is that
I don't really like difficult
mathematics. [laughter]
What I love is
simple, clear, easy to understand
arguments that prove a surprising
result. That's my favorite situation.
And actually, so the question of whether
it's a new result or not is somehow less
important to me. And and so that has to
do with this question of the great and
so on, whoever does it first. Because I
think for example, if you prove a new
result with a with a a bad argument or
complicated argument,
that's great because you prove something
new, but I still want to see the
beautiful simple
cuz that's what I can understand. Also,
I mean, I'm kind of naturally skeptical
about any complicated argument because
it might be wrong. And
>> if I can't really understand it fully,
like every single step all at once in my
head, then I'm just worried maybe it's
wrong. And so there's different styles.
Sometimes mathematicians get involved
with these enormous research projects
that involve huge numbers of working
parts and different technology coming
together. I mean mathematical
technology, not physical technology.
>> And sometimes it actually involves now
more and more something like the lean
programming language where some parts
are automated. So you have to
>> I see. Well, that's another issue
because maybe those things are you know
less subject to skepticism when it's
validated by lean. But I'm thinking
about the case where the arguments are
just extremely complicated and so I sort
of worry whether it's right or not.
Whereas you know I like the simple thing
and so so I tend to have often worked on
things that are a little bit off the
beaten path from what other people are
working on from that point of view.
>> Your curiosity draws you towards
simplicity. I want to work on the things
that I can understand and that are
simple. And uh and luckily I've found
that I've been able to make
contributions that other people seem to
like you know in this way in this style
and so I've been kind of fortunate from
that point of view. I mean my process
always though and um I've recommended
this always to my students uh
is just a kind of playful curiosity. So
whenever I have whenever there's an idea
or a topic
uh then I just play around with it and
change little things or understand a
basic case and then make it more
complicated or press things a little bit
on this side or apply the idea to my
favorite example
you know that's relevant or and see what
happens or you just play around with
ideas and this often leads to insights
that then lead to more methods or more
you know then pretty soon you're making
progress on the problem and so this is
basically my method is I just you know
fool around with the ideas until I can
see a path through uh towards something
interesting and then prove that um and
that's worked extremely well for me so
I'm pretty pleased with that method
>> you do like thought experiments where
you anthropomorphize like you mentioned
>> yeah yeah so this is a basic tool I mean
I use this all the time. You know, you
imagine a set theoretic model, a model
of ZFC as like a place where you're
living and you might travel to distant
lands by forcing and this is a kind of
metaphor for what's going on. Of course,
you know, the actual arguments aren't
anything like that because there's not
land and you're not traveling and you're
not
>> but you allow your mind to visualize
that kind of thing in actual
>> and it helps you to understand
particularly when there's parts of the
argument that are in tension with one
another. Then you can imagine that
people are fighting or something and
those kind of metaphors you know or you
imagine it in terms of a game theoretic
you know two players trying to win. So
that's kind of tension and and those
kind of metaphorical ways of
understanding a mathematical problem
often are extremely helpful in realizing
aha the enemy is going to pick this
thing to be like that because you know
it makes it more continuous or whatever
and then we should do this other thing
in order to so it it makes you realize
mathematical strategies for finding the
answer and proving the theorem that you
want to prove because of the the ideas
that come out of that anthropomorphy.
ization. What do you think of uh
somebody like Andrew Wilds who spent
seven years grinding at one of the
hardest problems in the history of
mathematics and maybe contrasting that a
little bit with somebody who's also
brilliant, Terrence Tao, who basically
says [snorts] if he hits a wall, he just
switches to a different problem. Maybe
comes [clears throat] back and so on. So
it's less of a focused grind for many
years without any
uh guarantee that you'll get there which
is what Andrew Wilds went through. Maybe
Gregory Pearlman did the same.
>> I mean wild proved an amazing theorem.
The formless theorem result is
incredible. Um this is a totally
different style than my own practice
though of working in isolation. I mean
for me mathematics is often a kind of
social activity. I have
>> I
>> counted I mean it's pushing towards a
hundred collaborators co-authors on
various papers and so on and you know
anybody has an idea they want to talk
about with me if I'm interested in it
then I'm going to want to collaborate
with them and we might solve the problem
and have a joint paper whatever you want
to have a joint paper
>> yeah exactly let's go
>> so my approach to like making
mathematical progress tends to involve
working with other people quite a lot
rather than just working on my own and I
enjoy that aspect very much.
>> So [snorts] I personally I couldn't ever
do what Wilds did. Maybe I'm missing
out. Maybe if I locked myself, you know,
in the bedroom and just worked on
whatever then I would sell it. But I
tend to think that no actually like
being
on math overflow so much and I've gotten
so many ideas, so many papers have grown
out of the math overflow conversations
and back and forth. someone posts an an
you know someone posts a question and I
post an answer on part of it and then
someone else has an idea and it turns
into a full solution and then we have a
three-way paper coming out of that
that's happened many times and so for me
it's I [clears throat] enjoy this kind
of social aspect to it and it's not just
the social part rather that's the nature
of mathematical investigation as I see
it is
putting forth mathematical ideas to
other people and they respond on to it
in a way that helps me learn, helps them
learn and I think that's a very
productive way of undertaking
mathematics. I think it's uh when you
work solo on mathematics from my
outsider perspective it seems
terrifyingly lonely
because you're especially if you do
stick to a single problem especially if
that problem has broken many brilliant
mathematicians in the past that you're
really putting all your chips in and
just the torment the roller coaster of
day because I imagine
you have these moments of hopeful break
many breakthroughs
And then you have to deal with the
occasional realization that no, it was
not a breakthrough and that
disappointment,
>> right?
>> And then you have to go like a weekly
maybe daily disappointment where you hit
a wall and you have no other person to
brainstorm with. you have no other um
avenue to pursue and it's I I don't know
the the mental fortitude it takes to go
through that but every everybody's
different and people are recluse and
just really
uh find solace in that lone grind uh I
have to ask about uh Gisha Gregori Pman
what do you think of him famously
declining the Fields Medal and the
Millennial Prize so He stated, "I'm not
interested in money or fame. The prize
is completely irrelevant to me. If the
proof is correct, then no other
recognition is needed." What do you
think of him turning down the prize? I
guess what I think is that mathematics
is full of a lot of different kinds of
people and my attitude is that hey, it
doesn't matter. Maybe they have a good
math idea and so I want to talk to them
and interact with them. And so I think
the Pearlman case, you know, is a is
maybe an instance where the, you know,
he's such a brilliant mind and he solved
this extremely famous and difficult
problem and that is a huge achievement.
Um, but he also had these views about,
you know, prizes and somehow I don't
really fully understand why he would
turn it down. I I do think uh I have a
similar thing just observing Olympic
athletes that are in many cases don't
get paid very much and they nevertheless
dedicate their entire lives for the
pursuit of the gold medal. I think his
case is a reminder that some of the
greatest mathematicians, some of the
greatest scientists and human beings do
the thing they do, take on these
problems for the love of it, not for the
prizes or the money or any of that. Now,
as you're saying, if the money comes,
you can use it for stuff. If the prizes
come and the fame and so on, that might
be useful, but the reason fundamentally
the grades do it is because of the art
itself.
>> Sure. I totally agree with that. I mean
I share the view that's you know that's
why I'm a mathematician uh is because I
find the questions so compelling and
I've spent my whole life thinking about
these problems and um you know but like
if I won an award
>> yeah it's great it's great I mean I'm
I'm pretty sure you don't contribute to
Math Overflow for the for the wealth and
[laughter] and the and the power
that you gain. I mean it's yeah genuine
genuine curiosity.
>> Well you asked who the greatest
mathematician is and um and uh and of
course if we want to be truly objective
about it we would need a kind of
objective criteria about how to evaluate
the relative
>> you know strength in the reputation of
various mathematicians and so of course
we should use math overflow score
because [laughter]
>> that you're definitively I mean nobody
objectively the greatest mathematician
of all time. Uh
>> I've also argued that tenure and
promotion decisions should be based on
math overflow.
>> So my daughter introduced me to uh her
boyfriend and told me that she had a
boyfriend
>> and I u
>> asked him
first of all what is his chess rating
and secondly what is his math overflow
score.
>> Oh man what is the only way to judge a
person I think that's I think
objectively correct. Yeah.
>> [laughter]
>> I mean, since you bring it up, chess, I
got to ask you about infinite chess. I
can't let you go. You've I mean, you
worked on a million things, but infinite
chess is one of them. Uh, somebody asked
on uh math overflow, the mathematical
definition of a chess.
>> So, can we talk about the math of chess
and the math of infinite chess? What is
infinite chess?
>> Oh, yeah. Absolutely. Infinite chess.
Fantastic. Chess ordinarily is played on
this tiny tiny board, this 8 by8 board,
right? So, when you play chess, normally
it's on the 8 by8 board. But we want to
play infinite chess. So on the on the
integer board, it's infinite in all four
directions, you know, but it still has
the chess board pattern. And maybe
there's pieces on this board, maybe
infinitely many pieces we allow. But one
difference from finite ordinary chess in
in infinite chess, we don't
play from a standard starting position.
Rather you
the interesting situation is that you
present a position where there's a lot
of pieces already on the board in a
complicated way and you say what would
it be like to start from this position
or from that one you know and we want to
produce positions that have interesting
features meaning mathematically
interesting features and so I can tell
you uh for example
probably a lot of people are familiar
with the um say the maiden two genre of
chess problem. You know, you have a
chess problem and it's white to mate in
two which means that white is going to
make two moves but the second move is
going to be a checkmate
>> or maybe maid in three or maid in five
or whatever we can have mate in n
positions for any n. I mean in infinite
chess
you can create a position which is not
mate in n for any n but
white has a winning strategy that will
win infinitely many moves.
So in other words,
>> let me say it again. There are positions
in infinite chess that white can
definitely win in finitely many moves.
White is going to make checkmate.
But there's no particular n for which
white can guarantee to win in n moves.
>> There's no n.
>> No n. So it's not made an end for any n.
But it's a white win infinitely many.
The way to think about it is white is
going to win, but black controls how
long it takes.
>> Ah, got it.
>> But it's doomed. Black can say, "Well, I
know you're going to win, but this time
it's going to you're going to take a
thousand moves at least."
>> Mhm.
>> And or maybe in a different way of
playing, black can say, "Well, I know
you're going to win, but this time
you're going to have to take a million
moves for any number." Black can say
that. So, it's these really interesting
positions. There's a position in my
first infinite chess paper. So, it's
black to play in this position. And if
black doesn't move that that rook there,
then white is going to checkmate pretty
quickly.
>> By the way, can we describe the rules of
infinite chess?
>> Right. So, the rules of infinite chess.
So, there there's just the ordinary
pieces and they move on this infinite
board which is just a chess board but
extended in all directions infinitely
with no edge. So, there's no boundary
>> but the pieces move just like you'd
expect. So the the knights move just the
same and the rooks move, you know, on
the ranks and files and the bishops move
on the same color diagonals and just
like you would expect except they can
move as far as they want, you know, if
there's no intervening piece in the way.
Um the one thing is that okay, so the
white pawns always move upwards and the
black pawns always move downwards, but
when they're capturing the pawns, you
know, capture on the diagonal. So I
think the piece movement is pretty
clear. There's a couple of differences
that you have to pay attention to from
ordinary chess. For example, um there's
this three-fold repetition rule in
ordinary chess. But we just we just get
rid of this for infinite chess because
of course three-fold repetition is just
a proxy for infinite play. The real rule
is infinite play is a draw, not
three-fold repetition is a draw. That's
just a kind of convenient approximation
to the what I view as the actual rule,
which is that infinite play is a draw.
So the only way to win is to make
checkmate on the board at a finite stage
of play. And if you play infinitely, you
haven't done that. And so it's a draw.
>> And the pawns can be converted.
>> And there's no promotion because there's
no edge. Right. Exactly. And this
position that we were just talking about
is a position with game value omega,
which means that because it has an
ordinal value, white is going to win.
But black can play as though counting
down from omega. What is the nature of
counting down from omega? If you're
black and you need to count down from
omega, then you have to say a finite
number
>> and then after that it's going to be at
most that many numbers afterwards to
count down. Right? So the nature of
counting down from omega is that you
take this giant step on the first count
and then after that you subtract one
each time. You can't subtract one from
omega because that's not an ordinal. So
if you come down from omega, you have to
go to some finite number. And then if
you just subtract one each time, then
that's how many more moves you get.
>> So that's the sense in which black can
make it take as long as he wants because
he can pick his initial number to be
whatever he wants.
>> By the way, I I just noticed that you're
citing a math overflow question, which
is really cool.
>> That's right. Yeah, this my interest in
infinite chess was born on math overflow
because someone asked this question.
>> Gnome Elkis asked this question. That's
so cool to see a math overflow citation
in a in an archive paper. That's cool.
How do you construct
the position
>> right
>> that satisfies this? Is there algorithm
for construction?
>> No, this is an act of mathematical
creativity really to come up with I had
a co-author my co-author Corey Evans.
He's a national a US national master
chess player. Uh very strong chess
player. Um he's also a uh philosophy
professor of law.
>> Your collaborations are wonderful.
That's great.
>> So I met him because he was a grad
student at Cooney where I was at the
time in New York. And uh also he was my
son's chess coach for when my son was
playing chess competitively in
elementary school.
>> Then Corey was the coach. Um and so we
knew him that way. And and that was
right around the time when I was getting
interested in infinite chess and I knew
I needed a chess knowledgeable
partner. And so Corey was invaluable uh
for the paper because
the proofs in this chess in infinite
chess are extremely finicky because you
create these positions
but the details of the argument have to
do with kind of chess reasoning you know
like and and my chess reading wasn't
quite up to it uh because I would create
the positions almost all the positions
are ones that I made But the initi but
this is like after many generations of
being corrected by Corey because Cory
would come and say, "Hey, you know, this
pawn is hanging and it breaks your
argument and or you know, this bishop
can leak out of the cage or whatever."
and and so uh and so the process was uh
I knew kind of in terms of these
ordinals what we needed to create with
the position and I would struggle to do
it and create something that sort of had
the features that I wanted and then I
would show it to Corey and he would say
look it doesn't work because of this and
that and so on and so this kind of back
and forth was extremely helpful to me
and eventually we you know converged on
arguments that were correct um and Uh so
it's yeah it's quite interesting also
maybe another thing to say is the the
follow-up paper to this one was a
three-way paper with also Corey and
myself and my PhD student Norman
promoter in which we improved the
bounds. So we were aiming to produce
more and more chess positions with
higher and higher ordinal values. So the
initial position was value omega and
then we made omega squared and omega
cubed in the in the first paper and then
in this three-way collaboration we made
omega to the 4th. The title of the paper
is a position in infinite chess with
game value omega to the fourth
>> right and so at the time this was the
best known result the sort of
state-of-the-art but since that time
it's been improved now dramatically um
and in fact we know now that every
countable ordinal arises as the game
value of a position in infinite chess so
it's fantastic result
>> before I forget let me ask
[clears throat] about
your views on AI and LLM that are
getting better and better mathematics
and we've spoken about collaborators and
you have so many collaborators. Do you
see AI as a potential great collaborator
to you as a mathematician and what do
you think the future role of those kinds
of AI systems is?
>> I guess I would draw a distinction
between you know what we have currently
and what might come in future years.
I've played around with it and I've
tried um experimenting you know but I
haven't found it helpful at all
basically zero it's not it's not helpful
to me and you know I've used various
systems and so on the paid models and so
on and um my typical experience is
interacting with AI on a mathematical
question is that it gives me garbage
answers that are not mathematically
correct. Mhm.
>> And
and so I find that not helpful and also
frustrating. Like if I was interacting
with a person, you know that the
frustrating thing is when you have to
argue about whether or not you know the
argument that they gave you is right and
you point out exactly the error and the
AI saying, "Oh, it's totally fine." And
>> and
>> you know, if if I were having such an
experience with a person, I would simply
refuse to talk to that person again. But
okay, one has to overlook these kind of
flaws. And so I tend to be a kind of
skeptic about the value the current
value of the current AI systems. As far
as mathematical reasoning is concerned,
it seems not reliable. Um, okay. But I
know for a fact that many that there are
several prominent mathematicians who I
have enormous respect for who are saying
that they are using it in a way that's
helpful and I'm often very surprised to
hear that based on my own experience
which is quite the opposite.
uh and so
maybe my process isn't any good although
you know I use it for other things like
you know for programming things or for
image generation and so on it's
amazingly powerful and helpful but for
mathematical arguments I haven't found
it helpful and maybe I'm not interacting
with it in a in the right way yet or it
could be um and so maybe I just need to
improve my skill but also Maybe I wonder
like these uh examples that are provided
by other people maybe involved quite a
huge amount of interaction and so I
wonder if maybe the mathematical ideas
are really coming from the person you
know these great mathematicians who are
doing it
>> rather than the AI and and so so I tend
to be kind of skeptical but also
I'm skeptical for another reason and
that
because of the nature of the
um the large language model approach to
AI doing mathematics.
Um
I recognize that the AI is trying to
give me an argument that sounds like a
proof rather than an argument that is a
proof. The motivation is misplaced.
And and so I I worry that this is a very
dangerous source of error because it
often happens in mathematics that I mean
if I think back to when I was in
undergrad, you know, here at Keltech and
I was a math major eventually and at
that time latte was a pretty new thing
and I was learning latte and so I was
typing up my homeworks in latte and they
looked beautiful. Actually, they looked
like garbage from my current standards.
I'm sure it was terrible. Except at the
time, you know, I didn't know anything.
I was in undergrad and uh latte was sort
of unheard of and so I was producing
these beautifully types set you know
problem set solutions and so on
>> and and I would print it up and submit
it and so on and the grades would come
back terrible grades
>> and and I realized what was happening is
that
the you know the the copy was so
beautiful mathematically types set in
this way it looked like the kind of
mathematics you find in a book, you
know, cuz basically that's the only time
you saw that kind of mathematical type
setting was in a
>> in a professional, you know, published
book.
>> And those ma that mathematics was almost
always correct in a book, right?
>> And so I had somehow, you know, lost my
>> critical
because it was so beautiful and I was
used to only seeing that kind of type
setting when an argument was, you know,
totally right. I wasn't enough and
making these sort of bonehead mistakes
in the proofs and and so okay so I I
corrected this of course
>> but this kind of effect is very much
real with the modern llam system and so
I think that that the chat programs and
so on are producing these arguments that
look really
>> they look like a that's what they're
striving to do that it's what they're
designed to do. They're not designed to
make a logically correct argument.
They're designed to make something that
looks like a logically correct argument.
And it's easy to get fooled if you're
not skeptical. And so that's why I worry
a bit when people rely on AI for
mathematical arguments. I mean, using
tying them to lean and the formal proof
um verification systems and so on, this
is a totally different way of operating.
But for the sort of ordinary person
sitting down and using chat to come up
with a mathematical argument, I think
it's a dangerous source of error if
you're not especially attuned to this
very issue that the the AI is going to
produce something that's not grounded in
mathematical understanding, but rather
something that is trying to look like
something that is grounded in
mathematical understanding. And those
are not the same thing at all.
And furthermore, I really wonder if one
can make a kind of system for producing
genuine mathematical insight that isn't
based in what I would view as
mathematical understanding as opposed to
the text generation systems. the methods
that are used, yeah, they don't seem
close enough grounded in understanding
of the underlying mathematical concepts,
but rather grounded in the way words
appear on a page in arguments about
those concepts, which are not the same.
So, there's a couple things to say
there. So, one, I think there is a real
skill in
providing the LLM system with enough
information
to be a good collaborator.
>> Yeah. cuz you really are dealing with a
different it's not a human being. You
really have to load in everything you
possibly can from your body of work from
the way you're thinking and that's a
real skill. And then the other thing is
you know for me if it's at all anything
like programming because I I have a lot
of uh colleagues and friends who are
programmers who kind of feel similarly
to you
and for me I have gotten better and
better and better at giving
uh as much information as possible to
the systems in in a really structured
way maybe because I just like natural
language as a way to express
my thinking. And then the benefit comes
from the inspiration that the system can
provide by its ability to
know a lot of things and make
connections between disperate fields and
between disperate concepts. And in that
way it provides not the answer but the
inspiration the handholding the
camaraderie that helps me get to the
answer because it does know a lot more
than me know like knowledge
and if you give it a lot of information
and ask the broader questions
uh it can make some really beautiful um
connections but I do find that I have to
be extremely patient like you that the
the the amount of times I'll do
something dumb where I feel like you
don't get this at all, do you? That's a
source of a lot of frustration for us
humans like this. Wait, this thing
doesn't understand at all. If you can
have the patience to look past that,
there might be some brilliant little
um insights that it can provide at least
at least for me in the realm of
programming. I should say programming
there's just so much training data.
There's so much there and
uh at least I see the light at the end
of the tunnel of promising
uh possibilities of it being a good
collaborator versus like
something that gives you really true
genius level insights,
>> right? It's probably true. I also find
it likely that a lot of the as far as
mathematical training data is concerned,
I just have to assume that math overflow
answers are part of the training data.
>> It's so uh
>> and you're [laughter]
>> I mean you're talking to yourself.
>> Yeah, maybe.
>> Um
sorry for the ridiculously big question,
but what idea in mathematics
is most beautiful to you? We've talked
about so many. The most beautiful idea
in mathematics is the transfinite
ordinals. These were the number system
invented by Gayorg Caner about counting
beyond infinity. Just the idea of
counting beyond infinity. I mean you
count through the ordinary numbers the
natural numbers 0 1 2 3 and so on and
then you're not done because after that
comes omega and then omega + 1 and omega
+ 2 and so on and you can always add one
and so of course after you count through
all those numbers of the form omega plus
n then you get to omega plus omega the
first number after all those and then
comes omega plus omega + 1 and so so on.
You can always add one. And so you can
just keep counting through the ordinals.
It never ends. Eventually you get to
omega * 3, omega * 4, and so on. And
then the limit of those numbers, the
first number that comes after all those
numbers will be omega squared. And this
one is the first compound limit ordinal
because it's a limit ordinal is one of
these numbers, an ordinal that doesn't
have an immediate predecessor like omega
and omega * 2, omega * 3. Those are all
limit ordinals.
But omega squar is a limit ordinal. But
it's also a limit of limit ordinals
because the omega * 3 omega * 4 and so
on. Those are all limit ordinals that
limit up to omega^ 2. And then of course
you form omega^ 2 + 1 and then omega^ 2
+ 2 and so on. And it never stops. And
it's just absolutely beautiful and
amazing and furthermore forms the
foundation for these transfinite
recursive constructions that came later.
I mean starting with the canerbendixen
theorem that I mentioned um and uh
continuing with uh the construction of
the of the v hierarchy and girdle's
constructible universe is built uh this
way and uh zerlo's proof of the
well-order principle using the exim of
choice is a transfinite recursive
construction and and so the idea of just
counting past infinity is so simple and
elegant and has led to so much
fascinating mathematics.
>> Yeah, the infinity is not the end. And
what about philosophy? What is the most
beautiful idea in philosophy?
>> So I do I have a foot in both fields,
philosophy and mathematics. And in some
context I seem to be required to choose
whether I'm a mathematician or a
philosopher. I mean my training is in
mathematics. My PhD, all my degrees are
mathematics. But somehow I turned myself
into a philosopher over the years
because my mathematical work was
engaging with these philosophical
issues. Um and so when I went in New
York I had appointments first in
mathematics only but then eventually I
was also joining the philosophy faculty
at the graduate center and when I went
to Oxford for the first time my main
appointment was in philosophy and that's
also true now at Notre Dame. Um although
I'm also concurrent professor in
mathematics and I have math PhD students
still and philosophy PhD students and so
I don't really
care to decide whether I'm a
mathematician or philosopher and my work
is engaging with mathematics and with
philosophical issues in mathematics and
with plain philosophy and there's this
ample
region between these reg between these
two subjects. So it's not necessary to
choose. I remember when I first went to
Oxford and I told my daughter that I was
going to become professor of philosophy
in Oxford. Um and she looked at me
plaintively and said uh but but papa
you're not a philosopher [laughter]
because in her mind you know her father
was the mathematician and her mother was
the philosopher because my wife Barbara
is a
>> [gasps]
>> um philosopher
now also at Notre Dame. together there
and uh okay but unfortunately I don't
really have to choose be between them.
So you ask about the most beautiful idea
in philosophy and I would have to say
that uh I think it's the distinction
between truth and proof the one that we
discussed already. Um it's it's so
profound and gets at the heart of so
many philosophical issues. I mean of
course this is a distinction that's
maybe born in mathematics or
mathematical logic but that's already
already phil philosophical to a degree
and it's you know fundamentally a
philosophical distinction.
that truth is about the
the nature of the world and the way
things are. It's about objective reality
in a sense. Whereas proof is about our
understanding of the world and about how
we come to know the things that we know
about the world.
>> And so to focus on proof is to focus on
the interaction that we have with the
objective reality. And okay, I'm talking
about the reality of mathematics, not
the physical world because as I said, I
live in the Platonic realm and I
interact with mathematical reality. And
so proof is about the interaction uh and
how we come to know the facts that are
true in this mathematical reality.
Whereas truth is about what's really the
case sort of apart from our knowledge of
it. And um and this is I think a such a
core way of that I have of understanding
the world and uh and the nature of logic
and reasonings
>> and the gap between the two is full of
fascinating mysteries both in the
platonic realm but also in the physics
realm and uh I would even say in the in
the human psychology, sociology,
politics, geopolitics, all of it. If you
think about proof more generally which
is the process of discovery
uh versus
the truth itself and that's our journey
uh whatever field we're in. Well, I for
one am grateful for uh for how marvelous
of a philosopher, mathematician and
human being you are. It's truly an honor
to speak with you today.
>> Well, thank you so much. It's such a
pleasure to be here and thank you for
inviting me.
Thanks for listening to this
conversation with Joel David Hamkins. To
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Thank you for listening. As always,
happy new year. I love you all.