0:00:17.498,0:00:19.767
Numbers are strange.

0:00:20.670,0:00:22.925
They are not physical objects.

0:00:22.925,0:00:28.535
No one has bumped into the number two[br]or tripped over the number three;

0:00:28.535,0:00:30.995
not even your crazy math professor.

0:00:32.451,0:00:35.974
They are not mental objects either.

0:00:35.974,0:00:39.231
The thought of your beloved [br]isn't your beloved

0:00:39.231,0:00:41.812
no matter how much[br]you might want it to be.

0:00:41.812,0:00:46.375
And no more is the thought [br]of the number three, the number three.

0:00:47.088,0:00:50.718
Nor do numbers exist in space or time.

0:00:50.718,0:00:55.027
You don't expect to find the number three[br]in the kitchen cupboard,

0:00:55.027,0:00:57.340
and you don't need to worry

0:00:57.340,0:01:02.527
that numbers once didn't exist[br]or might one day cease to exist.

0:01:03.260,0:01:06.540
But even though numbers [br]are far removed[br]

0:01:06.541,0:01:09.804
from the familiar world[br]of thoughts and things,[br]

0:01:10.892,0:01:17.566
they're intimately connected to that world[br]because we do things with numbers.

0:01:18.296,0:01:22.423
We count with them, we measure with them,

0:01:22.423,0:01:26.120
we formulate[br]our scientific theories with them.

0:01:27.475,0:01:31.213
So this makes it [br]all the stranger what they are.

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How can they be so far removed [br]from the familiar world

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and yet so intimately connected to it?

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In this talk I want to consider [br]three views about the nature of number

0:01:45.952,0:01:49.881
that were developed [br]by mathematicians and philosophers

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around the end of the 19th century[br]and the beginning of the 20th century.

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All of these views presuppose [br]that strictly speaking[br]

0:02:00.492,0:02:05.490
what we count are not things,[br]but sets of things.

0:02:05.490,0:02:11.527
A set is just many things, any things[br]you like, considered as one.

0:02:11.527,0:02:18.039
So for example, we have the set [br]of beer bottles that you drank last night.

0:02:19.683,0:02:22.938
The bottles are put in these braces [br]to indicate that the six bottles

0:02:22.938,0:02:25.672
are being considered as one object.

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Then we have the set consisting [br]of your two favorite pets, Fido and Felix.

0:02:33.402,0:02:38.203
Or we have a set consisting [br]of all the natural numbers,

0:02:38.203,0:02:40.425
so they're put together[br]in this very big set:

0:02:40.425,0:02:42.715
0, 1, 2, 3, 4 and so on.

0:02:44.234,0:02:49.002
And so what we do when we count [br]is associate a number with a set.

0:02:49.002,0:02:52.256
In the case of the beer bottles, [br]the number six,

0:02:52.256,0:02:55.775
assuming you're not too drunk[br]to count them.

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In the case of your pets, the number two.

0:03:00.041,0:03:04.667
And in the case of the natural numbers,[br]when we put them into one big set,

0:03:04.667,0:03:06.811
is going to be some infinite number.

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The first view I want to consider[br]about the nature of numbers

0:03:12.098,0:03:16.682
was developed independently [br]by two great philosopher mathematicians,

0:03:16.682,0:03:19.662
Gottlob Frege and Bertrand Russell.

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These two individuals[br]were very different from one another.

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Russell came from the English aristocracy;[br]

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Frege from the comfortable[br]German middle class.

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Russell was a crusading liberal;

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Frege, I'm sorry to say, was a proto-Nazi.

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Russell had four wives, [br]and innumerable mistresses;

0:03:45.479,0:03:47.300
Frege had a single wife, [br]

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and as far as I know, [br]enjoyed a happy, staid marital existence.

0:03:53.160,0:03:55.606
But despite these differences,[br]

0:03:55.606,0:03:58.679
they had more or less the same view[br]about the nature of number.

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So what was it?

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Well let's take the number two[br]as an example.

0:04:04.094,0:04:09.122
Two can be used to number[br]any two-membered set or pair.

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So it can be used to number the set[br]whose members are Frege and Russell.

0:04:15.554,0:04:18.038
Or it can be used to number the set

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consisting of your favorite pets, [br]Fido and Felix.

0:04:23.176,0:04:25.502
Or it can be used to number

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Dickens' famous two cities,[br]London and Paris.

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I insisted that London [br]be placed first there.

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(Laughter)

0:04:35.484,0:04:42.232
Now the idea of Russell and Frege was to[br]put all of these pairs into one big set.

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We pile them all into one big set, [br]and that would be the number two.

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So the number two [br]would be a set of sets,

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and these sets would just be all the pairs[br]that could be counted by the number two.

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Similarly, for all other numbers,

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the number three [br]would be the set of all triples,

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the number four the set [br]of all quadruples, and so on.

0:05:04.424,0:05:07.172
A simple and beautiful theory.

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Unfortunately, it led to contradiction.

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I can't give a demonstration [br]of the contradiction here,

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but I can give you a feel[br]for how it arose.

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You'll recall that the number two[br]was the set of all pairs,

0:05:24.828,0:05:27.058
all pairs of whatever.

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So in particular, it would include pairs[br]that themselves contain the number two.

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So let's look at the particular such pair,

0:05:35.943,0:05:39.196
the pair consisting of the number two[br]and the number one.

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Then that pair, the pair {1, 2},[br]would itself be inside the number two.

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So the number two would contain itself, [br]

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and that looks as if it's impossible.

0:05:53.308,0:05:55.289
So here's an analogy:

0:05:55.289,0:06:00.292
imagine a very hungry serpent [br]that tries to eat its own tail.

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Now, it could succeed in doing this

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- this is the best we could do[br]by way of illustration -

0:06:06.737,0:06:09.315
This is gross, but still possible.

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(Laughter)

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But imagine now[br]that the serpent is so ravenous

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that it attempts to eat itself[br]in its entirety.

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That's not even possible

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because then, the serpent's stomach[br]would have to be inside its stomach.

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And that's what happens [br]with the number two.

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The number two, as you see,[br]has itself inside of its very own stomach.

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What was to be done?

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The mathematician John von Neumann[br]came up with a brilliant solution;

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von Neumann was perhaps [br]one of the most versatile mathematicians[br]

0:06:48.470,0:06:50.156
who ever existed.

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He helped invent game theory[br]and the modern computer.

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He was a prodigy

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and had the most amazing[br]computational skills.

0:07:01.464,0:07:04.581
So what was his solution?

0:07:04.581,0:07:05.661
There he is.

0:07:05.661,0:07:07.206
He said: "Well look,

0:07:07.206,0:07:13.051
rather than take the number two[br]to be the set of all pairs,

0:07:13.051,0:07:16.478
take it to be a particular pair."

0:07:16.478,0:07:18.553
Well, which pair would it be?

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He suggested that the number two[br]should be the set of its predecessors.

0:07:24.043,0:07:28.827
Two has two predecessors, zero and one.

0:07:28.827,0:07:34.831
We take two to be the set[br]whose members are zero and one.

0:07:34.831,0:07:37.984
But we still have numbers;[br]we have zero and one.

0:07:37.984,0:07:42.752
Well, zero is the set of its predecessors.

0:07:42.752,0:07:45.866
Zero has no predecessors, [br]so it's what's called the 'null set,'

0:07:45.866,0:07:48.103
the set without any members.

0:07:48.103,0:07:52.778
And one has one predecessor, [br]which is zero.

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So one is the set[br]whose sole member is zero.

0:07:57.275,0:08:01.731
So there we have two defined, [br]one defined and zero defined.

0:08:01.731,0:08:05.679
If we put these definitions [br]together, we get the set.

0:08:05.679,0:08:10.110
The number two is the set[br]whose two members are the null set,

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which is the number zero,

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and the set whose sole member[br]is the null set, which is the number one.

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So that's according to von Neumann[br]what the number two is;

0:08:21.966,0:08:24.297
it's sets all the way down

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- sets, not turtles -

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And you actually hit rock bottom, too.

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And similarly for all other numbers,

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the number three would be [br]an even more complicated thing, and so on.

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Remember, the Frege-Russell view[br]gave birth to monsters.

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Here, we no longer have a monster;

0:08:46.000,0:08:47.939
the monster has turned into an angel,

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because although the number two[br]contains other numbers,

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it doesn't contain itself.

0:08:53.995,0:08:57.901
The monster is always eating [br]a smaller monster, so to speak.

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It doesn't get in the way of itself.

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This view is generally accepted [br]by philosophers and mathematicians today,

0:09:05.743,0:09:08.369
but it also has its difficulties.

0:09:08.809,0:09:11.039
One difficulty that especially bothers me

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is there's nothing special[br]about the number two.

0:09:14.340,0:09:19.338
We want the number two [br]to be what is common to all pairs,

0:09:20.022,0:09:25.253
but von Neumann's number two [br]is just one pair among many,

0:09:25.253,0:09:26.865
and there's no special way

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in which that pair is[br]what's common to all pairs.

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So it doesn't make [br]the number two special anyway;

0:09:34.387,0:09:37.200
it's just one pair among many.

0:09:37.200,0:09:42.508
We come now to the final view,[br]and the one I like most of all.

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It's a view that's generally [br]dismissed or ignored

0:09:49.742,0:09:52.634
by philosophers [br]and mathematicians of today.

0:09:52.634,0:09:58.435
It was developed by Georg Cantor[br]in the late 19th century.

0:09:59.406,0:10:04.815
Cantor was a multi-talented individual, [br]

0:10:04.815,0:10:08.194
a brilliant violinist

0:10:11.104,0:10:16.260
with wide-ranging interests[br]ranging from religion to literature.

0:10:17.444,0:10:20.911
But he's best known [br]for his theory of infinite number.

0:10:21.631,0:10:25.540
Cantor wanted to count [br]not only finite collections

0:10:25.540,0:10:29.430
- I know there are a lot of people here,[br]but it's still a finite number -

0:10:29.430,0:10:32.680
so not just the finite collections,[br]like the number of people here,

0:10:32.680,0:10:36.180
or the number of stars in the Milky Way,

0:10:36.180,0:10:39.403
he also wanted[br]to count infinite collections,

0:10:39.403,0:10:44.332
like the collection of all natural numbers[br]or the collection of all points in space.

0:10:45.456,0:10:50.288
And to this end, he attempted [br]to develop a general theory of number.

0:10:51.342,0:10:53.467
So what was his view?

0:10:53.467,0:10:55.816
Again, let's consider the number two.

0:10:55.816,0:10:59.699
Let's take two objects, Fido and Felix.

0:11:00.259,0:11:01.613
Now Cantor said:

0:11:01.613,0:11:08.205
"Look, let's deprive these two objects[br]of all of their individuating features

0:11:08.205,0:11:11.699
beyond their being distinct[br]from one another."

0:11:11.699,0:11:14.443
So we remove their fur,

0:11:14.443,0:11:18.607
we remove their flesh and blood,

0:11:18.607,0:11:22.062
until we're simply left [br]with two bare objects

0:11:22.062,0:11:25.479
- what he called units[br]without any differentiating features.

0:11:25.479,0:11:28.207
I hope there are no animal lovers[br]amongst you.

0:11:28.207,0:11:32.743
But anyway, this is what happens to pets[br]when Cantor gets hold of them.

0:11:33.973,0:11:36.205
So what are these units?

0:11:36.205,0:11:40.032
Well, take the two dollars [br]in your bank account

0:11:40.032,0:11:43.497
- I hope you still have two dollars left[br]after paying the admission fee.

0:11:43.497,0:11:47.362
These two dollars[br]aren't any particular dollars,

0:11:47.362,0:11:49.602
but when you go to the ATM machine,[br]

0:11:49.602,0:11:52.753
you can redeem them[br]for two particular dollars.

0:11:52.753,0:11:55.009
So they aren't those particular dollars,

0:11:55.009,0:11:57.653
but they can be redeemed[br]for any two particular dollars.

0:11:57.653,0:11:59.700
This is what Cantor's units are like.

0:11:59.700,0:12:03.812
But when you go to the Cantorian [br]ATM machine to redeem your units,

0:12:03.812,0:12:06.593
you get back any two objects or whatever.

0:12:06.593,0:12:08.871
It's the ultimate lucky dip.

0:12:09.633,0:12:12.145
Cantor's idea was this: [br]

0:12:12.145,0:12:17.090
we take the number two[br]to be the set of these two units.

0:12:17.090,0:12:19.230
So we take these two units,

0:12:19.230,0:12:22.359
which could be derived[br]from any two objects,

0:12:22.359,0:12:26.892
and the number two[br]is the set of those two units.

0:12:26.892,0:12:29.018
And similarly, for all other numbers;

0:12:29.018,0:12:31.411
the number three [br]would be the set of three units,[br]

0:12:31.411,0:12:33.811
and so on and so forth.

0:12:33.811,0:12:37.002
So we have three views on the table.

0:12:37.002,0:12:38.808
The Frege-Russell view

0:12:38.808,0:12:42.193
according to which the number two[br]is the set of all pairs;

0:12:42.193,0:12:44.043
the view of von Neumann,

0:12:44.043,0:12:47.976
according to which the number two[br]is the set whose members are zero and one;

0:12:47.976,0:12:51.781
and the Cantorian view,

0:12:51.781,0:12:55.587
according to which two[br]is the set of two units.

0:12:55.587,0:13:01.524
The Frege-Russell view breeds [br]monsters, so we can't have it.

0:13:02.984,0:13:05.927
The von Neumann view [br]doesn't properly account

0:13:05.927,0:13:09.890
for why the number two [br]is common to all pairs.

0:13:11.005,0:13:15.178
The Cantorian view suffers[br]from neither of those difficulties.

0:13:15.178,0:13:18.997
It doesn't breed monsters because[br]the number two only contains units;

0:13:18.997,0:13:21.932
it doesn't itself contain the number two.

0:13:21.932,0:13:25.097
And it's, in an obvious sense, [br]common to all pairs,

0:13:25.097,0:13:28.137
because it's derived [br]by this process of abstraction,

0:13:28.137,0:13:30.897
or stripping away, from each pair.

0:13:33.000,0:13:36.872
So thanks to Cantor,[br]we now know what numbers are.

0:13:37.735,0:13:38.804
Thank you.

0:13:38.804,0:13:40.090
(Applause)