What are numbers? | Kit Fine | TEDxNewYork
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0:17 - 0:20Numbers are strange.
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0:21 - 0:23They are not physical objects.
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0:23 - 0:29No one has bumped into the number two
or tripped over the number three; -
0:29 - 0:31not even your crazy math professor.
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0:32 - 0:36They are not mental objects either.
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0:36 - 0:39The thought of your beloved
isn't your beloved -
0:39 - 0:42no matter how much
you might want it to be. -
0:42 - 0:46And no more is the thought
of the number three, the number three. -
0:47 - 0:51Nor do numbers exist in space or time.
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0:51 - 0:55You don't expect to find the number three
in the kitchen cupboard, -
0:55 - 0:57and you don't need to worry
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0:57 - 1:03that numbers once didn't exist
or might one day cease to exist. -
1:03 - 1:07But even though numbers
are far removed -
1:07 - 1:10from the familiar world
of thoughts and things, -
1:11 - 1:18they're intimately connected to that world
because we do things with numbers. -
1:18 - 1:22We count with them, we measure with them,
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1:22 - 1:26we formulate
our scientific theories with them. -
1:27 - 1:31So this makes it
all the stranger what they are. -
1:31 - 1:36How can they be so far removed
from the familiar world -
1:36 - 1:39and yet so intimately connected to it?
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1:40 - 1:46In this talk I want to consider
three views about the nature of number -
1:46 - 1:50that were developed
by mathematicians and philosophers -
1:50 - 1:56around the end of the 19th century
and the beginning of the 20th century. -
1:57 - 2:00All of these views presuppose
that strictly speaking -
2:00 - 2:05what we count are not things,
but sets of things. -
2:05 - 2:12A set is just many things, any things
you like, considered as one. -
2:12 - 2:18So for example, we have the set
of beer bottles that you drank last night. -
2:20 - 2:23The bottles are put in these braces
to indicate that the six bottles -
2:23 - 2:26are being considered as one object.
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2:26 - 2:32Then we have the set consisting
of your two favorite pets, Fido and Felix. -
2:33 - 2:38Or we have a set consisting
of all the natural numbers, -
2:38 - 2:40so they're put together
in this very big set: -
2:40 - 2:430, 1, 2, 3, 4 and so on.
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2:44 - 2:49And so what we do when we count
is associate a number with a set. -
2:49 - 2:52In the case of the beer bottles,
the number six, -
2:52 - 2:56assuming you're not too drunk
to count them. -
2:57 - 3:00In the case of your pets, the number two.
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3:00 - 3:05And in the case of the natural numbers,
when we put them into one big set, -
3:05 - 3:07is going to be some infinite number.
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3:08 - 3:12The first view I want to consider
about the nature of numbers -
3:12 - 3:17was developed independently
by two great philosopher mathematicians, -
3:17 - 3:20Gottlob Frege and Bertrand Russell.
-
3:20 - 3:23These two individuals
were very different from one another. -
3:24 - 3:27Russell came from the English aristocracy;
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3:27 - 3:30Frege from the comfortable
German middle class. -
3:31 - 3:34Russell was a crusading liberal;
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3:35 - 3:38Frege, I'm sorry to say, was a proto-Nazi.
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3:39 - 3:45Russell had four wives,
and innumerable mistresses; -
3:45 - 3:47Frege had a single wife,
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3:47 - 3:53and as far as I know,
enjoyed a happy, staid marital existence. -
3:53 - 3:56But despite these differences,
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3:56 - 3:59they had more or less the same view
about the nature of number. -
3:59 - 4:01So what was it?
-
4:01 - 4:04Well let's take the number two
as an example. -
4:04 - 4:09Two can be used to number
any two-membered set or pair. -
4:09 - 4:16So it can be used to number the set
whose members are Frege and Russell. -
4:16 - 4:18Or it can be used to number the set
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4:18 - 4:22consisting of your favorite pets,
Fido and Felix. -
4:23 - 4:26Or it can be used to number
-
4:26 - 4:29Dickens' famous two cities,
London and Paris. -
4:29 - 4:32I insisted that London
be placed first there. -
4:32 - 4:34(Laughter)
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4:35 - 4:42Now the idea of Russell and Frege was to
put all of these pairs into one big set. -
4:43 - 4:47We pile them all into one big set,
and that would be the number two. -
4:47 - 4:50So the number two
would be a set of sets, -
4:50 - 4:55and these sets would just be all the pairs
that could be counted by the number two. -
4:55 - 4:58Similarly, for all other numbers,
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4:58 - 5:00the number three
would be the set of all triples, -
5:00 - 5:04the number four the set
of all quadruples, and so on. -
5:04 - 5:07A simple and beautiful theory.
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5:08 - 5:12Unfortunately, it led to contradiction.
-
5:13 - 5:16I can't give a demonstration
of the contradiction here, -
5:16 - 5:19but I can give you a feel
for how it arose. -
5:20 - 5:25You'll recall that the number two
was the set of all pairs, -
5:25 - 5:27all pairs of whatever.
-
5:27 - 5:33So in particular, it would include pairs
that themselves contain the number two. -
5:34 - 5:36So let's look at the particular such pair,
-
5:36 - 5:39the pair consisting of the number two
and the number one. -
5:42 - 5:46Then that pair, the pair {1, 2},
would itself be inside the number two. -
5:46 - 5:49So the number two would contain itself,
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5:50 - 5:53and that looks as if it's impossible.
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5:53 - 5:55So here's an analogy:
-
5:55 - 6:00imagine a very hungry serpent
that tries to eat its own tail. -
6:00 - 6:03Now, it could succeed in doing this
-
6:03 - 6:07- this is the best we could do
by way of illustration - -
6:07 - 6:09This is gross, but still possible.
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6:09 - 6:11(Laughter)
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6:11 - 6:14But imagine now
that the serpent is so ravenous -
6:14 - 6:18that it attempts to eat itself
in its entirety. -
6:19 - 6:21That's not even possible
-
6:21 - 6:26because then, the serpent's stomach
would have to be inside its stomach. -
6:26 - 6:29And that's what happens
with the number two. -
6:29 - 6:35The number two, as you see,
has itself inside of its very own stomach. -
6:36 - 6:38What was to be done?
-
6:39 - 6:44The mathematician John von Neumann
came up with a brilliant solution; -
6:45 - 6:48von Neumann was perhaps
one of the most versatile mathematicians -
6:48 - 6:50who ever existed.
-
6:50 - 6:54He helped invent game theory
and the modern computer. -
6:55 - 6:58He was a prodigy
-
6:58 - 7:01and had the most amazing
computational skills. -
7:01 - 7:05So what was his solution?
-
7:05 - 7:06There he is.
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7:06 - 7:07He said: "Well look,
-
7:07 - 7:13rather than take the number two
to be the set of all pairs, -
7:13 - 7:16take it to be a particular pair."
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7:16 - 7:19Well, which pair would it be?
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7:19 - 7:24He suggested that the number two
should be the set of its predecessors. -
7:24 - 7:29Two has two predecessors, zero and one.
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7:29 - 7:35We take two to be the set
whose members are zero and one. -
7:35 - 7:38But we still have numbers;
we have zero and one. -
7:38 - 7:43Well, zero is the set of its predecessors.
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7:43 - 7:46Zero has no predecessors,
so it's what's called the 'null set,' -
7:46 - 7:48the set without any members.
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7:48 - 7:53And one has one predecessor,
which is zero. -
7:53 - 7:57So one is the set
whose sole member is zero. -
7:57 - 8:02So there we have two defined,
one defined and zero defined. -
8:02 - 8:06If we put these definitions
together, we get the set. -
8:06 - 8:10The number two is the set
whose two members are the null set, -
8:10 - 8:12which is the number zero,
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8:12 - 8:16and the set whose sole member
is the null set, which is the number one. -
8:17 - 8:22So that's according to von Neumann
what the number two is; -
8:22 - 8:24it's sets all the way down
-
8:25 - 8:27- sets, not turtles -
-
8:27 - 8:30And you actually hit rock bottom, too.
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8:31 - 8:34And similarly for all other numbers,
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8:34 - 8:37the number three would be
an even more complicated thing, and so on. -
8:38 - 8:43Remember, the Frege-Russell view
gave birth to monsters. -
8:43 - 8:46Here, we no longer have a monster;
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8:46 - 8:48the monster has turned into an angel,
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8:48 - 8:51because although the number two
contains other numbers, -
8:51 - 8:53it doesn't contain itself.
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8:54 - 8:58The monster is always eating
a smaller monster, so to speak. -
8:59 - 9:01It doesn't get in the way of itself.
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9:01 - 9:06This view is generally accepted
by philosophers and mathematicians today, -
9:06 - 9:08but it also has its difficulties.
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9:09 - 9:11One difficulty that especially bothers me
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9:11 - 9:14is there's nothing special
about the number two. -
9:14 - 9:19We want the number two
to be what is common to all pairs, -
9:20 - 9:25but von Neumann's number two
is just one pair among many, -
9:25 - 9:27and there's no special way
-
9:27 - 9:31in which that pair is
what's common to all pairs. -
9:32 - 9:34So it doesn't make
the number two special anyway; -
9:34 - 9:37it's just one pair among many.
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9:37 - 9:43We come now to the final view,
and the one I like most of all. -
9:45 - 9:50It's a view that's generally
dismissed or ignored -
9:50 - 9:53by philosophers
and mathematicians of today. -
9:53 - 9:58It was developed by Georg Cantor
in the late 19th century. -
9:59 - 10:05Cantor was a multi-talented individual,
-
10:05 - 10:08a brilliant violinist
-
10:11 - 10:16with wide-ranging interests
ranging from religion to literature. -
10:17 - 10:21But he's best known
for his theory of infinite number. -
10:22 - 10:26Cantor wanted to count
not only finite collections -
10:26 - 10:29- I know there are a lot of people here,
but it's still a finite number - -
10:29 - 10:33so not just the finite collections,
like the number of people here, -
10:33 - 10:36or the number of stars in the Milky Way,
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10:36 - 10:39he also wanted
to count infinite collections, -
10:39 - 10:44like the collection of all natural numbers
or the collection of all points in space. -
10:45 - 10:50And to this end, he attempted
to develop a general theory of number. -
10:51 - 10:53So what was his view?
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10:53 - 10:56Again, let's consider the number two.
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10:56 - 11:00Let's take two objects, Fido and Felix.
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11:00 - 11:02Now Cantor said:
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11:02 - 11:08"Look, let's deprive these two objects
of all of their individuating features -
11:08 - 11:12beyond their being distinct
from one another." -
11:12 - 11:14So we remove their fur,
-
11:14 - 11:19we remove their flesh and blood,
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11:19 - 11:22until we're simply left
with two bare objects -
11:22 - 11:25- what he called units
without any differentiating features. -
11:25 - 11:28I hope there are no animal lovers
amongst you. -
11:28 - 11:33But anyway, this is what happens to pets
when Cantor gets hold of them. -
11:34 - 11:36So what are these units?
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11:36 - 11:40Well, take the two dollars
in your bank account -
11:40 - 11:43- I hope you still have two dollars left
after paying the admission fee. -
11:43 - 11:47These two dollars
aren't any particular dollars, -
11:47 - 11:50but when you go to the ATM machine,
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11:50 - 11:53you can redeem them
for two particular dollars. -
11:53 - 11:55So they aren't those particular dollars,
-
11:55 - 11:58but they can be redeemed
for any two particular dollars. -
11:58 - 12:00This is what Cantor's units are like.
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12:00 - 12:04But when you go to the Cantorian
ATM machine to redeem your units, -
12:04 - 12:07you get back any two objects or whatever.
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12:07 - 12:09It's the ultimate lucky dip.
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12:10 - 12:12Cantor's idea was this:
-
12:12 - 12:17we take the number two
to be the set of these two units. -
12:17 - 12:19So we take these two units,
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12:19 - 12:22which could be derived
from any two objects, -
12:22 - 12:27and the number two
is the set of those two units. -
12:27 - 12:29And similarly, for all other numbers;
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12:29 - 12:31the number three
would be the set of three units, -
12:31 - 12:34and so on and so forth.
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12:34 - 12:37So we have three views on the table.
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12:37 - 12:39The Frege-Russell view
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12:39 - 12:42according to which the number two
is the set of all pairs; -
12:42 - 12:44the view of von Neumann,
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12:44 - 12:48according to which the number two
is the set whose members are zero and one; -
12:48 - 12:52and the Cantorian view,
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12:52 - 12:56according to which two
is the set of two units. -
12:56 - 13:02The Frege-Russell view breeds
monsters, so we can't have it. -
13:03 - 13:06The von Neumann view
doesn't properly account -
13:06 - 13:10for why the number two
is common to all pairs. -
13:11 - 13:15The Cantorian view suffers
from neither of those difficulties. -
13:15 - 13:19It doesn't breed monsters because
the number two only contains units; -
13:19 - 13:22it doesn't itself contain the number two.
-
13:22 - 13:25And it's, in an obvious sense,
common to all pairs, -
13:25 - 13:28because it's derived
by this process of abstraction, -
13:28 - 13:31or stripping away, from each pair.
-
13:33 - 13:37So thanks to Cantor,
we now know what numbers are. -
13:38 - 13:39Thank you.
-
13:39 - 13:40(Applause)
- Title:
- What are numbers? | Kit Fine | TEDxNewYork
- Description:
-
This talk was given at a local TEDx event, produced independently of the TED conferences.
Numbers are neither physical objects nor mental ones. They don't exist in space or time, and yet, we are so intimately connected to them.
In this captivating talk, Kit Fine considers three views about the nature of numbers. - Video Language:
- English
- Team:
- closed TED
- Project:
- TEDxTalks
- Duration:
- 13:44
Denise RQ edited English subtitles for What are numbers? | Kit Fine | TEDxNewYork | ||
Denise RQ edited English subtitles for What are numbers? | Kit Fine | TEDxNewYork | ||
Denise RQ edited English subtitles for What are numbers? | Kit Fine | TEDxNewYork | ||
Denise RQ approved English subtitles for What are numbers? | Kit Fine | TEDxNewYork | ||
Denise RQ edited English subtitles for What are numbers? | Kit Fine | TEDxNewYork | ||
Denise RQ edited English subtitles for What are numbers? | Kit Fine | TEDxNewYork | ||
Denise RQ edited English subtitles for What are numbers? | Kit Fine | TEDxNewYork | ||
Denise RQ edited English subtitles for What are numbers? | Kit Fine | TEDxNewYork |