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y mission.
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00:00:02,603 --> 00:00:06,038
NARRATOR:
We live in an age
of astonishing advances.
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MAN:
Descending at about .75 meters
per second.
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00:00:09,243 --> 00:00:13,379
NARRATOR:
Engineers can land
a car-size rover on Mars.
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MAN:
Touchdown confirmed.
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00:00:14,482 --> 00:00:16,849
(cheering)
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NARRATOR:
Physicists probe the essence
of all matter,
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00:00:21,856 --> 00:00:26,825
while we communicate wirelessly
on a vast worldwide network.
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00:00:28,595 --> 00:00:31,130
But underlying
all of these modern wonders
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00:00:31,132 --> 00:00:35,934
is something deep
and mysteriously powerful.
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00:00:35,936 --> 00:00:38,670
It's been called
the language of the universe,
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00:00:38,672 --> 00:00:42,608
and perhaps it's civilization's
greatest achievement.
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00:00:42,610 --> 00:00:44,276
Its name?
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Mathematics.
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00:00:46,547 --> 00:00:48,947
But where does math come from?
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00:00:48,949 --> 00:00:52,418
And why in science
does it work so well?
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00:00:52,420 --> 00:00:54,353
MARIO LIVIO:
Albert Einstein wondered,
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00:00:54,355 --> 00:00:57,122
"How is it possible
that mathematics
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00:00:57,124 --> 00:01:01,860
does so well in explaining
the universe as we see it?"
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NARRATOR:
Is mathematics even human?
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There doesn't really
seem to be an upper limit
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to the numerical abilities
of animals.
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00:01:11,672 --> 00:01:15,507
NARRATOR:
And is it the key to the cosmos?
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MAX TEGMARK:
Our physical world
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doesn't just have
some mathematical properties,
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00:01:19,480 --> 00:01:21,480
but it has only
mathematical properties.
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NARRATOR:
"The Great Math Mystery,"
next on NOVA!
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NARRATOR:
Human beings have always
is provlooked at natureowing:
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and searched for patterns.
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Eons ago, we gazed at the stars
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and discovered patterns
we call constellations,
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even coming to believe
they might control our destiny.
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00:02:02,490 --> 00:02:08,494
We've watched the days
turn to night and back to day,
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00:02:08,496 --> 00:02:11,864
and seasons
as they come and go,
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and called that pattern "time."
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We see symmetrical patterns
in the human body
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and the tiger's stripes
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and build those patterns
into what we create,
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from art to our cities.
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But what do patterns tell us?
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Why should the spiral shape
of the nautilus shell
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be so similar
to the spiral of a galaxy?
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Or the spiral found in
a sliced open head of cabbage?
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When scientists
seek to understand
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the patterns of our world,
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they often turn
to a powerful tool: mathematics.
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They quantify their observations
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and use mathematical techniques
to examine them,
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hoping to discover
the underlying causes
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of nature's rhythms
and regularities.
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And it's worked,
revealing the secrets
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behind the elliptical orbits
of the planets
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to the electromagnetic waves
that connect our cell phones.
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Mathematics has even
guided the way,
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leading us right down
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to the sub-atomic
building blocks of matter.
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Which raises the question:
why does it work at all?
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Is there an inherent
mathematical nature to reality?
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Or is mathematics
all in our heads?
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Mario Livio
is an astrophysicist
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who wrestles
with these questions.
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He's fascinated by the deep
and often mysterious connection
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between mathematics
and the world.
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MARIO LIVIO:
If you look at nature,
there are numbers all around us.
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You know,
look at flowers, for example.
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So there are many flowers
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that have three petals
like this, or five like this.
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Some of them may have 34 or 55.
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These numbers occur very often.
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NARRATOR:
These may sound like
random numbers,
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but they're all part of what is
known as the Fibonacci sequence,
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00:04:31,872 --> 00:04:35,941
a series of numbers developed
by a 13th century mathematician.
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You start with the numbers
one and one,
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and from that point on,
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you keep adding up
the last two numbers.
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So one plus one is two,
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now one plus two is three,
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two plus three is five,
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three plus five is eight,
and you keep going like this.
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NARRATOR:
Today, hundreds of years later,
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this seemingly arbitrary
progression of numbers
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fascinates many,
who see in it clues
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to everything from human beauty
to the stock market.
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While most of those claims
remain unproven,
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it is curious how evolution
seems to favor these numbers.
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And as it turns out,
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this sequence appears
quite frequently in nature.
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NARRATOR:
Fibonacci numbers show up
in petal counts,
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especially of daisies,
but that's just a start.
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CHRISTOPHE GOLE:
Statistically,
the Fibonacci numbers
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do appear a lot in botany.
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For instance, if you look
at theottom of a pine cone,
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you will see often spirals
in their scales.
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You end up
counting those spirals,
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you'll usually find
a Fibonacci number,
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and then you will count
the spirals
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going in the other direction
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and you will find
an adjacent Fibonacci number.
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NARRATOR:
The same is true of the seeds
on a sunflower head--
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two sets of spirals.
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And if you count the spirals
in each direction,
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both are Fibonacci numbers.
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While there are some theories
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explaining
the Fibonacci-botany connection,
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it still raises
some intriguing questions.
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So do plants know math?
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The short answer
to that is "No."
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They don't need to know math.
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In a very simple, geometric way,
they set up a little machine
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that creates the Fibonacci
sequence in many cases.
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NARRATOR:
The mysterious connections
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between the physical world
and mathematics run deep.
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We all know the number pi
from geometry--
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the ratio between
the circumference of a circle
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and its diameter--
and that its decimal digits
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go on forever
without a repeating pattern.
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As of 2013,
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it had been calculated out
to 12.1 trillion digits.
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00:07:10,164 --> 00:07:14,433
But somehow,
pi is a whole lot more.
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00:07:14,435 --> 00:07:17,469
Pi appears in a whole host
of other phenomena
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00:07:17,471 --> 00:07:19,571
which have,
at least on the face of it,
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nothing to do
with circles or anything.
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In particular, it appears in
probability theory quite a bit.
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Suppose I take this needle.
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So the length of the needle
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is equal to the distance
between two lines
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on this piece of paper.
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00:07:34,922 --> 00:07:37,889
And suppose I drop this needle
now on the paper.
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NARRATOR:
Sometimes when you drop
the needle, it will cut a line,
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00:07:42,229 --> 00:07:44,930
and sometimes it drops
between the lines.
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00:07:44,932 --> 00:07:47,632
It turns out the probability
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that the needle lands
so it cuts a line
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is exactly two over pi,
or about...
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00:07:56,209 --> 00:07:59,211
...64%.
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Now, what that means is that,
in principle,
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I could drop this needle
millions of times.
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00:08:05,419 --> 00:08:08,353
I could count the times
when it crosses a line
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00:08:08,355 --> 00:08:10,922
and when it doesn't
cross a line,
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00:08:10,924 --> 00:08:12,991
and I could actually
even calculate pi
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00:08:12,993 --> 00:08:15,760
even though
there are no circles here,
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no diameters of a circle,
nothing like that.
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It's really amazing.
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NARRATOR:
Since pi relates a round object,
a circle,
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with a straight one,
its diameter,
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00:08:31,144 --> 00:08:35,180
it can show up
in the strangest of places.
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Some see it in the meandering
path of rivers.
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A river's actual length
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as it winds its way
from its source to its mouth
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compared to the direct distance
on average seems to be about pi.
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00:08:49,897 --> 00:08:52,497
Models for just about anything
involving waves
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will have pi in them,
like those for light and sound.
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Pi tells us which colors
should appear in a rainbow,
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00:09:02,676 --> 00:09:06,511
and how middle C should sound
on a piano.
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00:09:06,513 --> 00:09:09,047
Pi shows up in apples,
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00:09:09,049 --> 00:09:12,417
in the way cells grow
into spherical shapes,
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00:09:12,419 --> 00:09:17,455
or in the brightness
of a supernova.
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One writer has suggested
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it's like seeing pi
on a series of mountain peaks,
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poking out
of a fog-shrouded valley.
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00:09:26,900 --> 00:09:29,067
We know there's a way
they're all connected,
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but it's not always obvious how.
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Pi is but one example
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of a vast interconnected web
of mathematics
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that seems to reveal
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an often hidden and deep order
to our world.
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Physicist Max Tegmark from MIT
thinks he knows why.
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He sees similarities
between our world
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and that of a computer game.
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00:10:05,705 --> 00:10:09,240
MAX TEGMARK:
If I were a character
in a computer game
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that were so advanced
that I were actually conscious
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00:10:12,045 --> 00:10:14,512
and I started exploring
my video game world,
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00:10:14,514 --> 00:10:16,881
it would actually feel to me
like it was made
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of real solid objects
made of physical stuff.
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♪ ♪
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Yet, if I started studying, as
the curious physicist that I am,
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the properties of this stuff,
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00:10:33,433 --> 00:10:36,534
the equations
by which things move
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00:10:36,536 --> 00:10:40,205
and the equations that
give stuff its properties,
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I would discover eventually
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that all these properties
were mathematical:
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the mathematical properties
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that the programmer had
actually put into the software
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that describes everything.
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NARRATOR:
The laws of physics in a game--
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like how an object floats,
bounces, or crashes--
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are only mathematical rules
created by a programmer.
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00:11:02,829 --> 00:11:06,297
Ultimately, the entire
"universe" of a computer game
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is just numbers and equations.
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That's exactly what I perceive
in this reality, too,
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00:11:12,572 --> 00:11:13,905
as a physicist,
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that the closer I look at things
that seem non-mathematical,
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like my arm here
and my hand,
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the more mathematical
it turns out to be.
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00:11:21,081 --> 00:11:23,448
Could it be that our world
also then
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is really just as mathematical
as the computer game reality?
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NARRATOR:
To Max, the software world
of a game isn't that different
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from the physical world
we live in.
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00:11:36,463 --> 00:11:40,899
He thinks that mathematics works
so well to describe reality
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because ultimately,
mathematics is all that it is.
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00:11:45,172 --> 00:11:48,673
There's nothing else.
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00:11:48,675 --> 00:11:50,675
Many of my physics colleagues
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00:11:50,677 --> 00:11:54,412
will say that mathematics
describes our physical reality
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00:11:54,414 --> 00:11:56,848
at least in some
approximate sense.
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00:11:56,850 --> 00:12:02,687
I go further and argue that it
actually is our physical reality
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00:12:02,689 --> 00:12:05,557
because I'm arguing that
our physical world
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00:12:05,559 --> 00:12:08,293
doesn't just have some
mathematical properties,
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00:12:08,295 --> 00:12:10,261
but it has only
mathematical properties.
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00:12:13,265 --> 00:12:16,835
NARRATOR:
Our physical reality is a bit
like a digital photograph,
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00:12:16,837 --> 00:12:18,369
according to Max.
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00:12:20,306 --> 00:12:23,174
The photo looks like the pond,
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00:12:23,176 --> 00:12:27,045
but as we move in
closer and closer,
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00:12:27,047 --> 00:12:30,615
we can see it is really
a field of pixels,
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00:12:30,617 --> 00:12:33,551
each represented
by three numbers
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00:12:33,553 --> 00:12:38,490
that specify the amount of red,
green and blue.
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00:12:38,492 --> 00:12:43,728
While the universe is vast
in its size and complexity,
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00:12:43,730 --> 00:12:47,332
requiring an unbelievably large
collection of numbers
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00:12:47,334 --> 00:12:49,334
to describe it,
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00:12:49,336 --> 00:12:52,203
Max sees its underlying
mathematical structure
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00:12:52,205 --> 00:12:54,606
as surprisingly simple.
220
00:12:54,608 --> 00:12:57,642
It's just 32 numbers--
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00:12:57,644 --> 00:13:01,412
constants, like the masses
of elementary particles--
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00:13:01,414 --> 00:13:05,517
along with a handful
of mathematical equations,
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00:13:05,519 --> 00:13:08,853
the fundamental laws of physics.
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00:13:08,855 --> 00:13:11,890
And it all fits on a wall,
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00:13:11,892 --> 00:13:15,860
though there are still
some questions.
226
00:13:15,862 --> 00:13:17,762
But even though we don't know
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00:13:17,764 --> 00:13:19,631
what exactly
is going to go here,
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00:13:19,633 --> 00:13:22,834
I am really confident that
what will go here
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00:13:22,836 --> 00:13:25,036
will be mathematical equations.
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00:13:25,038 --> 00:13:28,206
That everything
is ultimately mathematical.
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00:13:28,208 --> 00:13:32,277
NARRATOR:
Max Tegmark's Matrix-like view
232
00:13:32,279 --> 00:13:35,547
that mathematics
doesn't just describe reality
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00:13:35,549 --> 00:13:39,584
but is its essence
may sound radical,
234
00:13:39,586 --> 00:13:42,253
but it has deep roots
in history...
235
00:13:44,223 --> 00:13:46,591
going back to ancient Greece,
236
00:13:46,593 --> 00:13:50,728
to the time of the philosopher
and mystic Pythagoras.
237
00:13:50,730 --> 00:13:54,499
Stories say
he explored the affinity
238
00:13:54,501 --> 00:13:57,235
between mathematics and music,
239
00:13:57,237 --> 00:14:00,772
a relationship that resonates
to this day
240
00:14:00,774 --> 00:14:03,241
in the work
of Esperanza Spalding,
241
00:14:03,243 --> 00:14:06,711
an acclaimed jazz musician
who's studied music theory
242
00:14:06,713 --> 00:14:10,682
and sees its parallel
in mathematics.
243
00:14:12,852 --> 00:14:15,320
SPALDING:
I love the experience of math.
244
00:14:15,322 --> 00:14:17,121
The part that I enjoy
about math
245
00:14:17,123 --> 00:14:19,858
I get to experience
through music, too.
246
00:14:21,627 --> 00:14:22,627
At the beginning,
247
00:14:22,629 --> 00:14:24,329
you're studying
all the little equations,
248
00:14:24,331 --> 00:14:27,265
but you get to have this
very visceral relationship
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00:14:27,267 --> 00:14:29,500
with the product
of those equations,
250
00:14:29,502 --> 00:14:31,402
which is sound and music
and harmony and dissonance
251
00:14:31,404 --> 00:14:33,037
and all that good stuff.
252
00:14:33,039 --> 00:14:35,240
So I'm much better at music
than at math,
253
00:14:35,242 --> 00:14:37,308
but I love math with a passion.
254
00:14:37,310 --> 00:14:38,610
They're both just as much work.
255
00:14:38,612 --> 00:14:40,879
They're both, you have to study
your... off.
256
00:14:43,249 --> 00:14:44,716
Your head off,
study your head off.
257
00:14:44,718 --> 00:14:45,650
(laughs)
258
00:14:47,553 --> 00:14:49,921
NARRATOR:
The Ancient Greeks
found three relationships
259
00:14:49,923 --> 00:14:52,657
between notes
especially pleasing.
260
00:14:52,659 --> 00:14:58,830
Now we call them an octave,
a fifth, and a fourth.
261
00:14:58,832 --> 00:15:00,865
An octave is easy to remember
262
00:15:00,867 --> 00:15:03,067
because it's the first two notes
of "Somewhere Over the Rainbow."
263
00:15:03,069 --> 00:15:04,702
♪ La, la. ♪
264
00:15:04,704 --> 00:15:06,437
That's an octave-- "somewhere."
265
00:15:06,439 --> 00:15:10,141
(plays notes)
266
00:15:10,143 --> 00:15:12,010
A fifth sounds like this:
267
00:15:12,012 --> 00:15:13,678
♪ La, la. ♪
268
00:15:13,680 --> 00:15:16,748
Or the first two notes of
"Twinkle, Twinkle, Little Star."
269
00:15:16,750 --> 00:15:18,716
(plays notes)
270
00:15:18,718 --> 00:15:21,052
And a fourth sounds like:
271
00:15:21,054 --> 00:15:23,721
♪ La, la ♪
272
00:15:23,723 --> 00:15:24,923
(plays notes)
273
00:15:24,925 --> 00:15:26,557
You can think of it
as the first two notes
274
00:15:26,559 --> 00:15:27,892
of "Here Comes the Bride."
275
00:15:27,894 --> 00:15:31,496
(plays notes)
276
00:15:31,498 --> 00:15:34,465
NARRATOR:
In the sixth century BCE,
277
00:15:34,467 --> 00:15:36,834
the Greek philosopher Pythagoras
is said to have discovered
278
00:15:36,836 --> 00:15:39,837
that those beautiful
musical relationships
279
00:15:39,839 --> 00:15:43,207
were also beautiful
mathematical relationships
280
00:15:43,209 --> 00:15:47,745
by measuring the lengths
of the vibrating strings.
281
00:15:47,747 --> 00:15:51,983
In an octave, the string lengths
create a ratio of two to one.
282
00:15:51,985 --> 00:15:54,652
(plays notes)
283
00:15:54,654 --> 00:15:57,889
In a fifth,
the ratio is three to two.
284
00:15:57,891 --> 00:15:59,724
(plays notes)
285
00:15:59,726 --> 00:16:03,094
And in a fourth,
it is four to three.
286
00:16:03,096 --> 00:16:06,364
(plays notes)
287
00:16:06,366 --> 00:16:08,800
Seeing a common pattern
throughout sound,
288
00:16:08,802 --> 00:16:11,302
that could be
a big eye opener of saying,
289
00:16:11,304 --> 00:16:13,504
"Well, if this exists in sound,
290
00:16:13,506 --> 00:16:17,408
"and if it's true universally
through all sounds,
291
00:16:17,410 --> 00:16:21,245
"this ratio could exist
universally everywhere, right?
292
00:16:21,247 --> 00:16:22,714
And doesn't it?"
293
00:16:22,716 --> 00:16:25,383
(playing a tune)
294
00:16:25,385 --> 00:16:28,753
NARRATOR:
Pythagoreans worshipped the idea
of numbers.
295
00:16:28,755 --> 00:16:32,590
The fact that simple ratios
produced harmonious sounds
296
00:16:32,592 --> 00:16:36,594
was proof of a hidden order
in the natural world.
297
00:16:36,596 --> 00:16:38,830
And that order
was made of numbers,
298
00:16:38,832 --> 00:16:42,333
a profound insight that
mathematicians and scientists
299
00:16:42,335 --> 00:16:46,904
continue to explore to this day.
300
00:16:48,974 --> 00:16:52,310
In fact, there are plenty
of other physical phenomena
301
00:16:52,312 --> 00:16:56,814
that follow simple ratios,
from the two-to-one ratio
302
00:16:56,816 --> 00:17:00,651
of hydrogen atoms
to oxygen atoms in water
303
00:17:00,653 --> 00:17:03,521
to the number of times the Moon
orbits the Earth
304
00:17:03,523 --> 00:17:07,425
compared to its own rotation:
one to one.
305
00:17:07,427 --> 00:17:11,329
Or that Mercury rotates
exactly three times
306
00:17:11,331 --> 00:17:16,234
when it orbits the Sun twice,
a three-to-two ratio.
307
00:17:18,837 --> 00:17:22,373
In Ancient Greece,
Pythagoras and his followers
308
00:17:22,375 --> 00:17:27,011
had a profound effect on another
Greek philosopher, Plato,
309
00:17:27,013 --> 00:17:29,981
whose ideas also resonate
to this day,
310
00:17:29,983 --> 00:17:32,483
especially among mathematicians.
311
00:17:32,485 --> 00:17:35,853
Plato believed that geometry
and mathematics
312
00:17:35,855 --> 00:17:40,358
exist in their own ideal world.
313
00:17:40,360 --> 00:17:42,794
So when we draw a circle
on a piece of paper,
314
00:17:42,796 --> 00:17:44,562
this is not the real circle.
315
00:17:44,564 --> 00:17:46,697
The real circle
is in that world,
316
00:17:46,699 --> 00:17:49,033
and this is just
an approximation
317
00:17:49,035 --> 00:17:50,368
of that real circle,
318
00:17:50,370 --> 00:17:52,637
and the same
with all other shapes.
319
00:17:52,639 --> 00:17:55,640
And Plato liked very much
these five solids,
320
00:17:55,642 --> 00:17:58,242
the platonic solids
we call them today,
321
00:17:58,244 --> 00:18:02,246
and he assigned each one of them
to one of the elements
322
00:18:02,248 --> 00:18:04,315
that formed the world
as he saw it.
323
00:18:04,317 --> 00:18:09,387
NARRATOR:
The stable cube was earth.
324
00:18:09,389 --> 00:18:14,692
The tetrahedron with its pointy
corners was fire.
325
00:18:14,694 --> 00:18:20,631
The mobile-looking octahedron
Plato thought of as air.
326
00:18:20,633 --> 00:18:26,904
And the 20-sided icosahedron
was water.
327
00:18:26,906 --> 00:18:29,240
And finally the dodecahedron,
328
00:18:29,242 --> 00:18:31,843
this was the thing that
signified the cosmos as a whole.
329
00:18:36,682 --> 00:18:39,016
NARRATOR:
So Plato's mathematical forms
330
00:18:39,018 --> 00:18:41,953
were the ideal version
of the world around us,
331
00:18:41,955 --> 00:18:45,389
and they existed
in their own realm.
332
00:18:45,391 --> 00:18:47,391
And however bizarre
that may sound,
333
00:18:47,393 --> 00:18:50,595
that mathematics exists
in its own world,
334
00:18:50,597 --> 00:18:55,166
shaping the world we see,
it's an idea that to this day
335
00:18:55,168 --> 00:18:59,003
many mathematicians
and scientists can relate to--
336
00:18:59,005 --> 00:19:01,105
the sense they have
when they're doing math
337
00:19:01,107 --> 00:19:03,441
that they're just
uncovering something
338
00:19:03,443 --> 00:19:05,543
that's already out there.
339
00:19:05,545 --> 00:19:07,879
I feel quite strongly
that mathematics is discovered
340
00:19:07,881 --> 00:19:09,447
in my work as a mathematician.
341
00:19:09,449 --> 00:19:11,682
It always feels to me
there is a thing out there
342
00:19:11,684 --> 00:19:13,718
and I'm kind of trying
to find it
343
00:19:13,720 --> 00:19:15,186
and understand it and touch it.
344
00:19:17,689 --> 00:19:19,957
JAMES GATES:
As someone who actually
has had the pleasure
345
00:19:19,959 --> 00:19:21,626
of making new mathematics,
346
00:19:21,628 --> 00:19:24,562
it feels like there's something
there before you get to it.
347
00:19:24,564 --> 00:19:26,164
If I have to choose,
348
00:19:26,166 --> 00:19:28,366
I think it's more discovered
than invented
349
00:19:28,368 --> 00:19:30,067
because I think
there's a reality
350
00:19:30,069 --> 00:19:32,436
to what we study in mathematics.
351
00:19:32,438 --> 00:19:34,438
When we do good mathematics,
352
00:19:34,440 --> 00:19:37,742
we're discovering something
about the way our minds work
353
00:19:37,744 --> 00:19:39,710
in interaction with the world.
354
00:19:39,712 --> 00:19:41,646
Well, I know that
because that's what I do.
355
00:19:41,648 --> 00:19:43,981
I come to my office, I sit down
in front of my whiteboard
356
00:19:43,983 --> 00:19:47,852
and I try and understand
that thing that's out there.
357
00:19:47,854 --> 00:19:50,154
And every now and then,
I'm discovering a new bit of it.
358
00:19:50,156 --> 00:19:52,356
That's exactly
what it feels like.
359
00:19:52,358 --> 00:19:54,892
NARRATOR:
To many mathematicians,
360
00:19:54,894 --> 00:19:59,397
it feels like math is discovered
rather than invented.
361
00:19:59,399 --> 00:20:01,899
But is that just a feeling?
362
00:20:01,901 --> 00:20:04,001
Could it be that mathematics
363
00:20:04,003 --> 00:20:08,072
is purely a product
of the human brain?
364
00:20:08,074 --> 00:20:12,610
Meet Shyam,
a bonafide math whiz.
365
00:20:12,612 --> 00:20:14,278
MICHAEL O'BOYLE:
800 on the SAT Math.
366
00:20:14,280 --> 00:20:15,680
That's pretty good.
367
00:20:15,682 --> 00:20:17,081
And you took it
when you were how old?
368
00:20:17,083 --> 00:20:17,949
Eleven.
369
00:20:17,951 --> 00:20:18,950
Eleven.
370
00:20:18,952 --> 00:20:21,085
Wow, that's, like,
a perfect score.
371
00:20:21,087 --> 00:20:23,521
NARRATOR:
Where does Shyam's math genius
come from?
372
00:20:23,523 --> 00:20:28,859
It turns out we can pinpoint it,
and it's all in his head.
373
00:20:28,861 --> 00:20:34,532
Using fMRI, scientists
can scan Shyam's brain
374
00:20:34,534 --> 00:20:37,034
as he answers math questions
375
00:20:37,036 --> 00:20:39,804
to see which parts of the brain
receive more blood,
376
00:20:39,806 --> 00:20:43,874
a sign they are hard at work.
377
00:20:43,876 --> 00:20:45,209
MAN:
All right, Shyam,
we'll start about now.
378
00:20:45,211 --> 00:20:46,210
Okay, buddy?
379
00:20:46,212 --> 00:20:47,144
SHYAM:
Okay.
380
00:20:48,914 --> 00:20:51,349
NARRATOR:
In images of Shyam's brain,
381
00:20:51,351 --> 00:20:56,120
the parietal lobes glow
an especially bright crimson.
382
00:20:56,122 --> 00:20:58,689
He is relying on parietal areas
383
00:20:58,691 --> 00:21:01,726
to determine these
mathematical relationships.
384
00:21:01,728 --> 00:21:04,729
That's characteristic of lots
of math-gifted types.
385
00:21:04,731 --> 00:21:08,065
NARRATOR:
In tests similar to Shyam's,
386
00:21:08,067 --> 00:21:10,568
kids who exhibit
high math performance
387
00:21:10,570 --> 00:21:13,638
have five to six times
more neuron activation
388
00:21:13,640 --> 00:21:16,407
than average kids
in these brain regions.
389
00:21:16,409 --> 00:21:20,544
But is that the result of
teaching and intense practice?
390
00:21:20,546 --> 00:21:25,783
Or are the foundations of math
built into our brains?
391
00:21:30,722 --> 00:21:33,157
Scientists are looking
for the answer here,
392
00:21:33,159 --> 00:21:36,994
at the Duke University
Lemur Center,
393
00:21:36,996 --> 00:21:38,629
a 70-acre sanctuary
in North Carolina,
394
00:21:38,631 --> 00:21:42,533
the largest one for rare and
endangered lemurs in the world.
395
00:21:45,337 --> 00:21:48,806
Like all primates,
lemurs are related to humans
396
00:21:48,808 --> 00:21:51,442
through a common ancestor
397
00:21:51,444 --> 00:21:54,478
that lived as many as
65 million years ago.
398
00:21:54,480 --> 00:21:56,847
Scientists believe lemurs
399
00:21:56,849 --> 00:22:00,651
share many characteristics
with those earliest primates,
400
00:22:00,653 --> 00:22:04,255
making them a window,
though a blurry one,
401
00:22:04,257 --> 00:22:07,758
into our ancient past.
402
00:22:07,760 --> 00:22:09,960
Got a choice here, Teres.
403
00:22:09,962 --> 00:22:11,429
Come on up.
404
00:22:11,431 --> 00:22:13,130
NARRATOR:
Duke Professor Liz Brannon
405
00:22:13,132 --> 00:22:16,300
investigates how well lemurs,
like Teres here,
406
00:22:16,302 --> 00:22:18,636
can compare quantities.
407
00:22:18,638 --> 00:22:21,839
BRANNON:
Many different animals
choose larger food quantities.
408
00:22:21,841 --> 00:22:24,608
So what is Teres doing?
409
00:22:24,610 --> 00:22:26,944
What are all of these
different animals doing
410
00:22:26,946 --> 00:22:29,580
when they compare
two quantities?
411
00:22:29,582 --> 00:22:32,083
Well, clearly he's not using
verbal labels,
412
00:22:32,085 --> 00:22:34,318
he's not using symbols.
413
00:22:34,320 --> 00:22:37,521
We need to figure out whether
they can really use number,
414
00:22:37,523 --> 00:22:39,924
pure number, as a cue.
415
00:22:41,993 --> 00:22:45,930
NARRATOR:
To test how well Teres
can distinguish quantities,
416
00:22:45,932 --> 00:22:49,333
he's been taught
a touch-screen computer game.
417
00:22:49,335 --> 00:22:52,570
The red square starts a round.
418
00:22:52,572 --> 00:22:55,506
If he touches it,
two squares appear
419
00:22:55,508 --> 00:22:58,743
containing different numbers
of objects.
420
00:22:58,745 --> 00:22:59,877
He's been trained
421
00:22:59,879 --> 00:23:02,847
that if he chooses the box
with the fewest number...
422
00:23:02,849 --> 00:23:04,348
(ringing)
423
00:23:04,350 --> 00:23:07,818
...he'll get a reward,
a sugar pellet.
424
00:23:07,820 --> 00:23:09,353
A wrong answer?
425
00:23:09,355 --> 00:23:10,287
(buzzer)
426
00:23:13,859 --> 00:23:15,526
We have to do a lot
to ensure
427
00:23:15,528 --> 00:23:18,462
that they're really attending to
number and not something else.
428
00:23:18,464 --> 00:23:22,266
NARRATOR:
To make sure the test animal
is reacting
429
00:23:22,268 --> 00:23:25,302
to the number of objects
and not some other cue,
430
00:23:25,304 --> 00:23:30,941
Liz varies the objects' size,
color, and shape.
431
00:23:30,943 --> 00:23:33,844
She has conducted
thousands of trials
432
00:23:33,846 --> 00:23:36,847
and shown that lemurs
and rhesus monkeys
433
00:23:36,849 --> 00:23:40,317
can learn to pick
the right answer.
434
00:23:40,319 --> 00:23:42,553
BRANNON:
Teres obviously
doesn't have language
435
00:23:42,555 --> 00:23:44,522
and he doesn't have
any symbols for number.
436
00:23:44,524 --> 00:23:47,458
So is he counting, is he doing
what a human child does
437
00:23:47,460 --> 00:23:50,594
when they recite the numbers
one, two, three?
438
00:23:50,596 --> 00:23:52,062
No.
439
00:23:52,064 --> 00:23:55,533
And yet, what he seems
to be attending to
440
00:23:55,535 --> 00:24:00,604
is the very abstract essence
of what a nuer is.
441
00:24:00,606 --> 00:24:04,475
NARRATOR:
Lemurs and rhesus monkeys
aren't alone
442
00:24:04,477 --> 00:24:06,210
in having this primitive
number sense.
443
00:24:06,212 --> 00:24:10,114
Rats, pigeons, fish, raccoons,
444
00:24:10,116 --> 00:24:13,584
insects, horses, and elephants
445
00:24:13,586 --> 00:24:16,754
all show sensitivity
to quantity.
446
00:24:16,756 --> 00:24:19,824
And so do human infants.
447
00:24:21,927 --> 00:24:24,361
At her lab on the Duke campus,
448
00:24:24,363 --> 00:24:28,065
Liz has tested babies
that were only six months old.
449
00:24:28,067 --> 00:24:30,768
They'll look longer at a screen
450
00:24:30,770 --> 00:24:33,704
with a changing number
of objects,
451
00:24:33,706 --> 00:24:35,773
as long as the change
is obvious enough
452
00:24:35,775 --> 00:24:38,309
to capture their attention.
453
00:24:38,311 --> 00:24:42,980
Liz has also tested
college students,
454
00:24:42,982 --> 00:24:44,949
asking them not to count,
455
00:24:44,951 --> 00:24:47,485
but to respond
as quickly as they could
456
00:24:47,487 --> 00:24:50,554
to a touch-screen test
comparing quantities.
457
00:24:50,556 --> 00:24:52,223
The results?
458
00:24:52,225 --> 00:24:56,927
About the same as lemurs
and rhesus monkeys.
459
00:24:56,929 --> 00:24:58,729
BRANNON:
In fact, there are humans
460
00:24:58,731 --> 00:25:01,098
who aren't as good
as our monkeys,
461
00:25:01,100 --> 00:25:03,100
and others that are far better,
462
00:25:03,102 --> 00:25:05,102
so there's a lot of variability
in human performance,
463
00:25:05,104 --> 00:25:08,239
but in general, it looks
very similar to a monkey.
464
00:25:10,308 --> 00:25:13,577
Substitute in the three,
you raise that to the four...
465
00:25:13,579 --> 00:25:16,046
BRANNON:
Even without any
mathematical education,
466
00:25:16,048 --> 00:25:19,149
even without learning
any number words or symbols,
467
00:25:19,151 --> 00:25:21,752
we would still have,
all of us as humans,
468
00:25:21,754 --> 00:25:23,621
a primitive number sense.
469
00:25:23,623 --> 00:25:27,558
That fundamental ability
to perceive number
470
00:25:27,560 --> 00:25:30,427
seems to be a very important
foundation,
471
00:25:30,429 --> 00:25:32,429
and without it,
it's very questionable
472
00:25:32,431 --> 00:25:36,200
as to whether we could ever
appreciate symbolic mathematics.
473
00:25:36,202 --> 00:25:38,335
NARRATOR:
The building blocks
of mathematics
474
00:25:38,337 --> 00:25:41,338
may be preprogrammed
into our brains,
475
00:25:41,340 --> 00:25:44,608
part of the basic toolkit
for survival,
476
00:25:44,610 --> 00:25:48,712
like our ability to recognize
patterns and shapes
477
00:25:48,714 --> 00:25:51,181
or our sense of time.
478
00:25:51,183 --> 00:25:53,517
From that point of view,
on this foundation,
479
00:25:53,519 --> 00:25:55,953
we've erected one
of the greatest inventions
480
00:25:55,955 --> 00:25:59,723
of human culture:
481
00:25:59,725 --> 00:26:02,693
mathematics.
482
00:26:02,695 --> 00:26:05,095
But the mystery remains.
483
00:26:05,097 --> 00:26:11,168
If it is "all in our heads,"
why has math been so effective?
484
00:26:11,170 --> 00:26:14,738
Through science, technology,
and engineering,
485
00:26:14,740 --> 00:26:17,274
it's transformed the planet,
486
00:26:17,276 --> 00:26:22,012
even allowing us to go
into the beyond.
487
00:26:24,182 --> 00:26:27,251
As in the work here, at NASA's
Jet Propulsion Laboratory
488
00:26:27,253 --> 00:26:28,919
in Pasadena, California.
489
00:26:28,921 --> 00:26:31,455
MAN:
Roger, copy mission.
490
00:26:31,457 --> 00:26:32,823
Coming up on entry.
491
00:26:32,825 --> 00:26:38,028
NARRATOR:
In 2012, they landed
a car-size rover...
492
00:26:38,030 --> 00:26:40,998
MAN:
Descending at about .75 meters
per second as expected.
493
00:26:41,000 --> 00:26:43,067
NARRATOR:
...on Mars.
494
00:26:43,069 --> 00:26:45,569
MAN:
Touchdown confirmed,
we're safe on Mars.
495
00:26:45,571 --> 00:26:50,507
(cheering)
496
00:26:50,509 --> 00:26:53,611
NARRATOR:
Adam Steltzner
was the lead engineer
497
00:26:53,613 --> 00:26:55,679
on the team that designed
the landing system.
498
00:26:55,681 --> 00:27:00,050
Their work depended
on a groundbreaking discovery
499
00:27:00,052 --> 00:27:02,620
from the Renaissance
500
00:27:02,622 --> 00:27:06,824
that turned mathematics
into the language of science:
501
00:27:06,826 --> 00:27:10,227
the law of falling bodies.
502
00:27:12,764 --> 00:27:15,899
The ancient Greek philosopher
Aristotle
503
00:27:15,901 --> 00:27:20,204
taught that heavier objects
fall faster than lighter ones--
504
00:27:20,206 --> 00:27:25,342
an idea that, on the surface,
makes sense.
505
00:27:25,344 --> 00:27:29,246
Even this surface:
the Mars yard,
506
00:27:29,248 --> 00:27:32,016
where they test the rovers
at JPL.
507
00:27:32,018 --> 00:27:33,584
ADAM STELTZNER:
So Aristotle reasoned
508
00:27:33,586 --> 00:27:37,655
that the rate at which things
would fall
509
00:27:37,657 --> 00:27:39,289
was proportional
to their weight.
510
00:27:43,294 --> 00:27:44,595
Which seems reasonable.
511
00:27:44,597 --> 00:27:46,830
NARRATOR:
In fact, so reasonable,
512
00:27:46,832 --> 00:27:50,868
the view held
for nearly 2,000 years,
513
00:27:50,870 --> 00:27:53,504
until challenged
in the late 1500s
514
00:27:53,506 --> 00:27:58,876
by Italian mathematician
Galileo Galilei.
515
00:27:58,878 --> 00:28:00,644
STELTZNER:
Legend has it that Galileo
516
00:28:00,646 --> 00:28:03,380
dropped two different weight
cannonballs
517
00:28:03,382 --> 00:28:05,983
from the Leaning Tower of Pisa.
518
00:28:05,985 --> 00:28:08,085
Well, we're not in Pisa,
we don't have cannonballs,
519
00:28:08,087 --> 00:28:11,255
but we do have a bowling ball
and a bouncy ball.
520
00:28:11,257 --> 00:28:13,290
Let's weigh them.
521
00:28:13,292 --> 00:28:17,361
First, we weigh
the bowling ball.
522
00:28:17,363 --> 00:28:19,463
It weighs 15 pounds.
523
00:28:19,465 --> 00:28:20,764
And the bouncy ball?
524
00:28:20,766 --> 00:28:23,734
It weighs hardly anything.
525
00:28:23,736 --> 00:28:24,868
Let's drop them.
526
00:28:24,870 --> 00:28:27,738
NARRATOR:
According to Aristotle,
527
00:28:27,740 --> 00:28:31,041
the bowling ball should fall
over 15 times faster
528
00:28:31,043 --> 00:28:32,943
than the bouncy ball.
529
00:28:35,880 --> 00:28:37,815
STELTZNER:
Well, they seem to fall
at the same rate.
530
00:28:39,918 --> 00:28:41,518
This isn't that high, though.
531
00:28:41,520 --> 00:28:44,455
Maybe we should drop them
from higher.
532
00:28:51,796 --> 00:28:54,898
So Ed is 20 feet in the air
up there.
533
00:28:54,900 --> 00:28:58,102
Let's see if the balls fall
at the same rate.
534
00:28:58,104 --> 00:28:59,536
Ready?
535
00:28:59,538 --> 00:29:03,207
Three, two, one, drop!
536
00:29:10,281 --> 00:29:11,815
Galileo was right.
537
00:29:11,817 --> 00:29:13,984
Aristotle, you lose.
538
00:29:13,986 --> 00:29:17,621
NARRATOR:
Dropping feathers and hammers
is misleading,
539
00:29:17,623 --> 00:29:21,525
thanks to air resistance.
540
00:29:21,527 --> 00:29:24,094
DAVID SCOTT:
Well, in my left hand,
I have a feather.
541
00:29:24,096 --> 00:29:26,697
In my right hand, a hammer...
542
00:29:26,699 --> 00:29:30,634
NARRATOR:
A fact demonstrated on the Moon,
where there is no air,
543
00:29:30,636 --> 00:29:34,338
in 1971
during the Apollo 15 mission.
544
00:29:34,340 --> 00:29:36,473
SCOTT:
And I'll drop
the two of them here.
545
00:29:38,343 --> 00:29:39,443
How about that?
546
00:29:39,445 --> 00:29:40,978
Mr. Galileo was correct.
547
00:29:40,980 --> 00:29:44,381
STELTZNER:
Little balls, soccer balls...
548
00:29:44,383 --> 00:29:45,916
NARRATOR:
So while counterintuitive...
549
00:29:45,918 --> 00:29:47,851
STELTZNER:
Vegetables!
550
00:29:47,853 --> 00:29:49,953
NARRATOR:
...if you take the air
out of the equation,
551
00:29:49,955 --> 00:29:54,124
everything falls
at the same rate,
552
00:29:54,126 --> 00:29:56,026
even Aristotle.
553
00:29:59,764 --> 00:30:01,765
But what really
interested Galileo
554
00:30:01,767 --> 00:30:04,134
was that an object
dropped at one height
555
00:30:04,136 --> 00:30:09,807
didn't take twice as long
to drop from twice as high;
556
00:30:09,809 --> 00:30:12,409
it accelerated.
557
00:30:12,411 --> 00:30:15,179
But how do you measure that?
558
00:30:15,181 --> 00:30:19,082
Everything is happening so fast.
559
00:30:19,084 --> 00:30:21,051
STELTZNER:
Oh, yes!
560
00:30:23,788 --> 00:30:26,790
NARRATOR:
Galileo came up
with an ingenious solution.
561
00:30:30,295 --> 00:30:36,600
He built a ramp,
an inclined plane,
562
00:30:36,602 --> 00:30:42,239
to slow the falling motion down
so he could measure it.
563
00:30:42,241 --> 00:30:44,775
STELTZNER:
So we're going to use this ramp
564
00:30:44,777 --> 00:30:49,746
to find the relationship
between distance and time.
565
00:30:49,748 --> 00:30:53,917
For time, I'll use
an arbitrary unit: a Galileo.
566
00:30:53,919 --> 00:30:55,452
One Galileo.
567
00:30:56,955 --> 00:30:59,890
NARRATOR:
The length of the ramp
that the ball rolls
568
00:30:59,892 --> 00:31:04,962
during one Galileo
becomes one unit of distance.
569
00:31:04,964 --> 00:31:07,231
So we've gone
one unit of distance
570
00:31:07,233 --> 00:31:09,266
in one unit of time.
571
00:31:09,268 --> 00:31:11,768
Now let's try it
for a two-count.
572
00:31:11,770 --> 00:31:14,137
One Galileo, two Galileo.
573
00:31:14,139 --> 00:31:15,973
NARRATOR:
In two units of time,
574
00:31:15,975 --> 00:31:20,077
the ball has rolled
four units of distance.
575
00:31:20,079 --> 00:31:25,482
Now let's see how far it goes
in three Galileos.
576
00:31:25,484 --> 00:31:29,286
One Galileo, two Galileo,
three Galileo.
577
00:31:29,288 --> 00:31:31,488
NARRATOR:
In three units of time,
578
00:31:31,490 --> 00:31:35,926
the ball has gone
nine units of distance.
579
00:31:35,928 --> 00:31:37,594
So there it is.
580
00:31:37,596 --> 00:31:39,630
There's a mathematical
relationship here
581
00:31:39,632 --> 00:31:42,232
between time and distance.
582
00:31:42,234 --> 00:31:45,135
NARRATOR:
Galileo's inspired use of a ramp
583
00:31:45,137 --> 00:31:49,306
had shown falling objects
follow mathematical laws.
584
00:31:51,342 --> 00:31:53,477
The distance the ball traveled
585
00:31:53,479 --> 00:31:58,382
is directly proportional
to the square of the time.
586
00:31:58,384 --> 00:32:03,353
That mathematical relationship
that Galileo observed
587
00:32:03,355 --> 00:32:07,124
is a mathematical expression
of the physics of our universe.
588
00:32:07,126 --> 00:32:09,326
NARRATOR:
Galileo's centuries-old
589
00:32:09,328 --> 00:32:12,696
mathematical observation
about falling objects
590
00:32:12,698 --> 00:32:15,999
remains just as valid today.
591
00:32:16,001 --> 00:32:19,369
It's the same mathematical
expression that we can use
592
00:32:19,371 --> 00:32:22,873
to understand how objects
might fall here on Earth,
593
00:32:22,875 --> 00:32:25,108
roll down a ramp.
594
00:32:25,110 --> 00:32:27,377
It's even a relationship
that we used
595
00:32:27,379 --> 00:32:30,881
to land the Curiosity rover
on the surface of Mars.
596
00:32:33,451 --> 00:32:35,285
That's the power of mathematics.
597
00:32:35,287 --> 00:32:39,957
NARRATOR:
Galileo's insight was profound.
598
00:32:39,959 --> 00:32:43,026
Mathematics could be used
as a tool
599
00:32:43,028 --> 00:32:47,898
to uncover and discover
the hidden rules of our world.
600
00:32:47,900 --> 00:32:49,900
He later wrote,
601
00:32:49,902 --> 00:32:54,304
"The universe is written
in the language of mathematics."
602
00:32:55,840 --> 00:32:58,508
Math is really the language
603
00:32:58,510 --> 00:33:00,644
in which we understand
the universe.
604
00:33:00,646 --> 00:33:02,579
We don't know why it's the case
605
00:33:02,581 --> 00:33:06,149
that the laws of physics
and the universe
606
00:33:06,151 --> 00:33:09,286
follows mathematical models,
but it does seem to be the case.
607
00:33:10,855 --> 00:33:13,056
NARRATOR:
While Galileo turned
mathematical equations
608
00:33:13,058 --> 00:33:15,392
into laws of science,
609
00:33:15,394 --> 00:33:19,696
it was another man,
born the same year Galileo died,
610
00:33:19,698 --> 00:33:23,734
who took that to new heights
that crossed the heavens.
611
00:33:23,736 --> 00:33:27,971
His name was Isaac Newton.
612
00:33:29,240 --> 00:33:33,710
He worked here at Trinity
College in Cambridge, England.
613
00:33:33,712 --> 00:33:37,247
SIMON SCHAFFER:
Newton cultivated the reputation
614
00:33:37,249 --> 00:33:39,182
of being a solitary genius,
615
00:33:39,184 --> 00:33:42,986
and here in the bowling green
of Trinity College,
616
00:33:42,988 --> 00:33:46,223
it was said that
he would walk meditatively
617
00:33:46,225 --> 00:33:49,993
up and down the paths,
absentmindedly drawing
618
00:33:49,995 --> 00:33:53,096
mathematical diagrams
in the gravel,
619
00:33:53,098 --> 00:33:57,100
and the fellows were instructed,
or so it was said,
620
00:33:57,102 --> 00:33:59,336
not to disturb him,
621
00:33:59,338 --> 00:34:02,572
not to clear up the gravel
after he'd passed,
622
00:34:02,574 --> 00:34:05,942
in case they inadvertently
wiped out
623
00:34:05,944 --> 00:34:09,780
some major scientific
or mathematical discovery.
624
00:34:09,782 --> 00:34:14,951
NARRATOR:
In 1687, Newton published a book
625
00:34:14,953 --> 00:34:19,056
that would become a landmark
in the history of science.
626
00:34:19,058 --> 00:34:22,192
Today, it is known simply
as the "Principia."
627
00:34:22,194 --> 00:34:23,693
In it, Newton
gathered observations
628
00:34:23,695 --> 00:34:24,861
from around the world
629
00:34:24,863 --> 00:34:29,433
and used mathematics
to explain them--
630
00:34:29,435 --> 00:34:33,570
for instance, that of a comet
seen widely in the fall of 1680.
631
00:34:33,572 --> 00:34:35,839
SCHAFFER:
He gathers data worldwide
632
00:34:35,841 --> 00:34:39,643
in order to construct
the comet's path.
633
00:34:39,645 --> 00:34:45,982
So for November the 19th,
he begins with an observation
634
00:34:45,984 --> 00:34:49,719
made in Cambridge in England
at 4:30 a.m.
635
00:34:49,721 --> 00:34:51,655
by a certain young person,
636
00:34:51,657 --> 00:34:58,462
and then at 5:00 in the morning
at Boston in New England.
637
00:34:58,464 --> 00:35:01,398
So what Newton does
is to accumulate numbers
638
00:35:01,400 --> 00:35:04,868
made by observers
spread right across the globe
639
00:35:04,870 --> 00:35:06,770
in order to construct
640
00:35:06,772 --> 00:35:09,739
an unprecedentedly
accurate calculation
641
00:35:09,741 --> 00:35:14,611
of how this great comet
moved through the sky.
642
00:35:14,613 --> 00:35:18,215
NARRATOR:
Newton's groundbreaking insight
was that the force
643
00:35:18,217 --> 00:35:21,918
that sent the comet
hurtling around the Sun...
644
00:35:21,920 --> 00:35:23,120
(cannon fire)
645
00:35:23,122 --> 00:35:24,988
...was the same force
646
00:35:24,990 --> 00:35:28,325
that brought cannonballs
back to Earth.
647
00:35:28,327 --> 00:35:33,964
It was the force behind
Galileo's law of falling bodies,
648
00:35:33,966 --> 00:35:39,402
and it even held the planets
in their orbits.
649
00:35:39,404 --> 00:35:44,741
He called the force gravity,
and described it precisely
650
00:35:44,743 --> 00:35:47,277
in a surprisingly
simple equation
651
00:35:47,279 --> 00:35:50,013
that explains how two masses
attract each other,
652
00:35:50,015 --> 00:35:55,919
whether here on Earth
or in the heavens above.
653
00:35:55,921 --> 00:35:59,089
SCHAFFER:
What's so impressive
and so dramatic
654
00:35:59,091 --> 00:36:02,292
is that a single
mathematical law
655
00:36:02,294 --> 00:36:05,762
would allow you to move
throughout the universe.
656
00:36:08,633 --> 00:36:14,437
NARRATOR:
Today, we can even witness it
at work beyond the Milky Way.
657
00:36:15,740 --> 00:36:18,742
This is a picture
of two galaxies
658
00:36:18,744 --> 00:36:21,311
that are actually being drawn
together in a merger.
659
00:36:21,313 --> 00:36:23,246
This is how galaxies
build themselves.
660
00:36:23,248 --> 00:36:24,714
Right.
661
00:36:24,716 --> 00:36:26,283
NARRATOR:
Mario Livio is on the team
662
00:36:26,285 --> 00:36:29,085
working with the images
from the Hubble Space Telescope.
663
00:36:29,087 --> 00:36:32,155
For decades,
scientists have used Hubble
664
00:36:32,157 --> 00:36:35,325
to gaze far beyond
our solar system,
665
00:36:35,327 --> 00:36:38,595
even beyond the stars
of our galaxy.
666
00:36:38,597 --> 00:36:41,998
It's shown us the distant
gas clouds of nebulae
667
00:36:42,000 --> 00:36:46,169
and vast numbers of galaxies
wheeling in the heavens
668
00:36:46,171 --> 00:36:49,406
billions of light-years away.
669
00:36:49,408 --> 00:36:51,541
And what those images show
670
00:36:51,543 --> 00:36:53,910
is that throughout
the visible universe,
671
00:36:53,912 --> 00:36:56,613
as far as the telescope can see,
672
00:36:56,615 --> 00:37:00,383
the law of gravity
still applies.
673
00:37:00,385 --> 00:37:02,519
LIVIO:
You know, Newton wrote
these laws
674
00:37:02,521 --> 00:37:04,254
of gravity and of motion
675
00:37:04,256 --> 00:37:07,657
based on things
happening on Earth,
676
00:37:07,659 --> 00:37:09,859
and the planets in
the solar system and so on,
677
00:37:09,861 --> 00:37:13,263
but these same laws,
these very same laws
678
00:37:13,265 --> 00:37:16,032
apply to all these
distant galaxies
679
00:37:16,034 --> 00:37:17,901
and, you know, shape them,
680
00:37:17,903 --> 00:37:21,137
and everything about them--
how they form, how they move--
681
00:37:21,139 --> 00:37:25,642
is controlled by those
same mathematical laws.
682
00:37:25,644 --> 00:37:29,813
NARRATOR:
Some of the world's
greatest minds have been amazed
683
00:37:29,815 --> 00:37:33,984
by the way that math
permeates the universe.
684
00:37:33,986 --> 00:37:35,885
LIVIO:
Albert Einstein, he wondered,
685
00:37:35,887 --> 00:37:39,756
he said, "How is it possible
that mathematics,"
686
00:37:39,758 --> 00:37:43,326
which is, he thought,
a product of human thought,
687
00:37:43,328 --> 00:37:47,530
"Does so well in explaining
the universe as we see it?"
688
00:37:47,532 --> 00:37:51,134
And Nobel laureate in physics
Eugene Wigner
689
00:37:51,136 --> 00:37:52,736
coined this phrase:
690
00:37:52,738 --> 00:37:55,872
"The unreasonable effectiveness
of mathematics."
691
00:37:55,874 --> 00:37:58,341
He said the fact
that mathematics
692
00:37:58,343 --> 00:38:01,378
can really describe
the universe so well,
693
00:38:01,380 --> 00:38:03,546
in particular physical laws,
694
00:38:03,548 --> 00:38:08,518
is a gift that we neither
understand nor deserve.
695
00:38:08,520 --> 00:38:11,154
NARRATOR:
In physics,
696
00:38:11,156 --> 00:38:15,492
examples of that "unreasonable
effectiveness" abound.
697
00:38:17,328 --> 00:38:19,329
When nearly 200 years ago
698
00:38:19,331 --> 00:38:22,065
the planet Uranus
was seen to go off track,
699
00:38:22,067 --> 00:38:24,734
scientists trusted the math
700
00:38:24,736 --> 00:38:29,606
and calculated it was being
pulled by another unseen planet.
701
00:38:32,410 --> 00:38:36,346
And so they discovered Neptune.
702
00:38:36,348 --> 00:38:38,782
Mathematics
had accurately predicted
703
00:38:38,784 --> 00:38:42,018
a previously unknown planet.
704
00:38:42,020 --> 00:38:46,856
SAVAS DIMOPOULOS:
If you formulate
a question properly,
705
00:38:46,858 --> 00:38:49,592
mathematics gives you
the answer.
706
00:38:49,594 --> 00:38:52,495
It's like having a servant
707
00:38:52,497 --> 00:38:55,832
that is far more capable
than you are.
708
00:38:55,834 --> 00:38:57,834
So you tell it "Do this,"
709
00:38:57,836 --> 00:39:01,004
and if you say it nicely,
then it'll do it
710
00:39:01,006 --> 00:39:04,107
and it will carry you
all the way to the truth,
711
00:39:04,109 --> 00:39:06,643
to the final answer.
712
00:39:06,645 --> 00:39:09,045
RADIO HOST:
WGBH, 89.7.
713
00:39:09,047 --> 00:39:12,916
NARRATOR:
Evidence of the amazing
predictive power of mathematics
714
00:39:12,918 --> 00:39:15,485
can be found all around us.
715
00:39:15,487 --> 00:39:18,455
I heard it took five Elvises
to pull them apart.
716
00:39:18,457 --> 00:39:24,627
NARRATOR:
Television, radio,
your cell phone, satellites,
717
00:39:24,629 --> 00:39:31,668
the baby monitor, Wi-Fi,
your garage door opener, GPS,
718
00:39:31,670 --> 00:39:34,471
and yes, even maybe
your TV's remote.
719
00:39:34,473 --> 00:39:38,708
All of these use invisible waves
of energy to communicate,
720
00:39:38,710 --> 00:39:41,444
and no one even knew
they existed
721
00:39:41,446 --> 00:39:43,980
until the work of James Maxwell,
722
00:39:43,982 --> 00:39:47,684
a Scottish mathematical
physicist.
723
00:39:47,686 --> 00:39:52,389
In the 1860s,
he published a set of equations
724
00:39:52,391 --> 00:39:56,393
that explained how electricity
and magnetism were related--
725
00:39:56,395 --> 00:40:01,431
how each could generate
the other.
726
00:40:01,433 --> 00:40:05,101
The equations also made
a startling prediction.
727
00:40:07,405 --> 00:40:09,773
Together, electricity
and magnetism
728
00:40:09,775 --> 00:40:12,008
could produce waves of energy
729
00:40:12,010 --> 00:40:16,413
that would travel through space
at the speed of light:
730
00:40:16,415 --> 00:40:19,849
electromagnetic waves.
731
00:40:19,851 --> 00:40:21,584
ROGER PENROSE:
Maxwell's theory gave us
732
00:40:21,586 --> 00:40:24,754
these radio waves, X-rays,
733
00:40:24,756 --> 00:40:27,657
these things which were simply
not known about at all.
734
00:40:27,659 --> 00:40:31,094
So the theory had a scope,
which was extraordinary.
735
00:40:34,064 --> 00:40:37,333
NARRATOR:
Almost immediately, people
set out to find the waves
736
00:40:37,335 --> 00:40:40,770
predicted by Maxwell's
equations.
737
00:40:40,772 --> 00:40:42,972
What must have seemed
the least promising attempt
738
00:40:42,974 --> 00:40:46,109
to harness them is made here,
in northern Italy,
739
00:40:46,111 --> 00:40:48,745
in the attic of a family home
740
00:40:48,747 --> 00:40:52,215
by 20-year-old
Guglielmo Marconi.
741
00:40:52,217 --> 00:40:56,386
His process starts
with a series of sparks.
742
00:40:56,388 --> 00:41:00,256
(buzzing)
743
00:41:00,258 --> 00:41:04,427
The burst of electricity creates
a momentary magnetic field,
744
00:41:04,429 --> 00:41:08,331
which in turn generates
a momentary electric field,
745
00:41:08,333 --> 00:41:11,367
which creates
another magnetic field.
746
00:41:11,369 --> 00:41:13,670
The energy cycles
between the two,
747
00:41:13,672 --> 00:41:16,739
propagating
an electromagnetic wave.
748
00:41:16,741 --> 00:41:19,943
(buzzing)
749
00:41:19,945 --> 00:41:23,813
Marconi gets his system
to work inside,
750
00:41:23,815 --> 00:41:27,317
but then he scales up.
751
00:41:30,588 --> 00:41:34,591
Over a few weeks, he builds
a big antenna beside the house
752
00:41:34,593 --> 00:41:38,394
to amplify the waves
coming from his spark generator.
753
00:41:38,396 --> 00:41:42,966
Then he asks his brother
and an assistant
754
00:41:42,968 --> 00:41:45,969
to carry a receiver
across the estate
755
00:41:45,971 --> 00:41:48,104
to the far side
of a nearby hill.
756
00:41:48,106 --> 00:41:49,973
They also have a shotgun,
757
00:41:49,975 --> 00:41:53,276
which they will fire if they
manage to pick up the signal.
758
00:41:58,048 --> 00:42:00,149
(buzzing)
759
00:42:04,655 --> 00:42:08,157
(buzzing)
760
00:42:08,159 --> 00:42:10,493
(gunshot)
761
00:42:10,495 --> 00:42:11,995
And it works.
762
00:42:11,997 --> 00:42:13,796
The signal has been detected
763
00:42:13,798 --> 00:42:17,867
even though the receiver
is now hidden behind a hill.
764
00:42:17,869 --> 00:42:19,669
At over a mile,
765
00:42:19,671 --> 00:42:22,739
it is the farthest
transmission to date.
766
00:42:22,741 --> 00:42:25,575
In fewer than ten years,
767
00:42:25,577 --> 00:42:30,146
Marconi will be sending radio
signals across the Atlantic.
768
00:42:30,148 --> 00:42:34,851
In fact, when the Titanic sinks
in 1912,
769
00:42:34,853 --> 00:42:39,155
he'll be personally credited
with saving many lives
770
00:42:39,157 --> 00:42:42,492
because his onboard equipment
allowed the distress signal
771
00:42:42,494 --> 00:42:46,195
to be transmitted.
772
00:42:46,197 --> 00:42:50,300
Thanks to the predictions
of Maxwell's equations,
773
00:42:50,302 --> 00:42:54,203
Marconi could harness
a hidden part of our world,
774
00:42:54,205 --> 00:42:58,274
ushering in the era
of wireless communication.
775
00:42:58,276 --> 00:43:01,210
(voices on radio overlapping)
776
00:43:03,147 --> 00:43:06,883
Since Maxwell and Marconi,
777
00:43:06,885 --> 00:43:10,920
evidence of the predictive power
of mathematics has only grown,
778
00:43:10,922 --> 00:43:14,557
especially
in the world of physics.
779
00:43:14,559 --> 00:43:19,395
100 years ago,
we barely knew atoms existed.
780
00:43:19,397 --> 00:43:22,298
It took experiments
to reveal their components:
781
00:43:22,300 --> 00:43:26,069
the electron, the proton,
and the neutron.
782
00:43:26,071 --> 00:43:28,538
But when physicists
wanted to go deeper,
783
00:43:28,540 --> 00:43:31,574
mathematics began
to lead the way,
784
00:43:31,576 --> 00:43:35,979
ultimately revealing a zoo
of elementary particles,
785
00:43:35,981 --> 00:43:41,250
discoveries that continue
to this day here at CERN,
786
00:43:41,252 --> 00:43:44,454
the European organization
for nuclear research
787
00:43:44,456 --> 00:43:46,356
in Geneva, Switzerland.
788
00:43:46,358 --> 00:43:51,194
These days, they're most famous
for their Large Hadron Collider,
789
00:43:51,196 --> 00:43:55,999
a circular particle accelerator
about 17 miles around,
790
00:43:56,001 --> 00:43:59,469
built deep underground.
791
00:44:01,672 --> 00:44:05,575
This $10 billion project,
decades in the making,
792
00:44:05,577 --> 00:44:09,212
had a well-publicized goal:
the search
793
00:44:09,214 --> 00:44:12,749
for one of the most fundamental
building blocks of the universe.
794
00:44:15,519 --> 00:44:17,253
A subatomic particle
795
00:44:17,255 --> 00:44:22,025
mathematically predicted
to exist nearly 50 years earlier
796
00:44:22,027 --> 00:44:26,696
by Robert Brout and Francois
Englert working in Belgium
797
00:44:26,698 --> 00:44:29,499
and Peter Higgs in Scotland.
798
00:44:29,501 --> 00:44:32,568
TEGMARK:
Peter Higgs sat down
799
00:44:32,570 --> 00:44:34,404
with the most advanced
physics equations we had
800
00:44:34,406 --> 00:44:35,738
and calculated and calculated
801
00:44:35,740 --> 00:44:37,874
and made this audacious
prediction:
802
00:44:37,876 --> 00:44:40,043
if we built the most
sophisticated machines
803
00:44:40,045 --> 00:44:41,377
humans have ever built
804
00:44:41,379 --> 00:44:43,246
and used it
to smash particles together
805
00:44:43,248 --> 00:44:45,081
near the speed of light
in a certain way
806
00:44:45,083 --> 00:44:46,983
that we would then discover
a new particle.
807
00:44:46,985 --> 00:44:49,819
You know, if this math
was really accurate.
808
00:44:49,821 --> 00:44:52,588
NARRATOR:
The discovery
of the Higgs particle
809
00:44:52,590 --> 00:44:55,324
would be proof
of the Higgs field,
810
00:44:55,326 --> 00:45:00,029
a cosmic molasses that gives
the stuff of our world mass--
811
00:45:00,031 --> 00:45:04,934
what we usually experience
as weight.
812
00:45:04,936 --> 00:45:08,571
Without mass, everything would
travel at the speed of light
813
00:45:08,573 --> 00:45:11,908
and would never combine
to form atoms.
814
00:45:11,910 --> 00:45:13,910
That makes the Higgs field
815
00:45:13,912 --> 00:45:16,579
such a fundamental part
of physics
816
00:45:16,581 --> 00:45:19,282
that the Higgs particle
gained the nickname
817
00:45:19,284 --> 00:45:21,818
"The God Particle."
818
00:45:23,554 --> 00:45:25,288
(cheering)
819
00:45:25,290 --> 00:45:27,590
In 2012, experiments at CERN
820
00:45:27,592 --> 00:45:30,927
confirmed the existence
of the Higgs particle,
821
00:45:30,929 --> 00:45:33,196
making the work of Peter Higgs
822
00:45:33,198 --> 00:45:35,364
and his colleagues
decades earlier
823
00:45:35,366 --> 00:45:39,902
one of the greatest predictions
ever made.
824
00:45:39,904 --> 00:45:42,171
And we built it and it worked,
825
00:45:42,173 --> 00:45:44,674
and he got a free trip
to Stockholm.
826
00:45:48,145 --> 00:45:50,680
(applause)
827
00:45:55,719 --> 00:45:58,554
LIVIO:
Here, you have
mathematical theories
828
00:45:58,556 --> 00:46:02,558
which make
very definitive predictions
829
00:46:02,560 --> 00:46:04,994
about the possible existence
830
00:46:04,996 --> 00:46:07,663
of some fundamental particles
of nature,
831
00:46:07,665 --> 00:46:11,667
and believe it or not,
they make these huge experiments
832
00:46:11,669 --> 00:46:15,271
and they actually discover
the particles
833
00:46:15,273 --> 00:46:17,006
that have been predicted
mathematically.
834
00:46:17,008 --> 00:46:19,675
I mean,
this is just amazing to me.
835
00:46:21,812 --> 00:46:24,080
ANDREW LANKFORD:
Why does this work?
836
00:46:24,082 --> 00:46:27,316
How can mathematics
be so powerful?
837
00:46:27,318 --> 00:46:31,053
Is mathematics, you know,
a truth of nature,
838
00:46:31,055 --> 00:46:32,989
or does it have something to do
839
00:46:32,991 --> 00:46:36,025
with the way we as humans
perceive nature?
840
00:46:36,027 --> 00:46:39,128
To me, this is just
a fascinating puzzle.
841
00:46:39,130 --> 00:46:40,897
I don't know the answer.
842
00:46:43,300 --> 00:46:47,136
NARRATOR:
In physics, mathematics has had
a long string of successes.
843
00:46:47,138 --> 00:46:51,207
But is it really
"unreasonably effective"?
844
00:46:51,209 --> 00:46:53,776
Not everyone thinks so.
845
00:46:53,778 --> 00:46:55,344
I think it's an illusion,
846
00:46:55,346 --> 00:46:57,180
because I think what's happened
847
00:46:57,182 --> 00:47:01,117
is that people have chosen
to build physics, for example,
848
00:47:01,119 --> 00:47:03,719
using the mathematics
that has been practiced,
849
00:47:03,721 --> 00:47:05,188
has developed historically,
850
00:47:05,190 --> 00:47:07,990
and then they're looking
at everything,
851
00:47:07,992 --> 00:47:10,726
they're choosing to study things
which are amenable to study
852
00:47:10,728 --> 00:47:13,329
using the mathematics
that happens to have arisen.
853
00:47:13,331 --> 00:47:16,732
But actually, there is a whole
vast ocean of other things
854
00:47:16,734 --> 00:47:20,002
that are really quite
inaccessible to those methods.
855
00:47:20,004 --> 00:47:23,940
NARRATOR:
With the success of mathematical
models in physics,
856
00:47:23,942 --> 00:47:27,777
it's easy to overlook
where they don't work that well.
857
00:47:27,779 --> 00:47:30,580
Like in weather forecasting.
858
00:47:30,582 --> 00:47:33,082
There's a reason meteorologists
predict the weather
859
00:47:33,084 --> 00:47:34,917
for the coming week,
860
00:47:34,919 --> 00:47:37,286
but not much further out
than that.
861
00:47:37,288 --> 00:47:42,491
In a longer forecast,
small errors grow into big ones.
862
00:47:42,493 --> 00:47:46,195
Daily weather is just
too complex and chaotic
863
00:47:46,197 --> 00:47:48,731
for precise modeling.
864
00:47:48,733 --> 00:47:50,166
And it's not alone.
865
00:47:50,168 --> 00:47:54,937
So is the behavior of water
boiling on a stove,
866
00:47:54,939 --> 00:47:58,040
or the stock market,
867
00:47:58,042 --> 00:48:01,911
or the interaction of neurons
in the brain,
868
00:48:01,913 --> 00:48:04,080
much of human psychology,
869
00:48:04,082 --> 00:48:06,449
and parts of biology.
870
00:48:06,451 --> 00:48:08,317
DEREK ABBOTT:
Biological systems,
871
00:48:08,319 --> 00:48:10,286
economic systems,
872
00:48:10,288 --> 00:48:13,256
it gets very difficult to model
those systems with math.
873
00:48:13,258 --> 00:48:15,725
We have extreme difficulty
with that.
874
00:48:15,727 --> 00:48:20,396
So I do not see math
as unreasonably effective.
875
00:48:20,398 --> 00:48:23,866
I see it as reasonably
ineffective.
876
00:48:26,937 --> 00:48:30,039
NARRATOR:
Perhaps no one
is as keenly aware
877
00:48:30,041 --> 00:48:32,575
of the power and limitations
of mathematics
878
00:48:32,577 --> 00:48:35,778
as those who use it
to design and make things:
879
00:48:35,780 --> 00:48:37,046
engineers.
880
00:48:37,048 --> 00:48:38,814
Look at that wheel!
881
00:48:38,816 --> 00:48:42,318
NARRATOR:
In their work,
the elegance of math
882
00:48:42,320 --> 00:48:48,658
meets the messiness of reality,
and practicality rules the day.
883
00:48:48,660 --> 00:48:51,227
Mathematics
and perhaps mathematicians
884
00:48:51,229 --> 00:48:53,629
deal in the domain
of the absolute,
885
00:48:53,631 --> 00:48:58,434
and engineers live in the domain
of the approximate.
886
00:48:58,436 --> 00:49:03,406
We are fundamentally interested
in the practical.
887
00:49:03,408 --> 00:49:06,575
And so frequently, we make
approximations, we cut corners.
888
00:49:06,577 --> 00:49:08,110
We omit terms and equations
889
00:49:08,112 --> 00:49:11,347
to get things
that are simple enough
890
00:49:11,349 --> 00:49:14,016
to suit our purposes
and to meet our needs.
891
00:49:17,621 --> 00:49:20,556
NARRATOR:
Many of our greatest
engineering achievements
892
00:49:20,558 --> 00:49:23,025
were built using
mathematical shortcuts:
893
00:49:23,027 --> 00:49:27,263
simplified equations
that approximate an answer,
894
00:49:27,265 --> 00:49:30,733
trading some precision
for practicality.
895
00:49:30,735 --> 00:49:35,671
And for engineers,
"approximate" is close enough.
896
00:49:35,673 --> 00:49:38,941
Close enough
to take you to Mars.
897
00:49:40,911 --> 00:49:42,445
STELTZNER:
For us engineers,
898
00:49:42,447 --> 00:49:44,113
we don't get paid
to do things right;
899
00:49:44,115 --> 00:49:48,117
we get paid to do things
just right enough.
900
00:49:50,887 --> 00:49:53,856
NARRATOR:
Many physicists
see an uncanny accuracy
901
00:49:53,858 --> 00:49:55,524
in the way mathematics
can reveal
902
00:49:55,526 --> 00:49:57,927
the secrets of the universe,
903
00:49:57,929 --> 00:50:04,867
making it seem to be
an inherent part of nature.
904
00:50:04,869 --> 00:50:09,538
Meanwhile, engineers in practice
have to sacrifice
905
00:50:09,540 --> 00:50:12,708
the precision of mathematics
to keep it useful,
906
00:50:12,710 --> 00:50:16,879
making it seem more like
an imperfect tool
907
00:50:16,881 --> 00:50:19,515
of our own invention.
908
00:50:19,517 --> 00:50:22,518
So which is mathematics?
909
00:50:22,520 --> 00:50:26,122
A discovered part
of the universe?
910
00:50:26,124 --> 00:50:29,358
Or a very human invention?
911
00:50:29,360 --> 00:50:33,929
Maybe it's both.
912
00:50:36,600 --> 00:50:39,068
LIVIO:
What I think about mathematics
913
00:50:39,070 --> 00:50:42,038
is that it is an intricate
combination
914
00:50:42,040 --> 00:50:44,874
of inventions and discoveries.
915
00:50:44,876 --> 00:50:47,410
So for example, take something
like natural numbers:
916
00:50:47,412 --> 00:50:50,346
one, two, three, four, five,
etcetera.
917
00:50:50,348 --> 00:50:52,114
I think what happened
918
00:50:52,116 --> 00:50:54,216
was that people were looking
at many things, for example,
919
00:50:54,218 --> 00:50:57,253
and seeing that
there are two eyes, you know,
920
00:50:57,255 --> 00:51:00,056
two breasts, two hands,
you know, and so on.
921
00:51:00,058 --> 00:51:02,191
And after some time,
922
00:51:02,193 --> 00:51:05,561
they abstracted from all of that
the number two.
923
00:51:07,297 --> 00:51:11,067
NARRATOR:
According to Mario, "two"
became an invented concept,
924
00:51:11,069 --> 00:51:14,804
as did all the other
natural numbers.
925
00:51:14,806 --> 00:51:17,306
But then people discovered
that these numbers
926
00:51:17,308 --> 00:51:20,276
have all kinds
of intricate relationships.
927
00:51:20,278 --> 00:51:23,779
Those were discoveries.
928
00:51:23,781 --> 00:51:27,550
We invented the concept,
but then discovered
929
00:51:27,552 --> 00:51:30,252
the relations
among the different concepts.
930
00:51:30,254 --> 00:51:33,756
NARRATOR:
So is this the answer?
931
00:51:33,758 --> 00:51:37,760
That math is both invented
and discovered?
932
00:51:37,762 --> 00:51:39,595
This is one of those questions
where it's both.
933
00:51:39,597 --> 00:51:42,398
Yes, it feels like
it's already there,
934
00:51:42,400 --> 00:51:44,700
but yes, it's something
that comes out of our deep,
935
00:51:44,702 --> 00:51:47,269
creative nature as human beings.
936
00:51:47,271 --> 00:51:50,940
NARRATOR:
We may have some idea
to how all this works,
937
00:51:50,942 --> 00:51:53,843
but not the complete answer.
938
00:51:53,845 --> 00:51:59,081
In the end, it remains
"The Great Math Mystery."
939
00:52:22,606 --> 00:52:26,475
This NOVA program
is available on DVD.
Major funding for NOVA
is provided by the following:
940
00:52:26,477 --> 00:52:32,348
To order, visit shopPBS.org
or call 1-800-PLAY-PBS.
941
00:52:32,350 --> 00:52:34,216
NOVA is also available
for download on iTunes.
78489
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