The Great Math Mystery (2015) | Full Documentary | NOVA

00:53:22
https://www.youtube.com/watch?v=8hl5uZT41RY

Résumé

TLDRThis video delves into the intriguing relationship between mathematics and the workings of the universe, posing essential questions about the origins, efficacy, and nature of mathematics in describing reality. It showcases the historical developments from Galileo's laws of motion to the modern exploration of the cosmos, emphasizing mathematics as a vital tool in science and technology. By exploring concepts like the Fibonacci sequence and pi, alongside insights from prominent scientists and mathematicians, the narrative suggests that mathematics may not only describe but fundamentally shape our understanding of the universe. Ultimately, it posits that mathematics is both a discovery and invention of the human mind, intricately entwined with the fabric of reality itself.

A retenir

  • 🌌 Mathematics is called the language of the universe.
  • 🔍 The Fibonacci sequence appears often in nature.
  • 🌀 Pi is found in diverse natural phenomena.
  • 🐒 Animals demonstrate a primitive number sense.
  • 🏗️ Engineers use approximations in mathematical models.
  • 📚 Mathematics is both invented and discovered.
  • 🌍 Galileo's laws help understand motion and gravity.
  • ✨ Mathematics has predictive power in physics.
  • 🎶 Pythagoras linked music and mathematics.
  • 🚀 Math has enabled groundbreaking scientific discoveries.

Chronologie

  • 00:00:00 - 00:05:00

    The video begins with a discussion of modern advancements facilitated by mathematics, emphasizing its role as a foundational element in scientific exploration and engineering feats, such as landing a rover on Mars. It raises questions about the origin and efficacy of mathematics in explaining the universe.

  • 00:05:00 - 00:10:00

    The narrative explores the historical quest of humans to identify and understand patterns in nature, leading to the realization that mathematics serves as a powerful tool in deciphering these patterns across various phenomena, from the orbits of planets to biological structures like flowers and seashells.

  • 00:10:00 - 00:15:00

    Introducing the Fibonacci sequence, the video illustrates how math is observed in natural patterns, particularly in plants. It discusses claims about the significance of these numbers in various contexts, although many of these claims remain unverified, leading to questions about plants' innate knowledge of mathematics.

  • 00:15:00 - 00:20:00

    The video highlights the deep connection between mathematics and physical reality, using the example of the number pi, which transcends its geometric definition and appears in diverse areas such as probability and wave phenomena. Pi captures the intricate relationships between seemingly unrelated elements in the universe.

  • 00:20:00 - 00:25:00

    Max Tegmark's perspective is introduced, suggesting that if the universe is akin to a computer simulation, then all physical properties might fundamentally be mathematical in nature. This radical view brings forth researchers’ acknowledgment of the mathematical essence in understanding our reality and the structures within it.

  • 00:25:00 - 00:30:00

    The video reveals the ancient philosophical origins of the idea that mathematics is not merely a human construct but an essential element of the universe, starting with Pythagoras's revelations about musical harmony being mathematically constructed ratios, reinforcing the connection between math and the natural world.

  • 00:30:00 - 00:35:00

    Continuing with Pythagorean beliefs, the narrative details Plato's concept of ideal mathematical forms, where these idealized geometric shapes influence the nature that appears in the physical world, thus reflecting mathematics' profound impact through history on our understanding of the cosmos.

  • 00:35:00 - 00:40:00

    The transition from geometric principles to the establishment of fundamental laws, as embodied by Newton's work, shows how mathematics can unify observations of motion and gravity into a coherent framework. This connection establishes mathematics as the foundation of scientific inquiry in the natural world.

  • 00:40:00 - 00:45:00

    Mathematics' predictive power is showcased through examples, from the discovery of Neptune to modern applications like wireless communication predicated on Maxwell's equations, illustrating how mathematical models yield incredible advancements in technology and our understanding of the universe.

  • 00:45:00 - 00:53:22

    The video concludes with a discussion on the dual nature of mathematics as both an invention and a discovery, suggesting that while humans have created mathematical systems and concepts, they also reveal truths about the universe, leaving the audience grappling with the mystery of mathematics' role in our understanding of existence.

Afficher plus

Carte mentale

Vidéo Q&R

  • What is the significance of mathematics in science?

    Mathematics is considered a powerful tool that quantifies observations and uncovers the underlying principles of nature.

  • What is the Fibonacci sequence, and where is it found in nature?

    The Fibonacci sequence is a series of numbers found in various natural patterns, such as the arrangement of leaves, flowers, and seeds.

  • How does pi relate to various natural phenomena?

    Pi is observed in many contexts beyond geometry, such as probability theory, the behavior of waves, and even the structure of galaxies.

  • Do animals have a sense of numbers?

    Yes, various animals, including lemurs and monkeys, exhibit an ability to compare quantities without language or symbols.

  • Is mathematics purely a human invention?

    The debate continues, but many believe that mathematics is a combination of human invention and discovery, reflecting both abstract ideas and inherent relationships in the universe.

  • What are some examples of mathematics predicting physical phenomena?

    Math has accurately predicted the existence of celestial bodies like Neptune and subatomic particles like the Higgs boson.

  • How do engineers use mathematics in their work?

    Engineers apply mathematical principles, often using approximations, to design practical solutions that work in real-world applications.

  • What did Pythagoras contribute to the field of mathematics?

    Pythagoras found a deep connection between mathematics and music, recognizing ratios that produce harmonious sounds.

  • What did Galileo discover using mathematical principles?

    Galileo established that falling objects accelerate at the same rate, regardless of their weight, leading to the formulation of gravitational laws.

  • What is the conclusion regarding the nature of mathematics?

    Mathematics appears to be both an inherent part of the universe and a human creation, reflecting our quest to understand the world.

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Défilement automatique:
  • 00:00:08
    MAN: Roger, copy mission.
  • 00:00:09
    NARRATOR: We live in an age of astonishing advances.
  • 00:00:12
    MAN: Descending at about .75 meters per second.
  • 00:00:16
    NARRATOR: Engineers can land a car-size rover on Mars.
  • 00:00:20
    MAN: Touchdown confirmed.
  • 00:00:21
    (cheering)
  • 00:00:23
    NARRATOR: Physicists probe the essence of all matter,
  • 00:00:28
    while we communicate wirelessly on a vast worldwide network.
  • 00:00:35
    But underlying all of these modern wonders
  • 00:00:37
    is something deep and mysteriously powerful.
  • 00:00:42
    It's been called the language of the universe,
  • 00:00:45
    and perhaps it's civilization's greatest achievement.
  • 00:00:49
    Its name?
  • 00:00:51
    Mathematics.
  • 00:00:53
    But where does math come from?
  • 00:00:55
    And why in science does it work so well?
  • 00:00:59
    MARIO LIVIO: Albert Einstein wondered,
  • 00:01:01
    "How is it possible that mathematics
  • 00:01:03
    does so well in explaining the universe as we see it?"
  • 00:01:08
    NARRATOR: Is mathematics even human?
  • 00:01:12
    There doesn't really seem to be an upper limit
  • 00:01:15
    to the numerical abilities of animals.
  • 00:01:18
    NARRATOR: And is it the key to the cosmos?
  • 00:01:22
    MAX TEGMARK: Our physical world
  • 00:01:24
    doesn't just have some mathematical properties,
  • 00:01:26
    but it has only mathematical properties.
  • 00:01:28
    NARRATOR: "The Great Math Mystery," next on NOVA!
  • 00:01:53
    NARRATOR: Human beings have always looked at nature
  • 00:01:56
    and searched for patterns.
  • 00:01:58
    Eons ago, we gazed at the stars
  • 00:02:01
    and discovered patterns we call constellations,
  • 00:02:05
    even coming to believe they might control our destiny.
  • 00:02:10
    We've watched the days turn to night and back to day,
  • 00:02:16
    and seasons as they come and go,
  • 00:02:19
    and called that pattern "time."
  • 00:02:23
    We see symmetrical patterns in the human body
  • 00:02:28
    and the tiger's stripes
  • 00:02:32
    and build those patterns into what we create,
  • 00:02:35
    from art to our cities.
  • 00:02:44
    But what do patterns tell us?
  • 00:02:46
    Why should the spiral shape of the nautilus shell
  • 00:02:50
    be so similar to the spiral of a galaxy?
  • 00:02:55
    Or the spiral found in a sliced open head of cabbage?
  • 00:02:59
    When scientists seek to understand
  • 00:03:03
    the patterns of our world,
  • 00:03:04
    they often turn to a powerful tool: mathematics.
  • 00:03:09
    They quantify their observations
  • 00:03:12
    and use mathematical techniques to examine them,
  • 00:03:16
    hoping to discover the underlying causes
  • 00:03:19
    of nature's rhythms and regularities.
  • 00:03:23
    And it's worked, revealing the secrets
  • 00:03:25
    behind the elliptical orbits of the planets
  • 00:03:28
    to the electromagnetic waves that connect our cell phones.
  • 00:03:34
    Mathematics has even guided the way,
  • 00:03:36
    leading us right down
  • 00:03:38
    to the sub-atomic building blocks of matter.
  • 00:03:43
    Which raises the question: why does it work at all?
  • 00:03:48
    Is there an inherent mathematical nature to reality?
  • 00:03:53
    Or is mathematics all in our heads?
  • 00:04:01
    Mario Livio is an astrophysicist
  • 00:04:05
    who wrestles with these questions.
  • 00:04:06
    He's fascinated by the deep and often mysterious connection
  • 00:04:11
    between mathematics and the world.
  • 00:04:14
    MARIO LIVIO: If you look at nature, there are numbers all around us.
  • 00:04:18
    You know, look at flowers, for example.
  • 00:04:20
    So there are many flowers
  • 00:04:22
    that have three petals like this, or five like this.
  • 00:04:24
    Some of them may have 34 or 55.
  • 00:04:29
    These numbers occur very often.
  • 00:04:31
    NARRATOR: These may sound like random numbers,
  • 00:04:35
    but they're all part of what is known as the Fibonacci sequence,
  • 00:04:39
    a series of numbers developed by a 13th century mathematician.
  • 00:04:47
    You start with the numbers one and one,
  • 00:04:49
    and from that point on,
  • 00:04:51
    you keep adding up the last two numbers.
  • 00:04:53
    So one plus one is two,
  • 00:04:56
    now one plus two is three,
  • 00:04:59
    two plus three is five,
  • 00:05:03
    three plus five is eight, and you keep going like this.
  • 00:05:08
    NARRATOR: Today, hundreds of years later,
  • 00:05:10
    this seemingly arbitrary progression of numbers
  • 00:05:13
    fascinates many, who see in it clues
  • 00:05:16
    to everything from human beauty to the stock market.
  • 00:05:20
    While most of those claims remain unproven,
  • 00:05:23
    it is curious how evolution seems to favor these numbers.
  • 00:05:27
    And as it turns out,
  • 00:05:29
    this sequence appears quite frequently in nature.
  • 00:05:33
    NARRATOR: Fibonacci numbers show up in petal counts,
  • 00:05:36
    especially of daisies, but that's just a start.
  • 00:05:41
    CHRISTOPHE GOLE: Statistically, the Fibonacci numbers
  • 00:05:43
    do appear a lot in botany.
  • 00:05:47
    For instance, if you look at the bottom of a pine cone,
  • 00:05:50
    you will see often spirals in their scales.
  • 00:05:54
    You end up counting those spirals,
  • 00:05:57
    you'll usually find a Fibonacci number,
  • 00:06:00
    and then you will count the spirals
  • 00:06:02
    going in the other direction
  • 00:06:04
    and you will find an adjacent Fibonacci number.
  • 00:06:08
    NARRATOR: The same is true of the seeds on a sunflower head--
  • 00:06:13
    two sets of spirals.
  • 00:06:15
    And if you count the spirals in each direction,
  • 00:06:18
    both are Fibonacci numbers.
  • 00:06:22
    While there are some theories
  • 00:06:25
    explaining the Fibonacci-botany connection,
  • 00:06:27
    it still raises some intriguing questions.
  • 00:06:32
    So do plants know math?
  • 00:06:34
    The short answer to that is "No."
  • 00:06:38
    They don't need to know math.
  • 00:06:40
    In a very simple, geometric way, they set up a little machine
  • 00:06:45
    that creates the Fibonacci sequence in many cases.
  • 00:06:53
    NARRATOR: The mysterious connections
  • 00:06:55
    between the physical world and mathematics run deep.
  • 00:06:59
    We all know the number pi from geometry--
  • 00:07:01
    the ratio between the circumference of a circle
  • 00:07:04
    and its diameter-- and that its decimal digits
  • 00:07:08
    go on forever without a repeating pattern.
  • 00:07:11
    As of 2013,
  • 00:07:13
    it had been calculated out to 12.1 trillion digits.
  • 00:07:17
    But somehow, pi is a whole lot more.
  • 00:07:22
    Pi appears in a whole host of other phenomena
  • 00:07:25
    which have, at least on the face of it,
  • 00:07:27
    nothing to do with circles or anything.
  • 00:07:29
    In particular, it appears in probability theory quite a bit.
  • 00:07:33
    Suppose I take this needle.
  • 00:07:35
    So the length of the needle
  • 00:07:37
    is equal to the distance between two lines
  • 00:07:40
    on this piece of paper.
  • 00:07:42
    And suppose I drop this needle now on the paper.
  • 00:07:45
    NARRATOR: Sometimes when you drop the needle, it will cut a line,
  • 00:07:49
    and sometimes it drops between the lines.
  • 00:07:52
    It turns out the probability
  • 00:07:55
    that the needle lands so it cuts a line
  • 00:07:58
    is exactly two over pi, or about...
  • 00:08:03
    ...64%.
  • 00:08:06
    Now, what that means is that, in principle,
  • 00:08:10
    I could drop this needle millions of times.
  • 00:08:13
    I could count the times when it crosses a line
  • 00:08:16
    and when it doesn't cross a line,
  • 00:08:18
    and I could actually even calculate pi
  • 00:08:20
    even though there are no circles here,
  • 00:08:23
    no diameters of a circle, nothing like that.
  • 00:08:26
    It's really amazing.
  • 00:08:32
    NARRATOR: Since pi relates a round object, a circle,
  • 00:08:35
    with a straight one, its diameter,
  • 00:08:38
    it can show up in the strangest of places.
  • 00:08:42
    Some see it in the meandering path of rivers.
  • 00:08:46
    A river's actual length
  • 00:08:47
    as it winds its way from its source to its mouth
  • 00:08:51
    compared to the direct distance on average seems to be about pi.
  • 00:08:57
    Models for just about anything involving waves
  • 00:09:00
    will have pi in them, like those for light and sound.
  • 00:09:06
    Pi tells us which colors should appear in a rainbow,
  • 00:09:10
    and how middle C should sound on a piano.
  • 00:09:14
    Pi shows up in apples,
  • 00:09:16
    in the way cells grow into spherical shapes,
  • 00:09:20
    or in the brightness of a supernova.
  • 00:09:25
    One writer has suggested
  • 00:09:27
    it's like seeing pi on a series of mountain peaks,
  • 00:09:31
    poking out of a fog-shrouded valley.
  • 00:09:34
    We know there's a way they're all connected,
  • 00:09:36
    but it's not always obvious how.
  • 00:09:43
    Pi is but one example
  • 00:09:46
    of a vast interconnected web of mathematics
  • 00:09:49
    that seems to reveal
  • 00:09:51
    an often hidden and deep order to our world.
  • 00:09:58
    Physicist Max Tegmark from MIT thinks he knows why.
  • 00:10:03
    He sees similarities between our world
  • 00:10:07
    and that of a computer game.
  • 00:10:13
    MAX TEGMARK: If I were a character in a computer game
  • 00:10:16
    that were so advanced that I were actually conscious
  • 00:10:19
    and I started exploring my video game world,
  • 00:10:22
    it would actually feel to me like it was made
  • 00:10:24
    of real solid objects made of physical stuff.
  • 00:10:28
    ♪ ♪
  • 00:10:35
    Yet, if I started studying, as the curious physicist that I am,
  • 00:10:39
    the properties of this stuff,
  • 00:10:41
    the equations by which things move
  • 00:10:44
    and the equations that give stuff its properties,
  • 00:10:47
    I would discover eventually
  • 00:10:49
    that all these properties were mathematical:
  • 00:10:52
    the mathematical properties
  • 00:10:54
    that the programmer had actually put into the software
  • 00:10:57
    that describes everything.
  • 00:10:59
    NARRATOR: The laws of physics in a game--
  • 00:11:02
    like how an object floats, bounces, or crashes--
  • 00:11:05
    are only mathematical rules created by a programmer.
  • 00:11:10
    Ultimately, the entire "universe" of a computer game
  • 00:11:14
    is just numbers and equations.
  • 00:11:18
    That's exactly what I perceive in this reality, too,
  • 00:11:20
    as a physicist,
  • 00:11:21
    that the closer I look at things that seem non-mathematical,
  • 00:11:24
    like my arm here and my hand,
  • 00:11:26
    the more mathematical it turns out to be.
  • 00:11:28
    Could it be that our world also then
  • 00:11:31
    is really just as mathematical as the computer game reality?
  • 00:11:36
    NARRATOR: To Max, the software world of a game isn't that different
  • 00:11:41
    from the physical world we live in.
  • 00:11:44
    He thinks that mathematics works so well to describe reality
  • 00:11:48
    because ultimately, mathematics is all that it is.
  • 00:11:52
    There's nothing else.
  • 00:11:56
    Many of my physics colleagues
  • 00:11:58
    will say that mathematics describes our physical reality
  • 00:12:02
    at least in some approximate sense.
  • 00:12:04
    I go further and argue that it actually is our physical reality
  • 00:12:10
    because I'm arguing that our physical world
  • 00:12:13
    doesn't just have some mathematical properties,
  • 00:12:16
    but it has only mathematical properties.
  • 00:12:21
    NARRATOR: Our physical reality is a bit like a digital photograph,
  • 00:12:24
    according to Max.
  • 00:12:28
    The photo looks like the pond,
  • 00:12:30
    but as we move in closer and closer,
  • 00:12:34
    we can see it is really a field of pixels,
  • 00:12:38
    each represented by three numbers
  • 00:12:41
    that specify the amount of red, green and blue.
  • 00:12:46
    While the universe is vast in its size and complexity,
  • 00:12:51
    requiring an unbelievably large collection of numbers
  • 00:12:55
    to describe it,
  • 00:12:57
    Max sees its underlying mathematical structure
  • 00:12:59
    as surprisingly simple.
  • 00:13:02
    It's just 32 numbers--
  • 00:13:05
    constants, like the masses of elementary particles--
  • 00:13:09
    along with a handful of mathematical equations,
  • 00:13:13
    the fundamental laws of physics.
  • 00:13:16
    And it all fits on a wall,
  • 00:13:19
    though there are still some questions.
  • 00:13:23
    But even though we don't know
  • 00:13:25
    what exactly is going to go here,
  • 00:13:27
    I am really confident that what will go here
  • 00:13:30
    will be mathematical equations.
  • 00:13:32
    That everything is ultimately mathematical.
  • 00:13:36
    NARRATOR: Max Tegmark's Matrix-like view
  • 00:13:40
    that mathematics doesn't just describe reality
  • 00:13:43
    but is its essence may sound radical,
  • 00:13:47
    but it has deep roots in history...
  • 00:13:52
    going back to ancient Greece,
  • 00:13:54
    to the time of the philosopher and mystic Pythagoras.
  • 00:13:58
    Stories say he explored the affinity
  • 00:14:02
    between mathematics and music,
  • 00:14:05
    a relationship that resonates to this day
  • 00:14:08
    in the work of Esperanza Spalding,
  • 00:14:11
    an acclaimed jazz musician who's studied music theory
  • 00:14:14
    and sees its parallel in mathematics.
  • 00:14:20
    SPALDING: I love the experience of math.
  • 00:14:23
    The part that I enjoy about math
  • 00:14:24
    I get to experience through music, too.
  • 00:14:29
    At the beginning,
  • 00:14:30
    you're studying all the little equations,
  • 00:14:32
    but you get to have this very visceral relationship
  • 00:14:35
    with the product of those equations,
  • 00:14:37
    which is sound and music and harmony and dissonance
  • 00:14:39
    and all that good stuff.
  • 00:14:40
    So I'm much better at music than at math,
  • 00:14:43
    but I love math with a passion.
  • 00:14:45
    They're both just as much work.
  • 00:14:46
    They're both, you have to study your... off.
  • 00:14:51
    Your head off, study your head off.
  • 00:14:52
    (laughs)
  • 00:14:55
    NARRATOR: The Ancient Greeks found three relationships
  • 00:14:57
    between notes especially pleasing.
  • 00:15:00
    Now we call them an octave, a fifth, and a fourth.
  • 00:15:06
    An octave is easy to remember
  • 00:15:08
    because it's the first two notes of "Somewhere Over the Rainbow."
  • 00:15:10
    ♪ La, la. ♪
  • 00:15:12
    That's an octave-- "somewhere."
  • 00:15:14
    (plays notes)
  • 00:15:18
    A fifth sounds like this:
  • 00:15:19
    ♪ La, la. ♪
  • 00:15:21
    Or the first two notes of "Twinkle, Twinkle, Little Star."
  • 00:15:24
    (plays notes)
  • 00:15:26
    And a fourth sounds like:
  • 00:15:28
    ♪ La, la ♪
  • 00:15:31
    (plays notes)
  • 00:15:32
    You can think of it as the first two notes
  • 00:15:34
    of "Here Comes the Bride."
  • 00:15:35
    (plays notes)
  • 00:15:39
    NARRATOR: In the sixth century BCE,
  • 00:15:42
    the Greek philosopher Pythagoras is said to have discovered
  • 00:15:44
    that those beautiful musical relationships
  • 00:15:47
    were also beautiful mathematical relationships
  • 00:15:51
    by measuring the lengths of the vibrating strings.
  • 00:15:55
    In an octave, the string lengths create a ratio of two to one.
  • 00:15:59
    (plays notes)
  • 00:16:02
    In a fifth, the ratio is three to two.
  • 00:16:05
    (plays notes)
  • 00:16:07
    And in a fourth, it is four to three.
  • 00:16:11
    (plays notes)
  • 00:16:14
    Seeing a common pattern throughout sound,
  • 00:16:16
    that could be a big eye opener of saying,
  • 00:16:19
    "Well, if this exists in sound,
  • 00:16:21
    "and if it's true universally through all sounds,
  • 00:16:25
    "this ratio could exist universally everywhere, right?
  • 00:16:29
    And doesn't it?"
  • 00:16:30
    (playing a tune)
  • 00:16:33
    NARRATOR: Pythagoreans worshipped the idea of numbers.
  • 00:16:36
    The fact that simple ratios produced harmonious sounds
  • 00:16:40
    was proof of a hidden order in the natural world.
  • 00:16:44
    And that order was made of numbers,
  • 00:16:46
    a profound insight that mathematicians and scientists
  • 00:16:50
    continue to explore to this day.
  • 00:16:56
    In fact, there are plenty of other physical phenomena
  • 00:17:00
    that follow simple ratios, from the two-to-one ratio
  • 00:17:04
    of hydrogen atoms to oxygen atoms in water
  • 00:17:08
    to the number of times the Moon orbits the Earth
  • 00:17:11
    compared to its own rotation: one to one.
  • 00:17:15
    Or that Mercury rotates exactly three times
  • 00:17:19
    when it orbits the Sun twice, a three-to-two ratio.
  • 00:17:26
    In Ancient Greece, Pythagoras and his followers
  • 00:17:30
    had a profound effect on another Greek philosopher, Plato,
  • 00:17:34
    whose ideas also resonate to this day,
  • 00:17:37
    especially among mathematicians.
  • 00:17:40
    Plato believed that geometry and mathematics
  • 00:17:43
    exist in their own ideal world.
  • 00:17:48
    So when we draw a circle on a piece of paper,
  • 00:17:50
    this is not the real circle.
  • 00:17:52
    The real circle is in that world,
  • 00:17:54
    and this is just an approximation
  • 00:17:56
    of that real circle,
  • 00:17:58
    and the same with all other shapes.
  • 00:18:00
    And Plato liked very much these five solids,
  • 00:18:03
    the platonic solids we call them today,
  • 00:18:06
    and he assigned each one of them to one of the elements
  • 00:18:10
    that formed the world as he saw it.
  • 00:18:12
    NARRATOR: The stable cube was earth.
  • 00:18:17
    The tetrahedron with its pointy corners was fire.
  • 00:18:22
    The mobile-looking octahedron Plato thought of as air.
  • 00:18:28
    And the 20-sided icosahedron was water.
  • 00:18:34
    And finally the dodecahedron,
  • 00:18:37
    this was the thing that signified the cosmos as a whole.
  • 00:18:44
    NARRATOR: So Plato's mathematical forms
  • 00:18:46
    were the ideal version of the world around us,
  • 00:18:49
    and they existed in their own realm.
  • 00:18:53
    And however bizarre that may sound,
  • 00:18:55
    that mathematics exists in its own world,
  • 00:18:58
    shaping the world we see, it's an idea that to this day
  • 00:19:03
    many mathematicians and scientists can relate to--
  • 00:19:06
    the sense they have when they're doing math
  • 00:19:09
    that they're just uncovering something
  • 00:19:11
    that's already out there.
  • 00:19:13
    I feel quite strongly that mathematics is discovered
  • 00:19:15
    in my work as a mathematician.
  • 00:19:17
    It always feels to me there is a thing out there
  • 00:19:19
    and I'm kind of trying to find it
  • 00:19:21
    and understand it and touch it.
  • 00:19:25
    JAMES GATES: As someone who actually has had the pleasure
  • 00:19:27
    of making new mathematics,
  • 00:19:29
    it feels like there's something there before you get to it.
  • 00:19:32
    If I have to choose,
  • 00:19:34
    I think it's more discovered than invented
  • 00:19:36
    because I think there's a reality
  • 00:19:37
    to what we study in mathematics.
  • 00:19:40
    When we do good mathematics,
  • 00:19:42
    we're discovering something about the way our minds work
  • 00:19:45
    in interaction with the world.
  • 00:19:47
    Well, I know that because that's what I do.
  • 00:19:49
    I come to my office, I sit down in front of my whiteboard
  • 00:19:51
    and I try and understand that thing that's out there.
  • 00:19:55
    And every now and then, I'm discovering a new bit of it.
  • 00:19:58
    That's exactly what it feels like.
  • 00:20:00
    NARRATOR: To many mathematicians,
  • 00:20:02
    it feels like math is discovered rather than invented.
  • 00:20:07
    But is that just a feeling?
  • 00:20:09
    Could it be that mathematics
  • 00:20:11
    is purely a product of the human brain?
  • 00:20:16
    Meet Shyam, a bonafide math whiz.
  • 00:20:20
    MICHAEL O'BOYLE: 800 on the SAT Math.
  • 00:20:22
    That's pretty good.
  • 00:20:23
    And you took it when you were how old?
  • 00:20:25
    Eleven.
  • 00:20:25
    Eleven.
  • 00:20:26
    Wow, that's, like, a perfect score.
  • 00:20:29
    NARRATOR: Where does Shyam's math genius come from?
  • 00:20:31
    It turns out we can pinpoint it, and it's all in his head.
  • 00:20:36
    Using fMRI, scientists can scan Shyam's brain
  • 00:20:42
    as he answers math questions
  • 00:20:44
    to see which parts of the brain receive more blood,
  • 00:20:47
    a sign they are hard at work.
  • 00:20:51
    MAN: All right, Shyam, we'll start about now.
  • 00:20:53
    Okay, buddy?
  • 00:20:54
    SHYAM: Okay.
  • 00:20:56
    NARRATOR: In images of Shyam's brain,
  • 00:20:59
    the parietal lobes glow an especially bright crimson.
  • 00:21:04
    He is relying on parietal areas
  • 00:21:06
    to determine these mathematical relationships.
  • 00:21:09
    That's characteristic of lots of math-gifted types.
  • 00:21:12
    NARRATOR: In tests similar to Shyam's,
  • 00:21:16
    kids who exhibit high math performance
  • 00:21:18
    have five to six times more neuron activation
  • 00:21:21
    than average kids in these brain regions.
  • 00:21:24
    But is that the result of teaching and intense practice?
  • 00:21:28
    Or are the foundations of math built into our brains?
  • 00:21:38
    Scientists are looking for the answer here,
  • 00:21:41
    at the Duke University Lemur Center,
  • 00:21:44
    a 70-acre sanctuary in North Carolina,
  • 00:21:46
    the largest one for rare and endangered lemurs in the world.
  • 00:21:53
    Like all primates, lemurs are related to humans
  • 00:21:56
    through a common ancestor
  • 00:21:59
    that lived as many as 65 million years ago.
  • 00:22:02
    Scientists believe lemurs
  • 00:22:04
    share many characteristics with those earliest primates,
  • 00:22:08
    making them a window, though a blurry one,
  • 00:22:12
    into our ancient past.
  • 00:22:15
    Got a choice here, Teres.
  • 00:22:17
    Come on up.
  • 00:22:19
    NARRATOR: Duke Professor Liz Brannon
  • 00:22:21
    investigates how well lemurs, like Teres here,
  • 00:22:24
    can compare quantities.
  • 00:22:26
    BRANNON: Many different animals choose larger food quantities.
  • 00:22:29
    So what is Teres doing?
  • 00:22:32
    What are all of these different animals doing
  • 00:22:34
    when they compare two quantities?
  • 00:22:37
    Well, clearly he's not using verbal labels,
  • 00:22:40
    he's not using symbols.
  • 00:22:42
    We need to figure out whether they can really use number,
  • 00:22:45
    pure number, as a cue.
  • 00:22:49
    NARRATOR: To test how well Teres can distinguish quantities,
  • 00:22:53
    he's been taught a touch-screen computer game.
  • 00:22:57
    The red square starts a round.
  • 00:23:00
    If he touches it, two squares appear
  • 00:23:03
    containing different numbers of objects.
  • 00:23:06
    He's been trained
  • 00:23:07
    that if he chooses the box with the fewest number...
  • 00:23:10
    (ringing)
  • 00:23:12
    ...he'll get a reward, a sugar pellet.
  • 00:23:15
    A wrong answer?
  • 00:23:17
    (buzzer)
  • 00:23:21
    We have to do a lot to ensure
  • 00:23:23
    that they're really attending to number and not something else.
  • 00:23:26
    NARRATOR: To make sure the test animal is reacting
  • 00:23:30
    to the number of objects and not some other cue,
  • 00:23:33
    Liz varies the objects' size, color, and shape.
  • 00:23:38
    She has conducted thousands of trials
  • 00:23:41
    and shown that lemurs and rhesus monkeys
  • 00:23:44
    can learn to pick the right answer.
  • 00:23:48
    BRANNON: Teres obviously doesn't have language
  • 00:23:50
    and he doesn't have any symbols for number.
  • 00:23:52
    So is he counting, is he doing what a human child does
  • 00:23:55
    when they recite the numbers one, two, three?
  • 00:23:58
    No.
  • 00:24:00
    And yet, what he seems to be attending to
  • 00:24:03
    is the very abstract essence of what a number is.
  • 00:24:08
    NARRATOR: Lemurs and rhesus monkeys aren't alone
  • 00:24:12
    in having this primitive number sense.
  • 00:24:14
    Rats, pigeons, fish, raccoons,
  • 00:24:18
    insects, horses, and elephants
  • 00:24:21
    all show sensitivity to quantity.
  • 00:24:24
    And so do human infants.
  • 00:24:29
    At her lab on the Duke campus,
  • 00:24:32
    Liz has tested babies that were only six months old.
  • 00:24:36
    They'll look longer at a screen
  • 00:24:38
    with a changing number of objects,
  • 00:24:41
    as long as the change is obvious enough
  • 00:24:43
    to capture their attention.
  • 00:24:46
    Liz has also tested college students,
  • 00:24:50
    asking them not to count,
  • 00:24:52
    but to respond as quickly as they could
  • 00:24:55
    to a touch-screen test comparing quantities.
  • 00:24:58
    The results?
  • 00:25:00
    About the same as lemurs and rhesus monkeys.
  • 00:25:04
    BRANNON: In fact, there are humans
  • 00:25:06
    who aren't as good as our monkeys,
  • 00:25:09
    and others that are far better,
  • 00:25:11
    so there's a lot of variability in human performance,
  • 00:25:13
    but in general, it looks very similar to a monkey.
  • 00:25:18
    Substitute in the three, you raise that to the four...
  • 00:25:21
    BRANNON: Even without any mathematical education,
  • 00:25:24
    even without learning any number words or symbols,
  • 00:25:27
    we would still have, all of us as humans,
  • 00:25:29
    a primitive number sense.
  • 00:25:31
    That fundamental ability to perceive number
  • 00:25:35
    seems to be a very important foundation,
  • 00:25:38
    and without it, it's very questionable
  • 00:25:40
    as to whether we could ever appreciate symbolic mathematics.
  • 00:25:44
    NARRATOR: The building blocks of mathematics
  • 00:25:46
    may be preprogrammed into our brains,
  • 00:25:49
    part of the basic toolkit for survival,
  • 00:25:52
    like our ability to recognize patterns and shapes
  • 00:25:56
    or our sense of time.
  • 00:25:59
    From that point of view, on this foundation,
  • 00:26:01
    we've erected one of the greatest inventions
  • 00:26:03
    of human culture:
  • 00:26:07
    mathematics.
  • 00:26:10
    But the mystery remains.
  • 00:26:13
    If it is "all in our heads," why has math been so effective?
  • 00:26:19
    Through science, technology, and engineering,
  • 00:26:22
    it's transformed the planet,
  • 00:26:25
    even allowing us to go into the beyond.
  • 00:26:32
    As in the work here, at NASA's Jet Propulsion Laboratory
  • 00:26:35
    in Pasadena, California.
  • 00:26:36
    MAN: Roger, copy mission.
  • 00:26:39
    Coming up on entry.
  • 00:26:40
    NARRATOR: In 2012, they landed a car-size rover...
  • 00:26:46
    MAN: Descending at about .75 meters per second as expected.
  • 00:26:49
    NARRATOR: ...on Mars.
  • 00:26:51
    MAN: Touchdown confirmed, we're safe on Mars.
  • 00:26:53
    (cheering)
  • 00:26:58
    NARRATOR: Adam Steltzner was the lead engineer
  • 00:27:01
    on the team that designed the landing system.
  • 00:27:03
    Their work depended on a groundbreaking discovery
  • 00:27:08
    from the Renaissance
  • 00:27:10
    that turned mathematics into the language of science:
  • 00:27:14
    the law of falling bodies.
  • 00:27:20
    The ancient Greek philosopher Aristotle
  • 00:27:24
    taught that heavier objects fall faster than lighter ones--
  • 00:27:28
    an idea that, on the surface, makes sense.
  • 00:27:33
    Even this surface: the Mars yard,
  • 00:27:37
    where they test the rovers at JPL.
  • 00:27:40
    ADAM STELTZNER: So Aristotle reasoned
  • 00:27:41
    that the rate at which things would fall
  • 00:27:45
    was proportional to their weight.
  • 00:27:51
    Which seems reasonable.
  • 00:27:52
    NARRATOR: In fact, so reasonable,
  • 00:27:54
    the view held for nearly 2,000 years,
  • 00:27:58
    until challenged in the late 1500s
  • 00:28:01
    by Italian mathematician Galileo Galilei.
  • 00:28:06
    STELTZNER: Legend has it that Galileo
  • 00:28:08
    dropped two different weight cannonballs
  • 00:28:11
    from the Leaning Tower of Pisa.
  • 00:28:14
    Well, we're not in Pisa, we don't have cannonballs,
  • 00:28:16
    but we do have a bowling ball and a bouncy ball.
  • 00:28:19
    Let's weigh them.
  • 00:28:21
    First, we weigh the bowling ball.
  • 00:28:25
    It weighs 15 pounds.
  • 00:28:27
    And the bouncy ball?
  • 00:28:28
    It weighs hardly anything.
  • 00:28:31
    Let's drop them.
  • 00:28:33
    NARRATOR: According to Aristotle,
  • 00:28:35
    the bowling ball should fall over 15 times faster
  • 00:28:39
    than the bouncy ball.
  • 00:28:44
    STELTZNER: Well, they seem to fall at the same rate.
  • 00:28:48
    This isn't that high, though.
  • 00:28:49
    Maybe we should drop them from higher.
  • 00:28:59
    So Ed is 20 feet in the air up there.
  • 00:29:03
    Let's see if the balls fall at the same rate.
  • 00:29:06
    Ready?
  • 00:29:07
    Three, two, one, drop!
  • 00:29:18
    Galileo was right.
  • 00:29:19
    Aristotle, you lose.
  • 00:29:22
    NARRATOR: Dropping feathers and hammers is misleading,
  • 00:29:25
    thanks to air resistance.
  • 00:29:29
    DAVID SCOTT: Well, in my left hand, I have a feather.
  • 00:29:32
    In my right hand, a hammer...
  • 00:29:34
    NARRATOR: A fact demonstrated on the Moon, where there is no air,
  • 00:29:38
    in 1971 during the Apollo 15 mission.
  • 00:29:42
    SCOTT: And I'll drop the two of them here.
  • 00:29:46
    How about that?
  • 00:29:47
    Mr. Galileo was correct.
  • 00:29:49
    STELTZNER: Little balls, soccer balls...
  • 00:29:52
    NARRATOR: So while counterintuitive...
  • 00:29:53
    STELTZNER: Vegetables!
  • 00:29:55
    NARRATOR: ...if you take the air out of the equation,
  • 00:29:58
    everything falls at the same rate,
  • 00:30:02
    even Aristotle.
  • 00:30:07
    But what really interested Galileo
  • 00:30:09
    was that an object dropped at one height
  • 00:30:12
    didn't take twice as long to drop from twice as high;
  • 00:30:17
    it accelerated.
  • 00:30:20
    But how do you measure that?
  • 00:30:23
    Everything is happening so fast.
  • 00:30:27
    STELTZNER: Oh, yes!
  • 00:30:31
    NARRATOR: Galileo came up with an ingenious solution.
  • 00:30:38
    He built a ramp, an inclined plane,
  • 00:30:44
    to slow the falling motion down so he could measure it.
  • 00:30:50
    STELTZNER: So we're going to use this ramp
  • 00:30:52
    to find the relationship between distance and time.
  • 00:30:57
    For time, I'll use an arbitrary unit: a Galileo.
  • 00:31:02
    One Galileo.
  • 00:31:05
    NARRATOR: The length of the ramp that the ball rolls
  • 00:31:07
    during one Galileo becomes one unit of distance.
  • 00:31:13
    So we've gone one unit of distance
  • 00:31:15
    in one unit of time.
  • 00:31:17
    Now let's try it for a two-count.
  • 00:31:19
    One Galileo, two Galileo.
  • 00:31:22
    NARRATOR: In two units of time,
  • 00:31:24
    the ball has rolled four units of distance.
  • 00:31:28
    Now let's see how far it goes in three Galileos.
  • 00:31:33
    One Galileo, two Galileo, three Galileo.
  • 00:31:37
    NARRATOR: In three units of time,
  • 00:31:39
    the ball has gone nine units of distance.
  • 00:31:44
    So there it is.
  • 00:31:45
    There's a mathematical relationship here
  • 00:31:47
    between time and distance.
  • 00:31:50
    NARRATOR: Galileo's inspired use of a ramp
  • 00:31:53
    had shown falling objects follow mathematical laws.
  • 00:31:59
    The distance the ball traveled
  • 00:32:01
    is directly proportional to the square of the time.
  • 00:32:06
    That mathematical relationship that Galileo observed
  • 00:32:11
    is a mathematical expression of the physics of our universe.
  • 00:32:15
    NARRATOR: Galileo's centuries-old
  • 00:32:17
    mathematical observation about falling objects
  • 00:32:20
    remains just as valid today.
  • 00:32:24
    It's the same mathematical expression that we can use
  • 00:32:27
    to understand how objects might fall here on Earth,
  • 00:32:31
    roll down a ramp.
  • 00:32:33
    It's even a relationship that we used
  • 00:32:35
    to land the Curiosity rover on the surface of Mars.
  • 00:32:41
    That's the power of mathematics.
  • 00:32:43
    NARRATOR: Galileo's insight was profound.
  • 00:32:48
    Mathematics could be used as a tool
  • 00:32:51
    to uncover and discover the hidden rules of our world.
  • 00:32:56
    He later wrote,
  • 00:32:58
    "The universe is written in the language of mathematics."
  • 00:33:03
    Math is really the language
  • 00:33:06
    in which we understand the universe.
  • 00:33:08
    We don't know why it's the case
  • 00:33:10
    that the laws of physics and the universe
  • 00:33:14
    follows mathematical models, but it does seem to be the case.
  • 00:33:19
    NARRATOR: While Galileo turned mathematical equations
  • 00:33:21
    into laws of science,
  • 00:33:23
    it was another man, born the same year Galileo died,
  • 00:33:27
    who took that to new heights that crossed the heavens.
  • 00:33:31
    His name was Isaac Newton.
  • 00:33:37
    He worked here at Trinity College in Cambridge, England.
  • 00:33:41
    SIMON SCHAFFER: Newton cultivated the reputation
  • 00:33:45
    of being a solitary genius,
  • 00:33:47
    and here in the bowling green of Trinity College,
  • 00:33:51
    it was said that he would walk meditatively
  • 00:33:54
    up and down the paths, absentmindedly drawing
  • 00:33:58
    mathematical diagrams in the gravel,
  • 00:34:01
    and the fellows were instructed, or so it was said,
  • 00:34:05
    not to disturb him,
  • 00:34:07
    not to clear up the gravel after he'd passed,
  • 00:34:10
    in case they inadvertently wiped out
  • 00:34:14
    some major scientific or mathematical discovery.
  • 00:34:18
    NARRATOR: In 1687, Newton published a book
  • 00:34:23
    that would become a landmark in the history of science.
  • 00:34:27
    Today, it is known simply as the "Principia."
  • 00:34:30
    In it, Newton gathered observations
  • 00:34:31
    from around the world
  • 00:34:33
    and used mathematics to explain them--
  • 00:34:37
    for instance, that of a comet seen widely in the fall of 1680.
  • 00:34:41
    SCHAFFER: He gathers data worldwide
  • 00:34:44
    in order to construct the comet's path.
  • 00:34:47
    So for November the 19th, he begins with an observation
  • 00:34:54
    made in Cambridge in England at 4:30 a.m.
  • 00:34:57
    by a certain young person,
  • 00:34:59
    and then at 5:00 in the morning at Boston in New England.
  • 00:35:06
    So what Newton does is to accumulate numbers
  • 00:35:09
    made by observers spread right across the globe
  • 00:35:13
    in order to construct
  • 00:35:15
    an unprecedentedly accurate calculation
  • 00:35:18
    of how this great comet moved through the sky.
  • 00:35:22
    NARRATOR: Newton's groundbreaking insight was that the force
  • 00:35:26
    that sent the comet hurtling around the Sun...
  • 00:35:30
    (cannon fire)
  • 00:35:31
    ...was the same force
  • 00:35:33
    that brought cannonballs back to Earth.
  • 00:35:36
    It was the force behind Galileo's law of falling bodies,
  • 00:35:42
    and it even held the planets in their orbits.
  • 00:35:47
    He called the force gravity, and described it precisely
  • 00:35:52
    in a surprisingly simple equation
  • 00:35:55
    that explains how two masses attract each other,
  • 00:35:58
    whether here on Earth or in the heavens above.
  • 00:36:04
    SCHAFFER: What's so impressive and so dramatic
  • 00:36:07
    is that a single mathematical law
  • 00:36:10
    would allow you to move throughout the universe.
  • 00:36:16
    NARRATOR: Today, we can even witness it at work beyond the Milky Way.
  • 00:36:24
    This is a picture of two galaxies
  • 00:36:27
    that are actually being drawn together in a merger.
  • 00:36:29
    This is how galaxies build themselves.
  • 00:36:31
    Right.
  • 00:36:32
    NARRATOR: Mario Livio is on the team
  • 00:36:34
    working with the images from the Hubble Space Telescope.
  • 00:36:37
    For decades, scientists have used Hubble
  • 00:36:40
    to gaze far beyond our solar system,
  • 00:36:43
    even beyond the stars of our galaxy.
  • 00:36:46
    It's shown us the distant gas clouds of nebulae
  • 00:36:50
    and vast numbers of galaxies wheeling in the heavens
  • 00:36:54
    billions of light-years away.
  • 00:36:57
    And what those images show
  • 00:36:59
    is that throughout the visible universe,
  • 00:37:02
    as far as the telescope can see,
  • 00:37:04
    the law of gravity still applies.
  • 00:37:08
    LIVIO: You know, Newton wrote these laws
  • 00:37:10
    of gravity and of motion
  • 00:37:12
    based on things happening on Earth,
  • 00:37:15
    and the planets in the solar system and so on,
  • 00:37:18
    but these same laws, these very same laws
  • 00:37:21
    apply to all these distant galaxies
  • 00:37:24
    and, you know, shape them,
  • 00:37:26
    and everything about them-- how they form, how they move--
  • 00:37:29
    is controlled by those same mathematical laws.
  • 00:37:33
    NARRATOR: Some of the world's greatest minds have been amazed
  • 00:37:38
    by the way that math permeates the universe.
  • 00:37:42
    LIVIO: Albert Einstein, he wondered,
  • 00:37:44
    he said, "How is it possible that mathematics,"
  • 00:37:48
    which is, he thought, a product of human thought,
  • 00:37:51
    "Does so well in explaining the universe as we see it?"
  • 00:37:55
    And Nobel laureate in physics Eugene Wigner
  • 00:37:59
    coined this phrase:
  • 00:38:01
    "The unreasonable effectiveness of mathematics."
  • 00:38:04
    He said the fact that mathematics
  • 00:38:06
    can really describe the universe so well,
  • 00:38:09
    in particular physical laws,
  • 00:38:11
    is a gift that we neither understand nor deserve.
  • 00:38:16
    NARRATOR: In physics,
  • 00:38:19
    examples of that "unreasonable effectiveness" abound.
  • 00:38:25
    When nearly 200 years ago
  • 00:38:27
    the planet Uranus was seen to go off track,
  • 00:38:30
    scientists trusted the math
  • 00:38:33
    and calculated it was being pulled by another unseen planet.
  • 00:38:40
    And so they discovered Neptune.
  • 00:38:44
    Mathematics had accurately predicted
  • 00:38:47
    a previously unknown planet.
  • 00:38:50
    SAVAS DIMOPOULOS: If you formulate a question properly,
  • 00:38:55
    mathematics gives you the answer.
  • 00:38:57
    It's like having a servant
  • 00:39:00
    that is far more capable than you are.
  • 00:39:04
    So you tell it "Do this,"
  • 00:39:06
    and if you say it nicely, then it'll do it
  • 00:39:09
    and it will carry you all the way to the truth,
  • 00:39:12
    to the final answer.
  • 00:39:14
    RADIO HOST: WGBH, 89.7.
  • 00:39:17
    NARRATOR: Evidence of the amazing predictive power of mathematics
  • 00:39:21
    can be found all around us.
  • 00:39:23
    I heard it took five Elvises to pull them apart.
  • 00:39:26
    NARRATOR: Television, radio, your cell phone, satellites,
  • 00:39:32
    the baby monitor, Wi-Fi, your garage door opener, GPS,
  • 00:39:39
    and yes, even maybe your TV's remote.
  • 00:39:42
    All of these use invisible waves of energy to communicate,
  • 00:39:46
    and no one even knew they existed
  • 00:39:49
    until the work of James Maxwell,
  • 00:39:52
    a Scottish mathematical physicist.
  • 00:39:55
    In the 1860s, he published a set of equations
  • 00:40:00
    that explained how electricity and magnetism were related--
  • 00:40:04
    how each could generate the other.
  • 00:40:09
    The equations also made a startling prediction.
  • 00:40:15
    Together, electricity and magnetism
  • 00:40:18
    could produce waves of energy
  • 00:40:20
    that would travel through space at the speed of light:
  • 00:40:24
    electromagnetic waves.
  • 00:40:28
    ROGER PENROSE: Maxwell's theory gave us
  • 00:40:29
    these radio waves, X-rays,
  • 00:40:33
    these things which were simply not known about at all.
  • 00:40:35
    So the theory had a scope, which was extraordinary.
  • 00:40:42
    NARRATOR: Almost immediately, people set out to find the waves
  • 00:40:45
    predicted by Maxwell's equations.
  • 00:40:49
    What must have seemed the least promising attempt
  • 00:40:51
    to harness them is made here, in northern Italy,
  • 00:40:54
    in the attic of a family home
  • 00:40:57
    by 20-year-old Guglielmo Marconi.
  • 00:41:00
    His process starts with a series of sparks.
  • 00:41:04
    (buzzing)
  • 00:41:08
    The burst of electricity creates a momentary magnetic field,
  • 00:41:12
    which in turn generates a momentary electric field,
  • 00:41:16
    which creates another magnetic field.
  • 00:41:19
    The energy cycles between the two,
  • 00:41:22
    propagating an electromagnetic wave.
  • 00:41:25
    (buzzing)
  • 00:41:28
    Marconi gets his system to work inside,
  • 00:41:32
    but then he scales up.
  • 00:41:38
    Over a few weeks, he builds a big antenna beside the house
  • 00:41:42
    to amplify the waves coming from his spark generator.
  • 00:41:46
    Then he asks his brother and an assistant
  • 00:41:51
    to carry a receiver across the estate
  • 00:41:54
    to the far side of a nearby hill.
  • 00:41:56
    They also have a shotgun,
  • 00:41:58
    which they will fire if they manage to pick up the signal.
  • 00:42:06
    (buzzing)
  • 00:42:13
    (buzzing)
  • 00:42:16
    (gunshot)
  • 00:42:18
    And it works.
  • 00:42:20
    The signal has been detected
  • 00:42:22
    even though the receiver is now hidden behind a hill.
  • 00:42:26
    At over a mile,
  • 00:42:28
    it is the farthest transmission to date.
  • 00:42:31
    In fewer than ten years,
  • 00:42:33
    Marconi will be sending radio signals across the Atlantic.
  • 00:42:38
    In fact, when the Titanic sinks in 1912,
  • 00:42:43
    he'll be personally credited with saving many lives
  • 00:42:47
    because his onboard equipment allowed the distress signal
  • 00:42:50
    to be transmitted.
  • 00:42:54
    Thanks to the predictions of Maxwell's equations,
  • 00:42:58
    Marconi could harness a hidden part of our world,
  • 00:43:02
    ushering in the era of wireless communication.
  • 00:43:06
    (voices on radio overlapping)
  • 00:43:11
    Since Maxwell and Marconi,
  • 00:43:15
    evidence of the predictive power of mathematics has only grown,
  • 00:43:19
    especially in the world of physics.
  • 00:43:22
    100 years ago, we barely knew atoms existed.
  • 00:43:27
    It took experiments to reveal their components:
  • 00:43:30
    the electron, the proton, and the neutron.
  • 00:43:34
    But when physicists wanted to go deeper,
  • 00:43:36
    mathematics began to lead the way,
  • 00:43:39
    ultimately revealing a zoo of elementary particles,
  • 00:43:44
    discoveries that continue to this day here at CERN,
  • 00:43:49
    the European organization for nuclear research
  • 00:43:52
    in Geneva, Switzerland.
  • 00:43:54
    These days, they're most famous for their Large Hadron Collider,
  • 00:43:59
    a circular particle accelerator about 17 miles around,
  • 00:44:04
    built deep underground.
  • 00:44:10
    This $10 billion project, decades in the making,
  • 00:44:14
    had a well-publicized goal: the search
  • 00:44:17
    for one of the most fundamental building blocks of the universe.
  • 00:44:23
    A subatomic particle
  • 00:44:25
    mathematically predicted to exist nearly 50 years earlier
  • 00:44:30
    by Robert Brout and Francois Englert working in Belgium
  • 00:44:35
    and Peter Higgs in Scotland.
  • 00:44:37
    TEGMARK: Peter Higgs sat down
  • 00:44:40
    with the most advanced physics equations we had
  • 00:44:42
    and calculated and calculated
  • 00:44:44
    and made this audacious prediction:
  • 00:44:46
    if we built the most sophisticated machines
  • 00:44:48
    humans have ever built
  • 00:44:49
    and used it to smash particles together
  • 00:44:51
    near the speed of light in a certain way
  • 00:44:53
    that we would then discover a new particle.
  • 00:44:55
    You know, if this math was really accurate.
  • 00:44:58
    NARRATOR: The discovery of the Higgs particle
  • 00:45:00
    would be proof of the Higgs field,
  • 00:45:03
    a cosmic molasses that gives the stuff of our world mass--
  • 00:45:08
    what we usually experience as weight.
  • 00:45:13
    Without mass, everything would travel at the speed of light
  • 00:45:17
    and would never combine to form atoms.
  • 00:45:20
    That makes the Higgs field
  • 00:45:22
    such a fundamental part of physics
  • 00:45:25
    that the Higgs particle gained the nickname
  • 00:45:27
    "The God Particle."
  • 00:45:31
    (cheering)
  • 00:45:33
    In 2012, experiments at CERN
  • 00:45:36
    confirmed the existence of the Higgs particle,
  • 00:45:39
    making the work of Peter Higgs
  • 00:45:41
    and his colleagues decades earlier
  • 00:45:43
    one of the greatest predictions ever made.
  • 00:45:48
    And we built it and it worked,
  • 00:45:50
    and he got a free trip to Stockholm.
  • 00:45:56
    (applause)
  • 00:46:04
    LIVIO: Here, you have mathematical theories
  • 00:46:06
    which make very definitive predictions
  • 00:46:11
    about the possible existence
  • 00:46:13
    of some fundamental particles of nature,
  • 00:46:16
    and believe it or not, they make these huge experiments
  • 00:46:20
    and they actually discover the particles
  • 00:46:23
    that have been predicted mathematically.
  • 00:46:25
    I mean, this is just amazing to me.
  • 00:46:30
    ANDREW LANKFORD: Why does this work?
  • 00:46:32
    How can mathematics be so powerful?
  • 00:46:35
    Is mathematics, you know, a truth of nature,
  • 00:46:39
    or does it have something to do
  • 00:46:41
    with the way we as humans perceive nature?
  • 00:46:44
    To me, this is just a fascinating puzzle.
  • 00:46:47
    I don't know the answer.
  • 00:46:51
    NARRATOR: In physics, mathematics has had a long string of successes.
  • 00:46:55
    But is it really "unreasonably effective"?
  • 00:46:59
    Not everyone thinks so.
  • 00:47:02
    I think it's an illusion,
  • 00:47:03
    because I think what's happened
  • 00:47:05
    is that people have chosen to build physics, for example,
  • 00:47:09
    using the mathematics that has been practiced,
  • 00:47:12
    has developed historically,
  • 00:47:13
    and then they're looking at everything,
  • 00:47:16
    they're choosing to study things which are amenable to study
  • 00:47:19
    using the mathematics that happens to have arisen.
  • 00:47:21
    But actually, there is a whole vast ocean of other things
  • 00:47:25
    that are really quite inaccessible to those methods.
  • 00:47:28
    NARRATOR: With the success of mathematical models in physics,
  • 00:47:32
    it's easy to overlook where they don't work that well.
  • 00:47:36
    Like in weather forecasting.
  • 00:47:39
    There's a reason meteorologists predict the weather
  • 00:47:41
    for the coming week,
  • 00:47:43
    but not much further out than that.
  • 00:47:45
    In a longer forecast, small errors grow into big ones.
  • 00:47:50
    Daily weather is just too complex and chaotic
  • 00:47:54
    for precise modeling.
  • 00:47:57
    And it's not alone.
  • 00:47:58
    So is the behavior of water boiling on a stove,
  • 00:48:03
    or the stock market,
  • 00:48:06
    or the interaction of neurons in the brain,
  • 00:48:10
    much of human psychology,
  • 00:48:12
    and parts of biology.
  • 00:48:14
    DEREK ABBOTT: Biological systems,
  • 00:48:16
    economic systems,
  • 00:48:18
    it gets very difficult to model those systems with math.
  • 00:48:21
    We have extreme difficulty with that.
  • 00:48:24
    So I do not see math as unreasonably effective.
  • 00:48:28
    I see it as reasonably ineffective.
  • 00:48:35
    NARRATOR: Perhaps no one is as keenly aware
  • 00:48:38
    of the power and limitations of mathematics
  • 00:48:41
    as those who use it to design and make things:
  • 00:48:44
    engineers.
  • 00:48:45
    Look at that wheel!
  • 00:48:47
    NARRATOR: In their work, the elegance of math
  • 00:48:50
    meets the messiness of reality, and practicality rules the day.
  • 00:48:57
    Mathematics and perhaps mathematicians
  • 00:48:59
    deal in the domain of the absolute,
  • 00:49:02
    and engineers live in the domain of the approximate.
  • 00:49:06
    We are fundamentally interested in the practical.
  • 00:49:11
    And so frequently, we make approximations, we cut corners.
  • 00:49:15
    We omit terms and equations
  • 00:49:16
    to get things that are simple enough
  • 00:49:19
    to suit our purposes and to meet our needs.
  • 00:49:26
    NARRATOR: Many of our greatest engineering achievements
  • 00:49:29
    were built using mathematical shortcuts:
  • 00:49:31
    simplified equations that approximate an answer,
  • 00:49:35
    trading some precision for practicality.
  • 00:49:39
    And for engineers, "approximate" is close enough.
  • 00:49:44
    Close enough to take you to Mars.
  • 00:49:49
    STELTZNER: For us engineers,
  • 00:49:50
    we don't get paid to do things right;
  • 00:49:52
    we get paid to do things just right enough.
  • 00:49:59
    NARRATOR: Many physicists see an uncanny accuracy
  • 00:50:02
    in the way mathematics can reveal
  • 00:50:03
    the secrets of the universe,
  • 00:50:06
    making it seem to be an inherent part of nature.
  • 00:50:13
    Meanwhile, engineers in practice have to sacrifice
  • 00:50:18
    the precision of mathematics to keep it useful,
  • 00:50:21
    making it seem more like an imperfect tool
  • 00:50:25
    of our own invention.
  • 00:50:28
    So which is mathematics?
  • 00:50:31
    A discovered part of the universe?
  • 00:50:34
    Or a very human invention?
  • 00:50:37
    Maybe it's both.
  • 00:50:45
    LIVIO: What I think about mathematics
  • 00:50:47
    is that it is an intricate combination
  • 00:50:50
    of inventions and discoveries.
  • 00:50:53
    So for example, take something like natural numbers:
  • 00:50:55
    one, two, three, four, five, etcetera.
  • 00:50:58
    I think what happened
  • 00:51:00
    was that people were looking at many things, for example,
  • 00:51:02
    and seeing that there are two eyes, you know,
  • 00:51:05
    two breasts, two hands, you know, and so on.
  • 00:51:08
    And after some time,
  • 00:51:10
    they abstracted from all of that the number two.
  • 00:51:15
    NARRATOR: According to Mario, "two" became an invented concept,
  • 00:51:19
    as did all the other natural numbers.
  • 00:51:23
    But then people discovered that these numbers
  • 00:51:25
    have all kinds of intricate relationships.
  • 00:51:28
    Those were discoveries.
  • 00:51:32
    We invented the concept, but then discovered
  • 00:51:36
    the relations among the different concepts.
  • 00:51:38
    NARRATOR: So is this the answer?
  • 00:51:42
    That math is both invented and discovered?
  • 00:51:46
    This is one of those questions where it's both.
  • 00:51:48
    Yes, it feels like it's already there,
  • 00:51:50
    but yes, it's something that comes out of our deep,
  • 00:51:53
    creative nature as human beings.
  • 00:51:55
    NARRATOR: We may have some idea to how all this works,
  • 00:51:59
    but not the complete answer.
  • 00:52:02
    In the end, it remains "The Great Math Mystery."
Tags
  • mathematics
  • science
  • Fibonacci sequence
  • pi
  • Galileo
  • Einstein
  • Higgs boson
  • engineering
  • nature
  • discovery