The Great Math Mystery

PBS Airdate: April 15, 2015

NARRATOR: We live in an age of astonishing advances: engineers can land a car-sized rover on Mars, physicists probe the essence of all matter, while we communicate wirelessly on a vast worldwide network. But underlying all of these modern wonders is something deep and mysteriously powerful.

It's been called the language of the universe, and perhaps it's civilization's greatest achievement. Its name? Mathematics.

But where does math come from? And why, in science, does it work so well?

MARIO LIVIO (Author, Is God a Mathematician?): Albert Einstein wondered, “How is it possible that mathematics does so well in explaining the universe as we see it?”

NARRATOR: Is mathematics even human?

ELIZABETH BRANNON (Duke University, Center for Cognitive Neuroscience): There doesn't really seem to be an upper limit to the numerical abilities of animals.

NARRATOR: And is it the key to the cosmos?

MAX TEGMARK (Massachusetts Institute of Technology/Author, Our Mathematical Universe): Our physical world doesn't just have some mathematical properties; it has only mathematical properties.

NARRATOR: The Great Math Mystery, next, on NOVA!

Human beings have always looked at nature and searched for patterns. Eons ago we gazed at the stars and discovered patterns we call constellations, even coming to believe they might control our destiny.

We've watched the days turn to night and back to day, and seasons, as they come and go, and called that pattern “time.” We see symmetrical patterns in the human body and the tiger's stripes, and build those patterns into what we create, from art to our cities.

But what do patterns tell us? Why should the spiral shape of the nautilus shell be so similar to the spiral of a galaxy or the spiral found in a sliced open head of cabbage?

When scientists seek to understand the patterns of our world, they often turn to a powerful tool: mathematics. They quantify their observations and use mathematical techniques to examine them, hoping to discover the underlying causes of nature's rhythms and regularities.

And it's worked, revealing the secrets behind the elliptical orbits of the planets to the electromagnetic waves that connect our cell phones. Mathematics has even guided the way, leading us right down to the sub-atomic building blocks of matter, which raises the question, “Why does it work at all? Is there an inherent mathematical nature to reality? Or is mathematics all in our heads?”

Mario Livio is an astrophysicist who wrestles with these questions. He's fascinated by the deep and often mysterious connection between mathematics and the world.

MARIO LIVIO: If you look at nature, there are numbers all around us. You know, look at flowers, for example. So there are many flowers that have three petals like this or five like this. Some of them may have 34 or 55. These numbers occur very often.

NARRATOR: These may sound like random numbers, but they're all part of what is known as the “Fibonacci sequence,” a series of numbers developed by a 13th century mathematician.

MARIO LIVIO: You start with the numbers one and one, and from that point on you keep adding up the last two numbers. So one plus one is two, now one plus two is three, two plus three is five, three plus five is eight, and you keep going like this.

NARRATOR: Today, hundreds of years later, this seemingly arbitrary progression of numbers fascinates many who see in it clues to everything from human beauty to the stock market. While most of those claims remain unproven, it is curious how evolution seems to favor these numbers.

MARIO LIVIO: And as it turns out, this sequence appears quite frequently in nature.

NARRATOR: Fibonacci numbers show up in petal counts, especially of daisies, but that's just a start.

CHRISTOPHE GOLÉ (Smith College): Statistically, the Fibonacci numbers do appear a lot in botany. For instance, if you look at the bottom of a pine cone, you will see, often, spirals in their scales. You end up counting those spirals, you'll usually find a Fibonacci number, and then you will count the spirals going in the other direction and you will find an adjacent Fibonacci number.

NARRATOR: The same is true of the seeds on a sunflower head: two sets of spirals, and if you count the spirals in each direction, both are Fibonacci numbers.

While there are some theories explaining the Fibonacci botany connection, it still raises some intriguing questions.

CHRISTOPHE GOLÉ (Smith College): So do plants know math? The short answer to that is, “No.” They don't need to know math. In a very simple, geometric way, they set up a little machine that creates the Fibonacci sequence in many cases.

NARRATOR: The mysterious connections between the physical world and mathematics run deep. We all know the number Pi from geometry—the ratio between the circumference of a circle and its diameter—and that its decimal digits go on forever without a repeating pattern. As of 2013 it had been calculated out to 12.1 trillion digits, but somehow Pi is a whole lot more.

MARIO LIVIO: Pi appears in a whole host of other phenomena, which have, at least on the face of it, nothing to do with circles or anything. In particular, it appears in probability theory quite a bit.

Suppose I take this needle and… So, the length of the needle is equal to the distance between two lines on this piece of paper. And suppose I drop this needle, now, on the paper.

NARRATOR: Sometimes, when you drop the needle, it will cut a line, and sometimes it drops between the lines. It turns out, the probability that the needle lands so it cuts a line is exactly two over Pi, or about 64 percent.

MARIO LIVIO: Now, what that means is that, in principle, I could drop this needle millions of times. I could count the times when it crosses a line and when it doesn't cross a line, and I could actually even calculate Pi. Even though there are no circles here, no diameters of a circle, nothing like that. It's really amazing.

NARRATOR: Since Pi relates a round object—a circle—with a straight one—its diameter—it can show up in the strangest of places. Some see it in the meandering path of rivers. A river's actual length, as it winds its way from its source to its mouth, compared to the direct distance, on average seems to be about Pi.

Models for just about anything involving waves will have Pi in them. Like those for light and sound: Pi tells us which colors should appear in a rainbow and how middle C should sound on a piano.

Pi shows up in apples, in the way cells grow into spherical shapes, or in the brightness of a supernova.

One writer has suggested it's like seeing Pi on a series of mountain peaks, poking out of a fog-shrouded valley. We know there's a way they're all connected, but it's not always obvious how.

Pi is but one example of a vast interconnected web of mathematics that seems to reveal an often hidden and deep order to our world.

Physicist Max Tegmark from M.I.T. thinks he knows why. He sees similarities between our world and that of a computer game.

MAX TEGMARK: If I were a character in a computer game that were so advanced that I were actually conscious, and I started exploring my video game world, it would actually feel to me like it was made of real solid objects made of physical stuff.

Yet, if I started studying as the curious physicist that I am, the properties of this stuff, the equations by which things move and the equations that gives stuff its properties, I would discover, eventually, that all these properties were mathematical: the mathematical properties that the programmer had actually put into the software that describes everything.

NARRATOR: The laws of physics in a game, like how an object floats, bounces or crashes, are only mathematical rules created by a programmer. Ultimately the entire “universe” of a computer game is just numbers and equations.

MAX TEGMARK: That's exactly what I perceive in this reality, too, as a physicist, that the closer I look at things that seem non-mathematical, like my arm, here, and my hand, the more mathematical it turns out to be. Could it be that our world also then is really just as mathematical as the computer game reality?

NARRATOR: To Max, the software world of a game isn't that different from the physical world we live in. He thinks that mathematics works so well to describe reality because, ultimately, mathematics is all that it is. There's nothing else.

MAX TEGMARK: Many of my physics colleagues will say that mathematics describes our physical reality, at least in some approximate sense. I go further and argue that it actually is our physical reality, because I'm arguing that our physical world doesn't just have some mathematical properties, but it has only mathematical properties.

NARRATOR: Our physical reality is a bit like a digital photograph, according to Max. The photo looks like the pond but as we move in closer, and closer, we can see it is really a field of pixels each represented by three numbers that specify the amount of red, green and blue.

While the universe is vast in its size and complexity, requiring an unbelievably large collection of numbers to describe it, Max sees its underlying mathematical structure as surprisingly simple.

It's just 32 numbers, constants, like the masses of elementary particles, along with a handful of mathematical equations, the fundamental laws of physics. And it all fits on a wall, though there are still some questions.

MAX TEGMARK: But even though we don't know what exactly is going to go here, I am really confident that what will go here will be mathematical equations, that everything is ultimately mathematical.

NARRATOR: Max Tegmark's Matrix-like view—that mathematics doesn't just describe reality but is its essence—may sound radical, but it has deep roots in history, going back to ancient Greece, to the time of the philosopher and mystic Pythagoras.

Stories say he explored the affinity between mathematics and music, a relationship that resonates, to this day, in the work of Esperanza Spalding, an acclaimed jazz musician, who's studied music theory and sees its parallel in mathematics.

ESPERANZA SPALDING (Jazz Musician): I love the experience of math, but the part that I enjoy about math, I get to experience through music, too. At the beginning, you're studying all the little equations, but you get to have this very visceral relationship with the product of those equations, which is sound and music and harmony and dissonance and all that good stuff. So, I'm much better at music than at math, but I love math with a passion. They're both just as much work. They're both you have to study your … off, your head off, study your head off.

NARRATOR: The Ancient Greeks found three relationships between notes especially pleasing. Now we call them an octave, a fifth, and a fourth.

ESPERANZA SPALDING: An octave is easy to remember because it's the first two notes of “Somewhere Over the Rainbow.” La, la. That's an octave. “Somewhere.”

A fifth sounds like this: la, la. Or the first two notes of “Twinkle, Twinkle, Little Star.”

And a fourth sounds like… You can think of it as the first two notes of “Here Comes the Bride.”

NARRATOR: In the sixth century B.C.E., the Greek philosopher Pythagoras is said to have discovered that those beautiful musical relationships were also beautiful mathematical relationships by measuring the lengths of the vibrating strings.

In an octave, the string lengths create a ratio of two to one. In a fifth, the ratio is three to two. And in a fourth, it is four to three.

ESPERANZA SPALDING: Seeing a common pattern throughout sound, that could be a big eye-opener of saying, “Well, if this exists in sound, and if it's true universally through all sounds, this ratio could exist universally everywhere, right? And doesn't it?”

NARRATOR: Pythagoreans worshipped the idea of numbers. The fact that simple ratios produced harmonious sounds was proof of a hidden order in the natural world. And that order was made of numbers,…

…a profound insight that mathematicians and scientists continue to explore to this day.

In fact, there are plenty of other physical phenomena that follow simple ratios, from the two-to-one ratio of hydrogen atoms to oxygen atoms in water, to the number of times the moon orbits the earth, compared to its own rotation, one-to-one, or that Mercury rotates exactly three times when it orbits the sun twice, a three-to-two ratio.

In ancient Greece, Pythagoras and his followers had a profound effect on another Greek philosopher, Plato, whose ideas also resonate to this day, especially among mathematicians.

Plato believed that geometry and mathematics exist in their own ideal world.

MARIO LIVIO: So when we draw a circle on a piece of paper, this is not the real circle. The real circle is in that world, and this is just an approximation of that real circle, and the same with all other shapes. And Plato liked very much these five solids, the platonic solids we call them today. And he assigned each one of them to one of the elements that formed the world as he saw it.

NARRATOR: The stable cube was Earth; the tetrahedron, with its pointy corners was fire; the mobile-looking octahedron, Plato thought of as air; and the twenty-sided icosahedron was water; and finally, the dodecahedron, this was the thing that signified the cosmos as a whole.

So Plato's mathematical forms were the ideal version of the world around us, and they existed in their own realm. And however bizarre that may sound, that mathematics exists in its own world, shaping the world we see. It's an idea that, to this day, many mathematicians and scientists can relate to, the sense they have when they're doing math that they're just uncovering something that's already out there.

MARIA CHUDNOVSKY (Columbia University): I feel quite strongly that mathematics is discovered in my work as a mathematician. It always feels to me there is a thing out there, and I'm kind of trying to find it and understand it and touch it.

S. JAMES “JIM” GATES, JR. (University of Maryland): As someone who actually has had the pleasure of making new mathematics, it feels like there's something there before you get to it.

DUSA McDUFF (Barnard College/Columbia University): If I have to choose, I think it's more discovered than invented, because I…there is a reality to what we study in mathematics. When we do good mathematics, we're discovering something about the way our minds work in interaction with the world.

MARIA CHUDNOVSKY: Well, I know that, because that's what I do. I come to my office, I sit down in front of my whiteboard, and I try and understand that thing that's, that's out there. And every now and then, I'm discovering a new bit of it. That's exactly what it feels like.

NARRATOR: To many mathematicians, it feels like math is discovered rather than invented. But is that just a feeling? Could it be that mathematics is purely a product of the human brain?

Meet Shyam, a bona fide math whiz.

MICHAEL O'BOYLE (Texas Tech University): 800 on the S.A.T. Math.

That's pretty good. And you took it when you were how old?

SHYAM NARAYANAN: Eleven.

MICHAEL O'BOYLE: Eleven! Wow, that's, like, a perfect score.

NARRATOR: Where does Shyam's math genius come from?

It turns out, we can pinpoint it, and it's all in his head.

Using f.M.R.I., scientists can scan Shyam's brain as he answers math questions, to see which parts of the brain receive more blood, a sign they are hard at work.

F.M.R.I. OPERATOR: All right, Shyam, we'll start about now. Okay, buddy?

NARRATOR: In images of Shyam's brain, the parietal lobes glow an especially bright crimson.

MICHAEL O'BOYLE: He is relying on parietal areas to determine these mathematical relationships. That's characteristic of lots of math-gifted types.

NARRATOR: In tests similar to Shyam's, kids who exhibit high math performance have five to six times more neuron activation than average kids in these brain regions.

But is that the result of teaching, and intense practice, or are the foundations of math built into our brains?

Scientists are looking for the answer here, at the Duke University Lemur Center, a 70-acre sanctuary in North Carolina, the largest one for rare and endangered lemurs, in the world.

Like all primates, lemurs are related to humans through a common ancestor that lived as many as 65-million years ago. Scientists believe lemurs share many characteristics with those earliest primates, making them a window, though a blurry one, into our ancient past.

LIZ BRANNON: Got a choice here, Teres. Come on up.

NARRATOR: Duke professor Liz Brannon investigates how well lemurs, like Teres here, can compare quantities.

LIZ BRANNON: Many different animals choose larger food quantities, so what is Teres doing? What are all of these different animals doing when they compare two quantities?

Well, clearly he's not using verbal labels, he's not using symbols. We need to figure out whether they can really use number, pure number, as a cue.

NARRATOR: To test how well Teres can distinguish quantities, he's been taught a touch screen computer game. The red square starts a round. If he touches it, two squares appear, containing different numbers of objects. He's been trained that if he chooses the box with the fewest number, he'll get a reward, a sugar pellet.

A wrong answer? [Buzzer sounds.]

LIZ BRANNON: We have to do a lot to ensure that they're really attending to number and not something else.

NARRATOR: To make sure the test animal is reacting to the number of objects and not some other cue, Liz varies the objects' size, color and shape.

She has conducted thousands of trials and shown that lemurs and rhesus monkeys can learn to pick the right answer.

LIZ BRANNON: Teres obviously doesn't have language, and he doesn't have any symbols for number. So, is he counting, is he doing what a human child does when they recite the numbers, one, two, three? No. And yet, what he seems to be attending to is the very abstract essence of what a number is.

NARRATOR: Lemurs and rhesus monkeys aren't alone in having this primitive number sense. Rats, pigeons, fish, raccoons, insects, horses and elephants all show sensitivity to quantity.

And so do human infants. At her lab on the Duke campus, Liz has tested babies that were only six months old. They'll look longer at a screen with a changing number of objects, as long as the change is obvious enough to capture their attention.

Liz has also tested college students, asking them not to count but to respond as quickly as they could to a touch screen test comparing quantities. The results? About the same as lemurs and rhesus monkeys.

LIZ BRANNON: In fact, there are humans who aren't as good as our monkeys, and others that are far better, so there's a lot of variability in human performance, but, in general, it looks very similar to a monkey.

Even without any mathematical education, even without learning any number words or symbols, we would still have, all of us, as humans, a primitive number sense.

That fundamental ability to perceive number seems to be a very important foundation, and without it, it's very questionable as to whether we could ever appreciate symbolic mathematics.

NARRATOR: The building blocks of mathematics may be preprogrammed into our brains, part of the basic toolkit for survival, like our ability to recognize patterns and shapes or our sense of time. From that point of view, on this foundation, we've erected one of the greatest inventions of human culture: mathematics. But the mystery remains. If it is “all in our heads,” why has math been so effective?

Through science, technology and engineering, it's transformed the planet, even allowing us to go into the beyond. As in the work here, at NASA's Jet Propulsion Laboratory, in Pasadena, California.

MISSION CONTROL 1: Roger, copy mission.

MISSION CONTROL 2: Coming up on entry.

NARRATOR: In 2012, they landed a car-sized rover,…

MISSION CONTROL 2: Descending at about point-75 meters per second, as expected.

NARRATOR: …on Mars.

MISSION CONTROL 2: Touchdown confirmed. We're safe on Mars.

NARRATOR: Adam Steltzner was the lead engineer on the team that designed the landing system. Their work depended on a groundbreaking discovery, from the Renaissance, that turned mathematics into the language of science: the law of falling bodies.

The ancient Greek philosopher Aristotle taught that heavier objects fall faster than lighter ones. An idea that, on the surface, makes sense, even this surface: the Mars yard, where they test the rovers at J.P.L.

ADAM STELTZNER (Jet Propulsion Laboratory): So Aristotle reasoned that the rate at which things would fall was proportional to their weight, which seems reasonable.

NARRATOR: In fact, so reasonable, the view held for nearly 2,000 years, until challenged in the late 1500s by Italian mathematician, Galileo Galilei.

ADAM STELTZNER: Legend has it that Galileo dropped two different weight cannon balls from the leaning tower of Pisa.

Well, we're not in Pisa, we don't have cannon balls, but we do have a bowling ball and a bouncy ball. Let's weigh them.

First, we weigh the bowling ball. It weighs 15 pounds. And the bouncy ball, it weighs hardly anything.

Let's drop them.

NARRATOR: According to Aristotle, the bowling ball should fall over 15 times faster than the bouncy ball.

ADAM STELTZNER: Well, they seem to fall at the same rate. This isn't that high though. Maybe we should drop them from higher.

So, Ed is 20 feet in the air up there. Let's see if the balls fall at the same rate.

Ready? Three, two, one, drop!

Galileo was right. Aristotle, you lose.

NARRATOR: Dropping feathers and hammers is misleading, thanks to air resistance.

DAVID R. SCOTT (NASA Footage): Well, in my left hand I have a feather, in my right hand, a hammer.

NARRATOR: A fact demonstrated on the Moon, where there is no air, in 1971, during the Apollo 15 mission.

DAVID R. SCOTT (NASA Footage): And I'll drop the two of them here.

How about that? Mr. Galileo was correct.

NARRATOR: So, while counterintuitive, if you take the air out of the equation, everything falls at the same rate, even Aristotle.

But what really interested Galileo was that an object dropped at one height didn't take twice as long to drop from twice as high. It accelerated.

But how do you measure that? Everything is happening so fast.

Galileo came up with an ingenious solution. He built a ramp, an inclined plane, to slow the falling motion down so he could measure it.

ADAM STELTZNER: So we're going to use this ramp to find the relationship between distance and time.

For time, I'll use an arbitrary unit, a Galileo, one Galileo.

NARRATOR: The length of the ramp that the ball rolls during one Galileo becomes one unit of distance.

ADAM STELTZNER: So, we've gone one unit of distance in one unit of time. Now, let's try it for a two-count. One Galileo, two Galileo.

NARRATOR: In two units of time, the ball has rolled four units of distance.

ADAM STELTZNER: Now let's see how far it goes in three Galileos. One Galileo, two Galileo, three Galileo.

NARRATOR: In three units of time, the ball has gone nine units of distance.

ADAM STELTZNER: So, there it is. There's a mathematical relationship here, between time and distance.

NARRATOR: Galileo's inspired use of a ramp had shown falling objects follow mathematical laws. The distance the ball traveled is directly proportional to the square of the time.

ADAM STELTZNER: That mathematical relationship that Galileo observed is a mathematical expression of the physics of our universe.

NARRATOR: Galileo's centuries-old mathematical observation about falling objects remains just as valid today.

ADAM STELTZNER: It's the same mathematical expression that we can use to understand how things might fall here on Earth, roll down a ramp. It's even a relationship that we used to land the Curiosity rover on the surface of Mars.

That's the power of mathematics.

NARRATOR: Galileo's insight was profound. Mathematics could be used as a tool to uncover and discover the hidden rules of our world. He later wrote, “The universe is written in the language of mathematics.”

JANNA LEVIN (Barnard College/Columbia University): Math is really the language in which we understand the universe. We don't know why it's the case that the laws of physics and the universe follows mathematical models, but it does seem to be the case.

NARRATOR: While Galileo turned mathematical equations into laws of science, it was another man, born the same year Galileo died, who took that to new heights that crossed the heavens.

His name was Isaac Newton. He worked here, at Trinity College, in Cambridge, England.

SIMON SCHAFFER (University of Cambridge): Newton cultivated the reputation of being a solitary genius, and here, in the bowling green of Trinity College, it was said that he would walk meditatively up and down the paths, absentmindedly drawing mathematical diagrams in the gravel. And the fellows were instructed, or so it was said, not to disturb him, not to clear up the gravel after he'd passed, in case they inadvertently wiped out some major scientific or mathematical discovery.

NARRATOR: In 1687, Newton published a book that would become a landmark in the history of science. Today it is known simply as The Principia. In it, Newton gathered observations from around the world, and used mathematics to explain them, for instance, that of a comet seen widely in the fall of 1680.

SIMON SCHAFFER: He gathers data worldwide in order to construct the comet's path. So for November the 19th, he begins with an observation made in Cambridge, in England, at 4:30 a.m., by a certain young person, and then, at 5:00 in the morning, at Boston, in New England. So what Newton does is to accumulate numbers made by observers spread right across the globe, in order to construct an unprecedentedly accurate calculation of how this great comet moved through the sky.

NARRATOR: Newton's groundbreaking insight was that the force that sent the comet hurtling around the sun was the same force that brought cannon balls back to Earth. It was the force behind Galileo's law of falling bodies, and it even held the planets in their orbits.

He called the force “gravity,” and described it precisely, in a surprisingly simple equation that explains how two masses attract each other, whether here on Earth or in the heavens above.

SIMON SCHAFFER: What's so impressive and so dramatic is that a single mathematical law would allow you to move throughout the universe.

NARRATOR: Today, we can even witness it at work beyond the Milky Way.

JENNIFER WISEMAN (NASA): This is a picture of two galaxies that are actually being drawn together in a merger.

MARIO LIVIO: This is how galaxies build themselves.

JENNIFER WISEMAN: Right.

NARRATOR: Mario Livio is on the team working with the images from the Hubble Space Telescope.

For decades, scientists have used Hubble to gaze far beyond our solar system, even beyond the stars of our galaxy. It's shown us the distant gas clouds of nebulae and vast numbers of galaxies wheeling in the heavens, billions of light years away. And what those images show is that throughout the visible universe, as far as the telescope can see, the law of gravity still applies.

MARIO LIVIO: You know, Newton wrote these laws of gravity and of motion based on things happening on Earth and the planets in the solar system and so on, but these same laws, these very same laws apply to all these distant galaxies, and, you know, shape them. And everything about them, how they form, how they move, is controlled by those same mathematical laws.

NARRATOR: Some of the world's greatest minds have been amazed by the way that math permeates the universe.

MARIO LIVIO: Albert Einstein, he wondered. He said, “How is it possible that mathematics,” which is, he thought, a product of human thought, “does so well in explaining the universe as we see it?” And Nobel laureate in physics, Eugene Wigner, coined this phrase, “The unreasonable effectiveness of mathematics.” He said the fact that mathematics can really describe the universe so well, in particular physical laws, is a gift that we neither understand nor deserve.

NARRATOR: In physics, examples of that “unreasonable effectiveness” abound. When, nearly 200 years ago, the planet Uranus was seen to go off track, scientists trusted the math and calculated it was being pulled by another unseen planet, and so, they discovered Neptune.

Mathematics had accurately predicted a previously unknown planet.

SAVAS DIMOPOULOS (Stanford University): If you formulate a question properly, mathematics gives you the answer.

It's like having a servant that is far more capable than you are. So you tell it, “Do this,” and if you say it nicely, then it'll do it. And it will carry you all the way to the truth, to the final answer.

NARRATOR: Evidence of the amazing predictive power of mathematics can be found all around us: television, radio, your cell phone, satellites, the baby monitor, Wi-Fi, your garage door opener, G.P.S., and yes, even, maybe, your T.V.'s remote, all of these use invisible waves of energy to communicate.

And no one even knew they existed until the work of James Maxwell, a Scottish mathematical physicist.

In the 1860s, he published a set of equations that explained how electricity and magnetism were related, how each could generate the other. The equations also made a startling prediction: together, electricity and magnetism could produce waves of energy that would travel through space at the speed of light: electro-magnetic waves.

ROGER PENROSE (University of Oxford): Maxwell's theory gave us these radio waves, x-rays, these things which were simply not known about at all. So, the theory had a scope, which was extraordinary.

NARRATOR: Almost immediately people set out to find the waves predicted by Maxwell's equations. What must have seemed the least promising attempt to harness them is made here, in northern Italy, in the attic of a family home, by 20-year-old Guglielmo Marconi.

His process starts with a series of sparks. The burst of electricity creates a momentary magnetic field, which, in turn, generates a momentary electric field, which creates another magnetic field. The energy cycles between the two, propagating an electro-magnetic wave.

Marconi gets his system to work inside, but then he scales up. Over a few weeks, he builds a big antenna beside the house to amplify the waves coming from his spark generator. Then he asks his brother and an assistant to carry a receiver across the estate to the far side of a nearby hill. They also have a shotgun, which they will fire if they manage to pick up the signal.

And it works.

The signal has been detected even though the receiver is now hidden behind a hill. At over a mile, it is the farthest transmission to date.

In fewer than 10 years, Marconi will be sending radio signals across the Atlantic. In fact, when the Titanic sinks, in 1912, he'll be personally credited with saving many lives, because his onboard equipment allowed the distress signal to be transmitted.

Thanks to the predictions of Maxwell's equations, Marconi could harness a hidden part of our world, ushering in the era of wireless communication.

Since Maxwell and Marconi, evidence of the predictive power of mathematics has only grown, especially in the world of physics.

A hundred years ago, we barely knew atoms existed. It took experiments to reveal their components: the electron, the proton and the neutron. But when physicists wanted to go deeper, mathematics began to lead the way, ultimately revealing a zoo of elementary particles, discoveries that continue, to this day, here at CERN, the European organization for nuclear research, in Geneva, Switzerland.

These days, they're most famous for their Large Hadron Collider, a circular particle accelerator about 17-miles-around, built deep underground.

This 10-billion-dollar project, decades in the making, had a well-publicized goal, the search for one of the most fundamental building blocks of the universe: a subatomic particle, mathematically predicted to exist nearly 50 years earlier by Robert Brout and François Englert, working in Belgium, and Peter Higgs in Scotland.

MAX TEGMARK: Peter Higgs sat down with the most advanced physics equations we had and calculated and calculated, and made this audacious prediction, and if we built the most sophisticated machine humans have ever built, and used it to smash particles together near the speed of light in a certain way, that we would then discover a new particle, if this math was really accurate.

NARRATOR: The discovery of the Higgs particle, would be proof of the Higgs field, a cosmic molasses that gives the stuff of our world mass, what we usually experience as weight.

Without mass, everything would travel at the speed of light and would never combine to form atoms. That makes the Higgs field such a fundamental part of physics that the Higgs particle gained the nickname the “God particle.”

In 2012, experiments at CERN confirmed the existence of the Higgs particle, making the work of Peter Higgs and his colleagues, decades earlier, one of the greatest predictions ever made.

MAX TEGMARK: And we built it, and it worked. And he got a free trip to Stockholm.

Here, you have mathematical theories, which make very definitive predictions about the possible existence of some fundamental particles of nature, and, believe it or not, they make these huge experiments and they actually discover the particles that have been predicted mathematically. I mean this is just amazing to me.

ANDREW LANKFORD (University of California, Irvine): Why does this work? How can mathematics be so powerful? Is mathematics, you know, a truth of nature, or does it have something to do with the way that we, as humans, perceive nature? To me this is just a fascinating puzzle. I don't know the answer.

NARRATOR: In physics, mathematics has had a long string of successes.

But is it really “unreasonably effective?” Not everyone thinks so.

STEPHEN WOLFRAM (Chief Executive Officer, Wolfram Research): I think it's an illusion. Because I think what has happened is that people have chosen to build physics, for example, using the mathematics that has been practiced, has developed historically, and then they're looking at everything, they are choosing to study things amenable to study using the mathematics that happens to have arisen. But actually there is a whole vast ocean of other things that are really quite inaccessible to those methods.

NARRATOR: With the success of mathematical models in physics, it's easy to overlook where they don't work that well, like in weather forecasting.

There's a reason meteorologists predict the weather for the coming week but not much further out than that. In a longer forecast, small errors grow into big ones. Daily weather is just too complex and chaotic for precise modeling. And it's not alone. So is the behavior of water boiling on a stove, or the stock market, or the interaction of neurons in the brain, much of human psychology and parts of biology.

DEREK ABBOTT (The University of Adelaide): Biological systems, economic systems, it gets very difficult to model those systems with math. We have extreme difficulty with that. So I do not see math as “unreasonably effective.” I see it as reasonably ineffective.

NARRATOR: Perhaps no one is as keenly aware of the power and limitations of mathematics, as those who use it to design and make things: engineers. In their work, the elegance of math meets the messiness of reality, and practicality rules the day.

ADAM STELTZNER: Mathematics, and perhaps mathematicians, deal in the domain of the absolute, and engineers live in the domain of the approximate. We are fundamentally interested in the practical. And so, frequently, we make approximations, we cut corners. We omit terms and equations to get things that are simple enough to suit our purposes and to meet our needs.

NARRATOR: Many of our greatest engineering achievements were built using mathematical shortcuts, simplified equations that approximate an answer, trading some precision for practicality.

And for engineers “approximate” is close enough, close enough to take you to Mars.

ADAM STELTZNER: For us engineers, we don't get paid to do things right, we get paid to do things just right enough.

NARRATOR: Many physicists see an uncanny accuracy in the way mathematics can reveal the secrets of the universe, making it seem to be an inherent part of nature.

Meanwhile, engineers, in practice, have to sacrifice the precision of mathematics to keep it useful, making it seem more like an imperfect tool of our own invention.

So which is mathematics? A discovered part of the universe or a very human invention?

Maybe it's both.

MARIO LIVIO: What I think about mathematics is that it's an intricate combination of inventions and discoveries. So, for example, take something like the natural numbers, one, two, three, four, five, etc., I think what happened was that people were looking at many things, for example, and seeing that there are two eyes, you know, two breasts, two hands, you know, and so on. And after some time they abstracted from all of that the number two.

NARRATOR: According to Mario, “two” became an invented concept, as did all the other natural numbers, but then people discovered that these numbers have all kinds of intricate relationships. Those were discoveries.

MARIO LIVIO: We invented the concept but then discovered the relations among the different concepts.

NARRATOR: So is this the answer? That math is both invented and discovered?

JIM GATES: This is one of those questions where it's both. Yes, it feels like it's already there, but yes it's something that comes out of our deep creative nature as human beings.

NARRATOR: We may have some idea to how all this works, but not the complete answer. In the end, it remains The Great Math Mystery!