eons ago we gazed at the Stars and discovered patterns we call constellations even coming to believe they might control our destiny we watch the day's turn tonight and back today and seasons as they come and go and call that pattern time we see symmetrical patterns in the human body and the tiger 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 cellphones mathematics has even guided the way leading us right down to the subatomic 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 [Music] mario livio is an astrophysicist who wrestles with these questions he's fascinated by the deep and often mysterious connection between mathematics in the world 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 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 thirteenth century mathematician you start with the numbers 1 and 1 and from that point on you keep adding up the last two numbers so 1 plus 1 is 2 now 1 plus 2 is 3 2 plus 3 is 5 3 + 5 is 8 and you keep going like this trust me 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 and as he turns out I mean the sequence appears quite frequently in nature tribune achi numbers show up in petal counts especially of daisies but that's just a start the typically the Fibonacci numbers do appear a lot in botany for instance if you look at the bottom of a pinecone you will see often spiral in their scale 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 if you go in actually 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 well there are some theories explaining the Fibonacci botany connection it still raises some intriguing questions 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 the mysterious connections between the physical world and mathematics run P we all know the number pi from geometry the ratio between the circumference of a circle and its diameter and that it's decimal digits go on forever without a repeating pattern as of 2013 it had been calculated out to twelve point one trillion digits but somehow pi is a whole lot 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 so that 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 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 ends so it cuts a line is exactly two over pi or about 64% now what that means is that in principle I could drop this little millions of time I could count the times when it crosses a line and when it doesn't cross a line and I could actually even calculate 5 even though there are no circles here no diameters of a circle nothing like that it's really amazing since pie 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 a meandering path of rivers a rivers actual length as it winds its way from its source to its mouth compared to the direct distance on average seems to be about pie models for just about anything involving waves will have pie in them like those four lights and sound hi tells us which color 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 pie on a series of mountain peaks poking out of a fog shrouded valley we know there is a way they're all connected but it's not always obvious how pie is but one example of a vast interconnected web of mathematics that seems to reveal an often hidden and deep order to our world [Music] physicist max tegmark from MIT thinks he knows why he sees similarities between our world and that of a computer game [Music] if I were a character in a computer game there were so advanced that I were actually conscious and I started exploring my video game world who would actually feel to me like it was made of real solid objects made of physical stuff yet if I started studying at the curious visitors than I am the properties are discussed the equations by which things move and the equations would give the metallurgy 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 the laws of physics in a game like how an object floats bounces or crashes our only mathematical rules created by a programmer ultimately the entire universe of a computer game is just numbers and equations that's exactly what I perceive in this reality - is a physicist that the closer I look at things that seem non-mathematical like my arm here in 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 to match 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 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 has only mathematical properties 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 [Music] while the universe is vast in its size and complexity requiring an unbelievably large collection of numbers to describe maxi's 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 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 max tegmark 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 studied music theory and sees its parallel in mathematics I love the experience of math 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 relationships with the product of those equations which is sound and music and harmony and distance 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 okay both you have to study it off your head off study your head off ha ha ha the ancient Greeks found three relationships between notes especially pleasing now we call them an octave a fifth and a fourth an octave is easy to remember because it's the first two notes are somewhere over the rainbow wah-wah that's an octave somewhere a fifth sounds like this la la or the first one of those twinkle twinkle little star and a fourth sounds like la la you can think of it as the first two notes of here comes the bride in the sixth century BCE 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 2 to 1 in 1/5 the ratio is 3 to 2 and in 1/4 it is 4 to 3 being a common pattern throughout sound that could be a big eye-opener saying well if this existence sound and if it's true universally through all sounds this ratio could exist universally everywhere right and doesn't it pythagoreans worship the idea of numbers the fact that simple ratios produce 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 continued 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 watch 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 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 in the same with all other shapes and Plato liked very much these 5 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 the stable cube was earth the tetrahedron with its pointy corners was fired the mobile looking octahedron Plato thought of as hair and the 20-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 his 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 I feel quite strong that mathematics is discovered in my work as a mathematician those feels to me there is a thing out there and I'm kind of trying to find it and understand it and touch it but someone who actually has had the pleasure of making new mathematics it feels like there's something there before you get to it if I have to choose I think it's more discovered and invented because I think there's a reality to what we study in mathematics when we do good mathematics we are discovering something about the way our minds work and interaction with the world 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 out there and every now and then I'm discovering a new bit of it that's exactly what it feels like 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 meat shop a bona fide math whiz 800 on the SAT math that's pretty good and you took it when you are how old 11 11 wow that's like a perfect score where does Sean's math genius come from it turns out we can pinpoint it and it's all in his head using fMRI scientists can scan Sean's brain as he answers math questions to see which parts of the brain receive more blood a sign they are hard at work alright Sean we'll start right now everybody in images of Sean's brain the parietal lobes glow an especially bright crimson he is relying on parietal areas to determine these mathematical relationships that's characteristic of lots of math gifted types or in tests similar to Shawn's kids who exhibit high math performance at 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 [Music] like all primates lemurs are related to humans through a common ancestor that lived as many as 65 million years ago [Music] scientists believe lemurs share many characteristics with those earliest primates making them a window though a blurry one into our ancient past got a choice here Terry come on up Duke professor Liz Brannon investigates how well lemurs like parries here can compare quantities many different animals choose larger food quantities so what is Terry is 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 to test how well Terry's can distinguish quantities he's been taught a touchscreen computer game the red square starts around 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 our wrong answer [Music] we have to do a lot to ensure that they're really attending to number and not something else to make sure the test animal is reacting to the number of objects and not some other cute Liz varies the object size color and shape she has conducted thousands of trials and shown that lemurs and rhesus monkeys can learn to pick the right answer Tereus 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 1 2 3 no and yet what he's going to be attending to is the very abstract essence of what a number is 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 6 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 touchscreen test comparing quantities the results about the same as lemurs and rhesus monkeys 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 substituting the 3 what to the 4 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 involves mathematics the building blocks of mathematics may be pre-programmed 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 Roger copy mission country in 2012 Scott Grady started they landed a car-sized Rover sending out about 0.75 meters per second that expected on Mars touchdown confirmed Adam steltzner was the lead engineer on the team that designed the landing system they're worth 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 make sense even this service the Mars yard where they test a Rovers at JPL so Aristotle reason that the rate at which things would fall was portion to their weight which seems reasonable in fact so reasonable the view held for nearly two thousand years until challenged in the late 1500s by Italian mathematician Galileo Galilei legend has it that Galileo dropped two different weight cannonballs from the Leaning Tower of Pisa well we're not in Pisa we don't have cannonballs 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 according to Aristotle the bowling ball should fall over 15 times faster than the bouncy ball well they seem to fall the same rate this isn't that high though maybe we should drop them from higher incomes and based in favor so II D is 20 feet in the air up there let's see if the balls fall at the same rate ready three two one drop [Music] Galileo was right Aristotle you lose dropping feathers and hammers is misleading thanks to air resistance my left hand I have a feather my right hand a hammer a fact demonstrated on the moon where there is no air in 1971 during the Apollo 15 mission you know drop - I'm here how about that after Calvin Louis track little ball soccer ball so while counterintuitive that's it's balls if you take the air out of the equation everything falls at the same rate even Aristotle [Music] 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 [Music] Galileo came up with an ingenious solution he built a rare an inclined plane to slow the falling motion down so he could measure so we're going to use this ramp to find a relationship between distance and time for time I'll use an arbitrary unit the Galileo one Galileo the length of the ramp that the ball rolls during one Galileo becomes one unit of distance so we've gone one unit of distance in one unit of time now let's try it for two counts one Galileo two Galileo in two units of time the ball has rolled four units of distance now let's see how far it goes in three Galileo's one Galileo two Galileo three Galileo in three units of time the ball has gone nine units of distance so there it is there's a mathematical relationship here between time and distance Galileo is inspired use of a ramp had shown falling objects follow mathematic laws the distance the ball traveled is directly proportional to the square of the time that mathematical relationship that Galileo observed is a mathematical expression of the physics of our universe Galileo's centuries-old mathematical observation about falling objects remains just as valid today it's the same mathematical expression that we can use to understand how objects might fall here on earth roll down a ramp even a relationship that we used to land the Curiosity rover on the surface of Mars that's the power of mathematics 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 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 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 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 absent mindedly 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 passed in case they inadvertently wiped out some major scientific or mathematical discovery 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 seemed widely in the fall of 1680 he gathers data worldwide in order to construct the Comets path so for November the 19th he begins with an observation made in Cambridge in England at 4:30 a.m. by certain young person and then at 5 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 unprecedented ly accurate calculation of how this great comet moved through the sky Newton's groundbreaking insight was that the force that sent the comet hurtling around the Sun was the same force that brought cannonballs back to earth it was the force behind Galileo's law of falling bodies and it even held the planets in their orbits a 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 what's so impressive and so dramatic is that a single mathematical law would allow you to move throughout the universe today we can even witness it at work beyond the Milky Way this is a picture of two galaxies that are actually being drawn together in a merger his whole dog is building alright 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 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 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 some of the world's greatest minds have been amazed by the way that math permeates the universe Albert Einstein he wondered he said how is it possible that mathematics which is he thought the 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 that the fact that mathematics can really describe the universe so well in particular physical laws is a gift that we neither understand nor deserve in physics examples of that unreasonable effectiveness about 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 [Music] and so they discovered Neptune mathematics had accurately predicted a previously unknown but if you formulate a question properly mathematics gives you the answer like having a certain that is far more capable than you are so you tell it do this and if you say it nicely then it will do it and it will carry you all the way to the truth to the to the final answer [Music] evidence of the amazing predictive power of mathematics can be found all around us I heard it took five Elvis's to pull them apart i television radio your cell phone satellites the baby model Wi-Fi your garage door Oh GPS and yes even maybe your TV'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 electromagnetic waves 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 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 twenty-year-old Giuliano Marconi his process starts with a series of sparks the burst of electricity creates the momentary magnetic field which in turn generates a momentary electric field which creates another magnetic field the energy cycles between the two propagating an electromagnetic 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 assuring in the era of wireless communication [Music] 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 their most famous for their Large Hadron Collider a circular particle accelerator about 17 miles around it built deeper 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 the subatomic particle mathematically predicted to exist nearly 50 years earlier by Robert Brout and Francois unclear working in Belgium and Peter Higgs in Scotland Peter Higgs sat down with the most advanced physics equations we had and calculated isn't I plated and made the audacious prediction than it we built the most sophisticated machines humans have ever built and used it to smash particles together near the speed of light in a certain way that we would their discover a new particle and if this math was really accurate 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 [Music] 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 have made we built it and it worked and he got a free trip to Stockholm [Music] [Applause] 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 why does this work how to mathematics be so powerful it's mathematics you know a truth of nature or does it have something to do with way we as humans perceive nature and it's to me this is just a you know a fascinating puzzle I don't know the answer in physics mathematics has had a long string of successes but is it really unreasonably effective not everyone thinks so I think it's an illusion because I think what's 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're choosing to study things which are mean able to studies in the mathematics that happens to have arisen but actually there's a whole lost ocean of other things that are really quite inaccessible to those methods with the success of mathematical models and 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 model and it's not a woman 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 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 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 look at how will in their work the elegance of math meets the messiness of reality and practicality rules today mathematics and perhaps mathematicians deal in the domain of the absolute and engineers live in the domain of the approximate now 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 they meet our needs [Music] 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 [Music] for us engineers we don't get paid to do things right we get paid to do things just right enough many physicists in 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 [Music] what I think about mathematics is that it is an intricate combination of inventions and discoveries so for example take something like natural numbers 1 2 3 4 5 etc I think what happened was that people were looking at many things for example in seeing with our two eyes below two breasts two hands you know and so on and after some time they abstracted from all of that the number two according to Mario 2 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 who discoveries we invented the concept but then discovered the relations among the different concepts so is this the answer that math is both invented and discovered this is one of those questions where it's both yes it is feels like it's already there but yes it's something that comes with our deep creative nature as human beings we may have some idea to how all this works but not the complete answer in the end it remains the great math mystery [Music] [Music] you