The Beauty of Math and Music | Marcus Miller | TEDxOttawa

[Music] [Music] hello I'm here today to talk to you about beauty in music and in mathematics however in order to get to the big takeaway we are going to have to take two cold showers together that is to say we are going to have to address two sticking points that typically arise in this discussion first cold shower most people are terrified of mathematics for many of you at some point someone pointed a finger at you and told you that you weren't smart enough to understand it or perhaps that you didn't really need math in the real world so you were fine not knowing it often that person was you this is common and it is false and it will keep you from understanding the really cool story I'm about to tell you so for the duration of the talk I'm going to need you to pretend that that conversation between you and yourself never happened second cold shower beauty is an ill-defined and oft maligned word as a matter of fact great author Toni Morrison says that physical beauty is one of the most dangerous concepts in the history of human thought and so instead of talking about beauty directly I'm going to tell you a story this is a story about infinity so if you start counting 1 2 3 4 etc you will never run out of numbers you can count forever but by a trick of language mathematicians can talk about the set of all the counting numbers instead of having to list them all which is nice mathematicians call this set the natural numbers and denote it with this special n when we talk about the number of items in a set mathematicians use the word cardinality which is just a fancy word for size and the size of the natural numbers is a left null we're using this hebrew letter instead of the usual sideways 8 for reasons that will become clear shortly well infinity is a funny thing suppose I were to dump in all of my negative natural numbers and zero into my original set I would get what mathematicians called the integers or the whole numbers and we denote it with this special Z and it turns out that the size the cardinality of the integers is in fact also a left null and now you're thinking to yourself well this is absurd I just added an infinite number of numbers into this set surely it must be a larger set well when mathematicians compare sizes what we do is we compare lists so let's draw up some lists on on the Left I have the natural numbers and on the right I have the integers and the way that we we say that two sets are of two infinite sets are of the same size if we can find any number in one list and match it to a single counterpart in the other list and here we have a nice rule that does that right like 1 goes to 0 2 goes to 1 3 goes to negative 1 4 goes to 2 5 goes to negative 2 etc we can continue this rule forever and in fact if we choose any number in say the right list the list of integers we can find a counterpart we can find a match in the natural numbers and in that way we say that the natural numbers account for all of the integers even though there are integers that are clearly not natural numbers and therefore they're the same size all right and if I just lost you what I basically told you is that infinity plus infinity is infinity right your eight-year-old is gonna love that that's how little kids talk right and and because that's so cool I'm going to play a little riff on the saxophone so this gets even better if I were to dump all of the fractions into my existing set of integers I get a new set that mathematicians call the rational numbers and denote with this special Q and it turns out that the size of the rational numbers is also a Alif know this is absurd of course you're thinking to yourself there are an infinite number of fractions just between 0 and 1 1/2 1/3 1/4 1/5 etc surely this must be a larger size this must be a set of a larger size well let's see if we can find a rule that associate saw love my rational numbers to all of my natural numbers if so then they would be in fact be the same size now we're talking about fractions and fractions are really just one number on top of another a numerator on top of a denominator and so I'm going to write all my numerators horizontally and all my denominators vertically and that's going to give me a grid and this grid in this grid in each box what I'm going to write is all of the fractions that they create there are some redundancies right some of them reduce don't worry about that it doesn't cause us a problem what we do want to do is we want to find a rule that associate Sall of these rationals to all of our natural numbers and this rule has to do two things first it has to enumerate all of the rationals right we want to say which box is first second third fourth fifth because that matches up with our natural numbers 1 2 3 4 5 right the other thing we need to do is we need to make sure that we cover every number in the grid we have to get everything we can't leave anything out so how do we do this oh I know draw a counting snake alright so here's a counting snake it's clear that if you know you were to let this counting snake go forever on this grid we would cover the whole grid and we can enumerate the boxes in the order that the counting snake eats them so alright so in that way we can say that once again the natural numbers account for all of the rational numbers now if I just lost you again what I basically just told you is that infinity times infinity is infinity right your eight-year-old is going to love this because that's how little kids talk and that's also cool so I'm going to play another blues riff now you may be asking yourself well is there a larger infinities everything just kind of a little and it turns out that there is the real numbers consist of everything we've talked about thus far plus another set of numbers called the irrational numbers which are characterized by having an infinite number of non repeating decimal places in case that sounded confusing you know some of these irrational numbers like the square root of 2 or pi all right and it turns out that the size of the real numbers is in fact larger than the size of the natural numbers well how is that what do you mean larger infinity that seems like a very uncomfortable concept well remember that this whole project is associating lists and it turns out that if we generate any list of natural numbers and associate them to any list of real numbers we can find and we can create construct a number that must be excluded from the list but is part of the set of real numbers and that was confusing to say and hard to remember so I'm just going to do it with you right now let's first pull up our list we have the natural numbers and we have the real numbers the real numbers are selected such that they have an infinite number of non repeating decimal places as indicated by the ellipses at the end of each number and they also have they also consist of all zeros and ones surely this doesn't cover the real numbers I know but it makes the trick that I'm about to show you really easy to explain so observe the prestidigitation in order to correct construct this excluded number we go to the first number in our list we go to the first digit of the first number we change it right from 1 to 0 and then we write that down as the first number in our excluded number we then go to the second number in the list we go to the second digit of the second number change it and write that change number down as the second digit of our excluded number and we can continue this process forever all right we are guarantee that the number in red is excluded because for every number in the list of real numbers in this infinite list of real numbers it will disagree with our excluded number at some point one digit somewhere is going to disagree at least because we've constructed it this way and that means that this particular list of natural numbers does not account for this particular list of real numbers but as I said before we can carry out this process with any possible list of natural numbers and real numbers which means that the natural numbers can't account for the real numbers at all in the same way that they accounted for our integers and our rational numbers but the natural numbers are also an infinite and infinite size set so that means that the real numbers must constitute a larger infinity now this gets even more fun there's Aleta there's an argument by leading set there's WH wooden that says that the cardinality oh by the way the cardinality of the real numbers we're gonna call a left one right it went up from zero to one and mathematicians sometimes call this infinity uncountable now there's an argument by leading set there is WH wooden which says that this Alif one is actually just two to the power a lift null but if we can do stuff like that then what's to stop us from considering that there might be an a lift to that's 2 to the power a loved one or a nail of three that's 2 to the power a love 2 and so on and what we're drawn to is the conclusion that there are actually infinite infinities each larger than the last now this might be a little bit too intense for your eight-year-old who is either crying or is a budding mathematician and because that's pretty heavy Liz and I are going to play some contemplative music [Applause] [Music] [Music] [Applause] [Music] thank you now in case you didn't follow that whole infinity discussion completely don't sit there trying to figure it out don't worry about the technicalities what I want you to do is just get in touch with how it feels to learn that there are actually infinite infinities how does it feel to know that when you contemplate the mysterious concept forever you contemplate but an infinitely small measure of existence that the world of the imagination is literally inconceivably vast the first time I learned this fact it felt a little bit like the first time I heard petrushka by Igor Stravinsky or rosewood by trumpeter woody Shaw I was awestruck that the human mind could conceive of such a thing grateful that my life in its ups and downs had brought me to the point that I could experience it and deeply desirous of sharing that feeling with others so math and music connect in several places the sciences of sound signal processing instrument design and acoustics are all shaped by principles of physics which are of course written in the language of mathematics ideas in rhythm and in western harmony are cousin to certain ideas in number theory and topology and sometimes there's even some direct cross-pollination fields medalists manju bhargavi notes that what we now know as the Fibonacci sequence was actually discovered in medieval India as a result of exercises in rhythm and poetry and mathematical physicist Stephon Alexander uses his studies in jazz saxophone to think outside the box on string theory equations but the platitude that math is music rings hollow to me while some working musicians are interested in these connections by and large the musicians that I've met are not particularly enamored of mathematical thinking and haven't relied on it heavily to home their art of their craft and conversely many mathematicians are not exceptional musicians or even entertainers furthermore the neuroscience on the math music connection is inconclusive we know that mathematics and music rely on complex systems utilizing both hemispheres of the brain and are thought to lean heavily on language processing and spatial reasoning but the full and true extent of the overlap is yet undiscovered and even some of the more famous studies that link math and music don't quite live up to their place in our popular imagination the Mozart Effect the idea that playing that playing Mozart for schoolchildren will improve their math scores is now thought to be a specific case of a larger effect called enjoyment arousal which says that doing anything that puts you in a positive mood will help you improve on any number of tasks is not math and music specific so let's review then it looks like the link between bodies of musical and mathematical knowledge are interesting but not particularly foundational in practice mathematicians and musicians lead starkly different lifestyles and talk about starkly different things and the neuroscience is far from settled it seems that unless we're to invoke kind of the sacred geometry of the ancient Greeks in their ancient Egyptian pedagogical ancestors we don't even have a mystical connection and yet in my experience I see a profound connection in the texture of thought and the aesthetic quality of these two disparate worlds I'm not alone international jazz pianist and topologist Rob Schneiderman notes that mathematicians jam on problems in the way that musicians jam on Tunes and I would add that there is a personal as well as social dimension to this for me the connection is in the inspiration the joy of learning the great ideas of those who came before me the transformation of dots and squiggles page into pictures and sounds in my mind and the ecstasy of solving a difficult mathematical puzzle or having a composition finally come together there's a sense of majesty of wonder of aliveness of beauty but I think the problem is that the word beauty is not quite potent enough and that's why it's often hard to see this particular connection between math and music it's hard to develop good science questions around a concept with no obvious moving parts and it's hard to develop good philosophy questions around an idea with no conceptual boundaries okay so let's set some by beauty I don't mean inspiring of lust or care nor do I mean inspiring of community or empathy in the way that many of these talks do nor do I mean simply pleasing to the senses the sensation I'm talking about has the following elements it's life-affirming paradigm-shifting and humbling it clears out the dust from everyday life it has to be earned it has a sense of looking out from the top of some mountain that you almost died attempting to scale or of meeting and mastering some dark part of the psyche or of surviving a cold shower there's a bitter sweetness that adds crucial flavor to the bright bursting joy the experience of your own personal infinity and I'm doing my best to evince this feeling in you right now with words to reveal this emotional color and metaphor because I don't know the word that does it I have an idea [Music] [Music] [Applause] [Music] [Applause] [Music] [Music] [Applause] [Music] so here's the opportunity princeton music psychologist Elizabeth Margolis denotes a relationship between practitioners philosophers and theorists and scientists in moving the field of music psychology forward she says that practitioners can talk about a phenomenon as is lived philosophers and theorists can make generalizations based on these anecdotes and scientists can tease out and test the relevant variables as a practitioner I hope that I can instantiate this math and music connection that kind of sloshes around in the cool sea Cove of our collective unconscious I hope that scientists can take my examples and work out the whole biology behind them I hope that educators can teach it systematically and offer our children a healthy escape from the general malaise of life I hope that leaders can make the beauty of this connection part of their daily ritual before they go out and make big decisions representing the people but if none of this comes to pass I hope that I've at least left you with a sense of the majesty and the Wonder and the triumph of the connection between math and music thank you [Music] you

Channel: TEDx Talks

Views: 9,412

Rating: 4.8793101 out of 5

Keywords: TEDxTalks, English, Life, Entertainment, Math, Music (performance), Physics

Id: K0jkbaJqL1s

Length: 20min 26sec (1226 seconds)

Published: Fri Dec 20 2019