 I'd like to introduce now Marcus Miller, who will be introducing tonight's speaker. Marcus is a jazz saxophonist, and I first met him because he loves math, and he came to Mo Math and said, we should do something together. And since that time, we've done a lot of things together. Marcus was a math major at Harvard University, and he's now pursuing a very successful career as a jazz musician. But he's also helping us kick off this February 17th, a new monthly series here called Quadrivium. He'll tell you more about it, but if you want a common experience, math, music, and really interesting conversation, I suggest you mark your calendar for February 17th. And with that, I'm delighted to bring out my good friend, Marcus Miller. Thank you so much. So a little bit about the Quadrivium series. I first became aware of Mo Math because my little sister took a class trip to the Museum of Mathematics. The family then told me that a Museum of Mathematics existed, and immediately I said, well, next weekend we're going. I came here and immediately I said, I want to work with these people. This is an amazing place. And in the past year, we've done a lot of work together. And it's been really exciting as a place to discuss mathematical ideas, kind of bring them to the mainstream and think about them differently, and hopefully to add kind of like a hip chic to technical thinking. In New York, we have Silicon Alley. We have a lot of very educated people who think about technical issues. And oftentimes, though, don't get to talk about the intersection between, say, mathematics and music and kind of different creative arts and all kinds of cool people that I've accessed to through my network as a jazz musician. So I'm hoping to bring kind of everybody together, have some cool conversations with the Quadrivium series. So hopefully you can come to that. That is February 17th. You can register on the MoMath website. It's going to be a great time. What's also going to be a great time is our speaker tonight, Dr. David Kung. I was here for the talk at 4 p.m. And it was really amazing. You're in for some really kind of cool special moments when he talks about some of these mind-bending paradoxes and the possibility of changing your mind. A little bit about Dr. Kung. He got three degrees at the University of Wisconsin-Madison, which is known for its excellent math program, doing his PhD work in harmonic analysis. Then he went to the, landed a faculty position, where he's now a professor at the University of Maryland, St. Mary's. And in addition to being an excellent mathematician, he does a lot of work in music. He played violin and now works with the community orchestra, plays with them and works with students in the music department, as well as work in activism, where he started programs to bring more young women, as well as more underrepresented minorities into mathematics. So you're getting kind of a broad spectrum of interests and a great intersection of mathematics with other things. And Dr. Kung, tonight, let's see, so without further ado, enjoy the talk on mind-bending paradoxes and the possibility of changing your mind. Dr. Kung. Thanks very much. Thanks for coming out. Thanks to the Museum of Mathematics for the invitation. I want to talk to you about changing your mind. I know that it's possible. I know that it's possible because I grew up in Wisconsin and I've been hanging out here in New York for a few days. And I know that at one point, I thought that like 10 degrees above zero wasn't that cold. Apparently, my mind's been changed about that. That was pretty cold a few nights ago. So thanks for coming out tonight. And especially, I'm glad we beat the snow. So we're going to have a lot of fun tonight. We're going to talk about changing your mind. But I do want to start with sort of a little bit of a hard story, a difficult story. When I think about changing your mind, I always think about this guy named Bob Eberling. And I think about this really bad day that Bob had. See, Bob was an engineer and he had this horrible day back in 1986. He spent the entire day going from one boss to the next, trying to convince them to change their minds about something. He did that the entire day. He failed the entire day. At the end of the day, he went back to his wife. He turned to her and he said, tomorrow the space shuttle will blow up. And it did. You see, Bob Eberling was an engineer at Morton Thyacall and they were in charge of, among other things, the rubber, the material in the O-rings that held the fuel in the space shuttle. And what he knew is that material was brittle when it got down near 34 degrees, which is what launch temperature was going to be. He could not convince any of his bosses to change their minds about that with fatal consequences. Now most of us, when we have instances like this where we're trying to change our mind or we're trying to change somebody else's mind, the stakes are not nearly so high. And so I want to talk about some of those instances where the stakes are not nearly so high. But I want to see these as places where we can practice changing our mind. And the first step in changing your mind is admitting that you're wrong. And you may think you're good at this, but if you talk to your close friends or spouses, they will set you straight on this. We, as humans, are really bad at admitting that we're wrong. And so today we're going to do something which is really unusual. Most of the time we celebrate being correct. We're happy when we're right. Today we are going to celebrate being wrong. And I know you in New York can celebrate like almost nobody else on the planet. So I want to just hear what it sounds like when New Yorkers celebrate. Let's hear it. If you're wrong, this is what I want to hear. Let's celebrate. Let's hear it. Come on. Yeah! All right. All right. That sounds great. So every time you're wrong, we're just going to celebrate being wrong. All right. So there we go. And the goal here is to be better at changing our minds. That's the goal. So I want to start with this. This is a very mathematical question. It'll follow up with a demo. Who's blown bubbles? Like little blown bubbles with kids, right? So sometimes when you blow bubbles, two of them come out attached, right? If you have two bubbles that are the same exact size, you can imagine that the interface between them in the center there is just a circle, right? It's just a flat circle. So that's an easy question. Here's a harder question. What if one of them is bigger than the other? So here the top bubble is smaller than the bigger bubble. And so the question here is, do you think that boundary, do you think it bends into the larger one? Do you think it bends into the smaller one? Or do you think it's just flat? No. So that's the question. And I actually, you know, this is sort of, I can't blow bubbles, but I can do another demo which is, it turns out will help us answer this. And so I have two balloons here. So these are identical balloons, and I've just, I've connected them with a valve which is currently closed. And so these, you know, I just blew up one with more air than the other, right? And so with these, the question here is if I open the valve, what do you think happens? Do you think air flows from the bigger one to the smaller one? Do you think air flows from the smaller one to the bigger one? Or do you think like no air flows despite the fact that the valve is completely open? Does everybody understand the question? Right. So the sheet that was on your, on your seat when you came in has a bunch of these questions. This is question number one. And every time I'm going to ask you to do the same thing, which is I'm going to ask you to answer this without talking it to anybody, for yourself. What do you think the answer is? And I really want you to mark down, circle the answer you think is correct. And I want you to do that because the research is clear, you will remember all of this better if I force you to take a stand. So take a stand, make an answer. It's the end. If you're wrong, you'll get to cheer, right? So, so take a stand. Take a minute. If you haven't marked, you're trying to delay to give you time to mark down an answer. Okay. Now turn to your neighbor and convince them that you have the right answer. Go. All right. So, so the valve is closed right now. I'm going to open the valve. Let's see what happens. Let's hear it. Were you wrong? Were you wrong? Excellent. Excellent. You know, the thing that I really love about this question is that almost every single person walked into this room tonight with the right knowledge in their head to answer this question for the right reason and yet the vast majority of you didn't. And I love that about this question, right? So here's the knowledge that you had in your heads that you may or may not have used. This is the graph of how hard it is to blow up a balloon as the balloon gets bigger. On the horizontal axis you have the volume. On the vertical axis you have the pressure. And we all know that when you grab a balloon it's really hard initially and then it gets easier. And if you've worked with kids, you know that there's an age at which the kids cannot produce enough pressure to get over that hump and what they end up doing, if you get over that hump for them, then you can hand them the small balloon and they can blow it up the rest of the way. And so if we have this curve, if we think about it, this is maybe where I put those two balloons initially on this graph. And if you look closely, it's the larger balloon here which has the lower pressure and it's the smaller balloon that has the higher pressure. As many of you probably thought, an equalization of pressure, but what you may have gotten wrong is it's the smaller one that has the higher pressure. And so what happens, the smaller one has higher pressure and so it blows air into the larger one and this is what happens on the graph when those two happen. And so it does actually equalize the pressure but it equalizes the pressure in a really, for most people, highly unexpected way. These two balloons now have the same pressure in them. And this demo doesn't end here. If you look up this demo on YouTube, most people, they end it right here. But if you notice, if we were dealing with soap bubbles, this would be the end of it because the curve just sort of goes down and then the soap bubble bursts. But you notice that toward the right on this graph, the graph comes up again. And what that means is that's properties of the rubber going on. That means that we can do something different. So yeah, first of all, the smaller bubble has higher pressure and so the interface goes down and bends into the bottom one. Were you wrong? Let's hear it. All right, all right. Excellent, excellent job. You are being wrong very well tonight. Because that curve goes up, you can actually set this experiment up to do any of the three answers that I had on the board. And so I want you to think through if I had put these two balloons in either one of these situations, I'd put those two balloons as the two dots on either one of these curves, what would happen if I had done each one of those? Go ahead and turn to your neighbor and explain. And I'm going to set this up. And the cool thing about this is I'm going to set this up and I'm not sure where it's going to be on these curves and then we'll see. So go ahead and tell your neighbor what's going to happen in each one of those cases. Go. I worry that I'm going to faint sometimes when I get in front of a crowd doing this. All right, I think I have these. So here's the answer to this. If I set this up and it's on the left, what happens, the larger one has higher pressure and so it blows into the smaller one and they meet in the middle. On the left, I'm sorry, on the right, what happens is they have the same pressure already. And so nothing happens. And here's the deal. I think everybody agrees that the balloon on your right is bigger. Is that true? Yeah. But I don't know where these two balloons are on this curve. This is completely unknown to me. So when I open the valve, I don't know exactly what's going to happen. So here it is with the valve open. So I felt just a little air move but essentially they're different sizes but they apparently have the same pressure. And so we're in this case on the right. We have two balloons that have different sizes but the same pressure like that. All right, so there's balloons and pressure and I hope you're excited about being wrong because I think a number of you are you excited about being wrong? Were you wrong? Let's hear. All right. Excellent work. A lot of the paradoxes in mathematics, the beautiful paradoxes that the math world has boiled down on some level to the Liar's Paradox. And the Liar's Paradox is this amazing thing. It's simply this sentence is false. And when you first come upon this, it's just an amazing, when you teach kids and the first time they read this, it's lovely to watch them sort of chase their tails around and around. If it's true then it's false and if it's false then it's true and back and forth and there's no resolution to this. And it's what Douglas Hofstadter calls a strange loop. And strange loops come in lots of flavors and lots of... and we'll actually hear one in a minute, but these strange loops happen in life too. I had my daughter and we were hiking in Hong Kong. We were hiking along and she was in a really bad mood. I had her in my backpack hiking along with this guy talking to him and I was talking to him in Chinese and so we were talking and she was just sort of ignoring us and at some point he looks at me and he says, he motions to her and says ta tingle dong putong huama does she understand Chinese? And she... I didn't say anything I look up at her and she turns to him and says, no. That's a strange loop, right? Somehow it's self-contradictory. Asher did wonderful art around strange loops. Here are the drawing hands. One hand draws the other, which is flat which turns into a three-dimensional thing to draw the first. Waterfall is a great example of continuously going down and then fall down the waterfall and then continue to float downward, right? And here we are, a similar thing the soldiers here in ascending and descending in one direction they're always walking up in the other direction they're always walking down and it turns out you can do a similar thing with sound. Everybody agree that that sound keeps going up in pitch? Everybody also agree that it comes back to where it started? Yeah, strange loops. You can do them with sound as well. Yeah, this New York water is actually really great. You have great water in this city. It's fantastic. Are we cheering great water too? Alright, to being wrong. Best in the country, yeah. So here I have a tube. I play the violin, I didn't bring my violin but I have this tube and when you spin it around it makes a sound, right? And so I have a question about this sound. So if I swing this tube faster what's going to happen? Do you think it's going to just get steadily louder like that? Do you think it's going to steadily get higher and higher like that? Do you think it'll do both? Or do you think something else happens? So those are the four options. I want you to, before talking to anybody circle one of those options, pick one for yourself what's going to happen and then I want you to turn to your neighbor and I also want to make sure nobody's left out in these discussions. Make sure everybody has somebody to talk to. What's going to happen? Convince them that you're right. Go. Talk. Alright, here we go. Here we are. Here's the pool tube. I love being in New York where I mean you all are like prize really nice instruments, like you know a Strat at Christie's is going for ten million or something. Like I paid four dollars for this at the pool store. So here we go. So it definitely gets louder but did you notice it didn't just sort of get steadily higher? It jumps. Did you notice this? It jumps up and so like in some sense the right answer is D. Let's hear it. Were you wrong? Yes! We're doing a great job of being wrong. The mathematics behind this the mathematics behind a tube of air like this. I love thinking about the mathematics of this in part because it ties together two of my loves mathematics and music. The math of a tube of air like this is identical to the math of a vibrating string and in both cases they tell you that you get only certain harmonics, certain frequencies that can be vibrated on this and a string you change the frequency of a string by putting your finger on it making the string shorter and a lot of wind instruments you make longer and shorter with things like valves or like a trombone you can really see it being longer or shorter but there is a brass instrument that is just a fixed length of pipe like this. You don't change the length of right? It's a trumpet without the valves what is it? A bugle and what do you hear played on a bugle? Taps Taps and so the mathematics says that if you can play tabs on a bugle you should be able to play tabs on this because the mathematics is the same. Thank you very much. The other wonderful thing about this demonstration is you remember in your high school chemistry class maybe you were like me you go through these p orbitals and s orbitals and they tell you that these electrons can only have certain energy levels and then like I remember asking my chemistry teacher why and he just said because which is a horrible answer right? But the answer is that because the mathematics says that that's only the only things that are possible and in fact it's the same mathematics the same exact mathematics that tells you that a violin string and a tube of air have this set of frequencies tells you that an electron has only a certain set of frequencies it tells you there's a reason for these p orbitals and s orbitals and we use that all the time we were just there with the Haydn planetarium like we use that fact all the time because it tells us things about the universe it tells us about red and blue shifts of light coming to us where we need that all the time and it's the same mathematics. Poincaré said it best he said mathematics is the art of giving the same name to different things it's the same mathematics in this case partial differential equations that allows us to see that there's different energy levels and these different frequencies and that's just a wonderful thing so I want to ask you a question about adding everybody who's added who can add right great alright I'm glad we're all on the same page about that so if you add positive numbers forever just keep adding positive numbers here's the question that some when you keep adding them do you think that will keep getting bigger and go to infinity do you think it'll stop at a finite number or do you think sometimes it'll do both right and so we're going to answer this question it's not on your sheet you don't have to take a stand on this right now but I want to give you sort of a visual of this like here's a box and we can just say this box is size one right and we can split this box in half and then it's a half and what's left and then we can split what's left in half and that's a half so now it's a quarter in that bottom part and then we can split what's left in half again and that's an eighth and then a sixteenth and then a thirty second and a sixty fourth on and on and on right but it's still all inside the box the box has size one and what that's telling us is that we must have one half plus a fourth plus an eighth plus a sixteenth plus a thirty second if you go forever you just get the whole box right and so mathematicians call this a geometric series and so clearly at least sometimes when we add up positive numbers we end up just getting a finite number like one all right so did you know you can do a musical version of this as well in fact like the language we use to describe the music is mathematics that's a half note followed by a quarter note followed by an eighth note and then a sixteenth and a thirty second and a sixty fourth I think my musical program would only allow me to go to a hundred and twenty eighth note but you could keep going forever and the amazing thing is in theory you would put an infinite number of notes inside of a single measure of music and how long would that measure take to play exactly as long as every other measure in that in that piece of music with an infinite number of notes and so that gives us one answer you don't always go to infinity but we know sometimes you do in fact this is the most famous series that does this in when you learn this in calculus it's called the harmonic series when I hear that there are calculus students who have learned about the harmonic series but their teacher has not tied to them that fact together with the fact that that series of numbers is the same sequence of numbers that came out of that swinging tube that's where the name harmonics comes from a little bit of me dies every time I hear that there are students who haven't heard that so it's harmonic series because it comes from harmonics and the thing we know about the harmonic series that when you add it up you get infinity that series does what mathematicians call diverge so some series converge to a finite number some series go to infinity so it's clearly we have some of both right so here's what I want you to do the summary related I want you to flip your paper over and just pick a number between zero and one any number between zero and one not zero not one any number in between write it down go ahead take a second to write it down alright everybody got a number great just share your number with your partner some people around you tell them what your number is alright and here's the question I have about your number is it rational in other words can you write it as a fraction remember that if you wrote down like 0.25 0.25 is a rational number it's not written as a fraction but you can write it as a fraction so is your number is your number rational can you write it as a fraction right so raise your hand if the number you picked was can be written as a fraction can be it's a rational number great and raise your hand if your number cannot be written as a fraction great what do we have one number pi minus three anything else 0.6 0.6 is 6 over 10 no sorry yeah root 2 over 2 root 2 over 2 any any others 1 over pi 1 over pi right we get a lot of pi a lot of root 2s right but it was something like 90% of you picked fractured rational numbers right yeah so here's an amazing mathematical fact fractions are these common things you could ask yourself sort of mathematically what are the chances that if we randomly pick a number between 0 and 1 like and I think about this is like you know you have like a really fine razor blade passing over 0 to 1 and it is equally likely to stop at any point on the number line and if you randomly stop that and get a particular number what are the chances that it's a rational number the answer mathematically is 0 which seems strange since around here it was more like 90% because the last time I checked 90% and 0 were different pretty different right and so I want to prove this to you so so we're going to prove that the probability of getting a rational number is actually 0 so what I wanted to I think we can all agree on this like the chances of picking a number between one half and one that's like half of the interval right and so the chances of picking one of those numbers should be one half everybody good with that right and if we shorten like between one half and three fourths that's a quarter of the interval and so the probability of picking a number in between those two numbers is all is going to be one fourth and in general like the probability of picking X in some interval is going to be the length of that interval everybody okay with that that seems like that that by itself seems it seems fine we haven't done anything mind-bending or crazy yet so here's what I want you to think about I want you to pick a C a positive number C that's really pretty small number and we'll get back to exactly what C might be later see some small number what I want you to think about is see that one half in the middle I want you to put a little interval around one half and the length of that interval I want to be C over two one half of C what I want to see what I want to impress upon you is now what's the probability of picking a number in that little interval it's the length of the interval right which is C over two right and in fact the probability of picking exactly one half has to be less than that because one half is inside that interval right okay so there's our interval of length C over two and now I want to take an even smaller interval half again is small C over four and put that one around the fraction one-third and so now the probability of picking a number inside of that one is going to be its length which is C over four the probability of picking a number that's in one of those two intervals the sum of those two lengths C over two and C over four and not only that but the probability of picking like either one half or one-third has to be less than the sum of those lengths because those two numbers are in those two intervals right and so let's keep going with this so now I want to cover two-thirds I want to cover two-thirds with an interval of length C over eight half again is big and now I want to cover one fourth with something half again is big I think I'm at C over sixteen and then I think I've already done two-fourths because I have that covered at one half so let's skip to three-fourths and put something of I think we're at C over 32 and we're just going to keep going those are the fourths and next I'm going to do the fifths one-fifth, two-fifths, three-fifths, four-fifths and then I'm going to do the sixths and the sevenths I'm going to think about keep and going I'm just going to enclose these in ever decreasing sizes of intervals and here's the deal once I've done this if I've done this forever I've covered all of the fractions I did all the fractions with a denominator of five and then six and then seven and eight I covered all of the fractions right and so now the probability of picking any fraction has to be less than the probability of picking any of the numbers in those intervals and that was the sum of those intervals and the sum of those intervals we started at C over two and so that's C over two plus C over four plus C over eight plus C over sixteen we can factor the C out we get one half plus one fourth plus one eighth which is exactly the sum we saw in that box half of it and then a quarter and then an eighth right and that sum to one and so we get a sum that's just C so if you read through this inequality it's saying that the probability of picking a fraction has to be less than or equal to the constant C and now here's the rub we can pick C as small as we'd like we can pick C to be point one point zero zero zero zero one and therefore the probability of picking a rational no matter what we pick C to be the probability of picking a rational number has to be less than it the only possibility is that the probability of picking a rational number is zero and that's despite the fact that ninety percent of you picked rational numbers here's the here's the crazy thing most numbers we can name are rational but most numbers in some sense are irrational there's a difference between something being just familiar you know it and something being actually common out there and that's a really radical difference and this showcases it more than anything else what's really going on here is that the infinity that is the irrational numbers is a bigger infinity than the infinity that is the rational numbers and so one infinity overwhelms the other and you actually get a probability of zero alright that was a little wacky let's turn to something I want to go to something a little bit a little bit more down to earth so here I have a bunch of books and we're going to stack these books and so the question is if I stack these books how far out can I get the top one this is a little hard for you to see I'm going to hold this up so I'm going to stack down the edge of the table is going to be right here and I'm going to stack these books out over and the question is about this top book how far out over the edge can this top book go and so I have some options here maybe you cannot get this top book to be over the edge of the table it just can't get that far maybe you can get it to just be barely off the edge of the table maybe you can get it well over the edge or maybe you can stack these all the way to New Jersey alright that's the question you have it on your sheet so take a stand circle and answer go ahead and circle and answer how far can you stack the top book go alright I'm going to start stacking we have some people who are right in line with this I'm going to leave it up to you to sort of judge whether or not we're out over the edge I think what everybody can believe without too much is that this top book can only go out halfway over the previous one everybody got that? if this were the edge of the table I can't get more than half it would just fall off the edge so that top one can only go halfway out half is sort of like the edge of that if I take it a little too far it won't be able to go so there's one half if you do it with two books so I'm going to do it here with just taking the top two books if you do that and you sort of do it carefully and you can do the calculation what you find is that that second book can extend a quarter past the edge of the table so now I have one half of a book and then a quarter of a book so that's not out over the edge yet if I do three I can get the third one one sixth of a book and again this is sort of just on the edge of what's possible and so there that's stable but I think you would all agree I'm not out over the edge yet and so I'm going to keep going and here's the thing if you keep going if you keep going what you find out is that you get a particular sequence of numbers and when you factor out one half what you end up with is the harmonic series and what do we know about the harmonic series the harmonic series diverges do you all agree that the top book is actually past the edge of the table that top book is definitely past the edge of the table with these six books I've managed to get that top book out over the edge and what does this tell you this tells you that the harmonic series diverges and so it goes to infinity so that tells you that at least in theory if I had enough of these books the correct answer to this question is that I should be able to stack them to New Jersey I'm going east I'm sorry I should be able to stack these as far out the parking lot that is the Long Island Expressway as far out as I want I should be able to stack these so in theory I should be able to stack these books as far as we want which is sort of an amazing fact it turns out that if you want to get far it's pretty difficult I did a quick calculation seven miles might get me to like flushing or something if you want to go seven miles if you want the top book to be seven miles out past the end of this you need roughly just roughly speaking you need about ten to the eight thousand books one followed by eight thousand zeros that's the number of books you need just as a rough gauge of how big that is if you look at the number of particles in the known universe it's about ten to the seventieth or so ten to the eight thousand is a little bit bigger than that alright let me hear it did you get this wrong excellent excellent work I have another demo for you and this involves a slinky and I need a hand with this and Greta has agreed to help me out Greta round of applause for helping me out so I'm going up here on this ladder and so I'm going to dangle this slinky and the bottom of the slinky I'm going to dangle it and so we're just going to try to get it so that the bottom you know what that's the bad end of the slinky you didn't know that slinkies had a good end and a bad end but this one's been around so we're going to dangle this I'll give you a little bit more slack let's see let's go see if it bounces oh I need a little bit more alright oh there we go that's the problem alright let's leave it there Greta I think that's going to do so the question is Greta's going to stabilize the bottom of that until it stops bouncing and then I'm going to drop the top and the question is about the bottom of the slinky maybe a little lower keep going until it stops are you pulling it don't pull it just let go let it go Greta okay we're going to let it just put your hand under it and don't hang on to it and just keep coming up until it stops bouncing don't hang on to it and now slowly let your hands down there we go just leave your hands right there and now just take away your hands there we go alright so the question is when I let this go what happens to the bottom does the bottom of the slinky does the power of the slinky pull up harder than gravity does it go down does gravity beat the slinky or does it not move somehow the bottom the gravity of the slinky that's okay so that's the question so circle your answer on your sheet circle your answer does the bottom of the slinky come up go down and I'm asking just initially because eventually the whole thing is going to fall I let go does the bottom of the slinky move up first before falling does the bottom of the slinky fall alright alright everybody have their answer so here we go I'm going to drop it on the count of three one two three would you notice like if you look closely it doesn't move and most people I'm going to do this one more time can you just I'll do this one more time I'll try to get this nice and high so you can see it I'll give you a little bit more the bottom of the slinky here alright there you go on the count of three watch carefully the bottom of the slinky doesn't move until the top hits it one two three do you see that alright were you wrong what I love about this demonstration is the best way to understand this demonstration is not about the slinky at all the best way to understand this is about information transfer you see here's what's going on you can imagine yourself being like one ring of the slinky and I'm holding on to the top all you know is that there's something pulling you up and there's an equal and opposite force which is the weight of the slinky below you pulling you down and those are equal and opposite and so you're going nowhere and here's the thing like when I let go the slinky above it doesn't move the only thing that changes for any of those rings of the slinky is the very top ring of the slinky when I let go the force pushing that up is gone and so that top ring of the slinky compresses and only then does the next ring of the slinky know something happened and so it compresses and only then does the third ring of the slinky know something happened so what's really going on the bottom of the slinky in some sense does not know that I dropped the slinky until the information gets there how does the information get there by the whole slinky compressing and only then does the bottom of the slinky go oh I got dropped and down goes the slinky right it's about we don't think about information taking much time anymore we think about information being instantaneous but you know the folks down in Wall Street they know how important it is to get the information transfer time right and that's why they build huge you know huge systems to be closer to Wall Street to make the time between trades even smaller so that they can make more money on that right so the information transfer this is a great demonstration of information transfer and how long it takes I want to go back to fractions we had some much friend fun with fractions and so I want to think about you know imagine you go to the doctor you go to doctor and well among other things your doctor says drink more water water is great right but maybe your doctor says you know I'm sorry you have some condition like kidney stones or something and like there's two treatments you might have treatment A or treatment B and so you have this conversation with your doctor and your doctor might say you know overall treatment A has better outcomes you might say okay let's go with treatment A but your doctor might say you know just to be safe or maybe just to make an extra charge on your file we'll do an image of your kidney stones right so we'll do an MRI and it comes back and the MRI says good news you have small kidney stones and it turns out for small kidney stones treatment B is better and you might say to your doctor oh well you know if I had large kidney stones then we would have gone with treatment A and your doctor might actually say well for large kidney stones treatment B is also better and in fact it might be the case that you know even if you put like if you put these into like five different bins kidney stones between zero and three millimeters three millimeters and four millimeters in each one of those bins treatment B might be better even though overall treatment A is better which seems to make no sense at all everybody agree with me on that it seems to make no sense at all so what's going on so I'm going to give you a different it turns out completely equivalent setting and you're going to just agree with me because the numbers are here so we have basketball fans in here when you're talking about basketball players shooting you talk about one out shooting the other it's in terms of the percentage and so maybe in the first half Steph Curry out shoots LeBron James and in the second half Curry out shoots LeBron James again in percentage terms it might be possible that overall LeBron number is up here so in the first half Steph Curry goes one for seven he's pretty cold but not as cold as LeBron James who goes 0 for three does everybody agree that one seventh whatever that works out to be percentage wise is higher than zero good in the second half Steph Curry goes three for three hits a hundred percent LeBron misses a couple shots but he goes five for seven everybody agree that a hundred percent is better than whatever five for seven is alright let's add those up for the game Steph Curry was four for ten LeBron James was five for ten Steph Curry outshot LeBron James in percentage terms in both halves and LeBron still outshot him for the game it's just the numbers right that's what the numbers say it is clearly possible and now all we have to do is we have to switch you know switch out Steph Curry and LeBron James for treatment A and B and you know the the and small and large kidney stones and you have exactly the case that I was telling you it might be the case that treatment A is better than treatment B in each one of those bins the small and large kidney stones but overall treatment B is better what's going on here is that we're thinking of these things as fractions we see them as fractions we think of them as fractions and so we think about the relationship that holds for fractions these are not fractions in that sense these are ratios when you added across to get Steph Curry's overall shooting the numerators and the denominators you treated them as ratios not fractions the inequality that we want to be true it does hold true for fractions but it does not hold true for ratios and that's why we get this so badly wrong our intuition is horrible about this and this actually happens so there are cases so here are some examples of this is what's called Simpsons Paradox Berkeley graduate students in the 70s Berkeley graduate school figured out that the admissions rates for women were 10% lower than for men and they went around looking for the culprit which department is guilty of discriminating against female applicants and the answer no department there was no department where the admissions rate for women was significantly lower than it was for men but overall it was and it's just an example of Simpsons Paradox if you look at test scores in this country when another test score the international benchmark comes out you see these graphs in the paper and what you see is overall US test scores are flat or maybe slowly declining and everybody says oh our schools our students blah blah blah if they were to split out those scores by race what you would see is that among white students scores are slowly rising among African American students scores are slowly rising among Hispanic, Latino, Latinx students scores are slowly rising in each one of the subgroups we are getting better overall we're not and that partly has to do with the fact that we have such a gap between white and Asian students and African American Latino students in this country that as we change the composition of the population that's what's getting us to have flat scores overall or slightly declining scores but it's this it's this quirk of statistics it's Simpsons Paradox that allows us this to happen in the sub scores amazingly this same paradox helps explain part of the phenomenon of white flight so white flight was typically especially in the 60s and 70s an inner ring suburb that was mostly historically white and that was historically by law it was white and then slowly a few African American families would move into that neighborhood and then some of the white families would start to leave and slowly over time that neighborhood would turn over from mostly white to mostly black and brown over the course of a few years and part of what's going on is actually this mathematical quirk you see this is what happens the African American families moving into the inner ring suburb were the ones who had enough money to move up in the world and they were moving to an inner ring suburb but they were coming from places with badly underfunded schools with teachers with much less preparation and what was happening is they were bringing their students those first students were coming into those schools they had with their much worse preparation they would drop the average test score a little bit and at that moment the education that kids were getting in that school hadn't changed a bit it might have even been getting better over time but the average test score went down and that is enough to propel some of those white families to say oh test scores are going down a little bit we better move out to a further suburb and then more African American more Latino families come in from inner cities and the test scores go down a little bit more and then there is a black loop that just continues to get more poorly prepared students into that school dropping the test scores more and at that point it's possible that the education given to the students in that school is just rising up and up and up even though the average is going down it's just an example of Simpsons Paradox of course in this case it's probably an example of racism but Simpsons Paradox and this sort of goes to something I think we need to face as a mathematics community we need to understand this myth that math is some sort of meritocracy where the brightest sort of rise to the top it's not only wrong but it's really damaging why is it not a meritocracy because there are different opportunities in terms of funding in terms of the preparation of teachers and there is just the historical exclusion of African Americans especially in education in higher education in a lot of our schools poverty plays an incredibly important role and here's how bad poverty is in this country I actually want you to write down a number here the median wealth of a white family in this country I will tell you this one is $111,000 if you take the total value of everything a white family owns in this country the median is $111,000 I want you to write down what you think the median wealth is for an African American family in this country everybody understand this is not income this is wealth the total value of everything you own including your property everything else white family $111,000 write down what you think an African American family is go ahead write it down the answer the newest the most recent figures I can find are somewhere a couple of years ago the answer is somewhere around $12,000 $111,000 $12,000 poverty affects us differentially especially differentially by race if you're wondering about Latinx families it is slightly higher like $13,000 or $14,000 poverty plays a huge role in addition there are much more subtle things in in effect here including teachers unconscious biases about their students this is especially true of girls in mathematics where teachers simply do not sort of see those students as bright in mathematics and those teachers expectations impact student performance and in fact students own unconscious biases about themselves and the groups they're from there's a lot of great research out there about stereotype threat students own biases have an impact all of these are suggesting that mathematics is not a meritocracy it is not just about the best rising to the top and we need to we need to change our thinking about that when we think when we act it's about a math gene or it's about the best and the brightest like we are doing damage and that's something we need to admit we're wrong about can we get on board about that we're wrong about that all right all right I've got another demo here and I need some help with this one and so let's let me set this up move some things aside somebody want to help me out here you want to help me out what's your name Joel so Joel's going to help me out here and so so Joel this is what I have so I just I want you to do this come on over here we're going to make you face so the cameras can see you and so what I want you to do is I want you to pick up these three blocks just lift them a little bit and then I want you to pick up just the top one and then I'm going to ask you to tell them what you feel okay all right that's relatively light okay and now just the top one that's definitely feels quite a bit heavier than the entire pile I want to I just want to like I want to be clear about this because when you were lifting the bottom three correct me if I'm wrong you were also lifting the top one is that true yes okay but then but then when you lifted okay help me again that's light this felt consider it felt lighter than a single one and so so the single one is heavier than yes it plus um I'm sorry I should have been more specific can I get a good volunteer up here I'm just kidding do you want to come up and try this let's see let's make sure let's confirm that he is not in fact crazy so yeah come around here so again just all three and then the top one and then tell them what you're feeling okay I think you got it I think you got it hand uh-huh yeah it doesn't feel very heavy okay and now just the top one yes it also feels heavier it's heavier it feels heavier than the entire three of them I'm guessing there's a magnet or some kind of thing going on are you wondering what's what's under it yeah oh let's see let's see okay so let's take these to the side what's under this okay why don't you pick that up like open it feels empty looks empty but it doesn't tell me about the materials but you're still claiming that hang on let's let's so you're still claiming that I'm guessing this is actually lighter than just the top one that's what it feels like okay all right all right thank you very much can you give them both a round of applause there's this thing that teachers do that the teachers need to stop doing and if teachers paid attention to the entertainment industry especially network television teachers would get this right so this is what teachers do all the time and I'm guilty of this too so we have a point to make for the day right and we get near the end of the hour and we look at our watches and we're like only ten minutes and then we speed up we go faster and we get to the point we get to the conclusion before the end of the hour and then we're done and classes dismiss and they go away right who's had that class and the closer you get to the bell the faster the professor the teacher goes and you can't really follow them anyway right and if network television work this way this is what the end of the season would look like there would be like a cataclysmic event like a plane crash and you would know that like oh and then they would speed things up oh there are five main characters on the plane crash and oh my gosh it's a plane crash some people are gonna die and then if network television work like teachers they would tell you right before the end oh these three lived these two died see you in the fall this is not at all what Hollywood does this is you see what Hollywood does is they like leave you with a good question right they're like there's a plane crash some people died we don't know who think about it all summer and we'll see you in the fall right teachers need to do this right teachers you leave with a question you don't answer everything everybody on board with that like we need to do the better job of that and I'm gonna demonstrate that so those are called gravity blocks we're moving on all right so I want to talk a little bit about dimension I feel evil when I do that how do these Hollywood people live with themselves so so I want to talk about dimension so when we scale something up if we take a cube and like double its size what we're talking about is we're double talking about doubling length and that's not what happens to the surface area in the volume and so if you can quickly calculate you know if you double the side length if you go from 1 to 2 on the side length of a cube you can quickly calculate that the surface area goes from say 6 square centimeters to 24 goes up by a factor of 2 squared or 4 and the volume of this cube goes up from from 1 cubic centimeter to 8 cubic centimeters it goes 2 times 2 times 2 so you get 8 cubic centimeters in the end and that's 2 cubed and the important thing here is that these exponents the 2 squared the exponent the 2 cubed are first of all linguistically that's why we call them squared and cubed it has to do with the dimension right but those those exponents are telling us the dimension of what it is that you're measuring you're measuring surface area which is 2 dimensional the exponent is 2 you're measuring volume which is 3 dimensional the exponent is 3 if you double a 1 dimensional object all 1 dimensional traits of it get 2 times bigger all 2 dimensional traits get 4 times bigger all 3 dimensional traits get 2 to the 2 to the 3rd times bigger right and so it's the exponent on the scaling factor that tells you the dimension and that in some sense is what dimension is right that's what dimension is and so I want to look at a particular object there's actually examples of it upstairs here I want us to take a triangle and split this triangle into 4 triangles that are all the same size and remove the middle one everybody got that and how many triangles do we have now we've removed the middle one it's gone oh we got 3 right and now each one of those smaller triangles I want you to do the same thing split it into 4 triangles remove the middle one how many triangles do we have now 9 very good oh wait we're celebrating being wrong 9 okay so now we take each one of those 9 triangles we split it up we remove the middle one and we keep doing this and we keep doing this and we keep doing this and we ask the question if we do this forever what's left and in particular the question I want to ask is what's the dimension of what's left and let me just give you 2 arguments here one argument is that we started with a triangle which is clearly 2-dimensional and at every stage what we have is a bunch of triangles that are all small but still 2-dimensional and so this has to be 2-dimensional here's another argument we keep taking out the middle we take 2-dimensional objects have to have some sort of middle and we keep taking the middle out and so maybe but we never touch the edge and so maybe like that edge is just 1-dimensional so maybe what's left is 1-dimensional and so what I want you to do is I want you to write on your paper this is question 5 what's the dimension of this which is called the Sierpinski triangle what is the dimension of the Sierpinski triangle go ahead and write down what you think the dimension is write it down on your paper and then and then compare notes with the people around you go ahead go ahead and write it down and then talk to them alright here we go we're going to figure out what the answer is together here's the thing if you take a square and you double its size you can actually see that you have 4 squares as big as the original in there right but when you take this object and you double its size right now we just have 3 of them which seems really weird right if you take the square what you get it's a 2-dimensional object and so you get 2 squared times the area of the original but with this thing we get 3 of the original we're not expecting the 3 we're expecting the scaling factor 2 to some power and that power was going to give us the dimension but now we just get a 3 so in some sense we were trying to get we responded just to get 2 to the dimension and so what has to happen is that 2 to the dimension has to equal 3 and you need some you need some algebra to do that you need logarithms you figure out your logarithms and the answer is that the dimension of the Spinski triangle is the log base 2 of 3 about 1.585 let's hear it if you were wrong come on let's hear it alright alright right you get a fractional dimension once you understand how to measure dimensions the inescapable conclusion is that dimensions can be fractional you can have dimensions in between 1 and 2 in between 2 and 3 you can have wonderfully fractal things which have dimensions anything you want in there and that's just an absolutely amazing fact alright one more one more demo here so I've got the spool the spool always reminds me of the age old toilet paper question like do you have the toilet paper hanging off the front or the back right and so we're going to do something similar this table is a little slippery so I'm just going to put this down to add a little bit of friction and so the question is here's an easy question if I have this if I have the spool like this with the thread coming off the top and I put this spool down and I pull it rolls in the direction that I'm pulling everybody okay with that that's exactly what you would expect the question I have for you is what happens if I flip it over if I flip it over so that the string is coming off the bottom right and then I'm going to put it down here on this table and then I'm going to gently pull in the direction I'm going to pull to your right and the question is do you think it rolls toward the puller do you think it rolls away from the puller right and I'm just to be clear I'm going to pull gently I'm not going to jerk this thing right everybody understand the question answer take a stand right down an answer and then turn to your neighbors and decide and convince them you're right go alright here we go here we go I'm going to pull this gently and we're going to see which way it goes here we go gently pulling I'm a little off center I'm sorry gently pulling and it definitely rolls toward the puller let's hear it let's hear it you were wrong alright I'm going to give you I'm going to give you two explanations for this one of these two explanations is clearly wrong here's two explanations for you here's one explanation is is that you know if I'm pulling the only force is toward the puller if the only force is toward the puller then the motion has to be toward the puller that's one explanation here's another explanation what's really going on has something to do with the only point with friction in here at all is actually the point where the spool touches the table and since I'm pulling above that I'm going to get it to rotate in that direction around that point which seems weird because you're thinking about it rotating around the center but if you rotate it around that then it's also going to go toward you those are two arguments which one of which has to be has to be junk but I can I can demonstrate why one of them is junk you see if I pull high enough I can get it to go the other way I can get it to go the other way I'm still sort of pulling in this direction but I'm pulling at a higher angle for those of you in the back if I pull at this angle it rolls away but if I pull at this angle it rolls toward you and in fact there's this cool demonstration you can do if you have a dog you might do this I can take my spool for a walk if I get the angle exactly right I can take it for a walk and it doesn't roll at all it just walks along like that without rolling at all and that's really fun so let's hear you were wrong you were wrong all right these strangest paradox in all of mathematics is called the Bonak Tarski paradox and I have to tell you exactly what this says the Bonak Tarski paradox says you can do the following with either a solid ball or just the shell of a ball either one you can split it up into just a small number of pieces like six pieces you can take those pieces you can take a few of them like three of them over here reassemble them into a ball the same size as the original and then take the remaining pieces and twist them into another ball the same size as the original let me say that again one ball split into pieces take some of the pieces over here get a second ball other pieces over here get another ball right so you start with one ball without adding or subtracting any material you end up with two it doesn't matter so the two the two balls at the end are the same size as the original the same size and shape they're both spheres and you can do this right and in fact if you felt like it you could do it again and again and again and again starting with from one you could get as many as you wanted I have to caution you what I've said is true if you're thinking that you should go down and buy a piece of gold apply Bonak Tarski to it and get two pieces of gold and if it's going to make you rich you're sadly mistaken the pieces that we're talking about are really complicated I mean they're sort of like if I said here's one piece of numbers between zero and one one piece is the fractions that are sort of sprinkled we can call that one piece that'd be an example of the kind of piece that we're talking about what you saw in this animation is just sort of an artist's interpretation of this you can't actually the amazing thing is you can think it not only can you think it but you can prove that it is true and mathematicians have done this and the end result of this is that not every set is measurable there's a branch of mathematics called measure theory where we deal with the consequences of this you can't measure everything and so some things you just have to agree you have to sort of give up the fact that they're measurable anybody wonder what this picture is looking at all night it's not my backyard it's a neighbor's backyard but this is true I want you to fill in the blanks for me here every autumn the leaves turn colors and then they yeah you're all wrong about that here so I have the pictures to prove it so this is a neighbor's tree and one of the branches on this tree came down in a thunderstorm in July and so there's the thunderstorm took down that branch it was cracked off but it was still hanging there and if you notice all the leaves on that branch are brown like they've dried up already and if you go through the summer so here we are a little later in the summer the rest of the tree is doing fine still brown leaves on that here we are into the fall and so a lot of the leaves on this tree have fallen down notice on that branch they're still hanging on and they're still brown you get into winter there is not a leaf on that tree and now I claim this proves that we're wrong about this and so I asked like leaves don't fall they're pushed this is what's really going on you know we're actually wrong about the turning colors thing too like when we see maple leaves that we think are turning red that red was already in there it was just completely overwhelmed with green chlorophyll we couldn't see the red for the green that was in there right and what's happening in the autumn the tree is sucking out everything it can get out of these leaves and when it's done sucking nutrients and everything else it can in chlorophyll out of these leaves it actually builds a layer of cells that are designed to cut off the leaf it's like thanks for all the nutrients see ya and off goes the leaf and in this branch that's hanging there it was cut off from the tree and so the tree didn't have the ability to extract all those nutrients and then cut off the leaves right and all of those leaves they didn't just fall they're pushed that's what happens in the autumn leaves get pushed I think all of you were wrong about that let's hear it here's the challenge I have for you and this is the challenge I give to my students I teach a class in mind bending math paradoxes at St. Mary's College in Maryland and what I really pitched these paradoxes is a little playground where we can practice changing our minds when we think about changing our minds we think changing other people's minds we put ourselves in the position of Bob Eberling the real challenge is what if you're in the position of Bob Eberling's bosses and you are wrong you are dead wrong about something what are you going to do are you going to have the ability to admit that you're wrong and change your mind or are you going to stick to your guns humans are incredibly good at ignoring evidence that comes in because it doesn't conform to what they already believe humans are incredibly good at dismissing the messenger like oh he's telling me that he's wrong but he doesn't know what he's talking about she doesn't know what she's talking about we are really good at sticking to what we already believe when it comes time and it's your moment will you change your mind and so I'll leave you with a challenge it's the hardest assignment I ever give it's my student semester project you know on some level lots of things find something important to you that you're wrong about admit it and change your mind thank you very much alright we have time for a couple of questions so if you could raise your hand I will bring you the microphone sorry about the little mess over here good I just want your thoughts on a thought I had but when you were talking about the rational numbers and the probability of picking in the graphic that came to mind was stars in the universe like the empty space and maybe the numbers being stars and the chance of getting started do you see any parallels? I like the fact that you're searching there for parallels and something that the idea of our universe and how mind-bogglingly large it is and how many stars are out there it definitely gets across the idea of huge numbers versus much smaller numbers the fact that our sun billions of stars within our solar within our galaxy thank you for catching me there within our galaxy and then that's just one of billions of galaxies so the numbers here are immense that doesn't go far enough to get zero that only goes far enough to get really really really small but mathematically speaking the chances of picking a rational are actually zero which in order to get to actually zero you need to be able to think about infinity which is what so much of mathematics grappled with and mathematics only figured out the idea of infinity in the 1800's 1870's 1880's and when mathematicians figured out how to understand infinity a lot of mathematicians didn't believe it they were like no that can't be right and so mathematicians had a hard time changing our collective minds about that you really have to grapple with infinity in order to get that but it's sort of along those lines yeah question over here hi thanks for a great talk to build on what you were just saying about mathematicians sometimes being changing their minds a little mathematics is a very certain subject where you are very definite about being right 100% right can you give us some examples when mathematicians were really wrong and changed their mind examples where mathematicians were really wrong so that's one of them I think that the example is Cantor put out his ideas about infinity and he was ridiculed by a lot of mathematicians for being completely wrong about that and it was only a small group of mathematicians who started to build confidence in that and that being right other examples where mathematics got it really wrong does somebody have one off the top of their head Euclid's parallel postulate great so Euclid had five axioms that all of geometry was built on and for a really long time people tried to prove the fifth one from the first four and most mathematicians thought you could thought you could prove the fifth one from the first four and there were lots of times when people thought they had proved the fifth one from the first four and then it turns out you cannot and in fact you can develop lots of systems where the fifth one is wrong and that's how you get spherical geometry or hyperbolic geometry and things like that so there are lots of cases where mathematicians get it wrong here's another case where mathematicians are intuition is wrong there's something called the axiom of choice so it's a fairly complicated thing but the axiom of choice is one of the axioms that we all agree upon most mathematicians agree to accept it sort of on the same level of line as like there is a zero axioms that we'll all start out with and we all the mathematicians know that there are all these other theorems out there that are equivalent to the axiom of choice because we have this list in our heads so let's see the well ordering principle is equivalent to the axiom of choice Zorn's lemma is equivalent to the axiom of choice all these other things are equivalent in the sense that if one of them is true all of them are true if one of them is false all of them are false but this is what most mathematicians think most mathematicians think the axiom of choice yeah that's true most mathematicians think the well ordering principle okay that's false and most mathematicians think Zorn's lemma I can never remember what that one says but at the same like we can hold these thoughts in our heads like intuitively it seems like the axiom of choice is true and the well ordering principle is false mathematicians can hold that in our heads at the same time holding in the fact that we know that they have been proven to be equivalent and so we are clearly wrong about this in some sense does that make sense? great do we have any other questions? yeah over here I'm not allowed but are you able to explain the gravity blocks to us? you know if I explain the gravity blocks then I'm like being every teacher and I told you like there's a problem with that and so this is what I'll say like I'm gonna be here for a bit and so like when we're done we're done with the Q&A come on up and you know you can try your hand at playing taps you can pick up the gravity blocks and see what you think I'm not gonna give anything away tonight but you know email me what you think in a few days and see how that goes and here's the thing I'm gonna keep thinking about it and it's gonna eat at you and that's the point like that's my whole point like keep thinking about that we have another question over here yeah one more question could you elicit what you were trying to state about the difference between fractions and ratios? oh fractions and ratios yeah so fractions when you add fractions you have to find common denominators right when you're adding a fourth and a third you have to find a common denominator right so you change those into common denominators and once you've done that you know what a fourth and the third right is three twelfths and four twelfths and then when you add them you only add the numerators and you get seven twelfths if you're thinking about those as ratios if somebody shot four out of twelve in the first half and three out of twelve in the second half what did they shoot for the whole game? they did not shoot seven out of twelve you don't keep the denominator the same for the whole game you add both the numerators the shots they made and the shots they they took and you get seven out of twenty four right and so when you're adding when you're talking about ratios the rules are different and the rules we have for fractions don't apply one of the rules we have for fractions is that you know if this fraction is bigger than this fraction this fraction is bigger than this fraction then when we add across the result will be bigger and that is totally true for fractions and not always true for ratios that's the point that makes sense? excellent thank you