Harvard Book Store; WGBH Forum Network; Robert and Ellen Kaplan: Hidden Harmonies of the Pythagorean Theorem

Good evening. My name is Ryan mead and on behalf of Harvard bookstore I'm thrilled to welcome you to tonight's event with Robert Kaplan and Ellen Kaplan. They're with us tonight to discuss their new book Hidden harmonies the lives in times of the tiger in there. Tonight's event is one of many interesting talks that Harvard bookstore is hosting this spring upcoming talks include appearances by game designer Jane McGonigal professor of psychology and neuroscience V.S. Ramachandran and MIT professor of technology and society. Sherry Turkle these events and more are listed online at Harvard dot com. The best way to find out about upcoming events is through our weekly email newsletter which you can sign up for by visiting Harvard dot com and clicking on subscribe. He can also follow us on Twitter become our fan on Facebook or pick up a paper event schedule at the information desk. After tonight's reading well have time for questions from the audience. And at the close of the talk will have a signing at this table. Of course you can

find copies of hidden harmonies at the registers behind you. I thank you for purchasing books from Harvard bookstore and attending tonight's event. Your participation supports the existence of not only this author series but of an independent and community focused bookstore as well. A gentle reminder that now is a great time to switch off or silence your cell phones. Tonight on behalf of Harvard bookstore I am honored to introduce Robert Kaplan and Ellen Kaplan to discuss their latest book Hidden harmonies the lives and times of the pet tiger. With the wit verve and clarity to Kaplan's trace the life of the Pythagorean theorem from ancient Babylon until the present. Inspired by their Harvard based math program these papers of mathematics transform the relationship relating the sides of a right triangle into an ancient oak in the landscape of thoughts. Anthony doer of the Boston

Globe claims hidden harmonies possesses an alluring lyricism and a good sense of humor. It's often fun to be around Robert and knowing Kaplan are founders of the math circle a school for students of all ages teaches the enjoyment of mathematics. Classes are held at Harvard University and Northeastern Robert Kaplan is the author of the bestselling book The nothing that is which has been translated into 10 languages. Helen Kaplan is co-author of chances are and Bozo sapiens why To err is human. Without further ado ladies and gentlemen please join me in welcoming Robert and Ellen Kaplan. We're not going to read a lot from hidden harmonies which is more a book for mulling over at leisure and letting your thoughts

take off from its pictures. I did the pictures. I will read enough to give you a sense of the great faggery an adventure. And then let's talk together about its pleasures and those of math in general and also about the scary parts. So I'll begin at the beginning. The Englishman looked down from the balcony of his villa outside Florence. Guido the peasant six year old son was scratching something on the paving stones with a burnt stick. He was inventing a proof of the Pythagorean theorem. Do just look at this dude he coaxed into Joel. It's so beautiful and so easy and Guido show the Englishman's son how the same square could be filled

with four copy. Of a right triangle and the squares on it sides or the same four triangles and the square on the hypothenuse so that the two squares of his first diagram must equal in area the square in the second and the Englishman for the vast differences between human beings. We classify men by the color of their eyes and hair in the shape of their skulls. Would it not be more sensible to divide them up into intellectual species. There would be even wider goals between the extreme mental types than between a bushman and a Scandinavian. This child I thought when he grows up will be to me intellectually what a man is

to a dog. And there are other men and women who are perhaps almost as dogs to me. But the child never grew up. He threw himself to his death and despair at being snatched from his family by a well-meaning señora who forced him to practise his scales and took away the Euclid that the kind Englishman had given him. True story. Emphatically not. It's all this Huxley's young Archimedes published three years after Sir Thomas Heath's history of Greek mathematics came out in 1921 with its quotation from ancient Colin MacOS by happy chance Beth ecclesia son found old faily scraping the ground and drawing the figure discovered by Pythagoras the falsity of the story isn't just in Huxley's having patched it

together from what he had read in he and from the legend of the brutal centurion who sent to fetch Archimedes killed him instead because he wouldn't stop drawing his diagrams in the dirt. It is much more deeply false false to the way mathematics is actually invented and falls to the universe ality of mind. It's certainly dramatic to picture super humans in our midst living putdowns to our little pretensions. Yet testimony to more things in heaven and earth but the actual truth has greater drama still woven as it is of human curiosity persistence and ingenuity with relapses into appeals to the extraterrestrial. This is the story we shall now tell. The great thing about math is that you never have to take anyone's word for it.

It's the only human undertaking founded on irrefutable proofs. Not surprising then that there aren't too many of these not too many foundations not too many theorems that is statements that have proofs that someone or other came up with attached to them. So the Pythagorean theorem is especially astounding because it has several proofs it has more than one proof. How many would you guess there are. Hundreds of other guesses do I hear. Tens. Well let me read from the book. Some people collect Tkachuk on beer coasters. Some stir me Archer three speed hubs others wives and elements. Jury whip are collected proofs of the Pythagorean theorem. He wasn't the first. Fifty nine years before him in one thousand twenty one Johann Joseph

Ignacio's Hoffman published more than 30 in 1778 A Frenchman named Fool Day included 38 among his curiosity Tasia which week and ahead glop had translated others from a Russian anthologist. Nor was jury whipper the last professor's yawning and called her head from the universities of Worcester and Ohio and curry in Pennsylvania. Gather together some hundred proofs between 1896 and eight hundred eighty nine. A lawyer at the District of Columbia barred him. Arthur Kober and published one hundred eight of his own starting in one thousand ten perhaps the currents of lead litigation ran more slowly before air conditioning. On the shoulders of these giants still stands. Alicia Scott Loomis the boy born in a log cabin in 1852 who rose to be a 30 second degree MD Freemason and quote cloud have

information grooves in the plastic brains of over 4000 boys and girls and young men and women. He tells us that of all the honors conferred upon him he prize the title of teacher more than any other quote either educational social or secret. Was he as Lenin as his portrait shows in Florida mustache and wing color. You'll have to buy the book to see that. Or should we believe a pencilled note in the Harvard library's copy of his book. He was somewhat high in manner but was in reality a good sport. I never met him. Then how did you know. G W Evans. Thank you re an investigations breed mystery. Here's another. How many proofs are there in Lewis's book. B of Babylonian you say and count them but counting as even a Babylonian do is one of the hardest of human tasks.

Lewis claims to have three hundred sixty seven proofs in his second edition though some are circular. Some defective some no more than variations on or subsume by others. Are the algebraic proofs which he says readily follow from this or that geometric demonstration to be thought of as different from them. He asks about a possible proof here. Quote can't calculate the number of others there and speaks of several A number of and countless different proofs from those he gives. Nine thousand seven hundred twenty eight proofs for example derived from his figures for algebraic proofs 6 and 7. He tells us. Sixty five thousand seven hundred eighty more from his figure 8. When he writes. As for his geometric proof one hundred ten of more cases extent. Does he mean more than he has given. Are two proofs really different. If a square in one has no more than slid sideways from

that in another. Or if a grid of lines is differently parsed. We conclude that his book contains three hundred sixty seven proofs minus a few plus several increased by number drivable but not in fact derived two which are added those that are other and different resulting in many plus a multitude increase by a limitless as well as an unlimited horde of the likely at a slew indefinitely great that will be discovered by he says the ingenious resources of light and ideas of the mathematical investigator giving us as an approximate total more than we should or could or may or want to count this absolute who does consonant with the generous spirit of Brotherhoods by thuggery and or Masonic and is an image of life itself in the earth below each tree of spreading order. The mice are somewhat normal while chaos in its foliage is

made by the insects etc.. Numerous credits his 14th such a small number geometric proof to miss a Coolidge a blind girl. He then gives us no more than a reference to the journal he found it in. What were his curiosity. His imagination his compassion not stirred. Did use compulsion to move on to the next in the next and the next leave him no time to wonder at the visions of the blind not being in such a hurry. We hunted out the Journal of education body of twenty eight eighteen eighty eight through the stacks in Harvard's Gutman library where it has who knows how many years before been mis shelved giving the

diligent librarian a dusty two hours before she cornered it down a stack receding to infinity. That made my day she said. Terrifying view of life. And there on page 17 was Miss Coolidge's proof. Across from the Notes and Queries. What is the difference between Belfast in Chicago Belfast in Maine and Belfast in Ireland. What is the guinea n sect. Try googling it it's not there for proof however was not as it appears in Loomis who had clearly exercised his editorial powers over it. He was somewhat high in manner. We give her original proof in the book with all its extraneous steps. What's particularly fascinating about this proof is that Miss Coolidge begins it

geometrically making and comparing shapes. But. Then turns to calculating with letters when perhaps she can't quite figure out what's left and what's wanting. You might think that being blind Miss Coolidge would have resorted to abstraction as early as possible. But in fact she delays it as long as she can trying to stay true to the spirit of geometry. Having followed for proof if we now follow her we may better understand the play between obstruction and different sorts of sensory information. But how find the woman behind Lewis's brief Miss Coolidge. It struck us that Coolidge is a good post a name and the journal in which her proof first appeared was

published in Boston. Might she not then have been a student at the Perkins School for the blind. We e-mailed Jan Seymour for their research librarian who answered. That was inspired guesswork. Emma a Coolidge was a Perkins student. She was born on August 4th 1857 in Sturbridge mass and she lost her sight from whooping cough when she was a year old. She could only detect light and shadow.

And none of the remedies her parents tried having her wear gold earrings putting talcum powder into her eyes blistering her temple with poison flies helped. After graduating from after graduating from Perkins she studied at Wellesley for a year then returned as a teacher to Perkins where one of her students is said to have been Annie Sullivan. Later Helen Keller's tutor Emma married had a daughter wrote children's books taught music in her New Hampshire village school sewed knitted could kill and dress a fowl for dinner. If her husband was away and boldly went out alone tying white rags to doorways so she could find her way from place to place. Isn't this what we see in her proof. Or what you'll see in her proof if you look in the book where the proof is

catching the cure a skewer of prominent shapes but navigating otherwise by those abstract relations with which practice and memory furnish the mind. What sort of imagination this involves may matter here. The blind mathematician Louis Antoine was led by his advisor. Oh you live a study two and three dimensional typology is quoting the bag in such a study. The eyes of the Spirit and the habit of concentration will replace the lost vision. His equally outstanding compatriot Bernard Morva has been completely blind since age 6 was asked how he knew the correct sign in a long and difficult computation by feeling the weight of the thing he said. More

tellingly more un distinguishes between what he calls time like and space like mathematical imagination and surprisingly says that he excels at the latter. A problem with picturing geometrical objects is that we tend to see only their outsides which hide what might be complicated within MOA who works with extremely intricate objects in three dimensions has taught himself how to pass from outside to inside from one room to another. Our spatial imagination he says is framed by manipulating objects. You act on objects with your hands not with your eyes. So being outside or inside is something that's really connected with your actions on objects. Think of Emma sewing and knitting or

killing and dressing fellow. Might Emma's extraneous calculations have come from an intrusion of the timelike into her sounder space like imagination and is the tactile yet one more intermediate between being and becoming. So. Spatial imagination. What does this mean. What does it mean when physicists talk of string theory and see these strings in leavened dimensional space 11 or 9. Which is it. What does it mean when that magicians toss around 12 and 200 dimensional space in their coffee fueled conversations. They aren't superhuman like us. They can only see three dimensions. So let's go back for centuries together to look for an answer. This is from the book in 17th century

the master regular Yohannes fellow Harper discovered a spatial generalization of the Pythagorean theorem and saw how it will lead him to a deeper understanding of six hundred sixty six which he already knew was divine rather than diabolic while the margins of our minds are too narrow to contain fall Harbor's cabalistic proof we will unveil to you his rectilinear pyramid. Analogy said his contemporary Yohannes Kepler is the greatest of my masters moving by analogy from squares on the lengths of a right triangle sides just squares on the areas of a right pyramids basis. He came up with this beautiful generalisation given such a pyramid three of its edges meeting at right angles as in the corner of the room. The sum of the

squared areas of these faces a B and C equals the square of the area on the last hypothenuse space d.. That is a squared plus B squared le c squared equals D squared. If not quite up there with 666 still pretty wonderful. In the winter of 16 19. Take heart. Passing through them. They will have met fell harder. A local legend has it that one night dazzled by Jake Hart's brilliance fall hover reached out. I picture them as sitting opposite sides of his fireplace in a stuffy comfortable room with weather freezing outside the harbor reached out and touched his guest to make sure that he was human and not an angel. Wouldn't you have done this much where your visitor to have mentioned that is there are also held in four dimensions so enough of the book conversation. What do you make of that

what do you make of this talk of dimensions beyond those we can see. Is this meaningful is it meaningless is it a game is it the most important thing in the world. Any Of The Above all or none or anything you'd like to talk about. Whether it's the Pythagorean theorem and its consequences it's applications. The still unsolved open problems in it is that a question Rick. Yes. First. Off. Just looks like you're right. It looks like so what. What is a proof. What constitutes a proof. Why are not the algebraic manipulations with letters and symbols an equal size Why are those also an optical illusion or a linguistic illusion.

Why are all proofs illusions. What isn't an illusion. Yes. Yes yes. 0 0 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. We do with 8 year olds. He's. Let.

Me read. This. But I so I see your point here is. That. You write your name right almost as an origami or using of origami like proofs for Angle trisection or supposedly of legitimate out of. Oh yes oh yes yes yes yes yes yes yes yes. Where oh oh oh oh oh there's a beautiful fruit the 64 65 if you take the right sort of book rectangle

apart and reassemble it like this. Not one that means very well. But remember there are there are cultural reasons behind at the time. This is the third century A.D. that the time they said this three and four were very important numbers 3 was the think of origami proof or think of

the proven president of an angle. Three was the circumference of a circle of Diana radius 1 and 4 was the circumference of the square of side once a three and four key numbers and thought of well as my favorite number. So. So if it's true of that key mystical triangle Well of course is going to be true of other triangles. And there was another factor in Chinese in Chinese thinking about this you'll see this in the book that for them the only triangle the head area was a right triangle because you have height times length of base give you area other triangles more or less head area. Rather like the problem with this 64 by 65 rectangle. But I think the you've touched on a real nerd in mathematics. How does one validate a proof. Chris custodian custodian steps us who takes who who's in charge of the people in charge. What finally validates anything

in the century that's passed and the one that we're in the midst of the foundational issues or devastating juggles seems to have proven that with nothing matics as we know it there will be true statements that you cannot prove and yet they're in some sense true. What does that mean. And when you take a proof which has all the paraphernalia of rigor attached to it with the questions and symbols reduced to definitions and axioms well why take these actions rather than those a big issue with a bit like me and theorem where a service is other than a plain analogues of the theorem holds but not a square of the speed squared equals C squared. Perhaps something with CO signs will hold instead on a sphere or on a and inverse fear looking like looking like well how can one see what it looks like other than give a picture there. Yeah

on that you get something with hyperbolic co signs which is the equivalent of the Pythagorean theorem. So which surface. Does the theorem hold. What is the mean. I mean we don't actually. Therefore the shorthand way of describing what's going on is with what's marvelous about anything like this is that we can hear a stance that Leader Reid would actually hinder to find out things from you. So we would love to hear more of your ideas about what constitutes proof in mathematics. It allows them to get the basic idea when you start something you know you know to be true. Oh this is this is how you perform operations that you accept. Yes right.

So I don't know Station of Lenny. You know. Whole nations. Well so you say well you say that I start from one place that I may be sure and I want operations there. He set the record right there the thing that I've come up with. But there are also I mean it's nice that you have two parts to your to your story. One no. You know the foundation and you accept the way of developing from it to different kinds of research a little bit of reason I guess that I've seen many run experiments where many lation come up with ridiculous Yes not like this or they're something. You basically say you're one of Galileo's patron Guido Bondo good Dante

came to Galileo one day and said I have a proof that God exists because God alone makes the universe the one out of nothing. And he was able to prove that zero is equal to 1. And the proof is wonderful and fat you'll go home believers. Zero is equal to 1 minus 1 plus 1 minus 1 plus 1 month is 1 forever right. That's 0. I'm just going to say again what came after the equal sign zero equals one minus plus one minus one plus one minus one plus one might as well. All I've done is just move for the C's around. But that equals 1. So 0 equals 1 so God exists now. What was wrong there what the. The foundation seemed pretty secure there is a zero there is one.

I think there has to be a zero because I wrote a book called the nothing that is I don't know about the one but what about the accepted ways of moving from that can you re associate sons. Well the associative law of addition says you can put the associative law of addition says you can for a finite sum were not for an infinite series though sometimes you can with a telescopic series. Sometimes a solution. Actually when you have a sentence which I think is relevant to you. Which is about why of all of mathematics I feel the only thing I have any understanding of is geometry and that geometry is the one open road into mathematics. You are seeing something you are drawing something you or it can be embodied and you feel that you're much less likely

to be due than you are in the middle of a calculus proof or suddenly everything goes to zero and you're left saying I guess which is what I spend a lot of time in my math classes saying when people say Isn't it obvious that and you said. We know in Parys Syrians what it is like to be there and you know if you see this is bigger than that you can look and you can measure and you can understand. And so we can draw I confidently into trying to prove because once you're convinced that something is true you're you. Amazing how much harder work you're willing to do then when you're not at all

sure that what you're working on is in the right direction. On the other hand the. Contrast of this is a sentence from the book. A proof can this is us by showing us something apparently surprising or unlikely. Lies in a matrix of relations we recognise and truths we acknowledge. Trusting is taking up several points made this evening. The fact that there are so many proofs of the Pythagorean theorem doesn't that just by sheer dismiss in a democracy doesn't make it true or that there's something different ways of coming out and that it it it is just not some isolated fact with one thin thread of a proof to it it's there woven into our thought. In fact the text of Lois Yes or take what. Down the line that the Titian said about the heft the weight of the equation he he felt with the sign of it would have to be. And

a late chapter in the book called The Deep point of the dream. Comes up says makes clear I hope makes clear the surprising truth that Euclidean geometry the geometry of fly planes which is distinguished from other geometries curved up in a sphere or like that strange shape the geometry of the flat plane is based on one postulate. The parallel postulate that through a point not on a line there is one at only one parallel to that line. That's the postulate which distinguishes our visual intuition and the geometry of flat planes and it's equivalent to the Pythagorean theorem which is quite astonishing. The two can be interchanged take away the parallel postulate put in the Pythagorean theorem

as your postulate. You'll get all you could in geometry back kind of a maze of other issues you'd like to bring up. Yes. Mrs. Brady we have her in the room with us one of the 10 percent of the mathematicians in the world who prefers the algebraic to the geometric. This is not true John. It's. Not counting is so hard to. Think of limits how many proofs of that book. Yeah.

Yeah. But. They still use my records. It seems to be it seems to be true of I haven't read a authentic survey but they'd let us do sociology read just how many people thought geometry easy comfortable and calculus. Yeah. Oh OK. All right algebra algebra. You may soon begin dividing by zero. OK invented this.

You ARE YOU are claiming that both of these were invented. Ha ha ha ha ha ha ha. All right so now. Let's let's let's say you make it you make a recession. See I see what you get. Yet you say no he has no say no more. Say yes say yes. Yes yes. But this time. Or maybe make a wrist magick. Not even algebra recently versus geometry. It was long about how many now

feel more comfortable with geometry than arithmetic. You are in fact a minority whereas arithmetic you're actually using seeing the results of that and the other confirmed by six plus seven is seven plus six six times seven which is harder is seven times six if you take a marching band which is six rows of seven high and just rotate it 90 degrees Well I just as a geometric you know the six times seven is still seven times six. Yeah yeah true yeah. If you think of the axioms of arithmetic the fundamental

the things we take for granted good commutativity that you can it doesn't matter the order in which you add the numbers or doesn't matter how you associate them unless you have an infinite sum NRA patron of Galileo's. Those those do ring with a deep truth unsay of beyond the visual. In fact you could you can close your eyes and do arithmetic. So it has a a dug deep ring of truth to it. I don't know proof says the note to end on. We don't know. Yes when you watch be sure you are. Right. Right. Now you know you write. Your own

money. Very good case. You know. No. Not at all when I look at the fruits of my house with me. This heat is going to suck. Well yes but that would be odd. I think where yes buy something locally as well that. Way is what we're doing. I forget do you think the notion truth is is an abstraction from the feeling of what was said you will have being right I think it was always so. Oh please you're really. Robert Byrd you are. Oh.

Yeah. Yeah yeah of course it was. I have a very poor sense of direction. I feel I know how to get home from here. I probably make a lot of mistakes with me but I feel I know it. This is why I put it into practice that it begins to fall apart. But that is wordlessly squared equals C squared. When those of the sides in the high ponds were right triangle I can't unthink it. I can't I can't make it not to be true no matter where I am. Ah that's very hard to do. Well that's of course what I'm on not of what I think. Thanks very much for coming.

We learned a lot. Thank you for coming. We have copies of it in harmony is that the cash registers behind you and will have a sign in here at this table. Thank you.