> [!cite]- Metadata > 2025-07-22 19:08 > Status: #secondary #course > Tags: `Read Time: 1h 5m 36s` > **Terence Tao** (born July 17, 1975) is an Australian mathematician awarded a [Fields Medal](https://www.britannica.com/science/Fields-Medal) in 2006 “for his contributions to partial differential equations, combinatorics, [harmonic analysis](https://www.britannica.com/science/harmonic-analysis), partial differential equations and additive number theory.” > [!Tip] [Class Guide Book](https://drive.google.com/file/d/1ZPk670cU6Xe1cNqUHMAdCA_dhE4bbcGD/view?usp=sharing) > [!Abstract]- 1. Meet Your Instructor > I find mathematics beautiful. The act of stripping things down to the bare essentials creates a certain elegance that you wouldn't have seen if you tried to keep the original problem in all its complexity. > > In 2006, Professor Tao won the Fields Medal. His single most famous result is a proof that you can find billions of evenly-spaced prime numbers. > > When I first learned mathematics, I didn't appreciate just how connected it was with the rest of the world. I like concepts that make unexpected connections. Anywhere where there's a pattern or some shape, it's there. And it's exciting and thrilling. > > I think people often have a fear of mathematics, that it's just too alien, too different from the everyday experience for it to be useful in their lives. To me, mathematical thinking is just an extension of ordinary everyday thinking. Mathematics is often about trying things you don't quite understand, and failing the first time. But in mathematics, failure is OK. You can just keep experimenting and keep playing. You may not achieve your original goal, but you often learn something. > > I like to think that this class is for anyone really. I think everyone has an innate ability for mathematics. In fact I think this class may be more suitable for people who have not had formal mathematical training, because they might get more out of seeing what math is actually like. > > I wanted the opportunity to show that knowing how to think mathematically can be a tool that you can use to solve problems in a systematic way, in a way which can give you reassurance and can make a complex world a little more manageable. > > A large-part of this class will be talking about problem-solving. There is a certain way in which mathematicians approach problems. We abstract them. We break them up into pieces. We make analogies. We try to find connections with other problems. So my class is not an advanced math class. I'm not going to write down any equations. I'm not going to give you homework. It's about what mathematics is like and hopefully how some of these problem-solving strategies and thought processes can be of use to you in your daily life. > > I would like to convey just the sense of, a little of what it's like to be a mathematician. You know, we are not wizards or aliens. We're just regular people solving problems. > > **Chapter Review** > > Mathematical thinking is actually an extension of ordinary, everyday thinking. It's not magic. > > It's okay to fail! Failing is how you learn to approach problems differently, and find creative solutions. > [!Abstract]- 2. Terence Tao's Journey > I can't imagine life without math for me. I've always grown up liking maths puzzles, and I've been a mathematician my whole adult life. I really can't imagine life any other way really, but I remember growing up and thinking, I should be a storekeeper. Because I thinking I can handle balancing the accounts and inventory. It seemed like something I could do, because I had no idea what a mathematician did. > > I thought that there was some authority that handed them big difficult problems to solve, and you just handed them back when you solved them. And you almost never see a mathematician portrayed in popular media, except maybe some slightly crazy genius or something. And for most of my teenage life, I was mostly with kids much older than me. I skipped a lot of grades. In fact, I skipped five grades. When I was in primary school, I would take some high school classes. My parents certainly tried very hard to balance my math and science type activities with everything else that a kid does. I participated in a lot of mathematics competition. They were almost like sports. There's a time limit. There's a certain number of questions you need to solve in a certain time, and it's quite different from the experience of mathematical research. > > The analogy I like to give is that mathematical competitions are like contributor sprints, and mathematical researchers are like marathoners. They take sustained effort and you need a lot of stamina. > > **Find a Mentor** > > I was lucky to have many mentors. My first was my mother. She was a high school math teacher. She went through the basics of math with me from a very early age. There was a retired math professor who I would spend weekends with. I remember he had a little journal, a little math journal, where he would write solutions to math problems and phrase them in terms of little stories. And I remember writing one of these stories myself. I was very proud contributing to that journal. > > As an undergraduate I had an excellent advisor who recommended me go into study abroad to do my PhD, and when I went to my PhD, my PhD advisor was very supportive. You know, he gave me a lot of tough love. He definitely motivated me to work harder and to actually impress him. As time goes by, I find more and more rewarding working with the younger generation. Mentoring them. It feels good to pass on my knowledge and tips to other people and see them take things further than I ever could. It's very satisfying. > > **Mathematicians Are Humans, Too** > > I think mathematicians are humans, like anyone else. You know, we have frequently imposter syndrome. We feel like we don't deserve to be doing math, and everyone around us seems to be doing better than us. Because we're always faced with problems that we can't solve, and sometimes, we wonder whether we have what it takes. We are intimidated by people more senior than ourselves. I remember as a graduate student I wanted to meet with a very famous professor once, and I walked all the way to the campus, where he was and I stood at his front door. I was going to knock. This was before email. I did not set up an appointment and I chickened out. It was just too intimidating. > > We're normal people, like everyone else. I think maybe a little bit more nerdy, more socially awkward maybe, but still pretty normal. > > **Navigating Math's Landscapes** > > So what life lessons can one draw from mathematics? One lesson that, I think, really hit me was just how connected everything is. So when I learned mathematics, as a student, every single aspect of mathematics was on a separate subject. I took a class in algebra. I took a class in geometry. I took a class in calculus. Then I also learned physics, chemistry and all the other subjects. They were somewhat connected, but it was only much later in my career that I realized that all these subjects are really part of a single body of knowledge. And there are so many analogies and connections between these subjects. > > The way that I've heard it described sometimes is that mathematics, when you learn it for the first time, it's like you're looking out at a landscape, which is full of mist, and all you see are a few mountain peaks. There's a mountain peak of algebra. A mountain peak of number theory. But as you keep learning this mist lowers and lowers, and you start seeing some valleys. There are more and more of these valleys that appear. And the more advanced your studies are you see deeper and deeper connections. It's only at the undergraduate and graduate level that you start seeing all these connections. It's a bit unfortunate that all the good stuff in mathematics, you don't get to see until you've done years and years of training. > > It was only later in my career that I really appreciated that math is not so much just about solving problems, or that's only part of it. But it's about gaining insight and connections, seeing that two things are related in ways that no one thought was possible before. Also it sometimes connects to the real world. > > I've done many things in mathematics. Most of which are very abstract and not super practical, but there were one or two things that ended up having practical value. I guess I'm just excited to know what happens next. > > **Chapter Review** > > The more you think about your surroundings, the clearer it becomes that everything is connected. > > Math is not just about solving problems. It's about gaining insight, finding connections, and explanations. > > The deeper you dive into anything in your life, the more you will discover and learn. > [!Abstract]- 3. Demystifying Math > There is a perception that mathematics is some sort of sorcery. You're taught this whole book of magic spells, that if you want to solve a quadratic equation you invoke the quadratic formula. And you write some arcane symbols, and you solve the equation. But often you are taught to apply these rules without really understanding why they work. And as a consequence, maybe you're afraid to deviate. If you do anything which just doesn't follow the recipe maybe it'll be a disaster. > > Mathematics gives you a way of solving problems. It's a way of thinking. It's a way of systematically taking a complicated problem, breaking it up into simpler pieces, working on each piece separately, and then putting them back together again, which is most effective for very abstract quantitative problems. But it's also a useful skill in the real world. > > **Is There a Math Gene?** > > Are there people who are naturally good at math and some people who are hopelessly bad at math? I don't think so. I think everyone has an innate mathematical talent. You can see it in children. They ask questions about numbers and shapes. > > There's a result in mathematics that all children or many children discover by themselves, and in fact they teach other children this one fact. And it's the fact that there is no largest number. And the way children realize this is at school they sometimes play this game, who can name the larger number. So one of them says 1 million and the other one says 1 billion. But eventually they figure out that no matter what enormous number the other child names, they can just say plus 1. You get that? Whatever you said plus 1, and that's a bigger number. > > Once they realize that, they actually realize no biggest number. And in doing so, they have actually discovered a very important technique in mathematics. It's called proof by contradiction. Proof by contradiction is established by showing that if the opposite were true, it would lead to an impossibility or contradiction. If you want to show that something can't happen, for example, that there's no largest number, you assume that there is a largest number, and then you show that that leads to a contradiction. > > That is an example of a mathematical technique which is often very challenging for even undergraduate students to grasp when they're taught it formally. But children in schoolyards can actually pick up this concept just by playing a game. Everyone has an inbuilt mathematical intuition. It's obvious to everyone if you have a pizza and you share it among four people, everyone gets a smaller piece than if you shared it among three people. That's a very basic mathematical fact and it's something we all understand. > > The main reason why people think they are bad at math or they don't have the math gene, whatever, is that mathematics is often taught in a very prescriptive manner, and it is not natural to think this way. Often we have to learn all these drills and rules, and it's not tied much to our intuition. There are different ways in which people can access mathematics. Some people are very visual, some people are very logical, some people are very systematic, and they are all valid ways of approaching mathematics. > > **Abstraction** > > One of the first steps when you take a real world problem and try to turn it into a mathematical problem is called abstraction. Abstraction is the process of taking away all the real world elements of a problem and stripping it down to its basic mathematical form. You can solve it using algebra or arithmetic or whatever other type of mathematics you wish. > > You learn this in school actually where you're given these word problems. You have so-and-so many dollars, and you need to get from A to B in a set amount of time, and you need to plan your route or whatever. And so you identify what aspects of the problem are important. If you're planning to get from A to B in the right way, maybe things like the speed of your vehicle, what stops you take, these are important variables. But other things like, what color is the car that you're driving there, that's not important. So you have to identify what features of the problem are essential, and which ones are not, and then try to translate all the data that you're given into mathematical language. > > Sometimes these are equations. Sometimes these are geometric relations. And then once you strip away the original context of the problem and you just have this abstract mathematical model, then you can start applying your mathematical tools - algebra, geometry, whatever. > > **Math Is a Language** > > People sometimes think of mathematics as a language, and I certainly do. All children naturally pick up languages. But imagine if English was taught where in your English classes, you didn't speak, or you didn't hear the words. You just practiced how to diagram sentences, how to distinguish nouns from adjectives, and you just did all this theory, which is an important part of English. But if that's all you learned, you might think, "oh, I'm bad at English because I don't know what a preposition is or whatever. So it's a little unfortunate maybe that school education over emphasizes rigor in mathematics, even though it is important. > > The purpose of mathematics, really, is to communicate ideas and concepts in a very precise way. And to strip out the essence of what is the real problem at hand. You may be studying a specific problem that relates to some physical objects or some economic data or whatever. But in mathematics we'll just say, okay, this object we're going to call x, this one we're going to call y, and these are the relationships between x and y. We're really stripping the problem down to its bare essentials. > > And when you first see that, it looks very abstract and weird. But by removing all the inner central components of a problem, you can focus on what's really going on. And it can help you see the way forward. So it's a very clear language for solving quantitative problems. > > **Math Makes the World Less Scary** > > Pretty much every aspect of modern technology has mathematics going on under the hood. If you want to send your credit card information securely over the internet, it is encrypted by mathematics. If you want to have multiple cell phones working in the same room without interfereing with each other we take it for granted that actually happens, but it happens because there are mathematical algorithms that separate the signals from each other. > > If you have a mathematical mindset, you can gain some confidence that something you don't understand, like how a cell phone works or how a computer works, or how the internet works, or how an economy works, it gives you enough tools that you can feel like, if you really had to, you could actually understand from first principles how these things actually work. > > And the world somehow becomes a less scary place. You don't have to resort to conspiracy theories or think that everything that you don't understand is magic. The world becomes a more rational place. Which I find very comforting. > > **Chapter Review** > > The purpose of mathematics, ultimately, is to communicate ideas and concepts with precision. > > Even in your day-to-day life, stripping down a problem to its bare essentials can provide clarity and insight that you wouldn't have had before. > > Math In Your Everyday Life > [!Abstract]- 4. Math In Your Everyday Life > I think a good way to connect to the mathematical side of your brain is to find a hobby or something that you are passionate about and are willing to tinker with. Some people like playing mathematically themed puzzles, like Sudoku puzzles are very popular. As a kid, I liked playing with computer games and logic puzzles. If you like fixing up old machinery or something, the challenge of making some broken piece of machinery work, that kind of problem solving that shows up in that is actually very parallel to mathematical thinking. > > I think a good way to connect to the mathematical side of your brain is to find a hobby or something that you are passionate about and are willing to tinker with. Some people like playing mathematically themed puzzles, like Sudoku puzzles are very popular. As a kid, I liked playing with computer games and logic puzzles. If you like fixing up old machinery or something, the challenge of making some broken piece of machinery work, that kind of problem solving that shows up in that is actually very parallel to mathematical thinking. > > I think we bump up against mathematics all the time. Now often you can kind of wing it. An experienced chef may have some rules of thumb. But having some mathematical training can help you avoid a disaster I think. At least give you some sort of first approximation as to how to adapt to an unexpected situation. I think we could all play a little bit more, with trying little tasks first before moving on to big high stakes tasks. > > In mathematics it's such an extreme example where it doesn't matter how many times you fail to solve the math problem that there's no real penalty other than waste of time. But it's not even really a waste as long as you learn something from it. > > **How to Hang a Curtain** > > The more you expose yourself to doing tasks in a fun, challenging way, enjoying just the challenge. You know, I enjoy the challenge of assembling furniture from very unreadable instructions. I don't want to do it for a living, but almost anything in real life can be turned into a little problem. And finding situations in life where the stakes are low, it's OK if you screw up, you just start over. And just getting into a mindset where the goal is not necessarily to solve the problem quickly or efficiently, but to enjoy yourself and draw lessons from it. > > One of the basic techniques in mathematics is, if you have a complicated problem, you isolate a simpler version of the problem. Solve that first, and then once you know how to solve various simpler sub cases of the problem, try to put them back together again and solve the full problem. > > Just to give you one example of my own personal experience, I once had to put up some curtains on one my windows in my house. And you had to stand on a chair and there's this heavy rod and there's a bunch of curtain rings and there's a heavy curtain. But maybe because of my mathematical training, the way I approached this was I first tried to assemble the curtain on the ground. And then I practiced how to assemble each piece. So I would take just the rod and see how to put the rod on the hooks and how to put the curtain on the rod. And only after I had practiced each individual piece of the problem would I try to put it all together. That took longer than if I had tried to assemble the curtain directly. But I'm pretty sure if I tried it directly I would have messed up one or two of the stages. And I think this process was actually better in the end. > > **Math = Creative Freedom** > > So is mathematics a creative subject? Definitely. When you see mathematics in a school context, it's often presented in a somewhat dry manner. That there's certain recipes that you have to follow in order to solve a problem. And if you deviate, then you get your marks deducted or something. > > When you're a research mathematician you are solving problems that standard techniques don't quite apply. Because it's so abstract and not necessarily tethered to reality, it allows you to be very creative and very flexible. In the real world, you may have a problem where you have say, some finite number of resources. You only have X amount of dollars to thwart a problem. You may only have so much time. But in mathematics, you can change the parameters. You may say, OK, what if I had a billion dollars? Could I solve this problem? Or what if I had an infinite amount of manpower? It gives you a lot of flexibility to change the problem into one that maybe you can solve first and then you can then from there, go back to solve the actual problem. And that's a freedom that you just don't have in the real world. The abstraction that mathematics has affords it a lot of creative freedom. > > **An Unexpected Approach** > > Many people, they had a problem to solve and they basically did the mathematics even without knowing that they did mathematics. There's one story I quite like. So there was this mathematician named Johannes Kepler in the 17th century, and once he had to purchase his daughter's wedding. And the way that the wine merchants would price these barrels was there there was a little bung hole in the middle of the barrel and there will be this stick. And they would poke the stick through the hole to the opposite corner of the barrel and measure the length. And based on that length, they would say "OK, this is how many ducats or whatever worth of wine." When Kepler saw this, he was amazed. I mean, normally, you would think that if you want to measure how much wine there is in a barrel, you'd have to measure how many gallons fill it up or something. And this was a very quick way. And for these wine merchants, it worked apparently. And they couldn't explain why it worked. They just empirically worked out what lengths of this stick corresponded to how much, what quantity of wine. > > So Kepler got very intrigued and he approached the bottle mathematically. He considered wine barrels of various widths and heights, and computed the lengths of the sticks and their volumes. And he did indeed see that there was actually this very nice relationship. The barrels that were made in Australia were all roughly the same shape. And it turned out that if you made the barrel 5% narrower or 5% wider, the relationship between volume and the length of stick was roughly the same. The dependence of volume on the shape of the barrel was actually at a near local maximum. > > Local maximum refers to a situation in which any small changes in the unknown variables, in this case the shape of the barrel, would lead to a reduction in the quantity being studied, which in this case, is the volume of the barrel. And so small changes in the shape did not actually make big changes in the volume. > > And in fact, Kepler developed some of the early tools of calculus in order to solve this problem. And it was actually quite important for the development of mathematics. So there's a lot of mathematical problems that we just encounter in our everyday lives, and we often just have to solve them with or without formal mathematical training. But once you have the training, you can explain why these funny tricks that people have in their various professions actually work. > > **Practical mathematical thinking requires:** > - Experimentation > - A willingness to fail > - Proactive deconstruction > - Remembering sometimes to zoom out, apply abstraction, choose a plan of action, then act accordingly. Keep an eye on the big picture! > [!Abstract]- 5. Choosing the Problem to Solve > If I were to summarize what makes a good problem, it should be beyond your current range of capability. It should have maybe just one difficulty that you don't quite know how to solve, but every other aspect, you have some idea of how to proceed. Hopefully, the question has some connection to some intuition that you already have. So like, many questions have a geometric flavor, and so you can use your special intuition. > > Some people are naturally very good at numbers, and so actually having a question with a lot of numbers in it, you can leverage your intuition. So it has to match your skill set. And it's not so much whether you succeed or fail at the question, but it's whether you can learn something from it. > > **Asking the Right Question** > > The ability to ask the right question and to frame it in the right way, I think, shows up all over human endeavor- not just in mathematics. Just to give you one example, there was this airport that had received a lot of complaints about people having to wait for their luggage to arrive. So they tried to solve the question of shortening the time between the landing of the plane and the deployment of the baggage. But they found that the solutions they implemented didn't decrease customer complaints that much. > > And it turned out what they really didn't like was waiting at the carousel for the luggage to arrive. And actually, the solution was to take longer to walk from the airplane to the luggage carousel. So they put some partitions, so that they would feel like they are making more progress. And by the time they arrived at the carousel, the luggage would arrive shortly afterwards. And this reduced the complaints by quite a bit. > > So it is possible for your thinking to get stuck in a rut. You are so certain that a certain approach is going to work or a certain answer is going to be what you expect. So sometimes, asking the right question is key to actually getting a satisfactory answer. > > **Problem-Solving and Rock Climbing** > > You don't want a problem that is too easy, that is too routine. You always want problems that are just barely out of reach of the known technologies. One analogy that many mathematicians use is to liken mathematical problem solving to climbing. In fact, many mathematicians are rock climbers. I myself am not, but many of my friends are. And actually, rock climbers- they use some of the same language that mathematicians use. A difficult cliff is called a problem. And when you climb the cliff, you solve the problem. If you want to get from the bottom of a cliff to the top of a cliff, and you know it's 30 feet high, you can't just go directly up there. That's not feasible. But you want to find some handhold or some ledge which is more feasible just almost within reach. You just need to stretch a little bit. And then once you're at that point, you identify the next step, which gets you closer to your goal-- but one which is just, again, just barely within reach. > > You know, even the best rock climber, there will be certain cliff faces that are just not climbable, even with the best tools and the best physical training. And that's OK. In mathematics, it's OK if you can just get some new progress that is just a little bit above what previous climbers were able to reach. And then the next person- maybe someone younger and more energetic, or just someone with a different mindset than you- can go and take the next step. It doesn't matter if you fail nine times. If you make partial success one time, and you can explain that success to someone else who can take it to the next level, that's how mathematics proceeds. > > You have to learn to be part of a bigger process that's been going on for thousands of years. And we have the advantage over climbers as that, you know, it's OK to fall. You can just start over. So you can take some risks that you wouldn't normally do if you actually had to climb a cliff. > > **Creating Bite-Size Problems** > > It often happens that a really complicated problem can be broken up into bite-sized pieces, and each piece is manageable. One historical example of a complex problem that was solved by breaking it into smaller parts was the problem of determining the orbit of the Earth and Mars around the sun. So in the 15th century, I believe, Nicholas Copernicus famously theorized that Earth and Mars and all the other planets went around the sun-- the so-called heliocentric model. Previously it was thought that the sun instead went around the Earth. > > He, in fact, theorized that the orbits were perfect circles. And that was approximately correct, but it didn't quite match the observations. Copernicus was also able to determine the period. He was able to work out that the Earth went around the sun once every year. Mars, for instance, took a little longer. It took 687 days to go around the sun. So that was correct. The periods were correct, but the orbits were a little bit off. > > But the whole problem was that we only had one fixed landmark, the sun. And Mars-- Mars also moved, and Kepler did not know what the orbit of Mars was, either. But what Kepler realized was that he could reduce the problem to a simpler problem. So if Mars was fixed, if Mars didn't actually move, then it could be used as a landmark, and you could use triangulation from the sun and Mars. And Copernicus had already worked out that every 687 days, Mars returned to its original position. So Kepler's idea was to take the measurements of Mars and the sun that were available, but only sample them in spacings of 687 days. So he took measurements that were 687 days apart. For those particular increments of measurement, Mars was in a single location. So Mars was effectively fixed. And then it was a landmark that he could use to triangulate. And he was able to use that to first work out the orbit of the Earth, and then there's a way to go backwards, once you know the orbit of the Earth, you can work at the orbit of Mars. Then the orbit of the other planets. And he eventually discovered what we now call Kepler's laws of planetary motion. > > For example, he discovered that, in fact, all the planets move in ellipses. But the key idea was to reduce a difficult problem where everyone was moving to a simpler problem where you had fewer moving parts. > > **Group Testing** > > Just one example of breaking up a complex problem into simpler parts is the story of group testing, which was a technique developed in the 1950s. The US Army wanted to test soldiers for a certain rare disease. And there were blood tests available, but the blood tests were in limited supply. So there was not enough tests to test all the soldiers individually. > > The solution they came up with was to take blood samples from many different soldiers and mix them together, so that if there was a positive test there, that would mean that someone in that group had the disease. And if it was a negative test, that meant that nobody in that group had the disease. The mathematical problem was what is the least number of tests you would need in order to determine who had the disease and who didn't? Now, this problem is complicated, because there are so many possibilities. Maybe there's 100 patients or 100 soldiers, and maybe one of them has the disease, maybe two, maybe three. We don't know. But you can start by simplifying the problem. > > Suppose that exactly one person had this disease, and no one else did. Then how can you isolate this one person out of 100? So we're going to demonstrate how group testing works, and we're going to demonstrate this visually using ping-pong balls. So instead of 100 soldiers, we will have 100 ping-pong balls, each with a number. This is the 89th soldier. And one of the balls has a little rattle in it, and it will make noise representing the face that it is the patient with the disease. So you can hear it if I shake the whole box. Ok? There's a little rattle you can hear. > > And the way we're going to test for this is that we will take several of the ping-pong balls and put them in a jar like this, to represent the mixing of samples. So it turns out that the correct thing to do is to split the sample into equal proportions, 50/50, and just test one batch of 50 at a time. OK. So we have divided, now, the population into two groups-- the balls 1 through 50, and the balls from 51 to 100. And we're going to test the balls in this sample. And one of two things will happen. Either I will hear a rattle, which means that the infected patient has a number between 1 and 50, Or I will not hear a rattle, in which case the infected patient must be in the other group. But in either case I will have cut down the number of possibilities from 100 to 50. I'm going to shake this. And I heard a rattle. So the patient is somewhere in this group, and I can eliminate all the patients from 51 to 100 and only now work with the first group. > > So that is how we use our first test. And now, we can just repeat the process. We're going to divide up this population from 1 to 50 into two equal portions. Now, we're going to use 1 to 25 and 26 to 50. So we're ready for our second test. OK, there was no rattle so the test was negative. Therefore the infected soldier must be in the range of 26 to 50. We have eliminated the first half, 1 to 25. So now we can repeat the process with our new population from 26 to 50, and we will, again, subdivide into two equal populations. > > Ok, so after seven tests, we have found that the infected patient is patient number 33. All right. So we have seen that in this particular case, it only took seven tests for us to narrow down the infected patient to a single candidate. > [!Abstract]- 6. Solving Problems With Story > One thing that surprised me, and I had to learn somewhat late in my mathematical career, is that the solution to mathematical problems often goes through narrative, through storytelling. I thought that mathematics was a very impersonal subject, where you just present equations and theorems and arguments without context. But actually the context is very important. The key is often to tell a good story about it. > > Mathematical narratives take abstract mathematical problems and add real-world context. It can help you see that your goal is more than just solving for x. Making a problem fit into the context of a relatable story can also give you insight or clues about it. It helps the viewer understand what's going on if there is a protagonist and an antagonist, if there is some goal or obstacle to solve, there's some obstacle framing things in a way which has a narrative to it, that activates certain parts of your brain. > > **Good Stories Come From Dumb Questions** > > If you want to develop your skills to find analogies to see connections, basically the most important thing is to ask questions. And the dumber the question, the better, really. So to give you one simple example from arithmetic, when you multiply two negative numbers together- let's say, negative 3, negative 4- you get a positive number, plus 12. But that's not intuitive to many people. To many people, two negative numbers should combine to form another negative number. And so you can ask, is there some analogy you can use to pursue this? > > So you can maybe use an economic analogy. So if you want to multiply 3 times 4, you could say, okay, suppose I am being paid $3 an hour to do something and you work for 4 hours. So every hour you get $3 and after 4 hours, you get $12. Fine. But now suppose, instead of being paid $3 an hour, something is costing you $3 an hour. Maybe you're running water or something or electricity, and it's costing you $3 an hour. So after 4 hours, this will cost you $12. So this is why minus 3 times 4 is minus 12. > > So you can ask, how can I push this analogy further? So then you can ask, how do you interpret minus 3 times minus 4. Well suppose your water is running or something, and it's costing you $3 an hour. But you manage to shut off the water for 4 hours, so you saved 4 hours of expenses. You have saved $12. And this is why minus 3 times minus 4 is plus 12. > > So by exploring this analogy and just asking questions- what does this mean, what does this mean in this analogy- you can gain a sense of what multiplication of negative numbers really means. And your intuition is now better than it was before. > > **Good Guys and Bad Guys** > > It's actually very useful to anthropomorphize the mathematical objects you deal with. They're not just x's and y's. Often you will hear, if you listen to a mathematician talk about a problem, they might say that this is the enemy, and certain things are good guys. Maybe there's a certain powerful tool that we want to apply. But in order to apply it, certain conditions have to be met. And so you start working on that instead. > > If you want to solve for some unknown X, you could think of it as almost a detective hunt or something. There is some mystery opponent who has, and you don't know the location of this opponent. He's located at X. And so you need to identify where X is. And this puts you in a mindset where you start thinking about clues. You can start using process of elimination. "Okay, so X can't be this because if X was equal to this, then this would happen, and you know this doesn't happen. So it can activate certain detective-type aspects of your thinking that help you realize how to solve your question. > > **Good Narratives** > > So I'm going to talk about an example of how you can use analogies or narrative to really understand a mathematical concept. So the example I'm going to use is that of polling. So if you have a large population, such as the population of the United States, several hundred million people, and you want to know their position on some topic, like how many people are voting for a political party. We have these polls. And when you first learn about how polling works, it is unintuitive. A poll may only sample say 500 people out of 200 million possible respondents. And that's a tiny, tiny fraction of the whole population. > > But nevertheless, polls can be incredibly accurate. One of the facts about polling is that the size of the population does not actually matter too much. What matters is the sample size. So a survey of 1,000 people will be more accurate than a survey of 500 people. Regardless of what the total population size is. This is unintuitive, but a good analogy can really clarify the situation. > > So the analogy that I like to use is "What is the salt content of the ocean?" You can tell the ocean is salty just by taking a single drop of seawater and just tasting it. And immediately, from one drop of seawater, you can tell instantly that the ocean is full of salt. And the reason for that is that the salt is quite well distributed within the ocean. Now the beauty of the analogy is that it not only explains the initial question of why the size of the population is not so important, but it also tells you the limitations of polling. If the ocean contains freshwater portions and saltwater portions, if you sample the wrong portion, you can get a misleading idea of the salt content of the whole. And so sometimes polls fail because they are not polling a sufficiently mixed portion of the population. > > It's important to sample as randomly as possible, try to poll people from different regions, different demographics, different social levels. If your population is 50-50 male and female, you should probably try to get a sample, which is also 50-50 male and female. If not, you can correct for it with some math, but it would be more accurate if it's closer to the representative mix. Analogies are an excellent way of using both halves of your brain, the rational half, the formal half which uses very precise, rigorous thinking, and the intuitive part, which is using analogies and intuition. > > Part of advanced mathematical training is learning how to somehow see it stereoscopically with both your rigorous perspective and your intuitive perspective and know how to translate back and forth. Going back and forth is actually one of the most fun parts of doing math because you really feel smarter when you make the connection and you see how the numbers in the equations match with your intuition. > > **Bad Narratives** > > Sometimes narratives can lead you astray. For instance, a lot of mathematics is centered around solving equations exactly, to the last decimal point. And it was only realized in the 20th century, really, that there are many questions, many practical questions where an exact solution is not possible. And what you really want is just an approximate solution. > > For example, Newton famously solved what's called the two body problem in gravity. That if you have two massive bodies, like the sun and the Earth, you described the orbits quite exactly, and people spend centuries trying to solve what's called the three body problem. And if you have three bodies, what is the exact formula for the orbits. And no one could find an exact solution. In fact, we still do not have an exact solution for this problem. But that turns out not to be the right question to ask. > > The question to ask is, can we compute the trajectories to a sufficient accuracy for a sufficient period of time. Like, if you want to send a rocket to the moon, It's okay to get within half a meter or something of the correct position of the moon. Once you frame things in the context of finding approximate solutions rather than exact solutions, there are whole different set of techniques that you can use. There's a whole different narrative. > > **Error Tolerance** > > You can tolerate certain errors as along as they don't get out of control. And so it all becomes a question of controlling errors. For example, polling is another good example that if you poll 1,000 people, there is a tiny, tiny chance that your poll will be completely off because all of the 1,000 people will be from a certain demographic, which has a bias. It's very unlikely, especially if you do enough randomization, but there is always a tiny chance of failure. > > But for many applications, it's okay to have a quick and cheap method that has a tiny chance of failure. Of course, if you want to fly a plane, you want near 100% chance that the plane will not fall apart. But if you just want to send an email from A to B, if it fails in delivery 1% of the time, that's not too bad. > > Knowing what error tolerance you have for your problem is important in finding the right framing. > > **Chapter Review** > > - Framing a problem with a narrative or analogy can make a huge difference. > - Mathematics is not black and white. > - A good narrative allows you to understand an abstract mathematical problem in a practical way. It can help you see the solution as more than just a numerical answer. > - A bad narrative, however, can make you think that you need a solution that's not actually necessary, such as a precise solution to a problem that might require only an approximate answer. > [!Abstract]- 7. Transforming Problems > One of the great insights of mathematics is that there are different frameworks, different ways of thinking abou the same mathematical problem, which may look quite different but are mathematically related. One of the great advantages is that we can then transform the problem and think of it in a new way, which has nothing to do with the original context. > > The human brain has got many different modes of thinking. So we have visual modes, we have symbolic modes, we have modes where we are trying to fight some sort of adversary. And by changing the language of your problem, you are activating different areas of your brain. > > So for instance, if you are transforming the problem into a geometric problem, then you are activating the visual centers of your brain. So transformation is a way of swapping your thought patterns. > > **Get Physical** > > For some problems, actual physical sensation can actually be useful. Many mathematicians, you will find, they wave their hands, or gesture somehow when thinking about a problem. Act that act of making the thoughts physical is often quite useful. I know people who, while they work on a pen and paper for a while, on a problem. If they get stuck, they take a walk. They force themselves to think about the question while walking without the pen and paper. And that's a lot harder, but it forces you to somehow only focus on the essence of the problem. Concepts are simple enough to keep in your head at one time without having to write down any computations. And that can sometimes lead to a better way of thinking about the problem. > > Talking about the problem to other people can help, even if they're not mathematicians. And the process of verbalizing the problem can often lead them to actually visualize what the problem is, when previously they were only just thinking about it internally in their head. I have occasionally used my own physical location as a way to transform. > > There was one time when I was trying to understand a very complicated geometric transformation in my head involving- I was rotating a lot of spheres at the same time. And the way I actually ended up visualizing this was actually lying down on the floor, closing my eyes, and rolling around. And I was staying at my aunt's place at the time. And she found me rolling on the floor with my eyes closed. And she asked me what I was doing and I said, I was thinking about a math problem, and she didn't believe me. > > You find whatever analogies, physical, or mental, or whatever, that work for you. And sometimes it makes you look silly but that's an occupational hazard. > > **Disrupt Your Thought Patterns** > > I sometimes find myself solving little mathematical problems on the fly. To give one example, I was once at an airport and I had to make a connection. I was at one end of the airport and I had to race to the other end to catch my flight. But my shoelace was undone. I didn't know when I should stop to tie the shoelace. But the whole way I was running down, some of the hallway had these moving walkways. And so the question that ran in my head while trying to run across, was it more efficient for me to tie the shoelace on the walkway or off the walkway? > > I ended up doing a little bit of algebra in my head and actually figuring out the answer, that it's actually better to tie the shoelace on the walkway. Now a little later on, I mentioned this in my personal blog. And someone commented that there was a much simpler way to solve this question, a much more conceptual way that didn't require any algebra. And for this was, you need to transform the problem. > > So you imagine, instead of one person racing to catch the connection, imagine that you had an identical twin running beside you. And you are both running side by side to get to the other end of the airport and you both had untied shoelaces. But suppose the only difference between the two twins was that, as you approach a walkway, one of the twins will stop just before the walkway to tie the shoelace. And the other twin will get on the walkway and tie the shoelace. > > If you think about it, the twin who is on the walkway will be moving forward a little bit while both twins are tying the shoelace. And so when they both get up again and both start running, the twin on the walkway had a little head start, and will therefore arrive sooner. It becomes obvious that the correct thing to do is to tie the shoelace on the walkway. > > It's important to keep trying different approaches to a problem. It's very common that you're thinking, you get into a rut, that you look at a problem and you think, I must solve it this way. Especially if you feel yourself getting stuck, you need to mix things up sometimes and be willing to let go of preconceived beliefs. Often, if you're really stuck, you come up with more and more desperate ideas. > > **Playing With Transformations** > > My next example of transforming a problem comes from a very nice exhibit at the Museum of Mathematics in New York. And it involves what's called the finding 15 game. The game involves nine numbers, the numbers from 1 through 9. And the finding 15 game works like this, the first player who can collect three numbers that add up to 15 wins the game. > > OK, so now I'll be playing a few rounds of finding 15 with my evil twin terry. So let me demonstrate- I will start by, maybe I'll start with the number 3. So evil terry plays number 6. I will play 5. Evil Terry plays two. And I will play 7. Notice that 3 plus 5 plus 7 is 15, and so I have won the game, hah. > > OK, so that's how the game works. And playing this game is quite challenging if you don't know the trick. But there is a transformation that transforms the game into the familiar game of tic-tac-toe. To do this, we'll be using what's called a magic square. A magic square is a square of numbers in which all the rows, columns, and diagonals of the square add up to the same number, in this case, 15. > > Magic squares actually have a long and ancient history. Some cultures thought they had mystical properties. They don't, unfortunately. Actually, they're mostly mathematical curiosity. They don't have too many applications. But, this 3x3 mathematic square that sums up to 15 turns out to be the perfect tool to solve this game. > > What I have here is the magic square. So these are the numbers from 1 to 9, the same numbers that we used to play the finding 15 game. Now you can use this magic square game to transform finding 15 into a much more familiar game, tic-tac-toe. So if we review the game that we just played. You can see the patten. It turns out that we are much better at tic-tac-toe than we are at finding 15. So if you apply a transformation while you play a game of finding 15, you will be much more effective. > > And so to demonstrate this, I have generously allowed my evil twin to use this representation. Let's see how he plays in a rematch. In tic-tac-toe the strongest opening move is actually to play the centerpiece. In the magic square this turns out to be the number 5. The great benefit of this transformation is that it transforms a mysterious problem into a very familiar and non-threatening game, of whether you can play tic-tac-tie. And almost all children are familiar with this game. > > It immediately makes the problem less scary. Maybe you can't always win at tic-tac-toe, but at least you have a lot of natural experience with how to play the game. So you can use your intuition about tic-tac-toe to help you play the finding 15 game. That's the power of transformation. > [!Abstract]- 8. Games, Cheats, and Puzzles > As a kid, I loved computer games, logic puzzles. I think I was always drawn to artificial scenarios where the rules were very clear and simple. There was a certain number of moves you could do. There was a sense of a right answer and a wrong answer. Certainly there's the thrill of solving them, especially if they have somehow fought against you. You get a sense that they're almost alive. And the same is true for a really good puzzle, or a good game. I'd like to talk about how a toy puzzle can lead to a very practical application, which had nothing to do with the original puzzle, at least at first glance. > > As a kid, I loved computer games, logic puzzles. I think I was always drawn to artificial scenarios where the rules were very clear and simple. There was a certain number of moves you could do. There was a sense of a right answer and a wrong answer. Certainly there's the thrill of solving them, especially if they have somehow fought against you. You get a sense that they're almost alive. And the same is true for a really good puzzle, or a good game. I'd like to talk about how a toy puzzle can lead to a very practical application, which had nothing to do with the original puzzle, at least at first glance. > > **The Counterfeit Coin Puzzle** > > So there's a classical puzzle, almost 100 years old, called the Counterfeit Coin Puzzle. You're given 12 coins, let's say, gold coins and they're all identical, except for one, which is counterfeit. And the counterfeit coin is either slightly lighter or slightly heavier than the other coins. You don't know which coin is the counterfeit one, and you don't know whether it's heavier or lighter. Your task is to work out which coin is counterfeit and the only tool you're given is a balance scale. And you can put some coins on one side of the scale, some coins on the other. And you can see whether the coins are evenly matched, or whether one is heavier and one is lighter. But the catch is you're only allowed to use the weighing scale three times. So if there's 12 coins, you can't weigh each coin separately. You're only allowed three weighing's. > > This is a purely mathematical puzzle. There's no conceivable practical situation where you have to determine a kind of a coin and you have this very limited number of weighing's. So the solution for this counterfeit coin problem is quite complicated. And we'll include a full solution in the class guide. It uses a mathematical object called a matrix. > > And the techniques we use to solve this problem actually turn out to be useful to solve a very different problem that I was involved in, actually, about 10-15 years ago, which was that of MRI scanning. So an MRI machine scans your organs for things like tumors or other anomalous objects. But the problem is that these machines are somewhat slow. And for certain patients, like young children, they may not be able to stay still for enough time for the scan to actually work. > > So there was a question. Could you figure out whether there was a tumor using fewer measurements than the traditional MRI scan? And this problem turns out to be mathematically very similar to the coin weighing problem. For instance, there is a way to solve the coin weighing problem using a field of mathematics called linear algebra. And maybe we will discuss this more in the written notes. > > And the same type of mathematics is also used to solve the compressed sensing problem. You put your body inside this machine, they measure these magnetic fields, and with enough measurements they can reconstruct an image of your human body. And you can see if there's something wrong. So the human body is the analog of these coins. A tumor might be the analog of a counterfeit coin. And each measurement of the MRI machine is like a single weighing. > > And so it turned out that some of the techniques that could be used to solve the coin weighing puzzle were also found to be of practical use in speeding up MRI scans. And now, actually, the latest models, they use this technique called compressed sensing, which I actually helped develop. They can speed up MRI scanning by a factor sometimes as much as 10. A scan that used to take three or five minutes maybe takes 20 seconds. > > You can start with a completely frivolous mathematical question and, many years later, it becomes of a very practical application. > > **Gaming** > > I'm from a generation where if you wanted to play a computer game, you would actually get a book, and it would give you the code for a game, and you would type it into your little Commodore computer and run these games yourself, and the benefit is that you also got to tweak the code sometimes. If you wanted to make the game faster or give yourself more lives, you just changed some numbers in the code. Sometimes this led to hilariously easy or difficult situations or maybe the game just didn't run at all. But I find that very instructive. > > Some computer games have difficulty levels. You can play on easy mode, where you have a lot more ammunition or a lot more lives, whatever. Or you can play on hard mode, where things are a lot more challenging. And often, if you want to get good at a game, you should first play the game on easy mode and then slowly turn up the difficulty. This is what makes computer games often a lot easier to solve than real life problems. In real life, the problem is often thrown at you. You can't adjust the difficulty level. You have to solve it as is. > > But mathematics is more like a computer game in that you have a lot of freedom to choose the difficulty of your problem. If the problem is too hard, make it easier by adding more hypotheses, research into a special case, assuming that something is true that you can't prove yet, but you suspect is true. If you have something which is only approximately true but not completely true, just pretend it is completely true and see if that helps you. > > **Cheat!** > > This maybe comes from my gaming background. But one thing that you often do in mathematics is you make the problem easier fist, often a lot easier, just to get started. And it's akin to trying to pass a computer game by turning on some sort of cheat, which maybe gives yourself infinite lives or the ability to teleport or something ridiculous. Then you go back, and you turn the cheat off. > > **Cheat Tip 1: Allow impractical answers** > > You may have a problem where the answer has to be a whole number. You need to know how many people does it take to dig a certain hole in a certain amount of time. You need 10 people, you need 12 people, but you can't use 3 and 1/2 people. It's often good to first suspend the rule that you have to have a whole number solution and get an answer which is a bit ridiculous, that you need 10.5 people to solve this equation. But then you fix it, and maybe you need to round up. Maybe you need 11 people rather than 10 and a half. > > **Cheat Tip 2: Assuming a Spherical Cow** > > In physics, this is sometimes called assuming a spherical cow. So you may have some physics problem that involves a real world object, like a cow. And you wonder how fast the cow can move and so forth. But the laws of physics that you're given, they're only nice if you assume the cow is frictionless or completely round or something. So the joke is you assume a spherical cow. So even though the cow is, of course, not a perfect sphere, you first assume it is, you solve the question. This will give you some approximate answer to your question. And then you ty to add back some of the complexities. > > **Cheat Tip 3: Linearization** > > Mathematics is a lot easier when things are completely straight. So you can take a curve and say, I'm going to approximately treat this curve like a straight line. And so if that is a simplified problem, it's actually a very powerful technique in mathematics. It's called linearization. And people often solve the linearized problem first, but it often gives you a lot of insight and it gives you a lot of clues as to how to solve the actual problem. > > There's no sharp dividing line between what is a puzzle, what is a problem, what is real world application. There's so many connections and analogies that I find it better to think fluidly. These are all just things to do. As you get more experienced, you don't pigeonhole individual tasks into separate boxes so much. It all becomes just one continuous organic whole. > [!Abstract]- 9. Math Fails > Trial and error is one of the most important aspects of problem solving. It's one that you often don't see when you see a problem solved that's presented in a textbook or class or research paper. The tendency is almost always to only present the correct solution, the one that worked. And you don't see the outtakes, which is a shame, actually, because that's often the most informative part. We don't often show our embarrassingly stupid first attempts at solving a problem. But it is extremely important. And you need to have the freedom to try things that are potentially stupid. Well, of course, A, they might actually work. But, B, because the way in which they fail is often very instructive. It gives you clues to figure out what the solution should be. > > I think the physicist Niels Bohr once said that an expert is someone who has made all the mistakes that can be made in a very narrow field. The only reason why experts seem so competent is because they've already screwed up in every conceivable way, that they know how to not do that in the future. > > **Embrace Failure** > > I must confess that my first one or two years in graduate school were not super productive. I did some math. And I took some classes. But I did spend too much time doing computer games and discovering the world wide web, which was very recent at that time. So about halfway through graduate school, there was this rather intimidating exam, which we call the general example, or the generals. It's a two or three hour oral exam where you pick three topics in mathematics and three faculty in your department - and they quiz you about these topics. And if you get the questions right, they ask you even harder questions. If you get them wrong, they ask you easier questions. > > Most of the other students in my class, you know, they spent months and months practicing this. They had quizzed each other. They had these mock general exams. I basically studied a little bit, maybe for a few weeks. But I'd figured I would wing it. I did not do very well. My knowledge was very superficial. Whenever there was a follow-up question, I could not provide a good answer. And the questions became easier and easier. And I could barely solve most of them. > > The only reason I even passed the exam was because the one topic which I thought was the most challenging for me was the only one I actually tried to study for. And that one I actually managed to answer enough correct questions that I could pass. But my graduate advisor afterwards took me aside and said he was quite disappointed in a very gentle and encouraging way and that I really should try to do better. > > I was used to doing well in mathematics without working super hard. And so it was a shock to me to realize that that mode of thinking had limits, that you had to think more systematically. You have to actually plan your study. You have to actually learn from your mistakes. And so, yeah, those were important life skills to learn. > > **Failures Are Clues** > > One famous example of an experiment that failed and led to an interesting discovery was the story of the Greek philosopher Eratosthenes. So many, many years ago, he lived in a city in Egypt called Alexandria. And he read that there was another city, another town, called Syene which had a well in it. And this well had the funny property that on one day of the year, the summer solstice, if you looked at high noon into this well, the sun was directly overhead, and you could see the sun reflected in the water deep below. > > Eratosthenes was intrigued by this story. And so he decided to wait until the summer solstice in his own town of Alexandria and go to their local well. And he found, at noon, that the sun did not shine straight down into the water, it was at an angle. So his experiment failed. And for most people, they would just say, okay, this was, I guess the modern term is fake news- that this story wasn't true. > > But what Eratosthenes realized is that because the experiment failed in his city but it didn't fail in this other city, that the earth must be curved. And furthermore, he realized that, in fact, he could now use this failure to measure the size of the Earth. And in fact, he was the very first person to actually get a pretty good measurement. Within 10% accuracy. And all he did was he measured the angle that the sun made at Alexandria, he knew the distance between Alexandria and Syene. It was a couple hundred miles, And the mathematics of the time was fairly primitive. But still good enough to get a pretty good measurement. > > So sometimes a failure can be extremely productive. This happens all throughout science. I think penicillin was discovered by accident. You know, that someone left a jar of bacteria and it got moldy or something. > > **Failure Is Cheap** > > The great thing about mathematics is that failure is cheap. In a practical discipline, if you fail at, say, building a bridge, you know, you may cost lives or lots and lots of money. But if you try and solve the problem, like you solve for x, and you don't get the right value of x, it's okay. You just go and fix your problem. You try again. > > There was a famous mathematician, Vladimir Arnold, that said that mathematics is the part of science where experiments are cheap. You can try almost anything. You just need a blackboard and a bit of time. And just getting to a mindset where the goal is not necessarily to solve the problem quickly or efficiently but to enjoy yourself and to draw lessons from it. > > If you start doing that in your life, I think maybe it will just naturally become part of your way of thinking, just another way of going through life. Sometimes a failure can be extremely insightful and productive. It can be a sign that there's something more going on than you expected, something unexpected. > > Isaac Asimov once said that the most exciting sentence that a scientist says is not "eureka" but "that's funny." If there's something unexpected, something that didn't go as planned, that's not a failure. It's often a clue that something new and exciting is going on. > > **Eureka!?** > > People often think that the way math or science works is that people work very hard on a problem, and there's no progress for a long time. And suddenly, there is a eureka moment. You know, some lightning bolt hits you, and suddenly everything gets illuminated, and you find the right way forward. That's not quite how it works, in my experience. You do work for a long time on a problem. And it's frustrating, because you're not making visible progress, but you're making a lot of progress underneath the surface. You are working out what doesn't work. You are finding partial solutions. You're realizing connections with other topics. And you're just setting the stage for putting the problem in exactly the right perspective. > > It often is just a very small thing that's the very last puzzle piece to put into place. And suddenly, everything makes sense. It's rarely a big lightning bolt of clarity. It's just-- it's more like a realization. That you are almost there already, that a lot of the inghts, the partial progress that you had, actually just takes one last ingredient to make it work. > > When you fail at a task, maybe the most natural thing, psychologically, says, okay, that sucked. I don't want to do this. I was-- maybe you blame yourself. When failure occurs, it's very natural to try to find the person to blame. But mathematical problems are not like that. Often, what you need to do is just not so much to assign blame but to figure out what was the specific aspect that was blocking this approach from working. > > You have to look at the precise point of failure. And it will often tell you a clue as to what the working solution should look like. It's actually not the solution to specific problems that are so important for us but more the process and what we learn from the journey in asking these questions. > > **Chapter Review** > - Don't be afraid to fail. > - Try to avoid blaming a failure on yourself or others. > - Remember that failure is often what provides us the clues to a correct answer. > - Figure out what doesn't work, and why. > - Persistence is key. Stay patient, it's all part of the process. > [!Abstract]- 10. Stumped > The natural state of a research mathematician is to be frustrated, to be stuck. We are always surrounded by problems that we would love to solve, but we can't. Sometimes, it just tells you the problem is it's not ready. It's before its time, and progress on some other problem has to happen first. > > This is one thing, by the way, that computer games have taught me. You know that your treasure lies behind the door, but the door is locked. You need a key, so you're stuck. But the moment you find a key, you know, then you race back to this door, and you can unlock the door hopefully. It is very satisfying sometimes to pick up a problem that you couldn't solve 10 years ago, and now, there's more tools. There's more technology, and the problem gets solved. > > There are problems, which I've spent years working on and not having the right idea for many, many months. But the process of trying things and seeing why they didn't work, again and again, was instrumental. When I, finally, found the approach that did work, all the partial successes I had before, which weren't enough to lead to a full solution, they often came in very handy at the end, and often, things snowballed. Sometimes, solving a tough math problem is like trying to budge a door that's stuck, and you keep slamming your shoulder against it. And it doesn't move, but every time you do it, it loosens the door a little bit. And then, when you finally, find the right way to hit it, it falls open, and you just stumble through. > > When I was a graduate student, I was always impressed when I worked on a problem. I would spend a week bashing my head against it, trying all kinds of things, and I would show what I did to my advisor. And he would think for a few minutes and say, oh, you know, this problem you're facing, it reminds me of what so-and-so did in this paper. And he would go to his filing cabinet, and fish out a little pre-print and say read this. This paper had the same issue that you had, and their solution would probably work for you as well. And I would go home and read it, and he was usually right. > > The techniques they had solved the problem that I spent hours working on, and what this showed me is that, often, experience trumps energy, that, if you know what to do, you can save yourself a lot of effort. As I get older, I find myself having less energy. I can't spend hours and hours on a single problem, like I used to. But I can often see the connections to an existing problem. I can use my experience a lot more effectively that I could before. > > **Overcoming a Mental Block** > > Sometimes, a task is just too big and scary to even get started, and you have to mentally break it up into smaller pieces. There was once a project, where the five of us were working on a quite hard problem, and we thought we had solved it very early on in the process. We worked together for just a few weeks, and we had this method. And it seemed to work, and we got very excited. And we spent a few weeks writing it up, but then one of us, when proofreading noticed that there was one case we had missed in one of the arguments. And there was a gap, and we panicked. And we tried fixing it. We tried many, many things, and it wasn't fixable. There was just this one piece that we had not considered properly at all, but by that point, we had already invested a lot psychologically into the problem. We had already experienced the high of solving it, so we kept working on it, in fact, for two years. > > But if we did not have the early false hope that we had solved it early on, we would probably have given up a lot sooner. But because you had that early psychological boost, we kept at it, and eventually, after two years, we found the right way to solve it. That's one of the papers that I'm most proud of actually, but it was important that we had made a mistake early on to give us this false hope that we were close to the solution when, in fact, it was much further away than we had expected. So sometimes, serendipity works in funny ways. > > **Getting Stuck Is Normal** > > If I were to advise students what to do, if they get stuck, I would say that this is completely normal, that this happens to everyone. And, in fact, if you are not feeling stuck regularly, you are not challenging yourself. People often think of success or failure in binary terms. You know, you build your bridge, or you don't build your bridge. You bake your cake, or you don't bake your cake. But in mathematics, it's really a spectrum. > > You may not succeed at your main task, but you have solved some special case. Or you would have demonstrated a proof of concept that some technique has some promise, but it doesn't yet have enough maturity to work for the full problem. And sometimes, you learn that the problem is just not ripe, that the techniques you have are just not suitable, and those are all valuable things to have. So even if you have no concrete tangible outputs, you often have a better understanding of the problem than when you started. That is the mindset that you need to take in these fields. > > When you're working on a difficult problem, it's not realistic to expect a perfect success rate, but you can often end up in a better place than when you started. And that's really what to shoot for. > > **Letting Problems Go** > > Sometimes, you can get too obsessed with a problem that is just too hard, but you are convinced that you can solve it. And you spend a lot of time working on that problem rather than working on more feasible questions. We call it a disease, a mathematical disease. So we all have to learn to, sometimes, take a breath, step back, admit that some problems are outside one's reach. It's not about necessarily being extremely smart or extremely knowledgeable. There are some problems for which, basically, mathematics is not ready to solve, and it doesn't matter how smart you are. So you have to learn to let some problems go. > > **Chapter Review** > > - Everyone gets stuck. If you're not hitting roadblocks, you're not challenging yourself. > - Tap into all your other resources if you get stuck. > - Experience often trumps energy. > - Finally, sometimes you need to learn to let your problems go, wait for a bit, and return with fresh eyes, so you can see something you didn't see before. > [!Abstract]- 11. Finding Strength in Numbers > > I think mathematics, like any other discipline, it's actually a very social discipline. In the past, maybe mathematics was conducted by isolated people in rooms, you know. We would just work for months or years on a problem. But nowadays, it's a much more social process. Problems in mathematics or in other disciplines are so interdisciplinary that we need to communicate with other people. > > One of the great advantages of working with someone who has a slightly different skill set than you is that you get to learn their toolbox. I didn't used to use numerical simulations very much in my work. I would much rather work things our by pen and paper than to write a little program to do things for me. But I've worked with people who are very, very good at simulations. And it was quite informative to actually see these graphs and figures come up, almost in real time. Some people were just extremely fast. That is the future of our field. > > **Divide and Conquer** > > Mathematics used to be a very individual activity. People used to just work in isolation. But nowadays, it's much more common to collaborate. I really enjoy collaborating with other mathematicians. It brings out, I think, certain modes of thinking that are more difficult to express when you're by yourself. It's good to explore complementarity. > > I like to say that a good collaboration should involve at least one optimist and one pessimist. The optimist keeps dreaming of new ideas and sort of these blue sky approaches to your problem. And the pessimist's job is to shoot down the crazy one, the crazy ideas, but keep the sane ones. And if you have too many optimists or too many pessimists on a project, it doesn't work. A mix is good. Diversity is always good in a collaboration. Sometimes in a collaboration, you try to assign precise roles to people. But I find it's best to just let things flow naturally. > > There were two famous mathematicians, Hardy and Littlewood, who wrote down what are called the Hardy-Littlewood rules of collaboration. One of them is that once you agree to collaborate, you do not try to ascertain whether the work was divided fairly. Everyone just does what they feel is appropriate. It's never very productive to try to ascertain, you know, oh, you did 30% of the work, you did 60%. If two people work on the same project, it's not like they get half the credit each. They each sort of get full credit for the paper, at least in principle. > > **Crowdsourcing** > > I think technology has really changed the way we can collaborate. But now we can crowdsource many types of mathematics. The problem can be split up into many, many different pieces. And you can have large groups of people work on each individual piece and almost have an assembly line. You see this phenomenon throughout science, actually. Amateur astronomers can discover comets. Biologists employ amateurs to try to solve protein folding problems. And we are just at the cusp in mathematics of also being able to harness the crowds of amateur math enthusiasts. > > One thing I've definitely noticed is that when you have many, many people working on a single project, there's always someone who can draw connections with an obscure field of mathematics or some other piece of literature that you might now otherwise have discovered. It used to be that if you weren't in the same university, the only way you could collaborate is through writing these very long letters to each other. It's so easy now that I can collaborate with people in different continents. They work on the problem while you sleep. And you wake up, and there's all this progress. And you work on it while they sleep. > > **The Erdos Discrepancy Problem** > > So the Erdos discrepancy problem was posed by the Hungarian mathematician Paul Erdos about 50 years ago. There hadn't been much progress made for many decades. But in 2010, a large collaborative project, what's called a polymath project, was launched to attack the problem. There was a mathematician named James Grime, who had a rather colorful interpretation of the problem as a sort of a logic puzzle. And I would like to present that version of it here. > > So in this formulation, you imagine that you are kidnapped by some sadistic torturer. And you're placed in this certain room. And you can move one step to the left, one step to the right. But if you ever move two steps to the right or two steps to the left, you fall to your death. So maybe on the right there are spikes here. And on the left, maybe there's a poisonous snake or something. > > You have to submit the list of moves beforehand. And each move can be one step to the left or to the right. So maybe moving right and then left and then left and then right, and then the torturer will make you move according to the instructions that you gave. Now, in this particular case, you would first go right and then left. Then left and then left and you would fall and be eaten by the snake. So this is not a good set of moves to submit. > > There's a twist. Your torturer can make you do every third move. So the torturer actually gets to pick the sequence. And to solve this puzzle, it's actually a little bit like solving a Sudoku puzzle, if you've seen that before. And you can continue this for a while. And you eventually find that by the 12th position, you actually have no choices left. No matter what you do you are going to lose > > But you can ask the same question where you're allowed a little bit more room. So suppose you can now move two steps in each direction, but if you move three, you lose. And in fact, with this setup, there's a sequence of over 1,000 moves you can submit that will keep you alive. But if you go over 1,160 moves, it turns out you will die. This was worked out in, I think, 2012. And it took a massive computer program to actually verify this. > > At the time, it was called the world's largest proof. But the actual Erdos discrepancy problem is the question, what if you extend even further? You can move up to four steps in every direction or up to 5 steps. Is there any way in which you can live forever? Or are you doomed to lose eventually, no matter how much room you have to move back and forth? And in 2015, I was able to actually solve this problem and say that, in fact, no matter how wide margin that you have, eventually, you are going to lose. > > We worked collaboratively online with a group of dozens of people. And we got stuck at some point. And eventually the project disbanded. But then a couple of years later, a commenter on my blog pointed out that a breakthrough on a different problem had sudoku-like elements as well. And he suggested maybe the two problems are related. And I remember responding on the blog, saying, no, it doesn't look right. I actually initially dismissed this comment. But then I came back a little later, I thought about it a little more, and I realized this person was right. And I was able to actually work it out. So it was a combination of working together with many other people but also working alone. > > **Sharing Victory** > > You know, with the best collaborations, you feel like you're almost a single mind when you're working on a problem. Because you have the same frame of reference, you can communicate extremely quickly to the other person. And one of the great joys of research is when you and your collaborator are both standing at a blackboard, this is a problem that's been your nemesis for months. > > And you finally find the right approach to crack the problem. And you both get excited, because you feel like it's close. And you write down the details on the blackboard, and everything checks out. And then you give each other high fives. And that is a great experience, more fun than if you're working on your own, because you have someone to share with who completely gets that feeling. > > I started out learning just one tiny slice of mathematics. But through my co-authors and my collaborators, I learned all these other fields, why they enjoyed it, and I started enjoying them too. And, in fact, I think I learned more mathematics after my formal education than before because of everyone who I worked with. > [!Abstract]- 12. Onward > > Mathematics is actually the most cumulative subject in human knowledge, really. We are using mathematics that was developed 2,000-3,000 years ago. We have-- the theorems of Pythagoras or Euclid, we still use today, the basis of modern mathematics. And that's more true in mathematics than almost any other discipline. > > For example, in science, you know, Aristotle thought that all matter was made up of four elements, Earth, air, fire, and water, which is completely incorrect. And modern physics is not really based on Aristotle in physics. But modern mathematics is based on ancient Greek mathematics and even more ancient mathematics than that. > > We stand on the shoulders of giants. Mathematics is famous for studying problems which has no practical application in mind. But many years later, a scientist or engineer or someone working on a practical problem realized that some mathematician studied this problem decades earlier and could use that mathematics to solve the problem. > > I think Eugene Wigner once called this "The unreasonable effectiveness of mathematics in the physical sciences." One classical example is in, I think, the 18th century. People started studying what are called non-Euclidean geometries. So the usual geometry, which is called Euclidean geometry, are straight lines and points. When you have two lines pointing in the same direction, which are parallel, they go on forever. They never get closer, or they get further apart. They just stay at the same distance forever. > > But people started asking, what if space was curved? And there are geometries where parallel lines eventually cross or where the parallel lines diverge? And this seemed like a purely theoretical pastime. Because, you know, clearly, the world was understood to be flat. > > But then centuries later, Albert Einstein realized that in order to make this theory of gravity work, space had to be curved. So he asked some mathematician friends, is there an existing theory of curved space? And there was. People had developed it earlier. And it turned out to be exactly what was needed to develop what we now call Einstein's general theory of relativity. > > There are insights and breakthroughs by very smart people, but they don't come from nowhere. They come from a very patient process of working our what everyone else has done, talking to other people, and then eventually, naturally, you see the way forward. > > There was a great mathematician, Alexander Grothendieck, who made this analogy that a hard math problem is like a walnut. The natural tendency for some people is to hit it with a sledgehammer. And some people actually are like sledgehammers. They can do that. But more often what happens, the process is more like you soak this walnut in water for a long time, and it gets softer and softer. And eventually, this shell becomes so soft, you can just peel it apart with your hands. And the solution is just so natural. > > **Becoming Part of the Story** > > Math is not so much just about solving problems, or that's only part of it. But it's about gaining insight and connections, seeing that two things are related in ways that no one thought was possible before, discovering phenomena and explaining them. It's like a TV series that's been going on for thousands of years. And you start watching, you know, partway through the series. And to begin with, it's all mysterious. You don't know who the main characters are. There's a lot of callback to things that you don't know. So you have to study literature of the past episodes. Maybe there's the Wikipedia or something. And you eventually learn the plot. You learn who the good guys are, the bad guys. > > You know why people get excited when there's some key plot development. And you gradually get invested in the story. And it's ongoing. Every few months, there is a new advance. There's a new twist in the story and you become part of the narrative. You also get to control the direction of the story. To begin with, you're just a passive observer. Eventually, you get to also move the story forward. You make some improvement to some problems and then someone else takes over. And that's the way it works. --- ### **References** [Terence Tao Teaches Mathematical Thinking | MasterClass](https://www.masterclass.com/classes/terence-tao-teaches-mathematical-thinking)