I find mathematics beautiful the act of stripping things down to their Bare Essentials creates a certain Elegance that you wouldn't have seen if you if you try to keep the original problem in all its complexity I guess tonight is now Terence cow Professor Terence Tao is the equivalent of a rock star in mathematics by nine he was doing maths alongside students twice his age I've always liked mathematics as long as I can remember my parents told me when I was two they found me trying to teach them older kids how to count it was mind-blowing he started doing things which I have never seen any other student come near until 2006 Professor Tau won the fields medal his single most famous result is the proof that you can find billions of evenly spaced prime numbers the Mozart of mathematics was named to President Biden's Council of advisors on Science and Technology 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 it's exciting and thrilling and I hope I can convey some of that to you I'm Terence tale and this is my masterclass [Music] I think that people often have a fear of mathematics that it is just too alien too different from their everyday experience for it to be useful in their lives to me mathematical thinking is just an extension of ordinary ordinary everyday thinking mathematics is often about trying things you don't quite understand and um and failing the first time but mathematics failure is okay you can just keep keep experimenting 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 maybe this class is even more suitable for people who have not had formal methodical training because they might get more out of seeing what math is actually like I would wanted the opportunity to to show that knowing how to think mathematically can 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 bit more manageable a large part of this class will be about 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 a symptom a little of a sense of what it's like to be a mathematician you know we are not Wizards or aliens you know we're just regular people solving problems [Music] so what are the key takeaways well firstly mathematical thinking is actually an extension of ordinary everyday thinking it's not Magic secondly it's okay to fail failing is how you learn to approach problems differently and find Creative Solutions [Music] 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 in the other way really but I remember growing up thinking maybe I should be a storekeeper because I think I can handle you know balancing the accounts and and inventory that seems 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 public media except maybe as 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 in high school I'll take some some college level classes my parents certainly tried very hard to to balance my math and science type activities with with everything else that a kid does I participated in a lot of mathematics competitions they're 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 is that mathematical competitions are like country meter Sprints and mathematical researchers are like marathons they take sustained effort and you need a lot of stamina [Music] foreign I was lucky to have many mentors um my first was my mother she was a high school math teacher she went through the basics of math with me when I have 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 I was very proud contributing to that journal as an undergraduate I had an excellent advisor who recommended me going to study abroad to do my PhD and uh 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 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 thank you 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 else around us seems to be doing better than us because we're always faced with problems that we can't solve and uh 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 to to meet with a very famous Professor once um and I walked all the way to um the campus where he was and I was I stood at his front door I was going to knock make a you know I this is before email I did not set up an appointment and I chickened out it was just too intimidating um I mean subsequently I talked to him much later and you know he's a great guy we were good friends yeah you know we are we are human too we're normal people like everyone else I think maybe a little bit uh more nerdy a little bit more socially awkward maybe but still pretty normal [Music] so what life lessons can one draw for mathematics um one of this and that I think really um hit me was just how connected everything is um so when I learned mathematics as a student um every single aspect of mathematics was uh on separate subject I took a class in Algebra I took a class in Geometry I took a class in calculus um and then I also learned you know physics and chemistry and all the other subjects and 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 um of a single um body of knowledge and there are so many analogies and connections between um these subjects the way I've heard it described sometimes is that um mathematics when you learn for the first time it's like uh you're looking out um at a landscape which is full of mist and all you see are a few mountain peaks there's a mountain peak of algebra it's a mountain become number Theory and so forth and they're not connected but as um as you keep learning the the um there's Miss lowers and lowers and you start seeing some valleys maybe you see a connection between algebra and geometry it's a connection between physics and differential equations and there are more and more of these values that appear and and the more advanced uh your um your studies are you you see deeper and deeper connections and then you start seeing sort of all the cities and and and and highways and all the the really um important connections between subjects that were obscured when you first learned the subject and so it's only at the undergraduate graduate level that yeah you start seeing um all these connections it's a bit unfortunate that I think we see all the best all the good stuff in mathematics is uh you don't get to see until you've been through years and years of of training [Music] it was only later in my career that I really appreciated that math is not so much just about solving problems so that's only part of it um but it's about gaining Insight finding connections seeing how two things are related in ways in ways that no one thought was possible before and also sometimes it connects to the real world I've done many things in mathematics most of which are very abstract and uh not super practical but they were one or two things that ended up having practical value and so that's also very satisfying I guess I'm I'm just excited to know what what happens next okay so what do we learn now so first the more you think about your surroundings the clearer it becomes that everything is connected secondly math is not just about solving problems it's about getting inside finding connections and explanations and finally the deeper you dive into anything in your life the more you discover and learn [Music] there is a perception that mathematics is some sort of sorcery your taught this whole book of magic spells you know that that if you want to solve a quadratic equation you invoke the quadratic formula and uh you know you write some clean symbols and you you solve the equation but often you're taught to apply these rules without really understanding why they work and as a consequence maybe you're afraid to deviate you know if you if you if you do anything with you it just doesn't follow the recipe maybe it'll be a disaster mathematics gives you a way of solving problems uh 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 in the real world [Music] other people who are naturally good at math and are there some people who and hopelessly bad at math um I don't think so I think everyone has an innate mathematical Talent you can see it in children they they ask questions about numbers and shapes there's a result in mathematics that all children or many children discover by themselves in fact in fact they teach other children this one fact and it's it's the fact that there's no largest number and the way children realize this is at school they sometimes play this game you know who can name the larger number you know so they say oh one million and then the other chart and it says 1 billion but eventually they figure out that no matter what enormous number the other child names they can just say plus one I get that whatever you said plus one and that's a bigger number and once they realize that they realize that there was there was actually no no biggest number and in doing so they have actually discovered a very important technique in mathematics it's called Truth by contradiction contradiction is when a proof 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 the largest number and then you you show that that leads to a contradiction that if someone says Ah the largest number is so and so you just say plus one okay it is not the largest number that is an example of a mathematical technique which is often very challenging for uh um for even undergraduate students um to grasp when they're taught informally but children in school yards can actually pick up this this this concept just by playing a game everyone has an inbuilt mathematical intuition it's it's obvious to everyone you know if you have a pizza and you share it among four people uh you everyone gets a smaller piece than if you shed it among three people that's a very basic methodical fact and it's something that we all understand the main reason why um people think that 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 um it's not natural um 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 uh some people are very systematic and that they are all valid ways of approaching mathematics [Music] one of the first steps when you take a reward problem and try to turn it into a mathematical problem is it's called abstraction abstraction is the process of taking away all the reward elements of a problem and stripping it down to its basic methodical form you can then solve it using algebra or authentic or whatever other type of mathematics you wish you learn this in school actually you're given these word problems you have sown so many dollars and and you need to get from A to B in a certain amount of time uh and you need to plan your route or whatever and so you you identify what aspects of the problem are important if you're planning how to get from A to B um in the right way maybe things like the speed of your vehicle um what stops you take these are important variables but other things like you know what color is the car that you're driving that's not important um so you you have to identify what features the problem are essential which ones are not um and then try to translate um all the data that you're given into mathematical language sometimes these are equations um sometimes these are geometric relations and then once you strip away the uh the original context of the problem and you just have this this abstract mathematical model then you can start applying your mathematical tools your algebra geometry whatever [Music] foreign people sometimes think of mathematics as a language and I certainly do all children naturally pick up languages but imagine if English was taught when in your English classes you you didn't speak 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 how to what a preposition is or whatever so it's a little unfortunate maybe that our school education maybe over emphasizes rigor in mathematics even though it is important the purpose of mathematics really is is to communicate ideas and and Concepts in a very precise way and to to 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 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 and these are the relationships between X and Y we're really stripping um the problem down to his Bare Essentials and when you first see that it looks very abstract and weird but by removing all the inessential components of a problem you can focus on on what's really going on and it can help you see uh the way forward so it's a very clear language for solving quantitative problems [Music] 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 interviewing with each other we take over granted that 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 know you 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 it gives you enough tools that you can feel like if you really had to you could actually understand from first principles how how these things actually work and the word sum up becomes a less scary place you don't have to resort to you know conspiracy theories or or you think that everything that you don't understand is Magic um yeah the world becomes a more rational place which I find very comforting so what have we learned okay so firstly the purpose of mathematics ultimately is to communicate ideas and concepts with precision secondly even in your day-to-day life stripping down a problem to his Bare Essentials can provide Clarity and insight that you wouldn't have had before [Music] I think a good way to connect to uh the mathematical side of your brain is to find a hobby or something that you are passionate about and are willing to to Tinker with some people like playing mathematically themed puzzles like Sudoku puzzles are very popular as a kid I like playing for computer games and logic puzzles if you like fixing up old Machinery or something the challenge of making some some broken piece of Machinery work that kind of problem solving that shows up in that it actually is very parallel to methodical thinking if you're a cook and you're following a recipe and the recipe is for four people but one day you have to cook for six people um and you have to change all the amounts of ingredients or maybe um the the recipe requires you to cook something for yeah for half or an hour and a half but you only have an hour that's a mathematical problem you know how can you use resources most efficiently I think we bump up against mathematics all the time now often you can kind of wing it you can just um an experienced chef may have some rules of thumb but um having some mathematical training can help you um you know avoid a disaster I think um well at least give you some sort of first approximation as to how to adapt to an unexpected situation I think we could we could all play a little bit more with trying little tasks first before moving on to to big high stakes uh tasks I think mathematics is just an extreme example where it doesn't matter how many times you fail to solve a little math problem that there's no real penalty of a wasted time but it's not even really a waste you know as long as you learn something from it [Music] thank you the more you expose yourself to doing tasks in a fun challenging way and enjoying just the challenge um you know I enjoy the challenge of assembling furniture [Music] you know um from very unbeatable instructions I mean I don't want to do it for a living or anything but almost anything in real life can be can be turned into a little problem and uh you know Finding situations in life where the stakes are low it's okay if you screw up you just start over and just getting into a mindset where um the goal is not necessarily to solve the problem quickly or efficiently but to enjoy yourself and to to draw lessons from it one of the basic techniques in mathematics is if you have a complicated problem you isolate a a simpler version of the problem solve that first and then once you know how to solve various simpler subcases of the problem try to put them back together again and and solve the full problem it's just to give you one example from my own personal experience I once had to put up um some curtains on on one of my windows in my house and you have to stand on a chair and there's a 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 um I practiced you know how to assemble each piece um on the chair so I would I would take just the rod and see how to put the rod um on on the hooks and how to put the curtain on the rod and only after I had practiced each individual um piece of the problem would I try to put it all together that took longer than if I had tried to to assemble the 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 [Music] yeah so is mathematics a creative subject definitely um 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 but when you're a research mathematician um you know you are solving problems that standard techniques don't quite apply um because it's so abstract and not necessarily Tethered to reality it allows you to be very creative and very very flexible you know in the real world you may have a problem where you have a say some finite number of resources you only have X amount of dollars to sort a problem you may only have so much time but in mathematics you can change the parameters you may say okay what if I had a billion dollars could I solve this problem or what have I had an infinite amount of Manpower it gives you a lot of flexibility to um to change the problem um into one that maybe you can solve first and then you can then from there go back to to solve the actual problem and that's a freedom that you just don't have in the real world you can't uh you can't just say oh I don't before I I solve this problem can I first have a billion dollars to to experiment yeah the abstraction that mathematics has afforded a lot of creative freedom [Music] many people they uh they had a problem to solve and they basically did mathematics even without knowing that they did mathematics there's one story I quite like um so there was this mathematician and named Johannes Kepler in the 17th century and he once had to purchase wine for I think his daughter's wedding and the way that um the wine Merchants would price these barrels was that there was a little bunghole in the middle of the barrel and there will be the stick and they would poke the stick through the back through the hole to the opposite corner of the the barrel and measure the the length and based on that length they would say okay this this is you know how many do cats or whatever worth of of wine when couples saw this he was amazed you know I mean normally you'd think that if you want to measure how much wine there is in a barrel you'd have to to measure how many gallons fill it up or something and this was a very quick way and for these wine emotions it worked apparently um 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 um so a couple of got very intrigued and he um he approached upon mathematically he considered wine barrels of various widths and Heights and computed these events of the sticks and their volumes and he didn't indeed see that there was actually this very nice relationship the barrels that were made in Austria they were all roughly the same shape and it turned out that that if you made the barrel you know five percent narrow or five percent wider the relationship between volume and um and the leftistic was roughly the same the dependence of volume on um on the shape of the barrel was was actually at near a local maximum local maximum refers to a situation in which any small change 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 tours of calculus in order to solve this problem and it was actually quite important for later development mathematics so there's a lot of of mathematical problems that we just encounter in our everyday lives and we often just have to solve them with or without formal methodical training but once you have the training you can you can explain why these funny tricks that people have in their various professions actually work so what are some components of practical mathematical thinking well they include experimentation a willingness to fail proactive deconstruction and remembering sometimes to zoom out apply abstraction choose a plan of action and then act accordingly so keep an eye on the big picture [Music] one thing that surprised me and I had to learn somewhat late in my mathematical career is that the solution to pathological 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 an antagonist if there is some goal to solve there's some obstacle framing things in in a way which has a narrative to it it activates certain parts of your brain foreign if you want to develop your skills to find analogies to to see connections basically the most important thing is ask questions and the dumb of the question the better really so to give you one simple example from arithmetic um when you multiply two negative numbers together let's say negative three negative four you get a positive number Plus 12. but that's not very intuitive to many people you too many people two negative numbers should combine to form another another negative number and so you can ask you know is there some analogy you can use to pursue this so you can maybe use an economic analogy so if you want to modify three times four you could say Okay suppose I am being paid three dollars an hour to do something and you work for four hours okay so every hour you get three dollars and then after four hours you get twelve dollars fine but now suppose instead of being paid three dollars an hour something is cost costing you three dollars an hour maybe you're running water or something or electricity and it's costing you three dollars an hour so after four hours your um this will cost you 12 so this is why minus three times four is minus twelve so you can ask how can I push this analogy further so then you can ask how do you interpret minus three times minus four well suppose uh you know your water's running or something and it's costing you three dollars an hour but um you managed to shut off the water for four hours so you have saved four hours of uh of expenses and so you have saved twelve dollars and this is why minus three times minus four is is plus twelve so by exploring this analogy and just asking questions but what does this mean what does this mean in this analogy you can gain a sense of what multiplication of negative numbers really means um and uh yeah your intuition your intuition is now better than it was before [Music] actually very useful to anthromorphize um these the mathematical objects you deal with you know they're not just x's and y's um often you will hear if you listen to a mathedition talk about a problem they might say that that this is the enemy and certain things are good guys you know like maybe there's a certain uh powerful tool um that that we want to apply uh but it's but in order for to apply it certain conditions have to be met and so you start working on on that instead if if you want to to solve for some unknown X you could think of it as like almost like a detective hunt or something there was there was some mystery opponent who has uh but and you don't know the location of his opponent he's located at X and so you need to identify um 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 copy this because if x was was equal to this then this would happen and you know this doesn't happen so we can activate certain uh detective type aspects of your thinking that help you realize how to solve your your question [Music] 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 you know if you have a large population such as the population of the United States um you know several hundred million people and you want to know their position on some topic like how many people are voting for a certain 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 um out of 200 million possible respondents and that's a tiny tiny fraction of the whole population but nevertheless posts can be incredibly accurate and 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 will 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 I like to use is what is the salt content of the ocean you can you can tell the ocean is salty just by taking a single drop of seawater okay and just tasting it and immediately from one drop of seawater you can tell instantly of the seed the ocean is full of salt and the reason for that is that the the salt is quite well distributed within the ocean now the beauty of this 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 you know fresh water portions and salt water portions if you sample the wrong portion you can get a misleading idea of assault content of a whole and so sometimes polls fail because they are not holding a sufficiently mixed portion of the population it's it's important to sample as randomly as possible try to pull people from from different regions different demographics different social levels you know 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 will be more accurate if it's closer to the representative mix analogies are an excellent way of using both halves of your brain you know the the rational half the formal half which which uses very precise rigorous thinking and the intuitive part which is using analogies and intuition part of advanced methodical training is learning how to to somehow see a problem serious topically with both your your rigorous perspective and your intuitive perspective and to know how to translate back and forth going back and forth is actually one of the most fun parts of of doing math because you really feel smarter when you make the connection and you see how the the the the numbers and the equations match with your intuition [Music] sometimes nowadays can lead you astray for instance a lot of mathematics is centered around solving equations exactly you know to the last decimal point and it was only realized in the 20th century really that um there are many questions um many practical questions where an exact solution is is not possible and what you really want is just an approximate solution for example um Newton famously a software is called the two body bomb in gravity that if you have two massive bodies like the Sun and the Earth you describe the orbits quite exactly and people spend centuries trying to sell what's called a three body problem and if you have three bodies like the the Earth's Moon and Sun 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 asked is can we compute the trajectories to sufficient accuracy for sufficient period of time like if you want to send a rocker to the moon it's okay to get within you know um half a meter or something of the correct position of the Moon once you frame things in in the context of finding approximate Solutions rather than Exact Solutions there are a whole different set of techniques that you can use there's a whole different narrative foreign errors as long 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 um you know if you pull a thousand people there is a tiny tiny chance that your pole will be completely off because all of the thousand 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 like 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 you know if it if it failures in delivery one percent of the time that's that's not too bad knowing what error tolerance you have for your problem is is important in framing in finding the right framing okay so what are the takeaways here honestly framing a problem with a narrative or analogy can make a huge difference secondly mathematics is not always black and white thirdly a good narrative allows you to understand an abstract methodical 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 only require an approximate answer [Music] one of the great insights of mathematics is that there are different Frameworks different ways of thinking about the same methodical problem which may look quite different but mathematically related one of the great advantages is that we can then um transform the problem and think of it in a in a new way which has maybe 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're transforming the problem into a geometric problem then you're activating the visual centers of your brain so transformation is a way of swapping your thought patterns for some problems you know actual physical sensation can actually be uh be useful many mathematicians you will find they wave their hands um or gestures somehow when uh when thinking about a problem and and that that act of of making their thoughts um physical uh is is often quite quite useful I know people who um they work on a pen and paper for a while on a problem if they get stuck they take a walk uh 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 um only focus on on the essence of the problem Concepts that are simple enough to keep in your head at one time without having to write down any computations and that can sometimes lead to um a better way of thinking about about the problem um 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 realize what the problem is previously they were only just thinking about 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 Arts 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 know you you find whatever analogy is physical or or um mental or whatever that that work for you and uh yeah sometimes it makes you look silly but that that's uh yeah that's occupational hazard [Music] I sometimes find myself solving little mathematical problems on the fly to give one example um house once at the airport and um I had to make a connection and 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 and and to tie the shoelace but the whole way I was running down um some of the hallway had these um moving walkways and so the question they've had in my head while trying to run across is it was it more efficient for me to tie the shoelace on the walkway or off the walkway I ended up running you know doing a little bit of algebra in my head and I keep figuring out the answer um that it's actually better to to tie the shoelace on the walkway now a little later on um I mentioned this in my personal blog and um 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 way you need to transform the problem so you imagine instead of one person racing to um to catch the connection imagine that you had an identical twin running beside you and you were both running side by side to get to the other end of the of the airport and you both had an untied shoelace but suppose the only difference um between the two twins was that um as you approach your walkway one of the twins will stop just before the walkway to have a shoelace and the other twin will get on the walkway and then tie the shoelace and 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 which was 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 some other commenter on my personal blog said this was the the most practical piece of advice he'd ever seen on my blog did you make a fight I made my flight so it's it's important to keep trying different approaches to a problem um it's very common and this still happens to me that you're thinking 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 um you need to to mix things up sometimes and be willing to let go of preconceived uh beliefs often if you're really stuck you you come up with more and more desperate ideas foreign of transforming a problem comes from a very nice exhibit at the Museum of mathematics in New York and it is it involves What's called the finding 15 game the game involves nine numbers the numbers from one to nine and the finding 15 game works like this first player who can collect three numbers that add up to 15. wins the game okay so now I'll be playing a few rounds of finding 15 with my twin evil Terry hey so let me demonstrate I will start by um maybe I'll start with number three so I will pick a number three okay evil Terry it's your turn so evil Terry plays number six I will play five okay Evo Terry Evo Terry plays two and now I will play seven and I will notice that three plus five plus seven is fifteen and so I have won the game ha okay 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. I take squares actually have a long in ancient history you know some cultures thought they had mystical properties they don't unfortunately actually they're mostly a mathematical curiosity they don't have too many applications but this magic square that sounds 15 turns out to be the perfect tool to solve this game what I have here is a magic square so these are the numbers from one through nine same numbers that we use to play the finding 15 game two seven and six add up to Fifteen six one and eight add up to Fifteen four five and six add up to 15 and the winning move from last game three five and seven also add up to 15. now you can use this magic square to transform the finding 15 game to a much more familiar game so if we review the game that we just played I started by choosing the number three then my evil twin counted with a six then I played five my opponent played two and then to win the game I played Seven and because three plus five for seven was equal to 15 I won the game in other words this magic square has transformed the final 15 game to the tic-tac-toe game and we are usually much better at Tic-Tac-Toe than we are at finding 15. so if you apply the transformation while you play a game of final 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 and it turns out the centerpiece corresponds to the number five and so actually the best opening move to play in the fighting 13 games to start with number five [Music] [Music] all right 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-toe and almost all children are familiar with this game it immediately makes the problem a lot less scary maybe you can't always win a tic-tac-toe but you at least have a lot of natural experience how to play the game so you can use your intuition about tic-tac-toe to help you create the finding 15 game as you can see evil Cherry I was able to transform this game into a game that he was much better at and his skill therefore at the game improved quite a lot and he got quite an unfair advantage so that's the power of transformation [Music] 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 there's the uh the feel 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 um a toy puzzle can lead to a very practical application which had nothing to do with the original puzzle at least at first class so there's a classical puzzle almost 100 years old um called the kind of a coin puzzle you're given 12 coins let's say gold coins say and they're all identical except for one which is kind of fit and the Cardiff coin is either slightly heavier or slightly lighter than the other coins you don't know which coin is is the counterfeit one and you don't know whether it's heavier or lighter and your task is to work out which coin is counterfeit and the only tool you're given is a balanced scale and you can put some coins on on one side of the scale once some coins on the other and you can see whether the coins are evenly matched or whether one is heavier one is lighter but the catch is you're only allowed to use the weighing scale three times so even though there's 12 coins you can't weigh each coin separately you only allowed three Wings this is a purely mathematical puzzle right there's there's no conceivable practical situation where you have to determine a kind of a coin and you have this very limited number of wings so the solution for this kind of a coin problem is quite complicated and we'll include a full solution in the class guide it uses a mathematical object order Matrix and the techniques we use to solve this problem actually turn out to also 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 um 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's a way to solve the coin main problem uh using a filter mathematics called linear algebra and maybe we'll discuss this more in the written notes and the same type of mathematics is also used to solve the compress 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 this 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 acute the latest models they use this technique called compressed sensing which I actually help develop they can speed up um MRI scanning by a factor sometimes as much as 10 a scan that used to take three five minutes you know maybe it takes 20 seconds you can start with a completely frivolous mathematical question and many years later it it becomes of a very practical application [Music] I'm from a generation where um if you wanted to play a computer game you would actually get a book and they would give you the code for a game and you type it into your little Commodore computer and and run these games yourself and the benefit also is that you you got to tweak the code sometimes you know if you wanted to make the game faster or give yourself more lives you just changed some numbers in in the code um sometimes this led to hilariously easy or difficult or maybe the game just didn't run at all um but I found that very instructive some computer games have difficulty difficulty levels you can play on easy mode where you have a lot more ammunition or a lot more lives or 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 uh turn up the difficulty this is what makes computer games often a lot easier to solve than real life problems in real life you're often thrown um the problem 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 freedom to choose the difficulty of your problem if the problem is too hard make it easier by adding more hypotheses searching to 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 this maybe comes from my gaming background but one thing that you often do in mathematics um is you make the problem easier first often a lot easier just to get started and it's akin to um trying to 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 and then you uh you go back and you you turn the cheat off 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 you can't use three and a half people it's often good to First suspend the uh the role that you have to have a whole number solution and get an answer which is a little bit ridiculous that you need 10 and a half people to solve this question but then you fix it or maybe you need to round up maybe you need 11 people rather than 10 and a half in physics this is sometimes called assuming a spherical cow you know so you may have some physics problem that involves a real world object like a cow and you want to know how fast the cow can move and so forth but the uh the laws of physics that you're given um they're only nice if you assume the car is frictionless or completely round or something so the joke is you 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 try to add back some of the complexities 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 that is a simplified problem um it's actually a very powerful technique in mathematics it's called linearization and people often solve the linearized bottom 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 machine what is a puzzle what is a problem what is a real world application there's so many connections and analogies that it's I find it better to think fluidly these are all just um things to do as you get more experienced you don't pigeonhole individual tasks into separate boxes so much because it all becomes just one continuous organic hole [Music] 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 presented in a textbook or a class or a research paper the tendency is almost always to only present the correct solution the one that worked and you don't see the the outtakes which is a shame actually because that's often the most informative part we don't often show are embarrassing these 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 because a they might actually work but B because the way in which they fail is often very instructive it gives you Clues to um to figure out what the solution should be I think the physicist news ball 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 I must confess that my first 100 years in graduate school was well not super productive I did some math and I took some classes but I did spend too much time doing computer games and uh 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 called the the general example the generals it's a two or three hour exam where you uh you pick three topics in mathematics and three faculty in the department 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 in my class you know they they spent months and months practicing this they had they quiz each other they had these mock General exams um I basically studied a little bit maybe for a few weeks but I figured I would wing it I did not do very well my knowledge it was very superficial Whenever there was a follow-up question I could not write a good answer and um the the questions became easier and easier and I could barely solve um most of them the only reason I even passed the exam is 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 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 that I really should should try to do better I was used to to doing water mathematics without working super hard and so it was a shock to me to realize that that mode of of thinking had limits that um you had to think more systematically more you had to actually plan your study um you have to actually um learn from your mistakes um and so yeah they're also important life skills to learn yeah one famous example of an experiment that failed and led to an interesting Discovery was a story of the the Greek philosopher eroticians so um many many years ago he lived in a city in Egypt called Alexandria and he read that there was another city another town called sayin which had a well in it and this war had the funny property that on one day of the year the summer solstice if you um looked at high at High Noon until as well the sun was directly overhead and you could see the sun reflected in the water deep below eratosis was intrigued by this story and so he decided to wait until the summer solstice um in his own town of Alexandria hand got it he's on local well and he found at noon that the water the sun did not shine straight down it was at an angle so his experiment failed and for most people um they would just say Okay this was I guess the modern was fake news okay that that is that the story just wasn't true but what else things realized is that because the experiment failed in his City but 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 I think within 10 accuracy and all he did was he measured the angle that the sun made at Alexandria he knew the distance between alexandrians and cyan 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 um I think I think penicillin was discovered by accident you know that someone left a job of uh of a bacterial moldy or something [Music] 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 you may cost lives or lots and lots of money but if you try and solve a 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 it was a famous mathematician of Vladimir Arnold that said that mathematics is the part of science where experiments are cheap you can you can try almost anything you just need a Blackboard um and a bit of time and just getting into a mindset where um the goal is not necessarily to solve the problem quickly or efficiently but to enjoy yourself and to to draw lessons from it if you start doing that in in your life I think maybe it would just naturally become just part of your way of thinking it's just another way of going through life sometimes um a failure can be it can be extremely insightful and productive it can be a sign that there's something um more going on than you than you expected something unexpected I think Asimov once said that the uh the most exciting sentence you should that a scientist says is not Eureka but that's funny okay it's there's something unexpected something that that didn't go as as planned that's not a failure it's often a clue that this uh something new and exciting is going on [Music] people often think that the way math or science works is that people work very hard on a problem and there's there's no progress for a long time and suddenly there's a Eureka moment you know some lightning bolt hits you and suddenly you everything gets illuminated and you find the uh the Right Way Forward that's not quite how it works in my experience you do work for a long time on a problem um 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 with other topics and you're just setting the stage for um putting the problem in exactly the right perspective it often says a very small thing it's a very last puzzle piece to put into place and suddenly everything makes sense it's rarely a big lightning bolt of uh Clarity it's just uh it's more like a realization that you're almost there already that that a lot of the insights the partial progress that you had actually it just takes the one last ingredient to make it work when you fail at a task um maybe the most Natural Things psychologically says okay that sucked I I don't want to do this I was maybe you blame yourself and failure occurs it's very natural to try to find the person to blame but mathematical problems are not like that often um what you need to do is just not so much to assign blame but uh 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 um 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 uh the journey in asking these questions so what are the takeaways firstly 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 [Music] 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 cut sometimes it just tells you the problem is is it's not ready it's before it's time and uh progress on some other problem has to happen first this is one thing by the way that computer games have taught me there are many computer games where you know that you're treasure lies behind the door but the door is locked so you need a key so you just 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 you there's 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 um but the process of trying things and seeing why they didn't work again and again uh 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 snowboard sometimes um 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 um the uh the door a little bit and then when you finally find the right uh 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 um bashing my head against it trying all kinds of things and I would show what I did to my advisor and you 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 filing cabinet and fish out a little pre-print as you read this this paper had this 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 it was usually right the uh um the techniques they had um solved the problem that I had spent hours working on and what this showed me is that often experience Trump's energy that if you know what to do you can save yourself a lot of effort as I get older I find myself I have less energy I can't spend hours and hours on on a single problem like I used to but I can often uh see the connections to an existing problem like I can use my experience um a lot more effectively than I could before [Applause] [Music] sometimes a task is just too big and scary to even get started and you have to mentally break it up into into smaller pieces there was once a project where um so five of us working on a quite hard problem and we had 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 uh considered properly at all um but by that point we had already um 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 and 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 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 proudest of actually um but it was important that we had made a mistake early on to give us this false hope that we were closer to the solution when in fact it was much further away than we had expected so sometimes uh Serendipity Works in funny ways [Music] to advise students what to do when they feel stuck [Music] I would say that this is completely normal that that this happens to everyone and in fact if you are not feeling stuck regularly you are not challenging yourself people often think of um 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 um 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 proof of concept that some technique has some promise but it doesn't yet have enough maturity to um to work for the full problem um and sometimes you learn that the problem is just not ripe that it's it uh that the techniques you have are just not suitable and those are all valuable things to 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 that you need to take in in these fields when you're working on a difficult problem it's it's not realistic to expect a perfect success rate but you can often be end up in a better place than when you started and that's really what to shoot for [Music] 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 uh 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 and so we all have to learn to sometimes take a breath step back admit that some problems are outside once once 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 and so you have to learn to uh to let some problems go all right so what have we learned today 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 and finally sometimes you need to learn to let your problems go just wait for a bit and return refresh eyes sometimes you can see something you didn't see before [Music] thank you take mathematics like any other discipline it's actually a very social discipline um in the past maybe we were mathematics was conducted by isolated people in rooms you know um 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 we need uh 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 is that you get to learn their toolbox I didn't used to use um numerical simulations very much in my work I would much rather work things out by pen and paper than to write a little computer program to to do things for me but I've worked with people who are very very good at simulations and it was quite informative Jack you see these graphs and figures come up almost in real time some people were just extremely fast that is the future of our field [Music] mathematics used to be very individual activity people used to just work in isolation that 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 exploit complementarity um 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 and sort of these Blue Sky approaches to a problem and the pessimist job is to shoot down The Crazy Ones and the crazy ideas but keep the uh the same 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 um I find it's best just let things flow naturally there are two famous mathematicians Hardy and littlewood who wrote down what are called the Hardy littlewood was a 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 what they feel is um is appropriate it's never uh very productive to try to ascertain your 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 before uh the paper at least in principle technology has really changed the way we can collaborate but now we can crowdsource many times mathematics um the problem can be split up into many many different pieces and you can have um large groups of people work on each individual piece and almost have an assembly line you see this phenomenon throughout science actually you know amateur astronomers can discover comets biologists employ amateurs to to try to solve protein voting problems and we're just at the cusp in mathematics of also being able to harness the crowds of amateur mathematic enthusiasts one thing I've definitely noticed is that when you have many many people working on on a single project there's always someone who can draw connections with an obscure field mathematics or some other piece of literature that you might not 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 to like I can collaborate with people in different continents they work in the problem while you sleep and you wake up in those progress and you walk in it while they sleep um and that's actually a lot of fun foreign [Music] about 50 years ago hadn't made 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 is a mathematician named James Gray who had a rather colorful interpretation of the problem as 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 torture 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 which we have so left you afford to your death so maybe on the right there are some spikes here okay and on the left maybe there's a poisonous snake or something I can't draw very well but still important you have to submit the list of movies beforehand and each move can be one step to the left or to the right it's like maybe moving right and then left and then uh left and then left on the 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 okay and then left and then left and then left but it should now get four down and you'll get poisoned by the snake so this is not a good uh um sort of move to submit and there's a Twist a torture can make you do every third move or every fourth move and so forth instead of every move in sequence so you can't just submit a list that goes left right left to right forever so the torture 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 truth position you actually have no choices left and no matter what you do if 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 a thousand moves you can submit that will keep you alive but if you go over 1160 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 that at the time it was called the world's largest proof but the um the actual urge discrepancy problem is um the question what if you extend even further you can move up to four steps in every direction or after five steps is there any way in which you can live forever or are you doomed to to lose eventually no matter how much room you have to move back and forth and in 2015 um I was able to actually solve this problem and um and say that in fact no matter how wide margin you have eventually if you are going to lose we work collaboratively online with a group of dozens of people and we got stuck at some point and eventually the project expanded but then a couple 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 uh saying no that it doesn't look right I actually initially dismissed uh this comment um but then I came back a little later I thought about it a bit more and I realized this this person was right and I was able to actually work it out um so it was a combination of working together with with many other people but also working alone [Music] here were the best collaborations if you like that you're almost a single mind when you're working on a problem because you have the same frame of reference you know you can communicate extremely uh quickly to the other person and one of the great joys of of research is when you when you and your collaborator are both standing at a Blackboard you there's this problem that's been you've been your Nemesis for for months and you finally find the uh the right approach to crack the problem and you get you you both get excited because you feel like it's close and you you write other details on the Blackboard and everything checks out and then you give each other high fives um and that 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 my co-authors my collaborators I learned all these other fields why the why they enjoyed it and I started enjoying them too and in fact I think I learned more mathematics after my my formal education and before because of everyone who I worked with [Music] mathematics is actually the most cumulative subject in in human knowledge really we are using mathematics that was developed two thousand three thousand years ago we have a you know the theorems are Pythagoras so Euclid we still use today um the basis of modern mathematics and that's that's more true in mathematics than almost any other discipline for example in science um you know Aristotle thought that matter was 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 had no practical application in mind but many years later um a scientist or engineer or someone working on a practical problem realized that some mathematicians studied this problem decades earlier and could use that mathematics to solve their 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 of straight lines and points when you have two lines pointing the same direction which are parallel they go on forever they never get closer or they get further apart they just stay the same distance forever but people just started asking what if space was curved like are there geometries where where parallel lines eventually um Cross or whether power lines diverge and this seemed like a purely theoretical Pastime because you know clearly the world was the actual world was flat um but then centuries later Albert Einstein realized that in order to make his theory of gravity work space had to be curved so he asked some mathematician friends you know 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 you know there are insights and breakthroughs um by very smart people but they um they don't come from nowhere they come from a very patient process of of working out whatever loss has done talking to other people and then eventually naturally um uh you see the way forward there is a great mathematician Alexander gutendik who made this analogy that a hard math problem is like a Warner the natural tendency for some people is to take it to open the Walnut you hit it with a sledgehammer and some people actually are like sledgehammers they can do that but uh 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 the the show becomes So Soft you can just peel it up out of your hands and the solution is just so natural [Music] math is not so much just about solving problems or that's certainly part of it but it's about gaining inside finding connections um seeing how 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 are you start watching you know part way 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 callbacks to things that you don't you don't know so you have to study literature on the past episodes maybe there's you know 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 advice there's a new twist in the story and you become part of of the um of the narrative you know you also get to control the direction of the story you know to begin with you 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 that's the way it works [Music]