Automatic Mathematics

Key Takeaways

blimey you fought the million dollar problem how a I hello world it's Suraj and did you know that you could earn 1 million dollars by solving one of seven math problems no matter where you live in the world they're called the Millennium problems and they were proposed 19 years ago by the clay mathematics Institute of Cambridge Massachusetts in this episode I'm going to explain how to a itools released in 2019 a neural equation solver and the Ramanujan machine can possibly help you solve these problems no matter what your background is to date only one of them has been solved by a Russian man based in saint-petersburg remarkably he refused the prize money and the Fields Medal the highest honor in mathematics for his solution you might be thinking why would he do that because in Soviet Russia problem solves you clearly he didn't desire an extrinsic reward at all his reward was completely intrinsic a deep fulfillment and joy from discovery alone while mathematics may seem scary and confusing to beginners to those who dedicate themselves to learning this foreign language a sublime experience of beauty awaits it's like in the movie The Matrix where if you stare at following green codes long enough you start to see a beautiful woman in a red dress math tells a story and mathematicians are students who listen to this story deriving joy from unexpected plot twists and new developments in ways that others don't the remaining millennium problems haven't been solved yet and although people have submitted dozens of potential solutions to the remaining problems none have held up to the peer review process like all intellectual tasks mathematical discovery has traditionally required years of training to acquire the necessary domain knowledge to make predictions but this is exactly what artificial intelligence can help us automate we can offload the training time onto computers instead of ourselves giving it mathematical data and having it make predictions for us before this is all over promise me you'll figure out which one of us is the machine definitely you Cortana just like AI has made it easier for people without years of training to generate songs medical diagnosis even legal advice it's starting to enable anyone to generate mathematical discoveries these tools have never before existed and if Gauss Newton and Ramanujan had access to them back then you can be sure they would have made many more discoveries let's first pick one of these millennium problems and I'll explain its premise then we'll look at how a I can help solve it the one I'm particularly fond of is the navier-stokes equations existence and smoothness problem besides earning you a million dollars to spend on an Iron Man suit proving it can open the way for more accurate weather modeling new types of aircrafts and even more realistic video games this problem deals with the properties of fluids fluids are substances that flow they can be liquid like water or gases like oxygen and there's a whole branch of physics that deals with fluids called fluids mechanics fluid mechanics helps modelers make assumptions about fluids measure their properties and then try to model their behavior this is actually pretty difficult if we were to drop a ball according to Newton's laws we could predict exactly what it would do in the future the mass of the ball the acceleration due to gravity all of these things help us make a prediction but fluids work very differently when Kendrick pours Hennessy into a cup it always flows differently so it's much harder to predict its movement this is because of its molecular makeup in a solid the molecules are very close together so there is little variability but fluid molecules aren't their variable making them harder to predict what the navier-stokes equations do is they take in several properties of a fluid and then use them to predict the speed of every molecule using this we can approximate how a fluid will flow they also take into account properties like pressure which could be water pressure or air pressure there's also viscosity a measure of how sticky something is put your finger in water versus molasses and you'll notice the difference in viscosity then there's a stress tensor if you imagine a fluid as a collection of very small layers you'll find that they keep rubbing against each other and this can be quantified these properties are all plugged into the equation and the result is a vector field this is a map of velocities of particles in a fluid the video game Crysis uses these equations to model its ultra-realistic water oh my God look at that rendering water gasm we can also use it to model ocean currents and even the flow of blood in the heart they work well by reliably describing fluid flows and have been confirmed in many experiments but mathematicians want more than anecdotal confirmation they want proof that these equations will work that no matter what vector field you start with and no matter how far into the future you play it the equations will always give you a unique vector field and that this vector field will provide the exact direction and magnitude of the current at every point in the fluid a solution that provides information at such infinitely fine resolution is called a smooth solution in a smooth solution every point in the field has an Associated vector that allows you to travel smoothly over the field without ever getting stuck at a point that has no vector a point from which you don't know where to move next this would be a complete representation of the physical world and that's the million dollar problem you just need to prove that solutions exist it's an existence proof that's it ha ha so simple so how do we prove it a proof is a series of statements each of which follows logically from what has come before it starts with things that we assume to be true and ends with the thing we're trying to prove it's like a story it has a beginning a middle and an end creating proofs requires mathematical domain knowledge creativity intuition and no distractions but computer-generated proofs are starting to become more common automated theorem proving is a field that is gaining more traction these days if AI is a collection of mathematical techniques then using AI to solve math proofs is like using math to solve math although deep learning technology has seen remarkable success in pattern matching tasks like image classification language translation and recommendation systems the ability to reason is still something that human intelligence excels at uniquely or ability to generalize in a mathematical domain is different from our ability to generalize in say a language translation domain consider this problem we have three equations f of X G of X and H of X we want to compute what the composite function here is G of H of f of X we'd have to plug f of X into the x value of H of X then plug the resulting equation into the x value of G of X then we have to compute all the operations and simplify it into a final equation consider the cognitive skills it requires to do this we have to parse the characters into entities like numbers operators variables and words we have to plan which order of functions to compose we have to use sub algorithms for function composition like addition and multiplication we have to use our working memory to store intermediate values like the composition h of f of X and lastly we have to apply our mathematical domain knowledge to understand rules like the order of operations to perform how do we get machines to do all of this for us there's a really promising paper in particular that caught my eye from deepmind at the annual ICL our machine learning conference this year called analyzing mathematical reasoning abilities of neural models the reason I like to this paper was that it was so ambitious they constructed a data set which they open-source by the way thanks Steve my love you K thanks bye love questions and answers from high school level mathematics mostly algebra problems check this out the input is the question a plain English problem statement like solve this equation or let this value be whatever is this value a factor of another things like that the output was the resulting answer we already have calculators that can perform hard coded operations like add subtract multiply and divide but since neural networks are excellent pattern matching tools why not just treat this as a pattern matching problem have it learn what basic math operations are by learning the relationship between input and output pairs they tested out a bunch of different models before ultimately finding that the Transformer model performed the best it's a variant of recurrent networks link in the video description for more they found that it was able to easily after training answer questions involving finding the place value in a number and rounding decimals and integers also in comparing lengths it was hardest to answer number theoretic questions like prime allottee and factorization though it was still able to give plausible looking answers it learned what addition and subtraction were which is really amazing looking at the data set on github we can extend it by adding even more data points to this this time the input is a statement like prove some equation and then the output would be the proof itself we could just add literally every known proof we have after training we then give it the statement prove the navier-stokes equation and see what it says get out of my way I have money to win although I think the ability to creatively come up with multiple conjectures first is a module that could be added to this that would definitely help it conjecture is something that is thought to be true but we haven't yet proven that it's true a proof is a formal way of using logic and mathematical manipulation to show that a conjecture is true just a few months ago a team at the Israel Institute of Technology built an algorithm that automatically generates conjectures for fundamental constants they called it the Ramanujan machine named after the Indian mathematician Romano John Romano John unlike his colleagues at Cambridge had an unconventional way of designing conjectures rather than progressively expanding the narrative of existing mathematics brick-by-brick he would use his intuition to make some inexplicable conclusion that was later proven it's like his formula that explains the behavior of black holes even before we had the math to describe black holes or even understand what they were this creative intuition was partly automated using the meet-in-the-middle algorithm it allows one to combine two recursive calls efficiently basically it breaks a given problem into two parts of equal length finds the solution to the two parts using brute force combines the result by picking an element in the first solution set and searching for a corresponding element in the second solution set then returns the result the results are variations of known equations so technically new ones but not semantically new ones you can form infinitely many equations defining any mathematical constant by redistribution of fractions hence why all the results are continued fractions which are the conjectures presented it's an algorithm to automate the painful process of finding and testing new representations by hand it's only applied to conjectures for fundamental constants but can be extended for other types of mathematical constructs we are starting to see the building blocks of a future where mathematical discovery is becoming accessible even to those who aren't formally trained mathematicians but have the motivation and drive to solve a problem using freely available open source tools the coding challenge for this week is to create a conjecture for pi using the Ramanujan machine submit your github links in the comment section and I'll announce the winner next week there are three things to remember from this video if you solve any of the six millennium problems you'll win 1 million dollars a transformer network can solve simple math problems after training on a data set of questions and answers and a Ramanujan machine can automatically solve conjectures for fundamental constants remember to tap the bell icon to get notified in my next video and until next time wizards [Music]