3 Discoveries in Mathematics That Will Change How You See The World

So today we're going to unlock Mysteries of the universe as we delve into three strange and surprising mathematical discoveries that are going to challenge everything that you thought you knew about maths from the Monty Hall problem which is a right brain scratcher to the Nash equilibrium to the idea that well maths might not actually be real this video is going to take a journey through the unexpected and the unknown join us as we explore how maths can prove something about the physical world without the need for an experiment yep it's confusing and like I said is it even real foreign [Music] that's fairly easy to understand if you haven't explained to you imagine you're on a game show the presenter points to three curtains on the stage and says behind each of these curtains is a prize one of the prizes is a million dollars another is a thousand dollars and the last one is a banana your task is to pick the carton with a million dollar prize so you could take home the jackpot now which of these curtains do you choose now let's suppose that you pick curtain number one however instead of showing you what's behind curtain number one the game show host cheekily decides to give you a free look at what's behind curtain number three he reveals the concealed behind curtain number three is a thousand dollars this means that one of the two remaining curtains conceals a million dollars and the other one is hiding a banana and now the host asks you another question do you want to stick with your original choice of character number one or would you like to switch your choice to curtain number two so one of these curtains hides a million dollars the other one hides a banana what are you gonna choose we'll give it a moment because you really do want to consider your answer here now common everyday intuition is going to contradict the right answer right now are most people when asked whether they want to change their choice they're going to stick with their original guess we have a tendency to favor the decisions we've already made even when new information becomes available you reason that you have a 50 50 shot of being right so you're gonna stick with curtain number one and if you made that choice I'm sorry that you're actually making the wrong choice because it's not 50 50. it's 66.6 and 33.3 if you had changed your choice to go to number two your chances of being right would have risen to 66 but well how is this possible it makes no sense I hear you type it in the comments section now this is the Monty Hall problem named after the host Monty Hall of Let's Make a Deal he often employed this trick in the show this is behind door number one or door number two or door number three problem according to probability experts is that we have a hard time understanding probabilities when the base assumptions of the such situation change in this scenario you begin with a third chance of getting the correct curtain on the first try simple enough but when a second curtain never the curtain you originally guessed is opened the probabilities change while your original guest still has a third chance of being correct the other remaining curtain now has a two-third chance of being the right one what has to be kept in mind is that the curtain the host shows you isn't the one you picked if the host had revealed the curtain you picked a show that it contained a banana and then asked you to choose from one of the other curtains you would have a 50 50 chart of getting it right if he's still confused at this point don't worry we'll explain a little bit more I also need a little bit of help to demonstrate this more clearly let's imagine a scenario where there are a hundred curtains 99 of which contain bananas and only one curtain which contains a million dollars you pick a random curtain say number 67. now imagine the host opens 98 of the other curtains only to reveal bananas now only your curtain 67 and let's say Curtin 81 are closed do you change your choice this time now the problem probably makes more intuitive sense to you the chances of you picking 67 correctly on the first try was one in a hundred the chances of number 81 being the correct curtain and 99 out of 100. you could try this yourself at home using a set of playing cards trying to guess which of three cards is an ace you'll find that when one of the other options is removed changing your choice Works more often than not it's pretty crazy isn't it it does kind of blow my mind I still like how is it changing but it's maths despite you playing a central part in the academy award-winning film A Beautiful Mind amazing movie the concept of the Nash equilibrium is never really explored in the film aside from a sort of lame example about the chaps in a pub getting dates with a hot blonde and that's really too bad not to mention that it's a needlessly sexist example that was insulting to the audience but well that's for another video because far better than helping nerds get laid some very smart people believe that this discovery might have actually saved the World perhaps it was because the film's creators thought the mathematics behind Nash's discoveries involving Game Theory were too abstract to be dealt with in-depth in a film that is really about an emotional journey of a brilliant man who suffers from debilitating schizophrenic episodes or perhaps the truth seemed a little too frightening for a Hollywood film but when you look into it the Nash equilibrium is actually a fascinating and fairly easy to comprehend principle of game theory that can help us to understand how many systems of the modern world function it explains for example example how we can use machine learning based AIS to train other AIS to win at chess it also helps explain much deeper problems of society such as the effort to lower carbon emissions or to reduce nuclear armaments maths is kind of amazing and it's simplest form the Nash equilibrium theory proposes that there exists scenarios in-game Theory as well as in the real world where all competing participants who know the optimal strategies of all other participants have no incentive to unilaterally change their own standing in order to win an advantage over the others we'll use the example of the coordination game which is a two participant game or common everyday situation which two people have a similar goal but must make strategic decisions that depend on the other person making a similar Choice imagine two cars driving down a one-lane road in opposite directions let's assume in each case that there's no way for the cars to avoid hitting each other unless both swerve in opposite directions both going to the right from the perspective of each car this is a Nash equilibrium in that there exists a universal binary choice that both participants must adopt for both to be equally successful if either player changes their strategy then both are going to lose while this may seem like a common sense idea when it is applied to a more complex system the Nash equilibrium Theory reveals that there often exists situations in which it is better to inform all the participants of the ideal strategies for everyone in order for everyone to follow the best strategy and equally benefit from the outcome so why does this matter let's imagine you're one of two cars driving down this one-way Street and there's enough room on the street for one of the cars to Swerve out of the way while the other simply continues driving in Game Theory this is called a mixed strategy in this case while there is an extremely minor benefit in time and effort saved to the person who barrels straight ahead without swerving it's severely reduces the chances of a successful outcome for both parties since both cars have potential to make the same decision to not swerve by making both drivers aware of their own responsibility to make room for each other as a matter of social convention we ensure that all such situations result in successful outcomes for everybody now look this may seem like obvious Common Sense today but there was actually a time in which governments did not fundamentally realize that driver education and roadway conventions were an essential part of Public Safety we now use these principles to design traffic laws along with innumerable other systems involving different participants who must make choices in places where the Nash equilibrium is not being applied other ad hoc strategies can arise that significantly endanger the public for example at certain times and places it was actually the convention for the larger corals to be given room by the smaller Cara horse but this creates certain problems as no two parties have perfect information what if I think my car is bigger than yours when it's actually not or what if our cars are the same size eyes now imagine that we aren't discussing cars driving down the street but instead we're talking about countries building nuclear weapons in a real world scenario where different countries are building world-ending weapons there exists a Nash equilibrium in which the participant countries gain all the benefits of being armed with nuclear weapons without you know ending the world in order to achieve that equilibrium it is required that all participants be aware of the ideal strategy for all other participants and that there'd be no Advantage as possible from failing to follow that strategy it breaks down if one of the countries can break those rules and get an advantage Nash's theorem helped nuclear strategy theorists to realize that the ideal approach to nuclear proliferation was to allow one's geopolitical enemies to know exactly what kinds of weapons one was developing and once country had developed nuclear weapons to allow those weapons to exist without interference as long as all countries maintain a doctrine of non-first use meaning that they would only consider using their nukes in response to a nuclear attack then the nuclear powers are in a default Nash equilibrium with each other this is why geopolitical strategists care so much more about Rogue States getting nuked than they do about established countries having them established countries are educated as to the ideal nuclear strategy and are not going to choose to violate the doctrine because it just doesn't confer any advantage to them in addition knowledge of the Nash equilibrium helps us make planning decisions for example it helped in the development of the induced demand theory of public infrastructure whereby the addition of new Lanes on a highway actually creates worse traffic problems than decreasing the number of lanes on the same highway it's grossly oversimplify by providing people with choices we actually cause them to make overall worse decisions that impact everybody negatively and that's not just a discovery that affects traffic patterns it may also have saved the world in the 1950s following the Perfection of tritium-fueled neutron bonds the Soviet Union discovered to their surprise that they're turned out to be no theoretical limit to the payload size of a tritium bomb this realization though it conferred a strategic Advantage actually led the Soviet Union to abandon further testing of the technology and possibly even to quietly inform the United States by way of known spies within the USSR of the potential for tritium bombs to destroy the world this was because Soviet an American strategists realized that having such bombs would upset the Nash equilibrium of the nuclear powers like adding Lanes to a highway adding bombs to that Arsenal would have led to decisions that benefited nobody for over 60 years the true potential of tritium bombs was not known to the public but Soviet and American leaders were aware of it and more importantly they chose to ensure that the other was also aware in this way paradoxically the race to build greater and greater nuclear weapons actually ended just as it had the gun with both sides adopting an ideal strategy not to play the game at all so look maybe you've smoked a little too much pot one time and been like dude what if maths isn't even real man well look have no fear even professional mathematicians have been there without the aid of THC Mad Magician and physicist Eugene wigner huge Stoner just just token I have no idea made the observation in a 1960 paper titled the unreasonable effectiveness of mathematics in the Natural Sciences to quote it is important to point out that the mathematical formulation of the physicists often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena what we can immense more broadly is that when one considers that numbers and equations on paper which are human Concepts that only symbolically represent actual things in the real world can actually be used to make concrete predictions with a stunning degree of accuracy is well just a little strange he continued the miracle of appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve we should be grateful for it and hope hope that it will remain valid in future research and that it will extend For Better or Worse to our pleasure even though perhaps also to our bafflement to wide branches of learning Winger was not arguing the math is not somehow representative of the real world according to our own experience but rather pointing out that there is no fundamental reason why the constants and laws of the universe should be possible to express in written formulas in the first place he was also pointing out that it's still very much possible that the way we conceive the laws of mathematics is fundamentally limited by the way we think and use symbolic representations of reality to theorize and draw conclusions if we have proved a scientific fact using maths alone have we really made a Discovery at all a simple way of thinking about this problem is to ask that question most of us are pondered at some point in childhood what if the color red looks red to me but to everyone else it looks like what I think of as blue how could I ever prove that everyone else sees the same red that I see if I can only use the color itself is a representative symbol of what I see the same is true of mathematics if maths can prove something about the physical world that we can't actually test then has it really been proved computer scientist Richard Hamming expanded on wigner's ideas in the 1980s proposing that a number of fundamental discoveries of science did not come from observation but were rather discovered through maths and only later roughly confirmed with observations he points out Galileo's discovery of the law of falling bodies the fact that all else being equal everything falls at the same velocity and it was not actually possible to derive this purely from experiments instead Galileo envisioned artificial scenarios in which the conditions were precisely controlled and it was these imagined experiments that actually proved his theory not the real world it is in the fact that no perfect experiment no all else being equal scenario is ever actually possible that we must consider that maths alone provides the proof of many of our ideas about the universe but math it's just symbols it's not reality how could we prove it was if no scenario is ever controlled in every possible way in essence maths is perfect but real life is not now if we go a little bit further with this idea we find that very often experiments that seem to prove mathematical laws are never actually as perfect as the maths itself there often remains a possibility however slight but the reality isn't exactly what the math says it is and historically this has been true for some pretty important discoveries Newtonian physics for example was mathematically proven in the 17th century but it was not until the 20th century that our observations proved it wasn't actually the whole truth einsteinian physics showed that Newtonian physics was incomplete the maths was perfect reality wasn't the same will probably be true for einsteinian physics thanks to quantum physics yet even quantum physics May one day prove to be incomplete Schrodinger's cat is famously both dead and not dead because that is why what maths tells us but we don't actually know that for sure just like I can't be sure that my blue is your blue I can't prove the Schrodinger equations using an experiment and yet at the end of the day this may not be something you actually need to worry about the more complex the mathematics the less often it has an impact on our daily lives the theory of relativity may be necessary to program satellites but it's probably not important for you on your drive to work Newtonian physics might be important for designing a building but it's not important to someone working in one for most purposes the human brain understands reality as we experience it without the need for any abstract symbols or mathematical Concepts you don't need to know calculus in order to throw a baseball even if you would need calculus to understand the baseball's movement your brain does all of this without any maths so sit back relax I know that everything's going to be fine even if maths isn't even real thanks for watching [Music]

Channel: Sideprojects

Views: 663,969

Keywords: megaprojects, construction, engineering, projects, sideprojects, mathematics, math discovered, math theory, nested square roots, math for fun, tededucation, math invented, philosophy of math, bubble problem, quanta magazine, tremendousness collective, math, explainer, quanta, educational video

Id: 9UBMEww7XLU

Length: 16min 46sec (1006 seconds)

Published: Sun Feb 26 2023