Max Tegmark - Is Mathematics Invented or Discovered?

max the question of mathematics what is it really is it's something that humans impose on the physical world much like the system of taxonomy and biology which seems to work very well and as we classify animals and plants and different kingdoms and all that or is it something that is is out there it's always existed and we're somehow let into little pieces at a time that we can we can access how do these two ideas work together and what's the significance of whatever answer you have it's very important to not conflate the language of mathematics which we do invent with the structures of mathematics which we discover for example when Plato and his contemporaries started getting really interested in how many regular three-dimensional shapes there were with flat sides they discovered that there were exactly five of them that they have trajeron that cube the octahedron dodecahedron and the icosahedron and they invented the name they were free to invent the name dodecahedron here before Island they could have called it the Schmo dekha he'd been there as oh they can he'd ruin if they wanted but they were not free to invent the sixth one there is no sixth one it just doesn't exist and it's exactly the same way in physics when me and discover things out there and then invent names for them my dad asked me what has a little kid he said max how can we know that the planet Jupiter is actually called Jupiter where no one has ever gone there and you know jacked I went back and thought about this Froylan I came back and really excited daddy so what I'm saying in my book is that the things that we have discovered in our external physical reality correspond to things that we discover in mathematics structures like the dodecahedron except of course much more complicated ones and this is a sentiment that the most of our mathematicians friends also shared David volgen at MIT for example has a beautiful poster he's put up on the wall of his office of e8 this is mathematical structure he spent the decade of his life studying and he would be really pissed at me if I insinuated that he just made this up he feels he's discovering it and of course if you take two people who both moved to New York and live there for a decade all right they're not gonna discover exactly the same streets but they're both gonna discover in Central Park and Times Square and the main through Ferris and it's the same way if you take our earth-based civilisation and some alien civilization eventually we're all going to discover that the basic things like the integers and + and multiplications and and the photonic solids and so on and then as we gradually map out the map of New York City or the map of the Platonic math landscape where we do what kind of obscure faraway places we find will depend on our cultural interests but we are still discovering rather than inventing so if we're discovering that that implicitly means that there's something there and and you define it platonically something in a platonic existence is there and we're uncovering how much stuff is in that photonic existence this platonic reality of all mathematical structures is vast but if this is advanced or is it infinite there are infinitely many different mathematical structures but what's so nice is that it's still not some kind of vague anything-goes that anything I can think about exists it's very hard to prove that a mathematical structure is actually self consistent and the famous mathematician David Hilbert said that mathematically existence really is you know freedom from contradiction and mathematicians work very hard and probably publish papers sometimes just to prove that something actually exists mathematically and it's consistent so you and you can imagine in the future of writing your program for a super advanced computer to generate an atlas which just has all the mathematical structures in there organized by you creasing complexity we're on page one you'll have some really simple stuff like the empty set and then you get eventually to the cube and the dodecahedron and then eventually you get to 3 plus 1 dimensional pursuit or ammonia manifolds and Hilbert spaces and the kind of stuff which we physicists work on now but it's so much more vast than that I mean this is just a tiny miniscule fraction of all the possibilities I'm just a number theory alone how many prime numbers if that if that itself is infinite that that's all out there in your Platonic space even though there are infinitely many counting numbers 1 2 3 4 5 6 late together form a single mathematical structure though that mathematicians like to call the integers but there are indeed there's also vast numbers of them and it raised this raises a fascinating question it seems like Nature prefers simplicity because it seems like the mathematical structure that we physicists seem to find ourselves in here is actually much simpler than a lot of things that you could cook up we live in a structure with enormous amounts of symmetry which strikes us is beautiful and elegant and this is a deep mystery which I feel we still don't fully understand why is this exactly so in your platonic existence of mathematics is it thing what exists there is it every number every County number every prime number or even every art or is it the concept of you know take n and add one and and do that dot dot dot n plus one then what what actually exists there the mathematical structures that exist in this Platonic reality that I call our level four multiverse consists of abstract elements with relations between them and this includes for example and so not all the specific generations the number-5 itself is not the mathematical structure and there's a the name doesn't mean anything but-but-but the concept of five things or so some missing parts um some popular examples of mathematical structures are different kinds of numbers the integers are a mathematical structure 1 2 3 4 5 etc the real numbers the complex numbers then there are a lot of mathematical structures and mathematicians call spaces of different kinds we have Euclidean space two dimensions or three dimensions four dimensions and so on we have Minkowski space which is Einstein's famous space-time we have curved spaces known as pseudo romani and manifolds which is the space that Einstein said we live in so there's a there's a vast variety of different kinds of but I'm trying to understand within each one of these structures is it just the description of the structure and a an algorithm for producing elements within that structure or do each elements exist in this platonic so I just take the counting numbers so what I'm positing is that we but our physical reality is one particular mathematical structure of a sort so for example Euclidean space is one mathematical structure which has within it infinitely many points which would correspond to all the different physical points so you would say at this point here corresponds to this mathematical point in this Euclidean space and this point there corresponds to that other point and this length here corresponds to what blah and we lit know of course that this is not really you're clearly in space because we realized that space is curved and so on but there's another kind of mathematical structure which known as a suitor romani and manifold which can take care of that and and what broadly what I'm saying is that for every single physical entity that we think of is something we can touch or measure with a detector there is a there is a corresponding mathematical entity there in a mathematical structure so for example if you take a thermometer you can measure a little number at each point and in the air here which we call the temperature and you can measure a pressure barometer etc and when we make weather forecasts what we do is we divide space in all these three-dimensional pixels we call voxels put this in a really big computer and trying to calculate whether it's gonna rain tomorrow these numbers are not fundamental but the magnetic field for example that you can measure by holding a compass there and check which direction the needle lines up to etc which is also described by a bunch of numbers throughout space is as far as we can tell something very fundamental similarly and the electric field and we have all these other fields which tells us about quarks and electrons and so on and it's very much like a weather forecast again at each point in space we think of areas being all these numbers there and by putting this into computer we've successfully managed to as physicist calculate all sorts of properties of protons and atoms and and most other things that we care about and this is a given this is again an example of how everything here can be described by a mathematics and therefore correspond to a a mathematical structure a mathematical structure I think is it most easily thought of as something which has no properties at all except mathematical properties if you specify all these numbers say how strong is a magnetic field here and there and there and you specify everything there is to say about the world

Channel: Closer To Truth

Views: 177,812

Keywords: Max Tegmark, Closer To Truth, Math, Mathematics, Science, Physics, Philosophy, Cosmos, education, lecture, University, College, is mathematics invented, Is Mathematics Invented or Discovered?, mathematics (field of study), philosophy of math, philosophy of mathematics

Id: ybIxWQKZss8

Length: 10min 23sec (623 seconds)

Published: Sat Sep 28 2019