Roger Penrose - Is Mathematics Invented or Discovered?

Roger I have been fascinated by mathematics my entire life and it is a pleasure to come to you to discuss two of its fundamental aspects one its incredible capacity to describe reality and then what mathematics really is so let's start with the first how accurately does math describe the physical world what is extraordinarily precise and in different areas more precise in some areas we know less about it but I think people often find it puzzling that something abstract mathematical really describe reality as we understand it I mean reality you think of something like a chair or something something made of solid stuff and then you say well what's our best scientific understanding of what that is well you say it's made of fibers and cells and so on and these are made of molecules and those molecules are made of atoms known as Athens that are made out of nuclei and electrons going around and then you say well what's a nucleus and you say well it's a protons and neutrons and they're held together by things called gluons and they're neutrons and protons and made of things called quarks and so on and then you say well what is an electron or what's a quark and at that stage the best you can do is to describe some mathematical structure you say they're things that satisfy the Dirac equation or something like that which you can't understand what that means without mathematics I mean the mathematical description of reality is where we're always led and these equations are it's fantastically accurate the Dirac equation which describes the electron or quarks is a very precise equation and for example there's a calculation which describes the magnetic moment that is it electrons behave like little magnets and the magnet magnetic moment the strength of that magnet can be described in terms of other parameters and there is a calculation which gets the accuracy of that well Fineman had a very good description he said it's you can it describes the distance between New York and Los Angeles to an accuracy of less than the thickness of a human hair so that's pretty precise unbelievable and that's describing the microstructure of atoms and but these are the particles about the electron and the gluons the electrons the courses specifically electrons and quarks the things which are called spin 1/2 particles but don't worry about that blue ones are slightly different now mathematics can also describe things in the ordinary physical world or gravitational attraction a magnetic attraction and with the same kind of descriptive accuracy well there's another yeah even more in a certain sense because gravity according to Einstein's theory I mean Newton's theory already had a precision of something like one part in 10 to the 7 so that's that's 10 millionths with one part in that and then there was discrepancy is seen and in the behavior of mercury and so on and that's where you start to see differences with Newton's scheme and then Einstein comes along and produces a theory which is now known to have a precision something like 10 to the power 14 and that and that precision is a measure of how accurate there's a particular system of two stars going around you special kinds of stars called neutron stars very dense objects and these stars one of them is a what's called a pulsar it emits pulses of signals which could be the time extremely precisely and over a period of when I suppose it's maybe more than 30 years and I can't remember they've been observing this thing and in that period of time the accuracy over that length of time is known to something like one part in 10 to the 14 and that the accuracy the agreement between Einstein's theory and the observations so it's telling you these are very very precise things so whether we're talking about the structure of very large entities neutron stars over great distances in the universe with gravity or the the structure of an electron that's right mathematics in both cases is able to describe it with that kind of incredible precision exactly yeah and these are small equations I mean they're not try and that's right they're relatively small equations I mean they're a little difficult to understand them Einstein's theory certainly subtle it's not complicated in the sense that the idea is okay you have to understand about curved space and that sort of thing which is not an easy thing to get your mind around once you get over there it's about the simplest thing you could write down well in in in term in that kind of term so we have this extraordinary precision between mathematics on the macroscopic level with neutron stars and at the microscopic level with the nature of the electron and mathematics incredibly precise in both cases so what does that now begin to tell us about what mathematics really is yes well in a sense this is telling us that that our picture of physical reality depends on something in the sense which is more precise at least in our understanding of it than then then how we think about the world and this precision really dates back to the ancient Greeks the time of Pythagoras and later where they developed the mathematical ideas as a field of study stimulated to some degree by physical reality because the geometry of Euclid which was very much part of the mathematics that was being studied then which we know nonno isn't extraordinarily precise and it is extraordinary precise but it's not as precise as Einstein's theory so one has to go a little bit beyond the geometry that they had I don't think they quite appreciated that they were doing physics because they didn't realize that the geometry of the world could have been anything else but they developed this mathematical scheme purely as a study on its own and so mathematics was studied as a pure intellectual activity and with that necessarily it being related to the structure of a physical world although geometry clearly was a big input but then the properties of numbers and how you and multiply and the notions of prime numbers the fact there are infinitely many prime numbers that goes back to Euclid and earlier and so these things about just about numbers were developed very much from the time of the Greeks and ever since then mathematics has been a subject which you can study for its own sake it has its own life in the sense and certainly mathematicians view it this way it's something out there it seems to have a reality independent of of the reality of the ordinary kind of reality like things like chairs and so on which which are what we normally think of as real but okay the mathematical reality is something different it's it's sometimes referred to as a platonic world brittonic reality and sometimes people have a lot of trouble thinking of that is real I mean philosophers worry about that and so on what is that what would that mean a platonic reality well I think it's a different kind of reality from the reality of the physical world I mean I thought end to think of there being different ways of looking at reality there's the reality of our mental experience which okay interrelates with the reality physical reality and so then there's the mathematical reality of this platonic world which gives reality to these notions so if you like mathematical facts like there is no largest prime number is if it's something independent of ourselves it's always been true it doesn't didn't some somehow become true as soon as somebody seems so hard to prove it it's always been true they're wonderful examples likely it would have been true of nobody if exactly yes and in a sense that had to be so because if the physical world depended so precisely on these mathematical laws I couldn't have known what to do in a certain sense if the mathematics hadn't already been there I mean it's not us that imposes this on the world that's it's out there come people think that you know maybe reason we have good mathematical laws of physics is that's the best way we can come to understand the world but it's something more than that it really is out there in the world well that's the argument whether mathematics is invented by us by human beings trying to impose our way of thinking on the physical world or whether it is discovered because it's already out there and we're finding it because it's already there those are the two polar views sometimes people do argue they say well you know it's just our way of organizing the the what we see about us but I really don't think that's good enough because Newton for example the observations probably had about three figures and three decimal places and he produced his theory which kept on working it's all about seven figures you see then there were discrepancies seen Einstein produced his theory mostly out of his head with that appealing to know things that were known to Galileo and so on but apart from that it was not much more empirical evidence but he produced his theory which extended far beyond anything that the observations at that time told us about and they keep on agreeing with the observation so that theory which is if you like a a platonic absolute thing and it's a mathematical thing seems to be inbuilt into the way the world operates it's not as though you see a new effect and say ok we never even think of a better theory to accommodate that one sometimes science is like that but these really good physical theories are not like that you're revealing something in the way the world operates which is there all the time and I I don't think there's any way of understanding that just in terms of are trying to understand what we see around us a critical fact really seems to be what you said that that when these these mathematical theories were discovered the accuracy that the observation that they had at the time was small compared to the accuracy that those theories then produced that's right that's right I mean Einstein's case okay seven figures of decimal were known perhaps in the planetary motions here but there's another seven out there the precision is over and above that yes that was a million ten millions it's ten minutes ten million ten million yes that's right yes yes I mean that's unbelievable yes so push it further what what does that mean because mathematics is it is almost infinite in terms of all the different relationships and expressions and things that we already know yes what does that mean in terms of how much mathematics is sitting out there well that's a good point because there's an awful lot of mathematics which doesn't seem to have any clear relation to the physical world way I like to picture it is there is this world of mathematics and only a small part of that and it's a very fruitful part it's an extraordinarily fruitful part has relevance to the physical world there's an awful lot out there which as far as we know has no relation to physical behavior well of course people said some of that in the past and then we've been surprised to find some other things that doesn't later but still there's so much math out there then so much bizarre I don't know how else to put it structures that it would seem impossible that that could relate to the physical world but well what does that mean about if it is out there in some Platonic world what is out there there was a these infinite ideas and structures and possibilities yes but sometimes people think of it these as mental creations you see but it doesn't really explain and there's just one example for the Mandelbrot set it's extraordinarily complicated the fractional yes and you can magnify a little bits of it you see all this incredible detail and that's all there in a very simple mathematical idea and it's in its encompassed by this very simple piece of mathematics how does that keep your your own sense of what mathematics really is well I think there two aspects to mathematics please how I look at it some people are just exploring the mathematics and that's their real interest and it's the beauty and the subject often and that's why they're doing it because they find it exhilarating something they find really wonderful to do but there's the other side of it which is how it relates to the physical world and there is this extraordinary precision that we find when you get the mathematics right it really mirrors the behavior of the physical world to an unbelievable degree and so there's these two sides to mathematics it has this reality which you can study quite independently of its role in physics and the other side which is how it really does seem to reveal how the real world operates in a certain sense what the world is as far as we can understand it

Channel: Closer To Truth

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Keywords: closer to truth, deepest questions, ideas of existence, life's big questions, pbs science show, robert lawrence kuhn, search for purpose, stem education channel, ultimate reality of the universe, vital ideas, roger penrose, Is Mathematics Invented or Discovered, origin of mathematics, closer to truth interview, is mathematics invented, is mathematics discovered, what is mathematics, mathematics, mathematics awareness month, philosophy of mathematics

Id: ujvS2K06dg4

Length: 13min 48sec (828 seconds)

Published: Mon Apr 13 2020

Reddit Comments

[u/kl0]

Having gotten a degree in mathematics long ago, I can honestly say that it probably didn't make me much better at basic math (maybe, but I doubt it), but it DID make me a substantially better writer. It's just rooted in such a fundamental level of philosophy that you become very good at being able to dismiss certain possibilities of things solely on the basis you can think of some example that counter your original premise - which is not permitted in mathematics, unlike many other scientific disciplines. Math is all about specificity and precision.

In fact, I had a college professor tell the class a joke once (it was in a thick Russian accent):

There were three scientists riding on a train across Australia. One was a biologist, the other a physicist. and the third a mathematician. They passed a field a sheep and the biologist exclaimed, look - there's a black sheep among a flock of white sheep. He went on, I suppose we can conclude that there are black sheep in Australia. The physicist said, well, we can really only conclude that there is at least ONE black sheep in Australia. Finally, the mathematician said, both of you gentlemen are incorrect. We can really only conclude that there exists a sheep in Australia that is black on at least one of its sides.

If that joke makes sense, then you get how math is different from other scientific disciplines.

[u/DAL82]

I asked this a while ago:

Are new knots invented or discovered?