Feynman's Lectures on Physics – The Relation of Mathematics and Physics

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“Richard Phillips Feynman (May 11, 1918 – February 15, 1988) was an American theoretical physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics for which he proposed the parton model.”

hi I can see the audience tonight so I can see also from the side of it which less time was a blacks lot in front of my eyes that there must be many of you here who are not thoroughly familiar with physics and also a number that are not too versed in mathematics and I don't doubt that there are some who know neither physics nor mathematics very well that puts a considerable challenge on a speaker who is going to speak on the relation of physics and mathematics a challenge which I however will not accept I publish say the title of the talk clear in clear and precise language and didn't make it sound like it was something it wasn't it's the relation of physics and mathematics and if you find that in some spots it assumes some minor knowledge of physics or mathematics I cannot help it this was named in thinking of the applications of mathematics to physics it's perfectly natural that the mathematics will be a useful when large numbers are involved in complex situations although for example if we took a biology the action of a virus on a bacterium is if you watch it under the microscope on mathematical as jiggling virus find some spot on this odd shaped bacterium and they're all different shapes and find some spot maybe it pushes its DNA in and maybe it doesn't and so forth and yet if we do the experiment with millions and billions of bacteria and viruses then we can learn a great deal about the viruses by taking averages and working with large numbers and we can use the mathematics involved in the averaging we can see whether the virus is developing the bacteria some new strains and in what percentage and so we can study the genetics the mutation and so forth to take another more trivial example you know imagine an enormous board a checker board to play checkers or draw and if the board is very large the actual operation of any one step is not mathematical it's very simple if it's mathematical or either goes one side of the other kind of diagonal or it reaches and becomes a king and can code backwards when it reaches the end in other words the statement the rules are very simple and do not really involve any mathematics but you could imagine that an enormous board with lots and lots of pieces some analysis of the best move or good moves or bad news might be made by a deep kind of reasoning which would involve somebody having gone off first and thought about a great great depth and that becomes mathematics the substrate for you another example is switching in computers that you have one switches either on or off and there's nothing very mathematical about that although mathematicians like to start there with their mathematics but with all the interconnections and wires to figure out what a very large system will do when requires a mathematics and I would like immediately to say that a mathematics has its primary application or as well it has a tremendous application in physics in a discussion of the detailed phenomena in complicated situations granting the fundamental rules of the game and that is something which if I were talking only about the relation of mathematics and physics I would spend most of my time discussing but since this is part of a series of lectures on a character of physical law I am NOT do not have the time to discuss the applications of mathematics in physics to calculating what happens in complicated situations but we will go immediately to another question which is the character of the fundamental law if we go back to our checker game the fundamental laws are the rules by which the checkers move the mathematics may be applied in a complex situation to figure out what happens in the next move what's a good move to make in a complicated set situation but very little mathematics is needed in the fundamental simple character of the basic laws now the strange thing about physics is that for the fundamental laws we still need mathematics for example well I give two examples one in which we really do not and one in which we do there's a law in physics called Faraday law which says that in an electrolysis the amount of material which is deposited is proportional to the current and to the time that the current is acting and that means that the amount of material is proportional to the charge which goes through the system sounds very mathematical but what's actually happening is that electrons going through the Y each carry one charge and to take an example a particular example it may be that if the deposit one atom requires one electron to come and so the number of atoms that are deposit is necessarily proportional to the number of electrons that flow and thus the charge goes to the wire so the mathematically peirong law has as its basis nothing very deep requiring no real knowledge of math a night that one electron is needed for each atom in order to earth the deposit itself that's a not a deep that's mathematic I had to say number one but it's not the kind of mathematics I'm talking about tonight now if we take on the other hand Newton's law for gravitation which has these aspects which I wrote down last time just to impress you with the speed at which mathematical symbols can convey can carry information we said that the force of proportional to the product of the masses of two objects and inversely as the square of the distance between them and that bodies react the forces by changing their speeds or changing their motions in a direction of the force by amounts proportional to the force and inversely proportional to their mass that sounds that's words all right and I didn't have to write the equation but nevertheless it's kind of mathematical and we would wonder how can this be a fundamental law how can this planet out there look what is it dude it looks at the song and it sees how far away it is and it decides to calculate on a internal adding machine the inverse of the square of the distance and that tells it how much to move this is certainly no explanation of the machinery of gravitation so you might want to look further and various people have tried to look further Newton was originally asked it doesn't mean anything it doesn't tell assign these as it tells you how it moves it should be enough I told you how it moves not one but people have often unsatisfied without a mechanism and I would like to describe one theory which has been invented for as among others of the type that you might want that this is the result of large numbers and that's why it's mathematical and I give this theory perhaps you've thought of it yourself every once a while some student comes running and he suddenly explains gravitation suppose that in the world everywhere there are flying through us at very high speed a lot of particles that come equally in all directions they're just shooting by shooting by shooting by and once in a while hit enable body but where we and the Sun are practically transparent for them nearly but some hit and so it's not completely transparent and look what would happen if the Sun is here and the earth is here then if the Sun weren't here there would be particles bombarding from all sides giving little impulses by the rattle of these bang bang the few that hit which would put not shake the earth in any particular direction because there is many coming from one side as the other from top from bottom however when the Sun is here the particles which are coming in this direction are partly absorbed by the Sun because some of them hit the Sun and don't go through therefore the number that are coming from this direction toward the earth is less than the number that are coming from the other side because here they have no opposition from the Sun there and it's easy to see at the Sun mental effort that the further the Sun is away the less in proportion of all of the particles are being taken out of the possible directions in which particles can come solar size appears smaller and in fact inversely as the square of the distance so there will therefore be an impulse toward the Sun on the earth that's inverse is the square of the distance and is the result of large numbers of very simple operations just just hit one after the other from all directions and therefore the strangeness of the mathematical relation will be very much reduced because the fundamental operation is much simpler than calculating the inverse of the square that this is this machine does the calculating these particles bounces only trouble with it is that it doesn't work for all the reasons every theory that you make up has to be analyzed against all the possible consequences and to see if it predicts anything else and this predicts something else if the earth is moving this way more particles will hit it from the front and from the back if you're running in the rain more rain hits you from the front of the face that in the back of the head because you're running into the rain and so as the earth is moving in this direction it's running into the particles rather and running away from the ones that are chasing it from behind so that more particles hit it from the front and from the back and there would be of course also sideways whenever there was any motion this sideways force would slow the earth up in the oven and certainly would not have lasted the at least three four billion years that has been going around the Sun so that's the end of that theory well you say I was a good one though it got rid of the mathematics for a while maybe maybe I can invent a better one and maybe you can because nobody knows the ultimate butt chewed up to today from the time of Newton no one has invented another theoretical description of the mathematical machinery behind this law which does anything else but say the same thing over a general make the mathematics harder and at the same time doesn't produce some wrong phenomena I mean they have like this model does it but it produces something which isn't true so there is no model of the theory of gravitation today other than the mathematical form if this were the only law of this character it would not be so it would be interesting and rather annoying but what turns out to be true is that the more we investigate and the more laws we find them the deeper we penetrate nature this disease that every one of our laws is a purely mathematical statement in rather complex and abstruse mathematics this is relatively simple mathematics it gets more and more abstruse and more and more difficult as mathematics and why I haven't the slightest idea why it is only my purpose in this lecture to tell you about this fact in other words it's my purpose in this lecture to explain really why I cannot satisfy you if you do not understand mathematics to well in trying to explain nature in any other way it is the burden of this lecture in fact to just tell you the fact that is impossible to answer the really – honestly the challenge of explaining in a way that a person can feel the beauties of the laws of nature without their having some deep understanding of mathematics I'm sorry seems to be the case you might tell like then there's no explanation of the law at least tell me what the law is why not tell me in words instead of in the symbols mathematics is just the language and I want to be able to translate the language and in fact I can and with patience I think I partly did I could go a little further and explain more details that this means that if it's quite as far away the forces one fourth is much and so on and can convert all these into words I would be in other words kind to the laymen as they all sit hopefully that you will explain them and various different people are get different reputations for their skill at explaining to the laymen in layman's language these difficult and abstruse subjects the laymen then searches for book after book with the hope that he will avoid the complexity so it ultimately set in even by the best exposit for of this type he reads the things hoping we have one after he finds as he reads are generally increased confusion one complicated statement after the other one difficult to understand thing after the other all apparently disconnected from one another and that becomes a little obscurity hopes that maybe in some other book there's some explanation which avoid which do you mean the man almost made it you see maybe another fellow makes it right and I don't think it's possible because there's another feature mathematics is not just the language mathematics is a language plus reasoning it's like a language plus logic it's mathematics is a tool for reasoning it's in fact a big collection of the results of some person's careful thought and reasoning by mathematics it is possible to connect one statement to another for instance I can say that the force is directed toward the Sun I can also tell you as I did before that the planet moves so that if I draw a line from the Sun to the planet and draw another one at some definite period like three weeks later the area that swung out by the planet is exactly the same as it will be the next three weeks in the next three weeks and so on as it goes around the Sun now I can explain both of those statements to you carefully but I cannot explain why they're both the same so that if you don't appreciate the mathematics and you cannot see that the VAP variety of facts the enormous apparent complexity of nature with all its funny laws and rules each of which have been can't be explained you are really very closely interwoven that logic permits you to go from one or to the other it may be unbelievable that I can demonstrate that equal x will be swept out if the forces are directed toward the Sun and just to if I may try I will show you one demonstration to show you that those two things really are equivalent and so that you can appreciate that there's more through merely the statements of the two laws that the two laws are connected such that reasoning alone will bring you from one to the other and the mathematics is disorganized reasoning and that it's good to know how to do that so they will appreciate the beauty of the relationship of the statements so I'm going to prove if I make the relationship that if the forces are directed toward the Sun that the equal areas are swept out in equal times so we start is a Sun and we imagine that at a certain time let's say the planet is here and it's moving in such a way that let's say one second later a one-hour pick any time it's a 1 second later it's moved in such a manner that has gotten to the position 2 now if the Sun did not exert any force on it then by a Galileo's principle it would keep right on going in a straight line so in the same interval of time later the next second it'll move exactly the same distance in the same straight line to the position 3 were there no force all right now first we're going to show that if there's no force equal areas are swept out in equal time I remind you that the area of a triangle is half the base times the altitude and that the altitude is the vertical distance to the base and that if put out if the triangle is sort of cockeyed there's a name for it which I forget obtuse obtuse then the altitude is this vertical height here see I know about the triangles I just don't know the names now let us draw the lines to these two points in the case that there was no motion whatsoever the question this doesn't draw very well I'm not accurate but these two distances are equal remember question is are these two areas equal well consider the triangle made from the Sun and the two points one and two it's this one what's its area it's this space multiplied by this height and what about the other triangle which is the triangle in motion from two to three it's this base times the same altitude the two triangles have the same altitude and as I indicated the same base and have the same area so so far so good if there were no force from the Sun equal areas would be swept out in equal times at two triangles of equal area but there is a force from the Sun and during this interval one to two to three the Sun is pulling and changing the motion in various directions this way this way that way to get a good approximation to that will take the central position or average position here and say that the whole effect during this interval was to change the motion by some amount in this direction toward the Sun that means that although the particles were moving this way and would have moved this way in the next second because of the influence of the Sun the motion is altered by an amount that's poking in this direction that's parallel to this exactly parallel these lines are parallel it's the direction in which this new motion is the new motion is a compound of what I wanted to do and the change that's been endorsed by the action of the Sun so it doesn't really end up at position but rather at position 4 so now we would like to compare it getting complicated in the diagram the triangle 2 4 s & 2 3 yes I show you that those are equal because they have the same base those two triangles this one here and the one that happened when we had no force the one with force and with no force have the same base and do they have the same altitude sure because they're included between parallel lines and so the a this have the same altitude and thus the area of the last triangle I drew is the same as the second one I drew this one free and that I had proved earlier was the same as the first one so in the actual orbital motion of the planet the area of the first in the first in second and in the second second are equal so by reasoning we can see a connection between the force the fact that the forces toward the Sun and that the areas are equal not M genius no this was I borrowed this from Newton that comes right out of the Principia diagram and all the letters are different that's all because he wrote in Latin these are our big numerals but incidentally Newton made his proof geometrical like this and made all his proofs in his book geometrical of this type today we don't use that kind of reasoning we use a kind of analytic reasoning with symbols which this requires this kind of reasoning requires an ingenuity to draw the right triangles the correct triangles I mean to notice about the areas and to figure out how to do it have to be clever but there have been improvements in the methods of analysis so that one can be quite more stupid and I write a much faster you're much more efficient then and I want only to show what it looks like in the notation of the more modern mathematics where you don't do anything with right a lot of symbols to figure out first we would like to talk about how fast the area changes and we represent that by area dot and area changes because of a it's when the radius is swinging it's the component of velocity at right angles to the radius times the radius that tells how fast the area changes so this is the component of the radial distance multiplied by the velocity or rate of change of the distance now the question is whether the rate of change of area itself changes the principle is it's not supposed to change the rate of change of area is not supposed to change so we differentiate so call this again and we put it'll means a little trick about putting dots in the right place and that that's all you have to learn the tricks no I'm not it's just a series of rules as people have found out that are very powerful for such a thing and this says the component of the velocity at right angles to the velocity it is none there is none the velocities in the same direction as itself and the acceleration which is this thing the second derivative or the derivative alhasan e is the force divided by the mass so this says that the rate of change of the rate of change of the area is the component of force at right angles to the radius but if the forces in the direction of the radius as Newton said then there's no force in at right angles to the radius and that means that the change rate of change of area doesn't change I just want to illustrate the different kinds of notation now Newton knew how to do this more or less the slight different notation but he wrote everything this way because he tried to make it possible for people to read his papers he invented the calculus which is this kind of mathematics and is a good illustration of the relation of mathematics to physics when the problems and physics get difficult we may often look to the mathematicians who have already studied such a thing and have reasoned about such an item before and have prepared a line of reasoning for us to follow on the other hand they may not have in which case we have to invent our own line of reasoning which will then pass back to the mathematicians because everybody who reasons carefully about anything Henri is making a contribution to the knowledge of what happens when you think about something and if you abstract it away and send it to the Department of Mathematics they put it in the book as a branch of mathematical mathematics then is a way of going going from one set of statements to another it's evidently useful in physics because we have all these different ways that we could speak of things and it permits us to develop consequences and analyze the situations and re change the laws in different ways and to connect all the various statements so that as a matter of fact the total amount that a physicist knows is very little he has only to remember the rules for getting from one place to another and he's able to do that do it then in other words all of the various statements about equal times the forces in the direction of the radius and so on are all interconnected by reasoning now an interesting question comes up is there some pattern to it is there a place to begin fundamental principles and deduce the whole works or is there some particular pattern or order in nature in which we can understand that these are more fundamental statements and these are more consequential space there are two kinds of ways of looking at mathematics which for the purpose of this lecture I will call the Babylonian tradition and the Greek tradition in Babylonian schools in mathematics the student would learn something by doing a large number of examples until he caught on to the general rule also a large amount of Jia Jia matauri for example was known a lot of properties of circles theorem of Pythagoras for example formulas for the areas of cubes and triangles and everything else and some degree of argument was available to go from one thing to another tables of numerical quantities were available so that you could solve elaborate equations and so on everything was prepared for calculating things out but Euclid discovered that there was a way in which all of the theorems of geometry could be ordered from a set of axioms that were particularly simple and you're all familiar without much geometry I'm sure but the Babylonian attitude was if I make my my way of talking what I call Babylonian mathematics is that you know all these various theorems and many of the connections in between but you've never really realized that it could all come up from a bunch of acts modern mathematics the most modern mathematics concentrates on axioms and demonstrations within a very definite framework of conventions of what's acceptable and not acceptable as axioms for example in geometry it takes something like Euclid's axioms modified to be made more perfect and then to show the deduction of the system for instance it would not be expected that a theorem like Pythagoras is that the sum of the squares of the areas of squares put on the sides of the triangle will equal the area of a square on a hypotenuse should be an axiom on the other hand from another point of view of method of geometry that of Descartes the Pythagorean theorem is an axiom so the first thing we have to worry about is that even in mathematics you can't start in different places because of all these various theorems are interconnected by reasoning there isn't any real way to say well these on the bottom here are the bottom and these are connected through logic because if you were told this one instead or this one you could also run the logic the other way if you weren't told that one and work out that one it's like a bridge with lots of members and it's over connected if pieces have dropped out you can reconnect it another way the mathematical tradition of today is to start with some particular ones which are chosen by come kind of convention to be axioms and then to build up the structure from there the babylonian thing that i'm talking about was not babylonian but it is to say all i know happen to know this and i happen to know that and maybe i know that and i work everything out from there and nick tomorrow i forgot that this was true but i remember that this was true and i reconstructed again and so on i'm never quite sure of where I'm supposed to begin and where I'm supposed to end I just remember enough all the time so that has the memory fades and the pieces fall out by repurchasing back together again every day the method of starting from the axioms is not efficient in obtaining the theorems in working something out in geometry you're not very efficient of each time you have to start back at the action but if you have to remember a few things in the geometry you can always get somewhere else it's a much more efficient to do it the other way and the what the best axioms are not exactly the same not ever the same as the most efficient way of getting around in the territory in physics we need the Babylonian method and not C you could be a Euclidian or Greek method and I would like to say why the problem in the Euclidean method is to make something about the axioms a little bit more interesting or important but there the question that we have is in the case of gravitation is it more important is it more basic is it more fundamental is it a better axiom to say that the forces directed toward the Sun or to say that equal areas are swept in equal time well from one point of view the forces is better because once I state what the forces are I can deal with a system with many particles in which the orbits are no longer ellipses because of the pull of one on the other and the theorem about equal areas fails therefore I think that the force law to be an axiom instead of the other on the other hand the principle at equal times is swept out in equal equal area that swept out in equal times can be generalized when there's a system of a large number of particles to another theorem which I had prepared to explain but I see I'm running out of time but there's another statement which is a little more general than equal areas in equal sign well I have to state what it is it's rather complicated to say and it's not quite as pretty as this one but it's obviously the the son of this one I mean it's the offspring if you look at all these particles Jupiter Saturn the Sun and all these things were going around lots of stars or whatever they are all interacting with each other and look at it from far away and projected on a plane like this picture then everything everything is moving this moving this way they're moving that way and so on then take any point at all at this point and then calculate how much each one is changing its area how much area is being swept out by the radius every particle and add them all together but wait those masses which are heavier count more strongly if this is twice as heavy as this one than this area it counts twice as much so that's doing the sweeping and the total of all of that is not changing in time that's the generalization obviously of the element incidentally the total of that is called the angular momentum and this is called the law of conservation of angular momentum conservation just means that it doesn't change now one of the consequences of this is but just to show what it's good for imagine a lot of stars falling together to form a nebula or galaxy as they come closer in if they were very far out and moving slowly so there was a little bit of area being generated but on very long on distances from the center then if the thing falls in the distances to the center is shorter now if all the stars are now close in the nice radii are smaller and in order to sweep out the same area they have to go a lot faster so as the things come in they swing swirl around and thus we can lovely understand the qualitative shape of the spiral nebulae and also understand in the same way back the same way way a skater spins when you start with in layout as moving slowly and as you pull the leg in and spins faster because when the leg is out its contributing when it's moving slowly a certain amount of area for a second and then when it comes in to get the same air you have to go around faster but I didn't prove it for the skater the skater uses muscle force gravity is gret is there a different force yet it's true for the skater now we have a problem we can deduce often from one part of physics like the law of gravitation a principle which turns out to be much more valid than the derivation this doesn't happen in mathematics that the theorems come out in places where they not supposed to be in other words if we were to say that the postulates of physics were thus law of gravitation we could deduce the conservation of angular momentum but only for gravitation but we discover experimentally that the conservation of angular momentum is a much wider thing now Newton had other PI postulates by which he could get the more general conservation law of angular momentum but Newtonian laws were wrong there's no forces it's all a lot of baloney the particles don't have all bits and so on yes the analogue the exact transformation of this principle about the area's the conservation of angular Menem is true with ionic motions in quantum mechanics and is still as far as we can tell today exact so we have these wide principles would sweep across all the different laws and if one takes too seriously is derivation and feels that this is only valid because this is valid you cannot understand the interconnections of the different branches of physics someday when physics is complete then maybe with this kind of argument we know all the laws then we could start with some axioms and no doubt somebody will figure out a particular way of doing it and then all the do all the deduction will be made but why we don't know all the laws we can use some to make guesses of theorems which extend beyond the proof so in auditory to understand the physics one must always have a neat balance and contain in his head all of the various propositions and their interrelationships because the laws often extend beyond the range of their deduction this will only have no importance when all the laws are known another thing that's interesting in the relation of mathematics to physics is this a very strange thing that by mathematical arguments you can show that you can start from very many different apparent starting points and come to the same thing that's pretty clear if you have axioms you can use some of the theorems but actually in the physical laws are so delicately constructed that the statements of them have such qualitatively different character that is very interesting so if you'll permit me I'm going to state the law of gravitation in three different ways all of which are exactly that turns out but they sound completely different one there's a forces between the objects as described before and each object when it sees the force on it accelerates or changes its motion either at a certain amount per second as I've described before the regular way I call it Newton's law now there's a completely different way that law says that the force depends on something at a finite distance away see it has a what we call non-local quality the force on this depends on where that one is over there now you may not like the idea of action at a distance but it can know what's going on over there well then is another way of stating the laws which are very strange and it's called the field way of representing the laws and is so very hard to explain but I just want to give you some rough idea of what it's like and it says different things completely differently that there's a number at every point in space I know it's a number it's not a mechanism it's the trouble with this whole physics that it must be mathematical this way there's a number every point in space here's a number here's an under thought and the numbers changing as changes rather when you go from place to place if an object is placed at one of these points at somewhere in space the force on it is in a direction in which that number I'll call it the name it's given called a potential is in a direction which that potential changes as quick its can and the force is proportional how fast it changes as you move that's one state that's not enough yet because I have to tell you now how to determine how the potential varies I could say the potential varies is one over the distance from each object but that's factory action resistance idea however the forces at a distance but you can't state the law in another way it says the following you don't have to know what's going on anywhere outside of a little ball if you want to know what's what the potential is here you tell me what it is on the surface of any ball no matter how small you don't have to look outside you just tell me what it's in the neighborhood and how much mass there is in the ball the rule is this that the potential at the center is equal to the Pataky average of the potential on a little ball surface – there's constant that's over there and the other equation divided by twice the radius of the ball let's suppose the radius of the ball is it called a and then multiplied by the mass that's inside the ball if the ball is small enough now you see that this flaw is different than the other one because it only tells what happens at one point in terms of what happens very close by Newton's laws tell what happened at one time in terms of what happens in MO the instant if you dip some instance in some how to work it out but in space it leaps from place to place but this thing is both local in time and also local impass in space because it depends only what's in the neighborhood and there's another way of representing that's another way now there's in a completely different way that see there's a difference in the philosophy in qualitative ideas involved you don't like action in existence you can get away without it now I'll show you one which is philosophically the exact opposite in which there's no discussion at all about how the thing works its way from place to place in which the whole thing is an overall statement as goes as follows when you have all the particles around and you want to know how this one moves from one place to another you do it as follows you calculate a certain quantity for you invent a possible motion that gets from one given place to some other place that you're interested in in a given amount of time say it wants to go from here to here in an hour and you want to know by what route it can get from there to there at an hour by what curve so what you do is you calculate a quantity getting the curve if you try this curve you calculate a certain number for this quantity I don't wanted to say what the quantity is but for those who have heard of these terms this quantity on this route is the average of the difference between the kinetic and potential energy now if you calculate this quantity for this route then for another route you'll get of course different numbers for the answer but there's one route which gives the least possible number for that and that's the route that the particle takes now we're describing the actual motion the ellipse by saying something about the whole curve we have lost the idea of causality that the particles here it sees the pull it moves to here point instead of that in some grand fashion it smells all the curbs around here and all of possibility and decides which one to take but this is an example of the wide range of beautiful ways of describing nature and that when people talk that nature must have causality well you could talk about it this way nature must be stated in terms of a minimum principle well you can talk about it this way nature must have a local field view it can do that and so on and the question is which one is is right now if these various alternatives are mathematically not exactly equivalent and if for certain ones there will be different consequences than fathers then it's a variant that's perfectly alright then because we got only to do the experiments to find out which way nature actually chooses to do it mostly people come along and they argue philosophically they like this one better than that one but we have learned from much experience that all intuitions about what nature's going to do philosophically fail it never works one just has to work out all the possibilities and just try all the alternatives now in this particular case that I'm talking about here these theories are exactly equivalent the mathematical consequences and every one of the different formulations of the three formulation Newton's laws local field method and this least this minimum principle give exactly the same consequences what do we do then you will read in all the books that we therefore cannot decide scientifically on one or the other that's true they're not equivalent they're not they are equivalent scientifically it is impossible to make a decision which there's no experiment the way to distinguish if to all the consequences is saying psychologically they're very different in two ways first philosophically you like them or don't like training is the only thing you can do to beat that disease second psychologically they're different because they're completely unequivocal as long as the physics is incomplete and we're trying to find out the other laws and understand the other laws then the different possible formulations give clues as to what might happen another circumstance and they've become not equivalent in a psychologically suggesting to us the guess as to what the laws might look like in a more in a wider situation for instance Einstein noticed that the law of gravity said that they he realized that signals couldn't scrapple propagate faster than the speed of light for light for electricity he guessed that it was a general principle the same print guessing game as taking this angular momentum and extending it from one case where you proved it to the rest of the universe he guessed that it was true of everything and he guess that it would be true of gravitation if the signals can't go any faster than the speed of light it turns out that the method of describing the forces instantaneously is very poor and in the Einstein generalization or of gravitation this method of describing physics is hopelessly inadequate and enormous ly complicated whereas this one is neat and simple and so is this one so we haven't decided between those two yet in fact it turns out that the quantum mechanics says that in exactly as I stated them neither as it is right but that the fact that a minimum principle exists is turns out to be a consequence of the fact that on a small scale particles obey quantum mechanics and the fact is is that the best laws at present understood is really a combination of the two in which we use minimum principles plus local force local laws and the present day believes that the laws of physics have to have the local character and also the minimum principle but we don't really know so it's this way that if you have a structure that's only partly accurate and something is going to fail if you write it with just the right axioms maybe only one axiom failed and the rest remain just changed one little thing but if you write it with another set of axiom they all collapse because they all lean on that but we can't tell ahead of time without some intuition and guesswork as to which is the best way to write it so we find out the new situation so we must therefore always keep all of the alternative ways of looking at the thing in our heads so the physicists do Babylonian mathematics and paint our little attention to the precise reasoning from fixed axioms one of the amazing characteristics of nature is this variety of interpretational schemes which is possible it turns out that it's only possible because the laws are just so in special and delicate for instance that the law is the inverse square is what permits it to become local if it with the inverse cube it couldn't be done this way that the other end of the equation that the force is related to the rate of change of the velocity that's a consequence that permits this kind of an interval ighting the laws the minimum principle because if for instance if the force were proportional to the rate of change of position instead of velocity you couldn't write it in that way but if you try to modify the laws much you find you can only write them in a very much fewer ways I always find that mysterious and I don't understand the reason why it is that the laws of physics always seem to be possible to be expressed in such a tremendous variety of ways they seem to be able to get through several wickets at the same time now I would like to make a number of remarks on the relation of mathematics and physics which a little more general the first is that the mathematicians only are dealing with the structure of the reasoning and they do not really care about what they're talking they don't even need to know what they're talking about or as they themselves say or whether what they say is true now I explained that if you state the axioms is say such-and-such a song in such constructor so in such and such a so what then then the logic can be carried out without knowing what the such-and-such words mean that is if there if the statements about the axioms are true I mean are carefully formulate and incomplete enough it is not necessary for the man is doing the reasoning to have any knowledge of the meaning of these words and will be able to deduce in the same language new concept new conclusions if I use the word triangle in one of the axiom there'd be some statement about triangles in the conclusion whereas the man who's doing the reason he might not know what the triangle is but then I can read his thing back and say oh a triangle that's just the three side of what have you that's its own salt and so I know this new fact in other words mathematicians prepare abstract reasoning that's ready to be used if you will only have a set of axioms about the real world but the physicist has meaning to all the phrases and there's a very important thing that the people who a lot of people who study physics should come from mathematics don't appreciate the physics is not mathematics and mathematics is not physics one helps the other but you have to have some understanding of the connection of the words with the real world if necessary to at the end to translate what you figured out into English into the world into the blocks of copper and glass that you're going to do the experiments with to find out of whether the consequences are true and this is a problem which is not a problem of mathematics at all I've already mentioned one other relationship that the cause it's obvious how the mathematical reasonings which have been developed our great power and use in for physicists that be on the other hand sometimes the physicists reasoning is useful for mathematicians mathematicians also like to make their reasoning as general as possible if you say I have a three-dimensional space of the ordinary space I want to talk about ordinary space you know you're in it that you measure distances and there are three what numbers you need to tell where something is you go Brett width and height three-dimensional space and you're beginning to ask them about theorems then they say now look if you had a space of n dimensions that here are the theorems well yeah but I only want the case 3 well substitute N equals 3 and then it turns out then it turns out that very many of the complicated serums they have are much simpler because it happens to be a special case now the physicist is always interested in a special case he's never interested in the general case he does he's talking about something he's not talking abstractly about anything he knows what he's talking about he wants to discuss then gravity law he doesn't want the arbitrary force case he wants the gravity and so there's a certain amount of reducing because the mathematicians have prepared these things for a wide range of problems which is very useful and later on it always turns out that the poor physicist has the compactors excuse me when you wanted to tell me about this four dimensions you now another item that's interesting in this relationship is the question of how to do new physics is it important to have a feeling of kind of into oh I must mention one other item when you know what it is you're talking about that these things are forces and these are masses and this is inertia and this is so on then you can use an awful lot of common-sense seat-of-the-pants feeling about the world you've seen various things you know more or less how the phenomenon is going to behave well the poor mathematician he translates it into equations and the symbols don't mean anything to him and he has no guide but precise mathematical rigor and care in the argument whereas the physicist who knows more or less how the answer can go is going to come out can sort of get partway and go right along rather rapidly they met the mathematical rigor of great precision is not very useful in the physics nor is the modern attitude and mathematics to look at axiom now mathematicians can do what they want to do one should not criticize them because they are not slaves to physics it is not necessary that just because this we be useful to you they have to do it that way they can do what they will it's their own job and if you want something else then you work it out yourself the next point is the question of whether we should get when we try to get a new law whether we should use the seat-of-the-pants feeling and philosophical principle I don't like a minimum principle or I do like a minimum vigil or I don't like action at a distance or I do like actually the question is to what extent models help and it's a very interesting thing very often models help and most physics teachers try to teach how to use these models and get a good physical feel for how things are going to work out but the greatest discoveries it always turns out abstract away from the model it never did any good Maxwell's discovery of electrodynamics was first made with a lot of imaginary wheels on idlers and everything else in space if you got rid of all the idlers and everything else in space the thing was okay Dirac discovered the correct laws or the lack of quantum mechanics for relativity quantum mechanics simply by guessing the equation and the method of guessing the equation seems to be a pretty effective way of guessing new laws this shows again that mathematics is a deep way of expressing nature and attempts to express nature in philosophical principles or in seat-of-the-pants mechanical feelings is not an efficient way I must say that there is possible in a No I've often made hypothesis that physics ultimately will not require a mathematical statement that the machinery ultimately will be revealed just to prejudice like one of these other prejudices it always bothers me that in spite of all this local business what goes on in a tiny is naive no matter how tiny a region of space and no matter how tiny a region of time according to the laws as we understand them today takes the computing machine an infinite number of logical operations to figure out now how can all that be going on in that tiny space that why should it take an infinite amount of logic to figure out what one stinky tiny bit of space-time is going to do and so I made a hypothesis often that the laws are going to turn out to be in the end simple like the checkerboard and that all the complexity is from size but that is of the same nature as the other speculations and other people make it says I like it you don't like it it's not good to be too prejudiced about the thing to summarize I would use the words of jeans which says that we said that the great architect seems to be a mathematician and for you who don't know mathematics it's really quite difficult to get a real feeling across up to the beauty of the deepest beauty of nature CP snow talked about two cultures I really think that those two cultures are people who do and people who will do not have had this experience of understanding mathematics well enough to appreciate nature 1 it's too bad that it has to be mathematics and mathematics for some people is hard when one of the kids reputed I don't know if it's true that when one of the Kings was trying to learn geometry from Euclid he complained that it was difficult and Euclid said that there's no Royal Road to geometry and there's no Royal Road it's not the JA if we cannot as people who look at this things a physicist cannot convert this thing to any other language you have if you want to discuss nature to learn about nature to appreciate nature it's necessary to find out the language that she speaks it she offers her information only in one form we are not so on humble as that's a demand that she change before we pay any attention it seems to me that that it's like them all the intellectual arguments that you can make would not in one in any way or very very little will communicate to deaf ears what music the experience of music really is and all the intellectual arguments in the world will not convince those of the other cultures the philosophers who tried to teach you by telling you qualitatively about this thing me who's trying to describe it to you which is not getting across because it's impossible I'm talking talking to deaf ears and it's when they it's perhaps that the horizons are limited which permit such people to imagine that the center of the universe of interest is man thank you you

12 Replies to “Feynman's Lectures on Physics – The Relation of Mathematics and Physics”

  1. 26:13 Unfortunately that is not an accurate statement. Simply because we can infer or connect one axiom to another does not rule out a possible non-equipotent, set-theoretic relationship. In other words, inference does not amount necessarily to equipotence, which is what he seems to be describing when saying one axiom can subordinate another and vice versa, depending on which one you start with and use to infer. I think there is an import to what he is saying but I am not sure what it is. It seems to be some more false claims, that the axioms all have equal ontological status and hierarchy does not enter into the equation. That's just to dismiss all the work done in higher order logic and set theory.

    I think what he is trying to say is that in the process of demonstrating the relation between mathematics and physics, as it pertains to the major axioms of mathematics and physics, it doesn't matter where you start and none of them are ontologically privileged.

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