MIT Godel Escher Bach Lecture 1

the following content is provided under a Creative Commons license your support will help MIT OpenCourseWare continue to offer high quality educational resources for free to make a donation or view additional materials from hundreds of MIT courses visit MIT opencourseware at ocw.mit.edu all right hello welcome to a griddle Escher Bach a mental Space Odyssey my name is Justin curry and I'm a senior in mathematics here at MIT I've spent the last year at Cambridge University at UK and the summer before that living in Germany so it's kind of a reverse culture shock coming back but I'm excited to teach gödel Escher Bach again I taught this course in spring 2006 it was a 10-week course then and we attempted the impossible task of trying to get through this thick monster all in one go and it's impossible most undergrads can't get through it in 13 weeks I got through it in about seven years so you're going to be attempting a feat here not to complete the entire book but to get the essence of griddled s your Bach out but I want to make sure we introduce everybody just to get people's names this will help me take attendance and it will also just I also want you to say what is it when you read the course catalog that interested you most and why essentially why you're sitting here today I'm curious so what is the idea behind this book I interviewed a good many of you this morning and just to make sure that you guys felt comfortable with mathematics this course isn't directly about mathematics there's a lot of mathematics being talked about yes you have a question ok so that's what I'm going to go through right now the idea here is that Douglas Hofstadter is interested in one primary question and that question is how does a self come out of things which have no selves how is it that all these carbon atoms and molecules and proteins which make us up and physical universe how they go from being meaningless to developing into an entity which can refer to itself like right now I'm saying I think this I think you like this I'm meeting all of you as individuals each one of you claim to have a self you might remember des cartes famous quote I think therefore I am so it seems like the I when I say the I I mean the things we call ourselves as a real existent thing but it's a complex question how do we how do we get eyes out of non eyes and that's that's kind of going to be the goal over over here so I'm just going to call it AI but how do you get to an AI you get to an AI by having a bunch of meaningless primitives things like atoms proteins molecules I should say if I want to etc like this this is what you're made up of but none of these things mean anything none of these things have eyes ourselves but but you do so what what's the what's the relationship here douglas hofstadter you wrote this book back in the 70s when he was doing graduate school in physics and this was after him doing a math undergrad at Stanford he believed that he saw that he saw the answer when he was playing around with with mathematics and in the very formal systems we play with like when we write down things like two plus two equals four these are just these are just symbols and as we go through today I'll show you completely equivalent ways of doing addition um which will look like this and and these are just logical primitives like if you've seen any set theory and you know don't feel scared if you haven't seen any of these symbols but like there exists an X for every we give these interpretations but the idea is that mathematics can be reduced to a bunch of meaningless operations just symbol shunting but what's interesting is that it within mathematics there exists an equivalent to self reference this is this is this is a bunch of atoms and proteins referring to itself calling itself an eye what happens here and this is this is going to be kind of underneath the name of girdle is we're going to get to some incompleteness theorems going to get to some statements which in mathematics refer to themselves and the question of how this happens we understand this rigorously mathematicians have worked out how do we go from meaningless symbols to something which refers to itself and which has meaning the claim then is is that these two systems are equivalent and this is really the profound idea I'm going to draw this symbol now we use a term called isomorphism an isomorphism it's basically in equals two and equals in a different sense but the idea here is in many ways we can link atoms and proteins to kind of logical symbolic primitives in mathematics and we understand how we get self-reference in mathematics so maybe we can use this to understand how we get eyes how self comes out of now non-self this is a really tall order but we're going to try to do it and that's what this book attempts to do and what I've done is isolate the chapters in this book which I think are most pertinent to pertinent to this string of thought but basically what we're going to do is we're going to learn how it works in mathematics we're going to go from logical primitives and workup self-reference and talk about Zen Buddhism consciousness etc but that's going to happen as we leap over here because we're going to we're going to work up down and then around and we'll conclude the course with some interesting questions about artificial intelligence and how intelligent things come out of unintelligent things so when I was teaching this course two years ago or two Springs ago I ran into kind of five things which I viewed as really important tools for thinking and this is kind of I've had to condense a little bit into my famous tools for thinking lecture the idea here is that girdle Escher Bach has an incredible number of conceptual tools for thinking about this complex problem of how do we go from a non self to a self and just outline these real quick we're going to have isomorphisms and i'll explain all these terms as we go on recursion I'm going to leave this one mainly up to current on the second lecture paradox and this is infinity which and all these concepts are very closely linked and finally the main subject of per today's lecture is going to be formal systems all righty so first let me go through kind of definitions of these terms and isomorphism I want you to all be very careful with this because when you start talking to mathematicians you know growing up professional mathematicians they're going to use the term isomorphism to mean something very very specific the way it's used in gertle Escher Bach the way it's going to be used in this class is very loose we're going to make very kind of intuitive statements like you know what's what's the isomorphism between a car I'm not a great artist here what's the isomorphism between a skateboard and a car and you know you might say lots of things like it carries a person it's it has four wheels so what we do is we construct a map which also has an inverse and that's that's the way you think of an isomorphism you can go either way and preserve information preserve Anam structure if you if you really feel like following along I've I've included actually a quote from Douglas Hofstadter and on a page seven of your of your lecture notes he says and this is in the middle of the page the word isomorphism applies when to complex structures can be mapped onto each other in a ways that to each part of one structure there's a corresponding part and the other structure where corresponding means that the two parts play similar similar roles and the respective structures this is how we're going to always use the term isomorphism in this class if you're taking the abstract algebra class it's going to mean something a lot more specific and you're going to have a lot more details you might actually think of these as kind of a what I'll say but don't worry about worry about it is a homomorphism and the idea what the homomorphism is that there are a lot more details here than there are here and your for example there's no steering well I'm just doing well in a car but there's no steering well specifically in a skateboard so if you were to go if you were to create a map from the car to the skateboard that detail would have to go somewhere else but don't worry about those necessities but when I say the term isomorphism think of equals and then I'll often use that symbol right there so this is going to be really important because it's going to be how we're going to get meaning out of things and you'll see it a lot coming up it over the book but first I want to hop on and talk about recursion recursion is basically it's seen everywhere but it's kind of a list of instructions which you follow but then repeat until you've reached kind of a final case so suppose you were you're cooking and you had a you could have a recursive algorithm for stirring eggs and that would be world and then world again and keep whirling until essentially everything looks mixed up that's a very loose way of understanding it but another way which you are probably familiar with a much more rigorous in the term of mathematics is the Fibonacci sequence this is where you start with two numbers one and one and then you construct the next number by summing the previous two so you have that and you have three you have five and you have eight and so on and you can create what's called a recursive definition where you define the nth Fibonacci number this is for n greater than equal to two and here you define the thing in terms of itself and this is a classic example of recursion what it is is really itself on a smaller level I think one of the most exciting applications of recursion are fractals because the way we create fractals is through recursion so I don't know if you all have seen this but the SAP in ski triangle or the sierpinski gasket it's kind of a classic fractal here you divide a triangle up into three and then you just repeat the process for infinitely number of an infinite number of times on each remaining triangle you create these very beautiful kind of mosaic forms but the nice thing about mathematics is that we can be very precise and do things that we can't do in the real world and that's repeat this infinitely and so on just for a quick digression and I really don't spend too much time on it because current will do more why is it called a fractal does anyone know nothing just like sure um that it was a term coined by benoît mandelbrot in 1977 I believe it actually refers to its number of dimensions so this might be kind of a mind-bending concept for most of you but we like to think we live in one two or three or four dimensions all integers right but my claim is that the sierpinski gasket actually lives in between one and two dimensions lives in like one point six three something dimensions but I want to help you kind of think about that and if you if you want to hop along to a page nine kind of got a recipe for for helping you think about dimension you know it's weird because only mathematicians would ever worry about rigorously understanding the concept of what a dimension means so here's one way to think about it if you take a line and you double it you have two copies of the line that you started with this guy's here in there if you have a square and you double the sides of the square you have four copies of the original square similarly and I'm not going to try to draw this because it will get too complicated way too fast if you take a cube and you double each of the sides you get if you think about it eight copies of the original cube so if you're perceptive enough you might kind of realize this action of powers going on here so here we had after our doubling process two copies we had two to the one here after our doubling process we had to the two after our doubling process here we had two to the three eight so this is weird because notice that the cube lives in three dimensions and the square lives in two dimensions the line lives in one dimension so this might suggest to you the relationship that two to the D or D is the dimension of the space you're living in equals the number of copies you have after the doubling process so let's return to our friend the sierpinski gasket if we start here and we imagine doubling each of the sides of the sierpinski gasket here and here we're very strangely led to the conclusion that whatever dimension the sierpinski gasket lives in it obeys this rule so take the logarithms and d times sorry this is getting crowded I take the logarithm both sides and solve for D you'll see that the dimension of the sierpinski gasket is log 3 over log 2 which it's approximately one point five eight five on to infinity so here's an exact example of something which lives somewhere between one and two dimensions and I think that's a really cool concept moving on for other tools for thinking we have paradoxes paradoxes come in all sorts of different flavors I don't know if some of you have heard of the birthday paradox where it's the idea of okay what's the probability that someone else in the room has your same birthday everybody thinks it's really small but if you actually work out the mathematics it turns out you actually have a good chance if you're in a room with over 40 people you have extremely high chance of finding someone else with your same birthday so I've actually listed out this is courtesy of a Wikipedia and mr. Quine we have sort of three variants of oops we have three variants of paradoxes this is a veridical and these are things which are true but they seem paradoxical at first there's political and I'll give an example of each of these and then kind of the classic the one which we are going to be interested in and these are real paradoxes our antenna means to give an example of another classic paradox and one which is visited in Goethe Ledger Bach very early on it's called Zeno's paradox and the idea is if I want to get from here to my laptop I first need to walk halfway across the distance and then if I wouldn't walk the remaining distance I need to walk half of that I want to walk the remaining distance and you know walk half of that and then half of that half of that and eventually I get stuck in this infinite loop where it seems like I'm not getting to my laptop a variant of this paradox is the idea that if I even want to move at all if my atoms want to pass in space first they have to go half way but before it can go half way it's got to go half way back half and half way of that half and that half of that half so Zeno back in Greece actually use this to prove that motion was impossible and that any motion we saw in the universe was an illusion so it's weird why and nobody really could answers you know for the longest time but then it took essentially the development of the understanding of limits in calculus to really get an idea of why this wasn't paradoxical what rigorously did we mean by an infinite number of steps what how could we actually get to the cross across the room it seemed paradoxical but we knew it had to be true we knew motion had to be possible um I'm sure when you're all were younger or even now you've seen all sorts of kind of false cynical paradoxes where somebody will write out a string of if you take one plus you take one minus 1 plus 1 minus 1 gothic dot and the person convinces you well look if you look in groups of this these are all zeros so if you just add a bunch of zeros together this is necessarily zero but I mean this is an infinite string right and we can repeat the pattern what happens if we add a 1 right so suddenly we get these weird conclusions where 0 equals 1 and they're usually built on kind of doing something illegal involving infinities and infinity is going to be a very important concept that we'll encounter again and again finally the antinomy ease these are the important paradoxes to think about I once went out to dinner with a bunch of mathematicians I don't know how I ended up in that but let me tell you it's kind of frightening um and there was this Korean mathematician who said well you know it like most these questions don't even matter we don't we don't understand some of the most fundamental things and the thing he was most interested in and I think which bothers mathematicians the most is the antenna me of the of the liar and Russell's paradox so the liars paradox you probably have heard before and it starts it's based on actually a biblical reference but it essentially says this sentence is not true so is it true or is it not true well if it's true then it says of itself that it's not true so true and Polly's not true contradiction so if it's not true then we know that we believe in the law of the excluded middle which means that things have to either be true or not true that it's negation is true so if it's not true then the sentence is true so not true implies true so we're stuck the liar paradox still hounds us today unlike Zeno's paradox it hasn't been solved we still don't know how to deal with it and when we talk about girdle's theorem the way he proves his result is actually going to be intimately linked with a variant on this so instead of saying I'm not true it's going to say I'm not provable and that's going to be a very interesting idea and we'll explore that a little bit later the other antinomy i want to look at is Russell's paradox also known as the barber's paradox and that's how i'm going to tell it it's the barber's paradox i think it's a little more friendly so you have a town and there's this male barber and he abides by the rule that he shaves all people and only people who don't shave themselves so what does the barber do when his beard is getting as thick as mine does he shave himself or does he not well let's see so by definition the barber only shaves those people who don't shave themselves so if he shaves himself then he doesn't if he doesn't shave himself then by definition he must shave himself a variant of this is which was coined by both Bertrand Russell Cambridge mathematician and philosopher and Zermelo great German magician is the idea that you can consider the set let's call it Omega which contains all sets that aren't members of themselves so remember a set is just a collection of objects and a mathematicians really believe that set theory was going to be what gave mathematics its ultimate shirin logical foundation so let's give an example of a set which contains itself so let's think of the set of all things which aren't Joan of Arc well sets aren't people I mean they're people not sets um so that set of all things which aren't Joan of Arc includes itself because the set can never be a person so that set is contained in itself so we have a bunch of things in here which are sets which aren't members of themselves and then we ask the question is Omega an element of itself and this means is in well if Omega contains itself but Omega by definition only contains things which don't contain themselves so it can't contain itself well if it can't contain itself it doesn't contain itself and that means it should contain itself contradiction this really really bothered a lot of mathematicians for a long time and it's it's an exact variant on the barbara's paradox so this isn't kind of interesting things to play around with finally is the concept of infinity I can't really talk too much about it we're going to look at it more but I want to introduce you guys to the idea that there are multiple types of infinity so you have the integers and you also have the real numbers and it is true that you cannot create a direct link you can't match every real number like 0.3 3 3 3 2 3 well wait three five something random PI that's the pick pi you can't put PI directly in connection with a natural number an integer and this is kind of famous Cantor's diagonal ization argument so somehow there are different degrees of infinity and the real numbers is a higher degree of infinity so that's that's an important thing to think about now we're going to jump ahead to our last tool for thinking and this is going to be the reason why we ignore the first three chapters of girdle Escher Bach and it's the idea of a formal system problem is as formal systems are boring and Douglas Hofstadter takes his sweet sweet time and introducing you to the concept of a formal system so I'm going to try to speed things up because I know you all are smarter than that and you can get through these concepts very quickly we're going to play a game it's called the move puzzle or mu and the way you play it is you start with you have a bag of three letters and you're going to have a rule you're going to start with pull two letters out you get MI and we have going to have four rules and these are completely strict typographical rules for thinking about for deriving new things that we can pull from our bag our first rule is that if we have an AI suppose we have MI or we could have anything and then an AI we can tack a yuan so aiyoo so right away we know that we can create em I you our second rule is suppose we have em and then a string of letters that are eyes and use since they're in our bag of alphabet our alphabet here then you're going to get for free M X X so just as an example suppose somehow you had em eye which we do you're going to get mi eye for free third rule suppose you have somewhere along the way you end up with a cluster of three eyes they don't have to be at the end they can be anywhere just needs to be three eyes all together and you can replace all three of those eyes they're equal to AU and our final rule is that if we have a double pair of views we can drop them and they just go away so somehow if we had em uu we could just have em now you have these rules you have these letters you start with one guy he's going to be our axiom an axiom is the starting point for reasoning for applying these rules and the game is can you get em you starting from MI and it's using only these two four rules can you get em you I will give $20 to the first person who can derive mu that's in this room only applying these four rules and starting directly from em I just to give you an idea of where you might be going where you might be playing just going off of our rules we already saw that we had mi we can get mi u we also saw that using rule 2 using were one we can get em I I we saw if we have anything like that we can repeat it twice so I can get MIUI you that's applying rule two again and so on leave this leave this as a puzzle take your time with it you'll be working on it for a few hours but first person that's in this room derive mu from this it's $20 yes yes fourth rule only applies to to use so yes if you have to use you can remove them and subtract them all right and once again I do urge everyone to buy the book these rules are listed explicitly in the chapter and you might gain some insight on how to derive what you want here so why is this interesting I mean it's a it's we're just playing with letters and strings and things like that well although this seems pretty meaningless and kind of dumb um does anybody feel like when they're just looking at this game looking at this rules that they're just essentially playing around with algebra that they learned you know in middle school or high school that really what we're doing here is we've got some statements like two plus two equals four and we all learn that we have a typographical rule for when we have an equal sign like that we can add one to both sides and preserve equality so something we have two plus three equals four five so really what mathematics reduces to is is just playing around with systems of this form and applying these rigorous kind of typographical rules except here there's doesn't seem to be any meaning it's just meaningless one of the important questions are in addressing this classes how do things gain meaning how do we go from meaningless to meaning um this obviously seems to have meaning but I want you to ask yourself why kind of before we proceed it's necessary it's my duty to do the boring task of writing down just a few definitions of things which which you can call these you have words so we already saw axiom that's that's the definition you call any of these guys a string so so a string is just any ordered sequence so in this case M is in use we already met an axiom an axiom is a starting point it's your first thing that you can apply the rules do so and this actually has a lot to do with mathematical logic because of math logic the idea is that we start from really primitive things which seem obvious like the successor of 0 is 1 and then we work from that concept and we derive all these truths of number theory and mathematics here your axiom is mi and you're trying to prove the theorem and that's kind of our next thanks guy here well you're trying to prove the theorem of MU so the theorem is basically a string which results at the end of a derivation derivation is like a proof for those of you have done geometry when you're in saying okay well this triangles can grow into this triangle because of side-angle-side and things like that those are you're deriving you're making rigorous justifications for your leaps and logic so here our rigorous justification that mi you was the theorem was that well we applied typographical rule number one that's a rigorous leap in logic and we got to this theorem and you can just call these four rules here these are rules of inference and logic and a lot of things that you'll play around with you know of interesting on SATs and things like that are you know if you have if you have the statement that P P implies a statement Q so if it's cloudy then it will rain you have you have that this is is kind of equivalent to you should use a different arrow here too not Q implies P and these are really nice because they're just typographical rules when you see something like when you have well I've got M followed by any string of letters well then I can double it that's a rule of inference just like this is a rule of inference if I have P implies Q I can always replace that it's completely equivalent to not Q implies not P so but for those of you who are scrambling away because you want $20 really fast I want you to take a break because once again you should focus on what we're what we're saying right now and we're going to talk a little bit about jumping outside the system this is kind of the cool renegade stuff that Hofstadter fills this book with and it's the idea that as you're playing around with this you're right now you're just playing a game and what mathematicians and what anybody human does is when they feel like they're caught in loops just cranking through pages of algebra and not getting anywhere humans are intelligent enough to stop they exit the system and they say I don't know I don't think this is going to go anywhere or well let me think about why I'm not getting or like how might I get mu you know maybe it has something to do with numbers of eyes and news or things like that I'm sure you start doing what I like to call meta thinking you're not thinking in the system applying typographical rules applying rules of inference to existing strings axioms and getting theorems that's thinking inside the system that's just thinking meta thinking involves you leaping outside the system and making judgments about it thoughts which cannot be expressed as any just normal typographical role within the system you're doing meta thinking what one of my favorite parts of this of this section in gertle Escher Bach is when Hofstadter says and if once again stop driving the drive in you try to turn to page 20 for your lecture notes oops some a syllabus get that that worries page 24 Hofstadter kind of uses this as like as a life lesson he says look of course there are cases when only a rare individual will have the vision to perceive a system which governs many people's lives a system which had never before even been recognized as a system then such people often devote their lives to convincing other people that the system really is there and that it ought to be exited from excessive our social customs and our kind of cultures are really just formal games you know we say hello we shake your hand that's an instance of a formal rule which we all follow but you know every once in a while you get somebody who says ah I don't want to shake your hand I'm going to exit the handshaking formal system but of course they're much more radical examples of this like I said Karl Marx and communism you know he viewed this idea of like well look you've got these people who are collecting money and property and you know they're they're getting someone else to do all the work and they're pressing this whole class of people can't people recognize the system so then people like Karl Marx and Freddie go like start writing and pamphlets encouraging people to overthrow governments etc because they viewed a system they said look we need to exit the thinking system for intelligent beings we can think on a higher level of course I'm not trying to promote communism here I'm just showing you an example historical interest you know anarchism socialism today working peoples the media nowadays I think it's one of most popular things to people sit for people to say is like well you know it's just the media trying to do this before we used to never like just refer to this entity as the media the media is trying to obscure understand this the media is trying to scare us also you know the government the government's responsible of course a classic example is also what Karl Marx said the you know the church they're they're the opiate of the masses has they said and also school schools my favorite example of you know a system which people have encouraged you to exit from it's like well you know it's just care that we have and we don't actually want kids to learn and grow up and this has inspired a lot of new free thinking educational movements like the Montessori and things like that and I really want you guys to think about in your daily actions and my living perhaps in a kind of formal system which is acting in a similar way try to do some meta thinking thinking on a higher level and is it worth being exiting that system Hofstadter kind of classifies these three levels of thinking and he likes to call it a mechanical mode when you're doing the normal games of the system an intelligent mode and then just an unmowed non modes when you just kind of reject the system he calls it this is a new way of approaching things and this is something we like to talk about a little more I want to quickly introduce you to another well first of all I want to talk about a concept of what we previously mentioned is you know we're eventually gonna be talking about artificial intelligence and it's weird because humans really like to say that their thoughts are logical we like to say that we do think in this manner but a lot of times we don't we we like to use kind of just inference about just collective events like one of our favorite tools of thinking is induction well you know the Sun has rised all these previous days sure it'll rise tomorrow and there's no real formal line of logic that's saying that well sunrise yesterday and it thus it will rise tomorrow and I want you to think of whether or not human our thoughts are actually just computations in a formal system much like mi U or P implies Q and things like that and that's going to bring me to another formal system which I have to mention just because in Chapter four he's going to refer to it and and it's going to lead us to this kind of interesting line of dialogue of when a formal system with meaningless symbols gains meaning and it's called the PQ system we're going to have three new letters or three new characters it's not going to be P Q and - and you've actually got an infinite number of axioms here and when you've got a definition and that's that if you know X P - and I'm going to kind of make sure I have just an underlined P Q X and this is going to be an axiom whenever X is just a string of hyphens so it's just some string of items so what's this saying it's saying that well if you have something like this well X here was two hyphens so we know that that's an axiom all right it's a little different than mi you seems just as meaningless and we're going to have different forms for manipulating and playing around with this and one rule is that if you have XY and Z which are just - strings X P Y Q Z then you can derive you're given for free the statement X P Y - Q Z - seems meaningless but what does it remind you of we've got this axiom we in fact have a whole infinite list of axioms and maybe you've noticed that we've got two hyphens here on one - here got three hyphens here and what does this do yeah exactly I mean what it does is it it says that well if this works right so let's let's apply this rule here and we'll apply this this rule here so we can take this and get for free that - - P - we can add another - Q and we had three hyphens here but this rule says we can tack on another - what does that say well this seems to say that two plus two equals four so I want you to realize that the symbolism which mathematicians have been using and what you've grown up learning is just shorthand meaningless notation yeah well yeah no what I meant to say here is that we seem to be inferring this rule that - string 1 + - string 2 always equals - string of 3 in the inside just 1 here refers to a whole string of hyphens and 2 refers to a string of hyphens like Y here or better yet I could say X plus y equals Z here and what makes this system different than than mi u does anyone have any ideas why do you something care a little more about this system than mi you other than the fact that you have 20 bucks going on the line for deriving mu anybody what about the fact that I just kind of showed you this equivalence here and now instead of applying these kind of typographical rules I've showed you that well you can also take this as two plus two equals four and then you're going to say aha well now I can do all sorts of things like now that I discovered the meaning of the PQ - system I can go ahead and just create all sorts of new theorems and starting from any of our axioms and you might even be tempted to say well I know it's obvious I know that two plus two plus two equals six and I've discovered this isomorphism between P's and Q's and and pluses and equal signs so I'm tempted to say that - - P - - P - - Q - - - - - - as a lot heightens what's wrong with this does anyone see a problem yes exactly exactly it doesn't follow the rule the rules I told you in the axioms which you start from you only ever have one P in one Q this is not even what we call so this is not what we will refer to as a well-formed formula so you have to be really careful with what meaning means and when you try to create an isomorphism between what you know about addition and the formal systems you play try to come up with an alternative interpretation we could have just interpreted these peas cubes and hyphens as you know we're going to call P we're going to say that's horse and Q and that's Apple and you know one - is happy and you know two hyphens is happy happy and so on so suddenly we have an interpretation for for this string it's not not two plus two equals four but it's happy happy horse happy happy Apple happy happy happy happy doesn't mean anything but it's an interpretation and there's no reason not to make that interpretation perhaps the horses this is actually more sensible in addition I mean first of all when we do addition we're representing these numbers in a in base 10 because we have 10 fingers but horses don't have 10 fingers and numbers written in base 10 don't mean anything to horses but perhaps happy horse Apple really makes much more sense to a horse so we're going to kind of throughout and I have to be a little rushed about this be thinking about where does meaning come from how do we actually assign meaning to meaningless symbols because that's the goal here we're going to go from meaningless symbols and mathematics to meaning and then we're going to try to create nice and morphism between the universe and our formal systems and this leads me you know perfectly into this idea you know is reality a formal system and if you go to page 29 in your notes you've got this kind of long quote stretches on 2:30 I'll go ahead and start reading it's at the bottom so scan all of reality be turned into a formal system in a very broad sense the answer might appear to be yes one could suggest for instance that reality is itself nothing but one very complicated formal system its symbols do not move around on paper but rather in a three-dimensional vacuum space they are the elementary particles of which everything is composed tacit assumption that there isn't into the descending chain of matter that the expression elementary particles make sense the typographical rules are the laws of physics which tell how we're on page 29 which is one kitchen the typographical rules are the laws of physics which tell how given the positions and velocities of all the particles at a given instant to modify them resulting in a new set of positions and velocities belonging to the next instant so the theorems of this formal grand formal system are the possible configurations of particles at different times in the universe the sole axiom is or perhaps was the original configuration of all the particles at the beginning of time this is so grandiose to conception however that is only the most theoretical interest and besides quantum mechanics and other parts of physics can at least cast at least some doubt on even the theoretical worth of this idea basically we are asking if the universe operates determinist deterministically which is an open question you know this it's I think it was Laplace we said well look if you were to give me the position and momentum of every particle in the universe I could tell you the rest of the future and this is leads to want to kind of the grand philosophical questions which you know we'll be investigating as part of this class as well which is you know if the universe operates terminus tickly if Newton's laws govern how my arm Falls and how all the atoms of my body interact where does free will creep into how do I know I have control over these actions and it's not the fact that at the Big Bang there was a denser cluster of atoms over here and a less dense over here and things evolved according to deterministic laws much like the formal systems we're playing with here so this question you can really think of on two levels one can the universe be thought of as being modelled by formal system having forces and solving equations for the particles here and it collides with another particle at this angle they go off like this and things like this but also I think likes to ask another question which is version two for those of you who are kind of matrix fans to what extent is the universe a formal system proper in the sense is it a program you know running in the background of some hyperdimensional alien who's playing wow and you know he's just running our universe as a simulation on his you know supercomputer cluster that he's got in his basement who knows I mean if the universe is deterministic or he can just he's just coated up you know hacking away in Python all of our rules of our universe and he said all right that's what the simulation go and here we are in his computer having all these kind of dramatic interactions with people cetera et cetera and he's just kind of interested up what bug came up and cetera it's kind of interesting to think about so we've now really kind of hit home these five tools for thinking and we're going to be revisiting all of these ideas throughout the entire book and I and one of the things that Bach one of the things that Douglas Hofstadter does is he he structures his book and its own kind of recursive fashion and you know I only gave you a few specific instances of where recursion shows up and this represents kind of my my bias for me I'm very much an art person and a math person but I'm not so much of a music person and I really encourage you guys to bring in different elements because GEB has such like a high dimensional structure to it everybody contributes their own slice to it and one thing which I would hate to deny for deny you guys from is is the music aspect of this book each one of Douglas Hofstadter's dialogues is actually structured and based upon a piece of box music if you listen to box music and you read the dialogue he might actually hint at some of the connections some of the isomorphism that Hofstadter is alluding to but first of all you should know why he chose Bach how recursion acts in music and that's why I have this whole speaker setup here so allow me to play so this is box little fugue in G minor just as a nice anecdote a who here has seen a beautiful mind in the movie alright so John Nash the mathematician who went crazy princeton cetera the story goes they used to actually stalk around the halls of the math department smoking cigarettes and whistling this song constantly and what were some of the things which you noticed about about this piece for those of you with good auditory abilities what did you notice okay elaborate a little bit on these patterns exactly so you heard it come in at a different tone at a different volume and you notice it was the same theme it's the same theme that he played stretched inverted backwards on higher levels on lower levels so GB is actually very much structured like a fugue Hofstetter lays out for us and what I did in this first lecture is I laid out the entire bond laying out the entire book for you all in one go so that way you understand it when I play it stretched out inverted backwards and at different volumes so this is nice you have a musical illustration you have artistic illustrations of the ideas we're talking about but we need to actually kind of settle into the book itself so current Kelliher and I or anyone else who's really excited about reading anybody really excited about volunteering for reading the dialogue anybody have the book with them right now oh good job would you like to read you don't have to you want to okay so we're going to spend the last kind of 15 minutes going through a dialog I actually have another copy good and so I need two characters one to be Achilles and one to be tortoise these are two characters we're going to meet this dialogue I'm going to play a prominent role throughout the entire book so let's does anyone else want to be well I'll see I like the tortoise so I'd like to be the tortoise but someone else can be the tortoise they want to be okay so we only have one so let's brave enough to do it all right already so page 79 so yeah sorry so I'm going to give you some some quick quick background on this dialogue so Hofstadter like me believes that it's important to introduce the idea of a topic conceptually first before we start really diving into it so he preface --is every chapter with a with a dialogue the dialogue is kind of conceptual introduction to the ideas we're talking about to go ahead and give you an idea of what this dialogue is based on it's up going to be the conflict of two mathematicians curt girdle and David Hilbert David Hilbert believed that mathematics could be put into a formal system very rigorously and it could also be proved to be consistent and complete those are two words which I'm going to define kind of at the end of this dialogue but let's go ahead and start it off and try to work quickly through this I'm going to ask that when you have the italics you go ahead and read it as part of your section so people have an idea what's going on in the book all right excellent so we don't have really any time you left oh but I want to say one thing it's a challenge pay attention to tortoises quote on page 81 when she talks about acrostics if you can find two acrostics in this dialogue you

Channel: jasonofthel33t

Views: 326,812

Rating: 4.9259439 out of 5

Keywords: godel, escher, bach, js bach, self reference, incompleteness theorem, recursion, philosophy, mit, computer science, quine, Math

Id: lWZ2Bz0tS-s

Length: 62min 34sec (3754 seconds)

Published: Sun Dec 02 2012

Reddit Comments

[u/xamdam]

Don't really love people re-uploading publicly available materials on private YT accounts. Here is the full official version http://ocw.mit.edu/high-school/humanities-and-social-sciences/godel-escher-bach/video-lectures/

[u/Hypersapien]

For those who are not aware, /r/geb exists.

[u/incaseyoucare]

I've tried before, but it's a struggle to watch this video because the presentation is so choppy and awkward. The guy clearly knows his stuff. It's unfortunate, though, how little emphasis is given to teaching and presentation skills in higher education.