Hello and welcome to my book club, this month we read Godel, Escher, Bach by Douglas Hofstadter.
Now this book is not the easiest book to read, it’s over 750 pages, when you include the notes and annotations and it’s about a difficult topic. It’s something that even me, having learned about a lot of these ideas in other classes and other places before, there were some sections that I had to stop and pause and re-read it to make sure I understood and it’s not because Douglas Hofstadter is bad at explaining things, it’s not because he’s not a sophisticated and eloquent writer, the difficulty is the ideas themselves.
However, I think the these ideas are very important and some of the most interesting ideas that I’ve ever read about. Now, what is the idea of the book? Well the core of the book is the idea of a strange loop. So what is a strange loop? Well I think it’s easiest to visualize what a strange loop is if we look at the second person in the Godel, Escher, Bach trilogy here of names: M.C. Escher.
M.C. Escher was an artist who was famous for doing mind-bending, paradoxical kinds of images where one setting which appears to be the foreground becomes the background or the subject becomes the object. So some of the classic ones you’ve probably seen before include the staircase that continually goes upwards. It’s a two-dimensional picture that shows a paradox that couldn’t exist in the physical world.
Another example is the drawing hands where he has drawn two pictures of hands which are holding pencils and drawing themselves. Other ones he has includes birds which turn into other animals and shift between the foreground and background. He even has one where there is a person looking at a picture in a museum and the scene is warping so that he is part of the museum itself.
So, this is something that I think is easy to see to visually, this kind of paradox, and it’s easy to dismiss it as well as being a bit of trickery. You might not even be a fan of M.C. Escher’s art, you might consider it too obvious. Perhaps you see it and you think, oh yeah, he’s trying to do a visual illusion, tricking us because it’s really a two dimensional picture and we’re operating with our three-dimensional brains.
However, I think it’s a really good entry point to grasping this idea of a strange loop. Now what I think Douglas Hofstadter calls a strange loop or this tangled hierarchy is where you have layers of something where something is built on top of something else.
We can think about this in the physical world: we know that the world is made out of atoms and quarks and basic elementary particles but then on top of that level we have chemistry and chemistry is a a greater abstraction (something built on top of physics) and on top of chemistry we have, let’s say, biology, and on top of that psychology and sociology, etc., etc., etc. So that’s a hierarchy.
And what a strange loop is, is where this hierarchy flips back in on itself. It’s where the top layer somehow also becomes the bottom layer. This is where the M.C. Escher idea comes in where the staircases just go up and up and up but they somehow end at the same starting point. Now, I have to admit, Johan Sebastian Bach was not someone that I knew a lot about prior to this book.
I am much less well versed than Douglas Hofstadter was so I’m going to take his word for it that a lot of the music conventions that Bach used had this similar quality of ideas that are repeating and turning on each other and having the same kind of self-referential paradoxical quality but in a musical domain as opposed to M.C. Escher who is in an artistic domain.
But what about the first name in this book? Godel. Now really it’s called Godel, Escher, Bach but those names don’t have equal weight. This book is mostly about the mathematician Kurt Godel or the logistician Kurt Godel.
Now, he has one of the most amazing discoveries in math, in logic, in just, the ability to reason, I think I’ve ever heard. I think it’s worth studying; it’s worth studying in and of itself because the ideas he presents are so mind boggling and it’s difficult to really fathom the implications of but also because the way he did it is a perfect example of this strange loop.
It’s a perfect example of this M.C. Escher idea of one idea being built on top of each other, now, the difference between M.C. Escher and Kurt Godel is kind of doing a bit of a trick. He’s drawing a situation which can’t possibly exist but we’re easy to dismiss it because we know, that, well, it’s really just a two-dimensional picture it doesn’t have to represent something that is physically realizable.
Whereas Kurt Godel I think his trick is much more impressive because he’s doing it within the field of math and logic itself. So he’s doing this bending, twisting, this hierarchy of one lower level being built on top and top and top of each other until it comes down and reaches the bottom again. Within the realm of logic, something you’d think would escape the ideas of paradox, would escape the ideas of self-referentiality, which is the work that we can find in Escher and Bach.
Now why is this important? So before I go into Godel’s actual proof and explaining why it’s so mind-bending and why honestly it’s one of the most interesting things I’ve ever learned, before that, I want to talk about the implications because at the end of the day, this might be interesting but maybe not enough for you to wade through 750 pages of a book to talk about some esoteric idea of strange loops.
Well the basic idea of where this is going is that Douglas Hofstadter believes that this very potent tool, a very potent idea that we can apply to human consciousness itself. So what are we actually? Is in Hofstadter’s mind a strange loop? Is it a situation where the hierarchies that were built upon flip back down onto themselves.
Now that may sound confusing and I hope as I explain the nature it will show how it was done and later draw links to human consciousness and it will make more sense. It is a bit of an abstract idea and it’s hard to wrap your mind around, not only is it a 750 page book and I read all of it, but I had to go back and re-read some parts especially when he’s talking about Godel’s proof and how it was executed technically because it’s very difficult to wrap your head around. So if you’re listening to this podcast and you think, I didn’t get any of that at all, that’s okay!
But keep in mind, I want you to just keep a kernel, even if what I’m saying sounds a little weird or you don’t get it, keep it in mind because you might want to couple back a few years later, a few months later, read this book again, maybe you will watch a documentary about Kurt Godel (there’s a really good one) you can read books about it, essentially I think that if you spend a little bit of time with this idea maybe over a couple years, it will really strike you as profound, it will strike you as something I think upheaves a lot of our ideas about how the world works, about who we are, about how we can be related to something that seems alien to us. In the sense that when we imagine the world our sense of consciousness seems so divorced, so alien from the quantum mechanical rules that physicists discover or the mechanical ideas of chemistry and molecular biology, how can we be related to that?
I think there’s a real profundity to this. In addition, I think this is an idea that is one of those analogies that if you really get it, it applies to many, many places. It certainly applies to human minds. It also applies to DNA — as we’re talking about in the book — the very machinery that makes us alive, this is an idea that underpins it. Computers, is a huge idea, I know that many people after reading this book decided to study computer science because they thought it was so fascinating this idea of a tangled hierarchy or a strange loop.
So, I’m going to start explaining the idea but I want you to keep this in mind because it may sound a little over your head or too esoteric or who cares about mathematical logic but if you invest in it and you put in the effort to get to the other end of the idea, you’ll see it has pretty broad implications.
Let’s get started: Kurt Godel’s mathematical idea comes together in what is known as his incompleteness proof. To understand this you have to understand the setting of the time. So it’s jus the beginning of the 20th century and there are a number of unsolved, outstanding problems in mathematics.
The mathematician David Hilbert even formulated a lot of these problems as a “To Do List” for mathematicians to solve and one of them was to prove consistency and completeness of axiomatized mathematics. So what does that mean? What on Earth does that mean?
Let’s think about it this way: in mathematical reasoning we all have some informal idea of what it means to prove something. Maybe in a math class we’ve all seen a proof of Pythagoras’ Theorem where a squared plus B squared equals C squared for the lengths of sides of a right angle triangle. Even if you’ve gone through a math calculation of that or maybe you’ve seen a visual representation where the put the grids on the sides of the triangles and they just happen to add up. Now we can understand this intuitive level of a proof but there was always a worry that perhaps we’ve making a mistake.
How do you know you’ve proved something? Maybe you’ve convinced yourself but there’s a secret glitch that undermines a lot of your reasoning. So one of the efforts to really make this more rigorous was to axiomatize mathematics which was to basically turn this process of coming up with elegant and intuitive proofs into something that was completely mechanical — something that a computer could do and assuming you didn’t write any bugs into the program, it would return a correct response.
So the basic idea is that you have a formal system where you have some axioms. These are things you take for granted as being true. It may seem problematic (how do you know where the axioms are) but think of them as things that characterize what you want to discuss. So if you want to discuss geometry you’d come up with different axioms then if you want to discuss number theory or something else.
Basically the axioms are going to tell you what areas of mathematics broadly conceived, you can explore. Alright so you come up with some list of axioms, hopefully it’s a small list or if there are a large number of axioms they can be represented by a rule for creating those axioms.
Next you want to come up with some rules for manipulating these symbols. So you write out the axioms with a list of symbols, just like you would in a computer program, and then you come up with straightforward rules that if you apply it to an axiom, you get something that’s true at the end.
Again, this may sound a little complicated but really we do this all the time. If you’ve ever studied algebra you know about, well, if you have multiply by 2 on one side you can divide both sides by 2 eliminated on one side and then have divided by 2 or multiply by one half on the other. This is pretty basic, this is something that we learn in grade seven or eight.
Now, this algebraic manipulation is very similar to what’s happening with axioms and theorems. Basically, what we’re doing is we’re starting with the axioms (our 1 + 1 = 2) or similar types of propositions. Then we’re manipulating them in a way where we can get other true statements in arithmetic. Basically if we’re following these rules eventually perhaps we can reach 2 + 2 = 4, for example.
Now, this is a purely mechanical process—I can’t stress this enough. This isn’t the case, well, what was happening here is we’re discovering the truth of math. What’s happening is we’re taking these axioms, we’re taking these basic rules and we’re just following very mechanical procedures and turning them into other statements which we hope, if the system is designed properly, will give us true statements.
So what was Hilbert’s problem? Well, Hilbert’s problem was that at the time there were no different axiomatic systems for mathematics. In particular a later one that was developed (for Godel’s proof) was principia mathematica which was a really exhaustive attempt to underpin all the foundations of mathematics. To codify, as it were, what it was.
This is essential idea is that you’re going to take these these axioms and you’re going to manipulate them with mechanical rules. What we would like to know are two things about this system:
So first off, is it consistent? What I mean by that is does it ever derive a contradiction. Now because we’re not talking about “real math” we’re talking about a mechanical system that’s taking about mechanical rules, there’s always the chance that you could have written a rule incorrectly and that could lead to a situation where you get 2 + 2 = 4 or if you follow a certain set of combinations you could get 2 + 2 = 5 or it gets, 2 + 2 doesn’t equal 4. So this is a situation that we want to avoid obviously because if we want our system to represent truth in mathematics (to find the answers of what are actual, real formulas) this is something we want to avoid.
Now, there’s another thing we’d like to do with this system. It’s possible to make a consistent system that nonetheless doesn’t follow certain rules. It doesn’t allow things that we consider to be important in order to get true statements. So if it’s too underpowered—let’s say it lacks the ability to represent multiplication—then there’s going to be a whole set of numbers that we agree are true that are going to be outside the realm of this system to solve them. We are going to have a huge set of numbers that we accept as true but are unprovable within this formal system.
So Hilbert’s idea was that if someone could come up with a proof that shows these axioms exists, was consistent and also complete, that would be a major triumph. We could feel very secure resting in our knowledge that this mechanical would work all the time. And what Godel did in the 1930’s, was that he showed that this was actually impossible.
Godel showed that any system that you have, any set of rules that you have, is powerful enough to represent the basics of natural numbers (the basics of arithmetic) will ultimately undermine itself by creating situations that we know are true but cannot be represented.
So this is a real fascinating result. What does this mean? So I think we can go through the proof now, and understand a bit about how it works. Now I’m going to be glossing over a lot of details. Douglas Hofstadter goes through these details in painstaking efforts so you can see from start to finish the entirety of the proof and I think there’s merit in that. I know this book is quite long but I think there’s merit in understand the proof because whenever someone presents you with a mind bending idea, it’s easy to dismiss it if you don’t fully understand.
I think that is the case here that in my cursory overview you might think, oh well what about this or what about that, I just want to caution you even if I’m not the best at explaining this idea, even if I skip a lot of parts, the what about’ism, is not going to work here.
There’s not going to be a situation where you say what about this and it’s going to undermine the idea of the proof. It really is that ironclad. However you have to go through all the pages to see how it works. It’s a summary but I am going to ask you be careful with your objections if you see a problem with the idea.
So here’s the idea of the proof and here is really the trick to understanding Godel’s strange loop: this is where Douglas Hofstadter decided to pin his book on this profound idea. Here is the idea: we take the formulas in this system so we take the formulas that say, you know, all the little symbols that make these things, and we represent them in a code just the way we would in a computer program, we make them into a code in actual numbers, now these numbers the rules of operating on these axioms to get new, true statements, to get new theorems of this arithmetic system the way of operation on this can get us a mirror version of this system in mathematics itself.
So you can imagine, like I was saying with algebra, this divide on one hand, and divide on the other hand, let’s say a rule of algebra, that if you have something that is multiplied by everything you divide on one side, you divide on the other, the equality still holds. Well what you could imagine doing is taking that rule of having that sort of progression of multiplying this and representing it as an actual statement about number.
So you have actual numbers representing the dividing sign and then these relationships between these things, the relationship between the statement after you divided by two, isn’t just a progression in the numbers in the actual symbols of math, it is an actual transformation of one number into another number.
This means that we can within the system itself ask questions about these numbers the same way we can ask questions about prime numbers or we can ask questions lets say which numbers are powers of two or is this number to this number a valid transformation within the existing system. Now the exact way of coding that up is complicated but I want you to leave with the essential insight here which is that the actual system we’re dealing with here is a bunch of systems that use numbers and what he is doing is turning those systems into numbers.
So you can think about how 2 + 2 = 4 so you have a two, you have a plus sign, you have a two, you have an equal sign, and you have a four. What Godel is doing is saying, let’s take that 2 and put it in a number, let’s take that plus sign, let’s take that 2 and put it in a number, let’s take that equal sign and put it into a number and let’s take that 4 and put it into a number.
So now you have one super long number that represents 2 + 2 = 4 as a just a number not as something that involves plus signs and equals signs and has separate numbers but as just one big number.
Okay so what can you do with this? Well if you go through a little bit of work what you can end up doing with this process is you can make a statement that is the mathematical equivalent of saying “this statement is false” or more specifically what it is saying is “this statement has no proof.”
Because you’re representing within themselves you can have a statement that refers to its own number so you can make a proof that says there is no proof for this particular statement. Now what does that mean? Well it’s interesting: it’s not saying that the statement is false. Let’s be clear about that. It’s saying that it can’t be proven.
And what that means is that a statement that says essentially this statement has no proof, well, if the statement is saying it has no proof then that means that if its true then it doesn’t have a proof or if it’s false then it means that there is a contradiction here.
So you can see if you go back to Hilbert’s problem of consistency and completeness that this is now impossible because you have a statement that gives you a forking path. On the one hand you choose inconsistency: a statement that is false and we know it to be true. Okay, how do we deal with that?
The other statement is that, let’s say we take it to be true, let’s say we say this is a true statement well now we know that it doesn’t have a proof so we know there are results in mathematics that are going to be forever outside the realm of our logic and our understanding. Now, there’s a lot of profound ideas in logic and mathematics and because of this there’s also a lot of related proof for instance, I think a little bit less elegant, but Touring’s (Alan Touring) halting problem from computer science. Basically it says it’s impossible to create a computer program that can analyze another computer program and say for sure whether they stop.
It’s also related to Canter’s diagonal argument which is the idea that if you make a list of all the real list (meaning numbers that have decimal and whatever numbers after that) that it’s impossible to list them out because it’s always possible to create a new number that’s not on your list. So all these cases, I think, hint at some limits of the ability to reason within these systems they show that it’s impossible paradox, it’s impossible to make a situation where you have complete rational control over the universe. That’s one of the really profound ideas. But if we follow Godel, Escher, Bach, that isn’t the crux of what the author is trying to argue that isn’t why he brought up this proof.
Instead what he wanted to talk about were two things: first the idea of formal systems. The idea of a system that just mechanically following rules and then building something more complicated out of that. This is the basis behind computers, yes, but also the basis behind the physical world. The idea behind objects that interact based on physical laws, the basis behind how DNA combines and replicates, all the machinery in our cells, so there’s something worth studying in the idea of formal system itself.
The second idea is that of the strange loop — the insight that Godel had which was to represent the rules of the system within the system itself. This kind of inception-like idea of taking a system and mirroring it within the system itself. And that might sound impossible but Godel was able to do this and you follow the proof to its end you can see it logically follows. It doesn’t involve magic it’s just a simple fact that when a system becomes sufficiently powerful it can eventually represent itself or a version of itself within it.
So what does that mean? Well what I think that means here is first of all understanding things in terms of formal systems is a potent idea and being able to think about things in terms of following ironclad rules is very powerful but the second idea is that once a system can reach a level of sophistication it’ll have this ability to loop back on itself.
What’s the extension here? One of them is the idea of human consciousness. He’s saying that what perhaps the self is, what we are, as not bodies but sort of abstract quality, what makes us different from rocks or computer programs that we have today is that our machinery for representing things is sufficiently complex and we can represent ourselves in that machinery and we can represent ourselves representing ourselves.
We can have thoughts about our own thoughts and about the person thinking the thoughts and about their relation to the world at large.
This kind of, turning back on itself, of the thought machinery that this can happen within a system which nonetheless is just obeying mindless rules and mechanical properties. Another idea that comes up here and this is a philosophical concept but it’s the idea between syntax and semantics.
What is syntax? It is the rules and mechanical structures of language and sentences and semantics and what those words mean. What Godel showed is that this boundary between syntax and semantics is not cleanly dividing, it’s not the case that there’s just syntax on one end of mechanical rules and the meaning is cut off from that. The reason that we know this is because Godel’s own proof was done within a system that was entirely syntactical. It’s not necessary for you—any interpretation that you have—on these rules saying that they correspond to real statements in math or real statements in number theory; that’s just your interpretation.
What you say they mean is what you’re applying on top of it. And yet what is happening here is that this system itself is creating its own substructures so that it can refer to itself even though it’s only a syntactic system.
By setting up the formula just right, you can get a system that talks about itself, the rules of itself and makes those inferences. You could argue that perhaps this doesn’t represent genuine meaning or genuine semantics in the sense that we can write out this proof but someone still has to know what it means. I think it provides an important bootstrapping; it shows how a system compose of meaningless symbols can acquire meaning and it can do it within itself. It can bootstrap itself.
I think this has really profound consequences for many areas. Now what’s the implication here for artificial intelligence and those kinds of things? Well, this book was written quite a while ago and because it’s been written quite a long time ago it was written prior to the recent developments in AI certainly deep learning and machine learning (the things that everyone is talking about these days) so in some sense this book is dated.
When we get to the end and the author is talking about AI he’s very much stuck in the old paradigm where humans thought well, if we’re going to make a computer that thinks it’s going to think the way that we do but in terms of here are some symbols and I’m going to manipulate them in a procedural way and get some answer. This was the dominant paradigm at the time and it was shown through a lot of starts and failures that this wasn’t a workable solution for artificial intelligence.
Now since then we’ve developed much more powerful computers. We’ve developed processors that can go far beyond what was there when Hofstadter was writing his book. And because of this certain algorithms that were prohibitive that we were not quite sure how to execute, we can now execute. So we have situations like Alpha Zero, the new deep mind program for playing chess, that can just operate with five thousand specially designed hardware units to test against itself many, many times so that it can learn the rules of chess incredibly well. Now this is just something that is not really programed into the computer but it is something that we’re developing with it.
So there’s some anachronisms later on in the book when we talk about AI however, with that in mind, the same fundamental rules apply. AI has grown more sophisticated and some of the tools we’re using are different, but we’re not at a point where we’d consider any of the AI’s we’ve seen to be conscious in the way a human is and not developing the kind of self-awareness where that’s even up for debate. That’s still restricted to TV and movies where we have robots that wax poetically about this own humanity.
We’re still in an era where AI is a limited tool. But what I think Hofstadter gets right (and what is still correct today) is that the future goal of AI and how it relates to what we are as human beings and what we can possibly achieve, namely, that if this strange loop idea is to be fully realized, is to be fully understood, then it means that creating an AI robot that has consciousness that has that self-awareness that ability to think for itself and decide for itself and is functionally similar to human beings is at least possible in theory. Whether or not we’ll reach it, whether or not we’ll make something that is similar to ourselves or alien to it, whether or not the AI we end up developed is going to be handicapped in some way is yet to be seen.
But, whether it’s possible in principle is still a debate amongst many philosophers — many philosophers think that this type of AI therefore resulting in some higher level semantics, higher level intelligence consciousness, is impossible.
I think if you follow the idea in Godel, Escher, Bach it’s at least suggestive if not iron clad proof that that’s not the case. That we can get systems that have meaning and the meaning is bootstrapped out of something that doesn’t have meaning. That you can have a system in Godel’s case of symbols being manipulated by very straightforward rules that nonetheless if you get complicated enough can reach back and refer to itself.
Now one of the things I think is interesting is that Godel’s proof is somewhat pessimistic: it’s proving something about the limits to human reasoning. It’s about our inability to find a complete and consistent set of mathematics. But one of the statements (I’m blanking on the name of the mathematicians named in the book) but one of the statements is the idea of a very different but very slightly different statement which is instead of having a statement that says that this statement cannot be proven you change it to say this statement can be proven. In this case that actually is enough to bootstrap the truth; you can have something that merely states that this statement can be proven. Without actually knowing what the proof is, you can know that that’s the case.
So this is something I think is more analogous to human consciousness. It’s a set of symbol manipulation yet sophisticated enough and robust enough that it can say about itself these qualities that it has they become true sort of as an incantation almost a magical ritual where merely stating that this is the case becomes true—it becomes a self-fulfilling prophecy.
Now I’m not saying that if a computer were to merely output the text, “I am conscious” it would become a conscious machine but merely that the mechanism for underpinning our own consciousness is very likely similar to this strange loop idea. I think there’s a lot of very interesting ideas in this book.
It goes beyond Kurt Godel’s famous proof although a lot of the book is devoted to really understanding it so you can get an intuitive idea of exactly how such a feat was done. But there’s also a lot of interesting ideas of logic itself, about what constitutes knowing something, one of the interesting results discussed in the book is that a corollary to Godel’s incompleteness proof is that it’s actually not possible to say that truth—whether or not something can be true or not—can actually be represented in mathematics for a very similar reason.
You can get a situation that says “this statement is false” in mathematics (you can make it say that idea) if you assume the idea that truth and falsity can be directly inferred from these mechanical statements. This also points to the idea that what makes something true or false is quite elusive. It’s something that is maybe not possible for mechanical systems to fully realize truth or falsity. And just before that makes you think of a distinction between humans and machines I think there’s a very important section where he talks about, because of these complications, it’s not the case that human beings have some transcendental power to lift ourselves above the power and see outside of it and say well this system is too weak to prove it but I can see this is true because well, this statement is true, this is false, it actually continues up higher and higher up the levels.
So as human beings we may be a somewhat more sophisticated system for recognizing these intuitive flaws then a computer but because there’s an infinite chain of these types inconsistencies and what you accept as true or false it goes up and up and up—it’s actually impossible for us as human beings to assess the same thing. So the implications of that—the idea that truth and falsity in this very mathematical sense, forget knowing about the world, forget doing scientific experiments and knowing physics, just even in logic, even in mathematics itself—we’re not able to know with certainty or arrive at this certainty of truth and of falsehood is also I think quite profound.
There’s many ideas of this nature; there’s a huge discussion about genetics and about how molecules again following simple mechanical rules bootstrap themselves into existence. How DNA replicates and then creates its own machinery for replicating itself. It goes on and on and on. I think there’s a lot to discover in this book that I have not talked about but I think if you’re interested in this idea—if the idea of strange loops, of things that loop back on themselves, sort of paradoxical self-references that must be true—I highly recommend reading this book.
Now it’s certainly the case that this is not an easy book to read. As I said myself earlier there were some passages I had to re-read to make sure I understood everything but I would’t say it’s not an unenjoyable read. Douglas Hofstadter has this unique way of mixing topic which are about these esoteric ideas and logic and computation with dialogues—with these invented characters who themselves mirror some of the concepts by talking about perhaps more ordinary everyday things.
Additionally because the book is somewhat about how the form or structure or syntax of things eventually the meaning I think it’s quite playful how he’s put the book together in that it involves many acts of self-reference, puns, names for things double standing for something else and so you can see even in the form of the book that it very much exemplifies the philosophy of the strange loop that the form and the substance are not separable that they can link up and mirror each other.
Even for that, this book is really I think a work of art even if you decided after you read it that you fundamentally disagree with you have to admire how the book was composed and indeed in its era it one a Pulitzer for that very reason.
Now I think this is an important book; I certainly would like to discuss more ideas related to cognitive science and philosophy of mind in this review. If you have any suggestions or ideas that you think I should cover or you think I got wrong, I’d be happy to get some emails from you and listen to your own thoughts because this is a subject that fascinates me and I’d be happy to discuss it more.
Next month we’ve going to be doing the book Seeing Like A Sate by James C. Scott and this is a fascinating book about the limits of organization and indeed how many of the evolutions that we’ve gone through as a society and civilization perhaps were not done with the best intentions. So thank you very much for that, this was December’s book, Godel, Escher, Bach. Thank you very much.