(A book review & a mimic of the dialogues in the book, borrowing Mr T. and Mr.A.)
Disclaimer: this only shows my partial opinion of an amazing book which has a lot of ideas interwoven within them. There is so much to this book, and to the Godel’s theorem, that it is not quite realistic to capture them all accurately in one article. I just hope to recommend this book and to provide an relaxing and fun read.
Achilles and Tortoise are visiting the Escher Museum in The Hague
Achilles: Why, will you look at these wonderful paintings! They combine different perspectives in one frame and create a paradoxical world. Tangled hierarchy and strange loops are everywhere.
Tortoise: Yes, I can see why Escher is your favourite painter.
Achilles: Speaking of Escher, were you not reading a book a few months back about Escher? You were fascinated by the book and promised to tell me about it.
Tortoise: Why yes. The book is called Godël, Escher, Bach—an Eternal Golden Braid, written by Douglas R. Hofstadter.
Achilles: My my, what a long title! And what a long name the author has!
Tortoise: The book itself is long as well. It is seven hundred pages long. You can see why it took me so long to read.
Achilles: Seven hundred? That definitely won’t make a relaxing read. But as long as it is interesting, I guess it is worth the time.
Tortoise: I assure you it is deeply intriguing. I have never read any book like this.
Achilles: What kind of book is it?
Tortoise: It is a marvellous book.
Achilles: …Mr Tortoise, you are known for your wise remarks and clarity of mind. I beseech you to offer me a more explicit explanation.
Tortoise: I would be willing to do that, but it would take quite a while to explain. Say, why don’t we sit down at the nice little cafe and have a chat?
Achilles: Yes of course! (sitting down in one of the bean bags offered by the cafe) Wow, comfy! And the cafe is playing music by your favorite Baroque composer, J.S.Bach! What a wonderful coincidence.
Tortoise: (straining his ears to listen) Yes… It is the cello suite No 2 in D minor, I always find it particularly calming.
Achilles: Yes, it does have a pleasing air. Waiter! (motioning to a waiter in a smart white shirt) A cup of cappuccino, please. And for you, Mr Tortoise?
Tortoise: Tea, please. No sugar.
Achilles: Now, you were telling me about this book…what IS it really about?
Tortoise: Dear Achilles, you do pose for me a hard question to answer. Even the venerable author himself admits in the preface that, quote, “This question hounded me”. It is a book that explores different layers of interpretations that arise in life and all fields across maths (hence Godël), art (Escher) and music (Bach). It is also about strange loops and paradoxes that we encounter, and how, when used correctly, things that arise from these things will seem…well, how do I put this…seem to have a life of their own. It is as if we are creating animate things from the inanimate.
Achilles: This reminds me of the endeavour of creating Artificial Intelligence.
Tortoise: AI is indeed mentioned with great zeal in the book, as it uses self-reference to improve itself. See, the way AI learn is from the past experiences and the information that it had already processed. It has the ability to self-repair. We haven’t achieved it yet, clearly, but the magic of self-reference is in use in that branch of study. The book also touches on topics like neuroscience, genetics, physics and linguistics. They all involve strange loops and paradoxes. It would seem that we could never do without self-reference. The author used art, music, and more importantly, maths to demonstrate this point. It actually ties in with a new and exciting branch of maths called “Metamathematics”.
Achilles: Very deep and philosophical. I hope this is not one of you maths books. You know how bored I get, reading those books. No offence, Mr T., but the book you gave me the other day on Statistics nearly killed me. (yawn)
Tortoise: (The waiter came on with the cup of coffee) Here, some caffeine will wake you up. No, I assure you, the book is extraordinarily well written, with amazing illustrations and playful dialogues inserted between each chapter. To make things easier for the reader to understand, the writer borrowed two characters from Lewis Carroll, Mr Tortoise and Mr Achilles, to explain the concepts.
Achilles: Who, you?
Tortoise: Why, you!
Achilles: Coincidence upon coincidences! The two characters share our name!
Tortoise: Yes, I wonder why I have never noticed it before. Anyways, the two characters debate in miniature dramas to explain things to the reader. The drama are sometimes a mimic of a piece of music, very likely a Bach piece, a piece of Escher artwork or a maths story.
Achilles: Much like the conversation we are having now?
Tortoise: Yes, just like what we are doing right now. The dialogues are one of the parts I enjoy the most in the entire book. They are ingenious and awe-inspiring… No really, the book is completely readable, the author took his time in explaining the maths, and invented a new system to explain in detail Godël’s theorem. It is not at all hard. Afterall, this is not a book on maths.
Achilles: Still, I fear that the book might be too long and boring for me. I am not the best reader of maths books.
Tortoise: No matter, I personally took delight in the parts that are not about maths as well. You can always skip around the chapters, and that would be totally enlightening too.
Achilles: You mentioned a Doodle theorem, what is that?
Tortoise: You mean Godël theorem. It is a theorem put forward by the logician, mathematician and philosopher Kurt Godël. It is quite intricate and complicated, but in its core, it states that not all theorem can be proven.
Achilles: Not all theorem can be proven? Is that you being pessimistic and cynical again, Mr T.?
Tortoise: Unfortunately, no. this fact has been proven already, fair and square.
Achilles: You mean… there are conjectures that cannot ever be proven no matter how hard and how long we try?
Achilles: (trembling, takes a sip from his cup) This is too frightful! I feel like I am approaching the end of the world at the speed of light. Would you like to give an outline of how he proved it?
Tortoise: Very gladly, but I’m afraid it would take far longer than necessary, and besides, you can find out about it as you go through the book. At the moment I’ll give the basic idea. The proof involves creating a strange loop in any formal system using self reference. An example of this is the sentence “This sentence is false”.
Achilles: Ha! I know this trick. Don’t try to trick me with this. This is the classic paradox, if you say that this is correct, then that means that this is wrong. If it is wrong, then it is correct. Such a decease of a sentence is obviously not relevant and completely trivial.
Tortoise: Maybe, but trivial or not, it is a valid statement. And if it is a valid statement in a formal system, we must be able to decide if it is true. And we would have to prove it, true or false. However, as this sentence is paradoxical, it is undecidable. What Godël did was to create a self-referent sentence like that inside a mathematical system and concluded that this is a theorem that is undecidable—therefore the system is incomplete.
Achilles: Hmm…you do sound convincing. But I still am skeptical. To me, this sentence is not true–nor false– it is pointless. Self reference does not lead to anything and it is better just to not ask this question!
Tortoise: You don’t realise, but you just touched on another part of the book! It is another interpretation and approach to these kind of problems. It is the Zen Buddhism approach to the problem, by “unasking” the question.
Achilles: Unasking? How can you unask a question once it is spoken?
Tortoise: That’s the one of the core ideas of Zen. In order to understand and able to interpret the hole in the mathematical system that creates that paradox, we must ascend to a more powerful level, a “metalevel”, if you will. But then again this level face the same problem as the ordinary level, so it also need a “meta-metalevel” to mend its hole, and so on and so forth. You will need an infinite amount of “meta”s to be able to describe the final level that gives us a complete system without defect. This is impossible to contain in our finite world, so the Zen approach to transcend infinity is to “unask” the problem instead.
Achilles: (trembling, takes a big gulf from his cup) Pft! It is cold. We have talked for so long. I fear my head might burst.
Tortoise: Do relax a little, Achilles, and listen to the cello sonata, it will clear your head immediately.(calmly sips his tea)