Gödel, Escher, Bach: an in-depth explainer
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This is only half of it!
Gödel's Incompleteness Theorem states that any sufficiently rich formal system, together with an interpretation, either has strings which are true but unprovable or has strings which are provable but untrue. Either is possible! In practice people prefer to have true things that can't be proven rather than provable things that are untrue. But either is possible.
Truth is defined by a model or interpretation of the system, not by the system. As TFA explains, the Incompleteness Theorem showed that truth and provability must be considered as separate notions.
No. You can build a really trivial system and know things about it "from the outside", so to speak, even if they can't be proved within the system itself.
Godel's work is exactly about showing the difference between something being actually true, and something being provable within a system.
Let's take a not-real example: we know there are infinitely many primes. If you define arithmetic formally, you can fairly easily prove that there are infinitely many primes. But, in theory, it could be something that isn't provable, but is still true; true because you can always actually generate more primes, but unprovable because there is no way to build a proof out of the axioms of arithmetic to show this.
Now, this isn't true about the infinitude of primes, because it can be proved within arithmetic. But there are lots of other statements that we have't proved yet - and theoretically, any one of them could end up being unprovable. E.g. Goldbach's conjecture or the 3n+1 theorem - they might be actually true but something we can't prove within the way we've defined arithmetic.
E.g.: if ZFC is consistent, then ZFC+~Con(ZFC) is consistent as well, where ~Con(ZFC) is the statement that there is a contradiction in ZFC.
Now, ZFC+~Con(ZFC), which is a perfectly good axiom system, consistent and can talk about anything, can prove that the Turing machine that searches for a contradiction in ZFC halts (just apply the axiom ~Con(ZFC)), but it doesn't make it true.
The opposite a system which has true statements we can't prove is still useful, even if there might be some problems we won't ever be able to solve. But a system which can prove both a statement and its negation loses meaning.
Or does it? Naïve set theory, despite Russel's paradox, and language in general, despite the local equivalent of "This statement is a lie." is still a useful tool, so maybe the same applies even to more formal systems?
I always had the impression that unprovable means you could add either the statement or its negation as an axiom, and both resulting systems are as consistent as the system you started with
Without biasing for a particular model, we could simply say that every sufficiently expressive proof system admits either zero models or more than one model.
However, Gödel made explicit in the very first statement of his incompleteness theorem that assuming more axioms cannot ever eliminate incompleteness (except by the extremely undesirable event of eliminating consistency). Gödel shows that, if a set of axioms (including additional axioms on top of your original set) doesn't allow you to prove everything (by virtue of being inconsistent, something called the "principle of explosion"), then there are necessarily statements that those axioms can't resolve the truth of.
So you can add more axioms and that will either make your system more powerful or inconsistent (depending on whether you chose inconsistent axioms to add), but you still can't get rid of the incompleteness phenomenon that way!
If you are a platonist and believe in some preferred model where every statement is decided this makes perfect sense. For the rest of us, this just means that in a sufficiently complex system there will be undecided statements. Which is not such a big surprise – but a rather awesome technical exercise!
I'm somewhat on thin ice on the mathetematical formulation, but I suppose something could be undecidable but true with respect to all standard models of a theory, while not true for some non-standard model.
For instance, there may be statements acting on infinite sets (such as N) that may not be compressed into a recoursive formulation, which means that a proof of the valididity of the statement (if it is true) would require an infinite number of steps.
The term "standard model" is interesting, because it pushes the problem a bit. So, let us take the natural numbers. Sure, we can prove in set theory that some of the undecided statements of Peano arithmetic are true for the standard model. But seen from the outside this means that we have changed our axiomatic system from Peano arithmetic to set theory. But there are still undecided statements – even arithmetic ones – in this new system! These will hold in some models of set theory, while being false in some other.
So, saying “Oh, I mean the standard ℕ, not any of those other ones” does not get us off the hook, because it raises the question: “The standard ℕ... in which model of ZF?”
GEB is a marvellous work that is accessible to anyone with reasonably good school grade maths. I chanced upon it by accident in the school library one day and was hooked after a few pages.
Anyway the crux of the matter is that you can very carefully construct a statement about a system that can't be either proven or disproven by that system! I don't have anything like the formal knowledge to really get to grips with it and the discussions on infinities and so on are pretty mind blowing. However, you feel that DH is imparting glimpses into the sheer beauty of the ideas he covers.
Wait 'til you discover what ricercar and quining is all about - bloody lovely. Just read it but take your time. There is something in there for everyone. You often get told by clever people about the links between maths, music and art. Mr H easily gives the best argument I've ever seen that attests to that being true, whilst giving your brain a right good kicking.
There was a Dutchman, a German and an Austrian who walked into a book ...
Close but not quite as that's an inconsistent statement.
"This statement is unprovable." is the approach Godel takes and eliminates the inconsistency. Either that statement is true, in which case it's unprovable, or it's false in which case there exists a proof of a false statement.
I just picked a classic to start off my prior comment which morphed more into praise for GEB than Goedel's brain breaker.
I have dim memories of a cracking constructive argument starting off with some very basic axioms where one was written as S, and two as SS and so on, then it all went a bit mad but the genius of Hofstadter is to make all that impenetrable palava nigh on accessible to the layman (with a bit of effort from the reader).
The horrendous thing about Goedel is that the final flourish is clearly correct (I assume that responsible adults have filled in the formal bits, I'm sticking to the lies to children version). It is both terrifying and perhaps obvious at the same time. I can imagine the sense of dread when mighty edifices such as Herr Hilbert's suddenly looked a bit shady and then anger, followed by disbelief and finally acceptance as the big hitters really got to grips with it. The world hasn't come to an end because of Goedel but it certainly got a bit more interesting.
I think that it is almost comforting that we have a system that can be complicated enough be to capable of saying things about itself that can't be proven within itself. I think that there is a good chance that our system of mathematics will eventually become complicated enough and no more. Obviously there will always be spherical cows and some absolutely mad numbers and when I say eventually - that will take forever (nearly).
My take on that statement is that it is not saying anything about the world. It only refers to itself, making it a self-contained mini-universe with no relation to the real world.
So, since it's not saying anything, it's neither true nor false. Only something confusing that feels like it should have some meaning.
Now, how do we attach meaning to the actual statement. Do the English words actually mean the same thing to me as they do to you? Probably.
Should any statement be applicable to anything - world or otherwise? It is just a statement and not my best man's speech at my brother's wedding 25 odd years ago, which I'm sure you can appreciate that I consider that being rather more important.
So here we have a statement that is unable to be consistent and as you say, it sounds like it should have meaning but doesn't.
So we have a way of saying things with spoken language that are logically inconsistent that are also grammatically correct and that is a sort of flavour of what Goedel proved with his incompleteness theorem.
> the links between maths, music and art.
Mr. H. is by reputation a very competent violinist; even though he's a mathematician, he can pronounce with some authority on subjects like Bach.
Kruskal's tree theorem is a more interesting example, you should look into that. You can prove it undecidable, so you can add either it or it's negation like you say, in Peano's arithmetic, but you can prove it ZFC or I think even less expressive axiomatic systems for set theory.
At 13:46 it gets into the "is there a way to prove every complete statement?" and furthermore goes into the Godel math for "there is no proof for the statement with Gödel number g" where that statement itself has a number "g".
The entire video is a good watch though.
And the related problem is that with Godel we don't know if math is consistent either.
Goldbach's conjecture is that "every even number bigger than 2 is the sum of exactly two prime numbers", so 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, etc.
For this statement to be *true* it just means that every even number there must exist two primes that add to that number. This is a statement about infinitely many integers.
A *proof* of Goldbach's conjecture consists of a finite number of formal reasoning steps that start with some axioms and end up at the statement of the result.
To this day, it seems as though Goldbach's conjecture is true. It holds for every number we've been able to test. However, no one has proved it is true or false yet (or proved that it is unprovable).
The proof of Gödel's result's involves very carefully formalizing what statements and proofs mean so that they can be encoded as statements about arithmetic. He then shows there is a statement with encoding G that says "The statement with encoding G cannot be proved" – if it is true, then it cannot be proved.
It's kind of confusing at first, but there is an easier way to get an intuition why there might be true statements that cannot be proved. Think of each statement about the natural numbers as a subset where each number in the subset makes the statement true. There are uncountably many subsets of the natural numbers (by Cantor's diagonalization argument). Proofs are finite chains of finite statements so there are only countably infinitely many of these. Therefore there must be subsets/statements that are true that do not have a matching proof.
The approach that Gödel's proof takes is not too different to the above argument – it is essentially a diagonalization argument – the complexity is in making the encoding of statements are numbers very precise.
Whether or not a visual demonstration like that is actually a “proof” is a separate question. It definitely wouldn’t satisfy Hilbert, and doesn’t meet this definition:
> A proof of a statement S is a finite sequence of assertions S(1), S(2), … S(n) such that S(n) = S and each S(i) is either an axiom or else follows from one or more of the preceding statements S(1), …, S(i-1) by a direct application of a valid rule of inference.
I also don’t know of any visual “proof” like that which can’t be explained much more rigorously and powerfully with a formal set of assertions. But pulling threads like this and really asking what makes a proof a “proof” are some of the deepest questions I think a person can ask. It’s worth doing if only to appreciate what an incredible accomplishment all of the formal set theory work is in unifying and attempting to define meta concepts like “proof” itself.
For such proofs to be contained in a finite space, the verifying person or machine needs to be able to distinguish between arbitrarily minute differences between proofs.
Again, that doesn’t really count as a “proof” by modern standards, but it’s how the ancient greeks thought (they used more than just visual intuition/they also used more rigorous and formal propositions than that water thing, but they were visual and didn’t involve finite sets)
Of course, this is not an argument that our brains are absolutely finite and classical. Perhaps analog computations is required for an appreciation of beauty, or quantum physics is necessary for us to fall in love. But we seem to be able to check math proofs without them.
A proof (there) is just a finite number of symbols which happens to have a specifix form (A=>B AND A), where A and B are sentences, which are finite sequences of symbols having a slecific form… Wait, I am telling you a half of what Gödel did to prove his result.
But one of those two must be true, you just can't prove it. Of course you can add a new axiom that allows you to prove one or the other (or accept one of those statements as an axiom), but you will still have other statements that you can't prove.
https://en.wikipedia.org/wiki/Continuum_hypothesis#Independe...
Also:
https://en.wikipedia.org/wiki/Axiom_of_choice#Independence
And thanks to Wikipedia for this elaborate list:
https://en.wikipedia.org/wiki/List_of_statements_independent...
Three alternative accounts:
* Both the CH and ¬CH mathematical universes really exist, so we just have to choose which one we're more interested in at a given time. Like one might say there are the "reall numbers" and the "realle numbers", both valid and interesting constructions which humanity was just slow to recognize the distinctions between (because they were initially less relevant to our interests and our day-to-day lives).
* We are actually ultimately thinking about one or the other of them, or are in some sense in one or the other mathematical universe, but we don't know enough about our intuition about the reals to be able to specify or explain which one. (Maybe we need other properties whose obviousness or relevance humanity is not smart enough to notice?)
* Some finitist or ultrafinitist approach is actually right: the real numbers are a formalism that, while reasonably motivated by historical attempts to "complete" mathematics in various ways, doesn't correspond to anything Platonically real or to anything intellectually relevant to humanity. (In this account, there is potentially no answer to the question because the real numbers don't exist at all. Neither CH nor ¬CH refers to a mathematical reality, just to games about formalisms.)
It's also a provable statement (via excluded middle)
I do believe this opinion places you very high on the 'confidence' axis, but not especially far along the 'competence' axis.
> The proof of Gödel's result's involves very carefully formalizing what statements and proofs mean so that they can be encoded as statements about arithmetic. He then shows there is a statement with encoding G that says "The statement with encoding G cannot be proved" – if it is true, then it cannot be proved.
Sorry I meant to quote this bit at the beginning of my comment. Parent comment which I was replying to talks about both.
that is disrespectful and very very very short sighted.
also, go read up (...on Gödel, Cantor, Turing, Tarski, etc...)
My training is as an applied physicist. We physicists have an interesting relationship with math. Obviously math is essential to the work that we do, but the physical world decides whether the math is right, not the other way around. Our mathematical models technically permit things like negative mass, time flowing backwards, or magnetic monopoles. But that doesn't mean tachyons, time machines, or fundamental magnetic particles exist--they don't, so far as we know. So I'm trained to actively disregard non-physical, not relevant mathematical implications. I'm sorry if this offends a pure mathematicians sensibilities, but pragmatically it is very useful.
Or take a different field: in computational semantics, a branch of formal linguistics, there are many models for inferring a formal logical statement from an example written sentence or spoken utterance, and then determining the validity (truth) of the statement. These models get caught up on stuff like "This sentence is false." What's the truth value for that sentence? If it is true then it must be false, and if it is false then it must be true. Error, validity of this statement can't be determined! But hey, it turns out that in practice this basically never happens unless the speaker is really confused, misspeaks, or deliberately evasive. Real sentences don't have this self-referential, circular logic structure because that's not how people think or communicate.
Now "this sentence is false" goes back to the greeks, IIRC, and Gödel's theorem is slightly different. Gödel's main work is in the formalization of proofs and proof systems, and I don't want to take away from that in any way. But the incompleteness theorem always seems to be explained through these sorts of self-referential examples and I have yet to ever see it reduced to a practical problem with real-world implications. Hence my question. Does Gödel's incompleteness theorem actually constrain a real world application of proof systems, where we tend to be interested in non-cyclical logical arguments?
Never mind then, I’m sorry I bothered.
That was the belief before the 1800s. But with the discovery of non-euclidean geometry, math has been divorced from the physical world. Math is simply a system of axioms and proofs. Math is purely abstract and logical. Whatever math that physicists use just simply happens to align with the physical world.
Also, the physical world doesn't confirm whether the math is "right". Math is deductive, not inductive. As long as the math is derivable from the axioms, it is right. The physical world/experiments determine whether the mathematical model aligns with the physical world. The physical world has no say in math. Not anymore.
> So I'm trained to actively disregard non-physical, not relevant mathematical implications.
In the past, when a mathematical model predicted something ( relativity to quantum physics to elements ), experiments were conducted to determine whether the models aligned with the physical world. If the mathematical models make predictions that physicists currently can't verify with experiments, should we disregard it? Do we need technology to advance to where we can conduct experiments. Do we need physics to become more abstract? Perhaps model the physical world in the virtual world. Would experiments in the virtual world be applicable to physics? Seems like physics is both at a dead-end and on the cusp of a revolution.
One of the things it inspired was Turing's work on uncomputable numbers. It turns out that computability and incompleteness are intertwined, which at least I find interesting. And without the "uncomputable numbers" malarkey, I wonder if the Turing Machine formalism would exist (answer, "probably, but maybe looking different and with another name"). And, well, the Halting Problem is essentially Gödel Incompleteness (imagine handwaving here).
As for proof systems, again, the answer is "probably". Knowing that there are true, unprovable, statements in a formalism is something that informs how you approach it, you need to put a limit on how far to go before you say "I don't know" and taht is in and of itself important.
The proof of Gödel's result uses the paradoxical statement "This statement is false", but that's being used to prove this very general result about all systems. So the hunt is then on to find "natural" statements that are True but Unprovable.
But the "unprovable" bit should more completely be stated as "unprovable in a specific axiomatic proof system". If we want to prove that statement S is "True but Unprovable" then we must actually prove that it's true. So if we've proved it's true, what does it mean to say it's unprovable? We just proved it! What's going on?
So let's take a specific example.
Peano Arithmetic (PA)[0] is an axiomatic proof system intended to capture Natural Numbers and their behaviour.
The "Goodstein Sequence" G(m) of a number m is a sequence of natural numbers ... you can find the definition here[1]. It's not hard, but it's longer than I want to reproduce here.
Goodstein's Theorem (GT) says that for every integer m greater than 0, G(m) is eventually zero.
It has been proven that GT cannot be proved in PA, but it can be proved in stronger systems, such as second-order arithmetic.
So the statement of GT is not self-referential, along the lines of "This Statement Is False" sort of thing. It's an actual statement about the behaviour of integers, so it's not a self-referential trick.
Your question now is: What's the point? How is this useful or relevant?
Much of modern (pure) mathematics is chasing things because the mathematicians find them interesting. The vast, vast majority will never, of themselves, be useful by (what I expect are) your standards.
But it was once thought that factoring integers was of no practical use, and only pursued or investigated by cranks. Imaginary Numbers were thought to be bizarre, useless, and dangerous. Non-Euclidean Geometry was thought to be utter nonsense, and held up as part of the "proof" that the fifth postulate was unnecessary and was deducible from the other four. All three of these now form critical components in modern technology.
Even more, to the average person on the street, anything to do with algebra is completely pointless.
For you, Gödel's theorem is completely pointless and useless and probably of no interest at all, but it helps us understand the limitations of formal systems. The techniques that have been developed in the time since it was proved have helped us understand more about what computer verification systems might or might not be able to accomplish.
Of itself, Gödel's theorem might not be of direct, immediate, and practical use, but the work it has inspired has tangentially been useful, and may yet be moreso.
But not for everyone. After all, some people don't care about the Mona Lisa, or Beethoven's Fifth Symphony, or Michaelangelo's David, or the fact that people have walked on the Moon, so why should people care about results in Pure Mathematics?
That's the thing about Pure Mathematics. Sometimes it ends up being useful in ways we never expected.
[0] https://en.wikipedia.org/wiki/Peano_axioms
[1] https://en.wikipedia.org/wiki/Goodstein's_theorem#Goodstein_...
It is fated to be proved as a trivial corollary to some more important mathematics; a corollary that no one would have bothered with if not for the historical importance.
The unsolved Twin Prime Conjecture, of roughly the same age, is expected to lead to much more interesting mathematics if it is proved.
But in general, Gödel applies to all formal systems that satisfy certain properties, it's just that the exact unprovable sentences will be different (since you may just add that sentence as an axiom) that's why the general example is very abstract. It shows that sufficiently rich theories are not only incomplete, but also incompletable.
The CH is undoubtedly meaningful, natural, and of huge interest to (a subset of) mathematicians.
There are huge branches of mathematics that don't find immediate practicality in physics. Often these end up being practical in cryptography or quantum mechanics, but sometimes they don't. But to dismiss the entire field that relates to the cardinality of real numbers as uninteresting if someone can't give you a practical application of it shows a simple disregard for other fields of study.
IMO the most weird thing is the following, intertwining consistency of ZFC with Diophantine equations[2]:
> One can write down a concrete polynomial p ∈ Z[x1, ..., x9] such that the statement "there are integers m1, ..., m9 with p(m1, ..., m9) = 0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent). [...] the polynomial is constructed so that it has an integer root if and only if ZFC is inconsistent.
[1] https://en.wikipedia.org/wiki/List_of_statements_independent...
[2] https://en.wikipedia.org/wiki/List_of_statements_independent...
My understanding comes from Keith Devlin’s wonderful book “Mathematics: A new golden age”.
Goodstein's theorem and the Paris-Harrington theorem are some examples of this for ZFC. There are several more, maybe a logician could chime in
Because if you show that the converse of Goldbach's conjecture is not provable, you have actually proven Goldbach's conjecture (since you have shown that there is no counterexample!).
>Think of each statement about the natural numbers as a subset where each number in the subset makes the statement true. There are uncountably many subsets of the natural numbers (by Cantor's diagonalization argument).
Don't we only care about the countable set of statements that can written down in a given logical system? Say second order logic + ZFC.
By “converse” I think you mean “negation”.
A statement being unprovable means we will never know whether it is this or false. > Because if you show that the converse of Goldbach's conjecture is not provable, you have actually proven Goldbach's conjecture (since you have shown that there is no counterexample!).
Nope: demonstrating that (the opposite of Goldbach’s conjecture) is unprovable is logically equivalent to demonstrating that (Goldbach’s conjecture) is unprovable. It means we’ll never know either way.
I think this is false. If you find a number N that is not the sum of two primes, you did disprove Goldbach's conjecture. Any such number would be smaller than infinity, and so there would only be a finite set of primes smaller it to check for.
So basically, if Goldbach's conjecture turns out to be false it IS going to be PROVABLY false.
Only if Goldbach's conjecture is actually true will it be the case that it is impossible to prove its negation. But if it is ALSO unprovable (but STILL true) you will NOT be able to prove that the negation is unprovable, because that would mean that there doesn't exist ANY number N that disproves the conjecture, so would prove the unprovable original conjecture....
Consider the statement "All swans are white", but you live in a universe with an infinite number of swans.
Lets assume that there exists at least one black swan. Proving the statement false is trivial once you find the first black swan.
However, if all swans in the given universe ARE white, and you have no way of inspecting every one, you can never PROVE that they are all white. Also you can NOT prove that it is impossible to prove the negation.
Yes, you can. Math is not physics. Pythagoras was able to prove his theorem about every triangle in the universe without inspecting every triangle in the universe. The trick is that "the universe" is not the physical universe, it's just a choice of axioms.
My understanding is that statements about members of infinite sets can be decidable if there exists a method to set up use recoursion/induction to cover all of them. For the set of right triangles, this is relatively straight forward (if we ignore the complexity introduced by real numbers).
For statements on infinite sets where it is not possible to reduce a proof to such recoursion/induction, you quickly end up needing an infinite number of steps to cover all cases, meaning the problem is undeciable/uncomputable.
See the Church-Turing thesis: https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis
Lets hypothetize that Goldbach's conjecture is true and unprovable (let's call it S1). In that case, the following statments are both true: C1) It is impossible to prove S1. C2) It is impossible to prove the converse of S1. (since in this case S1 is true, the converse is actually false, so cannot be proven).
Now, if we go into meta-proofs, we may be able to prove C1 (whether or not S1 is true). But we will never be able to prove C2 (if S1 is true and unprovable, C2 will also be true and unprovable).
Now, if S1 is actually false, We may actually at some point find a proof of that. But that's not really so interesting.
The interesting part is that there DOES exist statements like this that ARE true AND unprovable (even if we don't know which statments), we just don't know WHAT statments those are.
We can define a notion of complexity for any given integer as being the size of the smallest program that returns that integer. Obviously, I'm being imprecise here, but it should hopefully be clear that it is possible to get the details right and the precise nature of those details aren't going to be relevant for what follows.
Now suppose we have a program that can find the complexity of a given integer. Then, we can write the following program:
for (i = 0; ; i++) {
if (complexity(i) > 2*K) {
return i;
}
}
As a result, we have some integer that has a complexity--it's still well-defined (at least if you have the axiom of choice, but I'm not sure if that axiom is necessary)--but we can't necessarily prove that any integer has a given complexity.
However we can write a program that does a brute force search through all proofs from ZF looking for the complexity of i. As you just showed, this program will not establish the complexity of all integers.
But how does it fail? It fails because there are programs which do not halt, that it can't prove don't halt, which if they halted COULD return i.
And therefore the complexity cannot be determined from ZF.
The prevailing philosophy of mathematics says it is well-defined. But there are alternate (entirely consistent!) philosophies under which complexity is not well-defined at all.
See https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... and https://en.wikipedia.org/wiki/Peano_axioms#Nonstandard_model...
Godel's "completeness" theorem gives a converse. If the statement were true about all possible models, then it would be provable.
Concerning multiple models of ZFC: I'm always confused by such statements about the foundations of set theory itself, they seem weirdly self-referential. ZFC certainly can't prove that it has multiple (or even any) models. Does such a statement need additional axioms, or is there a general theorem like "If a first order theory has any model, then it has multiple ones."?
Löwenheim-Skolem implies the existence of a countable model of ZFC. https://en.m.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skole...
"What I meant to say is that multiple models are not the only reason for something to be true but unprovable, the incompleteness theorem also holds in more general conditions." I can't give a clean rebuttal for this, but I believe this to be profoundly mistaken. It might be technically correct though, depending on how you'd formalize this statement. To formalize math you need a logic that has some properties: it should be decidable whether a proof is correct, you should be able to write it down, it should not be contradictory. If you take these together the only way a statement is unprovable, is if it is independent from the axioms, i.e. there exist multiple models.
Edit: however, consistent second order theories don't always have a model. In first order, if S is an undecideable statement in theory T, then both T+S and T+!S have models, both of which are models for T, so undecideable statements always come from multiple models. But that does not need to be the case in second order, so your claim "it is independent from the axioms, i.e. there exist multiple models" is not neccessarily true, i.e. there may be cases when decideability of some statement fails in a theory with a unique model. Or is there another argument for your claim?
Forgive me for spamming questions, I just try to understand how these things fit together. But maybe we should just stick to first order, since everything else is too weird anyway.
Somewhat oddly, this is actually technically correct - a "sufficiently rich" system is one that can distinguish true versus false statements of Peano Arithmetic, and a inconsistent system, by the principle of explosion, can prove any statement, so cannot distinguish statements of Peano Arithmetic (eg, 1+1=2 and 1+1=3 are both provable), and hence is not sufficiently rich.
That's a somewhat unintuitive way of looking at it in practice, though.
In the end, Gödel is actually giving us a choice: Either accept incompleteness or accept inconsistency. Of course it's true that historically incompleteness has been perceived as the only viable choice, but at least a few paraconsistent logicians like Graham Priest have argued for (non-explosive) inconsistency instead.
Where are you getting this from?
But I've yet to figure out if the book actually has a specific thesis. I know it's all about the power of interpretation and the way in which interpreting a formal system as self-referencing has the effect of completely blowing up the intended design of that system. But can someone sum up how this connects to consciousness beyond the obvious way in which consciousness plays the role of the interpreter? Or is that itself the thesis of the book? Or maybe that consciousness is the originator of self-reference?
Bottom line is that I think the thesis can be summed up in a few sentences, but apparently it takes a whole 800 page book to get to the point. Would love other people's thoughts on what that point is ultimately. By the way, as salty as I sound about it, I love GEB. So don't get me wrong :).
Few people get it like that and I think it's a problem with the book. It's taken as more of a funny and poignant popular science dive into some aspects of logic and computability. Hofstadter expands on "I Am a Strange Loop" which I think you should read if you are interested into what was his take with GEB.
It is for sure reaching, but his core conceit is about pattern recognition and emergent behavior and he throws everything he's got at the wall there through the eyes of his interests because he KNOWS there is something there, even if Godel could describe it in a 19 page paper decades before.
That some of this is obvious in hindsight almost 50 years later does not discredit the bizarrely singular nature of his effort, maybe the same way Seinfeld seems corny now.
Seinfeld holds up brilliantly.
"GEB is a very personal attempt to say how it is that animate beings can come out of inanimate matter. What is a self, and how can a self come out of stuff that is as selfless as a stone or a puddle?"
Godel, quines, the phonograph ship of theseus stuff, Bach harmonies, all the self-aware dialogues. He's looking for the vital spark, the thing that makes us greater than the sum of parts. But he can't capture the essence (who can?), and settles for running around the outskirts.
p.s. GEB is ultimately arguing that mind arises from complexity in structure and algorithm. ENM counters that by noting (apparent) non-deterministic aspects of consciousness, proposing quantum effects as a possible source of consciousness. (That was my overall take on the these two books.)
p.s.s. What is not convincing about GEB's argument is the lack of a reasonable explanation for meaning. What it does argue well is how meaning unfolds (self-reference, recursion, composition) but just as Cellular Automatas (CAs) deceptively look like natural phenomena but entirely ignore the extra-CA substrate that simulates the CA world, GEB too ignores the missing mind that is supposed to traverse these layers of abstraction based on self-reference, recursion, and composition. So it is ultimately a purely aesthetic argument (even though the machinery of logic and mathematics are thrown in,) just like certain branches of modern theoretical physics.
Yet Godel's theorem is entirely formal and understandable by a computer, which can draw the same conclusions about any such statement as a human.
"My belief is that the explanations of 'emergent' phenomena in our brains--for instance, ideas, hopes, images, analogies, and finally consciousness and free will--are based on a kind of Strange Loop, an interaction between levels in which the top level reaches back down towards the bottom level and influences it, while at the same time being itself determined by the bottom level. ...The self comes into being at the moment is has the power to reflect itself."
The fundamentals of CS (strings, automata, graphs) are elementary building blocks. This is by design. You can apply them to almost anything. Almost everything "is a graph", or "recursion" if you formulate them to be so.
This is a deep philosophical insight - it rhymes with the Buddhist idea of no-self - the problems that arise because we hold onto a false sense of self.
Its as much a weird work of nerd art than a manual, like _whys guide to ruby_.
I guess I'm just too much of an applied engineering kinda guy.
When I'm more focused on the here and now, I think I would find it a chore to get through. When I feel like I have less on my plate and looking for "interesting observations" it would work really well.
But at the same time, when it really gets going I can't help but being swept away and entertained by the unfiltered enthusiasm he displays and the work he has done to put it to paper. It's like a conversation with a very exhausting but at the same time very interesting person. It's like trying to put that tingling in the back of the brain when some kind of insight looms in words.
I feel like it's probably best read as a curious psychograph of a nerd in the seventies who is deeply interested in theoretical computer science and its implications on awareness and (artificial) intelligence. Not as a scientific non-fiction-book which is trying to educate the reader or trying to make a strong argument for something
The second half is weird too, because it describes the academic work he does in the future, but is a stab in the dark compared to the documentarian aspect of the first half.
Think of it like forming a hypothesis:
GEB: This is what we can see.
EGB: This is what I think it means.
Every generation or so a book comes along that, in retrospect, seems almost clairvoyant. This is one of those books.
Part 1 - math proof that a formal system when evaluated by an agent "breaking out of the system" can use that system to prove itself inconsistent. ("f* up" the internal logic of that system by feeding it into itself)
Part 2 - ironically, to be conscious you have to have an awareness of your own system (a "strange loop") but you'll never be able to understand yourself as proven in part 1.
That's precisely what it is. And what makes it valuable too: such works of blatant self-indulgence are not very common in mathematics (at least, while remaining enjoyable to read).
> than a profound intellectual achievement in itself
Sure. I would call the book an artistic achievement in the intellectual realm: it talks about ideas, but is art. To put it another way: everyone knows "mathematics is not a spectator sport" — this is usually said in the sense of "you can't learn mathematics unless you do it yourself", which is true enough, but it's also a loss in the sense that there's no equivalent of sitting back and watching a game, watching someone play with ideas and do mathematics. This book comes close. How good the quality of the game is, and whether you'll learn anything from watching the game, is debatable (probably not much), but the experience itself is unusual, which makes the existence of this book a happy circumstance.
Hofstadter's actual thesis/intent with the book, and what he has to say about Gödel numbering (note: the book goes over Gödel's theorem and some implications, but Hofstadter's real interest is less in Gödel's theorem than in the proof technique—Gödel numbering—that Gödel used, because he is tickled by the self-reference), or Escher or Bach for that matter, are not really what make the book most worthwhile. What matters is the marriage of form and content, the perfect puns that seem to be set up several hundred pages apart, the way you'd be reading another dialogue before suddenly realizing that the lines are spelling out an acrostic, the satisfaction of things working themselves out perfectly (even the puns) (a feeling similar to the satisfying tying-together of the plots of certain novels, some of Wodehouse's included), and so on.
Very few authors can write about technical topics in a way their personality shines through. (Knuth is another IMO, except with the personality staying in second place to the technical content, while with Hofstadter it's the other way around!)
(Or non-technical writing: did you know about Hofstadter's translation of Eugene Onegin? He learnt Russian just to be able to read and translate that poem (see https://archive.nytimes.com/www.nytimes.com/books/97/07/20/r... https://www.tralalit.de/en/2018/07/25/a-tale-of-two-or-so-tr... https://www.york.ac.uk/depts/maths/histstat/pml1/onegin/ for the story, and https://archive.nytimes.com/www.nytimes.com/books/99/09/12/r... for the very negative NYT review) and the result is amazing for exactly the reasons the reviewer complains about: yes it can't stand alone and it's best read alongside a "real" translation say Falen's, but throughout you can see Hofstadter having fun with his word games, and somehow his personality shines through even in just the way he's choosing to translate Pushkin; it's so amazing how he manages to have so many things just work out. “Till his creditors cast their nyet's”)
The original review post focuses on Gödel, but you cannot truly review GEB without Escher and Bach. Yes, self-referentiality is a key theme of Hofstadter, but also recursion, symmetry and the foreground/background ambiguity in visual arts and music, after all the whole book is essentially a recursive, self-similiar poem (alternating chapters with prose and dialogues, one dialogue has palindrome form).
Despite that I disagree about his fundamental view of AI, namely that AI is an emergent epiphenomenon, it is still the most-read and most-appreciated book ever for me. I read it first in German and then in English, and given the playfulness with form versus function/content, it is also the best-translated work that I have ever encountered. I'm grateful that I bought it when I was sixteen, simply because it was mentioned in a computer magazine and in a completely different context, so I got curious. I know many people that deeply appreciate it, or that went into AI careers because of it.
Anyone read the Phenomenology of Spirit and notice the same ideas?
Can you elaborate? I've had some friends talk about Hegel with me but I never understood much of it
Hegel and Hofstadter seem to be trying to find "spirit". To keep it simple, spirit can be defined by the "I" aka self-consciousness (but it could also be an ant colony, the Church, the State, the biosphere, etc... all the way up to "God").
Hegel tries to notice the becoming of the notion of the Concept (which could be noticed as the basic thought-tool) into (and through) the notion of the "I".
Hofstadter is trying to do something similar but instead of the Concept as the base (or starting point), he uses the Axiom. Hofstadter then tries to discover if "I" (or "spirit") can emerge from a system of axioms.
Please Note: In order to keep it short I gloss over about 1500 pages of Hegel's & Hofstadter's works.
I have a linkdump on this https://github.com/adamnemecek/adjoint
I also have a discord https://discord.gg/mr9TAhpyBW
So x is a fixed point of a function f if f(x) = x. This can be generalized to various kinds of things in mathematics, but as an algebraic example:
f(x) = x^2
f(1) = 1^2 = 1
f(0) = 0^2 = 0
If you have two rulers and and compress one, then you when you lay them together there will be one point which is exactly at the same point.
Likewise, if you have two paper chessboards, and scrunch one up, there will be at least one square* which is exactly over the top of the intact one.
This also gives an intuition for the difficulty of actually finding the fixed point, even though you know there is one.
* more precisely not only a square, but an exact, dimensionless point. (afaik)
Just read the proofs. They're not that difficult. Their consequences are wonderful. Enjoy brilliance
Can we formalize what "stepping outside" means? How can we be sure we have done it correctly? When we step "outside" and do reasoning, are we doing it following the formal logic? Which logic? Are there rules as to what kind of inferences are NOT allowed when reasoning "outside of the system"?
In other words when we "reason outside of the formal system" we are using English. Can we trust the logic of English is "correct"?
Can a formal system really do self-reference? It's just strings. WE can give an interpretation to formal sentences but when we interpret a given sentence as referring to "itself" doesn't it really refers to the interpretation, not to "itself" but its interpretation?
When natural language refers to the formal system doesn't it really just refer to what the formal system "means" in the natural language, not to the formal system itself as such?
> WE can give an interpretation to formal sentences but when we interpret a given sentence as referring to "itself" doesn't it really refers to the interpretation, not to "itself" but its interpretation?
Indeed it can't - see e.g. Löb's theorem. A formal system can manipulate a set of rules that might happen to be its own rules, but it can never "know" that those are in fact its own rules. Which makes sense when you think about it.
So the term "unprovable" should probably best be always qualified with: in which system.
There are facts about systems which the system itself can not prove. Statements like that can be expressed in some formalized subset of English and proven following the rules of that formalized English. What is that "formalized English" that we use in our reasoning? Or does it not matter?
Again, the normal, routine thing is to just use a slightly larger formal system to work with them. But it really doesn't matter.
> Getting hit on the head
Well - there are recorded cases of brain damage unlocking abilities and aptitudes (albeit with great rarity). So probably not recommended.
But rest assured, psychedelics are less painful, cause less damage and have a higher probability of beneficial effects.
If you think drinking alcohol or taking cocaine has the same mental effects as psychedelics, then, I guess all I can say is don't talk about what you don't know.
Hofstadter doesn't own thinking about consciousness, thinking deeply, or thinking philosophically. As a matter of fact these sorts of things, to many people - many more than who have even heard of Hofstadter - associate them with psychedelics. So it's not unreasonable, it's not an insult, to think Hofstadter going pretty deeply into questions of meaning, putting mind-bending Escher works in his book, referring to the person that wrote GEB as different from Hofstadter himself, etc... is psychedelic. These are things that most people find eccentric, and what psychedelic users find psychedelic.
The only way I could find insult in such comparisons is if I felt psychedelics themselves were morally inferior or insulting. I would examine that.
So from my point of view, it devolved into flamebait the moment you said comparing GEB to psychedelics is insulting. Because to me, it makes sense on a lot of personal levels.
As I've said in another comment. If you think drinking alcohol and taking psychedelics have similar mental effects. Then I suggest you stop talking about what you don't know.
I did not struggle with 'looking behind the curtain' while reading GEB, nor did I find the concepts silly - I was rapt with curiosity the whole reading and came to many profound conclusions about the book and it's ideas - and I was not on LSD. Whatever helps you personally understand the world in a more meaningful light is wonderful, but it certainly was not needed for me and I doubt (based on what I know about the author) it was needed for him when writing this book.
What I find insulting is saying that calling GEB trippy is an insult. It's genuinely trippy and there's nothing wrong with that.
Because it is. And your trying to move the goalposts after the fact is just digging that hole deeper.
The suggestion that Hofstadter has tripped isn't insulting because GEB is a trippy book. Even if it weren't it wouldn't be insulting, only confusing, unless you think that trippy is a bad quality. Examine that.
GEB is whimsical, fun and mind-bending. If it were written any other way it wouldn't be trippy. There is nothing wrong with that.
At risk of asking a leading question: is it some idea that deep truth should come from reason rather than the noise and chaos of psychedelics? (I don't do them myself, but hold them in very high regard as a force in the world)
It's true that many people feel a deep connection to and understanding of consciousness during a psychedelics trip. Did they actually gain any knowledge about how their own brains work in the same way that a neuroscientist or ML expert might understand it? I contend that they did not, even if they feel that they did.
In a similar way a drunk might suddenly feel that he has the ability to dance. That's hardly a replacement for salsa classes.
Of course there are exceptions. Many brilliant people take psychedelics and credit them with breakthroughs in their fields, that can't be ignored. But you'll notice that its always _in their field_, ie they were already thinking about a problem and psychedelics offered them a new perspective.
i might feel a bit differently, but I think I hear where you're coming from: that it does a disservice to great people when we flippantly imply their greatness originates from simplistic things outside themselves (if I'm still way off, feel free to correct, but pls don't feel obligated to engage :) )
There are several famous mathematicians known for their avid use of amphetamines.
You think this is saying drugs sometimes lead to commitment? It sure reads like an absolute statement to me.
His 2007 book, "I Am A Strange Loop" is a bit of an answer to this. It's mostly the same ideas in GEB about the self and strange loops, but without all the meandering and interludes. It's been a while but I remember it being less logic/mathy than GEB. I did enjoy it.
But the meandering and the interludes are part of the enjoyment of GEB. So if you're looking for another book to capture that spirit of GEB, I'm drawing a blank, hopefully someone else knows one.
Barely related recommendation, but I remember reading "A Brief History of Time" around that age. Very doable for a young teen if they have a passion for that sort of topic.
Ah - interesting. It took me several readings to tease-out what I thought the central theme was. As it happens, it led to a position that I objected to (and still object to). So discovering that central theme brought the book down a peg or two in my estimation.
I still think it's one of the most entertaining books of popsci that I've ever opened.
+1
Penrose's The Emperor's New Mind Timothy Ferris Coming of Age in the Milky Way and the Zen and the Art of Motorcycle Maintenance, previously mentioned.
Maybe not the most current thinking (but perfectly fine as an introduction), but the writing is more accessible than GED.
(Chrome on a Chromebook). Gosh, I yearn for the times when people used HTML to create web sites ...
But then again I tried to read it when I was in high school.
So maybe I understand more in older age.
Interesting and seemingly not widely known is that Hofstadter's "Fluid Concepts and Creative Analogies" was the first book sold on Amazon.com.
Short take...our reality is a mesh between two different math desc, one is micro and one is not. The only problem is we still have no idea how nature bridges between those two different mathematic systems and that leads us down some very interesting rabbit holes:
1. String Theory and multiple worlds, except they are on micro basis, i.e. you cannot see them even with an electron microscope.
2. Tim travel at electron part level but not atom level.
among other interesting things.