(Left to right, top to bottom: Gödel, Hilbert, Chaitin, Turing.)
Such a heading deserves a longer and more detailed post than that which follows, but I would just like to bring a few topics and a tantalising paragraph to your attention.
In 1931 the mathematician Kurt Gödel showed that there are true statements in mathematics which cannot be proved. Prior to this, mathematicians worked with the hidden assumption that ultimately everything could be mathematized and proved. It was a vital supporting idea to David Hilbert’s plan to completely formalise mathematics: when Gödel kicked the support away, Hilbert’s dream came crashing down. (It is a curious fact that mathematicians essentially still operate as they did before Gödel’s result, a point quite rightly highlighted by Greg Chaitin, who has picked up Gödel’s and the great Alan Turing’s work and run with it.)
An article which today popped into my inbox – an actual physical inbox, I might add (some of you may be curious as to what such a thing is) – contains the paragraph:
[Gödel's] theorem was general and true of any logical framework. As Palle Yourgrau – a philosopher from Brandeis University in the US – explains in A World Without Time, Gödel overturned the idea that the “intuitive” and the “formal” worlds are one and the same . . . Gödel revealed the chasm between knowing and being. (My emphasis)
(Pedro Ferreira, Physics World, December 2005, pp44-45). It seems that human intuition cannot be excised from the scientific process. There is something in thinking which is not in computation. What is it?
Filed under: big picture, mathematics | Tagged: chaitin, godel, hilbert, incompleteness, intuition, logic, math, mathematics, maths, proof, turing
Have you read Roger Penrose’s “The Emperor’s New Brain”? If you haven’t you should. As a physicist Penrose has an interesting take on the difference between thinking and computing. I have to admit I haven’t finished the book. I gave up on it during my first year at university. (When you’re studying maths all day you don’t want to read stuff about the Incompleteness theorem when you come home…)
Godel Escher Bach by Douglas Hofstadter is another book that I’d heartily recommend reading if you are interested in this sort of stuff. Hofstadter’s newer book “I am a strange loop” might also be interesting, though I haven’t read that one yet.
Hi, incompetnce, and thanks for your input, your username, and your toons.
I have Penrose’s book on my shelves awaiting my attention. I’ve been sidetracked by some startlingly good books lately, so it has been languishing. I’ll try to give it a go soon. I also never finished his mammoth “Road to Reality” although it is a truly wonderful book – have you read it? It’s like a condensed maths and physics degree in book form.
Hofstadter’s books are wonderful, I quite agree. Thoroughly engage the three modes of thinking. Have you read his “Fluid Concepts and Creative Analogies“? That’s another one lying forlornly and unthumbed on my shelves.
Road to Reality is one of those books that I bought but subsequently decided that I’d need to set some time aside specifically to read it. It’s hardly light bedtime reading or holiday book material… (its enormous size is a hindrance there.) Similarly Hawking’s “and god created the integers”; can’t imagine sitting down and reading that at the moment…)
I forgot to mention a couple of books which take the opposite view; that there is nothing to thinking but computation. Daniel Dennett is the big one here as far as I’m concerned. He’s written a few books on that sort of theme. (He’s also “done a Dawkins” and written some rather strongly worded books on his atheism. More philosophically sound than Dawkins, but still missing the point I feel…)
I’ve left RtoR sitting on my desk and I actually use it as a reference book for the classes I teach. I don’t have the Hawking book you mention, although I would love to own it, it looks beautiful. His “On the Shoulders of Giants” is fantastic, and a real inspiration.
I have Dennett’s “Consciousness Explained”, ironically lying atop “The Emperor’s New Mind”, and both unread. I find the obnoxiously arrogant and misleading title of the one and the subtle and literary title of the other quite interesting.
I haven’t read Dennett’s “Breaking the Spell”, but I read and enjoyed Dawkins’s “God Delusion” and some of his other work. He is such a wonderful craftsman of the English language; a joy to read. But I agree that he misses the point. Well, he makes a very valid and important point that there is a large number of people who choose to bolster their bigotry with bronze age texts, who revel in ignorance and manipulation, and he quite rightly argues against them. But there’s a phrase concerning babies and bathwater which springs to mind.
I heartily recommend the Hawking book (despite not having read much of it). He has picked some cracking papers. The Riemann one (which I have read) is wonderfully philosophical about maths in parts.
Thanks, incompetnce: it has been duly added to my list of books to borrow from the library (along with 28 others, sheesh).
can someone explain to me onething:
so from godels theorem – the consequences of certain rules applied to a system can never be completely predicted? did i understand it right? so there will always be uncertainity in any system and we have to live with them?
also, would that mean the system reveals about itself to us thats something comprehensible by us humans due to the consciousness?
Hi truthnconsequences, and thanks for your interesting questions.
Godel took the simplest possible system – something that looks just like basic arithmetic – and showed that there are true statements in it which cannot be proved. He showed that all mathematical systems will contain true statements not provable within the systems.
This is not quite the same as saying that we cannot always predict what will happen. Talk of prediction usually means talk of physical systems which we certainly describe in mathematical terms. So it is true that there may be some mathematical statement about the real world which may not be provable, but sources of uncertainty in mathematical physics commonly arise from extreme sensitivity to initial conditions (chaos theory), the inherent unknowability of the very small (quantum theory), and the whole question of why mathematics should be any good at describing the universe in the first place.
Godel’s work, alongside that of Turing, Chaitin, and others, certainly has established that human (or at least, some kind of) cognition and intuition is essential to the discovery of mathematical truth. This apparent necessity for consciousness, alongside a similar apparent requirement in quantum mechanics, is perhaps one of the most intriguing aspects of modern science. I know of at least one scientist who claims that consciousness is more fundamental than the physical universe.
thanks for the response Phil.
1. i personally dont beleive tht math exists independent of physical world. i have a ques regarding one thing you said:
“but sources of uncertainty in mathematical physics commonly arise from extreme sensitivity to initial conditions”
lets take a abstract system (for example a computer – the computations though are dependent on the behaviour of the transistors the behaviour of the transistors themselves are abstracted) (another example – say a formal system is coded in a computer or as an another example US federal law or any nations law) so that its working itself is independent of natural world – and its initital conditions are purely controlled by us humans – even in such a system we will never be able to predict the complete consequences of the rules. wud i be right to assume so?
2. godels theorem (which i havent read in detail but will do it as soon as i get a chance) takes a special place in my life. since my childhood i was always averse of argument – coz i thought any possible outcome in any system should be provable so no argument is necessary. but when i came across godels theorem i felt really upset as well as happy – upset because of his incompleteness results and happy coz something is proved (this way or that way) abt it.
3. one important question (which cud be stupid frm ur perception
) I have is: frm my understanding godel used some kind of self-referential statement. so this incompleteness result is just an exception for self-referential statements? wud there exist really a meaningful statement which cannot be proved?
4. one favorite question of mine – there is materialistic world and nonmaterialistic world. nonmaterialistic world wud include signalling like in neural signalling or any other cell signalling, force-field based interactions as in the planetary bodies, electromagntic waves across the universe and with these waves – within their charecteristics like the wavelength there is some form of information. the information-based interactions appeared only in the biological life and with the humans the degree of this consciousness all of a sudden increased to a large extent. what mechanism in the brain achieves this information-based interactions?
well ……. it gets crazy to keep thinin abt these
Hi, truthnconsequences, and thanks for your questions. Apologies if you had trouble logging in before.
1. I think you’re asking about making predictions of the behaviour of model systems instead of using the model systems to predict the behaviour of the natural world. Any such model system will be based on mathematics which by its nature, Godel showed, contains true but unprovable statements. It might be that the result you want your model system to prove is one of those unprovable statements. But what I meant by “sensitivity to initial conditions” is that small changes in the data you give a model can create huge changes in the answer the model gives, even when the model performs perfectly. This is part of chaos theory.
2. I know what you mean about Godel’s result: it’s a bit of a slap in the face, isn’t it? But there’s no denying it and no use crying over it, we have to be adult and wake up to the beautiful and compelling world it reveals.
3. There are no stupid questions! Godel constructed a particular mathematical object which was true but unprovable within the system. This was enough to show that such statements exist. Constructing “meaningful” unprovable statements, as opposed to “merely” showing they exist, is a different kettle of fish. I don’t know of any. This is the sort of thing constructivists worry about.
4. The relation between what we perceive and measure, and what is “information”, is an intriguing one, and you’re right to highlight it. There aren’t any easy answers – thankfully! Maybe you can provide some headway here. Of course, some of the examples you mention are mediated by physical things like electrons, photons, or the physical nature of space-time, but even then there is a deeper question of what those things are, and whether modelling them as information is more useful in some sense. Good questions!
I believe that what is missing from formal systems but present in intuition is just lack of respect for the rules.
If we find that our current rules aren’t working, then we break them! And if we find they are inconsistent, then we simply ignore the inconsistency. In this way, we find rules that work for whatever we need them for, but which would fail if you took them to a “logical” conclusion.
Let’s say you believe “You can’t trust men”. Then you find one man that’s ok. Then you change the rule to “You can’t trust men except this one”. Or you simple ignore the contradiction and continue to believe mutually exclusive truths. Either response is equally common (the “ignore” probably more common).
So, take a formal system, and add “If there’s a contradiction, either ignore it or rewrite the rules of the system until it works again”. Then you get human logic.