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Achieving Mathematical Proofs Via Computers
eldavojohn writes "A special issue of Notices of the American Mathematical Society (AMS) provides four beautiful articles illustrating formal proof by computation. PhysOrg has a simpler article on these assistant mathematical computer programs and states 'One long-term dream is to have formal proofs of all of the central theorems in mathematics. Thomas Hales, one of the authors writing in the Notices, says that such a collection of proofs would be akin to the sequencing of the mathematical genome.' You may recall a similar quest we discussed."
I honestly think they need to stop teaching the halting problem to freshmen CS majors. They're just too inexperienced to understand that theory and practice are two different things.. so this whole "limits of computation" thing stifles their enthusiasm.
How we know is more important than what we know.
In fact, Godel proved the exact opposite: that you can make a list of all true statements of mathematics. Godel's completeness theorem states that every statement that follows from the axioms is in fact deducible from the axioms in finitely many logical steps. It is thus possible to enumerate all true statements by enumerating all deductions. Godel also has proved an "incompleteness" theorem. That more famous (and less important) result is that there are statements that are true in specific models yet not provable from the axioms. It implies that there is no algorithm to decide whether a given statement is true -- but this has nothing to do with enumerating all true statements. (Yes, I am a mathematician)
Not only that, but people should stop using this as a crutch in general. The journey is worth the effort, even knowing that you can never reach the end. This is why I agreed with Godel, Escher, Bach by Hofstadter and disagreed with The Emporer's New Mind by Penrose.
One of Penrose's conclusions was that any attempt at artificial intelligence is necessarily incomplete, so it won't be possible, while Hofstadter said that it is possible to successively approximate something intelligent, and we can learn a LOT about ourselves in the attempts, and that in itself is worth it.
At least that is one of the many things I got from the two books.
That Gödel's Completeness Theorem for first-order logic--predicates, individuals, quantifiers ("for all," "exists"), and truth connectives (not, or, and).
The incompleteness theorem is about arithmetic: natural numbers defined in terms of 0 and the successor relation, addition and multiplication. For any set of axioms you pick for arithmetic, there are true statements of arithmetic that cannot be proven from those axioms.
I'm not completely sure of how relevant the incompleteness of arithmetic is for what you're saying, but I am sure of this: first-order logic is complete, but the validity of a statement in first-order logic is undecidable. Therefore, you don't need to bring in Gödel's incompleteness theorem for arithmetic to conclude that in many important cases.
Are you adequate?
> Godel of course proved that you can never have a complete list of all true statements in mathematics.
Bzzzt. Wrong.
What Godel proved was that all mathematical thruths cannot be described by a finite axiom system. The statement that there is a single empty set cannot be proved. Similarly continuum = aleph_1 cannot be proved; in both cases one may argue that these are so "obvious" that they need to be accepted as axioms.
Note that continuum = aleph_1 was a strongly accepted conjecture as much as the Riemann hyp today.
I studied set theory and models. It is fascinating that you can have a countable model of set theory! This model is basically built from a countable set with countable many subsets accepted as sets, etc. The rest of the subsets an "outsider" can see are not known inside the model. We denote this model (which is a counatble set for the ousider aka meta theory) by M. The natural numbers are in M, denote it by A. A is known in the model and A is an infinite (countable) set.
The set containing the subsets of A, denoted by P(A) is different depending on whether you look inside or outside the model! Outside it is an uncountable set, of course. In the model world it is Pm(A) and it appears uncountable again, but for the outside observer it is actually the intersection of P(A) and M. Bear with me another minute!
In the model world of M the Pm(A) appears not countable. What does that mean? It means that there is no 1-1 mapping between A and Pm(A). In the outside there is a mapping of course, since both sets are countably infinite. This mapping (a function) is actually a set (everything is a set: function, relation etc.) so a mapping denoted by F between A and Pm(A) exists outside. However F is not an an element in M so it does not exist inside the model.
There is a way to extend the model M into something called M(F) by 'forcing' F to be in M. Basically we add F and everything else that need to be added to still have a valid theory inside M. It is not trivial to do this, it was discovered and proved by P. Cohen an analysis guy (not a set theorist). Set theorists of course run with this ever since. The careful picking of F allows different M(F)-s to be created and that leads to results showing that continuum = aleph_1 is not the only possibility. One can 'force' a bunch of other alephs under the continuum. (What aleph exactly can continuum be is also interesting to set theorists at least.)
Now there is one model called L which consists of the 'constructible' sets and nothing else. In L continuum=aleph_1. There are in fact theorems based on the assumption that V = L, that is V the world consists of only constructible sets.
You may wonder how this L can be defined. I cannot go into the details, but one eye-opening fact though is that you can define Lm inside the above M model!!! This Lm is a countable model of L and in some sense a minimal model of set theory if I recall correctly.
So one may take the position that we want mathematical thruths in L the minimal system. (This resonates with Occam's razor.) Of course note that there is no axiomatic system describing L.
I want to emphasize that if we fix a model then each statement has a truth value; but there exist truths which cannot be proved working inside the model only.
For the usual applied math we could say we assume V=L. (Or any other well defined model for that matter.) Note againg that aleph_1 == continuum in L, because there are no other alephs that could fit "between". The mathematicians inside L see this as this: we cannot prove that aleph_1 == continuum in fact there could be something between, but it is not constructible from what we have.
I hope this helped.
Why should AI be impossible? Surely it must be possible, with the human brain as evidence? One might argue that the brain is merely a biological computer not fully understood. Unless there is a "higher intelligence" giving humans beings thoughts, then AI must be possible. I suppose this is analogous to a Human "telling" a machine what to think, though, and gives rise to the "who created the creator" argument, but that's getting a little offtopic... :)
This comment is somewhat similar to one below for which I apologise, but I wanted to expand on the point slightly
But wouldn't Godel's theorem imply that it'd be impossible to build a flawless, all-encompasing intelligence, not necessarily an imperfect one?
Gödel's theorem's concludes, simplified a bit, that it's impossible to know the truth, the whole truth and nothing but the truth about sufficiently complex math. In this context, sufficiently complex means "anything that includes addition and multiplication of natural numbers".
An AI may be able to prove that sum(range(1, n+1)) == n*(n+1)//2 for all natural numbers n; that is, it may output a string that's a valid proof. Smart ones can, dumb ones can't. Just like humans. Intelligence is what limits you.
But if a set of axioms and inference rules don't allow for a proof of a given theorem T, no matter how smart you are, even if your silicon brain is smarter than all of mankind, monkeykind and birdkind added up, you can't transcend any limitation that's found wholly outside your intelligence. As in this case, where it's the nature of the mathematics you've made that prevents T from being proven, and not your inability to find the proof.
Similarly, if we agree that no evidence or reasoning can prove or disprove the existence of god, then no AI can know whether god exists: it's not your knowledge or ability to reason that limits you, it's the nature of knowledge and reasoning itself.
Looked upon that way, Gödel's theorem doesn't say anything about AI. It says something about the world.
I don't know what a "perfect" intelligence is. One that can solve all NP problems in O(1) time and space? Or s/NP/Recursive/? Or just something that can solve all recursive problems in finite time? To implement that, all we lack is the ability to store unbounded information in a world of finitely many atoms. Can intelligence do something more than Turing machines? Does that mean the answer is no? Or that we need to connect nerves to the PCI bus?