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Are Computers Ready to Create Mathematical Proofs?
DoraLives writes "Interesting article in the New York Times regarding the quandary mathematicians are now finding themselves in. In a lovely irony reminiscent of the torture, in days of yore, that students were put through when it came to using, or not using, newfangled calculators in class, the Big Guys are now wrestling with a very similar issue regarding computers: 'Can we trust the darned things?' 'Can we know what we know?' Fascinating stuff."
> 'Can we know what we know?' Fascinating stuff.
Reminds me of Rumsfeld... "Reports that say that something hasn't happened are always interesting to me, because as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns -- the ones we don't know we don't know."
The headline does a slight disservice in describing the article that way: Whether or not computers can create proofs isn't an issue. The problem comes when the resulting proof is too involved to be verified by a human, and so the computer's work has to be trusted.
Depends whether it's a Pentium with an FDIV bug, I imagine...
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and there are already programs out that help with this. Here's one for example...
Computers are a human creation...it's not a matter of whether we can trust the computer, but rather a matter of can we trust that the people who built the computer and coded the software it runs knew what they were doing and didn't make any errors. Computers can only do what we tell them to...so really it was humans who indirectly made the proofs by producing a system capable of doing so. All it really boils down to is whether the folks who made the system and it's software knew what they were doing or not and whether they made any errors or not.
It's not hard to *see*. It's hard to *prove*. Very little of mathematics is consumed with proving deep, mystical statements that no one would ever anticipate to be true. Much (maybe most) of mathematics is built around proving (relatively) obvious things. Why bother? because sometimes, relatively obvious things turn out to be false, and there's no way to know that they won't until you've prooved them true. In general, showing that discrete (or semi-discrete) phenomena are optimal is fairly tricky; you can't just appeal to calculus to optimize some function. You often have to somehow break the search space up into a bunch of disjunct cases that span all the possibilities, and be able to prove that they span all possibilities. Then, if you're lucky, you can use some kind of calculus-type argument on the continuous spaces you're left with.
Theorem provers have been around for a long time. A net search should turn up a ton of hits. The key is to implement a system that can be verified by hand, and then build on it.
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20th century mathematics has seen some pretty amazing things, but at the same time, there are very real questions as to what constitutes "proof" any more.
consider this: the hypothesis of the famous Riemann Zeta problem has been tested for trillions of different solutions, and it has held true in every case. (If you want an explanation of the Zeta problem, look elsewhere, I don't have the time)
Now that means that it's *probably* true, but nobody accepts that as mathematical proof.
On the other hand, the classification problem for finite simple groups has been rigorously solved, but the collected proof (done in bits by hundreds of mathematicians working over 30 years) is tens of thousands of pages in many different journals. given the standards of review, it is a virtual certainty that there is an error somewhere in there that hasn't been found. So, again, the solution to this problem is *probably* right, but it has been accepted as solved.
What's the difference between these two cases really? What's the difference between these and relying on computer proofs that are, again, *probably* right?
In this light, the math of the late 19th century and early 20th century was something of a golden age, modern standards of logical rigor were in place, but the big breakthroughs were still using elementary enough techniques that the proofs could be written in only a few pages, and the majority of mathematically literate readers could be expected to follow along. These days proofs run in the hundreds of pages and only a handful of hyper-specialized readers can be expected to understand, much less review them.
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Automated theorem provers have been around for a long time, if you can express your thoughts using first order logic. Here's a program from 1986... lisp code
I don't see what the big deal is; not only is the general problem of proving mathematical statements undecidable (even without considering Godel's theorem) but even solvable problems require a lot of human intervention to get solved. Most problems (ie - examples out of math textbooks) aren't going to come up with a proof in any reasonable amount of time by simply dropping it into a theorem prover and pushing "go".
"Automated theorem proving" is less an automatic process (like you'd get with an automated production line) and more of a mechanical assistance to the job (like using a fork-lift to move heavy things faster than you could by hand).
It not only takes work to convert a problem into a good representation, but then you have to structure the problem statement in such a way that a theorem prover can make optimal use of it. Often times, you're forced to, upon following the output, prove lemmas (sub-proofs).
Then, when you finally get a proof, you get the joy of trying to simplify it to something that -can- be understood by a person; again, this is part of the process that can't really be automated well.
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Change the title to: "Are Computers Able to Verify Mathematical Proofs Beyond All Doubt?"
The answer is left as an exercise for the reader.
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Though Dr. Hoare danced around that question a little, presumably that aspect of the project would have to be done by hand, a monumental task to say the least.
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The obvious solution is to have the computer create a new proof that shows that the algorithm it used to create the original proof is, in fact correct.
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as a graduate in the fields of mathematics, i spent a large portion of my five undergraduate years doing proofs. there are a great many ways to prove things, sometimes applicable sometimes not. (e.g. using inductive proofs for numeric theorems is all well and good but completely useless for any sort of ring-theory or spatial proofs)
there are also several levels of depth for proofs, ranging from "i've found a counter example so i can write the whole thing off as garbage" to "i have exhaustively and rigorously proved this starting with the basic axioms of number theory and worked my way on up"
the latter is really the only acceptable way to prove anything seriously. sometimes when you are reworking an already- done proof to illustrate a point, other mathematicians will allow a bit of latitude when it comes to cutting corners, but for a proof as far-reaching as the one in the article, i would only be interested in a "rigorous" proof, that is, one that started with the foundational tenets of mathematics and combined those to form and prove other postulates, etc. very much a form of abstraction, not unlike large development projects.
the problem arises when one (or several) humans have to be able to objectively check the whole thing. to use my prior example of a large development project, no one developer at microsoft understand the whole of windows. it's too big for a single human to understand. each developer knows what he needs to do to complete his part, and so on and so forth.
traditionally, for proofs, a single mathematician (or a small group) would hammer out the whole proof, so the level of complexity remained at a human-understandable level. (even if tedious) my concern, as a mathematician, with using an automated solution would be the rapidly growing order of complexity needed to properly back up increasingly complex proofs. as stated in the article, it's like trying to proofread a phonebook. (only, you must also consider that for every element of the proof (a particular listing) there are several branching layers of complexity (fundamental, underlying proofs many layers deep) underneath. this gets more complicated in an exponential fashion) obviously this approach will only remain human-checkable for relatively small problems. (programmers: think of some horrible nondeterministic- polynomial problem like the "traveling salesman" problem. systems, like humans, are a finite resource, but if you increase the size of the problem, the complexity will quickly grow far beyond your ability to compute. large proofs suffer from the same difficulties, if not quite as concrete and pronounced as NP algorithms)
in closing, i would have to agree that proofs, no matter the effort and computing time put into them, really should not be viewed as being as rigorous as those provable "by hand" and human- understandable, even if the computer has arrived at a satisfactory conclusion, because we have no way of KNOWING if the computer has built up the proof correctly, except that it says it has.
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I think the "as we know it" is the key word here. It creates a new juggling act that teachers have to deal with. On one hand, calculators are extremely useful for the things at which humans are error prone. If students can use calculators or Mathematica or something, they can check their arithmetic much more reliably, which is great for isolating problems with learning concepts from basic mathematical errors. Granted, some people don't see this as a bonus since it's good to be able to do arithmetic reliably without aid. For smaller numbers I agree, but in the real world people use calculators for arithmetic with numbers that have a lot of sig figs. Making students do this arithmetic by hand is just distracting them from learning the concepts they are supposed to be learning.
On the other hand, many students are prone to using these devices and applications as crutches and try to get away with doing things like using their calculator's implementation of Newton's Method instead of solving the problem themselves.
Some professors have found solutions to this problem, others havent. When I was at college, I think our math department had achieved a pretty good level of harmony with Mathematica - we were expected to do a lot of stuff by hand or in our heads - Gaussian elimination, for example - but in order to make the math seem useful, we were also exptected to be able to solve real-world problems with the stuff we learned. Not contrived "real-world" problems from your high-school textbook, but stuff like interpreting large and dirty scientific datasets where the specific technique we would have to use to solve the problem was something we could figure out, but not something that had been explicitly laid out by the textbook or in lecture. We had to apply the concepts we had learned to figure out the problem, but there was no way we were going to chug out that arithmetic by hand - when was the last time you tried to work on a 16x16 matrix using a pencil and paper? How about a 100x100 one?
I professor showed me the Robbins Algebra proof a while ago. I was going to link here, but first I searched the page for (Score:5, Informative), and there you were :)
Here's an excerpt:
In 1933, E. V. Huntington presented [1,2] the following basis for Boolean algebra:
x + y = y + x. [commutativity]
(x + y) + z = x + (y + z). [associativity]
n(n(x) + y) + n(n(x) + n(y)) = x. [Huntington equation]
Shortly thereafter, Herbert Robbins conjectured that the Huntington equation can be replaced with a simpler one [5]:
n(n(x + y) + n(x + n(y))) = x. [Robbins equation]
Robbins and Huntington could not find a proof. The theorem was proved automatically by EQP, a theorem proving program developed at Argonne National Laboratory.
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once upon a time, only advanced mathematicians knew calculus, but now we learn it in high school. Just wait until warp theory is an entry level college engineering course
Once upon a time, the majority of adult males knew how to trap a rabbit (or similar creature), gut it, skin it, start up a fire, cook it, and eat it.
I don't.
Heck, I couldn't even look up at the sky at night and tell you which way was north.
Once upon a time, most people could.
All I'm saying is that the amount of knowledge and skills the average human being can possess will not increase expontentially over time (barring artificial manipulation). We gain new skills as a population and lose old ones.
Here's a link to the Flyspeck Project, briefly mentioned at the end of the article, which aims to give a formal proof of the theorem.
Some things have to be taken as Axioms. A=A is one of them. All the things you prove that rely on A always being equal to A can be taken as true PROVIDING you accept that axiom. We like to pick axioms that we have a gut feel 'must' be true - but you can do interesting mathematics by denying some of those axioms and see where they take you. The classic geometric case of denying that parallel lines never meet produced a whole range of interesting geometries that don't exactly represent the real world but none-the-less have interesting and useful consequences.
So - when you come across a theorem that you can't prove but are pretty damned certain is true - you COULD choose to simply make it be an axiom. The problem is that the theorems you are subsequently able to come up with are only as reliable as your initial assumption of that axiom.
If your axioms eventually turn out not to match the real world - then all you have is a pile of more or less useless theorems that don't mean anything for the real world.
It's therefore pretty important to stick with a really basic set of axioms to reduce the risk that an axiom might turn out not to be 'true' for the real world and bring down the entire edifice of mathematics along with it.
If we ever found some sense in which A!=A then every single thing we thought we knew about math would be in doubt.
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Most people probably knew the "four quadrants of knowledge" thing, but didn't know they knew (DK). That is, they have enough to put it together, but have probably never put it into words before. Intuitive knowledge is one way of putting it. The bulk of most people's knowledge probably falls into this category, which is fine - language is often overrated as a conduit of knowledge (not that it's not incredibly useful and important, but other means exist and are constantly used).
;)
I don't actually believe particularly firmly in that model, though, because I don't agree with the D-K dichotomy that underlies it. It's your usual classical Greek quadrant, which means it springs from a dual dichotomy, or in this case a dual-aspect single one. Dichotomy (or even one-dimensional spectra) is not the only way to look at things, but it is a dangerously compelling model - that is, when people have been presented with a dichotomy, they typically become unable to consider without it. And the defence of a dichotomy is usually a tautology - I mean it's obvious, isn't it, you either know something or you don't?
Still useful and interesting if you can get it out of your head when needed, though.
I believe its called the fallacy of the false delemma.
He who knows not and knows he knows not is a wise man. He who knows not and knows not he knows not is a fool.