The End of Mathematical Proofs by Humans? 549
vivin writes "I recall how I did a bunch of Mathematical Proofs when I was in high school. In fact, proofs were an important part of Math according to the CBSE curriculum in Indian Schools. We were taught how to analyze complex problems and then break them down into simple (atomic) steps. It is similar to the derivation of a Physics formula. Proofs form a significant part of what Mathematicians do. However, according to this article from the Economist, it seems that the use of computers to generate proofs is causing mathematicians to 're-examine the foundations of their discipline.' However, critics of computer-aided proofs say that the proofs are hard to verify due to the large number of steps and hence, may be inherently flawed. Defenders of the same point out that there are non computer-aided proofs that are also rather large and unverifiable, like the Classification of Simple Finite Groups. Computer-aided proofs have been instrumental in solving some vexing problems like the Four Color Theorem."
Science by AI (Score:2, Insightful)
Re:Science by AI (Score:5, Interesting)
Functional genomic hypothesis generation and experimentation by a robot scientist [nature.com]
The question of whether it is possible to automate the scientific process is of both great theoretical interest and increasing practical importance because, in many scientific areas, data are being generated much faster than they can be effectively analysed. We describe a physically implemented robotic system that applies techniques from artificial intelligence to carry out cycles of scientific experimentation. The system automatically originates hypotheses to explain observations, devises experiments to test these hypotheses, physically runs the experiments using a laboratory robot, interprets the results to falsify hypotheses inconsistent with the data, and then repeats the cycle. Here we apply the system to the determination of gene function using deletion mutants of yeast (Saccharomyces cerevisiae) and auxotrophic growth experiments. We built and tested a detailed logical model (involving genes, proteins and metabolites) of the aromatic amino acid synthesis pathway. In biological experiments that automatically reconstruct parts of this model, we show that an intelligent experiment selection strategy is competitive with human performance and significantly outperforms, with a cost decrease of 3-fold and 100-fold (respectively), both cheapest and random-experiment selection.
New Scientist also had an article on it: "Robot scientist outperforms humans in lab." [newscientist.com]
Re:Science by AI (Score:5, Insightful)
Re:Science by AI (Score:5, Insightful)
Re:Science by AI (Score:5, Informative)
It has been PROVEN (and it's a well-known fact) that it's impossible to create a Turing machine which will determine if a given expression is true or false (see Incompleteness theorem [wikipedia.org] for details).
For example, it's impossible to find answer to CH [wikipedia.org] (continuum hypotesis) in ZFC (Zermelo-Fraenkel + Choice axiomatics).
In short: some problems can't be solved in existing theories, they require creating a new theories with new axioms. It's non-formalizable process (it's also proven), so no computer can do this.
Re:Science by AI (Score:3, Insightful)
Surely you mean it's impossible to create a turing machine that will determine if all expressions are true or false - ie, that there will always exist an expression that cannot be proved or disproved? I don't see how this prevents a computer proving or disproving a statement where such a proof exists.
Also I don't see how a turing machine is different
Re:Science by AI (Score:3, Insightful)
In the case of the continuum hypothesis, mathematicians are hoping to come up with and axiom that "sounds true" and makes sense that will settle the question. This is Godel's legacy -
Re:Science by AI (Score:3, Interesting)
Axioms don't usually have opposites. That's what makes deciding on good axioms non trivial. The parallel postulate has two possible replacements, leading to two different geometries. Neither is the opposite of the other. You could try to come up with more axioms (through every point not on a line there are exactly three lines parallel to the original line), but they might not lead to a logically
Re:Science by AI (Score:3, Interesting)
This
Re:Science by AI (Score:5, Insightful)
This actually is more about the limitations of logic than the limitations of computers. Indeed, Godel's Incompleteness Theorem has nothing to do with computers--it is a proof that in any system of logic (that meets some very broad criteria) there must exist statements that are true but that cannot be derived from the postulates of the system by any sequence of logical steps. Adding additional axioms does not solve this; there always remain unprovable propositions. This limitation applies to proofs by humans as well as proofs computers. However, the fact that there are some theorems that cannot be proved does not mean that there are not many others that can be.
However, the fact that there are some truths that are literally inaccessible from the postulates certainly suggests that there may be others that are accessible only by a very large number of steps, effectively requiring computers. I wonder if anybody has ever attempted to prove this?
Re:Science by AI (Score:3, Informative)
The continuum hypothesis, due to Cantor, has been shown, partly by Godel, to be consi
Re:Science by AI (Score:4, Informative)
The theorem says that there are either true unprovable things or things that are both provable and provable to be false. An interesting formal system is either incomplete or contradictory (it can be both, but it doesn't have to).
Re:Science by AI (Score:3, Interesting)
What you and others fail to grasp is that computers are evolving rapidly, human brains aren't. Our current computers are still far from having the data processing capability of a human brain.
In rough orders of magnitude, a human brain has 1e11 neurons, with 1e3 synapses each, doing 1e2 operations per second. Considering that a neuron can be emulated by a multiply-add operation, we would need 1e16 such operation
Re:Science by AI (Score:3, Informative)
Re:Science by AI (Score:3, Interesting)
Re:Science by AI (Score:3, Funny)
what about the software (Score:4, Insightful)
Secondly, how fast does software progress ? Suppose we all had computers 60 billion times faster than we do now. What would we do with them ? run SWING based java applications with tolerable responsiveness, play solitaire faster, run doom 5... [although the frame rate might be a bit low] ok... great,
Intelligence and computing power are orthogonal concepts: suppose you communicated with aliens who were a 100 light years away, would they be less intelligent because it too 200 years to get an answer. Anything you can do on todays supercomputers, you can do on pocket calculator [with enought memory].. it just takes longer.
Lastly, in the long run, computers wont outgrow our brains, they will be integrated with our brains.
Re:Science by AI (Score:3, Funny)
On the flip side, according to my calculations, I lost e3 neurons, e1 synapses, and e0.5 operations per second to beer last weekend alone. That almost never happens to a computer.
Re:Science by AI (Score:4, Interesting)
Who's to say that neurons operate in the same way as a computer's multiple-add operations? Another little problem is that you'll need additional programming to tell the computer how to emulate the communication and interaction between neurons. I imagine that this would take far more processing power than we could ever achieve.
We may be able to emulate the parts, but you can't just throw the parts together in a heap and expect it to work. The sum of the parts is far more complicated than the parts themselves.
Re:Science by AI (Score:3, Insightful)
Capability vs reality... (Score:3, Informative)
Kjella
Metaphor (Score:5, Insightful)
This takes a ridiculous amount of pattern recognition skill (which is one area where computers tend to be outperformed by all comers) and the ability to find new ways to abstract data. A computer could possibly come up with an idea like more-than-3-dimensional space on its own, but I'd be very surprised if even the best one could think of something like topology or tensors on its own.
Production of unusual metaphors for things we thought we knew is a major driving force for the most important mathematical developments. It's not something I can see computers managing at any time in the near future.
Re:Science by AI (Score:5, Funny)
Yeah, right. The great AI machine will be delivered in the same week as my flying car. Taking orders now, please form an orderly queue.
According to rumors it will be bundled with Duke Nukem Forever.
Re:Science by AI (Score:3, Informative)
Critics Reaction... (Score:5, Insightful)
So basically what they are saying is that if the proof is too long to be checked, then it is flawed? WTF?
Re:Critics Reaction... (Score:5, Insightful)
If you can't independently examine and verify your "proof" then how can it be considered proof of anything?
Re:Critics Reaction... (Score:5, Informative)
If you can't independently examine and verify your "proof" then how can it be considered proof of anything?
That's easy. Speaking as a PhD mathematician, there's nothing disturbing at all about these computer proofs. They're examples in which a computer was programmed to generate a perfectly standard proof, except that it's extremely long.
Checking the proof is not hard: it suffices to verify that the program emits only correct inferences. That's nothing more than a standard human-generated proof. In addition, a verifier can be coded by someone other than the original author, to check the validity of the inferences generated by the first program. The checker's algorithm can also be verified using a standard human proof, and would be used to confirm that a bug didn't result in an incorrect proof.
Note that Gödel's incompleteness theorem has nothing to do with these programs: they don't generate all possible proofs. They only generate one specific type of proof per program. Each program is easy to verify.
You could call the software correctness proofs "meta-proofs", but that's just being coy. They're perfectly legitimate proofs, and they are sufficient to prove the correctness of proofs generated by the program.
Re:Critics Reaction... (Score:3, Insightful)
Re:Critics Reaction... (Score:3, Insightful)
One should of course ensure that the programm is correct and all as good as possible, but I don't see much difference between a proof verified by a bunch of independently written computer programms and a proof verified by a bunch of humans.
Re:Critics Reaction... (Score:3, Interesting)
For example, if some random student would come and say "I looked at the proof, and I think it is OK", then you wouldn't give it as much weight as if some mathematician who clearly has proven his profound use of logic in several papers goes through the same proof and says "everything is OK." And for at least some time, the computer program would be more like
Re:Critics Reaction... (Score:5, Interesting)
> properly, the instructions for a computer to verify a proof can be a lot simpler
> than verifying the proof itself.
But even multiple computers performing a verify isn't _truly_ a verification.
After all, how long did the Pentium division bug go _unnoticed_???
Looks like the chip was released on March 22, 1993 [wikipedia.org]
and the bug was reported on October 30, 1994 [wikipedia.org]
Over a year and a half worth of time any/all such verifications obtained with the newest intel computers at the time were WRONG.
And any guesses how they even found this bug??
It was a human, not another buggy computer, that had to verify the data.
Yes computers can do things faster, but ever underestimate the power of truly knowing what your doing, which so far, a computer can't grasp at all, let alone do as well as the human mind.
Re:Critics Reaction... (Score:3, Insightful)
Read The Man Who Mistook His Wife [amazon.com]
Re:Critics Reaction... (Score:4, Insightful)
"peer" review (Score:5, Funny)
Why not have it verrified by other computers?
Re:Critics Reaction... (Score:3, Interesting)
The question is if the output of the program is equivalent to the problem you want to prove. Your proof actually consists of verifying that "Program Output is A" is only true if "Statement B is true" is true.
(It's not necessary to prove also the reverse notion. "Statement B is true" doesn't need to implicate that "Program Output is A". Imagine a program that prints out "A" if it is 8p
Re:Critics Reaction... (Score:2)
Re:Critics Reaction... (Score:5, Insightful)
Re:Critics Reaction... (Score:4, Insightful)
I could assert that 2+2=4. You may believe me, but have I really proven it to you? Not yet, so you don't need to believe me. If I instead say that the the cardinality of the union of two disjoint sets, each of cardinality 2, is 4, then I've (sort of) showed you that 2+2=4, assuming you accept my definitions of disjoint sets, set union, and set cardinality (which presumably I've proven elsewhere, or taken from some other source). Now do you believe me that 2+2=4?
I could assert anything. You may or may not know if it's true. A proof is just something to back up my assertion and convince you that I'm right. Hence, if a proof is unintelligable, it's pretty darn worthless.
Re:Critics Reaction... (Score:5, Informative)
- An abc (not THE abc, just one)
- formulas
- derivation rules
An axiom is just a derivation rule which derives from the empty set.Basically, a proof is, according to the axiomatic method, is just a non-infinite sequence of formulas, which can be created by the allowed derivation rules. The whole point is that a proof for A HUMAN, and mathematical proof is different. The axiomatic system is not perfect, either. The whole Hilbert [wikipedia.org]-plan is proven to be impossible to be done, thus it is not possible to prove that there are no contradictions in the axiomatic system
I think the "MUI" axiom system is commonly used to demonstrate how it works, basically. It is too lengthy, and i'm lazy.
Re:Critics Reaction... (Score:5, Informative)
Fully formal mathematical proofs depend on nothing but ability to distinguish characters, to compare text strings, and to perform substring substitution.
To your example (2 + 2 = 4). In formal arithmetics, based on Peano axioms, there is one primary operator, let's call it s:N -> N, and s(n) is interpreted as (n + 1). "2" is _defined_ as s(1). 3 is defined as s(2), and 4 is defined as s(3). So one has to prove that s(1) + s(1) = s(s(s(1))).
By definition of addition (remember, addition is not fundamental notion in the formal arithmetics, it's defined in terms of s-operator), a + s(b) = s(a + b), and a + 1 = s(a), so we have
s(s(1) + 1) = s(s(s(1))),
s(s(s(1))) = s(s(s(1)))
Q.E.D.
So, where proof above depends on anything but mechanically verifiable string manipulations?
P.S., of course mathematical formulae are not strings, but rather trees, but this doesn't change a bit.
Re:Critics Reaction... (Score:4, Informative)
Re:Critics Reaction... (Score:5, Informative)
So the question is - how are we going to prove/disprove a computer program that proves a theorem? Program complexity has meant that all but the most trivial programs cannot be 'proven'.
The solution, it seems to me, is per the article: get the s/w to output a series of steps using formal logic. Any series of formal logic steps should be confirmable by a 'formal logic validator', and that is the only program we need to prove correct. That will be non-trivial, but will then open the floodgates to any hacked up piece of code to be used to generate provable logic.
Re:Critics Reaction... (Score:5, Insightful)
Basically, at a certain point, you just have to "believe" that your axioms, logic, whatever you call it, is consistent. Because to prove it, you'd again need axioms, a logic, etcetera, ad infinitum.
Re:Critics Reaction... (Score:3, Interesting)
We don't know whether this abstract formal logic thing actually has any resemblance to the real world - that is not Goedel's theorem, which assumes logic to be valid and discusses potentially interesting frameworks for mathematics. Whether formal logic 'works' is simply something you have to believe, you cannot argue it,
Another Lomborg case? (Score:2, Interesting)
The Economist is the same magazine that supported Bjørn Lomborg [economist.com], thereby proving their utter incompetence in environmental science. They defended him in spite of clear and very detailed indications [lomborg-errors.dk] from the scientific environment that he was nuts.
I suppose they will know economics when they talk about it, but they demonstrated an inappropriate habit of pontificating on things they don't have a clue about. I for one think they burned a lot of karma.
Re:Another Lomborg case? (Score:4, Interesting)
Creativity (Score:5, Insightful)
I for one welcome our new robotic theorum proving overlords.
Re:Creativity (Score:4, Funny)
Hmpth (Score:2)
Source [bartleby.com]
* A physical alteration thought to occur in living neural tissue in response to stimuli, posited as an explanation for memory.
Um. Yeah. Or something like that.
Re:Creativity (Score:5, Insightful)
To some extent that's true, except in areas where human understanding has reduced mathematical proof to a mechanical process. For example, verifying algebraic identities, or even geometric proofs. A more advanced example is the Risch algorithm for elementary integration. It amounts to a proof that an integral either is or is not expressible in terms of elementary functions. Eventually we come to understand an area to such an extent that we can implement mechanical algorithms and move on. The proper role of the computer is to carry out these algorithms, so that we can use them to discover something else.
If computers could write proofs... (Score:5, Insightful)
As a tool to produce vast quantities of precise logical porridge quickly, computers have no equal in today's world, yet that is not what real mathematical proofs should be about.
Mathematical proofs should show short, clever ways of connecting otherwise disparate concepts that are only obvious in hindsight. This is where computers will always be weaker.
Re:If computers could write proofs... (Score:2, Insightful)
So you're saying that even a theoretical sufficiently advanced computer would be unable to match the logic and creativity of a human being? I think a simple brute force counter written in Mathematica (unlimited integer lengths) whose output was executed by a CPU would prove you wrong.
Computers can separate wheat from chaff. That'
Re:If computers could write proofs... (Score:4, Funny)
Re:If computers could write proofs... (Score:4, Insightful)
clever - yes, short - no (Score:2)
Mathematical proofs should show short, clever ways of connecting otherwise disparate concepts that are only obvious in hindsight. Modern mathematics is very complex, important and useful theorems proven recently have huge and difficalt proofs, built from a lot of steps. Imortant results hardly have a short proofs.
Take a look at the concepts used in the one of the most famous proofs of XX centery : the Feramt last theorem [wikipedia.org]. It's proof followed from the Taniyama-Shimura theorem [wikipedia.org] which establish non-obvi
Re:If computers could write proofs... (Score:2)
Actually, I was officially a math major for about a year (before my transition to the much easier computer science). Some parts of it were pretty simple, but the proofs always threw me
Seems simpler to prove proffs-by-computer (Score:3, Insightful)
To do that.... well, just make sure the program was designed by a correct computer.
Re:Seems simpler to prove proffs-by-computer (Score:5, Informative)
- Logical problems with proofs for correctness. For example, it has been proven that no program can prove itself correct.
- Correctness proofs are hard to do and incredibly tedious. Have you ever tried it? And no, you can't have a program do them, because you'd have to prove this program correct, which sends you right back to square #1.
- You'd have to prove all sorts of other factors correct, including the operating system and hardware your program is running on. This leads to another set of interesting problems, including that "correct hardware" is useful only as a theoretical concept. What's a "correct computer" if there's a small probability that bits will spontaneously flip in memory, for example?
In short: while it might seem elegant to prove the prover, then have everything else proved by this prover, this approach has little value in practice.Re:Seems simpler to prove proffs-by-computer (Score:5, Interesting)
With (2), the program can reduce the tedium of proving the original proof in some cases. That's what computers are good at and are better at than humans. Proving the program may well be much easier. I would think that's why the researchers in the article used computers in the first place. If you program in C++ you will have a problem, but a functional or logic language program is straight-forward to prove (PROLOG programs are essentially the execution of a proof).
With (3) you could show by running it on different hardware and software that the platforms did not affect the result by a huge probability. If you don't like the 'probability' bit, who says there isn't a human trait or gene that causes any human to get a proof wrong? Humans are imperfect too, but if enough of them agree, and they are qualified, then we agree that what they agree on is true. This is the same situation as the potentially flawed platforms problem.
Here's a good theorem prover (Score:5, Informative)
The best math is always elegant. (Score:5, Interesting)
What about Fermats last theorem? Fermat wrote in the margin of his note book that he had a proof, but it was too large to fit there, so he'll write it on the next page. Trouble was, the next page was missing from the book.
The modern proof for FLT took hundreds of pages of dense math and went through some math concepts that AFAIK hadn't even been invented in Fermats time.
What was Fermats proof (if it existed)? It would surely have been far more elegant than the modern version.
That doesn't make the modern version wrong, just less pure, I feel.
The problem with modern computer aided proofs is they allow the proof to become unwieldy and overly verbose, compared to what it would have to be if just a human produced it.
Such is progress I guess.
Re:The best math is always elegant. (Score:2)
Re:The best math is always elegant. (Score:3, Funny)
Re:The best math is always elegant. (Score:5, Insightful)
Re:The best math is always elegant. (Score:5, Informative)
Mathematicians think they know what Fermat thought was the "proof" that he could not fit in the margin, since Fermat used a similar strategy for another problem. Euler was the one who used Fermat's strategy on Fermat's last theorem explicitly, and showed that it did not give a full proof as Fermat had hoped. It might be that Fermat himself tried and then gave up, or that he was happy to have "solved it" and looked for other things to prove.
I think you (and most people) misunderstand the reason Fermat's last theorem has such a central place in math history. But first lets discuss the reason why the problem became so well known; it is because it is such an easy problem to state and to understand, still no one has been able to use "simple" math to prove it. Even Fermat himself thought the problem should be fairly straigth forward to solve, and it has made generations of people with curiosity for math look for proofs and even thinking they have found one. It is also a problem some of the greatest minds of math did not manage to solve. Fermat, Euler, Lagrange, Gauss, Abel, Riemann, etc have all had a try and did not solve it. Which math wanna-be wouldn't want to solve something this group of people did not manage?
Now, even though this has made Fermat's problem something that has created a lot of publicity, the number one reason Fermat's problem has been important, is because of all the beautiful maths that have been discovered by mathematicians trying to solve it. Fermat's theorem in itself is not interesting. It is not like the Riemann hypothesis, which if proven to be false, will make much of modern maths not true or at least force mathematicians to look for new proofs. This is because you can prove much interesting stuff if you assume the Riemann's hypothesis is true, problem is of course, this is not yet proven. If Fermat's theorem was been shown to be not true, that would have been suprising, but would not made large parts of maths collapse.
So, the modern proof of Fermat's theorem is the end of a long journey which has lead to some very deep mathematical knowledge, and in a way, Wiles' proof is much more interesting in its own right than that it also proofs that Fermat was right in his guess. A "simple" proof (watch out when mathematicians use the word simple or trivial) of Fermat's problem would give undeniable bragging rights, since you could say you solved a problem Gauss couldn't solve with the maths Gauss knew. But again, it probably would be more of a huge accomplishment for one person than a huge breakthrough in maths.
A last comment; the reason Wiles' proof is long is not because math is verbose, far from it . It is because Wiles is able to connect what would seem to be two unconnected branches of mathematics, showing that problems in one of the branches can be restated as problems in the other. This is not something you do in a few pages. And the importance of it becomse clear, if you consider that what can be an unsolveable problem in the one branch of maths might be reformulated as a solveable problem in the other. Math is always about trying to find ways to solve something as simply as possible. Not something computers is very good at, so no Abel prize to Big Blue for a while I think.
Re:The best math is always elegant. (Score:3, Informative)
Unfortunately, as Hardy pointed out, that proof assumed that all Quadratic extensions of the rational numbers are Unique Factorisation domains - which isn't true
It seems very likely that Fermat's proof was probably of a very similar sort.
Computer _aided_ proofs (Score:5, Insightful)
Computer proofs, like the graph color proof, are not proofs that are completely generated by a computer. The computer is merely used to brute force a fairly large number of 'special' cases which together account for all cases. The construction of the proofing method is and will remain human work, lest we create AI that matches our own I.In short, they are computer aided proofs only.
Further and more importantly, at this point we do not have and are not likely to have a machine that can prove any provable theorem (and fyi, not all truths in mathematics are provable!).
Re:Computer _aided_ proofs (Score:2, Informative)
Incomputable math theorems (Score:2)
This book basically describes Godel's incompleteness theorem [miskatonic.org] in an entertaining way for a general audience.
Re:Computer _aided_ proofs (Score:3, Insightful)
(and fyi, not all truths in mathematics are provable!)
This is wrong. It is based on a misunderstanding of Godel's proof that has been popularized by various authors.
All truths in mathematics are provable. However, assuming our system of mathematical axioms is consistent, there are some statements that are neither true or false. These statements have not been determined by the axioms. Furthermore, for any computable set of mathematical axioms we choose, there are always some statements which are und
Are computer-aided proofs really proof? (Score:3, Insightful)
No progress without understanding (Score:5, Insightful)
Although I discovered easier ways to do the arithmatic, I still knew the underlying theory of the equations & what the numbers were actually doing, not just what a computer was telling me.
Students should learn this, they are the basic building blocks of a science that dictates pretty much everything on this planet & although they won't have a use for everything they are taught they will have enough knowledge to "problem solve" which is what most of high school maths is designed to do, it trains our brains to think logically & be able to work out complex problems.
How are people going to be able to further phsyics, medicine, biology if they get into their respective tertiary courses without understanding the basic principals of all science & have to learn it all over again??
Or what about when computers just won't work & things have to be done by hand??
Its fair to integrate comuters into maths but not at the expense of the theory that makes us understand how things work, we should not put all our faith in technology just because its the easy thing to do.
Theorem Proving (Score:2, Interesting)
FOL Theorem provers jump through a number of hoops to make the whole bit a little more practical, but realistically speaking, having a computer that just runs through mathematical proofs in the manner that a human does is a long way off.
An interesting thing about the article is that the first proof was done with an FOL prover... it was a long, non-intuitive proof, but an FOL prover has performed that proof.
The second was done with code, a human had to write the program. Th
Elegant proofs (Score:2, Interesting)
unverifiable (Score:3, Interesting)
Nonsense. If people can't verify proofs, then what you need to do is to verify them by computer
I'm not entirely joking here - so long as the verification program is written independantly of the thoerem-proving program, and it can reliably distinguish between valid and invalid logical inferences, what's the problem? It's simple automated testing methods really.
Nothing too new? (Score:2, Informative)
IIRC - back in the early days of AI (1960s), some people in the field thought that in relatively short order computers would be a major soucre of such mathematical proofs. It hasn't happened yet, but that doesn'
Blowhard critics could use a logic course... (Score:4, Insightful)
Because if there's one thing that humans are better at than computers, it's performing large numbers of repeated steps. Flawlessly.
Re:Blowhard critics could use a logic course... (Score:5, Insightful)
Anyway, the problem isn't the ability for the computer to perform flawlessly, the problem is in our ability to accurately specify to the computer what we want it to do. It's the whole "fast working idiot" thing, mechanical perfection isn't much good if we wind up just directing the computer to perfectly, flawlessly do the wrong thing very quickly. We have enough trouble convincing ourselves real-world software is going to do what we wanted it to after it compiles; and in that case we at least have the advantage we can run it and test it to see if it does what we expect. With software-generated proofs, not so much, since the program IS the test.
I think computer aided logic can be useful if we just think of a proof-generating software program as a funny, mechanically verifiable sort of abstraction, but you find yourself making an argument that rests on assuming that a computer program you wrote does what you think it does then this is problematic.
here y'all go again, panicking... (Score:3, Insightful)
It's easy to check a proof. It's hard to come up with a proof. Computers are great at checking proofs - all the program needs to do is verify whether the steps are logically correct or there's a discrepancy. Coming up with a proof, on the other hand, is a very hard task (being NP-complete, unless defined in a certain way) and thus usually requires a human (or sometimes, a lot of humans) to work on the problem.
A computer would not be able to come up with new principles of mathematics in order to tackle a given problem, it would only try to use every trick that has been discovered to the point of creation of the program (of course that doesn't have to be the case, but my point is that human intervention would be required to "teach" the computer about the new concept so that it would try to use it for the proof)
That is not to say that computers are useless in proofs. Obviously, they're often used as assistants in proving something-or-other, but there's also a direction in computer science where your computer would take a program that you wrote in a certain manner, and prove certain properties about it, e.g. that it is not possible to get out of array bounds in your C program...
*yawn*
time to sleep
A proof needs to be intuitive (Score:4, Informative)
I doubt the human proof system will go away completely - even if we can check nasty theorem proofs using computers, we still need humans to sit and explain what they mean.
BTW, as a geek I want to know (Score:4, Interesting)
Re:BTW, as a geek I want to know (Score:5, Informative)
Re:BTW, as a geek I want to know (Score:3, Informative)
I love proofs (Score:3, Interesting)
Issac Azimov story (Score:5, Interesting)
A janitor at a science lab rediscovers the 'ancient knowledge' on his own. The military quickly gets ahold of it and immediatly puts it to use in weapons research, whereapon the janitor promptly takes his own life in shame.
Anyone think there might be a future where humans rely on computers so much that they don't bother learning math at all any more?
Great mathematicians (Score:3, Insightful)
An insight that enabled a five page proof of Fermat's Last Theorem using nothing more than high school algebra would be a major result, even though much more complex proofs exist.
Learn this! (Score:3, Insightful)
Proof generation can only be partially automated. It still requires massive human intervention, from choosing what to prove and then how to go about proving it. When you have computers automatically generating a proof of, say, Godel's Incompleteness theorem, it is actually a computer that was fed an intricately encoded version of of the theorem, along with some form of hint as to how to go about proving the theorem.
Not to mention that there is little use in a computer re-proving something that has already definitively been proven.
The difficulty arrises from the fact that there are absolute limitations to what computers can do. These limitations have been proven many decades ago... BEFORE THE CREATION OF ELECTRONIC COMPUTERS! But to the ignorant, these theoretical limitations do not matter... simply because they do not understand how absolute they are.
So again, learn this, you will time and again hear AI snake oil salesman spouting off crap about automating mathematical research and automating programming, etc. (Both are in a sense equally difficult as proofs can be seen as programs and visa versa.) In reality, at best, automated theorem proving is actually a tool that can be used to help mathematicians and computer scientists to do what they have been doing already. The level of automation varies, but in general is very low.
A human still has to drive the entire process, similar to how a car automates walking, but doesn't automate where to go and how to get there.
My Proof To The Four Color Theorem (Score:3, Interesting)
Imagine that you have a map that has five colors where every color is touching every other color.
It's on a piece of paper beneath a coffee can.
One color must be on the edge. There may be only one color, or there may be many, but at least one will exist.
Move the coffee can until you see that color. Lets say red.
Now imagine taking a marker and covering the rest of the page in red. (Alternatively, stretching the red area around the rest of the page).
Now you have four colors who are all touching one another, and who are all touching the outer red page.
Imagine that you have a tetrahedron. This is a 3D shape which has four sides. Imagine that each side of the tetrahedron has a color. Say, green, white, blue, and yellow.
Imagine that you poke a hole in the tetrahedron and spread it out flat - placing it on the red piece of paper.
No matter where you poke the hole, all four sides will touch one another.
However, you must poke a hole so that all four colors are on the border, i.e. so that they all touch the red paper.
Look a the green side. All of the sides are equivalent.
If you poke a hole in one of the corners, the opposite side will be in the center when you lay it down, and it won't touch the red. It won't work.
If you poke a hole on one of the edges, it's even worse. Two of the opposite sides will be in the center, and neither will touch the red. Again, it won't work.
Worst of all, if you poke a hole directly in the center of the green side. Then all three other sides will be in the center when you lay down the tetrahedron and none of them will touch the red. That won't work either.
Therefore, there's no way to poke a hole such that all four sides will touch the red when the tetrahedron is laid down, so it was impossible to have a map with 5 colors, all of which were touching one another, in the first place.
Came up with this in 1990. Never formalized it mathematically. It may not be a proof, but it makes logical sense to me.
doug@dougdante.com
Re:Godel/Turing/Cohen... (Score:2, Informative)
Why can't human brain functions be copied and improved upon in circuitry, other than the currently prohibitive costs involved, and our failure to accurately determine the exact neuron mapping of anyone's brain. Even the dynamic neuron connections could be pot
Re:Godel/Turing/Cohen... (Score:2, Funny)
Re:Godel/Turing/Cohen... (Score:3, Insightful)
Re:Godel/Turing/Cohen... (Score:3)
Actually, it was never thought to be obvious that parallel lines could never meet. Euclid tried to get rid of the postulate, and mathematicians for centuries after tried to show that the postulate was dependent on the others (the other postulates were never controversial in the same sense). In the qwest for a fundament for the parallel postulate, it was shown that the postulate is independent of the others. Then Gau
Re:Godel/Turing/Cohen... (Score:3, Insightful)
Turing applied Godel's work to Turing machines and showed that there are non-recursive functions. No talk of formal systems here.
I have no idea why you're mentioning Cohen, since his most famous work had to do with showing that there are models in which the Continu
Re:What about feigenbaum constant? (Score:3, Interesting)
Re:Seems the computer is wrong (Score:2)
Re:Seems the computer is wrong (Score:2)
Re:Consider the source (Score:5, Funny)
What does Slashdot know? It's a left-wing rag.
Re:Consider the source (Score:2, Informative)
Re:Consider the source (Score:5, Insightful)
Secondly, the claim that a magazine that opposes the death penalty and supports gay marriage is right-wing rag (which presumably you meant in US terms, is kinda amusing.
The Economist, correctly stated, is a liberal magazine. It supports liberal economics and liberal social policy. Unfortunately the word 'liberal' in the US has been badly distorted.
Re:If there are no proofs (Score:3, Interesting)
Geometry wasn't the most useful of my high school math classes, but it was the most fun, because we all worked together to come up with proofs, showing that those lousy formulas we had to memorize all had their root in Euclid.
Similarly, anyone who wants to go anywhere with calculus should at least try to understand the proof of the existence of limits, at least for a few minutes, because that's what makes calculus something other than voodoo.