The End of Mathematical Proofs by Humans?

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."

Comments

Critics Reaction...

[Suhas]

...From TFA if a computer is used to make this reduction, then the number of small, obvious steps can be in the hundreds of thousands--impractical even for the most diligent mathematician to check by hand. Critics of computer-aided proof claim that this impracticability means that such proofs are inherently flawed.

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...

[damieng]

Yes, thats right.

If you can't independently examine and verify your "proof" then how can it be considered proof of anything?

Re:Critics Reaction...

[Zork the Almighty]

Whether or not something is a proof is entirely our distinction to make. We choose the axioms on which the proof is based. To paraphrase Bill Klem (a famous umpire): when asked whether a pitch was a ball or a strike, "It isn't anything until I call it".

Re:Critics Reaction...

[spagetti_code]

Many proofs have been thought to have been found, only to be proven wrong years later. And that was only after years of study my mathematicians of the time. For example, from the article: this particular puzzle was twice 'solved' only to be 'unsolved' 11 years later. Consider also Wilkes proof for fermats conjecture - it was proven wrong and had to be redone.

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...

[A beautiful mind]

Um. A mathematical proof is just a sequence. It doesn't have to be accepted to be true. According to the axiomatic method of proving ,you just need these things:

• 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-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...

[nikitad]

If mathematical proofs were "nothing but a manner of convincing someone", how could it be verifiable by a computer?

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...

[g1t>>v]

Yes but a "correct mathematical proof" only establishes truth relative to the axiom system used. Because how will you ever prove that the axioms are true? (That's why they're called axioms---you cannot prove them and just have to assume the're "true".) In other words, there's no such thing as "absolute truth". (This is what Hilbert meant when he defined mathematics as "Mathematics is that subject in which we do not know what we are talking about, nor whether what we say is true.")

Re:Critics Reaction...

[dissy]

> By programming my computer to independently examine and verify the proof. Done
> 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
and the bug was reported on October 30, 1994

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...

[g1t>>v]

In general, it is impossible to prove that such a 'formal logic validator' is correct since it is not possible to prove that an axiom system is correct inside that axiom system (one of Goedel's theorems). So if you would find a proof that your validator is correct, you'd have used reasoning techniques outside the logic of your validator, and do you believe those? (If so, why didn't you include them in the validator, since now your validator clearly does not support a reasoning technique you considered valid in the first place!)

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...

[Len Budney]

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.

Creativity

[Daxx_61]

Much of mathematics isn't just grunt power, there is also a lot of creative work going on there. Without humans to drive the computers doing the work in the right directions, it could take a long time before a computer would be able to get its proof - it simply doesn't know what it is looking for.

I for one welcome our new robotic theorum proving overlords.

Re:Creativity

[R.D.Olivaw]

especially when we already know the answer it's looking for. 42

Re:Creativity

[Zork the Almighty]

Without humans to drive the computers doing the work in the right directions, it could take a long time before a computer would be able to get its proof - it simply doesn't know what it is looking for.

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...

[John Allsup]

Short, sweet, beautiful proofs of interesting and useful theorems, I would welcome them to do so with open arms.

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...

[Hideyoshi]

Computers can separate wheat from chaff. That's what AI is all about.

Which explains the glorious successes that subject has enjoyed in the last few decades ...

Here's a good theorem prover

[carnivore302]

In the past, I've used the HOL Theorem Prover. It's a nice toy to play with if want to get started in this area.

The best math is always elegant.

[Pants75]

That's what my math teacher always said, back in school. I'm sure he would extend that statement to include proofs, given that they are also just math.

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.

[PaschalNee]

For every complex problem there is an answer that is clear, simple, and wrong. H. L. Mencken

Re:The best math is always elegant.

[kisak]

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.

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.

Computer _aided_ proofs

[irexe]

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!).

No progress without understanding

[Freaky Spook]

I remember how much I hated learning alegebra, trig, calculus etc & how much the theory sucked, I never saw any point to it & loved it when I discovered my TI-83 could do pretty much everything.

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.

Re:Science by AI

[FleaPlus]

This reminds me of a Nature paper from last year:

Functional genomic hypothesis generation and experimentation by a robot scientist

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."

Re:Science by AI

[Zork the Almighty]

Well at least in regards to math, I stongly doubt that this will ever be the case. Mathematics is developed over decades and centuries. With a few notable exceptions, it doesn't just fall out of the sky in textbook form. Most areas of math started out as a giagantic mess (ex; calculus, linear algebra, even geometry), and it has taken the work of countless researchers, authors, and teachers to distill and refine it. This process will continue, and it is inevitable that the subjects which baffle us today will be hammered out and taught to grade school students eventually. Well developed theory makes mathematics easier, and this in turn fuels new discoveries.

Blowhard critics could use a logic course...

[geekpuppySEA]

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.

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...

[mcc]

One could argue that if a proof ever contains repetitive elements, then this is a bad thing and you want to be adding some sort of abstraction to the proof anyway.

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.

Re:Consider the source

[Trogre]

What does The Economist know? It's a right-wing rag.

What does Slashdot know? It's a left-wing rag.

Re:Consider the source

[sien]

Umm. So does this mean right wingers can't do maths?

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.

A proof needs to be intuitive

[nigham]

If you take a grad school AI course, they'll make you do proofs the way a computer does it... maybe using propositional logic. The idea is to break up the problem into a set of statements that looks quite ridiculous (e.g. NOT engine AND train AND NOT moving), and then taking pairs of these statements and mixing and matching. The result is that you determine your sequence steps by simple trial and error or by trying to combine the propositional symbols (AND, NOT etc) and the variables (train etc). Once you generate a proof, its just a list of such statements which evaluates to a FALSE or a TRUE value but if you want to understand the proof, its hopeless.

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.

Re:Seems simpler to prove proffs-by-computer

[rxmd]

Three major obstacles with this approach (which has been tried BTW):

• 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.

"peer" review

[lucason]

How can the review of proof generated by computer by a human be considered "peer" review?

Why not have it verrified by other computers?

Re:Seems simpler to prove proffs-by-computer

[darkov]

I agree with (1), I believe Godel had a hand in that one.

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.

Re:BTW, as a geek I want to know

[Maljin Jolt]

A functional Haskell language is the tool of the day. However, there is nothing in Haskell that could not be done in Prolog with proper problem domain language defined by grammar. H. is just fancy looking, out of the box. Unlike Lisp.

reverse :: [a] -> [a]
reverse = foldl (flip (:)) []

applied to a finite list of any type, returns a list of the same elements in reverse order.

Re:Science by AI

[rookworm]

I concur. Math will always be about insight. The best math is simple and shows why the result is true. Most mathemeticians are unsatisfied by the four-colour proof because it does not satisfy these two conditions. Even if computers are eventually able to discover such proofs, mathematicians will still have to ask the computers to search for them. We must remember that problems like solving certain differential equations used to be difficult and involved, but now thanks to computers, we don't have to worry about them as much. The same will apply for very specialized results. The big theorems will still be up to humans to prove. Think of computer- assisted math as a kind of spellchecker or Googe suggest. Computers replacing mathematicians completely is about as far-off as computers replacing poets or historians.

Re:Science by AI

[Cyberax]

Well, Economist should learn some REAL math. The first thing they should learn is math logic.

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 for details).

For example, it's impossible to find answer to CH (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.

Issac Azimov story

[Conspiracy_Of_Doves]

I recall a story I once read by Issac Azimov about a future culture where all knowledge of mathematics has been lost to humans, who have to rely on computers and calulators to do even the simplest math problems (older computers make the new computers and humans are left completly out of the process).

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?

Re:Science by AI

[Rune Berge]

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.

Metaphor

[Lifewish]

As a maths degree student I can confirm that a very large portion of mathematics is devoted to finding new metaphors and angles of attack for a given situation.

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

[tgibbs]

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).

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?