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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."
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.
Quoth the server, "404."
Yes, thats right.
If you can't independently examine and verify your "proof" then how can it be considered proof of anything?
[)amien
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.
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.
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."
- 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.As a state gets corrupt, its laws multiply; the most corrupt states have the most numerous laws. (Tacitus, Annales 3:27)
How can the review of proof generated by computer by a human be considered "peer" review?
Why not have it verrified by other computers?
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.
The toad can't burp - and for some reason can't fart either, so it swells up and eventually explodes. --Anonymous Coward
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.
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.
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.
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.
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.