Are Computers Ready to Create Mathematical Proofs? 441
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."
Create vs. Verify (Score:5, Informative)
I think so, yes. (Score:3, Informative)
It was a big help with the 4-color map proof... (Score:5, Informative)
Text of the article (Score:2, Informative)
In Math, Computers Don't Lie. Or Do They?
By KENNETH CHANG
Published: April 6, 2004
leading mathematics journal has finally accepted that one of the longest-standing problems in the field -- the most efficient way to pack oranges -- has been conclusively solved.
That is, if you believe a computer.
The answer is what experts -- and grocers -- have long suspected: stacked as a pyramid. That allows each layer of oranges to sit lower, in the hollows of the layer below, and take up less space than if the oranges sat directly on top of each other.
While that appeared to be the correct answer, no one offered a convincing mathematical proof until 1998 -- and even then people were not entirely convinced.
For six years, mathematicians have pored over hundreds of pages of a paper by Dr. Thomas C. Hales, a professor of mathematics at the University of Pittsburgh.
But Dr. Hales's proof of the problem, known as the Kepler Conjecture, hinges on a complex series of computer calculations, too many and too tedious for mathematicians reviewing his paper to check by hand.
Believing it thus, at some level, requires faith that the computer performed the calculations flawlessly, without any programming bugs. For a field that trades in dispassionate logic and supposedly unambiguous truths and falsehoods, that is an uncomfortably gray in-between.
Because of the ambiguities, the journal, the prestigious Annals of Mathematics, has decided to publish only the theoretical parts of the proof, which have been checked in the traditional manner. A more specialized journal, Discrete and Computational Geometry, will publish the computer sections.
The decision represents a compromise between wholehearted acceptance and rejection of the computer techniques that are becoming more common in mathematics.
The debate over computer-assisted proofs is the high-end version of arguments over using calculators in math classes -- whether technology spurs greater achievements by speeding rote calculations or deprives people of fundamentals.
"I don't like them, because you sort of don't feel you understand what's going on," said Dr. John H. Conway, a math professor at Princeton. But other mathematicians see a major boon: just as the computers of today can beat the grand masters of chess, the computers of tomorrow may be able to discover proofs that have eluded the grandest of mathematicians.
The packing problem dates at least to the 1590's, when Sir Walter Raleigh, stocking his ship for an expedition, wondered if there was a quick way to calculate the number of cannonballs in a stack based on its height. His assistant, Thomas Harriot, came up with the requested equation.
Years later, Harriot mentioned the problem to Johannes Kepler, the astronomer who had deduced the movement of planets. Kepler concluded that the pyramid was most efficient. (An alternative arrangement, with each layer of spheres laid out in a honeycomb pattern, is equally efficient, but not better.) But Kepler offered no proof.
A rigorous proof, a notion first set forth by Euclid around 300 B.C., is a progression of logic, starting from assumptions and arriving at a conclusion. If the chain is correct, the proof is true. If not, it is wrong.
But a proof is sometimes a fuzzy concept, subject to whim and personality. Almost no published proof contains every step; there are just too many.
The Kepler Conjecture is also not the first proof to rely on computers. In 1976, Dr. Wolfgang Haken and Dr. Kenneth Appel of the University of Illinois used computer calculations in a proof of the four-color theorem, which states that any map needs only four colors to ensure that no adjacent regions are the same color.
The work was published -- and mathematicians began finding mistakes in it. In each case, Dr. Haken and Dr. Appel quickly fixed the error. But, "To many mathematicians, this left a very bad taste," said Dr. Robert D. MacPherson, an Annals editor.,
To a
Re:Can someone elaborate on... (Score:5, Informative)
Theorem Provers (Score:4, Informative)
Theorem provers [wikipedia.org] 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.
Re:Are Computers Ready to Create Mathematical Proo (Score:3, Informative)
So many times I see people use programs like MAPLE to show something mathematical, and it ends up a disaster.
Problems is the brain on the chair, not brain on machine.
Godel? (Score:2, Informative)
regfree link (Score:3, Informative)
Wrong Summary (Score:2, Informative)
Re:Can someone elaborate on... (Score:3, Informative)
You beat me to it (Score:5, Informative)
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 [anl.gov], a theorem proving program developed at Argonne National Laboratory.
Re:Rumsfeld, anyone? (Score:5, Informative)
KK | KD
___|____
|
DK | DD
So, you can:
Know that you Know something
Know that you Don't Know something (the second most common)
Don't Know that you Don't Know something (most things fall in this category for most people)
Don't Know that you Know something (the most interesting of the categories)
Big, huge, red tape operations can easily fall into the latter category (DK)... since it is rather easy for a group to obtain knowledge, yet be unaware of it [political commentary omitted].
Flyspeck Project (Score:5, Informative)
Re:indeed (Score:5, Informative)
The resulting proofs are still hairy enough that they have to be checked by machine, but the size and complexity of the proof-checker is much less than that of the compilation toolchain. That means that while there's still some code that has to be trusted, it's much less. Here's my informal scariness hierarchy:
Normal model (you have to trust everything) > type safe languages (you have to trust the compilers / interpreters) > proof-carrying code (you have to trust the proof-checker*).
If you haven't already, you should definitely read Ken Thompson's Turing Award lecture, "Reflections on Trusting Trust" here [uoregon.edu].
* - Pedantry point: If you're talking about Necula's original PCC work, you also have to trust the verification condition generator, which is some fairly deep voodoo. Appel's Foundational PCC addresses this to a signficant extent.
Re:identity (Score:5, Informative)
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.
Re:new facet of an old issue (Score:2, Informative)
The classification problem for finite simple groups is more similar. In theory, a complete proof has been found - but it's too daunting for any one person to collect and verify the entire thing.
an Eternal Golden Braid (Score:2, Informative)
Godel, Escher, Bach: an Eternal Golden Braid [amazon.com]
I am only half way through it, and it handles this topic far more gracefully than the original article. Very entertaining if you happen to be a math, music, or art geek. Strange mix, but Douglas Hofstadter really nailed it.
Re:identity (Score:1, Informative)
Humans behind the scenes (Score:3, Informative)
Computers nowadays can handle symbolic calculations and prove identities and likewise, but for identifying what is interesting to have proved or not, a human may still be there with interpreting that, no matter how sophisticated computers or software can get...
Risc Institute: Theorema (Score:3, Informative)
Re:Ok I am always confused about the difference. (Score:4, Informative)
Re:Canada's former prime minister's "proof" (Score:2, Informative)
Re:What did godel say? (Score:2, Informative)
We know that given any collection of nontrivial axioms there will always be a statement that is consistent with all axioms and valid proofs that cannot be prooved or disprooved within that set of axioms. In other words, no matter how many axioms we select no nontrivial mathematical system with a finite number of axioms is complete. Secondly, as you said there are mathematical constructs that can not be proved or disproved, and so it is impossible to show that all existing proofs are consistent with the chosen axioms.