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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."
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.
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."
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.
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> 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.
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?
Technoli