Posted on 11/08/2005 8:48:52 AM PST by RightWingAtheist
Our brains have become too small to understand math, says a rebel mathematician from Britain. Or rather, math problems have grown too big to fit inside our heads. And that means mathematicians are finally losing the power to prove things with absolute certainty.
Math has been the only sure form of knowledge since the ancient Greeks, 2,500 years ago.
You can't prove the sun will rise tomorrow, but you can prove two plus two equals four, always and everywhere.
But suddenly, Brian Davies of King's College London is shaking the foundations of certainty.
He says our brains can't grasp today's complex, computer-generated math proofs.
"We are beginning to see the limits of our ability to understand things. We are animals, and our brains have a certain amount of capacity to understand things, and there are parts of mathematics where we are beginning to reach our limit.
"It is almost an inevitable consequence of the way mathematics has been done in the last century," he said in an interview.
Mathematicians work in huge groups, and with big computers.
A few still do it the old-fashioned way, he says: "By individuals sitting in their rooms for long periods, thinking.
"But there are other areas where the complexity of the problems is forcing people to work in groups or to use computers to solve large bits of work, ending up with the computer saying: 'Look, if you formulated the problem correctly, I've gone through all the 15 million cases and they all are OK, so your theorem's true'."
But the human brain can't grasp all this. And for Davies, knowing that a computer checked something isn't what matters most. It's understanding why the thing works that matters.
"What mathematicians are trying to get is insight and understanding. If God were to say, 'Look, here's your list of conjectures. This one's true, then false, false, true, true,' mathematicians would say: 'Look, I don't care what the answers are. I want to know why (and) understand it.' And a computer doesn't understand it.
"This idea that we can understand anything we believe is gradually disappearing over the horizon."
One example is the Four Colour Theorem.
Imagine a mapmaker wants to produce a colour map, where each country will be a different colour from any country touching it. In other words, France and Germany can't both be blue. That would be confusing.
So, what's the smallest number of colours that will work?
A kid can work out you need four colours. But can you prove it? Can anyone be certain, as with two-plus-two?
The answer turns out to be a hesitant Yes, but the proof depends on having a computer to work through page after page of stuff so complex that no single person can take it all in.
And it's getting worse, Davies writes in an article called "Whither Mathematics?" in today's edition of Notices of the American Mathematical Society, a math journal.
Math has tried to write a grand scheme for classifying "finite simple groups," a range of mathematical objects as basic to this discipline as the table of the elements is to chemistry -- but much bigger.
The full body of work runs to some 10,000 difficult pages. No human can ever understand all of it, either.
A year ago, Britain's Royal Society held a special symposium to tackle this question of certainty.
But many in the math community still shrug off the issue, Davies says. "Basically, mathematicians are not very good philosophers."
Yes, in Base 3.
A few milennia ago, I was working extensively in Motorolla 68000 assembly language, and I began balancing my checkbook in octal. Really screwed up my checking account. I had to go back and search for the '8's and '9's that were missing!
Mark
Heinlein wrote about computers that used "trinary" or base 3. It's an interesting thought... Circuits that used both positive and negative charges, as well as no charge. Of course this negates the possibility of using differential signaling...
Mark
Anybody else here remember doing a two page mathematical proof, getting something screwed around, and winding up with the original equation?
Mark
On the other hand, that's one way to make yourself a millionaire a lot faster.
Borrowed from another Freeper's tag-line...
(cogito, ergo Freepum) I Think, therefore I Freep!
I also like, "Freepito, ergo sum, "I Freep, therefore I am!"
Mark
I love that quote. I left it written on my whiteboard at my last job for over six months.
Don't build them too good!
That's right! You'll get the phillosopher's union all over you!
Mark
Colossus, the Forbin Project
We have a winner, and it only took 76 posts!
Mark
actually, I could believe it.
I am only in Calc 1 right now, but I figure that as I get on into my Engineering degree, that I will see some things that will defy everything else I have learned as well.
I figure there are branches of math out there that HAVEN'T been discovered yet.
Please take out a piece of paper, and proove that between every 2 rational numbers, there's an irrational number, and between every 2 irrational numbers, there's a rational number.
I still have nightmares about that class, even though I did get a 'B' in the class.
Mark
I raise my hand. I also remember a couple of unorthodox solutions that I had to defend, but which, on balance were correct.
The problem was when I went back to decimal without realizing I was changing base!
Mark
This is the entire reason we build computers! Not to get porn (thats what the personal computer was) but to solve math that would take so long it'd be impossible for a human to solve.
Our first homework assignment in non-Euclidean geometry was to prove 2 points determine a line. Oh, and the teacher thought having a textbook made the class too easy, so we had no reference material.
I love impossible homework assignments.
Yeah. The fun part was with punch cards. Every see a grown man cry?
Computers can self-program. Granted, they need to start somewhere.
That's why G-d invented rubber bands!
Mark
If only they didn't break.....
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