Skip to comments.Math problems too big for our brains
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."
I wanted to impress my grandpa with my math skills so I told him the formula Pi-r-square.
He laughed at me and said "No, boy. Pie are round. Cornbread are square!"
2+2 = 11? only if you redefine = or + or 2 or is...
"2+2 = 11? only if you redefine = or + or 2 or is..."
Nope. Base 3.
How can the article say "Four Colors"?
Russia touches at least 10 other countries. If you only used four colors, you would have to have the same color touching at least thrice.
I suppose if you just limited the scope to "map of Europe", but even then, you'd have to leave out the Balkans to use just four colors.
If a computer can "prove" that only four colors are needed, then the computer is clearly in error.
redefine is ? The stainmaker tried to do that
Try it in a "Base-3" number system.
Yes, in Base 3.
This writer and perhaps his subject need a course in logic. Deductive reasoning can prove truth statements - not only mathematics.
And much of what the articles gives as examples are inferences, inductive reasoning which are not considered proof.
His point about computers is ok, but the rest of this article..
That was my science teacher's favorite bad joke. Wonder if we're talking about the same guy....
Yup. That always gets 'em. It also makes the statement in the article false.
And for those who think there's no point in using math in anything other than base 10, I give you the computer you're typing on right now.
There are 10 types of people: Those who understand binary math, and those who don't.
Then that's redefining what the 11 is.
I think we're giving away our age...
Remember when big fat Rosie O'Doughnut said that nobody needs to learn math any more - we have computers. What a retard.
The computer has to be programmed, does it not?
You could use the same color for Finland, Poland and China, as they don't border each other.