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."
Computer is OK as long as all countires are contiguous. Since they aren't you are right, the computer is in error when talking about maps of the world.
Given a=b, prove that 1=2.
PROOF:
a = b Given
a*b = b^2 Multiply both sides by b
a*b - a^2 = b^2 - a^2 Subtract a^2 from both sides
a(b-a) = (b+a)(b-a) Factor
a = b+a Divide both sides by (b-a)
a = 2*a Since a=b as originally given
1 = 2 Divide both sides by a
My favorite, though, is to define "regular" to mean words that describe themselves, "irregular" to mean words that do not describe themselves.
BTW, the barber was a Cretin.
Of course you know your assumption is faulty. a cannot equal b in an ordered set of numbers.
Hey, what are you doing on this mathematician thread?
It must be Dark's fault!
That's a good one. LOL
Isn't '42' the answer to the Ultimate Question of Life, the Universe, and Everything?
Yup, but the problem is that nobody can remember exactly what the question was...
Mark
I can trisect an angle with nothing but a compass and a straight-edge, but apparently it is impossible.
When someone says prove "2 + 2 = 4", unless explicitly stated otherwise they mean base 10.
Incidently, Godel believed in the existence of God.
Starting with N = 4... Picture a "four corners" scenario like the states in the US Southwest. No problem... even though they all intersect at a single point, you have four colors. However, if you add one region/state/whatever that surrounds all four (or also intersects at the same point), the you would need a fifth color. Repeat as necessary for every N + 1.
The problem is only solvable for N = 4 if "adjacent" requires more than one point of border intersection.
Nope. Base 2. Answer = 3
yes- a map of one state needs only one color, two if you include bordering states (that do not also touch each other)
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