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."
Calculus -- absolutely fascinating. Though one thing I found guaranteed to make you wonder how your left brain hemisphere is talking to your right brain hemisphere is datawarehousing design (in the midst of one right now -- trying to squeeze in another source with a radically different system logic)
I dunno. I threw them in because they had to be noted for something!
What question? :) LOL
I see math teacher's fingers stutter on the keys... :)
That is so cool... It explains why 1 + 1 does not always equal 2. (Limit theorems)
That is so cool... It explains why 1 + 1 does not always equal 2. (Limit theorems)
And Zeno's paradox.
Mark
Infinity and Zeno's paradox is what hooked me on going to grad school in math. Stupid move, but it was fun.
I never went to grad school, but my brother was sure calculus was completely useless, until he needed to stain a large, irregular, curved deck... I think that he now (sort of) appreciates limits.
Mark
I've always heard that 'calculus' separates the men from the boys. LOL
I like this so much. Would you mind if I used it in a tagline someday. Or did you borrow it from someone else? Thanks.
geez, i too sit in my room for long periods, thinking... about lunch
AmishDude, Ph. D. in mathematics.
The fact is, the old areas are drying up. (But they still have large amounts of tenured faculty.) I wouldn't be surprised if this guy was an algebraic topologist.
That's just not true. I agree. The author is full of it. He's probably from the school who still does long division.
What is your field of study, AmishDude? Just curious. Know anything about profile monitoring and generalized least squares?
I'm an extremal and probabilistic combinatorialist. I don't know what you refer to by "profile monitoring" and "generalized least squares" but I suspect I know what it is under a different name.
My masters is in Prob & Stat from the math dept at UCSB. I think I had one course in combinatorics. My PhD dissertation uses statistical profile monitoring, which is an extension of statistical process control (SPC). Least squares relates, of course, to linear regression. Generalized least squares is looking at nonlinear regression such as logistical regression. I am looking at a response variable and using GLS to estimate the deviation from the expected value. Usually the response is a process as in SPC. My contribution will be the human response, such as a soldier response. I am trying to formulate the question I want to answer this quarter.
Does this make any sense to you at all? Did you study any mathematical statistics?
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