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."
You've divided by zero, which is why you reach an impossible result.
Repeat after: math is consistent, math is consistent, math is consistent. If I reach a nonsensical result, I did something wrong.
Ummm, build better computers?
Don't build them too good!
True, but it's fun to play mind games with those that forgot the "rules".
"We are animals..."
Well, there's your first mistake, Professor Nimrod.
The whole reason for the exercize is in order to print maps with the fewest colors (least printing costs)
You dont want the same color for any two countries that touch each other (like canada and mexico can both be green, but not US and mexico)
It is ALWAYS possible to do with 4 colors or less
Working as a programmer, I may have warped my kids forever while they were in elementary school. They just couldn't believe that I started from 0 and counted ...8, 9, A, B, etc. They just knew that 10 came after 9. Daddy had to be wrong. (My use of 24-hour time also threw them for a loop.)
Not true. When expressing the equation as 2 + 2 = 4, then it is implied that we are working in a base for which the symbol 4 is defined, which would be base 5 or higher. In any of these defined systems 2 + 2 = 4. To say that 2 + 2 = 4 is false is never logically consistent, because for the systems in which this is so the symbol 4 is undefined, therefore the statement is not provable.
"Base 8 is like base 10, really... if you're missing two fingers.
"
And there ya go, except for one small detail. One of the fingers has to represent zero in Base 8. That's a problem if you count on your fingers.
Assume x=y
x+x2 = x2+y
x-x2-y = x2
x-x2-y-xy = x2-xy
(x-y)(x+1) = x(x-y)
x+1=x for all x
divide by zero error.
Math is not a form of knowledge. It is an intellectual construct, useful in obtaining knowledge.
You can't prove the sun will rise tomorrow, but you can prove two plus two equals four, always and everywhere.
Apparently the reporter has never heard of Godel's Incompleteness Theorem. To wit: "In any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system."
..or less?
"Not true. When expressing the equation as 2 + 2 = 4, then it is implied that we are working in a base for which the symbol 4 is defined, which would be base 5 or higher. In any of these defined systems 2 + 2 = 4. To say that 2 + 2 = 4 is false is never logically consistent, because for the systems in which this is so the symbol 4 is undefined, therefore the statement is not provable."
Hah! You make a good point. Just as I can definitively say that 2+2=11 means that I'm using base 3, 2+2=4 means I must be using base 5 or higher. You win!
Majikthise and Vroomfondel...representatives of the Amalgamated Union of Philosophers, Sages, Luminaries and Other Thinking Persons...declare that they stand in solidarity with the Pencil Pushers Union!
Is that always true?
"Is that always true?
"
Yes, except when it isn't. [grin]
Which finger do you use to represent 0 in base 10? If you don't need it in base 10, you don't need it in base 8 either.
Mathematics is a subset of statistics where the variance equals zero.
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