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."
Really? I can prove that 2 + 2 = 11"
Yep, I questioned immediately whether that statement really came from a serious mathmetician. I am but a mere physicist and I know that the answer depends on what base you are using.
In other news, 8 Year enters...
My brain hurts!
I don't want to start a flame war, but in my academic experience(chiefly in Logic seminars), this was generally true.
Of course, the converse is true, as well.
"Then that's redefining what the 11 is."
Nope. It's not. The article said that 2+2=4. It does, in several number bases. It does not equal 4 in base 3. Without defining the number system you are using, such statements are false.
As several people noticed, 2+2=11 ONLY in base 3, but the sum is correct as written. Since the number base was not defined in the first example, I have changed nothing about the definition. I simply used a different base and came up with a different answer.
The article said that 2+2=4 always. That is clearly not true.
Now, I do not know of a practical use for Base 3 math, although they may be one, if there is a physical system somewhere that has three states.
Yes, but those 10 (may be more than that now...) don't all touch each other. Thus, Russia is color 1, and all the others touching it are 2, 3, or 4.
Well, now we know where the French surrender to the Muslims has gone - it's in Ottawa surrendering to the EVIL genius of Dr. Computer Math!!!
I'll bet the ancient Greeks, Mayans and Aztecs would argue the premise that human brains can't handle math.
I'll also bet that medical science will dispute the argument that our brains have shrunk and can't handle today's math problems.
Finally, I'll bet that this idiot has become lazy and has grown too reliant on his computer and calculator to perform the tasks that manually solving math problems used to require.
You didn't read it clearly: all that's required is that Russia be a different color than any color touching it, not that all countries touching it be a different color, unless they themselves touch. So, Mongolia and Finland can be the same color.
A few? Most mathematics papers are still single-author. 75% or 80% if memory serves. A lot of highly significant results in mathematics are still created by lone wolf thinkers.
There are 10 kinds of people in the world. Those that can count in binary and those that can't.
"If you don't define your number system, then you're talking about base 10. "
Not so. That is your assumption, and it is not accurate. If I am reading a book on programming, there will be a different assumption, depending on the type of computer being programmed.
The moment I write 2+2=11, everyone should know I'm in Base 3. If I write 2+2=10, then I'm in Base 4. In all bases from 5 up, 2+2=4.
iThink, therefore, iAm.
I thought it is always my fault, except when it isn't.
Human computer ping ;o)
Base 8 is like base 10, really... if you're missing two fingers.
- Tom Lehrer
Ha. I can show that 5 + 4 = 4.