I don’t think I’ve ever been less engaged by a presidential election including the one in 1980 when I was only one year old (like most children in those days, I cried whenever Carter was on TV), so I’m going to take a break from politics to rant about math. I hope that’s okay with everyone, and if it isn’t, then @#$% you. My blog.
Anyway, there is Goldbach’s conjecture which is that every even number greater than 2 can be expressed as the sum of two primes. No proof yet, but it’s been verified up to about 10^19 through brute computer force.
What I find more fascinating is Goldbach’s weak conjecture. It states that every odd number greater than 5 can be expressed as the sum of three primes. It’s the “weak” conjecture because a proof of the “strong” conjecture would prove it quite trivially (if you know you can express ever odd number greater than 2 as the sum of 2 primes, then just add 3 to each of those sets and you have every odd number greater than 5 expressed as the sum of three primes). What interest me is that it’s proven for all but a finite set. Someone has proven it’s true for odd numbers greater than about 2 * 10^1346, and simple brute computer force has shown there are no counter examples up to about 10^18. So, if no one is able to come up with an elegant proof, the conjecture will be completely proven or disproven when computer power catches up.
But what’s the point? Does any mathematicians honestly believe there’s a counterexample sitting out in the no man’s land between 10^18 and 2 * 10^1346? Have you ever heard of any special number that wasn’t somewhere around 3 (like e and pi)?
When I learned proofs in college, we were warned about ellipsis proofs. That’s where you say for an equations, “It’s true for x = 1, x = 2, x = 3,… so it’s true for all x.” The things is, though, the ellipsis proof tends to be correct, as patterns just don’t break down suddenly for no reason. And, let’s be real for a moment: When you’ve tried it up to x in the quintillions, I think you’re pretty much okay going “… it’s true for all x.” And there’s all these “unsolved” problems in mathematics that everyone knows is true even though no one has come up with a formal proof. It took hundreds of years for someone to finally prove Fermat’s last theorem, but it’s not like during those hundreds of years anyone thought it wasn’t true. And no one really thinks there’s an odd perfect number, but we’re supposed to pretend there might be one since no one has proved there isn’t one — same as no one has proved unicorns don’t exist.
I’m through pretending.
It’s time mathematicians stop wasting their time trying to prove things that no one actually thinks isn’t true. It helps no one and it’s stupid. Instead, they should put their energy into more useful, concrete things like finding even larger Mersenne primes. We still need to break the ten million digit barrier, people!
All mathematicians should be put to work finding ways to stop global warming.
some numnvts rated that post a half-star. Can you believe the cojones on that guy?!? What is the world coming to…
ps. I hate math too. That’s why I liked that post. Cause it justifies my hating of math.
My brain just screamed.
That’s the beauty of math. None of that “innocent ’til proven guilty” crap. Everything is wrong until we say it’s right!
Algebra is fun. I enjoyed adding the alphabet so much I decided to major in English.
Interesting.
Educational.
Non-political.
Where am I???
In all seriousness, though, that was pretty interesting. Good post, and a welcome change of pace, if even for a moment.
I suppose you really like that inelegant brute force proof of the four-color theorem too, huh? How about we kinda sorta solve the NP-complete problem and call it good? Go square a circle, you wanker.
😉
@3: this is actually one of the simpler-to-grasp unsolved mathematical problems. It’s really easy to understand the pretense: think of any even number. Now, think of two prime numbers that add up to that number. It might take a while, but you can probably do it for just about any number under 100 pretty quickly — it would probably help to have a list of primes under 100 handy. Now, the theory is that this is always true. But until we can explain why it’s always true, we can’t call it a law, only a theory.
Capice?
Math?? What about pants? Don’t they piss you off?
#8 – Posted by: James on June 23, 2008 02:07 PM
Thanks, but it’s a waste of time. I always hated math.
Don’t you want to rant about Pants?
I think people need to stop trying to put mathematicians to practical use.
As any mathematician will tell you, they’re all about correctness, not utility.
Ron Paul!
It was a bit of a shock to me when I realized that calculus was still basic math. I stopped at that point, in any serious way, from going farther in math, but the big questions are still pretty cool, and the little tricks make life easier.
So contrary to the algebraphobics above (ok, so I made up that word), I’m good with a bit of geeky conjecture about numbers.
I am also good with a bit of geeky conjecture.
I am not a mathmatician but it makes sense to me that we might learn something in how we prove it, not just that it is now proven.
In solving some problems you often develop tools for solving other problems.
That being said I tend to think we need to focus on the unknown more too. Good Post!
Ah yes, math the antisocial weapon of choice.
Why use bellicose and threatening language when a little math will drive people away in a hurry. Want to get people to shut up. Take out your trusty chalk and board. Want to spoil someone’s day. Bring up their deceased relatives, dead pets, or talk about math, either will suffice. Even the simplest of equations will cause a prison riot.
Yes, I do like math!
It actually makes me sad about how little I understand of algebra, but it does prod me to learn more.
Now if we could just get the left to understand math & basic economics (FairTax.org), things would improve exponentially.
Sadly, I have a better chance of solving the Riemann hypothesis with an abacus & a box of crayons than that ever happening. In fact, Ron Paul has a better chance of winning in November than that happening.
The real FrankJ is being held hostage somewhere in Wyoming. Nobody knows who really authored this post.
IT WAS HARVEY! We been lead to believe that Harvey is Frank’s alter-ego, but the reverse is true!
Ben has your answer, Frank, you innumerate. The reason you try to prove stuff like these wacky conjectures is the tools you develop and general insight into numerical analysis you get by doing so. The latter stuff is far more useful than the proof itself.
Go look up the origin of calculus, for example. Newton invented it so he could describe the orbits of the planets perfectly, so he could feed his fascination with astrology ans assorted related hermeneutical crap, wherein everything you want to know about your history, or history in general, is determined by precisely where the planets were on day X at hour Y. (I won’t speak of Leibniz, except to note he was into similar phantasmagorical Theories of Everything.)
Just leave the mathematics to those who understand it, mkay? If only your demographic peer group would exercise a similar humility about what the f*** they do and do not know enough about to propose broad judgment upon, we wouldn’t have this Obama problem.
I thought I had my fill of math when homeschooling my kids this morning. I couldn’t figure out the formula for an algebra problem which was disguised as a cute word problem.
At least now I know…when I can’t get the answer quickly on Sinapore Math help line, I can come here. I like a multi tasking blog!
An example of why we don’t trust ellipsis proofs:
The logarithmic integral Li(x) is larger than pi(x) for “small” values of x. This is because it is (in some sense) counting not primes, but prime powers, where a power p^n of a prime p is counted as 1/n of a prime. This suggests that Li(x) should usually be larger than pi(x) by roughly Li(x^1/2)/2, and in particular should usually be larger than pi(x). However, in 1914, J. E. Littlewood proved that this is not always the case. The first value of x where pi(x) exceeds Li(x) is probably around x = 10^316; see the article on Skewes’ number for more details.
#18 … I propose that it’s John Derbyshire of NRO.
Infiltrated by Buckleyites. Eww.
Oh, the upside is that Derb coined the phrase “Rubble don’t make trouble”, which I have to admit is right up there with “Nuke the moon”.
Duh … one fish, two fish, red fish, blue fish?
I heard once that the ancient chinese thought a man should marry a woman who was half his age, plus 1 year.
A man of 40 should marry a woman of 21, and so on.
The hardest math problem I ever set myself was trying to express that rule of thumb as an algebraic equation.
I came up with: (W + x) = (M + x)/2 + 1, where W is the womans’ age, M is the mans’ age, and x is the number of years they’d have to wait if she’s too young.
Is that right?
God I hate you guys.
#25 I don’t think that it can be expressed by a single formula.
By your formula if a man is 20, the half age plus one then says woman of 11. Unless you are Mohamed (may fleas be upon him) that is too young. So if the man waits 5 years for her to be of age (3 in Kentucky) he is then 25 and she is 16 and she is not 1/2 his age plus one.
So you need 2 formula, the simple one if
M/2 +1 > Age of consent (AOC) of W = M/2 +1
and if M/2 + 1 = Jailbait then
W = M – (AOC – 2)
So back to the example Man is 20, woman must be 16 to marry so he targets a “woman” of 6 waits 10 years then she is 16, he is 30 and M/2 + 1 is true.
As an engineer, I have to say that mathematicians are nitwits.
I’ll explain why in a joke (my other engineer joke).
So a mathematician and an engineer are in a bar, it’s nearly last call so they’re alone with the hot bartender. She goes and locks the door and tells the two that she is going to take off her clothes and stand against a wall. They have to stand against the opposite wall and come halfway toward her, then halfway again, and so on and when they reach her, they can do what they want with her.
So the two think about that. After a minute, the mathematician gets up, puts on his jacket and starts to leave. He sees the engineer taking off his clothes and putting his back to the wall.
The mathematician says, “Don’t you get it? It’s Zeno’s Paradox, you’ll never reach her.”
To which the engineer replies, ‘Maybe not, but I’ll get close enough to do what I want to do.”
Remember, 6.853698365897 * 2 = 14
Until you prove it, you don’t know. And computer power can never “catch up” to an infinite set. It reminds me of an old joke (since #28 broke the ice on math jokes):
Several students were asked the following problem:
Prove that all odd integers higher than 2 are prime.
Mathematician: 3 is prime, 5 is prime, 7 is prime, and by induction, we have
that all the odd integers are prime.”
Statistician: 100% of the sample 5, 13, 37, 41 and 53 is prime, so all odd
numbers must be prime.
Mechanical Statistician: 3 is prime, 5 is prime, 7 is prime, 9 is an
outlier, 11 is prime, 13 is prime, …. all odd numbers are prime.
Measure nontheorist: there are exactly as many odd numbers as primes
(Euclid, Cantor), and exactly one even prime (namely 2), so
there must be exactly one odd nonprime (namely 1).
Physicist: 3 is prime, 5 is prime, 7 is prime, 9 is … uh, 9 is an
experimental error, 11 is prime, 13 is prime… Well, it seems that
you’re right.”
The joke goes on, but you get the idea. The statistician might also be the one who fires at the moose, but three yards to the left, then again, but three yards to right, declaring, “We got him! We got him!” Which somehow reminds me of our foreign policy, but I suppose this was supposed to be an apolitical post.
Anyway, many conjectures that have lasted decades were later disproved. And this can have practical effects. Internet security is based on the fact that some problems have, for a long time, been thought difficult to solve. No one can prove it, and, in 1994, it is shown that if we could create quantum computers, they actually would be easy to solve, thus enabling public “secure” interactions to be broken. (Still, there could still be a way to prove that solving the problem on classical computers is either hard or easy.) So proofs are always nice, especially since one thing that distinguishes mathematics from other disciplines is the ability to say something and be 100% sure about it.
“And computer power can never ‘catch up’ to an infinite set.”
True, but we ARE rapidly approaching the point where computer power can “prove” a theorem for all numbers with a value that exists in the physical universe. There is a finite amount of everything – there must be, because if there were an infinite amount of anything, the universe would collapse in on itself. So, how much is there?
I’ve heard some theoretical physicists make statements throwing around numbers like 10^80 or so when talking about the total number of all subatomic particles in the universe.
At the current rate of improvement in computer technology, we might be able to start “proving” theorems for all numbers “And computer power can never ‘catch up’ to an infinite set.”
True, but we ARE rapidly approaching the point where computer power can “prove” a theorem for all numbers with a value that exists in the physical universe. There is a finite amount of everything – there must be, because if there were an infinite amount of anything, the universe would collapse in on itself. So, how much is there?
I’ve heard some theoretical physicists make statements throwing around numbers like 10^80 or so when talking about the total number of all subatomic particles in the universe.
At the current rate of improvement in computer technology, we might be able to start “proving” theorems for all numbers <10^80 in just a few years.
Now, in all seriousness, even though we know that 10^80 is far, far less than, say, a “googol,” and that even a googol does not begin to approach infinity, we have to ask ourselves:
DOES IT MATTER IF ANYTHING BREAKS DOWN FOR NUMBERS GREATER THAN THE THEORETICAL MAXIMUM NUMBER THAT CAN REPRESENT ANY QUANTIFIABLE ASPECT OF THE UNIVERSE?
Just something to think about.
That’s not what math is about, and it’s not even true. If the universe is just a big computer, then no subset of it could make an identical computation to the entire thing. It is somewhat’s useful to show something’s true for certain numbers, but to show it’s true for any possible combination of inputs is another thing entirely.
weird – the middle of my post was eaten. oh well, after the word “numbers” that ends a paragraph without a period after it, I wrote a bunch of really deep shit. now it is gone.
but you get the gist of what I was saying – in a few more years, we will be able to prove theorems by brute computer force for all numbers up to and possibly over 10^80. since there is no practical use for larger numbers, because there is nothing “real” that needs larger numbers (you could still USE larger numbers, but it would be possible to simplify everything – for instance, if you were measuring the number of tenths of neutrinos, you could divide by ten and just use the number of neutrinos), there is no reason to try to extend the proof to anything bigger.
There was more, but I am too lazy and disinterested to attempt to recreate it now.
“That’s not what math is about”
Funny, I thought math was about solving real problems. How many sheep do I have today? Seven. Didn’t I have eight yesterday? Yes. So one is gone.
That’s what math is about. It’s what math has ALWAYS been about, ever since some primative ape-like being first noticed that some other primative ape-like being had one more banana that he had.
“and it’s not even true. If the universe is just a big computer, then no subset of it could make an identical computation to the entire thing.”
You are a complete idiot who doesn’t understand the basic principles of mathematics.
How many dots are there below?
………..
How many is that?
11.
See what I just did? I just used two digits to show something that took up 11 spaces.
That’s why it is absolutely possible to do things like, say, count the total number of all possible outcomes of something and express it using a number that can it inside this hear dialog box. You know, like 10^80. You can’t fit 10^80 period-sized objects inside the physical universe, but you can express the value mathematically, and even multiply it by a billion for effect, using only 5 characters.
You dink.
“one thing that distinguishes mathematics from other disciplines is the ability to say something and be 100% sure about it.”
Hmmm…
Elvis is dead. I am 100% sure about it.
No math involved.
Um no.
Part of science is the ability to continue asking why and questioning the assumptive result even through overwhelming “evidence”. See Global Warming.
An unproven theory is, and will remain, a hypothesis. Regardless of how badly one “wants” it to be true. See Global Warming.
Liberals would agree with you though.
Please. Stop.
“Have you ever heard of any special number that wasn’t somewhere around 3 (like e and pi)?”
c (299,792,458 m/s) although it still isn’t close to 10^18, it has disproved the big bang theory (at least from my understanding) assuming Einstein’s theory of general relativity.
You may say that the big bang theory really isn’t a mathematical theory. However, this still proves that a really large number, say 3c or 4c, can cause major problems in trying to prove something.
Another number that isn’t close to 3 is G (6.673 x 10^-11 m^3/kgs^2).
My understanding is that Simon Davis published a proof that there are no odd perfect numbers in 2004.
http://arxiv.org/abs/hep-th/0401052
One thing I’ve heard about Goldbach, is that, because there are so many combinations of two (or three) prime numbers as you get larger and larger, the fact that it’s true for small numbers makes it very, very unlikely (statistically speaking, though of course “statistics” on numbers can be abused) that there would be any counterexamples. It would be like saying, “Every American who has met more than 100 others has met someone with an ‘a’ or ‘e’ in his or her name.” It’s very impractical to prove, but it’s hard to imagine that it’s untrue.
Gullyborg, you seem to get awfully angry and offended when any mathematics rising about the level of elementary school is being discussed. Maybe this isn’t the comment thread for you.
I’m not sure what you mean there. I don’t think it has. What seems to be true is that the universe itself has itself expanded at “speeds” that are greater than the speed of light, even if no object within the universe can expand that quickly. Is that what you’re referring to?
Given that the work was never published in a peer-reviewed source, I think there was likely an error. (Googling, math forum folks seem skeptical, and nothing came of it after 2004.) Even the best of us make them.
“I’m not sure what you mean there. I don’t think it has. What seems to be true is that the universe itself has itself expanded at “speeds” that are greater than the speed of light, even if no object within the universe can expand that quickly. Is that what you’re referring to?”
I’m no physicist, but the way I see things is that nothing can travel faster than light. Any object in motion has to be set in motion, so take a particle of light which is accelerated to c upon creation. By the equation a=F/m, the acceleration of a particle of light is ∞ no matter what F is because the mass of a particle of light is virtually zero. So light has an infinite acceleration but can only travel with a velocity of c. If light cannot reach speeds beyond c, there is no way anything else can either. Thus there is no way the universe could expand at a velocity > c.
I think mathematicians should put all their focus on finding pi to the umpity-kazillioneth digit…because when I want to know the area of a circle, I REALLY want to know the area of the circle!
Roy
‘There are no odd perfect numbers in 2004.’
Why? What was so special about 2004?
“…finding pi to the umpity-kazillioneth digit…”
We already know how to do that. They’re called digit extraction algorithms. They are limited to bases that are powers of two, but if you want to know what the 10^295th digit of pi in hex is, then you can find it.