quote:To a topologist, a rabbit is the same as a sphere. Neither has a hole. Longitude and latitude lines on the rabbit allow mathematicians to map it onto different forms while preserving information.
Clearly the author of this statement knows more about mathematics than lepan anatomy. We Rabbit's definitely have a hole, as do humans and other mammals, making us toroids (donuts) and not spheres.
There is an interesting article in today's New York Times link about a recent break through proof of one of the central tenets of Topology (Poincar's conjecture).
quote:Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space.
After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world’s mathematicians to pick up the pieces and decide if he was right.
Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them.
quote:In a speech at a conference in Beijing this summer, Shing-Tung Yau of Harvard said the understanding of three-dimensional space brought about by Poincaré’s conjecture could be one of the major pillars of math in the 21st century.
But perhaps even more intrigueing than this break through in mathematics is the man who made it.
quote:In a series of postdoctoral fellowships in the United States in the early 1990’s, Dr. Perelman impressed his colleagues as “a kind of unworldly person,” in the words of Dr. Greene of U.C.L.A. — friendly, but shy and not interested in material wealth.
quote:at the moment of his putative triumph, Dr. Perelman is nowhere in sight. He is an odds-on favorite to win a Fields Medal, math’s version of the Nobel Prize, when the International Mathematics Union convenes in Madrid next Tuesday. But there is no indication whether he will show up.
quote:Dr. Perelman returned to those woods, and the Steklov Institute, in 1995, spurning offers from Stanford and Princeton, among others. In 1996 he added to his legend by turning down a prize for young mathematicians from the European Mathematics Society.
Until his papers on Poincaré started appearing, some friends thought Dr. Perelman had left mathematics. Although they were so technical and abbreviated that few mathematicians could read them, they quickly attracted interest among experts.
quote:Recently, Dr. Perelman is said to have resigned from Steklov. E-mail messages addressed to him and to the Steklov Institute went unanswered.
The story is reminiscent of John Nash (It's a beautiful mind), and Ted Kazincsky (The Unibomber). It begs the question of why so many truly brilliant mathematical minds seem so far removed from the reality most of us live in.
In this context, I found the following quote from the article intriguing.
quote:“Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide,” explaining that curiosity is tied in some way with intuition.
“You don’t see what you’re seeing until you see it,” Dr. Thurston said, “but when you do see it, it lets you see many other things.”
Back in the day, I used to hang out with a number of mathematicians and they frequently got into these really abstract discussions of topology which always made topology sound like a mystic religion.
Curiously, a number disproportionate number of the truly gifted mathematicians I've known have suffered from serious mental illness.
I wonder what, if anything, this says about the nature of the human mind and or the nature of reality.
[ August 16, 2006, 11:33 AM: Message edited by: The Rabbit ]
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posted
It tells us that it's entirely possible to drive yourself beyond the point where your brain can keep up. You know, crossing that fine line between brilliance and insanity.
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It's a VERY interesting question. Another interesting piece of the puzzle is the cases of so-called "idiot savants" (really, a form of autism) some of whom have genius-level skill in math or music (arguably another form of math).
Ultimately we may need to change our standard view of the human mind and what is both normal and possible.
Meanwhile, how long is the coast of Maine?
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First off, TOPOLOGY RULES! I'm kinda sad that he proved the Poincare conjecture true (I think things would have been much more interesting if it was false).
However, I TOTALLY understand the "dropping off the planet" thing. Perelman had a goal - a goal that had stumped numerous mathematicians for a very long time. He solved it. It's time to retire now ;-)
As far as the insanity of mathematicians - yeah, all my favorite ones went crazy.
Also, as an edit to the OP, the story is NOT John Forbes in "A Beautiful Mind", it's John Nash (who proved my most favorite theorem EVER)
If anyone needs a crash course in what topology is, what the poincare conjecture asserts, which mathematicians I know of went crazy, or what John Nash's most amazing theorem ever is (in my opinion), just ask. I don't wanna bore you if you don't wanna know.
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quote:Originally posted by Bob_Scopatz: Meanwhile, how long is the coast of Maine?
I'm going to say infinite.. Probably helps that I was just reading Chaos (by Gleick). Actually speaking of Gleick, I would also recommend Genius, his biography of Richard Feynman, in terms of trying to understand what is going on in the mind of a genius.
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quote:Originally posted by katharina: I wanna know. All four, if you could.
Allright, you asked for it:
1. Topology is the study of continuous functions. Now, from high school, we know that a continuous function is "any function which you can draw without picking up your pencil"
Now, that's a great highschool definition, but it really doesn't help when it comes to real mathematics. The mathematical definition is defined more in terms of distance. If I change my x value a little bit, and the y value changes in small amount, we consider that continuous.
Now, topology is 2 sided:
1. What happens to continuous functions if we change our notion of "distance"? Suppose we just arbitrarily declare that if we measure the distance between any 2 distinct points, we get a distance of 1. Surprisingly, this fits all the rules for what it means to mathematically be a distance functions. With this definition of distance, what do continuous functions look like? (It turns out any function you draw becomes continuous with this measure of distance)
2. What sort of mathematical structures are "preserved" by continuous functions? Like, for instance, if I have a single connected piece of stuff, and I apply the continuous function to it, the resulting picture will neccesarily be a single connected piece. There are numerous example of things like this.
Further, topology looks at when 2 things have the same topology. That is, when all the continuous functions from one thing, are "the same" as the other thing. If this is true, it means that topology alone isn't enough to tell a difference between the two things.
2. As for the pioncare conjecture, first we have to introduce the idea of homology. Because it's easier to picture, we'll talk about homology of 1 dimension first. So, suppose you're working on a piece of paper. Draw a loop (that is, a curve that loops back on itself). In your head, if you kept shrinking the curve, could the curve eventually shrink down to a single point? Sure it can. To see this, think of your curve as a lasso that you're pulling tighter and tighter. But you didn't really lasso anything, so nothing prevents you from entirely closing the lasso.
Now, consider the same example, except this time, put a hole somewhere in the paper. Now we have 2 types of loops - loops that go around the hole, and loops that don't. For the loops that don't, we can still close the lasso all the way down, they collapse into points. But what if I lasso the hole? No matter how tight I pull, the rope must ALWAYS go around the hole. And since a single point can't go around the hole, our lasso will never entirely close. Thus, this is an entirely different kind of lasso.
The homology of dimension 1 of a space (either a piece of paper, a piece of paper with a hole in it, or something much more exotic), is related to the different types of lassos. In our first example, we found everything goes to a point - there is only 1 type of lasso. In our second example, we found 2 types.
For homology of 2 dimensions, we think of "lassoing" the space with 2 dimensional curved surfaces, and ask whether or not we can contract that to a point.
For homology of 3 dimensions, we think of lassoing the space with 3 differential curved surfaces, and ask whether or not we can contract that to a point.
Rinse repeat.
The poincare conjecture says this: If I have some random space, and it's homology in each dimension is EXACTLY the same as the homology of the 3 dimensional sphere, is it topologically the same as the 3-sphere? That is, if I look at all the continuous functions coming from the random space, to the match up somehow with the continuous functions coming from the 3-sphere?
Perelman answered this with a "yes"
I think I'll break this up into 2 posts....
*edited for grammar, clarity*
Mistake: the poincare conjecture was NOT one of Hilbert's 25 (or 23, or 29 depending on the source) millenium problems
posted
My favorite crazy mathematicians (in order)
1. Gregor Cantor. He developed the mathematics that today is used to analyze different kinds of "infinities". He developed a VERY powerful method of proof with one flaw: it never produced any explicit results.
For instance, there is something called a transcendental number. A number is transcendental if it is NOT the solution to a polynomial with integer coeffecients. That is, things like x^47 + 25 x^46 - 15x^45 + ... + 2 x - 1 = 0.
In Cantor's day, only 1 or 2 numbers were known to be transcendental. Cantor proved the following: There are vastly more transcendental numbers than non-transcendental numbers. In fact, if you were to pick a number at random every second forever, your probability of picking a number that WASN'T transcendental is 0.
Cantor's results were very nonintuitive, and because of that, he was VERY ostracized by the rest of the mathematical community.
Couple this with Cantor trying to prove the existene of God using his theory of infinities, and it's not hard to see why he went crazy.
In fact, he developed a pattern: Write a math paper, get committed, get out, write a math paper, get committed, etc.
Sadly, today we recognize his genious, and use his results and methods ALL the time.
2. Kurt Godel. He worked more at the logic side of mathematics that actual doing mathematics. For some history, in 1900, David Hilbert listed 25 "millenium problems". These were currently (then) unsolved problems of such importance that he thought mathematicians should be spending all their time on them. (One of them, I believe, was the poincare conjecture). Anyway, one of the questions was basically, "We all know that once we truly formalize the basics of mathematics (that is, logic), we'll be able to prove anything we already "know" to be true. Also, we want to be sure that mathematics is consistent (that is, we can't derive 2 contradictory things). Can someone please prove this?"
Russel and Whitehead come along and write "Principia Mathematica", which purpots to do this. To give you an idea of the rigor, somewhere after page 100 we come to the phrase, "and finally, with this all built up, we can prove 1 + 1 = 2". They never finished proving everything that needed proving, admitting "intellectual exhaustion".
Fast forward 30 years or so. A young mathematician named Kurt Godel comes along and TOTALLY screws everything up.
He publishes 2 proofs in one paper, called "Godel's first incompleteness theorem", and "Godel's second incompleteness theorem".
To summarize them:
Godel's first incompleteness theorem: Suppose you have a system strong enough to develope multiplication (and a few other more technical things). Suppose your system is consistent (in otherwords, it won't lead you to a contradiction). Then there are an infinite number of statements which are true and which you cannot prove to be true.
In other words, there are things which just can't be proven. Some of my favorite proofs involve showing that you can't prove something.
Godel's second incompleteness theorem: Assume a system like in the first incompleteness theorem (is consistent, has multiplication), then the system cannot prove it's own consistency.
In otherwords, if you're working with something, and you prove that it's consistent, it must have ACTUALLY been inconsistent (remember, if somethings inconcsistent, it leads to contradictions, like that it's both inconsistent and consistent).
What this means is that we will NEVER know absolutely that the mathematics we do is consistent. (I do want to stress that this doesn't keep mathematicans up at night. We all certainly BELIEVE our mathematics is consistent, we just know we'll never be able to prove it).
Anyway, I'm not exactly sure why Godel went crazy. I do know that he was a friend of Einsten (they were at Princeton together). Godel had this thing....he would only eat food prepared by his wife. His wife died before him, so you can see how this would cause problems. Einsten actually prepared food for him and told him his dead wife had prepared it. This suited Godel, apparently. Anyway, clearly crazy.
3. Einstein. I won't go into how revolutionary his ideas are. The fact that "Einstein" is now synonymous with "genious" proves it enough. But Einstein did kind of go crazy. He tried to stick general relativity and quantum mechanics together (both are fields which he initiated). The problem is that mathematically, they just don't fit together. (Even today, people are still trying to fit the 2 together by changing the math here and there). As soon as he recognized this problem, he kinda went into seclusion to focus all his time and energy on it. A lot of scientists of his day thought he had kinda lost it. Eventually, Einsten married his (second?) cousin (although, in Einstein's defense, he said he did it just to care for her; they never slept together). That took the cake for most everyone.
Finally, John Nash's most amazing theorem ever (in my opinion). First, a few definitions.
A manifold is something that up close looks flat, but you don't prejudice yourself on how it looks far away. So for instance, a sphere is a manifold (consider the earth. Looking out your window, it looks flat, but we know from far away it's definitlely not flat).
It turns out, you can precisely measure distance on a manifold (remember though that there are a ton of ways to measure distance), but you have to be careful. The curvature can really throw a monkey wrench in some problems. For instance, you can create thing where there are 2 points which have NO shortest path between them, or where 2 points have infinitely different shortest paths between them, etc etc.
Our final definition is the notion of an embedding. A manifold can be "embeded" into a space, meaning it gets shoved into in a such a way that we can pull it back out with all it's original properties. So, for instance, if you shove a sphere into a point, you can't pull it out with all it's original properties, but if you shove it into normal 3 dimensional space (like the earth in the universe), you can still pull it out.
Here's what Nash proved. Pick any manifold at all. Pick any way of measuring distance on it. Then there is some n-dimensional flat space and a particular way I can stick it in so that measuring distance on my manifold (now stuck into flat space) is just the "usual" way of measuring distance in the flat space.
It's kind of hard to communicate what makes this SO amazing without some exremely technical details, so I'll just focus on the breadth of it. It works for ANY manifold period. Anything that looks flat no matter where you zoom in. Also, it works for ANY meausre of distance, and there are a TON of them (an infinite number).
I dunno, it's beautiful stuff.
Sorry for the 2 long posts!
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I, of course, haven't given you the rigourous definitions or anything. Thus, there are a several statements I made which aren't ENTIRELY correct from a strictly mathematical viewpoint, without that additional rigour added. For the laymen who has no interest in mathematics, this should be good enough. I hope. (PLEASE DON'T MAKE ME TYPE ANYMORE!)
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And you have a bit of the gift that my mother has (and my dad has trouble with) -- making technical things comprehensible to the less-technical.
As far as the question between insanity and mathematical (and other) brilliance, I suspect that the same sorts of slightly-off brain chemistry and/or physiology that lead to (or are caused by) various types of insanity also lead one to see the universe from a perspective that is just different enough that one can come up with the brilliant insights of a John Nash.
As evidence, I present my parents. They will both freely admit to being slightly off. Posts: 32919 | Registered: Mar 2003
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Sorry about the John Forbes/Nash confusion. I don't know what came over me. I've always been terrible with names.
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Good stuff, Mathematician. Can you do a similar rundown on the Rienmann Hypothesis? I just finished "Prime Obsession" by John Derbyshire, but I don't feel up to trying to explain what's going on.
With analytical number theory and topology, I can grasp what's being discussed in a superficial way. With the former, I can form a sense - largely incomplete and superficial - of the implications, but I can't articulate them.
With topology, I can't even do that.
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posted
Of course, his full name is John Forbes Nash, so it's an understandable mistake.
Mathematician-
Your characterization of the beauty of Nash's ebbedding theorem is great. I would say exactly the same of his proof of a unique equilibrium in two-player, zero-sum games. I don't know that I've ever seen a more elegant proof.
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Topology and geometry (i.e., manifolds) are my interests. I know VERY little about the Riemann Hypothesis, but I'll do my best. I haven't read prime obsession, so I honestly don't really know what it covers.
So, here we go:
To even approach this, we must first discuss complex numbers. To start this, we'll start with a very simple equation.
Consider the equation x^2 = -1. Can this have a solution? Intuition says no, because if I multiply anything by itself (even if it's negative), you get a positive answer (or 0). So, we say that at least when x is considered as a "real" number, this equation has no solution.
Mathematicians HATE it when equations have no solutions, so we often times (when we can without creating havoc or destroying the universe) just make them up. In this case, mathematicians have decided to call the solution to that equation "i". That is, i^2 = -1 by definition. Clearly i isn't a real number, so we give it a new designation. We call it an "imaginary number".
Here's the kicker: There's nothing really "imaginary" about this number. It just took us much longer to stumble upon it than other kinds of numbers. For instance, the Greeks (around the time of Pythogarius) found that the number satisfying the equation x^2 = 2 (today we call it the square root of 2), didn't exist in the number system they used (rational numbers, which are just fractions where the numerator and denominator are numbers like ...- 2, -1, 0, 1, 2, ...). In otherwords, there is no fraction a/b = sqrt(2) where a and b can be any usual numbers you use in every day life.
They had a fit over this new kind of number - some of them even refused to believe they really existed. Today we call it an "irrational number" and use them without hesitation. Things like "pi" or the golden ratio "phi", as well as Euler's numer "e = 2.71...." are known to be irrational. Does anyone alive today doubt that the number "pi" is actually a number?
the "imaginary" number i is in the same boat. Upon first exposure, many people refuse to believe it "really exists", but it does. It is as "real" as the numer square root of 2 = 1.414....
Ok, so we have "imaginary numbers". Then what are "complex numbers"? A complex number is of the form a + b*i, where a and b are real numbers (think of a real number as anything with a decimal expansion, even if it goes on forever and never repeats).
Allright, so now that we got complex numbers covered, we move on to the Riemann zeta function.
We start by definining the Riemann zeta function. Let z = x+i*y be any complex number, then we define
f(z) = 1/1^z + 1/2^z + 1/3^z + 1/4^z + ...
The first thing people have issues with is this: What was, say, 1/1^i even mean? How in the world do you evaluate that? For now, just trust that mathematicians have solved it. We'll just grab a big theorem from them now.
Theorem: Suppose z is a complex number and you're going to add it, subtract, divide, multiply, take nth roots, etc to it (or any combination of things like that). Then your answer is a complex number.
In otherwords, scenarios like x^2 = -1, requiring the addition of "new" numbers don't come up with complex numbers. There are enough complex numbers to carry out ANY sort of algebra on them.
What this means, is that we really can figure out what f(z) is, and it will ALWAYS be a complex number (with one caveat. A real number IS a complex number. For instance, 5 = 5+0*i, pi = pi+0*pi).
Here's a few interesting facts to get started. f(1) = 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + ... = infinity, and this is one of the slowest growing infinite sums known.
If I plug in any complex number in for z, and f(z) is 0, then either z was one of those negative even numbers, or z is of the form z = 1/2 + a*i where a is some real number and I is the sqrt(-1). In otherwords, if z isn't one of the negative even numbers, then the real part of z must be 1/2.
So if the Riemann hypothesis is true, then every number that makes f(z) 0 is either -2, -4, -6, ..., or it's real part is 1/2.
If the Riemann hypothesis is false, then there is some number z so that f(z) = 0 but the real part of z isn't 1/2.
This problem was initially stated in 1859 by Bernard Riemann, was one of David Hilbert's 25 Millenium problems (I mentioned those above), and still hasn't been solved. We have found over 1,000,000,000,000 (a trillion) examples of numbers of the form 1/2 + a*i where f(1/2 + a*i) = 0, but none with real part different form 1/2.
Here's the kicker. There is an alternate way to write f(z). We chose the tradiational one.
f(z) = 1/1^z + 1/2^z + 1/3^z + 1/4^z + ...
but there's ANOTHER one.
It just takes a few steps. First, write a_n = (1-p_n^-z), where p_n is the nth prime number. (Recall, a number is prime if it's only divisible by 1 and itself. So, 2 is prime because we can only write 2= 2*1, but 9 isn't prime because we can write 9 = 3*3. The first few prime numbers are 2,3,5,7,11,13,17,...) So, for instance, a_1 = (1-2^-z), and a_5 = (1-11^-z).
Then f(z) = 1/(a_0 * a_1 * a_2 * ...). In otherwords, with both defintions (the one with the sum and the one with the products of primes), no matter which z we plug in, we get the same answer.
These two functions are the same in the same way that g(x) = x and g(x) = x + 1 -1 are the same (though this one is much easier to understand)
It turns out that if the Riemann hypothesis is true, we learn something about how often primes numbers appear in the list of all numbers. (this is the part that I don't really understand at all, so I won't go any further)
Why do we care about knowing about how prime number are distributed among the regular numbers? For years, it was just a mathematical curiousity, but today, a lot of encryption algorithms rely on primes. For instance, if you choose 2 incredible large prime numbers (I mean, like, with more than 100 digits a piece), a computer would probably never be able to tell you what those 2 original numbers were. This is (I think) known as the factoring problem. So here's how the encryption thing could work (VERY ROUGHLY):
Before I ever THINK of encoding something, we get together and we pick a HUGE prime number. Let's just call it b for now. So b = some really huge prime number.
Now, we part ways and I decide I want to send you an encrypted message. So I pick some other incredibly huge prime number c, and use it as my password for encrypting the message. I email you the encrypted message, as well as the product b*c. Since you already know b, you can easily do the division to get c, and hence, the password for unlocking it. But because it's SO hard to figure out what b and c are if you don't already know them (even with a computer), even if someone intercepts b*c and the entire message, they probably won't be able to figure out b or c (and hence, your password), so, your message is safe. In practice, there are many other issues (maybe the message was encrypted poorly, or if I send many messages and we use the same b on all of them, there are tricks that will help determine b, etc etc), but that's the general idea.
If the Riemann hypothesis is true, we know more about where prime numbers are. The more we know about where prime numbers are, the easier it is to factor things like b*c to find b and c.
That's about all I know.
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quote:Originally posted by SenojRetep: Of course, his full name is John Forbes Nash, so it's an understandable mistake.
Mathematician-
Your characterization of the beauty of Nash's ebbedding theorem is great. I would say exactly the same of his proof of a unique equilibrium in two-player, zero-sum games. I don't know that I've ever seen a more elegant proof.
I didn't realize his full name was John Forbes Nash. That's interesting.
Sadly, my knowledge of game theory is 0, but that's what getting more education is about!
And, personally, I do mathematics for the beauty of it. I find that many people have trouble comprehending that mathematics CAN be beautiful, much less see it. I want to share the beauty!
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posted
Mathematician- Thank you. My studies have been in physics, so some of these concepts are things I've touched upon (reimann sums, for example) but really have no understanding of.
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bump, because I don't want all that work to go to waste...yet ;-)
Also, feel free to ask other math questions. I certainly don't have all the answers, but I'll try my best!
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The annoying thing about the book was that it kept stressing that the Reinmann hypothesis was very important, but it never really made clear what it meant.
When they proved the four color map theorem, I understood that. (Not really, I was 6, but when I read about it later.)
With this, the hypothesis, if true, says something very important about the distribution of primes, which is very important about the fabric of numbers themselves, since a complete real number system can be constructed out of primes (maybe only rational - I can't remember). But I can only catch glimpses of what that the theorem means.
The book said many times that hundreds of theorems begin, "assuming the truth of the Reinmann hypothesis..."
One thing that fascinated me was that the distribution of the spacing of successive nontrivial zeros of zeta is statistically identical to with the distribution of eigenvalue spacings of Gaussian Unitary Ensemble operators.
The GUE operators happen to allow the modeling of quantum-dynamical systems. This means that there seems to be some connection between the zeta function, which somehow relates to the fabric of numbers themselves, and these operators, which relate to the fabric of the universe itself.
What that connection is, I have no idea. But it's interesting. Posts: 26071 | Registered: Oct 2003
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posted
Mathematics is definitely beautiful. Mathematician, your explanations are fantastic! You should teach this stuff.
Do you know anything about tensor calculus? I remember wishing I understood it long ago, for working problems in relativity or something. Probably not important anymore.
I was so excited when they proved the four color map theorem!
Did you ever read the column Mathematical Games by Martin Gardner in Scientific American magazine? That's where I found out most of my fun math. I took lots of math courses in college, mostly because they were an easy 5 hour A. But never really got very deep into what's going on in math that way, of course. I learned much more about that from Martin Gardner.
What is the cool thing about polynomials and Pascal's triangles? Can you tell the coefficients of (x + 1)^n or something when it's expanded by getting them from Pascal's triangle?
I love this thread and want to read more explanations like those here, but I can't think of any really good questions to ask.
Do you understand the math behind string theory?
Did you know that the universe is made of math? That's what Feynman said in "The Character of Physical Law". Go read it, if you haven't already (and if you have any interest in physics).
What is the question you would most like to answer here? Tell us the question, then answer it for us, please?
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Oh, I thought of one more! Didn't someone prove Fermat's Last Theorem a few years back? Tell us about that one. Posts: 6246 | Registered: Aug 2004
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posted
About brilliant mathematicians being insane, for my dad's funeral program, we had a Latin phrase that translated "There is no great genius without a certain admixture of madness."
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Martin Gardner is amazing. Every so often, I come across something new by him--this last year a short story in a mathematical fiction book and the Annotated Alice, in middle school the best book on codes and ciphers I've read yet (from practical and historical standpoints, plus some explanation of how to crack the ciphers and why)...and every time I'm just astounded. Sounds like the Mathematician, in a way: interesting cool things about math explained in a way that makes sense.
But:
quote:Does anyone alive today doubt that the number "pi" is actually a number?
Ooh, me, me! (Just to be contrary.) I mean, if you can't actually count sheep with it or pile pebbles with it it's not *really* a number, it's just some crazy thing mathematicians made up to justify extra funding for the math department. I mean, really, when do you use pi? Just when you're talking about circles, and you've already admitted that there are no perfect circles in real life, which are the only ones it really works for...so you're using made-up numbers to talk about made-up things! It's like Plato with his stupid Ideas...if I can't touch it or see it or sense it at all in real life, it's not really there. You're irrational.
O.K., in real life, I'm with Tatiana. Tell us more cool things!
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quote:Originally posted by Tatiana: Mathematics is definitely beautiful. Mathematician, your explanations are fantastic! You should teach this stuff.
Do you know anything about tensor calculus? I remember wishing I understood it long ago, for working problems in relativity or something. Probably not important anymore.
I was so excited when they proved the four color map theorem!
Did you ever read the column Mathematical Games by Martin Gardner in Scientific American magazine? That's where I found out most of my fun math. I took lots of math courses in college, mostly because they were an easy 5 hour A. But never really got very deep into what's going on in math that way, of course. I learned much more about that from Martin Gardner.
What is the cool thing about polynomials and Pascal's triangles? Can you tell the coefficients of (x + 1)^n or something when it's expanded by getting them from Pascal's triangle?
I love this thread and want to read more explanations like those here, but I can't think of any really good questions to ask.
Do you understand the math behind string theory?
Did you know that the universe is made of math? That's what Feynman said in "The Character of Physical Law". Go read it, if you haven't already (and if you have any interest in physics).
What is the question you would most like to answer here? Tell us the question, then answer it for us, please?
Wow, that's a ton of stuff your asking. Here goes!
First, with regards to teaching it, I'd LOVE to. Currently working on it (getting my Ph. D. so I'm allowed to teach it). So far I've only taught a single course, and it was ONLY computational, so I didn't have much fun with it (nor do I think I can teach computational mathematics well).
About Tensor Calculus....that's a big one. I'll let this entire post just cover that.
First, let's cover the idea of a vector space. A vector space is like a box that holds 2 different kinds of objects: vectors and scalars. There are 2 things I'm allowed to do. I can add 2 vectors together (and get another vector already in my box) or I can multiply a scalar and a vector and get a vector already in my box. These operations have to satisfy a few rules just to make sure they behave allright. Just so we have a picture in our heads, consider a plane. Our vectors will be arrows we draw. The base of the arrow will lie at the origin and the head of the arrow will lie at the different points in the plane. Adding vectors will be done pointwise. Scalars will just be regular real numbers. Multiplying by scalars will correspond to lengthening our arrows. For instance, if we multiply a vector by 2 we double the length.
For the rest of this post, assume x and y are vector and a is a scalars.
think of a function as something that accepts some vector, and spits out a (probably different) vector.
We call a function f "linear" if f(x+y) = f(x) + f(y) AND f(a*x) = a*f(x). We term them linear because in our picture above, such functions trace out lines.
It turns out that if you look at ALL possible linear functions on a vector space, this TOO is a vector space (we call the objects in here "covectors" and "scalars") (called the dual space). This is kind of hard to wrap your head around the first time you see it.
Here's the definition of a tensor: A tensor is a function which accepts lots of vectors and covectors and spits out a scalar. But it must be linear in EACH of the (co)vectors.
To try to give an example.....
Assuming you're somewhat familiar with a matrix, consider an n x n matrix. Think of the matrix as n vectors, each containing n numbers. Then the determinate of the matrix is a tensor when you view the inputs as each of the columns. This example is peculiar in that covectors don't enter the picture at all.
Moving on....Suppose we have a manifold (something that looks flat up close, but far away, we have no idea). There is a way to put a vector space (and hence, a dual vector space) at EVERY POINT of the manifold. Thus, we can have tensors at every point of the manifold. But since we have a ton of tensors, we can ask questions about how the tensor varies as I move about from point to point in the manifold.
It turns out, Lie derivatives become very useful here in analyzing the rates of change of the tensors. I don't really know much about the details here, so I'll leave it at that.
As far as the relationship to general relativity....
Einstein's equation for gravitation:
G = 8*pi *T is a tensor equation. Here, G and T are tensors which accept 4 vectors and 4 covectors. G relates to the curvature of space and T relates to how much matter/energy is present in the space.
The thing is, I'm not exactly sure what our "vectors" and "covectors" are in the context of general relativity. 4-velocity and 4-momentum, respectively, would be my guess, but I'm not sure at all.
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I actually HATE the 4 color theorem. That is not to say that I in any way doubt the validity of it, I just believe that there must be a better way than splitting it into 3000 cases and having a computer check them all. I just think that such solutions are ugly.
Martin Gardner:
I have heard of him, but to my knowledge, have no experience of anything he's done. This will probably change in the future.
Polynomials and Pascals triangle:
There isn't a strong link between arbitrary polynomials and pascals triangle, but there is between polynomials of the form (something)^n.
First, to clue everyone else in, Pascal's triangle is easy to generate. You put 1's on the edges of a triangle and fill in the rest (starting from the top) by looking at the 2 above numbers and adding them together.
(google search it, that's probably the best way to see it).
Here's the cool part.
Suppose I want to expand (1+x)^47. Using normal precalculus, I'd have to do a TON of multiplying to figure it out. No one would EVER want to figure it out. It would probably take several hours (by hand at least).
But there's a trick.
I know my final answer is gonna look something like a0 + a1*x + a2*x^2 + ... + a47 * x^47, so really I just need to figure out what a0, a1, ..., a47 are.
Here's the quick and easy way. Draw out 48 lines of Pascal's triangle. Just read off the last line, those will be your answers, in order.
Here's a quick example to illustrate. (1+x)^3 = 1 + 3x + 3x^2 + x^3.
Writing out pascal's triangle out to 4 lines, we have:
1 1 1 1 2 1 1 3 3 1, exactly the coeffecients we wanted (I hope the formatting works!).
Something even cooler, notice if we interpret the first row as digits, this is just 11^0. Interpreting the second as digits, we have 11^1, the 3rd as digits is 11^2, and the fourth as digits is 11^3. This pattern continues forever (though you hvae to be VERY careful when the "digits" get bigger than 10).
String theory: I've had a single course on something called "super algebra". It laid the foundations for the mathematics behind such things as "super gravity", "super string theory", and "super symmetry", but that's all I've had so far. String theory intrigues me from a mathematical standpoint, but I think it's garbage as a physical theory (at least until it actually predicts something).
Feynman: I have read no Feynman, but I definitely plan on it...
Posts: 168 | Registered: Jul 2006
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Hey, you never answered me when I asked in the class schedule thread. Have you ever used/read a Reed-Simon textbook?
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" String theory intrigues me from a mathematical standpoint, but I think it's garbage as a physical theory (at least until it actually predicts something)."
This is a favorite of mine because it DOESN'T require paragraphs and paragraphs of explanation to the non-mathematical (or as I term them, the mathematician wannabe's).
First some background, though. We've all heard the equation a^2 + b^2 = c^2 for right triangles. Many people know that, for instance, 3^2 + 4^2 = 5^2, as just one example.
In the 1600's (I think), Pierre de Fermate wrote (roughly)this in a margin of a book he was reading: "Consider the equation x^n + y^n = z^n, where n is bigger than 2. Then this has no solution if x,y, and z are integers (i.e., -2, -1, 0, 1, 2, ...). I have a found a marvelous proof of this, but it's too long to fit in this margin". Today, this is known as Fermat's last theorem.
In otherwords, if I tell you x^3 + y^3 = z^3 with x y and z integers, you know I'm lying. Same idea if I tell you x^3434 + y^3434 = z^3434 with x y and z integers. There just aren't any solutions.
Now, people made some quick progress on this. They showed pretty quickly that there were no solutions if n = 3, 4,5,6, or 7. But there were still an infinite number of n's to rule out!
Lets take a step back and look at how this problem can be handled geometrically. First, lets go back to n = 2.
We have x^2 + y^2 = z^2, or equivalently, (x/z)^2 + (y/z)^2 = 1. If we fix z, then this is the equation of a circle of radius z. Now we're looking for points on the circle where x and y are both integers. So in a sense, we've made the problem more geometric.
Fast forward 300 years....
The problem was STILL unsolved. Sure, a ton of specific n's had been ruled out (and also, all possible working solutions for n = 2 were found), but there were still an infinite to go!
Some guy (I forget his name) was studying curves in space and he made a conjecture about the possible behaviors of those curves. Shortly afterwords, someone showed that if this guy's conjecture was true, then fermat's last theorem was true - there are no solutions for n>2.
Enter Andrew Wiles. What he did is prove a large chunk of the conjecture, not enough to fully prove the conjecture, but enough that he showed Fermat's theorem was true....almost.
He had to have his findings approved by other mathematicians, and they found a flaw. A journalist once asked Andrew Wiles to describe the flaw. His response, "It would take me a month to teach you enough mathematics to understand what went wrong". I certainly have no clue what went wrong.
What is important is that he was able to find a way around that error and fix up everything. Thus, Andrew Wiles proved that there are no solutions to x^n + y^n = z^n unless n<3.
Interestingly, his proof is over 100 pages long and relied on the most current mathematics available. Because of this, many people doubt that Fermat REALLY had a proof of the theorem.
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Finally, "what is the question I most want to answer?" That's EASY!
The nature of infinities. What is infinity? Can one infinity be larger than others? How can you tell?
My favorite subject - the mathematicians conception of infinity. To approach this, we'll use an example inspired by Gwen. She mentions this: (I paraphrase, probably) "if you can't count sheep or a pile a pebbles with it, it's not a number". This EXACTLY highlights what counting is. It's matching things up. For each "number" you assign a pebble. Once you exhaust the pebbles, that's the "number" of pebbles.
More to the point, there's a big issues of "what exactly is a number" The technical definition is quite irrelevant. But here's the basics. Consider boxes (called sets by mathematicians). Just so we're on the same page, we declare that 2 boxes are the same if they hold exactly the same items. Now we're ready to define numbers.
0 is by definition a box with nothing in it.
1 is by definition a box that has a box with nothing in it. That is, 1 is the box that has 0 in it.
English wise, this is gonna get really messy, so lets abbreviate.
2 is by definition the box which has 0 and 1 in it.
Rinse repeat.
So 8324 is by definition the box that contains 8323, 8322, ..., 3,2,1,0.
Here's the important thing. The box we call 2 has "2" things in it. The box we call 45 has "45" things in it.
So what is counting? Well, lets go back to counting pebbles. Suppose I really have 5 pebbles, how do I determine this? I look at my 5 box (which contains the boxes for 0,1,2,3,4) and I notice I have a correspondence. For every "number" in the 5 box, I have a pebble. Thus, I declare that I really do have 5 pebbles.
Notice that if I try to stuff the pebbles in the 4 (or less) box, I'll always have a pebble left over. Likewise, if I try to stuff them in the 6 box (or higher), I'll always have boxes left over.
That is the essence of counting - it's matching things up.
We are almost ready to venture into infinities....just one more thought experiment.
Suppose you're touring the earth, and you come upon a tribe of people who can only count to 3. They just don't have a conception of things higher than 3. Suppose some mother has 5 children and 5 beds. Without being able to count them all, how can the mother determine if all her children are there?
Simple. Stick one child in each bed. If all the beds get used up, she has all her children. Does she know how many she has? No. Does she know that she has the same number of children as beds? Yes.
The importance of this is as follows: just because you can't count that high doesn't mean you lose the ability to tell if 2 things are the same size.
Apply this SAME reasoning to yourself. You can only count to finite numbers, but if you had an infinite number of apples and oranges, you could still tell if they were the same size. You just match them up. If no matter what, you have oranges left over, then you have fewer apples than oragnes, period.
Enter mathematics. Most people are ok with the notion of infinity, it exists. There are an infinite number of real numbers, there are an infinite number of regular (or natural) numbers (1,2,3,4, ...). Are these the same size of infinity? It turns out the answer is no. Cantor proved that no matter how you try to pair up the natural numbers with the real numbers, you just can't do it. There will always be real numbers left over. Hence, the size of all the real numbers is bigger than the size of the natural numbers.
It turns out, its quite easy to prove that the size of the natural numbers is the smallest possible infinite size.
Before moving on, I want to emphasize that infinities don't behave in any sort of intuitive way. Probably the most famous example of this is "Hilbert's Hotel". (This idea is not my own, I intend to take no credit for it)
Suppose you are the owner of an infinitely large hotel. Suppose further, that every room is filled. Now, late at night, a person shows up and asks for a room. What do you do? Well, you can just ask everyone of the tenants to move down a room. Now you have room for this new person!
But wait! All of a sudden a bus full of an infinite number of people arrives! Now what do you do? If a person is in room n, you tell them to move to room 2n. So person in room 4 moves to room 8, the person in room 50 moves to room 100. Then end result? You have an infinite number of free rooms available, so everyone on the bus gets a room.
It turns out that even if an infinite number of buses carrying an infinite number of people arrive, there are ways to to fit them in. This is CRAZY!
Infinities get even crazier. Before moving on, remember 2 things - Godel proved that in any sufficiently interesting system, there will be things you just can't prove to be true.
Second, the size of the natural numbers is the smallest infinity and is smaller than the size of the real numbers.
A very natural question arises: Is the size of the reals the second largest infinity? It turns out that this is one of those statements Godel warned us about. There is NO WAY to prove one way or another whether or not the size of the reals is second largest infinity.
In 1935, Godel proved that there's no way to disprove the fact. In 1965, Paul Cohen (using a BEAUTIFUL method called "forcing", which he invented to do the proof) proved that there's no way to prove it, and thus, it will forever remain out of the reach of mathematics.
Just to give a basic hint of how you can prove that you can't prove something....
Suppose this is all the information I give you:
1. There is a person named OSC. 2. All of the books writeen by OSC are awesome.
Here's the "unprovable" statement: There are books which aren't written by OSC but which are awesome. (I'll call it Q in the following discussion)
Now, ask yourself "Is it possible to envision a universe in which there are no other awesome books beyond those written by OSC?" The answer is clearly yes. What this means is that from 1 and 2, we cannot conclude that Q is true, because there are possible universes where Q is false.
Similarly, ask yourself, "is it possible to envision a universe in which there are other awesome books beyond those written by OSC" Again, the answer is clearly yes. This means that from 1 and 2, we cannot conclue that Q is false, because there are possible universes in which Q is true.
Thus, we have that we can't prove Q to be true or false.
That's the idea with asking if the size of the reals in the second smallest infinity. Is it possible to envision a universe where it is? To mathematicians, yes. Is it possible to envision a universe where it isn't? TO mathematicains, again, it's a yes. (Though you must be very careful about exactly what it means to "envision" in a mathematical sense, as well as what exactly a "universe" is in a mathematical sense.
That's enough typing for the evening. I think I set the Hatrack "most typing in one day" record...by far.
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quote:Originally posted by Paul Goldner: " String theory intrigues me from a mathematical standpoint, but I think it's garbage as a physical theory (at least until it actually predicts something)."
The problem is exactly what that site highlights: Right now string theory makes NO experimentally verifiable results, unless you add supersymmetry (and then, only maybe).
My problem with it is this: it is entirely possible to have supersymmetry without having strings. In fact, supersymmetry came about from trying to unify the electroweak force with the strong force (no mention of strings at the time)
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I think there is a difference between not predicting anything, and not predicting anything we can currently demonstrate experimentally.
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Ug. And I struggled like crazy just to make some inverse kinetics code to make a 3d robot walk. My brain hurts trying to read this stuff. And once upon a time I could understand calculus...
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This is all very interesting Mathematician. Thanks so much for sharing it with us . I found your last post on the nature of infinities very easy to understand, and I've so far only finished Honors Al2/Trig (Ok, I still don't quite get the whole reason why the size of the natural numbers is the smallest possible infinite size, but I'm sure being up at 3 during summer vacation isn't helping either, because I didn't remember what natural numbers were). I haven't completely read everyone one of your other posts here, but they sound like they'd be something to think about.
I've strangely enough also been interested in the nature of infinities. Most prominently, I've always wondered about what I think of as "ending infinities". I'll give you an example. At some point after 0, the number 1 begins. And at some point after that, the number 1 ends and the number 2 begins. First of all, I should be asking, is that even true? Is there a point where the number 1 ends ands and 2 begins? And if so, this is where what I think of as an ending infinity comes in.
If there is actually a point where a number begins and ends, then how can the number of points in between be called infinite, even if there IS a never ending number of them? I guess what I'm saying is that 1 obviously starts at a point somewhere after .999999999999... (Or I guess after .999999999...[infinity]...8, since .999 repeating is the same as 1), so using that same logic, 1 also ends before 1.99999 repeating. If that's true, and the number 1 does indeed end, how can there be an infinite amount of numbers between 1 and 2? (Like 1.2, 1.23, 1.899238923, 1.99999993, and so on).
Forgive me if this is addressed at some point not too much further along down my road of math classes.
Any kind of answer would be wonderful Mathematician. You’re great at explaining things. Thank you so much! Posts: 879 | Registered: Apr 2005
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I'm so glad you liked my questions enough to answer them at such length! A veritable cornucopia of delicious mathematics! You really are great at explaning things.
I remember Cantor's diagonal argument from Godel, Escher, Bach (you *have* read GEB, haven't you?), but I want to know more about infinities. I seem to remember that the points on a plane are the same infinity as the points on a line. Are there higher infinities than that? What are they?
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Mathmetician, Can you explain to me the axiom of choice? I understand that this is an unproved axiom which is generally assumed by most mathemeticians to be true. I'm told that assuming this axiom to be true, greatly simplifies mathematical proofs but I don't clearly understand why perhaps because I don't clearly understand the axiom.
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quote:Originally posted by Eaquae Legit: Out of curiosity, Mathematician, where did you go to school?
I'm currently attending the University of Pennsylvania to get my Ph.D (and maybe a master's in Physics while I'm at it). They say the program will take 4 or 5 years. I'm hoping for 4.
I did my undergrad at North Carolina State University (BS in math and physics) and got my M.S. in mathematics at North Carolina State as well. My Master's topic was basically deriving my Junior level physics course. (They had just handed us equations and said, "use these". I HAD to konw where they came from).
Posts: 168 | Registered: Jul 2006
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quote:Originally posted by Paul Goldner: I think there is a difference between not predicting anything, and not predicting anything we can currently demonstrate experimentally.
Agreed. Maybe it would have been more accurate for me to say that until we reach a point where we have the ability to falsify it, I think string theory is wonderful mathematics/philosophy, but shouldn't be regarded as physics. As soon as we, technologically, reach a point where we can test it, we can talk about it being physics.
P.S. I have both of Brian Green's books (one is signed), as well as the NOVA DVD "The elegant universe" which is ALSO signed by Brain Green (the narrator). I'm such a nerd!
Posts: 168 | Registered: Jul 2006
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quote:Originally posted by Hamson: This is all very interesting Mathematician. Thanks so much for sharing it with us :) . I found your last post on the nature of infinities very easy to understand, and I've so far only finished Honors Al2/Trig (Ok, I still don't quite get the whole reason why the size of the natural numbers is the smallest possible infinite size, but I'm sure being up at 3 during summer vacation isn't helping either, because I didn't remember what natural numbers were). I haven't completely read everyone one of your other posts here, but they sound like they'd be something to think about.
I've strangely enough also been interested in the nature of infinities. Most prominently, I've always wondered about what I think of as "ending infinities". I'll give you an example. At some point after 0, the number 1 begins. And at some point after that, the number 1 ends and the number 2 begins. First of all, I should be asking, is that even true? Is there a point where the number 1 ends ands and 2 begins? And if so, this is where what I think of as an ending infinity comes in.
If there is actually a point where a number begins and ends, then how can the number of points in between be called infinite, even if there IS a never ending number of them? I guess what I'm saying is that 1 obviously starts at a point somewhere after .999999999999... (Or I guess after .999999999...[infinity]...8, since .999 repeating is the same as 1), so using that same logic, 1 also ends before 1.99999 repeating. If that's true, and the number 1 does indeed end, how can there be an infinite amount of numbers between 1 and 2? (Like 1.2, 1.23, 1.899238923, 1.99999993, and so on).
Forgive me if this is addressed at some point not too much further along down my road of math classes.
Any kind of answer would be wonderful Mathematician. You’re great at explaining things. Thank you so much! :)
First off, don't feel too bad about understand that the natural numbers are the smallest infinity. I just kind of stated it with no justification. Let me try to explain it....
Suppose you have another infinite set. Let's just call it A.
Because it's infinite, it has 1 thing in it. So let's pair that one thing up with 1. Now, think about A with that first thing removed. It's certainly still infinite, so there's another thing in it. Pair that up with 2. Continue this process. Suppose at some stage, you ran out of things in A to pair up with natural numbers. Then that exactly tells you your size of A, and that it is finite. Therefore, this process never ends.
Now, we don't know if we end up using all of A, so we don't know that the naturals are as big as A. All we know is that A is AT LEAST as big as the naturals. But since A is ANY infinite set at all, it must be true that ANY infinite set is at least as big as the naturals. Thus, the naturals are the smallest.
As far as the other things you've said. You have to be careful. There is no such thing as .999999 (infinite number of 9's)99998. Because, then there would a "last 9" on which I can stick an 8. But if there's a last 9, there's only a finite number of them. But this contradicts our assumption that there were and infinite number of them! Thus, there is NO last 9. So I can't just stick an 8 at the end of the sequence of 9s.
Also curious is this following property: Every natural number can be written down in 2 ways, not just 1.
here they are: For 1 = .99999..., 2 = 1.999999...., 45 = 44.999999, etc.
Why does this work? Let's just check it for 1=.99999999.
Here's 2 alternate ways to see it.
First, consider the fraction 1/3. It is not too hard to see that this can be written in decimal form as 1/3 = .33333333333333333....
Now, multiply both sides by 3, you get
1 = .9999999999999999...., as desired.
Here's another way to see it, that I find more clever:
Let .9999999999.... = x.
Multiply both sides by 10 and you get
9.9999999999.... = 10x.
Now subract "x" from both sides. You get
9 = 9x, which tells us x = 1!. Therefore .9999999999..... = 1.
Getting back to the infinities a bit, you have to be VERY careful when messing with the real numbers. There is ALMOST NEVER a "next number" or a "previous" number. That's because between any 2 numbers (no matter how close) there are an infinite number of numbers.
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This reminds me a lot of this thread from last March. I especially like the thought question about the funnel on the second page.
I don't believe in infinity and infintesimals. Their nice mathematical conceits, but I don't believe they have any physical reality to them. Cantor (and, to a lesser extent (haha), Newton) was a kook.
Posts: 2926 | Registered: Sep 2005
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quote:Originally posted by Tatiana: I'm so glad you liked my questions enough to answer them at such length! A veritable cornucopia of delicious mathematics! You really are great at explaning things.
I remember Cantor's diagonal argument from Godel, Escher, Bach (you *have* read GEB, haven't you?), but I want to know more about infinities. I seem to remember that the points on a plane are the same infinity as the points on a line. Are there higher infinities than that? What are they?
I have read GEB. I LOVED the first half, and HATED the 2nd half. Though, I'm not one to accept hand wavy arguements, so shortly after reading GEB, I purchased a copy of Godel's original groundbreaking paper (and a book on logic). I'd really like to understand all the nuts an bolts of what Godel did (I still don't see how Godel used Godel numbers in his proof).
More about infinities... YAY!
Lets first talk about the number of the infinites. Earlier we have discussed 2 - the size of the natural numbers and the size of the reals. The question is, are there more?
Turns out...the answer is VERY surprising. When Cantor proved the natural numbers are smaller than the reals, he didn't actually use the diagonal argument (which I detest as the mathematical notion of "list" isn't rigourous enough), but a more abstract method. The thing about this method is, it was MUCH harder to show specifically that the naturals were smaller than the reals, but it was VERY easy to show a method to create a bigger infinity given one of any size.
From here, it is easy to see that there are at least as many infinities as there are natural numbers. What's even crazier is this: You can show (if you kick a mathematician hard enough), that there are SO many infinities that we don't have an "infinity" to describe that number!
As far as points on the line matching up with the points on the plane. You are correct!
To see this, I'll go back into the decimal expansion (which we know has some problems, but I'm hoping you'll just trust that these are easy to work around).
we want to take a real number r and match it up with a PAIR of real numbers (s,t).
We have to be careful how we do it - we don't want 2 different real numbers going to the same ordered pair. Likewise, we have to make sure that EVERY ordered pair gets hit by some real number.
Here's how I'll do it. First, I'll show that the line between 0 and 1 (NOT including 0 and 1) can be matched up with the square of length 1 (again, not including the end points).
So, let r be a real number between 0 and 1. So r = 0.982749832779234 or whatever. I'm going to make "s" by taking all the even digits of r. I'll make t by taking all the odd digits of r.
So in this example, t = .92482723 and s = .8793793
It's not TOO hard to show that this pairing has the desired properties. Thus, the infinity of the line between 0 and 1 is the infinity of the square of length 1.
Now, to extend this to the WHOLE real line, we can use the "tangent function" from trig. It turns out that if you slightly modify the tangent function, then apply it to everything between 0 and 1, you get ALL the REAL numbers! This means that a finite length line has the SAME number of points as an infinite lenght line. More to the point, we can use 2 modified tangent functions to show the number of points in the square is the same as the number of points in the whole plane. Here's our end result: The number of points in the infinite line is the number of points in the finite line is the number of points in the square is the number of points in the whole plane.
Thus the number of points in an infinite line is the same as the number of points in the infinite plane. In fact, even if you have a natural number number of dimensions, that things still only has as many points as a finite line (though showing this is MUCH harder).
The more you study infinities, the crazier they (and you?) become.
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quote:Originally posted by The Rabbit: Mathmetician, Can you explain to me the axiom of choice? I understand that this is an unproved axiom which is generally assumed by most mathemeticians to be true. I'm told that assuming this axiom to be true, greatly simplifies mathematical proofs but I don't clearly understand why perhaps because I don't clearly understand the axiom.
This is one of my FAVORITES!
Here's the axiom of choice in a nutshell:
Suppose I have an infinite collection of seperate boxes. Now the things in the boxes don't have anything distinguishable about them. The axiom of choice says that there is a "choice function", something which allows me to choose one thing out of each box.
To loosely quote Bertrand Russel, if I have an infinite number of shoe boxes, I DON'T need the axiom of choice, because I can say "choose the left shoe out of each box". If instead of shoe boxes, I have "sockboxes" (boxes containing pairs of sox), I need the axiom of choice to choose a sock out of each box, because there's not something that makes one sock better to pick than another.
Seems harmless, right? Turns out that you can prove a TON of nonintuitive things using this idea. For instance, Banach and Tarski proved, using the Axiom of Choice, that I can start with a sphere, but it into a specific 7 pieces, rotate and shift the pieces, and end up with 2 spheres EXACTLY the same size as the 1 sphere. (Note, there was no stretching, just rotating and moving)
Beyond that, the axiom of choice is still more problematic. For instance, it tells you you have a function, but it doesn't tell you what ANY of the values of the function are. That is, you may use the axiom of choice to KNOW you have a way of pairing up 2 sizes of infinity, but if I ask what thing "1" matched up with, you can't tell me.
In otherwords, the axiom of choice is nonconstructive - it tells me I have something with desired properties without it accidentally having any other (possibly useful) properties.
Here's the kicker - there are a TON of intuitive results that can only be proved using the Axiom of Choice. As a general rule, mathematicians aren't willing to give up these intuitive results just to get rid of the non-intuitive ones that come with it.
Beyond this, there are TON of equivalent formulations (meaning, if we accept an equivalent formulation we can prove the AoC true and if we accept AoC as being true, we can prove the equivalent formulation), some of which are AMAZINGLY intuitive, others which aren't.
As an example, the axiom of choice is equivlent to the following statement - "Every box has a size (though it may be infinite)". That's a VERY 'obvious' statement.
Beyond this, Godel (and someone else, perhaps), proved that AoC can't be proved true or false, using all the other axioms of mathematics (same idea as with question of whether or not the size of the reals is the second largest infinity).
The AoC is probably the least standard of all the axioms, but for the most part, mathematicians use it without a second thought. It's just too powerful to leave out.
There are some weakened version of AoC. For instance, instead of giving me a choice function for arbitrary infinite collections of boxes, there's something called "countable choice", which says that if I have a natural number number of boxes, then I have a choice function, but if I have an infinity bigger than that, I don't know if I do or not.
I haven't explored the implications of this weaker form, so I'll leave it at that.
Posts: 168 | Registered: Jul 2006
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quote:Originally posted by SenojRetep: I don't believe in infinity and infintesimals. Their nice mathematical conceits, but I don't believe they have any physical reality to them. Cantor (and, to a lesser extent (haha), Newton) was a kook.
I really like the way your phrased it. Because I DO believe in infinity and infinitesimals, but I have no feelings as to whether or not they have any physical reality.
I consider myself a "Platonist". Loosely, I think mathematics exists independent of humanity/society. Thus, I don't think we "create" mathematics, but that we "discover" it.
But that's metamathematics/metaphysics for another day ;-)
Posts: 168 | Registered: Jul 2006
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posted
It's great if the Poincare conjecture has been resolved, but I'm a bit cheesed off at how all these mathematicians are treating Perelman as a god on earth. Good for him coming up with the key ideas needed to resolve the conjecture. Maybe he really did make enough progress on the problem to deserve the Fields Medal and/or the Clay Institute prize. But he stuck the rest of the mathematical community with the job of unpacking a bunch of cases that he called "trivial" or "analogous to previous proofs" that turned out to be neither. I've read that his papers would not have been accepted in a high-level journal without major revisions. And a man with this sort of attitude is getting such unqualified praise?
I have no problem with Perelman; he did what he did, and he hasn't mocked anyone for not following his proof. (If he wasn't going to work on the problem anymore after posting his papers, it would have been nice of him to publicly say something like, "Sorry, these papers are the best I can do. There's nothing any further discussion with me will reveal, and I'd rather do something else for a while." But I'm willing to accept his lack of comment on the papers as tacit admission of this.) I'm just angry at the mathematicians who seem to think that Perelman has no flaws, that he speaks in the language of God, which we mere humans should be blessed to be able to understand.
Posts: 781 | Registered: Apr 2005
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posted
It's odd how there is room for ego and bitterness, even in this most rarefied of human endeavors.
I also believe in infinity and infintesimals, just like I believe in a lot of other things that don't necessarily have any particular physical manifestations. Like Beethoven's Ninth Symphony, for example, which exists even when nobody is listening to it or performing it and would even if all the copies were destroyed along with everyone who had ever heard it.
I think this is my favorite thread on Hatrack ever in ten years!
Will you explain the AoC some more? It is just saying that if you have an infinite number of boxes, with any number of things in the boxes, that there is a way to pick one specific thing from each box? What if one of the boxes has zero things in it? Is that not allowed?
Oh, and I totally believe Fermat had an elegant solution to his last theorem. We will figure it out eventually. Did he lie? Did he make a mistake? Neither of those things seems very likely. Is there an elegant proof that nobody else has thought of in all these years of trying? That seems perfectly likely to me. After all, these things are only obvious AFTER you see them. Posts: 6246 | Registered: Aug 2004
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I have been reading "The Road to Reality" by Roger Penrose, which first gives and mathematical foundation, and then goes on to explain how they explain the physical laws. The problem is, he writes so densly and jumps from a simple subject (at the beginning of the book when he's just trying to bring readers up to speed) to much more complex examples and descriptions. I find much of it is over my head.
But there are two things I would really like to know and, unfortunetly, the online explanations have not really helped.
1) what is a Line integral? I've had up to calculus 2, which included a preliminary on vectors. I don't remember much of that, but I like your explanations.
2) What is complex-analysis?
thanks for anything you can offer.
this thread is awesome.
Posts: 1346 | Registered: Jun 1999
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It really is as simple as just allowing you to pick an arbitrary thing out of each box. But you are right (and I should have specified), the boxes must be non-empty for any of this to make sense.
The deal with the axiom of choice, is that it is just that...an axiom.
So here we go, a discussion on axioms.
Logic is a process by which we can true statements and conclude true statements from them. In essense, logic is the study of which forms of reasoning preserve truth.
For example, consider this line of reasoning.
1.Either A or B is true. 2.A is not true. 3.Therefore B is true.
If we assume the first 2 statements are true, then it logically follows that the 3rd is true.
That is what logic studies. So what is an axiom then? An axiom is like 1 or 2 above, it's a statement we choose (by vote, by monarchy, whatever) to be true.
Some care must be excercised, however:
Suppose we accept the following as our axioms:
1. Either A or B is true 2. A is not true 3. B is not true
It should be clear that we have a contradiction, for all 3 axioms can't be satisfied at the same time.
Mathematics can be thought of (and is by some) as a list of axioms + all logical deductions from those axioms.
There are several schools of thought on what the "proper" axioms for mathematics are. By far the most common is something calld ZF for Zermelo-Fraenkel axioms. There are 7 axioms and 2 collections of an infinite number of axioms. Most of these are VERY intuitive. For instance, Axiom 0 states "there is a set", or the in the language I'm been using "there is a box". More or less everyone agrees that if we're gonna build all of mathematics out of the boxes, we'd better have at least one ;-).
Others are just as intuitive. If I have 2 boxes, I can pour their contents into another box. If I have 2 boxes, then I have another box which contains exactly those 2 boxes.
What about that issue above about never being able to satisfy all the axioms at once? Godel proved that the only answer we'll ever be able to prove is that they aren't consistent. That is not to say that they ARE inconsistent, just that there can't be a proof that they ARE consistent.
Now, the thing with the axiom of choice is that it's MUCH less intuitive, but it's also MUCH more powerful. Godel (and maybe someone else too) has proved that assuming all our previous axioms are consistent, we don't generate an inconsistency by adding the AoC to our list of axioms.
Because of it's nonintuitiveness, some (definitely not most) mathematicians are against using it.
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