quote:

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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.

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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.

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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.

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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.

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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.

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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.

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Recently, Dr. Perelman is said to have resigned from Steklov. E-mail messages addressed to him and to the Steklov Institute went unanswered.

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“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.”

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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 by Eduardo St. Elmo (Member # 9566) on :
 
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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Posted by Bob_Scopatz (Member # 1227) on :
 
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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Posted by Mathematician (Member # 9586) on :
 
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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Posted by katharina (Member # 827) on :
 
I wanna know. All four, if you could.
 

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Posted by Angiomorphism (Member # 8184) on :
 

quote:

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Originally posted by Bob_Scopatz:
Meanwhile, how long is the coast of Maine?

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quote:

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Curiously, a number disproportionate number of the truly gifted mathematicians I've known have suffered from serious mental illness.

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quote:

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Originally posted by katharina:
I wanna know. All four, if you could.

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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

[ August 16, 2006, 01:27 PM: Message edited by: Mathematician ]
 

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Posted by Mathematician (Member # 9586) on :
 
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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Posted by Mathematician (Member # 9586) on :
 
addendum:

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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Posted by rivka (Member # 4859) on :
 
Cool stuff.

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.
 

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Posted by TrapperKeeper (Member # 7680) on :
 
Problem:

How far would a stationary spherical rabbit fly if hit with a baseball bat?

If the rabbit was frozen?
 

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Posted by The Rabbit (Member # 671) on :
 
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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Posted by Dagonee (Member # 5818) on :
 
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 by SenojRetep (Member # 8614) on :
 
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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Posted by Mathematician (Member # 9586) on :
 
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.

f(2) = 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + ... = pi^2/6 (where pi = 3.1415926535....)

f(-2) = f(-4) = f(-6) = f(-8) ... f(-even number) = 0.

The Riemann hypothesis simply states this:

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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Posted by Mathematician (Member # 9586) on :
 

quote:

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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 by Paul Goldner (Member # 1910) on :
 
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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Posted by Mathematician (Member # 9586) on :
 
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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Posted by Dagonee (Member # 5818) on :
 
Mathematician, thanks for the rundown.

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.
 

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Posted by Tatiana (Member # 6776) on :
 
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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Posted by Tatiana (Member # 6776) on :
 
Oh, I thought of one more! Didn't someone prove Fermat's Last Theorem a few years back? Tell us about that one.
 

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Posted by Tatiana (Member # 6776) on :
 
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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Posted by Gwen (Member # 9551) on :
 
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:

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Does anyone alive today doubt that the number "pi" is actually a number?

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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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Posted by Mathematician (Member # 9586) on :
 

quote:

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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?

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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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Posted by Mathematician (Member # 9586) on :
 
4 color theorem:

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...
 

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Posted by rivka (Member # 4859) on :
 
No Feynman, and no Gardner!?!

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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Posted by Paul Goldner (Member # 1910) on :
 
" 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)."

http://www.superstringtheory.com/experm/index.html
 

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Posted by Mathematician (Member # 9586) on :
 
Fermat's last theorem.

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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Posted by Mathematician (Member # 9586) on :
 
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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Posted by Mathematician (Member # 9586) on :
 

quote:

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Originally posted by rivka:
No Feynman, and no Gardner!?! :eek:

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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quote:

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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)."

http://www.superstringtheory.com/experm/index.html

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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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Posted by Eaquae Legit (Member # 3063) on :
 
Out of curiosity, Mathematician, where did you go to school?
 

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Posted by Paul Goldner (Member # 1910) on :
 
I think there is a difference between not predicting anything, and not predicting anything we can currently demonstrate experimentally.
 

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Posted by human_2.0 (Member # 6006) on :
 
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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Posted by Hamson (Member # 7808) on :
 
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!
 

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Posted by Tatiana (Member # 6776) on :
 
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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Posted by The Rabbit (Member # 671) on :
 
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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Posted by Mathematician (Member # 9586) on :
 

quote:

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Originally posted by Eaquae Legit:
Out of curiosity, Mathematician, where did you go to school?

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quote:

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Originally posted by Paul Goldner:
I think there is a difference between not predicting anything, and not predicting anything we can currently demonstrate experimentally.

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quote:

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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! :)

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quote:

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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?

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quote:

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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.

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quote:

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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.

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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 ;-)
 

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Posted by Omega M. (Member # 7924) on :
 
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.
 

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Posted by Tatiana (Member # 6776) on :
 
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.
 

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Posted by IanO (Member # 186) on :
 
This thread is so cool.

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.
 

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Posted by Mathematician (Member # 9586) on :
 
Glad you like it!

the axiom of choice....

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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Posted by Mathematician (Member # 9586) on :
 

quote:

---

Originally posted by IanO:
This thread is so cool.

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.

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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.

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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)

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e^(pi*i) + 1 = 0

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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.

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What does it mean for something to exist?

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With Beethoven's symphony, you also have to determine if the score is the symphony, a performance is the symphony, or a recording is the symphony. Is a symphony "sound"?

If one decides that a recording is a symphony and the score isn't, then the reason for the distinction needs to be clarified. A recording and a score can both be thought of as instructions for performing the symphony, the former followable by machines, the latter an orchestra.

Same thing with books. Do I have "Ender's Game" on my shelf, or a copy of Ender's Game? My copy is the one with the racial slur still present. Is it still Ender's Game? Did Ender's Game change when OSC changed it for a new reprint?

This is a subject that fascinates me. It comes up on the first day of Contracts in law school. A contract is not a piece of paper. It can't be "ripped up." A writing is evidence of or a memorialization of a contract. A contract comes into existence full-born upon the completion of certain acts by the parties. One instant, no contract. Next instant, contract.

I realize I'm making no point here. You've simply sparked philosophical ramblings that I find interesting.
 

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Posted by Nighthawk (Member # 4176) on :
 

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Is "sound" defined by the physical process we call sound waves, or are sound waves only sound when they are perceived by a sentient being.

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1.a. Vibrations transmitted through an elastic solid or a liquid or gas, with frequencies in the approximate range of 20 to 20,000 hertz, capable of being detected by human organs of hearing.

1.b. Transmitted vibrations of any frequency.

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Originally posted by Gwen:

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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)

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I seem to recall Wikipedia claiming you can do it with just five pieces (when I was looking up paradoxes for the ask-the-next-person-anything thread).


Lots of fun.

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Originally posted by Nighthawk:

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1.a. Vibrations transmitted through an elastic solid or a liquid or gas, with frequencies in the approximate range of 20 to 20,000 hertz, capable of being detected by human organs of hearing.

1.b. Transmitted vibrations of any frequency.

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Nice of them to contradict themselves...

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Originally posted by Tatiana:

Mathematician, I probably didn't phrase my question about the AoC very well, because you answered a different question. I guess what I need to know is, what could we do if it were true (state how we can select uniquely exactly one item from each box?) that we couldn't do if it weren't true (?). I need a more concrete example to understand what the AoC means. Like say we had boxes of pencils. If the AoC were true, I could select the longest pencil in each box, and that would uniquely specify a particular pencil in each box. But if it weren't true, there would be some boxes with pencils which were exactly the same length? Is that how it goes? I still feel confused about what the AoC is actually saying.

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Originally posted by Mike:
Ooh, can I ask a question? Speaking of infinities, what is the largest number of non-intersecting topological figure-eights that can be placed in the plane? (And what about topological circles?)

And while we're at it, can you partition 3-space into non-degenerate circles?

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I'll answer the second 2 first, because I actually have seen the proofs with regards to those. The maximum number of circles that can be fit in the plane? The same number as the number of real numbers!

In fact, it's possible (this UBER uses the AoC) to partition 3-space into non-degenerate circles (and therefore all higher n-spaces). Interestingly, it is NOT possible to partition 2-space into non-degenerate non-intersecting circles.

As far as the topological figure 8's, the answer is the same as with the circles. I can fit the same number as the number of reals. Here's quick outline of the proof.

Let "N" be the number of topological figure 8's I can fit. Let c be the number of real numbers.

To see that c is less than or equal to N, note that there are c distinct planes, and I can easily just put a single figure 8 in each one. They certainly are nondegenerate and obviously don't intersect (since the planes don't intersect). Thus, I've stuck AT LEAST c figure 8's, and thus, N is at least c, that is, c is less than or equal to N.

To see the reverse, notice that each figure 8 is really just a curved line, and so has c points. If there where more than c figure 8's possible, and all of them were non-interesection (say, d of them), then I'd have number of points in each figure 8 * number of figure 8's points. That is, I'd have c*d > c points, an obvious contradiction (since I know there are only c points).

From this I can conclue that N is less than or equal to c.

By the Schroder-Cantor-Bernstein theroem, N = c.

Thus, I can place c nonintersection figure 8's. That is, I can place as many non-intersecting figure 8's as there are real numbers.

(I still haven't worked out whether or not 3-space can be partitioned into figure 8's. My gut instinct is yes, but that probably means the answer is no ;-). I KNOW that I can't partition 2-space into figure 8's, for more or less the same reason I can't do it with circles).

Edited to add a line or 2
 

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Posted by Mathematician (Member # 9586) on :
 
Oh no, Mike, you said we're sticking the topological figure 8's in the PLANE, not in 3-space. Give me some time to work on that....
 

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Posted by Gwen (Member # 9551) on :
 
How do you figure that you can partition 3-space into non-degenerate circles? (And what does non-degenerate mean?) And why can't you partition 2-space into non-degenerate non-intersecting circles? And why can't you do it with figure 8s either?

-Gwen, who still doesn't see how this will help her count sheep or pile pebbles.
 

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Posted by Mathematician (Member # 9586) on :
 
Gwen, the proof (at least, the one I've seen) that you can partition 3-space into non-degenerate circles is quite difficult. Non-degenerate simply means the circles aren't just points; they have some finite but non-zero radius.

The proof that you CAN'T do it for 2-space isn't too bad. Draw a circle (call it C1). If we're gonna be partitioning all of the plane, the center of C1 must be contained in some circle which is completely contained in C1 (otherwise, they'd interesect). But C2 has a center which must be contained, so suppose C3 contains it. Rinse repeat. Now consider the sequence of the centers of the circles. It turns out, the centers converge to a specific point (I'll call it p). What convergence means is this: Pick your favorite positive number n. I can find a point in my sequence so that all the centers after than are a distance of less than n from the point we're calling p. In otherwords, our centers get closer and closer to p (in fact, infinitely close, in some sense).

But we're covering the entire plane in circles. So p must be contained in a circle C of a postive radius. It will turn out that since the centers of the other circles get arbitrarly close to this one, our circle C MUST have intersected one of our other circles, so we don't actually have a partition. This contradiction means that we could never partition it in the first place.

Figure 8's work the same way. Look at the sequence of points which are the intersection points of the the figure 8s.

Back to work on the number of figure 8's in a plane. It's kicking my butt!
 

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Posted by Mathematician (Member # 9586) on :
 
I think I got the number of figure 8's. I had actually heard before that it was the same as the number of natural numbers, but I just worked out a proof (I think).

It's surprisingly simple....

Pick your 2 favorite ordered pairs, then there can be only 1 figure 8 that goes around them both (that is, 1 point in one of the circles, another point in the other). I don't really know how to PROVE this, though it seems quite obvious.

Lets look at all the ordered pairs of rational numbers (i.e., fractions). Since the rationals are countable (meaning same size as the naturals), so are the the ordered pairs of rationals. The rationals are important because they are dense, meaning they're practically everywhere (in between any 2 numbers, there is a rational number)

Lets assume for a contradiction, that we have MORE than a natural number of fig 8's. Pair each of these up with their 2 "defining" pair of ordered pairs of real numbers.

Now, fix a figure 8 and it's 2 defining ordered pairs. Since the rationals are dense, I can find a 2 rational ordered pairs so that each ordered pair is contained in one of the 2 circles of the figure 8. But then I can use these 2 rational ordered pairs as my defining ordered pairs!

But the set of all pairs of ordered pairs of rational numbers is countable. So, I have more than countable fig 8's matched up with a countable set. That means that at least 2 different fig 8's have the SAME defining 2 ordered pairs of rational numbers. But we already argued that any 2 fig 8's encompassing the same 2 ordered pairs MUST intersect. This contradicts our assumption that they are non interesecting.

The fact that we were able to derive a contradiction from assuming we had more than a natural number of fig 8's means that that assumption was wrong. Thus, there can be at most a countable number of fig 8's.
 

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Posted by Mike (Member # 55) on :
 
[edit: simultaneous posting! reading yours now...]

I guess I could've been clearer — the two questions aren't really related to each other. It's just that one made me think of the other.

Yeah, the figure eights in a plane is a tricky one. (Now if they were snakes...) I'll let you stew on it for a bit.

Partitioning 3-space with circles is actually not too difficult. I know a couple of different constructive proofs, one of which is really slick. I'll describe it if anyone's interested in hearing it. Unless you guys want to figure it out yourselves.

Hmm. How about this: can you partition 4-space into 2-spheres? (A 2-sphere is the object that most of us know as the surface of a three-dimensional ball.) I haven't given any thought to this one, but I suspect the answer is no.
 

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Posted by Mike (Member # 55) on :
 
Nice. That is essentially the proof that I'm familiar with.
 

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Posted by Mathematician (Member # 9586) on :
 
I'd definitely be interested in a constructive proof. The one I know relies on transfinite induction.

4-space into 2-spheres, huh...

If a 2-sphere not lying entirely in a particular 3-space can only intersect that 3-space in a finite number of places (or countable), then I can do it. The problem is, I can't convince myself one way or another.
 

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Posted by Mike (Member # 55) on :
 
Damn. I shouldn't've opened that can of worms — my brain hurts.

I'll see you your constructive proof in a bit.
 

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Posted by Hamson (Member # 7808) on :
 
Thanks so much for the answers Mathematician. I'm certainly going to take anything here that I can grasp.

This is definitely one of the best threads ever.
 

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Posted by Mathematician (Member # 9586) on :
 
Glad you like it! Keep the questions coming! (But I'm going to bed now, I'll answer whatever I can tomorrow morning).
 

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Posted by Mike (Member # 55) on :
 
OK, for this construction we'll build things up step by step. We'll start with a spherical shell (a 2-sphere) which we'll call A: can you partition it into circles? Well, no, but you can come pretty close. Take, eg., the set of planes parallel to the xy-plane. The intersection of these planes with A is a set of circles, plus two points at the north and south poles.

It turns out you can do the same thing with any two distinct points p and q on A. (To do this explicitly, note that the tangent planes of A at p and q intersect in a line L that lies outside of A. The set of all planes containing L, intersected with A form a set of circles that partition A - {p,q}.)

OK, now that we have a way of dealing with these shells, let's try to construct a solid sphere, or ball. A ball can be thought of as a union of shells, plus the point at the center. Each shell can be partitioned into circles except for two points, which we can choose arbitrarily. Consider the ball B centered at the origin with radius 1. Let C be a circle of radius 1/2 that is tangent to B and contains the center of B (eg. the circle in the xy-plane centered at (1/2, 0, 0)). Let's also choose a point p on the surface of B, say (-1, 0, 0). Can we partition B - C - p into circles? Sure we can: every "shell" of B - C - p is missing exactly two points, and so can be partitioned into circles; these shells themselves form a partition of B - C - p. Of course, it's a piece of cake to add C back in (it's a circle after all), so B - p is partitionable into circles. Pretty nice.

Now we're ready to extend to all of 3-space. Consider the set of circles in the xy-plane of radius 1/2 centered at x = ..., -11/2, -7/2, -3/2, 1/2, 5/2, 9/2, ..., y = 0. Let U be the union of these circles.

U:

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... O O O O O O O ...

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Originally posted by Nighthawk:

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Is "sound" defined by the physical process we call sound waves, or are sound waves only sound when they are perceived by a sentient being.

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Yes.

http://dictionary.reference.com/browse/sound

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1.a. Vibrations transmitted through an elastic solid or a liquid or gas, with frequencies in the approximate range of 20 to 20,000 hertz, capable of being detected by human organs of hearing.

1.b. Transmitted vibrations of any frequency.

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Nice of them to contradict themselves...

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It is a different thing for vibrations to be "capable of being detected by human hearing" and to actually be detected by human hearing. These definitions both adhere to the objective definition of sound. Neither recognizes the possibility for a subjective definition.

[ August 18, 2006, 05:19 PM: Message edited by: The Rabbit ]
 

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Posted by Mathematician (Member # 9586) on :
 
Just thought I'd attempt resurrect this (I confess, I'm enjoying this, perhaps a bit too much), and try to turn it more basic ideas. (That last bit about filling 3-space with circles and all....It's extraordinarily hard stuff).

I wanna start with an interesting tidbit. We've already talked about how the "infinity" of the natural numbers and the "infinity" of the reals are different sizes (with the reals bigger than the natural numbers).

Now, here's where it gets hard to wrap your head around (in case it's not there already!), how many sentences (in english) are there (where we require that sentences be of finite length)?

Lets work it out. First, let's order the sentences alphabetically (we'll define a space as being less than a. That is, "aba" comes after "a ba"). Let's just start counting them and see what happens. Can you convince yourself that given any sentence, after some finite number of steps, we'll reach that one (since there are only a finite number of letters/spaces)? It turns out, that this means that the box having all possible sentences in will have the same size as the natural numbers.

Note that we haven't even talked about the fact that sentences like "a dog" or "akbla kasld" aren't even sentences! Let's assume, just for ease, that these ARE "valid" sentences.

Anyone see where we're going with this?

We have at most a natural numbers number of sentences, but we have MORE real numbers than that. What this means is that even if EVERY sentence describes a unique real number (which, of course, they don't), there will be some real numbers which are inherently indescribable. They transcend all possible description. This rocks my world!
 

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Posted by Gwen (Member # 9551) on :
 
Except that in English, sentences can, oh I don't remember the word Language Log used in that one post, loop or nest or iterate or something...so you could potentially describe every number with something like "this is the number you get when you add one and one and one and one and one..."
And if there's a real number which is inherently indescribable, you could just call it "this is the first real number which is inherently indescribable," which is a unique description. Except it would then be false, so it would be something like "this is the first real number which would be inherently indescribable if it weren't for the sentence 'this is the first real number which is inherently indescribable'" and then the next number would do it, and so on.
Come on, Mathematician: 27 digits, if we count only letters and spaces, vs. 11 digits, if we count decimals and numbers--which do you think will win?
 

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Posted by fugu13 (Member # 2859) on :
 
Gwen: no, there are lots of real numbers that go on forever with no pattern, something that definitely can't be described by listing numbers one at a time, since any given sentence is, by nature, finite.

Now, some of that variety of number can be described in other ways (pi, anyone?), but the theorem does conclusively prove there are others that can't.

Once we know that any given sentence is finite, we know that the total (infinite) number of sentences (even arbitrarily long ones) is the same as the number of natural numbers, by Mathematician's proof.

Since there are many more real numbers than natural numbers, we know there are oodles of real numbers out there indescribable with sentences.

edit: and there is no beginning to the real numbers (excluding the natural numbers), nor would there exactly be a 'first' one even if there were, so no, your sentence doesn't work.
 

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Posted by Mathematician (Member # 9586) on :
 
Gwen,

The term is escaping me as well. I believe in GEB they're referred to as strange loops. The thing is, irrelevent of intended meaning, there are always less sentences than real numbers, period. It doesn't matter if the sentence has self references (even if every sentence is allowed to refer to as many (but finite) sentences as you'd like)

With regards to "this is the number you get when you add one and one and one and....", I agree, we can generate all the natural numbers (using exactly those types of sentences), but there will still be numbers you can't get to using that method (for instance, fractions, negative numbers, real numbers which are neither).

As far as calling a real number "the first real number which is inherently indescribable", the real numbers aren't well ordered. What this means is that I can pick collections of real numbers that DON'T have a smallest number. Consider this example:

Think of all x between 0 and 1, but not including 0 or 1. Does this have a smallest number in it? No! How do I know? If you hand me a number and claim "this is the one!", I can halve it. It's still between 0 and 1, but it's definitely smaller. Thus, no smallest number.

Because the real numbers aren't ordered in a nice way, there may be no "first number which is inherently dindescribable", meaning that sentence DOESN'T describe a real number.

I'll admit that if we're only trying to describe countable things, sentences such as "the first number not describable in 47 words or less" would create havoc (because then I have described it in less than 47 words). When doing hardcore mathematics, it turns out that statements like this simply can't be formulated. (And so if we restrict to PURELY mathematical sentences, written in the language of logic, my arguement goes through, no rebuttals are really possible).

As far as the 27 digits vs 11, you're forgetting one important detail. Looking at numbers (decimals), we allow for an infinite number of decimal places. I stressed at the begining that our sentences must be of finite length. It is exactly restriction that allows the proof to go through. If I allow sentences of infinite length, I can easily describe any real number - just write out it's digits one by one.

Why restrict sentences to be finite length, while nubmers can be any length? Because we understand how to interpret decimals of infinite length - the length just gives us more and more precision. How do you interpret a sentence of infinite length? Is it possible that given an arbitary (but finite) length sentence, I can construct a larger one which starts with that sentence, but which has a totally different meaning? I'd suggest that it's yes. And this means letting our length go to infinity doesn't help us intepret the sentence- in fact, it may make it worse (depending on the sentence).

edit *Fugu beat me to it* ;-)
edit 2 - grammar, clarity
 

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Posted by fugu13 (Member # 2859) on :
 
BTW, here's what I mean by there not really being a 'first' number even if the real numbers had a beginning.

Imagine we're only looking at positive numbers, so there's sort of beginning to the numbers; at least, there's a point where there's nothing before and all the numbers are immediately after (0).

Lets try to pick the first number. Clearly it must be a specific number.

Now divide that number by two -- the new number is clearly before the previous number, and also positive, therefore the previous number cannot be the first number. Perhaps this new one is? But no, the trick works again and again.

We never reach a first number. There is no first number.

edit: and then I'm beat to my explanation

BTW, there's a way out of your paradox. The sentence doesn't describe a number, even though it says it does.

edit again: that is, this paradox -- "the first number not describable in 47 words or less", with the same resolution for the other paradox tackled.
 

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Posted by Lalo (Member # 3772) on :
 
Wow, I really like Mathematician. Cool guy.
 

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Posted by Tatiana (Member # 6776) on :
 
Yes, me too. Maybe he will accept us as acolytes.
 

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Posted by The Rabbit (Member # 671) on :
 
If we are already Anne Kate's acolytes, and she becomes and acolyte of Methematician, would this make us sub-acolytes.
 

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Posted by The Rabbit (Member # 671) on :
 
If I understand correctly, the number of rational numbers is the same as the number of natural numbers. The rational numbers between 0 and 1 can be arranged in order using the following pattern, 0 , 1, 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, 3/5, 4/5. We could then create a similar pattern to order all the rational numbers between 1 and 2, if we repeat this to infinity we will have all the rational numbers. The number of irrational numbers, however, can not be counted and is therefore a larger infinite set than the number of rational numbers.

Also, there are an infinite and uncountable number of irrational numbers between any two rational numbers. So the number of irrational numbers between for example (1/100) and (1/101) is greater than the total number of rational numbers.

So now here is my problem. Is the number of irrational numbers between (1/100) and (1/101) the same size infinite set as the set of all irrational numbers? Since neither set can be arranged in a counting order, is there anyway to tell whether the two sets are the same size? Are there any infinite sets which are known to be larger than the set of irrational numbers? Is so, how can we know that they are larger when they can not be counted?
 

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Posted by The Rabbit (Member # 671) on :
 
My favorite Mathematical "proof", is the proof that there are no known uninteresting numbers.

First we shall prove that there are no uninteresting rational numbers. We begin by arranging the rational numbers into some logical order, such as the one I described in my previous post.

The first number, 0, has all kinds of interesting properties. The second number, 1, also has numerous interesting properties. Now let us assume that there are rational numbers that have no interesting properties. As we progress along our ordered list of rational numbers, we will eventually come to the first rational number that has no interesting properties. This would, however, be in itself an interesting property. Therefore, there can be no uninteresting rational numbers.


This can be extended to all known numbers, both rational and irrational. Consider the set of known numbers. We can order this set based on the time the number was discovered. We would start with the oldest known number and move up to the most recently discovered number. Now let us assume that numbers have been discovered which have no interesting properties. These would of course have to be irrational numbers since we have already demonstrated that there can be no uninteresting rational numbers. As we searched through the list of known numbers, we would eventually come to the first irrational number discovered which had no interesting properties, which would of course be a very interesting property.

Ergo, we will never discover an uninteresting number.

[ August 21, 2006, 08:44 PM: Message edited by: The Rabbit ]
 

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Posted by Gwen (Member # 9551) on :
 
But--but--doesn't fugu13's and Mathematician's reason for mine being wrong count for yours too?

Or do I just have to define what I mean by "first" to be the way you meant it (the number whose existence we learned of chronologically first)?
 

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Posted by The Rabbit (Member # 671) on :
 
Gwen, My definition is not a definition of the set of all irrational numbers. It can only work as a definition of the set of all known irrational numbers. The set of known irrational numbers, is a countable infinte set. It took me a minute to persuade myself that it is indeed an infinite set but consider the following example. Take the irrational number pi. If I multiply pi by any rational number, I will get an irrational number. This would create an infinite set of irrational numbers all of which are known. Since the set of irrational numbers is countable, this sub set of the irrationals (pi*x) where x is any rational number, is also countable.

The point is, that the set of known irrational numbers is not and can never be the same as the set of irrational numbers. There will always be irrational numbers that we don't know about.


What this means is that we can never determine whether or not there are uninteresting irrational numbers. All we can prove is that we will never know of any uninteresting numbers.
 

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Posted by Mathematician (Member # 9586) on :
 

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Originally posted by The Rabbit:

So now here is my problem. Is the number of irrational numbers between (1/100) and (1/101) the same size infinite set as the set of all irrational numbers? Since neither set can be arranged in a counting order, is there anyway to tell whether the two sets are the same size? Are there any infinite sets which are known to be larger than the set of irrational numbers? Is so, how can we know that they are larger when they can not be counted?

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