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Unfortunately, someone discovered an error.
And the paper was withdrawn.
Since the corresponding harmonic series is less than 2,
perhaps the old fox will prove that it's unprovable.

*unregistered_Pi

"What makes him a kook?"
I did not mean this is a mean or unkind way. I like your term "ecentric" but you have realize that he has a big ego and I am not certain that "ecentric" includes this.
I am rooting for his proof to be correct but I am extremely skeptical I am rooting for him because so many "good" mathematicians (e.g. Paul Cohen), have wanted to prove the Riemann Hypothesis for a very long time. It would be nice if a "little guy" got the prize.
(By the way, you should realize that the proof of the Bieberbach conjecture turned out to be fairly easy; compare this with Fred Almgren's BIG (1200 pages?) paper.)

See equation 18 at MathWorld for an expression of the analytic continuation to the complex plane. Sure, it's expressed as an infinite series, but who says there has to be a simple closed-form definition (I don't know if there is one).

Cool. Yesterday I searched Mathworld for April fool's jokes, and found this:
The "Ramanujan Constant"
Today, I see it in your username. Happens.

The truth (or falsity) of the Riemann Hypothesis is intimately related to the distribution of the primes. Specifically, if the RH is true, then the primes are distributed about as regularly as possible.[1]

Carl

[1] See, for example, equation (2) of Riemann Hypothesis

First: complex numbers, explained. You may have heard the question asked, "what is the square root of minus one?" Well, maths has an answer and we call it i. i*i = -1. If the real number line ...-4, -3, -2, -1, 0, 1, 2, 3, 4... is represented as a horizontal line, then the numbers ...-4i, -3i, -2i, -i, 0, i, 2i, 3i, 4i... can be thought of as the *vertical* axis on this diagram. The whole plane taken together is then called the complex plane. This is a two-dimensional set of numbers. Every number can be represented in the form a+bi. For real numbers, b=0.

Right. Now the Riemann Zeta Function is a function/map (like f(x)=x^2 is a function) on the complex plane. For any number a+bi, zeta(a+bi)will be another complex number, c+di.

Now, a zero of a function is (pretty obviously) a point a+bi where f(a+bi)=0. If f(x)=x^2 then the only zero is obviously at 0, where f(0)=0. For the Riemann Zeta Function this is more complicated. It basically has two types of zeros: the "trivial" zeroes, that occur at all negative even integers, that is, -2, -4, -6, -8... and the "nontrivial" zeroes, which are all the OTHER ones.

As far as we know, *all* the nontrivial zeroes occur at 1/2 + bi for some b. No others have been found in a lot of looking... but are they ALL like that? The Riemann Hypothesis suggests that they are... but until today nobody has been able to prove it.

First: complex numbers, explained. You may have heard the question asked, "what is the square root of minus one?" Well, maths has an answer and we call it i. i*i = -1. If the real number line ...-4, -3, -2, -1, 0, 1, 2, 3, 4... is represented as a horizontal line, then the numbers ...-4i, -3i, -2i, -i, 0, i, 2i, 3i, 4i... can be thought of as the *vertical* axis on this diagram. The whole plane taken together is then called the complex plane. This is a two-dimensional set of numbers. Every number can be represented in the form a+bi. For real numbers, b=0.

Right. Now the Riemann Zeta Function is a function/map (like f(x)=x^2 is a function) on the complex plane. For any number a+bi, zeta(a+bi)will be another complex number, c+di.

Now, a zero of a function is (pretty obviously) a point a+bi where f(a+bi)=0. If f(x)=x^2 then the only zero is obviously at 0, where f(0)=0. For the Riemann Zeta Function this is more complicated. It basically has two types of zeros: the "trivial" zeroes, that occur at all negative even integers, that is, -2, -4, -6, -8... and the "nontrivial" zeroes, which are all the OTHER ones.

As far as we know, *all* the nontrivial zeroes occur at 1/2 + bi for some b. No others have been found in a lot of looking... but are they ALL like that? The Riemann Hypothesis suggests that they are... but until today nobody has been able to prove it.

First: complex numbers, explained. You may have heard the question asked, "what is the square root of minus one?" Well, maths has an answer and we call it i. i*i = -1. If the real number line ...-4, -3, -2, -1, 0, 1, 2, 3, 4... is represented as a horizontal line, then the numbers ...-4i, -3i, -2i, -i, 0, i, 2i, 3i, 4i... can be thought of as the *vertical* axis on this diagram. The whole plane taken together is then called the complex plane. This is a two-dimensional set of numbers. Every number can be represented in the form a+bi. For real numbers, b=0.

Right. Now the Riemann Zeta Function is a function/map (like f(x)=x^2 is a function) on the complex plane. For any number a+bi, zeta(a+bi)will be another complex number, c+di.

Now, a zero of a function is (pretty obviously) a point a+bi where f(a+bi)=0. If f(x)=x^2 then the only zero is obviously at 0, where f(0)=0. For the Riemann Zeta Function this is more complicated. It basically has two types of zeros: the "trivial" zeroes, that occur at all negative even integers, that is, -2, -4, -6, -8... and the "nontrivial" zeroes, which are all the OTHER ones.

As far as we know, *all* the nontrivial zeroes occur at 1/2 + bi for some b. No others have been found in a lot of looking... but are they ALL like that? The Riemann Hypothesis suggests that they are... but until today nobody has been able to prove it.

First: complex numbers, explained. You may have heard the question asked, "what is the square root of minus one?" Well, maths has an answer and we call it i. i*i = -1. If the real number line ...-4, -3, -2, -1, 0, 1, 2, 3, 4... is represented as a horizontal line, then the numbers ...-4i, -3i, -2i, -i, 0, i, 2i, 3i, 4i... can be thought of as the *vertical* axis on this diagram. The whole plane taken together is then called the complex plane. This is a two-dimensional set of numbers. Every number can be represented in the form a+bi. For real numbers, b=0.

Right. Now the Riemann Zeta Function is a function/map (like f(x)=x^2 is a function) on the complex plane. For any number a+bi, zeta(a+bi)will be another complex number, c+di.

Now, a zero of a function is (pretty obviously) a point a+bi where f(a+bi)=0. If f(x)=x^2 then the only zero is obviously at 0, where f(0)=0. For the Riemann Zeta Function this is more complicated. It basically has two types of zeros: the "trivial" zeroes, that occur at all negative even integers, that is, -2, -4, -6, -8... and the "nontrivial" zeroes, which are all the OTHER ones.

As far as we know, *all* the nontrivial zeroes occur at 1/2 + bi for some b. No others have been found in a lot of looking... but are they ALL like that? The Riemann Hypothesis suggests that they are... but until today nobody has been able to prove it.

First: complex numbers, explained. You may have heard the question asked, "what is the square root of minus one?" Well, maths has an answer and we call it i. i*i = -1. If the real number line ...-4, -3, -2, -1, 0, 1, 2, 3, 4... is represented as a horizontal line, then the numbers ...-4i, -3i, -2i, -i, 0, i, 2i, 3i, 4i... can be thought of as the *vertical* axis on this diagram. The whole plane taken together is then called the complex plane. This is a two-dimensional set of numbers. Every number can be represented in the form a+bi. For real numbers, b=0.

Right. Now the Riemann Zeta Function is a function/map (like f(x)=x^2 is a function) on the complex plane. For any number a+bi, zeta(a+bi)will be another complex number, c+di.

Now, a zero of a function is (pretty obviously) a point a+bi where f(a+bi)=0. If f(x)=x^2 then the only zero is obviously at 0, where f(0)=0. For the Riemann Zeta Function this is more complicated. It basically has two types of zeros: the "trivial" zeroes, that occur at all negative even integers, that is, -2, -4, -6, -8... and the "nontrivial" zeroes, which are all the OTHER ones.

As far as we know, *all* the nontrivial zeroes occur at 1/2 + bi for some b. No others have been found in a lot of looking... but are they ALL like that? The Riemann Hypothesis suggests that they are... but until today nobody has been able to prove it.

First: complex numbers, explained. You may have heard the question asked, "what is the square root of minus one?" Well, maths has an answer and we call it i. i*i = -1. If the real number line ...-4, -3, -2, -1, 0, 1, 2, 3, 4... is represented as a horizontal line, then the numbers ...-4i, -3i, -2i, -i, 0, i, 2i, 3i, 4i... can be thought of as the *vertical* axis on this diagram. The whole plane taken together is then called the complex plane. This is a two-dimensional set of numbers. Every number can be represented in the form a+bi. For real numbers, b=0.

Right. Now the Riemann Zeta Function is a function/map (like f(x)=x^2 is a function) on the complex plane. For any number a+bi, zeta(a+bi)will be another complex number, c+di.

Now, a zero of a function is (pretty obviously) a point a+bi where f(a+bi)=0. If f(x)=x^2 then the only zero is obviously at 0, where f(0)=0. For the Riemann Zeta Function this is more complicated. It basically has two types of zeros: the "trivial" zeroes, that occur at all negative even integers, that is, -2, -4, -6, -8... and the "nontrivial" zeroes, which are all the OTHER ones.

As far as we know, *all* the nontrivial zeroes occur at 1/2 + bi for some b. No others have been found in a lot of looking... but are they ALL like that? The Riemann Hypothesis suggests that they are... but until today nobody has been able to prove it.

Wonder what he'll do with the money? Replace the stack of pencils he depleated, or the batteries in the calculator?

He states in the proof on page 23 that he wants to use the money to restore the chateau de bourcia as a mathematical institute.

Kind of like Wolfram... He got a huge grant, and used it to develop an awsome math package.

Cool idea, but let's hand out some credit:

The statician Hermann Chernoff was first to developed the idea of using faces to display multi-variable data.

Actually, if someone just wants a simple metaphor, faces probably are the best choice, given that our brains are hard-wired to do face recognition especially well.

While the turing machine is an amazing creation, I find the more recent work on Cellular Automata to be an interesting addition to the discoveries that worlfram made years ago.

Cellular automata are desceptively simple rulesets that produce extremely complex patterns - through a rule that can be encoded into a 8 bit number, you can produce Turing machines, as well as chaotic patterns.

To learn more about cellular automata, visit the MathWorld page

This is some weird combination of revisionist history (the ancient Greeks knew about irrational numbers) and just plain making shit up (irrational means it can't be expressed as a ratio of integers). See Mathworld's definition of irrational number for one more credible, and more researched, version.

Imaginary numbers under a variety of names were discussed at least as early the 16th and 17th centuries and credible sources claim references back to the ancient Egyptians; this reference also says the term "imaginary" was in common use at the time of Descartes (though makes no reference in the online material as to who coined the term). Many less credible online sources place the name as coming from Descartes and claim it to be dergatory, but many of those sites appear to be copying from some common source of unknown origin. So, the guess that imaginary was derogatory may be correct.

I suppose one of four (calling the fact of the name and date of discovery each as guesses at the truth)possibly correct speculations isn't bad for just spewing stuff that sounds credible

Well we can make a perfect number with it.

Every Mersenne prime gives rise to a perfect number.

To answer your question a little more seriously the number is not much use in itself but like many peices of research the route to the goal often turns out more interesting information than the goal. GIMPS pushes back the bounds on many levels such as highly optimised coding and mathematical DC.

Occam's Razor (...) Don't make things more difficult then they have to be. Black holes are the simplest explanation

With all due respect to the advantages that Occam's Razor has given to the advance of science, this was exactly the key factor that made the leading scientist of late XVIII century like Antoine Lavoisier to judge that stones cannot fall from the sky. In 1768, 1794 and 1795 there were substantial sightings of meteorite showers in France, Italy and England - yet according to the Occam's Razor, it was easier to explain them by assuming the witnesses just lie. Use Occam's Razor as any razor - with extreme caution.

Catalan's conjecture is not that, it's a conjecture regarding the solutions of a very specific Diophantine equation:
Mathworld entry: Catalan's Conjecture

Yes, it was proven in 2002, but the twin prime conjecture scores higher (IMO) because it's a very general problem in number theory, not one devious equation. (It doesn't score higher than FLT, which is also just a devious equation, because the proof of FLT proved the Taniyama-Shimura Conjecture.)

As for the famous AKS algorithm, I would classify that into computer science, not math... Mathematicians already knew it's possible to test numbers for primality (any integer is either prime or not!), it was up to the computer scientists to find how to do it efficiently.

And yes, these proofs are not (paraphrasing Erdos) "taken from God's book of mathematics", but until such a Godly proof is known, they will suffice...

Waring's Problem provides good examples. For example, the only numbers that cannot be written as a sum of 7 cubes are 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, and 454.

What you just stated, basically, is a conjecture. Something that seems right, intuitively, but isn't proven.

Another example that seems just as "obvious" might be Goldbach's which seems right and has been tested up to very, very large numbers. But it's still not proven.

To most people, the difference is tenuous, but it's there; and sometimes the difference between having a proof or not has critical applications outside pure math.

If anyone were, for instance, able to find a way to easily factor large primes them a great deal of today's best encryption would become moot. So, any additional knowledge about the properties of prime numbers is potentially important.

-- MG

This work is the outcome of about twenty years of "on and off" search and research on this and the related binary Goldbach problem; in the interim having been lured onto various misleading paths or frustrated by (for me) insurmountable difficulties, before ultimately recognizing and constructing a workable approach.

In the mode of some car-insurance commercial running in the US, I ran into my wife's office and said, "I've got great news!". Somehow, she didn't share my enthusiasm.

When I was in high-school in 1978, my math teacher, Alan Crokall (sp?) gave me the programing/math assignment of either proving Goldbach's Conjucture or finding a counter example. He later explained that he wanted me to find the counter example so that it could be called "Goldberg's rejecture of Goldbach's conjecture".

And you can find out about Goldbach's conjecture if you don't already know what it is.

OK, I'm too lazy to RTFA, but, if there are infinite numbers, why would there not be infinite prime numbers, and infinite prime twins? Are there also not infinite perfect numbers, or ...?

That's a good question.

The fact that there are an infinite number of numbers doesn't immediately imply that there are an infinite number of primes, but Euclid figured out how to prove this is true in about 300 BC. It only takes a few sentences to explain it; here's one example.

It is definitely not obvious that there are an infinite number of twin primes. It has been an open question for more than a century, and some of the greatest minds in mathematics have worked on it. If this proof is correct, it will be a major result.

I was trying to think of a good example of something that there is not an infinite number of. Browsing through MathWorld, I found Truncatable Primes - there are only 83 of these.

Can anyone think of any other examples of a type of number that only has a finite number of them, even though at first glance it seems like there might be an infinite number of them?

if any of the twins are sexy. ... Yes, I KNOW no sets of twins are sexy because one is p+2 and the other is p+6, but come on, it's a JOKE people. *sigh*

Something I read in Science the other day: There's a new proof in review that there are infintely many sequences such as 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 -- primes that differ by a constant offset. See also Mathworld.

Poker, Blackjack or other such games are the only sort of gambling I would be remotely willing to participate in because it involves much more than straight chance as involved in slots, roulette or craps. Sure statistics come into play, but nothing forces the stats to hold consistently.

Taking blackjack as an example, if you come up with a strategy that gives you better odds than the house, a little something called the central limit theorem guarantees that in the long run, you're going to win. What you should be concerned about instead is that your bankroll is big enough to let you play long enough for that to happen -- if you run out of money too quickly due to a run of bad luck (something math guys call the "gambler's ruin" problem), then you're screwed. Then it becomes trying to figure out what the odds are of that not happening, as a function of how much money you start out with.

The popular Bringing Down The House is an easy read that actually does a good job of discussing a lot of these issues, in the guise of an entertaining story.

Poker, Blackjack or other such games are the only sort of gambling I would be remotely willing to participate in because it involves much more than straight chance as involved in slots, roulette or craps. Sure statistics come into play, but nothing forces the stats to hold consistently.

Taking blackjack as an example, if you come up with a strategy that gives you better odds than the house, a little something called the central limit theorem guarantees that in the long run, you're going to win. What you should be concerned about instead is that your bankroll is big enough to let you play long enough for that to happen -- if you run out of money too quickly due to a run of bad luck (something math guys call the "gambler's ruin" problem), then you're screwed. Then it becomes trying to figure out what the odds are of that not happening, as a function of how much money you start out with.

The popular Bringing Down The House is an easy read that actually does a good job of discussing a lot of these issues, in the guise of an entertaining story.

Talk about not knowing what you're talking about. Geostationary orbit is at 42,245km, can only be above the equator (so you wouldn't get a signal at the poles) and means (depending on how you define it) either one or no rotation per day. The GPS satellites are not in geostationary orbit.

That usage is deprecated. You are refering to a tebibyte.

After reading through the comments here, it is obvious that there are some misconceptions about what Apple is doing.

Latent Semantic Indexing (LSI) was invented by Deerwester et. al. [1] as a method of reducing the dimensionality of a text corpus by finding a low-rank approximation of the term-document matrix.

The singular value decomposition (SVD) [2] factors a matrix A into the product of two orthogonal matrices and a diagonal matrix, A = U'SV. To find a rank k approximation of A using this factorisation, create matrices U^, S^ and V^ where S^ contains the first k rows and columns of S, U^ contains the first k rows of U and likewise for V^. Then, let A^ = U^'S^V^. The difference in Frobenius norms [3] of A and A^ is minimal for a rank-k approximation of A (least squares).

Rather than storing the full matrix, A^, in practice it is much more common to save U^ and S^ and project the columns and rows of A into a k-dimensional space. This allows both terms and documents to be clutered together and helps to associate keywords with documents.

You can do many things with these approximated document vectors, clustering, classification, document retrieval. Apple is probably using a k-nearest neighbour classifier [4] to determine how a message is to be filed.

I would be most interested to see Apple's updating strategy. There are several algorithms that allow you to add new rows and columns to a matrix where you know the full SVD, but none that I know of for the truncated SVD.

For one of my graduate-level courses, I wrote a little search engine that uses LSI to cluster 1000 newspaper articles. You can play with it here. My favourite query is "Rowan Gorilla." The Rowan Gorilla is an oil rig that frequents Halifax harbour. The search engine returns articles on the oil and gas industry that contain neither the word "Rowan" nor "Gorilla" but are still topical.

[1] Scott Deerwester, Susan T. Dumais, George W. Furnas, Thomas K. Landauer, Richard Harshman. Indexing by Latent Semantic Analysis. Journal of the American Society of Information Science, 1990.

[2] Singular Value Decomposition -- from MathWorld. http://mathworld.wolfram.com/SingularValueDecompos ition.html

[3] Frobenius Norm -- from MathWorld. http://mathworld.wolfram.com/FrobeniusNorm.html

[4] Artificial Intelligence Wiki: NearestNeighbour. http://www.ifi.unizh.ch/ailab/aiwiki/aiw.cgi?Neare stNeighbor

After reading through the comments here, it is obvious that there are some misconceptions about what Apple is doing.

Latent Semantic Indexing (LSI) was invented by Deerwester et. al. [1] as a method of reducing the dimensionality of a text corpus by finding a low-rank approximation of the term-document matrix.

The singular value decomposition (SVD) [2] factors a matrix A into the product of two orthogonal matrices and a diagonal matrix, A = U'SV. To find a rank k approximation of A using this factorisation, create matrices U^, S^ and V^ where S^ contains the first k rows and columns of S, U^ contains the first k rows of U and likewise for V^. Then, let A^ = U^'S^V^. The difference in Frobenius norms [3] of A and A^ is minimal for a rank-k approximation of A (least squares).

Rather than storing the full matrix, A^, in practice it is much more common to save U^ and S^ and project the columns and rows of A into a k-dimensional space. This allows both terms and documents to be clutered together and helps to associate keywords with documents.

You can do many things with these approximated document vectors, clustering, classification, document retrieval. Apple is probably using a k-nearest neighbour classifier [4] to determine how a message is to be filed.

I would be most interested to see Apple's updating strategy. There are several algorithms that allow you to add new rows and columns to a matrix where you know the full SVD, but none that I know of for the truncated SVD.

For one of my graduate-level courses, I wrote a little search engine that uses LSI to cluster 1000 newspaper articles. You can play with it here. My favourite query is "Rowan Gorilla." The Rowan Gorilla is an oil rig that frequents Halifax harbour. The search engine returns articles on the oil and gas industry that contain neither the word "Rowan" nor "Gorilla" but are still topical.

[1] Scott Deerwester, Susan T. Dumais, George W. Furnas, Thomas K. Landauer, Richard Harshman. Indexing by Latent Semantic Analysis. Journal of the American Society of Information Science, 1990.

[2] Singular Value Decomposition -- from MathWorld. http://mathworld.wolfram.com/SingularValueDecompos ition.html

[3] Frobenius Norm -- from MathWorld. http://mathworld.wolfram.com/FrobeniusNorm.html

[4] Artificial Intelligence Wiki: NearestNeighbour. http://www.ifi.unizh.ch/ailab/aiwiki/aiw.cgi?Neare stNeighbor

Now that I'm perplexed by this sniveling cashin, I've got a C&D letter of my own!

If you just project the line out, LNUX goes to zero around late summer.

The rise and fall of stock prices generally follow the Normal Distribution. So the value will tend to level off to zero as time approaches infinity.

I have discovered a truly marvelous demonstration of this proposition that this monument is too narrow to contain.

Apologies to Fermat.

If you're measuring the inside angle (as I was) then the lower the number, the harder the turn. This, of course, reverses if you measure the outside angle.

A fairly detailed description.

the distance between holes is inversely proportional to the distance between the interference fringes. The closer the holes, the further apart (and thus wider and easier to see) the interference maxima/minima. Also, the spacing for more than 2 hole should be as equal as possible to maximize interference effects.

for more info on this kind of interference google for 'diffraction grating' (some basic math and pictures here).

the probability is so unlikely ... which is caused by the de Broglie wavelength beeing so small.

Basically, light behaves as a wave, and since waves can constructively and destructively interfere with one another (cast two stones simultaneously in a pond and oberve the resulting interference pattern) light will form a funny looking pattern that one would not intuitively expect on a screen some distance from the slit.

1. Excel is an extremely poor tool for doing anything other than basic graphs and calculations. For engineering purposes, it's near useless.

I suppose that they did some marketing research and found that most people wouldn't tolerate jumping straight to $10.00 per track. If they do the frog-in-boiling-water thing, then at least some of the math-challenged will still be with them.

As for 3/2 being "a nice round number," doesn't a "round number" at least need to be an integer? I was interested to read this definition of "round number"--like every other folk term for numbers, this apparently has a precise mathematical definition (for some value of "precise").

> My training in physics is quite elementary, but
> I was led to believe the proton is relatively
> massive on the atomic level, especially when
> compared to an electron. How could a wimp be so
> large and yet unnoticed?

The key is the "weakly interacting" in the name. At the microscopic level, these particles (if they exist) can only interact via the weak force, which is both weak and short-range.

In particle physics the size of a particle has no relation to a physical size or a particle's mass. It is defined in terms of how strongly a particle interacts with other matter. (See the definition of cross section at PhysicsWorld.) So since the WIMP particle interacts only weakly, it is by definition "small," even if it is massive.

If the WIMP hypothesis is correct, then the WIMPs have hardly been "unnoticed." One of the chief motivations for looking for them is to explain the rotation of various galaxies which appear to be much more massive than can be calculated by adding up the mass of all the stars and dust in them. So if this missing mass does consist of WIMPs, then they have already been noticed!

> Speed of time? Excuse me, but can I get some of what you're smoking? How would you define a concept like that?

You're right; "speed of time" is nonsense.

Maybe he's trying to say is that object's world line would become space-like rather than time-like. (Which is true.) Mathematical definition here. Some information about the consequences here. (Not much, though. But I'm too tired to find a better link and much too tired to think independently.)

Seeing that the "d" in BBC's Dirac is actually a Greek lowercase delta, I think it is named after the same guy. The Dirac delta function shows up a lot in the maths used in digital signal processing.

Funny, I thought they would name it Kronecker.

While it's great to see that there are all sorts of free tools and software libraries that handle various types scientific computation, visualization and analysis, it is disappointing that there doesn't seem to be a 'free', integrated tool that can compete with Mathematica.

While Wolfram and his team have done some truely amazing things and produced a product that is worthy of the $1880 price tag, I am astonished that the mathematic and scientific communities have not pooled their resrouces to produce something like it (please tell me I'm wrong about this... if there's something better than Mathematica I'd love to know, especially if it can do symbolic tensor calculus).

There seem to be lots of computer science and mathematics researchers who churn out papers on computational methods for various 'hard' calculations, analysis, symbolic manipulation and visualization. C libraries, produced by their graduate students, for doing these things seem to be abundant.

As mentioned by other posters there are plenty of free graphing, plotting and analysis packages that can deal with specific areas of interest, but there doesn't seem to be a general purpose, extendable, package that can do all of that stuff the way that Mathematica can. I'm sure that Universities all over the world have enough demand for Mathematica licenses from their mathematics and physics professors alone to justify some colaborative effort to create an open tool that can do the same. In addition, a co-ordinated effort like that would provide a platform for those grad students to extend rather than just toss out another computation or analysis library that will gather dust.

Just last night I printed off a bunch of polyhedra polyhedra for my six year to cut out and assemble for fun.

I remember before the Dungeon Master's Guide, Player's Footbook and Monster Manual (which our DM forbade us to read), there was only a thick pamphlet-like book with a few monsters (giant rats, hobgoblin, gelatenous cube), and a sample 1/2 level. There sure were a lot of gelatenous cubes for level 1 ...

Well, assuming that the function of heat is continuous, the Intermediate Value Theorem tells us that it will be just right at some point between too hot and too cold.

I thought I might be the only one thinking Root Mean Square.

Well, now I'm confused. Changing the rules after the fact is confusing you see. Your definition is good, and if it's correct then I humbly apologise for being so trollish sir. Until some suitably respected authority confirms your definition, I shall have to call shenanigoats.

This is true as long as your school is non-competitive (i.e., no class rank or curve).

I actually did a pseudo-mathematical model of this, using Prisoner's Dilemma. Summary: It either benefits you or doesn't affect you to let someone copy your work at a noncompetitive school. It can hurt you at a competitive school, though.

--

Logic is poor problem solving technique for real world situations.

Logic is exactly what we use (and should use) for solving problems in the real world; it's just that our premises are far more numerous, and our inductive steps far more elaborate and involved than a simple logical exercise like "Socrates is a man. All men are mortal. Therefore Socrates is mortal." Are you telling me that you operate under some principle other than logic when trying to make decisions? Either you decide using some logical system, or you decide randomly (which is itself encompassed by a logical system, in which the premise is that all conclusions shall be generated randomly).

While you can debate vigorously the nature of gravity, at the end of the day if you drop a pencil it will hit the floor. Logic doesn't tell you that, experience tells you that.

Would experience tell that to someone who had grown up in zero gravity? Imagine a person who is born and raised on a space station orbiting Earth. They never leave the station. Gravity is unknown to them on a personal level; the space station is far short of the mass necessary for objects within it to be noticeably subject to its gravity. They never experience gravity the same way someone standing still on Earth's surface does.

Certainly, if they were educated, they'd be aware that gravity existed, that all matter has gravitational attraction to all other matter, and that gravity is a force that increases with the product of the masses involved and decreases with the square of the distance.

So if you asked this educated person what would happen if two objects were placed near each other -- say, a pencil placed 5 meters away from Earth -- they could and would quite rightly and reasonably use logic to determine that the pencil and Earth would be mutually attracted to each other, that from the point of view of an observer on Earth's surface, the pencil would fall... even if they had never seen it happen before.

The logical proof for one plus one is several hundred pages.

No, it only takes 5 simple rules for the basis of arithmetic. Proving that 1+1=2 is as simple as defining the operation of addition, then demonstrating that since the 1st successor of 1 is 2, if we take the 1st successor of 1 (a translation of 1+1 into Peano's rules), we get 2. QED.

Experience gives you the answer instantly.

Experience certainly is useful when performing simple arithmetical operations in your head. Does experience instantly tell you what 87598136410 + 562142346819 is? Not likely; you'll use logic (arithmetic) to find the answer. How about 293 + 4476? At what point does experience take over from logic? Even when it does, so what?

Even in matters of arithmetic, logic is superceeded by tradition and rules to work around the lack of precision. (Logically currency would be a real value, in practice you have 2 decimal places to work with.)

I'm beginning to think that you're not all that clear on what logic really is. I hope I'm wrong, so please correct me if so. When you say, "Logically currency would be..." you mean logically according to who? Who says that it would be more logical to handle all currency as real numbers rather than limiting it to some arbitrary precision? Clearly, since humans are far less capable of dealing with real numbers than simple integers when it comes to currency, and since efficiency is a primary goal of currency systems, it would be incredibly illogical to use a real number system for currency.

(Of course, if you were dealing with computers, it might be logical, and in fact a lot of