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tommy
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Is it true that mathematics have demonstrated the existence of elements that fall outside the physical world?
If there is a single truth about Magic, it is that nothing on earth so efficiently evades it.

Tommy
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? mathematics is about things that exist in the realm of ideals.
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Tommy, as far as I'm aware, mathematics/chemical theory predicted elements that were not known to exist in nature. Then we went and created them, and found the theory seemed to be correct. Of course, the created elements lasted only a fraction of a second, but it was still long enough to be observed.

There's still some room on the periodic table for more elements- and there's a point further up the atomic scale where there's an "island of stability" where extremely heavy elements will stick around a bit instead of decaying into smaller ones in a few milliseconds. Of course, nobody's exactly certain what "stick around" actually means- it may be only a few seconds. It's not likely they'd be stable enough to build something out of.
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balducci
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Oh, is that what Tommy meant by elements? In that case, I agree that the answer does appear to be yes (depending on how exact / precise / accurate you require the predictions to be in order to count).

http://en.wikipedia.org/wiki/Mendeleev%2......elements

Some predicted elements, BTW, last much more than an instant. See link above.
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MobilityBundle
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There are a lot of different meanings that one can ascribe to the word "existence." In a narrow sense, something can't exist "outside the physical world" by definition.

In the realm of pure mathematics, mathematicians talk about things that exist all the time, without being able to actually write them down. To clarify, it's not just that we aren't smart enough at the moment to write them down. It's worse: in principle one can't write these things down, even though they "exist." (The details are somewhat technical. For those familiar with the Axiom of Choice in mathematical logic, you probably already know the story. For those that don't, it's fascinating, but probably the topic of another thread. Smile )

Anyways, then there's applied mathematics. Applied mathematicians used math to predict stuff that hasn't yet been observed. For example, in the early 1900's there was an explosion of mathematics that predicted the existence of new particles. But many of those particles were eventually observed in the lab, so they are pretty solidly "in the physical world," even if they weren't at the time they were predicted.

Antimatter falls into that category. It was first predicted theoretically, and only a while later was it observed in the lab. Same for neutrinos.
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Yes, the question certainly does depend on how one is interpreting all of the key words like "predict", "existence", "elements", and "physical world". In my earlier response, I suppose I was substituting "known physical world" for "physical world".

I have some math friends who do geometry in 5+ dimensions. What they imagine certainly does not fall in the physical world as something they can see or touch, but it exists in their minds which are bound to the physical world. So it is like MobilityBundle says, in a sense nothing we can ever be aware of or think of or predict can exist outside the physical world.
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Balducci, you don't give yourself enough credit. 5+ dimensional geometry is something you're actually pretty accustomed to. You just don't realize it's 5+ dimensional geometry.

Visualize yourself sitting in a chair, at a table. Your arms are at your sides. There's a glass of water on the table. Now visualize grabbing that glass, taking a drink, and placing it back on the table. That is easily a 5 dimensional problem.

What are the tools you have to work with? Basically, a bunch of joints. Your shoulder has a ball-and-socket joint, which yields three dimensions of freedom. The first two of those dimensions define the area that you can sweep out with your fingertip if everything else in your arm were completely rigid, and the third dimension covers your ability to rotate your arm (while keeping your fingertip in a fixed position). Then you have your elbow, which is good for another rotational degree of freedom. Then your wrist, which is good for another two or three dimensions, then each of the joints in your fingers, which are each good for another dimension.

The total "configuration space" of your arm is a strange collection of sections of spheres and circles in something like 20 dimensions. From a particular configuration (say, your arm at your side), you can intuitively trace a path through some key points: one point describing your arm holding the glass, one point describing tilting the glass to your mouth, etc. And not only can you find one solution of this problem, but your natural solution is actually pretty good. For example, you move several of those joints at a time, and your hand travels (in 3D space) in more or less a straight line to and from the cup.

And of course, that's just your arm. When you consider what the configuration space of your whole skeletal system is, it's gotta be in the hundreds, if not thousands of dimensions. Yet, we can very easily do complicated movements like catching a football while running. Well, some of can at least.

When you think of it, 5D geometry is actually pretty easy. Smile
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Yes, but that's really not what I'm talking about. For one thing, your example exists in the physical world as something we can see and touch. Not the case with the example I mentioned.
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tommy
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Yes I guess its true in the realm of ideals, thanks.

If I understand; We know that 1 + 1 = 2 but we don't know it by using any sense.
If there is a single truth about Magic, it is that nothing on earth so efficiently evades it.

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MobilityBundle
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I understand what you're talking about. I was a mathematician for many years. Some of my focus even involved, roughly speaking, geometry in infinite-dimensional spaces. But even infinite dimensional spaces have roots in the physical world. They're abstractions, sure. But there's always (at least historically) something physical that's driving the whole thing.

Also, to be sure, my example doesn't exist in the physical world. My example is the configuration space of your arm. That is, a single point of configuration space corresponds to 20 or so "settings" of the different joints. I.e., joint #1 oriented at such-and-such angle, joint #2 oriented at so-and-so angle, etc. A line in configuration space -- a humble, one dimensional line -- corresponds to potentially complicated movements of all the joints simultaneously.

I'm not sure what your friend studies specifically, but one early source of 5D geometry problems was something in physics called Kaluza-Klein theory. This was a way to unify electromagnetism and general relativity. The idea was to model the universe in five dimensions: length, width, depth, time, and phase. The "phase" dimension had a circular geometry and was designed to keep track of electromagnetic stuff. That's no less rooted in the physical world as my arm example. He11, in Kaluza-Klein theory, the physical world itself was thought to be five dimensional!
tommy
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Forces such as gravity and electric we know exist but we don't know that by using any sense to know it. We know they exist by using our sense to observe their effects. Like we know magic exists by its effects. Isn't that so?
If there is a single truth about Magic, it is that nothing on earth so efficiently evades it.

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MobilityBundle
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I dunno, Tommy. It's true that we don't "directly" observe gravity, we only observe stuff falling in gravitational fields. We don't "directly" see an electric field, we just see how it effects charged objects.

From there, I guess it's a philosophical question whether we really observe gravity or an electromagnetic field. I'm kind of a knuckle-dragger when it comes to philosophical questions -- once the practical implications disappear from a question, I kind of lose interest. So stay tuned for further discussion from some of the more enlightened folks in the forum. Smile
balducci
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Quote:
On 2012-04-19 02:42, MobilityBundle wrote:

I understand what you're talking about. I was a mathematician for many years. Some of my focus even involved, roughly speaking, geometry in infinite-dimensional spaces. But even infinite dimensional spaces have roots in the physical world. They're abstractions, sure. But there's always (at least historically) something physical that's driving the whole thing.

Yes, however I said "exists in the physical world as something we can see and touch". Having roots there isn't quite the same thing. Not in the sense I mean, anyway.

BTW, the friends (plural, not singular) I spoke of are still mathematicians (as am I, at least by some definitions, according to one of the work 'hats' I wear) and I was reporting what they have told me about the geometrical topics they work on in 5+ (5 and higher) dimensions. I'll quiz them further on this, the next time we are in the local pub. Smile
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The first law of thermodynamics states that energy can neither be created nor destroyed,
unless met by Chuck Norris.
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tommy
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But if you tied a piece of string together and formed it into a circle you would get more in it than if you formed it into a square. Why is that?
If there is a single truth about Magic, it is that nothing on earth so efficiently evades it.

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Devious
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Less resistance on the toroid shape due to the momentum shift at the 90 degree angle.
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MobilityBundle
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Because of a topic called "Calculus of Variations," Tommy. I could tell you, but I'd have to kill you. Or, more accurately, you might want to kill yourself. So click the following link with caution. Smile

http://en.wikipedia.org/wiki/Calculus_of_variations

The non-technical answer doesn't really provide a lot of insight: because the area inside a perimeter is determined by more than just the length of the perimeter.

This kind of stuff occurs a lot. It's only loosely related, but I guess it's vaguely like the following observation: If you add 1 to a number, then the percentage by which it increases changes, depending on the size of the original number. For example, if you add 1 to 1, then you double what you had. But if you add 1 to 100, then you only increase what you had by 1%.

Of course, here the statement is completely obvious: the percentage increase doesn't just depend on what you add, but also what you're adding it to. I'm not sure why it would be less obvious with respect to the area inside a perimeter.
S2000magician
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Quote:
On 2012-04-19 02:30, tommy wrote:
If I understand; We know that 1 + 1 = 2 but we don't know it by using any sense.

Actually, we do: Alfred North Whitehead and Bertrand Russell proved it in their Principia Mathematica (1910 - 1913), though not until somewhere in (if memory serve) the second of three volumes.
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That sounds very strange, Russell and Whitehead tried to ground mathematics in logic, not in the senses. Hence the introduction of set theory to prove 1+1=2. Do you have the exact quote?
S2000magician
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Quote:
On 2012-07-25 09:26, Stromberg wrote:
That sounds very strange, Russell and Whitehead tried to ground mathematics in logic, not in the senses. Hence the introduction of set theory to prove 1+1=2. Do you have the exact quote?

"From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2."
—Volume I, 1st edition, page 379 (page 362 in 2nd edition; page 360 in abridged version).

(The proof is actually completed in Volume II, 1st edition, page 86, accompanied by the comment, "The above proposition is occasionally useful.")

Wikipedia: Principia Mathematica
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