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What Makes an Equation Beautiful?
New York Times ^

Posted on 10/25/2004 1:46:25 AM PDT by accipter

CONSIDER a verbal description of the effect of gravity: drop a ball, and it will fall.

That is a true enough fact, but fuzzy in the way that frustrates scientists. How fast does the ball fall? Does it fall at constant rate, or accelerate? Would a heavier ball fall faster? More words, more sentences could provide details, swelling into an unwieldy yet still incomplete paragraph.

The wonder of mathematics is that it captures precisely in a few symbols what can only be described clumsily with many words. Those symbols, strung together in meaningful order, make equations - which in turn constitute the world's most concise and reliable body of knowledge. And so it is that physics offers a very simple equation for calculating the speed of a falling ball.

Readers of Physics World magazine recently were asked an interesting question: Which equations are the greatest?

Dr. Robert P. Crease, a professor of philosophy at the State University of New York at Stony Brook and a historian at Brookhaven National Laboratory, posed the question in his Critical Point column and received 120 responses, nominating 50 different equations. Some were nominated for the sheer beauty of their simplicity, some for the breadth of knowledge they capture, others for historical importance. In general, Dr. Crease said, a great equation "reshapes perception of the universe."

(Excerpt) Read more at nytimes.com ...


TOPICS: Front Page News; Miscellaneous; Philosophy
KEYWORDS: 1plus1equals69; fun; india; math; thisisthis
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To: spunkets
Objectivity does not depend on God. It depends on the person doing the observation only.

See "Theory of Relativity", sub-reference: "How fast is that thing moving?"

121 posted on 10/26/2004 9:14:43 AM PDT by r9etb
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To: r9etb
" My rough translation: What "A" is, is in part a function of the observer -- or, "A isn't really A unless somebody sees it being A." Or, as per the example above, our definition of A, is part of what A is. "

What A is, does not depend on the observer. Neither Bohr, nor anyone else can describe what A is. They work with representations, such as points and waves. A is neither a point, nor a wave. The representations are used to know, describe and understand the observables only, not what has never been observed. If you jump off a building your path is desribed by a point and the kinetic energy you aquire is calulated by considering you as a point. You are not a point, however. The same is found by considering you as a wave. You're not a wave either.

122 posted on 10/26/2004 9:23:14 AM PDT by spunkets
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To: spunkets
What A is, does not depend on the observer.

An assumption on your part, which (for physical things) appears not to be supported by current science. You need to read the entire article to which I linked above.

But just to jump-start it a bit, if A has velocity, what velocity does it have? The theory of relativity says that the velocity (and therefore many of its other properties, including mass, length, and time) depend on the reference point of the observer. You have length. If you're moving with respect to me, then I will measure your length to be something different than what you will measure it to be. IOW, in my reference frame, "you" are different from what you are in "your" reference frame. What is your "true" length?

123 posted on 10/26/2004 9:36:32 AM PDT by r9etb
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To: r9etb
" See "Quantum Mechanics", sub-reference "Heisenberg." See "Theory of Relativity", sub-reference: "How fast is that thing moving?""

Neither of these changes what I said.

"Any rational and honest being can observe what exists and identitfy it. All my not be know at any particular point, but there are no inherent limitations to what can be observrved and known."

In fact they are examples of things we know, because they have been observed and shown to be true, by experiment. You may have a problem in not grasping that these observables are know and understood by using representations for what is not known and understood yet.

124 posted on 10/26/2004 9:37:16 AM PDT by spunkets
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To: accipter
What Makes an Equation Beautiful?

A nice hairstyle, good makeup, and flatering clothing.

125 posted on 10/26/2004 9:39:13 AM PDT by kevkrom (Power corrupts. Absolute power corrupts absolutely. But it rocks absolutely, too.)
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To: spunkets
Any rational and honest being can observe what exists and identitfy it.

As the linked article points out, what the "rational and honest being can observe" apparently depends on the frame of reference of the observer. What I observe is not necessarily what you will observe, even if we're both rationally and honestly observing the same thing. (See, e.g., Special Relativity. NB the discussion on Simultaneity and Causality.)

In fact they are examples of things we know, because they have been observed and shown to be true, by experiment.

As my previous link discussed, it is not clear that one can make such a sharp distinction between the observer and the observed.

You may have a problem in not grasping that these observables are know and understood by using representations for what is not known and understood yet.

Please read the offered links.

126 posted on 10/26/2004 9:46:24 AM PDT by r9etb
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To: r9etb
Me:
The axiom says only that each individual "A" is equal to itself. Why is that an issue for you?

You:
It's not an "issue" for me -- it's just a fun discussion where I'm trying to take a contrary view, and seeing where it goes.

It goes nowhere. If things aren't what they are, what is there to discuss? Hint: that's why A=A is an axiom.

For grins, let's plug "PatrickHenry" into the equation. Thus, "PatrickHenry is PatrickHenry." Fair enough, but does that tell us anything? No, not really, because we don't know what "PatrickHenry" really is.

I'm going to make one more attempt. Let's not work with an "A" as complicated as a PatrickHenry. Instead, tell me this (I'm using numbers, not alphabetical characters here):

Do you agree that 1 = 1?

127 posted on 10/26/2004 9:48:06 AM PDT by PatrickHenry (Hic amor, haec patria est.)
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To: CarrotAndStick

The men split the $5 three ways, or $1.66666... each. Each man chipped the .66666.... toward the tip, which is the $2.

Misleading question because the numbers are skewed to present an untenable solution. Hmmm. Democrat math.


128 posted on 10/26/2004 9:53:14 AM PDT by SlowBoat407 ("Hey, Terrorist!... Terrorize THIS!")
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To: kevkrom

Since you have answered the question with the least words, Muttly has another deep and important question for you:

If Muttly accidentally eats a roast beef hero sandwich that a careless telephone company lineman placed next to the chain-link fence, why is Muttly still hungry ?


129 posted on 10/26/2004 9:57:33 AM PDT by PoorMuttly ("The right of the People to be Muttly shall not be infringed,")
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To: PatrickHenry
Hint: that's why A=A is an axiom.

Hint: That's what the Copenhagen interpretation seems to call into question by requiring that context of the observer is important. I.e., A = A1 (for observer 1), and A = A2 (for observer 2), and A1 does not necessarily equal A2.

I'm going to make one more attempt. Let's not work with an "A" as complicated as a PatrickHenry.

Why not?

Do you agree that 1 = 1?

Sure. But does that allow us to consider "A is A" to be a generally self-evident statement? No. It's only self-evident for the particular case you've defined. For "A is A" to have general applicability, you'd have to be able to show it exhaustively. And for that, you cannot.

You used an example of a natural number, which is suggestive of something: see this interesting discussion of Goedel's Incompleteness Theorems. From the discussion, it appears that it is not always possible to show that "A is A," from within a given system.

130 posted on 10/26/2004 10:08:38 AM PDT by r9etb
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To: r9etb
"What is your "true" length?"

The length in the local frame is the true length, because only the local frame reflects the correct dimensional metrics in combination with their energy and momentum values. The other observers length is different, because their dim. metric differs-due to different E/p values. Length is a dimension, as is time. Energy and momentum determine the metric for the local dimensions.

You can envision the E of the particle itself as having a spread, or flucuation in time that depends on Planck's constant divided by the time interval. The wavelength, +/- the range for the time interval, for the particle itself is Compton's wavelenth. The frequency is normally high. Momentum adds a modulating wave on top of that, which depends on v. That modulating wave is only seen by observers in other frames, not within the particles frame itself. Within the particles frame, only Compton's wavelength is observed. That's because the metric of the local space is different, to maintain the same vacuum interactions.

h/dim, Planck's constant/dim, is a measure of the power flow between this universe and the vacuum per unit dimension. In order to see h as a constant only the local frame counts.

131 posted on 10/26/2004 10:14:55 AM PDT by spunkets
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To: spunkets

Your local frame, or my local frame? Are you in an accelerating frame? How do you know?


132 posted on 10/26/2004 10:27:13 AM PDT by r9etb
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To: PoorMuttly
f Muttly accidentally eats a roast beef hero sandwich that a careless telephone company lineman placed next to the chain-link fence, why is Muttly still hungry ?

Because Muttley would have been better off eating the lineman.

Especially if he was a union guy endorsing Kerry.

133 posted on 10/26/2004 10:27:48 AM PDT by kevkrom (Power corrupts. Absolute power corrupts absolutely. But it rocks absolutely, too.)
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To: r9etb
Sure [I agree that 1 = 1]. But does that allow us to consider "A is A" to be a generally self-evident statement? No. It's only self-evident for the particular case you've defined. For "A is A" to have general applicability, you'd have to be able to show it exhaustively. And for that, you cannot.

Ah. I see what's going on. You would agree, I assume, as to any specific "A." You might agree, for example, that this toad is this toad. Or that this rock is this rock. But if we attempt to rise above the level of specific, concrete examples, you are unable (or unwilling) to grasp the abstract principle that this [thing] is this [thing]. Okay. Now I understand.

Virtual Ignore is now on!

134 posted on 10/26/2004 10:30:06 AM PDT by PatrickHenry (Hic amor, haec patria est.)
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To: PatrickHenry
Now I understand.

Obviously you don't understand. But hey, if you want to ignore Einstein and Bohr, that's fine with me....

135 posted on 10/26/2004 10:34:07 AM PDT by r9etb
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To: r9etb
"Are you in an accelerating frame? How do you know?"

Particles only know their local frame. That's why c, h, particle lifetimes, spectoscopic lines are all constant. Physicist's use tensor equations that are independent of coordinate systems, accelerating frames and moving frames to maintain the same laws of physics in all frames. The particles themselves, regardless of whatever their nature really is, just behave according to their nature and the unique, definite and constant laws of nature.

136 posted on 10/26/2004 10:36:34 AM PDT by spunkets
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To: accipter
I see you're discussing Britney's Guide to Semiconductor Physics.
137 posted on 10/26/2004 10:38:24 AM PDT by Professional Engineer (...time for a change. Killed the computer and it's peripherals. Hard Drive screamed in vain.)
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To: r9etb
...A = A1 (for observer 1)...I don't think this is what the Copenhagen interpretation claims. What is claimed is that a measurement produces the result A1. Another measurement may be different but one ought not call that other measurement "A."
138 posted on 10/26/2004 10:42:44 AM PDT by Doctor Stochastic (Vegetabilisch = chaotisch is der Charakter der Modernen. - Friedrich Schlegel)
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To: spunkets
The particles themselves, regardless of whatever their nature really is, just behave according to their nature and the unique, definite and constant laws of nature.

Assumptions on your part, both as to the behavior of particles (see again the link to the Copenhagen interpretation), as well as the "constancy" of the laws of nature (current theories speculate that the laws as we know them may have been "formed" as part of the Big Bang).

Note also that you're making all sorts of metaphysical assumptions about the nature of reality.

139 posted on 10/26/2004 10:47:47 AM PDT by r9etb
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To: spunkets
The wavelength, +/- the range for the time interval, for the particle itself is Compton's wavelenth. The frequency is normally high. Momentum adds a modulating wave on top of that, which depends on v. That modulating wave is only seen by observers in other frames, not within the particles frame itself. Within the particles frame, only Compton's wavelength is observed.

The "Compton wavelength" isn't a wavelength at all. It doesn't mean that the particle has a wavelength associated with it. It's just the coefficient in front of the relativistic Doppler effect for an charged particle in an electromagnetic wave.

140 posted on 10/26/2004 10:55:33 AM PDT by mikegi
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