At least for your bank...
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 ...
BTTT
Crude perhaps, but I like this set of equations ....
http://www.strangecosmos.com/content/item/3410.html
robt there is something seriously wrong in that pic....
Bush + FL + (OH or PA) = Victory
Excuse me, but were you being sarcastic?
That equations means 'nothing' to me!
;)
Gosh! I thought the dollar went towards the NFL stadium that is to be built in Arlington for the Cowboys!
It's really a stunning result.
Another "beautiful" set of equations is Maxwell's Equations, which relate electricity and magnetism. What's most amazing about them is that Maxwell developed the relationship out of a desire to make the equations "more symmetric." So his "mathematical aesthetics" drove him to an incredibly important result.
Just for fun, you ought to try proving Ayn Rand's old saw, "A is A" from first principles. You can't do it without making assumptions such as "A existed before the Big Bang."
"A is A" is an axiom. Axioms are accepted as such because they can't be proven.
They each paid $9 and the clerk kept $27 ($25+2).
The hotel kept it. It should have been 9$ each, which is 27. They had already payed the 30 dollars and then told the clerk to keep 2 dollars so we are up to 29 right. So the additional dollar must have been kept by the hotel?
"Unprovability" is not a good basis for calling something an axiom, especially when the topic at hand is ripe for disproof, at least as a general statement.
Webster provides three definitions for axiom:
1 : a maxim widely accepted on its intrinsic merit
2 : a statement accepted as true as the basis for argument or inference
3 : an established rule or principle or a self-evident truth
When you look at it, "A=A" is not an axiom except in the very limited sense of the second definition. In essence, assume that "A=A. If so, then...." The logic doesn't require the axiom to actually be true, except within the context of a specific argument.
We can't really accept "A=A" as being axiomatic in the sense of being self-evident, nor even having intrinsic merit, for a couple of reasons.
First, the reason I picked on that one, is that it's possible to demonstrate that there was a time (i.e., before the Big Bang) when it is very likely that "A != A." E.g., consider the conditions ascribed by physicists to initial moments following the Big Bang. Does it make sense to state that the full set of possible "A's" is invariant under transformation during the Big Bang?
Of course, there is the additional problem that we don't really know which "A" Ms. Rand is talking about -- it's obviously stupid to generalize on "A=A", because we can pick some "A's" that were not always "A's." (Living entities or scientific theories being two examples.)
Now, it is possible to use A as a variable, and to deem certain principles as axiomatic, as is done in mathematics -- but in those cases the "intrinsic merit" or "self-evident" properties are clear.
Not so with Rand's general statement, the truth of which depends on a static universe filled with something akin to Plato's "forms".
A thing is what it is. If you disagree with "A is A" that's fine with me.
Are you "what you were" 1 second ago? Will you be "the same as now" in 1 second? No, on both counts. And in the world of "things" that's a general condition. It presents a problem for those who want to try to derive absolutes from "A is A." The statement is useless if we are constrained to considering only the very present time, knowing that past and future states of that thing are different.
That's the essential problem with the general statement "A is A." If I can point to a time when some thing "A" was not A, then one cannot designate that thing as an absolute.
It's true, OTOH, that there are some ideas for which the term "axiomatic" is clearly called-for. However, Rand's stated "axioms" do not qualify, there being logically consistent alternatives to her choices.
Not quite any. One needs finite variance for this to work (or a bunch of funny limits, see Feller's books.) Then the mean values are normally distributed with variance proportional to the original variance divided by the number of samples. Convergence is very slow (about 1/cuberoot(N)); see the Barry-Essen bounds.
To see how things fail, one can use the Cauchy distribution with density 1/pi(1+x**2). Of course, this distribution fails to have a mean or a variance; the average of the sums doesn' not converge.
However, for distributions with finite variance, the result is striking. Another striking statistical result is the Glivenko-Cantelli theorem which says that the distribution function of a (randomly generated) sample converges to the correct distribution function with probability one. Polling uses this fact.
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