"The total passion for the total height"
who is chatting on SOLO ChatThe Free RadicalPopular contentWho's onlineThere are currently 1 user and 13 guests online.
Online usersPollWhat should the government do about ailing financial institutions? Nothing, except to back off and get out—as any Objectivist knows, intervention is treating the disease with the disease 84% Intervene judiciously—enough to avert a catastrophe that is otherwise imminent 3% Intervene massively—as it's doing 3% Nationalize the whole economy and be done with it. Bring on the USSA! 1% Something else (specify) 9% Total votes: 76
 Basic Economics by Thomas Sowell
 Economics In One Lesson by Henry Hazlitt
|
[e to the (pi * i)] minus 1 equals zero Submitted by milesian on Mon, 2005-12-05 20:51.
IN DEFENSE OF THE ABSURD: THE PRIMACY OF EXISTENCE [This was originally posted to the Philosophy conference of The Well, the old Whole Earth Known in modern times as the Analytic-Synthetic Dichotomy, the fallacy says that some things can be proved logically even though they cannot be validated by our senses and that empirical observations can be contrary to logic. On the other hand, objective metaphysics (as developed from Parmenides of Velia to Ayn Rand of New York) asserts this: that which is true is both logical and observable. So, any apparent difference between a "rational" statement and an "empirical" perception results from incorrect logic and/or incomplete observation. A good example of this is the first (and strictest) discovery of the "absurd", i.e., the square root of two. It is said that when Pythagoras (or a later disciple) discovered that there is no fraction ("rational number") to express the hypotenuse of an isoceles right triangle, all were sworn to secrecy on threat of death. Later Greeks accepted the existence of the square root of two (the "absurd") but they were not happy with it. Ancient mathematicians were also unhappy with the idea of negative numbers, another kind of absurdity. And then there was the square root of a negative number. For centuries, the square root of minus one was offered as an example of how purely theoretical ideas in mathematics have no referents in reality. And then came electricity. Alternating current was transmitted through wires and then through the ether. Frequencies and amplitudes were modulated in precise, predictable ways because the algebra of trigonometry opened the door to transcendental functions. When you describe electro-magnetic fields, you use the number i. The square root of minus one is objective and practical -- and it is integral to the nature of pi and to the nature of e, as well as to the sine of any angle. The square root of minus one _seemed_ like an "absurdity" only to those who doubted the primacy of existence. The correct perspective is that there is this number, the square root of minus one, that we can derive in several ways, but we do not have a way to use it. In order to manipulate the equations of Einstein's electrodynamics of moving bodies, you need to consider 6 dimensions. These are real. We just do not perceive them. North America was real, even though people in Asia and Europe did not know about it. The number i existed before our solar system. It is of the fabric of reality. Therefore, it has application to anyone who can understand it and use it. This is true of any logical or synthetic or rational or abstract truth. If an assertion is logically consistent, it works. Capitalism is both moral and practical. Individualism is both moral and practical. Heroic art not only feels good, it _is_ good, objectively good. When you start with a truth, anywhere it leads must be real. We can construct complex ideologies of fallacy and from a wrong assumption argue logically for all kinds of silly ideas. Most political ideology and all of religion is of this class. These fallacies begin with one or more errors. The errors are wrong. They do not correspond to reality. All Black people do not have the same economic interests; 28% of gross is not a fair tax rate; the bishop of Rome both is and is not the vicar of Christ, depending on whom you ask, and the Jews both are and are not God's chosen people. When you start with an error, no amount of logic can bring reality into accord with your assertions. There is no such thing as a "purely theoretical truth." Our experience is in and of the world. Our imagination, however fanciful, derives from our experience of an objective reality. You can create a chimera, but only by having experienced the snake, lion, and eagle. Topology, n- dimensional geometry, abstract algebra, etc., etc., are as real as the concrete slabs of a sidewalk. We call photons "wavicles" and we let college freshmen compute the "wavelength" of a baseball in motion. Quantum mechanics is not a doorway into ignorance but our best understanding of how the universe works. The greatest challenge in physics has been to demonstrate how Quantum Mechanics and Relativity require each other. When demonstrated, The Grand Unified Theory will be both logically consistent and empirically verifiable -- as were the transcendental equations containing i.
( categories: )
|
User loginFeatured BookNavigationSyndicate  The Fountainhead by Ayn Rand
 Atlas Shrugged by Ayn Rand
|
To Milesian: Six Dimensions?
Minkowski space has 3 + 1 dimensions. The electromagnet stress energy tensor is a 4x4 anti symetric tensor with six independent components: three for the electric field, three for the magnetic field.
The Minkowsky four space (a hyperbolic geometry) is the natural home for the Maxwell Field Equations. For details consult a standard text on electrodynamics such as Schwartze's book.
Bob Kolker
sign this
I think you'll find that exp(iπ)+1=0, not exp(iπ)−1=0.
Imhotep wouldn't understand a thing ;)
Yes, in this case, I partly misunderstood the article.
It's also interesting to note that a cosmologic constant that Albert Einstein introduced in his early works (but rejected later on) is now partially proven correct, because it perfectly fits into a theory on universe expansion.
I think this shows that there is a correlation between abstract mathematics and reality, we just have to find the connecting points.
What would Imhotep do?
Max, it is not that "mathematics is very exact and definite, reality is not." It is that as our mathematics improves and as our understanding of reality improves, these necessarily come together.
For example, as you know, before positional notation ("Arabic" numerals) computation was grueling. The Egyptians relied heavily on fractions; the Greeks used geometry; Babylonians swore by, or swore at, sexigesimal counting. Today, we have better tools. We know pi to a zillion places and a $100 box will graph transcendental functions for you. A scholar from Newton's time would be astonished by the way college freshmen handle simple algebraic expressions of his complicated geometric proofs. In fact, Richard P. Feynman could not reconstruct a key proof from Newton (conservative forces in a plane by the equal areas rule) and had to re-do it with simpler geometry.
The mathematics of lubricated bearings is grueling today -- as was the computation of specific gravity in 500 BC. It is true that some problems are more tidily expressed and solved, but again, only because of new tools to do the work. There is no analytic-synthetic dichotomy.
________________________________________________________
"I have slipped the surly bonds of Earth
and danced the skies on laughter-silvered wings."
Engineering and Exactness
Yes, but mathematics is very exact and definite, reality is not. If you look how engineers are translating technical mechanics to real-life problems, you get a notion that mathematics is exact and definite, but only in its abstract space. It is statistics that gives us a better impression of the reality than pure mathematics.
One example: We have a roller bearing and we want to use it with an oily additive. Therefore we not only have to calculate the static and dynamic load, but also a life-time calculations of the bearing and here the reality and mathematics diverge. We are only making an educated guess really, instead of being definitive and I think this is the rift that divides reality and mathematics a bit....
But that's just a thought.
Clarifying...
Mr. Marotta,
Please don't get me wrong; as I wrote, I was only making a technical point. Nit-picking, in a sense.
As to the theme of the article itself, I found the large number of examples/specific instances somewhat distracting. That's perhaps my weakness -- a carry over from my dislike for (text)books with too many worked out examples and illustrations.
Sanjay (Please use my first name... it is a lot shorter!)
It's a Sine of the Thymes
Dr. Velamparambil, I apologize for allowing a common shortcut to disturb the flow of logic. Everyone knows that there are two square roots to every number. √4 = ±2
I suppose that everyone knows that every number is a complex number: 2 is really 0i+2, of course.
I am surprised, therefore, that the point of my essay eluded you. It was only this:
"... that which is true is both logical and observable. So, any apparent difference between a "rational" statement and an "empirical" perception results from incorrect logic and/or incomplete observation. ... The number i existed before our solar system. It is of the fabric of reality. Therefore, it has application to anyone who can understand it and use it. This is true of any logical or synthetic or rational or abstract truth. If an assertion is logically consistent, it works.
"
____________________________________________________
"I have slipped the surly bonds of Earth
and danced the skies on laughter-silvered wings."
Mr. Marotta,
Mr. Marotta,
Without getting into a debate on the central theme of your article (which, respectfully, is rather difficult to deduce), let me sharpen your mathematical description of 'i':
'i' is *not* the square root of -1; it is 'a' solution to the equation x^2+1 =0. If we don't make this distinction, we can end up with a paradox as follows: \sqrt{-1} = \sqrt{-1}
==> \sqrt{-1/1} = \sqrt{1/-1}
==> \sqrt{-1}/\sqrt{1} = \sqrt{1}/\sqrt{-1}
==> \sqrt{-1}*\sqrt{-1} = \sqrt{1}*\sqrt{1}
==> -1 == 1 ?
The mistake made in the above 'derivation' is in ignoring the 'sign' of the two roots of -1 (choosing two different 'branches' in the language of complex analysis).
Sanjay