DrRocket

But two photons may, through a somewhat subtle process, generate a pair. http://en.wikipedia.org/wiki/Two-photon_physics

Conjectures and speculation aside, current theory is unable to make predfictions much earlier than about 10^-33 seconds after the big bang. So nobody really knows what happened at t=0 or whether or not the universe "came out of nothing". That of course does not stop all sorts of people from making statements and selling books dealing with the ultimate origin of the universe. Buyer beware. One of the characteristics of a good scientist is the ability to utter the words "I don't know."

The general rule in mathematics is to be clear. "48÷2(9+3)" is not clear. To make it clear one needs to use parentheses. So one would write either 48÷(2(9+3) or (48÷2)(9+3). There is no point in arguing over what 48÷2(9+3) is "supposed to mean" as that reqiuires divining the intent of the writer. The fundamental problem is the symbol "÷" which is not normally used in scientific circles, or polite company, anyway. I have not seen it in regular use since about the third grade (caculators using algebraic notation (ugh!) excepted). I have been a professional mathematician for over three decades. I understand arithmetic pretty well, but I flunk tests of clairvoyance with regularity. When in doubt ask the author.

I don't think that any such criticism was intended by Feynman in that particular quote. What you imply is more closely reflected in this quote: The theoretical broadening which comes from having many humanities subjects on the campus is offset by the general dopiness of the people who study these things...” – Richard Feynman

This is vsimply wrong. The cosmological model of spacetime is an intrinsic Lorentzian 4-manifold. It is not embeddd in anything. What is called "space" is any of the leaves of a timelike one-parameter foliation into 3-dimensional spacelike hypersurfaces. These hypersurfaces may be compact (aka finite) or non-compact (aka infinite) but in either vcase they have no boundary. Belief has nothing to do with it. With what is known today space may be either finite or infinite. No one knows. The question of a boundary is irrelevant to the issue. You seem to be arguing against the cosmological model of general relativity, while sinultaneously ignoring the meaning of the mathematics in which that theory is formulated. There is no "elsewhere" and there is no boundary to the spacetime manifold, no matter whether space is finite or infinite. If you want to argue otherwise you must of necessity put your arguement into some context other than general relativity. So, what is your alternative to general relativity ?

If you are studying a subject that interests you, pursue it as far as you are able. If your interest is solely n making money, go to work.

Sequences of functions with the properties that you state are called "approximate identities". They can play a role similar to the "delta finction" in some contexts. The Dirichlet kernel and the Fejer kernel are such creatures and are important in the theory of Fourier series. But approximate identities are not a substitute for distributions. I went into mathematics after a graduate degree in electrical engineering. Yes, some engineers use mathematics like a hammer. One reason that I decided to learn mathematics from experts was the tendancy of some of my engineering professors to smash their fingers. Repeatedly.

An Introduction to the Theory of numbers by Hardy and Wright Basic Number Theory by Andre Weil

I don't know who gave you that definition of a distribution, but find another source. Laurent Schwartz is spinning in his grave. A distribution is a continuous linear functional (a linear function with scalar values) defined on the space of "test functions". A test function is an infinitely differentiable function with compact support. The topology on the space of test functions is a bit subtle, so we will not worry about what a continuous functional is for now. One can view ordinary functions and measures as distributuins as follows (with an abuse of notation intended to lend clarity): [math]f( \phi ) = \int_{- \infty}^ \infty f(x) \phi (x) \ dx [/math] [math] \mu ( \phi ) = \int_{- \infty}^ \infty \phi \ d \mu [/math] where [math] \phi [/math] is a test function, [math]f[/math] is an ordinary function and [math] \mu [/math] is a regular Borel measure. The "Dirac delta" is the case where [math] \mu [/math] is the point mass at 0. Integration by parts motivates the definition of the derivative of a distribution : [math] A'( \phi) = A(\phi ')[/math] for any distribution [math]A[/math]. To really understand the theory of distributions, particularly the topology on the space of test functions, requires some knowledge of topological vector spaces and functional analysis. That topology is neeeded to understand what a continuous linear functional is and to understand limits of distributions. An excellent source is Walter Rudin's book Functional Analysis. Aless rigorous but accessible treatment of distributions can bev found here: http://www.stanford.edu/class/ee261/reader/all.pdf One can also extend the notion of test functions to infinitely differentiable functions that "vanish rapidly at infinity" (go to zero more quickly than polynomially) and get into "tempered distributions". Tempered distributions are important in some aspects of the theory of the Fourier transform. The reason that you cannot see the limit that you are seeking is that it does not exist in the sense to which you are accustomed -- there is no such function [math] \chi [/math] in the usual sense of a function -- only as a linear functional on the space of test functions; i.e as a distribution. But you can't get to an understanding of whjat is going on with the misleading, hand-waving "definition" of a distribution that you were given.

That is what Bondi called the "perfect cosmologicalrinciple". It leads to s steady state theory of cosmology as advanced by Hoyle or Bondi and Gold. These theories intuitively appealing, relatively simple and wrong.

No. At least in the sense of "to exist" in my personal lexicon. There is a large and increasing body of evidence showing that space is expanding at an increasing rate, No one knows why. The cause has been dubbed "dark energy" so in that sense it exists. But no one understands what it is. The energy of the vacuum shows up as a negative pressure term in the stress-energy tensor of general relativity in some analyses. That acts as a positive cosmological constant and would be an explanation for dark energy --except that attempts to perform the necessary calculation over-estimate the observed effect by about 120 orders of magnitude (by a factor of [math]10^{120}[/math]). That is a mistake so large as to be ludicrous. Another approach is to simply insert a positive cosmological constant into the Einstein field equations, selected to match the observed expansion of space. That is what is done in the standard [math]\Lambda CDM[/math] model. But there is no explanation for the value of the cosmological constant "lambda". So we know that space is expanding at an accelerated pace, but are clueless as to why. The "why" has been given a name -- dark energy. No one knows what dark energy is. If that counts as "exists", then yeah, it exists. I have somewhat higher standards for the term.

http://www.scienceforums.net/topic/33180-cosmo-basics/

This is simply false. A finite space is simply a compact manifold, a manifold of finite volume. The surface of a sphere is such a 2-dimensional manifold. It has no boundary. The question in cosmology is whether space-like slices of spacetime are compact or non-compact. In either case they are 3-manifolds without boundary, submanifolds of 4-dimensional spacetime, which also in GR has no boundary. It certainly aids the understanding of those with sufficient mathematical competency to comprehend the formulation of general relativity in terms of differential geometry. I can see why that is of little comfort to the attendees of your conference. This is just bizarre. Maybe before ontologists ask questions about spacetime they should acquire some knowledge of the actual content of the theory. Spacetime is equipped with a metric that determines the geodesics. An object under no influence other than gravity follows a geodesic trajectory in spacetime -- so in that sense geodesics are "ruts". The whole content of the theory lies in the relationship between the distribution of mass/eneregy (the stress-energy tensor) and geodesics. And THAT is why you need to understand the mathematics. There is no substitute for understanding the language of the subject. The manifold in general relativity is not 4-D space. It is a 4-dimensional Lorentzian (not Riemannian) manifold -- spacetime. It is quite simple to construct manifolds and vector spaces of any dimension, but the spacetime of physics is 4-dimensional with a metric of signature (+, -,-,-) or equivalently (-,+,+,+) -- and this conveys a great deal of understanding to those who understand mathematics. I find your reference to "stuffed shirt relativists" more than a bit ironic -- downright hysterical is a better description. "Philosophy of science is about as useful to scientists as ornithology is to birds." -- Richard P. Feynman

The trampoline analogy usually creates as much or more confusion as enlightenment. That is clearly the case here. Objects under the influence of no force other than gravity follow geodesic paths in spacetime. They do NOT follow geodesic paths in space -- an elliptical orbit is not a spatial geodesic, not by a long shot. Unfortunately there is no simple intuitive explanation of the curvature tensor. I don't know of any good explanations outside of real no-kidding texts on general relativity or texts on differential geometry.. The best is probably Gravitation by Misner, Thorne and Wheeler. It requires quite a bit of background.

You need to read and study a good book on relativity. There are many. One good one is Wolfgang Rindler's Essential Relativity, Special, General and Cosmological. An Introduction to Special Relativity, also by Rindler is another good one. Xeno's "paradox" (there are at least three but my comments apply to all of them) is not a paradox at all but simply an example of faulty logic. It is quite possible to add up infinitely many positive quantities and get a finite number. This is exactly what is needed to resolve the usual Xeno's paradox. 1/2 + 1/4 + 1/8 + .... = 1 So if you walk at a speed of 1 ft/sec towards a wall, starting from 1 foot away you cover 1/2 ft in the first 1/2 second, cut the remaining distance in half in the next 1/4 sec cut it in half again in the next 1/8 sec etc. So you reach the wall in 1/2 + 1/4 + 1/8 + ... = 1 seconds just as you would expect. To see that 1/2 + 1/4 + 1/8 + .... = 1 you note that [math]\displaystyle \sum_{n=1}^N x^n = \dfrac {x-x^{N+1}}{1-x} [/math] For [math]x=\frac{1}{2}[/math] this becomes [math] \frac {1}{2} + \frac {1}{4} + ... + \frac {1}{2^n} [/math] [math] = \dfrac { \frac {1}{2} - \frac {1}{2^{n+1}}}{1-\frac {1}{2}}[/math] And [math] \displaystyle \lim_{n \to \infty} \dfrac { \frac {1}{2} - \frac{1}{2^{n+1}}}{1-\frac{1}{2}} = 1 [/math]

Is that legal in Mississippi ?

He did procreate in the academic sense -- producing new PhD mathematicians. I am one of his descendents. http://genealogy.math.ndsu.nodak.edu/

Exactly the same way that if you start with an empty plate and then put a pea on it, you now have one pea. Get a copy of Landau's Foundations of Analysis and see all of the usual numbers built from just the Peano Postulates.

Often wrong. Never uncertain.

What you don't believe in is entirely irrelevant. What is relevant is the accepted theory on which cosmology is based -- general relativity. In GR there is no "fabric of space". What there is is a 4-dimensional Lorentzian manifold without boundary -- spacetime. "Without boundary" means that all points have a neighborhood that is diffeomorphic to Minkowski 4-space; i.e. there is no edge. That does not mean that spacetime is an open (i.e. non-compact, aka infinite) manifold. It may well be compact (aka finite), but still without boundary. Philosophy is a good place for your ideas. They certainly do not meet the standards required for science.

In mathematics "=" means "is". Equivalence is an entirely different concept, but it is quite possible for two equivalence classes to be equal, and quite often mathematicians talk of equality when what is implicitly understood is that what is equal are equivalence classes. If one does not adopt this point of view the discussion quickly becomes pedantic.

In short it is acceleration, not speed, that one can sense, and the centripetal acceleration for the roughly circular orbit of the earth around or the sun around the galactic center are rather small. Even the effect of the rotation of the earth is not ordinarily felt, but see "Foucault Pendulum"or Coriolis force for a concrete effect. http://en.wikipedia.org/wiki/Foucault_pendulum http://en.wikipedia.org/wiki/Coriolis_effect

Believe it or not 1 + 1 =2 is essentially a definition of 2. You need to read a good treatment of the development of arithmetic and number systems, through the complex numbers, starting from the Peano Axioms. A good sourrce is Foundations of Analysis by Landau. Another is Principles of Mathematical Analysis by Rudin. Starting from the natural numbers, one defines addition, subtraction, multiplication and division, along the way constructing the integers, rationals, reals and finally the complex numbers.

Take a look at the definition of "keernel". Then see if you still think that {t,1} is a basis for ker(L).

What definition are you using for matter ? http://en.wikipedia.org/wiki/Matter