Aeschylus

Why dko you need to approximate anyway? the area of a circle is pi*r^2, you shpould always keep the pi in, unless for some reason you need to know where it approximately lies on the real number line, but generally there's no need to know.

The simple answer is we can't.

The CMBR is not the after glow of the big bang per se but it is from 300,00 years after the big bang when the unievrse's density became low enoguh that any photon emitted wasn't instantly reabsorbed, during a short period of time all areas of the univere (because at this time the universe was very, very, very homogenous) emitted alot of light within a very short period of time. Nothing to do with antimatter. The CMBR does pass us and it doesn't come back due to the effects of gravity, BUT there is always more CMBR coming to us. From our postion, the CMBR that we see at any insatnt is like viewing a giant spherical shell with a radius of about ~13 billion light years taht we are at the centre of, but this is only due to the effects of the unievrse we are exapniodn as by looking at the CMBR we are really looking at an area of space, much, much smaller than a spherical shell of radius 13 billion light years. If you've ever performed an inversion on a circle you might understand what I mean.

Then that may be the case, I don't know beacuse I haven't done alot of the really pure maths.

Basically can not be a member of the field of real numbers. I'll show ou if you want, but you might like to try and construct the proof yourself by looking at the field axioms.

The empty Set symbol is a capital letter phi, I looked it up.

Divison by zero is simply not defined, so your proof is worthless.

You know I never relaized that it was phi as capital phi is usually printed like this: [math]\Phi[/math].

All numbers exist as only abstract entities, maths is an abstarct subject which has no intent to describe 'physical reality' (if your interested in why maths does seem to be useful in describing physical relaity look up Wigner's essay on the subject).

Again you are wrong (besides which there is no great division between using a metric and not using a metric, it dopepnds entirely on context), (vector) spaces without metrics are used in physics too. No I am specifically talking about vector spaces. No, spacetime comes specifically from Minkowski, nothing to do with Gallileo. What Minkowski pointed out basically it would be rather useful in the context of Einstein's STR to use to combine space and time into spacetime, due to the invariance of the interval. Lorentz formulated his transformations, Eibstein then formalised the theory, recognizing the failure of simulatenity, Minkowski then came up with the iodea of spacetime as a useful tool. The discovery of the failure of simukataneity porecedes the use of lightcones. Curvilinear coordinate systems can be used in non-Euclidean spaces. I'm not sure where your getting this from, but a finite and unbounded universe is a specifc FRW cosmology. I think you need to review exactly what relativity says about mtter gravity and other dimensions, because staem,ents like 'electricity is the 5th diemnsion' really don't make any sense in the context of relativity.

(Interestingly this needn't be finite or even countable). A simpler less formal defintion would be the number of numbers needed to describe every point in space. Lets say we take our familair 3 dimensional Euclidean space and define each radius unit vector as a 'direction', how many directons are there? The answer is there are an infinite amount of them. However from this subspace of unit vectors we can pick an orthonormal basis of three unit vectors (and no more) that are linearly independent, this is why we say the space is has 3 dimensions (of course again it's worth saying that the number of orthonormal bases is infinite so each 'dimension' is not related to any specific one direction). So the concept of 'dimension' and 'direction', whilst being related are most certainly not the same thing. Minkowskian spacetime is an interesting, example of a space that is flat but non-Euclidean, as it has a pseudo-Riemmanian metric. Of course spacetime, needen't be and isn't flat.

A dimension and direction are not the same thing, by playing semantic games, doesn't change their actual mathematical definition. The number of dimensions of a (vector) space is the number of lineraly indpendent 'directions' if you like (or more formally the cardinality of the maximal set of lineraly independent vectors).

What we call 'Arabic numerals' are actually callled 'Indian numerals', by the Arabs.

The Sumerians were not Arabic (the descendnats of the sumerians almost certainly are though), infact they used a (broadly speaking base-60 number system). The Arabs rose as a force thousands of years later in the 7th century AD. Arabian traders imporetd the numerals of the Inidans into Arabian civilisation, this syatem was further developed by the Arabs with such inovations as decimal fractions.

EPR paradox does not violate causality or relativty as no information is transmitted faster than c. What it does illustrate is the non-local nature of quantum mechanics.

But the Arabs got their numerals originally from India.

Like all our other numerals, it comes from India via the Arabs.

No it doesn't. it's just an example of quanutm tunneling, a phenoumenum that's been known abou for mnay years. You have to ask what is specifically travelling faster than the speed of light? Is causality violatd?

A direction and a dimension are not the same thing.

From the point of view of classical wave equations, the use of complex numbers is simply convenient. From the point of view of the wavefunctions of quantum mechanics the wavefunction is fundamentally complex. The wavefunction though doesn't have any direct physical significance, so Im(Psi) and Re(Psi) don't have any direct physical signifcance either.

Differential geometry deals with this. What happens when you change the size of the circle that you use to measure the angle? The answer is according to the way you've defined the angle, the size of the angle changes! So you can't define an angle from any old circle as on such a surface the ratio depends on the size of the circle (and possibly where the circle is). However as the radius of the circle tends to zero the ratio always tends to pi, so you can deine the angle from the point where the two lines intersect.

What I mean is that it's like talking about 'north' of the North pole, by the way we define the north and the North pole there is nothing north of the North pole, simlairly by the way we define reference frames and photons, photons don't have reference frames. It's not a gap in our knowledge, it's just that our knowledge tells us that such questions are meaningless.

The question becomes menaingless in SR, as you cannot define reference frame for the photon so you cannot talk about whetehr or not time passes for a photon. The problem is that peole use theb formula for time dialation and input the value 'c' and think the answer is meaningful as it tells you nothing about the photon's reference frame.

Massless particles do obey special relatvity, but they also produce some reuslts that may be counterintutive. For exmaple the 4-velpcity is not defiend for a massless particle and they also give rise to null vectors (a null vector is not necessarily a zero vector in SR; a null vector is a vector whose square norm is zero and a zero vector is a vector whose components are all zero), which by the defintion of orthogonality are orthogonal to themselves (of course zero vectors which are the only kind of null vectors in basic 3-vector algebra can also be considered to be orthogonal to themselves anyway)! But none of these are actually problems becasue photons don't have rest frames.