matt grime

What does any of that mean? 'Established' is not as far as I am aware a well defined mathematical term. Seriously, that is almost totally incomprehensible. Don't confuse a firm definition in abstract geometry with a hand-wavy idea in physics. If you want to make it rigorous then let's talk atlases on manifolds which is the proper setting for this, if you are familiarwith them. Dimension is a very loose term with many meanings in different contexts that all haver roughly the same idea.

I'm not going to explain entanglement (I can't) but I can try to explain why it isn't as surprising as you think. Firstly, throughout history people have dismissed things as impossible, yet we now see them as silly for so doing. Perhaps this is just another thing? Secondly, I'm not asking you to accept it just because the equations say so, I'm saying you should accept it if the equations predict it and it is experimentally verifiable. Otherwise it is just speculation. But remember the maths is just a model for predicting what should happen, the important things are how well it works compared to real life. Thirdly, make sure you have a correct idea of what entanglement actually is (not what a popular science book says it is), and the experiments that verify it (look at the Niels Bohr Institute's website for instance). I suspect that what entanglement acutally 'is' is not what a popular science book paints it as, just as anyone who bases their idea of chaos theory or the riemann hypothesis on the popular science books would be surprised if they actually saw the 'real' maths people do on the subject. The analogies were supposed to be analogies by the way, and you were supposed to imagine that you hadn't discovered air pressure and (in)compressibility of fluids.

try thinking of it this way. i join together two objects with a brass rod, i move one and the othre object moves, nothing 'caused' them to become joined by a brass rod. too easy? how about I have a room with a two doors one at either end, both open inwards, one is slightly ajar, as i sharply open the other door the ajar one slams shut, it didn't 'know' i opened the other door and nothing 'caused' there to be air in the room to transmit the 'message' quantum particles do weird stuff, let's face it. i think we should stop being surprised and start accepting it happens if it happens.

GR/SR applies purely to the physical realm. i wonder what the mass of a 'sin' is in this crackpot's view. yep, that's right. laugh away, the bloke's being intellectually dishonest in drawing spurious analogy.

Sit back, take a breath, and read what you typed. What are the formal mathematical definitions of the words you're using? What does 'require 2 points of reference' mean? What I'm trying to make you realize is that you're asking questions that in and of themselves, whilst seeming reasonable, are not actually answerable. You're attempting to make concrete your initial vague notions of what these things are in a way that they probaby weren't intended to be used. The phrases just don't make sense to me, for example: "1 dimension (to me) requires 2 points of reference" "0 dimensions (x) + 0 dimensions (y) = 1 dimension. x (0) + y (0) = 1" "you can't have 1 dimension until you establish two points to plot the direction" don't make sense to me because you're being, oh, what's the word, too 'physical' about these things. Let us presume you're talking about R^3 as the ambient space in which you're thinking of your points living. Points are 0 dimensional by definition, lines are 1 dimensional, planes 2 dimensional. They take 0,1, and 2 parameters (not points) to describe them, that is where the dimension comes from, if we let x,y,z be position/direction vectors then a point is just x, a line is x+ry, r in R being one parameter a plane is x+ry+sz for two parameters r,s in R. each of the last two is naturally the same as R and R^2 respectively (x+ry maps to r, and x+ry+sz maps to (r,s) There is nothing about points of reference generating anything and no one is saying anything about addition of dimensions.

explain what you think generate means. A line in R^n is specified uniquely by any two (distinct) points on it, if that's what you mean. given x and y the equation of the line passing through them is (parametrically) x+t(y-x), though i have no idea if that satisfies your notion of generation.

of course not, i cannot call my dining table a law because of that description you can just limit the circumstances to any parameters you want. yep, that's how it works. Moore's law would certainly fail if we suddenly ran out of silicon, and other materials, but it is a tacit assumption that we won't. the point is that there is a ***non-vacuous*** set of circumstances under which the law appears to hold very well. However, I would personally not say that Moore's law is actually a law in the scientific sense. It is a law like murphy's law or parkinson's law is a law. An aphorism that holds true. It is surprising that it still remains true despite many people thinking that it would start to fail as we hit smaller and smaller dimensions. I don't think anyone is saying that there is some causal reason for it to be true like newton's laws of gravitational attraction, or kepler's laws of planetary motion.

I should have course have said 'which company would not want to break moore's law by making improvements significantly faster than it predicts'. No company would want to break moore's law by failing to double at least at the proposed rate.

If you look at my sig there is a link to my webpages where versions of those papers have been around for a year or two. There is a copy of my thesis too which gives more results in that area if you want to read it (tate cohomology, virtual subcategories, induction as a faithful functor between stable categories under some conditions). Why is it surprising or noteworthy?

That makes no sense. Which computer company would not want to be the first to break Moore's law?

The chaos thing.... here's what 'chaos' is: the study of dynamical systems that display sensitive dependence on initial conditions and topological transitivity. The former of these might get a mention in the likes of Gleick (which I too read before going to university and thought was 'amazing'). It is about homoclinic and heteroclinic orbits, and the like, solving annoying PDE's, all kinds of things. Summing it up as "butterfly effect" is completely unhelpful and mathematically inaccurate: even without any chaotic behaviour a butterfly flapping its wings in CHina can cause a hurricane in Guatemala, but that is not chaos just amplification of causes. Jeff Goldblum rambling on about nature always finding a way is not chaos theory. Drawing Mandelbrot sets and reading Arthur C Clarke is not learning about chaos theory. Sadly, I know, discussing mathematics can be dry and dull, but it is possible, as Korner, Gowers, Hardy, Baez, Devlin, and a multitude of others have shown to avoid this trap. Chaos theory books have a hint of 'jumping on the bandwagon' about them that makes me sick. To a certain degree white lies when explaining maths are necessary, even to other mathematicians, yet the white lies in books about chaos theory have become gospel truth.

fortunately amazon isn't the only seller of books in the world: http://www.cambridge.org:80/uk/catalogue/catalogue.asp?isbn=0521568234 admittedly i did get the title slightly wrong. to rephrase my question: do you want something that is mathematically rigorous or just a handwavy and inaccurate load of waffle? or general ethos of mathematics, perhaps, like Hardy's Apology, or Polya's How to Prove It (perhaps it's how to solve it), then there's Tim Gowers's VSI book.

That makes most sense, but I did also give the method of proving the correct version, albeit without stating it: the form of **primitive** pythagorean triples is a well known one and can easily be found. the vey first hit on google for primitive pythagorean triples gives the answer, for instance.

Tom Korner's The Pleasure of Counting springs to mind. Gleick's was the coffee table book for Chaos when I was applying to university. The former book has exercises I believe and makes you think, and the latter is a handwavy load of crap about nothing (as you come to appreciate if you ever actually come to do mathematics). Do you mean that you don't want something with exercises? You won't actually learn any maths from the Gleick type of book.

I can't prove it; it is false: there are infinitely many counter examples. Perhaps you omitted some other condition on the primitivity of the triangle (6,8,10 or 9,12,15 etc are the counter examples).

It is a covariant one form wedged with a contravariant one form, and is a indeed a joke.

we qualify our statements to define random, and number, and then get a generator that provides statistically indistinguishable numbers from the ideal we so define.

Pi doesn't have an e in it. (I've has an apostrophe, u isn't a word etc, other old man comments...) Otherwise use the formula you were given for arc length (or sector area) in term of angle.

Nothing like resurrecting very old dead threads; check the dates, lads.

Just work mod 10, that's easier than my first suggestion, actually.

The product is even. If it ends in 2 we're done, and the answer is 0. If not, well, just think about how to divide out to make it end in 2.

There are an infinite number of even integers, not every integer is even. So what's the problem? Incidentally, we are not 'going from 1 to 2' in the real number system, or the rationals by 'going from one to the next', there is no such things as the 'next' number in the rationals. If you are thinking about in terms like that then you have a misconception. Incidentally, it is spelled 'infinite' and something being infinite merely means it is not finite. If you are struggling to understand what something 'is' in mathematics then it is good to remember its definition.

I gave you the mathematical usage. 'Arbitrary' is used in proving things 'for all' objects of some kind, it has nothing to do with parameters. Further example: suppose i let x be in N and prove some result (say x^2+x is even) depending on nothing to do with *which* x i picked, then I would say that the result is true for all x in N since my choice of x was *arbitrary*. This is analogous to the choice of the first motel. There was no compelling reason to pick 'the first', the choice of the first was arbitrary, my choice of x was arbitray.

Now there's an interesting function: F(n) = sup { lcm(p_1,p_2,..,p_r) : p_1|p_2|...|p_r is a partition of n} No idea if there's a nice formula for it. What can we say about it though? It is increasing but not strictly: F(5)=F(6)=6. Obviosuly F(n)=>n, with possible equality as we've just seen. But beyond that nothing really springs to mind.