matt grime

Did I mention the word "internet"? Which books? And what level is this course at UCSC? Graduate? There is a vague and fuzzy notion that chaos theory is that explanation in Jurassic Park. It is non-linear dynamics. The model of throwing a die will be chaotic, it will be non-linear and display TT and SDIC. That isn't the same as a question of probability. Indeed, I do something completely deterministic and write down a number. I tell you it is in the range 1-10. You guess. I do it again, you guess again. We do it 1,000 times, what do you reckon is the approximate number of times you've got it right? Any mention of Chaos?

f is diffible, so f is continuous, hence (Riemann) integrable.

First, what mathematics have you done? Standard topics would include RSA encryption if you've done basic ring theory, knots if you've done any topology, stability of models of physical situations if you've done analysis and differential equations. Quite a nice one, if you ask me, and you've done some metric space stuff, is to explain how the CONTRACTION MAPPING THEOREM implies that differential equations indeed have the solutions you think of and no others. If the essay is not proof based, more expository, then how about the classification of all compact surfaces according to genus and orientability, or even the classification of finite simple groups. What about some foundational questions: the implications of the (mis)use of the axiom of choice and other axioms in set theory (every set can be well ordered with these things and this leads to the banach tarski paradox - that a sphere can be "cut" into some pieces, 6 I think, and these pieces may be rotated and translated and reassembled into a shape of strictly smaller volume - as well as other philosophical questions such as defining a game which apparently both players win with probability 1). A general discussion on the nature of mathematics might start off with any of the issues raised in these articles: http://www.dpmms.cam.ac.uk/~wtg10 and look at his expository articles. A good one to think about might be "the definition of the Real Numbers", or how maths is often shows that something exists (indeed many things exist) yet doesn't give any way of constructing them.

Doron, I cannot prove you wrong **within your system**, and have no intention of even trying. Indeed there is little I can prove or disprove in your system because you've not managed to explain anything clearly enough to allow me to make any certain deductions, or even state that deductive reasoning is allowed (deductive reasoning states that (Anb)=>C is equivalent to (A=>(B=>C)). You have for instance failed to define "complete", that adjective that we cannot apply to infinite sets, apparently. I can say you're wrong about Cantor's argument. It may fail in your system but that is because your system rejects the objects upon which it is built. Nothing wrong with that, but then to imply, as you do, that Cantor needs to get off the stage because, with *your* ideas about infinite things it is a nonsensical argument is wrong. It just means that in *your* theory it doesn't hold, not that our theory is wrong. If you can in any way demonstrate anything in your theory that is consistent and useful then perhaps some may take you seriously, if however all you're going to do is to say that results that weren't formulated in your system are wrong then no one will care. In fact you will only serve to dissuade people from examining it if the only results about it are that everything in mathematics currently needs to be thrown away. As I keep saying, there is a model of ZF where in the power set of the naturals and the naturals are isomorphic. It is called Skolem's paradox. This is all down to your idea of "actual" and "potential" infinity that is no part of mathematics as we know it.

Cantor at no point even requires, or states the the real numbers exist. The real numbers aren't a line, Doron, and are not important in this discussion, can you please get past this high-school misapprehension?

Erm, no, that's not actually what Chaos theory is. A dynamical system is chaotic if it displays topological transitivity and sensitive dependence on initial conditions (or at least that is the more usual definition). Chaos theory has no need to be used here since we aren't treating this as a dynamical system, we are treating it purely as a probability question. The combinatorics model of dice throwing doesn't ask you to model the throw with a dynamical system, does it?

I agree that I do not share your point of veiw on infinity, however since you're claiming that Cantor is wrong and Cantor's result is that which uses *my* definitions, then your definitions are completely irrelevant and have no bearing on the result. If you're saying that Cantor's theorem is false in *your* system, then so what. It isn't a statement about anything in your system. Why should it be true if you just reassign meanings at will. You ought to at least show Cantor is wrong within the "Cantorian" model, otherwise you're just spouting nonsense.

Chaos theory has nothing to do with this question; let's not obscure the important probability theory involved. (Conditional probability.) Your grasp of chaos seems informed by popular and misleading sources. Indeed your conclusion that chaos theory tells you to switch is utterly false. I don't know what topological transitivity has to do with this question, do you? The probability of obtaining a 6 in two throws of a die given the first is a six is 1, rather obviously, and has nothing to do with chaos but with the assumptions of probability.

You can't say what they are all equal to exactly in decimal expansions. Anyway, I think you should pick the ones you care about and google for them.

What has this to do with the pigeon hole principal, and what contradiction are you talking about? That in order to evaluate the sum of the first n-1 integers you put n-1 in the formula? That's what the formula is, how on earth can that be a contradiction?

Can you do it for two unknowns? Then you can do it for n unknowns. Or look up Gaussian elimination.

Seeing as GB are losing to NZ I'll amuse myself with this instead. Doron, your "disproof" of Cantor assumes that the power set is a binary tree that can be embedded in the plane. It isn't. That is the first point at which your argument fails. The fact you do not accept universal quantifiers is of no interest to anyone. Cantor does not at any point use "the set of all sets that contain themselves", and so doesn't not need "Russell's paradox", that is another point where you are wrong. Your diagonal argument pdf still contains the assumption that you can arrange all these things into an array with countable rows and columns in the case of the natural numbers since you can do so for the finite numbers that is obviously rubbish. Moreover the two large arrays you start writing out clearly only contain elements in them that, if the columns were enumerable and a complete list, contain only finitely many zeroes or finitely many 1s. It still presumes that 2^{aleph-0} is countable in the model of ZF where the power set is the set of functions to {0,1}. Note there are models where that isn't the definition of power set and where P(N) is countable, this is called Skolem's Paradox. So, just because you do not accept that complete, or all are words that cannot be used for infinite sets doesn't mean that anyone else agrees with you. Please, if you can, write down some integer that isn't in the complete set of integers, or the set of all integers.

erm, in what sense is the inverse of a functio the opposite of the reciprocal? The inverse of x^2 as a function from R+ to R+ is not -1/x^2 is it?

It is very strong, and very useful, though from what I recall it also doesn't have a proof that's at all worth memorizing, but it turns out to be very important in lots of areas.

Ah, that makes it a little harder then. Let G be any group, and suppose |G|=m.p^r, where p is a prime, and p doesn't divide m. Then G has a subgroup of order p^r called the Sylow p-subgroup. In fact it has potentially several of them, but the number of them satisfies a rule I can't recall precisely, they're all conjugate, and any subgroup of order p^s for some s is contained in some Sylow subgroup.

Actually, thinking about this (i've never heard of regulated functions so was interested in doing this): [math]|f(x_n)-f(x_m)|=|f(x_n)-\varphi(x_n)+\varphi(x_n)-\varphi(x_m)+\varphi(x_m)-f(x_m)|[/math] use the triangle inequality to make that into three terms, and that [math]\varphi[/math] is a step function to conclude that since x_n converges to a, that all those terms can be made either less than e, and the whole thing less than 3e, or each less than e/3, and the whole less than e, for all n,m greater than some N. Is that the proof you found? I only write it out to check because that style of argument is very useful, and is what you'll see when you show that a continuous function on a closed bounded interval is uniformly continuo

Well, why take my word for it? Let's prove it: if f(x) = 1 for all x, what is f^{-1}, the nominal inverse function? A function is invertible iff it is bijective. So obviously it fails to be a group. If G is a finite group with even order, let S be a Sylow 2 subgroup. Let v be any non-identity element in S, then v has even order, say 2*r for some r. Then v^r has order 2.

I think there are too many to list here, or anywhere really. Some of the more important ones: h, or h-bar used in quantum mechanics, alpha the fine structure constant, ditto, e for differential equations amongst many things, avagadro's constant, phi whihc approximately occurs in nature, for a good explanation about it and its implications as to why it occurs in nature see http://www.maa.org and look up some of devlin's old articles. the maths symbols are mark up thingummies enclosed between some tags. Next time you see one you want to look at click on in and you ought to get the code for it.

The set of maps from R to R isn't a group under composition though, and even if it were there are still elements that square to the identity (if G is finite and |G| is even there are always non-identity elements that square to the identity).

Doron, I've said my bit, ok: the ability to draw on a piece of paper somthing that you want to call a binary tree of an "equation" is nonsense to everyone else. We have a simple and consistent system that you clearly do not understand and no amount of explanation seems to make you understand us or even want to understand it. Perhaps if you were to explain to the public why you have this obsession with binary trees we could understand why you want to do these things, but most of us apparently just can't see through your unclear writing. The cantor set is uncountable, the nodes in your tree are countable, but then these are things that you don't accept, or do you eventually understand the proof of cantor's theorem that no set is in bijection with its power set when taken as the set of functions from the set to {0,1}? The cantor set can be put in bijection with the paths through the tree.

post 8, you asked how to manipulate e^(0.5log3)... and got some long post off 123 rock, or rather a post containing a long string of numbers. well, i'm not sure what they were getting at, but the answer, which you don't see how to get, is what I just wrote if you apply the log laws to the original expression.

How about this example: suppose S is a finite set and f a function from S to itself. If f is injective then f is surjective. Or one I saw recently: Suppose that you have six numbers, none of them divisible by 6. Show that at least two of them have difference that is divisible by 6. Or, given any six people, there wll either be 3 people who all know each other, or 3 who are all mutual strangers (assuming that if x knows y, then y knows x). The principal is just something that is useful in proving many theorems, especially those in combinatorics.

You mean the pigeon hole principal? As in assigning n letters to n-1 letter boxes means someone gets two letters at least? Yes, it is a very important lemma in mathematics that crops up all over the place. I doubt that it is of little practical value to the "real world" directly in the sense you appear to be geting at.

you do know that rlogx = log(x^r)? so that your thing to simplify is [math]\sqrt{3} + \frac{3}{\sqrt{3}} = 2\sqrt{3}[/math]

if a is in Hb, then there exists some element h in H such that hb=a, Then Ha=Hhb=Hb. since h permutes the elements of H. Of course if you really want to do it the longer way: if x in Ha, then x=ka some k in H, since a=hb for some h, we see x=ka=khb since kh is in H (H is a group and hence closed under composition), it follows that Ha<=Hb Similaly if x in Hb, then x=kb some k, but since a=hb => h^{-1}a=b x=kb=kh^{-1}a which is in Ha since H is a group and closed under inverse and composition.