matt grime

f(s)->s isn't the correct notation for a map, Doron. Tut, and you think you know something about mathematics?

Lets do an example, let x_n = 1/n, which converges to 0 and satisfies the first one. is the second one of these true (using x=0): there is an N such that for all e>0 and all n> N, |1/n| < e given that this N exists (and is fixed), what if I now pick e=1/2N?

"Completed" means that its end can be found. define end. do you mean max or sup or something? what if it weren't an ordered set? Is the set of all elliptic curves one with an end?

google for Cardan, or do i mean cardano?

In the first one we impliclty are saying that N depends on epsilon, in the second we are saying that N is independent of e, so in effect we are saying that x_n=x for all n>N

Ah, but I'm not claiming to be a genius, Artotius. I am English, but incredibly lazy at typing. And I can spell grammar, when I feel like it.

Sod it, one more, eh? "If you disagree with me, then you have to show how a universal quantification can be related to a collection of infinitely many elements." Define "related", define "completed". It is not up to us to show anything because you've not asked of us anything that is written in or about mathematics as is understood by anyone else. Until you phrase your posts in such a way as someone else can make sense of them then we cannot do anything but point out that with the ordinary meanings to all the words you're talking complete rubbish. For all x in R x^2 is positive. What's wrong with that? I'm well aware of the different schools of thought in math, Doron, only I seem to understand them and you appear wilfully ignorant of them.

I don't think this breaks my promise to respond only once to Doron as it is a reply to bloodhound. (and i must resist the temptation to pooint out his mathematical errors again). Bloodhound, Doron doesn't actually know what a bijection is, or any of the mathematical objects he may use. He may point out a wolfram link as if that explains he understands it, but that isn't the case. His is a simple premise, that any statement that includes a "for all" is automatically false for an infinite set. We know that to be nonsense. For instance where does he define 'endless'? We may have a vague notion that helps explain it, but that doesn't define it se we can use it. Just lookat his attempt at defininf invariant proportion. One need only look at his idea that Cantor defined transfinite arithmetic to see that he's completely ignorant - Conway did that, and there is Robinson's variation. Why, for instance is it important to him that we haven't defined aleph-0 -aleph-1? They are not natural numbers, or a ring for that matter, in any obvious non-trivial way. Of course there is a philosophical issue here that is fundamental: Cantor took the axiom of choice as true in order to well order the cardinal numbers. The cardinal numbers are of course too big to be a set, so there's another important delciate philosophical issue. After all aleph-0 is just a symbol, defined bty the use: a set has card aleph-0 if ther is a bijection with N. Very simple. Very hard is the question of whether 2^aleph-0 which is by definition the cardinality of the power set of N has cardinality aleph-1, the smallest uncountable cardinal. In fact it is independent of ZFC. But none of what has been written in post 1 makes any coherent sense.

ok, here is the only reply you'll get: 1. what has the fact that you can draw pictures got to do with provin anything about taking a limit, and then proving that something in the limit at infinity makes any sense? 2. who says we calculate the sum of infinitely many things? ONce more you're confiusing applying mathematical operations with a definition of a limit. 3. the cardinality of the natural numbers is not a natural number, so there is no reason to suppose it behaves as one 3. aleph-0 isn't a "value" it is an equivalence class of sets 4. yo'uve failed to define an arithmetic for cardinal numbers so your last bit about addin epsilon, an invariant proportion, to a cardinal amkes no sense 5. you've not actually defined this invariant proportion, merely offered an example, and nto said why this is a cardinal number that can be "added" to aleph-0 6. do you even know the definition of cardinal numbers? 7. cantors definition of cardinals is about bijections of (infinite) sets, so why don't you write a mathematical argument as to why it;s flawed other than saying that infinite sets can't be complete, which is not a mathematical statement, by the way. 8 why don't you actually look up the definitions of terms you use so you don't make mistakes? 9. why do you think that everything must behave as you wish rather than actually proving what is true?

Oh, it is, lama was one of his aliases in another forum that got banned.

Doron, you grace us with your presence, and inane ramblings, again. Glad to see you've not learned anything about mathematics in your absence.

Hey! don't provoke a guy who's figured out, apparently, how to force a win at chess, as well as nec and suff conds to solve all games. Of course, that is about te same as showing p=np so maybe he can get a million dollars. Mind you, I'd've expected such a genius to know the difference between "your" and "you're".

So what about the fact that, for instance, the whether or not we take the parallel postulate as true or false in geometry (it is independent of the other axioms) leads to 3 totally different theories all of which are useful in the "real world"? Isn or isn't this one of the inherent problems with mathematics. When we say it is `just' a study of results that may be derived from axioms that we may assume true or false (as long as we're not inconsistent) we are not claiming anything about the absolute truth of them at all. Do you think that the axiom of choice is in some physical sense? Or false? Please give me a physical model to verify this. You didn't answer my question about what the arbitrary choice of the number 10 as our radix has to do with the philosophical issues at hand. Nor about Tarski's completeness theorem. You do understand what that is? After all you are presuming to tell us what we ought to think about what mathematics is, so I'd presume you're familiar with this statement. I must admit I know very little about O-minimal structures (very very little indeed.) Moreover, I trust you don't believe the article's assertion that base 10 and base 20 are the only systems developed by 'ancient people' or whatever sill phrase they used. I could also take issue with the entirely unjustifed premise that mathematicians and scientists are attempting to abuse geometry (reduce nature to points etc) to create an ontological explanation for the nature of the universe. We do not. We create models with the best consistent tools we have and use them to see if they are good enough MODELS. No one will deny mathematics' origin, or that for many years it was developed intuitionally. However the author has this point completely backwards, what the author fails to realize is that it is precisely the division of mathematics from its origins that has led to great advances and cross fretilization of ideas in many disciplines. Particularly sickening is the implied statement that imgainary numbers are some how bad by placing the name in quotation marks. Agreed the name is misleading, not to say downright unhelpful, but they are nto a matter for distaste like that. As a standard challenge I like to ask people who have philosophical issues with them to define the real numbers. Usually they cannot, since if they can they know that complex numbers are just ordered pairs of reals with an algebraic structure that makes them a field. We could also make the point that, along with many people, the author (and by extension you) might not fully understand what a mathematician means when they say theirs is an abstract subject: they mean that it is not *necessary* to refer to any underlying physical situation that may have led to something being studied, even if there were one. Look at the transition from abstract to concrete groups. For instance, what is the quantitative meaning of the axiom: given any line and a point not on that line there is a line passing through that point and not intersecting the initial line (where by line we mean geodesic)? What is quantitative about saying that x_n is a homoclinic orbit (apart from the fact we indexed it by something you think is a number, though I could easily, if inelegantly, have witten it in set terms). Or how about if G is a group and H a subgroup then any cosets gH and kH intersect trivially or are the same. To make a wholly crass analogy, language (we may posit) was devised to express basic desires and needs, such as to say we are hungry, and explain how to get food. Should we have to remember this when reading, to quote on critic, the saddest tale ever told: "The good soldier" by Ford Maddox Ford/Hueffer. But thank you, a self admitted neophyte, for pointing out an article containing errors factual and possibly philosophical. I shall try and remember that when I'm sat in the lectures in my next conference: epistemiology and philosophy of mathematics, next month.

Have read the article now, or at least a large chunk of it before I gave up in digust at the ignorance of the author. Do you want a list of the mistakes in it, Daymare?

The excerpt is riddled with errors as anyone who understands, say, the continuum hypothesis will know. It seems based upon a high school idea of mathetamatics. In particlar Tarski has shown that geometry is complete, thus it cannot be quantitative, if by that we mean strong enough to necessasrily define the integers. That is of cousre on problem with these debates - what exactly do you mean by quantitative? Moreover, how many real pure mathematcians do you know? I'm sure a lot of them would like to know what on earth the decimal system has to do with this argument, cos I'm buggered if I can see the author's point.

0+0=0 which is divisible by three, bloodhound. I dont' see why the "general" disproof was offered, since it only requires one counter example, eg 0+1. i'm also mystified as to why the original 1st question limits us by some positvity condition - if the question is to make sense there is no reason for this.

the first question makes no sense. i think the word produc is missing. the third one is obvious from the definition.

You may draw these things in the complex plane, but you are simply describing a set of complex numbers, unlike the real plane where you are describing a relation between pairs of real numbers. You can have equations in complex variables such as w=cos(z) and so on but to draw them requires two complex planes, ie 4 dimensional real space. Functions and "graphs" of complex variables are very important (Fermat's Last Theorem is proved by considering certain fucntions of complex vaiables called modular forms). Loci of points in the complex plane are also of interest. Look up Julia Sets and Benoit Mandelbrot

A unit in a ring is *any* element that has a multiplicative inverse (ie divides 1). This is a very unfortunate thing really given the other meanings of unit/unital/unitary.

The order of an element doesn't tell you whether it is even or odd. A transposition has sign -1. The sign of a permutation is the product of the signs of any way of writing it as a a product of transpositions. (123) = (12)(23) so its sign is +1 as it is the product of an EVEN number of transpositions. In general if we write an element as a product of disjoint cycles its sign is the product of the signs of the cycles. A cycle of length r has sign (-1)^{r+1}

Avoid saying unit when you mean identity.

His E thing probably was a sigma *my* E thing was an expectation sign.,

The E I meant is in fact [math]\mathbb{E}[/math] not a sigma as bloodhound says, though apparently in response to a different remark, n and n-1 are used in different contexts depending on whether you want an unbaised estimator of the population variance or not. ah, he's added a further explanation now....

What do you mean, how is it derived? One wishes to have a measure of how spread out the data is. There are several ways of doing this (eg interquartile range), one is the standard deviation. It is a measure of how far things are away from the mean. Sadly one cannot just add up the (directed) distances from the mean of the sample points, as the answer will just be zero. One could add up the abs values of all the distances, indeed this is a recognized statistic. However, the preferred measure is the variance: E(X^2) - E(X)^2, It is the average of the sum of the squares of the distances from the origin. E meaning expectation, and X being the r.v. This has the benefit of giving more weight to those numbers which are further from the mean than those closer.

There is nothing to stop you inventing your own notation, as long as you explain it - that after is all the usual binomial expression is. Here's a naive way of writing it: [math](a+b)^n = \sum_{s \in S(a,b,n)} s[/math] where S(a,b,n) is the set of all words in a and b of total length n. You can refine that since S is the disjoint union of, say S(a,b,n,p) which is the words in a and b of length n with exactly p a's. Can you see how to prove that all this is true? Think about showing how every element in the expansion is some word of length n and how every word of length n can be gotten from the expansion.