Cantor Confusion
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From: Dik T. Winter on 17 Jan 2007 20:01 In article <MPG.201861e1fc338409989b62(a)news.rcn.com> David Marcus <DavidMarcus(a)alumdotmit.edu> writes: > Dik T. Winter wrote: .... > > I didn't receive it and I didn't enjoy it (how could the last happen if > > the first did not happen?). > > It couldn't. Although, I guess you could have enjoyed not receiving it. Oh. We have in our librarys a few publications that prove the squaring of a circle or the trisection of an angle. It actually is (in my opinion) enjoyable reading. The reason is that it is like a puzzle: where do you spot the first error. It is extremely like reading some of the articles by JSH, when he tries to prove something. Archimedes Plutonium is a bit more easy. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Michael Press on 17 Jan 2007 20:13 In article <MPG.201731bcec6d9f73989b5c(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Ah, you are at least a hundred years behind the times. No, a function is > most definitely not a formula. A function is a rule which assigns, to > each of certain real numbers, some other real number. For example, the > rule that assigns to each number a the number 0 if a is irrational and > the number 1 if a is rational is a function, but you will have a hard > time coming up with a formula (nor is a formula required). Hello. I agree. At least one-hundred-fifty years. Is this a formula? lim_{m -> oo} lim_{n -> oo} [cos (m! * x * pi)]^{2 * n} -- Michael Press
From: MoeBlee on 17 Jan 2007 21:17 Andy Smith wrote: > You are comfortable in the formalism and paradigm that you have been > taught, I surmise you mean 'formalism' not necessarily to refer to the philosophy of formalism but, I suspect, rather to refer to such formal theories (and the notation that goes with them) as Z set theory. And it is true that I am pretty comfortable with first order logic, set theory, and mathematical logic, though I have a staggering amount to learn just to catch up to a certain basic level. But it's not so much a matter of my having been taught in any sense of having been intellectually raised in a certain context from which I had no choice but to receive it. Instead, it's more like I got intererested in the subject and did, of course, first come across the dominant approach of classical predicate logic and I did, very quickly, find it to make great sense and to be, conceptually, extremely useful. Yet, in that process I have always QUESTIONED and put to SCRUTINTY everything I've learned, as well as I have done a lot of work myself, devising many of my OWN formulations (within first order logic and Z set theory), of many details that are usually not subjected to such scrutiny in the textbooks. And meanwhile, I always have my ears open for ideas not in the mainstream, including intuitionism, finitism, ultra-finitism, non-standard logics, paraconsistent logic, as well as the many philosophical approaches such as constructivism, realism, structuralism, fictionalism, and even as farflung as certain mystical views of logic and mathematics. (But that doesn't entail that I don't also exercise my prerogative to skewer postings by cranks.) And though I am not even close now to being well versed in all these, I do hope to become much better informed as I continue my amateur studies. And meanwhile I am perplexed by certain philosophical problems that I find to come from classical mathematical logic and set theory. > but everything that you know rests on the shoulders of giants. > Given a clean slate, could you create infinite set theory and a systemic > formalism from the ground up? If not, you can cut me some slack. Oh, but I am not AT ALL expecting you to devise your own formal systems. I am just telling you some of what mathematics has to offer and I would suggest also that you avail yourself of the well written textbooks. > Here's a thought for you. Possibly you are so locked into your paradigm > that you cannot think of any thought not expressed in its terms. Possibly, but I doubt it highly. Meanwhile, plain old mathematics does offer an approach that cuts right through so much confusion and vagueness such as is found in the homespun mathematical ruminations of people who have not availed themselves of the basics of a mathematical education. Thus, I believe it is very constructive to offer formal mathematical definitions as an alternative to floundering in picture-word mathematics. > So > maybe the only way in which you get something new is when some > neanderthal like me blunders in and asks some stupid questions. I haven't found your questions to be stupid. But I do think your questions would benefit by being put in the light of mathematical definitions and some familiarity with certain basics of the subject. > Of > course intelligence comes into it and I cheerfully accept that the > probability of me making any useful contribution has a lot of 0's > following the decimal point. I hold even less hope for myself in that regard. Mathematics is among the very least of my talents and I am really not good at it when it comes to coming up with clever proofs and things like that (just check it out when I post requests for help with proofs that are not, for a talented mind, difficult at all). > I only discovered usenet a week or so ago, but the sci.math forum is > something of a revelation. Snakes on a plane! You have sects, schisms > and heretics, massive egos ... no quiet, polite, thoughtful exchanges > coming to a common consensus, but ad hominem the norm. All very amusing > to an idealist. It is indeed a real trip! And, you're right that it is not at all an oasis of intellectual colloquy or bon homie. On the other hand, there are some very knowledgable and generous people from whom one can learn a lot. It is a beautiful thing that, just by typing, I can get help with problems and questions, for no charge, from experts around the world. And it's my impression that if one has questions, then one will be treated with an indifferent (usually not even what could be described as 'cordiality') but still a fair working respect if one has at least first done the prior leg work of thoughtfully reading a basic textbook. On the other hand, what often meets, as is to the good, sarcasm, ridicule, and derision are the postings of cranks - especially the most strident, self-righteous, and self-satisfied ones - who are virtually ignorant of the subject of discussion and who won't give even a hint as to the norms of whatever special personal logic they use to reach their errant conclusions. MoeBlee
From: MoeBlee on 17 Jan 2007 21:33 David R Tribble wrote: > Be fair. It's not immediately obvious that > f is a function <-> > (f is a relation & Axyz((<x y> in f & <x z> in f) -> y = z)) > is read as > f is a function > implies that > f is a relation and for all x,y,z it holds that > <x y> in f and <x z> in f implies y = z I'm not at all saying that it is immediatey obvious. I'm just saying that it is basic stuff and that, whatever is obscured by crude ASCII notation is also rendered very clearly in many a textbook. > Things like "Axyz" look like words upon first scan. It does take > a little experience to know that it is "for all x,y,z" and that "A" is > an upside-down "A" (for all). I quite agree. My point was not that the poster should already know the ins and outs of ASCII notation, but rather that whatever is not clear in such notation can be cleared up quickly by many a basic textbook. > Likewise "xeN" is "x in N", where > that little "e" is the non-ASCII "member-of" symbol similar to > an epsilon (but isn't). And then there is epsilon, also written > as "e". Usually, I preface by stating what 'e' stands for, though this time I went with plain 'in'. > You can see how novices can be overwhelmed by this at first, > can you not? Yes, I can. But then it is up to the novice to find out about the subject so as not to be overwhelmed. Sure, if one never laid eyes on a book in upper division mathematics, then my formulas could very look like what an electrical wiring diagram looks like to me. But the point is that the formulas stand so that they can be rendered in English by me or another poster if the novice just asks, and the formulas, put in a tidy list do stand for future reference if the novice does ever decide to read about such things in textbooks. MoeBlee
From: David Marcus on 17 Jan 2007 21:38
Michael Press wrote: > In article <MPG.201731bcec6d9f73989b5c(a)news.rcn.com>, > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > Ah, you are at least a hundred years behind the times. No, a function is > > most definitely not a formula. A function is a rule which assigns, to > > each of certain real numbers, some other real number. For example, the > > rule that assigns to each number a the number 0 if a is irrational and > > the number 1 if a is rational is a function, but you will have a hard > > time coming up with a formula (nor is a formula required). > > Hello. I agree. At least one-hundred-fifty years. > Is this a formula? > > lim_{m -> oo} lim_{n -> oo} [cos (m! * x * pi)]^{2 * n} These days or in Euler's day? Notice I said a "hard time". I didn't say it was impossible. -- David Marcus |