Cantor Confusion
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From: Jonathan Hoyle on 15 Dec 2006 14:19 > And too bad Aristotle's approach to syllogistic inference doesn't > produce the truth he hoped it would yield. The best we can say for it > is that it produces a series of truisms which in turn produce the > empirical approaches to math and science which belabor the present. > That mathematics and science rely on it even today after two millenia > stands in mute testament not to its universal validity but to the fact > there is no other better formal system of logical inference available. I'm not following. What "truth" is not being produced, and what is missing in logic?
From: cbrown on 15 Dec 2006 14:29 mueckenh(a)rz.fh-augsburg.de wrote: > cbrown(a)cbrownsystems.com schrieb: > > > > Every edge is divided into shares, namely in as many shares as there > > > are paths to which this edge belongs. > > > > Therefore /you/ assert that an edge /can/ be subdivided into "as many" > > parts as there are paths; which is to say, "as many" parts as there are > > real numbers (by the bijection (2)). > > > > This implies that you assert that the set of all parts of an edge can > > be bijected with the real numbers; so "x" is the cardinality of the set > > of real numbers. > > It is the cardinality of the set of paths witch can be obtained by > repeated multiplication by 2. No, according to your original statement, you ask us to: > Consider a binary tree which has (no finite paths but only) infinite > paths representing the real numbers between 0 and 1 as binary strings. So the paths are not "those which can be obtained by repeated multiplication by 2"; it was /given/ by you at the beginning of your argument that a path exists for each real number in [0,1]. > > > > > Note that I have /defined/ what I mean by "uncountable" in this > > context. > > Note that we are investigating how many paths exist in the tree. Note that you have already given that there are as many paths in the tree as there are real numbers in [0,1]. We are investigating how we can construct a surjection of the naturals onto the reals. <snip> > > I will give you a simple example: > Instead of giving examples, why not simply say exactly what you mean by "share of an edge" in the situation of your proof? Your edges are not being broken into 2 "shares" each; they are being divided into an infinite number of shares, and that is very different from 2 shares! > The notion "rational relation" is a further development of the notion > "relation". While a relation connects elements of a domain to elements > of a range, the rational relation connects shares of the elements of > the domain to (shares of the) elements of the range. A simple example: > > > A) Consider the relation: > > > 1 ---> a > 2 ---> b > Is this drawing supposed to represent a bijection {1,2} -> {a,b}? > > B) If "1" is divided into two parts, 1#1 and 1#2, we can write (A) also > in the form > > > 1#1 ---> a > 1#2 ---> a > 2 ---> b > Is this drawing supposed to represent a bijection {{1#1,1#2},2} -> {a,b}? > > C) Another rational relation could be: > > > 1#1 ---> a > 1#2 ---> b > > > 2#1 ---> a > 2#2 ---> b > > Is this drawing supposed to represent a bijection {{1#1,2#1},{1#2, 2#2}} -> {a,b}? > If 1#1 and 1#2 are the only shares 1, i.e., if 1#1 U 1#2 = 1 and if 2#1 > and 2#2 are the only shares of 2, i.e., if 2#1 U 2#2 = 2, then one full > element of the domain is available for every element of the range. And what is the domain now? Is it {1#1, 1#2, 2#1, 2#2} or is it {1,2}? > Therefore (C) proves that a surjective mapping like (A) is possible, > even if it cannot be found. How does it prove it? Do you mean that the existence of a bijection {{1#1,2#1},{1#2, 2#2}} -> {a,b} proves the existence of the bijection {{1#1,1#2},{2#1, 2#2}} -> {a,b} which proves the existence of {1, 2} -> {a, b}? Your example, which has 2 "things" being split into 2 "shares" leaves me guessing. Is this an example of a rational relation? (C) 1#1 -> b 1#2 -> b 1#3 -> b 2#1 -> b 2#2 -> a 2#3 -> b 2#4 -> a Does this (C) prove the surjection {1,2} -> {a,b} "is possible, even if it cannot be found"? If instead 1#3 -> a, then does it prove it? Could you explain why? Wouldn't it be simpler to just give a clear /definition/ of what you mean by your "rational relation", instead of endless examples? > > > > I am only interested in your "rational relation" argument. > > Fine. Why are you interested only in that argument. And why are you > interested in it at all? I am interested in that argument because it has a surface plausibility of "mathiness" while being obviously wrong. I am only interested in that argument, because to pursue your other arguments regarding |N| = |R| would be a distraction from the argument in question. Cheers - Chas
From: Jonathan Hoyle on 15 Dec 2006 14:34 > > True, but you are ignoring Virgi's adjective "sound". Calculus existed > > way back during the time of Newton and Leibniz, but you could hardly > > call their use of the infinite and infinitessimals at all "sound" by > > today's standards. It wasn't until Bolzano and Weierstrass made things > > truly rigorous in the 19th century was Calculus anywhere near sound. > > Allright. And they should have _stopped_ at this point in time. Why? Are you stuck in the 19th century? You don't like progress? > > Bolzano and Weierstrass gave way to more rigor in numbers by Cantor, > > and then rigor in Set Theory by Zermelo and Fraenkel. > > There was a pre-emptive war. Set Theory invaded Calculus. ??? Please tell me you're joking. > > With the > > exception of Aristotle's Logic and Euclid's Geometry, much of > > mathematics would not be considered acceptable by today's standards. > > Who's "standards"? The problem with standards is that you have so many > to choose from. Are you imposing _your_ standards upon the rest of us? The standard of rigorous logic applied to proofs. Logic: beginning with assumptions A1, A2, etc., we yield conclusions C1, C2, ... This was lacking in most of mathematics prior to Weierstrass. Today, rigor is required. No assumption goes unchecked. Are these the very things you are unhappy with? You want the assumptions to be hidden and muddled? > Even Guass and Euler played a bit fast and loose (although they were > > considered impeccably precise in their day.) > > Huh? Would you say that absolute nitwits like Zermelo and Fraenkel, who > have contributed nothing to actual mathematics, are greater individuals > than Gauss and Euler? Did I really read this? Can't believe my eyes ... I didn't say "greater individuals"; you did. Zermelo and Fraenkel contributed greater rigor to mathematical thinking than Guass and Euler. I am not saying that Guass and Euler did not contribute, or even that they contributed less. Combined, they represent some of the most important mathematics in history. But their work was not always as rigorous as it should be (particularly Euler). Their proofs today are taught to young mathematicians, but only through the more precise language and integrity required today. > > In ancient times, arithmetic was discovered in much the same way > > physical laws were. "Hey, notice that when we do this, that always > > happens..." As centuries of very hard work, mathematicians have boiled > > arithmetic assumptions down to some basic axioms, and all of the > > remaining theorems flow forth. > > According to the Holy Gospel of Modern Mathematics, I suppose ... According to those who wish to be accurate. To those who wish to play fast and loose and are not concerned about accuracy and precision, then your method may be fine. (It would not be called mathematics today though.) Jonathan Hoyle Eastman Kodak
From: Math1723 on 15 Dec 2006 15:54 Han de Bruijn wrote: > > True, but you are ignoring Virgi's adjective "sound". Calculus existed > > way back during the time of Newton and Leibniz, but you could hardly > > call their use of the infinite and infinitessimals at all "sound" by > > today's standards. It wasn't until Bolzano and Weierstrass made things > > truly rigorous in the 19th century was Calculus anywhere near sound. > > Allright. And they should have _stopped_ at this point in time. I agree. And what's with these people saying the Earth is round? Geez, they are going too far. Next thang you know, they'll be pushin' them thar evolution to our kids! > There was a pre-emptive war. Set Theory invaded Calculus. A sneak attack it was too. Like Pearl Harbor. Damn ruskies. > > Even infinitessimals were consistently defined by Robinson. > > I've never seen such useless things as Robinson's infinitesimals. Anything too small to see ain't real, anyone knows that. > > With the > > exception of Aristotle's Logic and Euclid's Geometry, much of > > mathematics would not be considered acceptable by today's standards. > > Who's "standards"? The problem with standards is that you have so many > to choose from. Are you imposing _your_ standards upon the rest of us? Yeah, it's just discrimination, I tell ya. Like wearing shoes or counting teeth! We don't need standards in mathematics! Speaking of which, what's the latest definition of pi these days? > > Even Guass and Euler played a bit fast and loose (although they were > > considered impeccably precise in their day.) > > Huh? Would you say that absolute nitwits like Zermelo and Fraenkel, who > have contributed nothing to actual mathematics, are greater individuals > than Gauss and Euler? Did I really read this? Can't believe my eyes ... None of them fer-ners ain't no damn good nohow! > > In ancient times, arithmetic was discovered in much the same way > > physical laws were. "Hey, notice that when we do this, that always > > happens..." As centuries of very hard work, mathematicians have boiled > > arithmetic assumptions down to some basic axioms, and all of the > > remaining theorems flow forth. I prefer the good ol' days, where the men were men, the sheep were nervous, and Mathematics had no axioms. Tracking your assumptions seems like an invasion of privacy to me anyway. (It's the NSA doin' it, I tell ya.) What business is it of yours what assumptions I make? Hell, Russell's paradox never harmed me none.
From: Jonathan Hoyle on 15 Dec 2006 16:11
> Therefore I fault him. Except for lenses Aristotle had every contrivence > and technology that was available to Galileo. Greek shipwrights could > have made straight smooth wooden ramps to be used as inclined planes. > The Greeks had Egyption drip clocks at their disposal. So it wasn't lack > of tools or technology that prevented Aristotle from checking. It was > attitude. I see your point. I'm a big fan of Aristotle and the work he did, but his physics does appear to lack the attention to detail that his other works included. If he is deserving of the praise for his other works, he likewise deserves the criticisms of his failures. > We would be a millenium ahead or extinct. Aristotle's followers cost us > over a thousand years of progress. And the worst part to this was his "followers", for all those centuries, not *one* amongst the untold millions even attempted to coroborate his work on "Natural Philosophy" to see if it held true. This is more than simply "being sheep". This is not caring enough to even try. :-/ |