Cantor's Diagonal?
in [Math]
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From: Transfer Principle on 12 Feb 2010 18:35 On Feb 10, 9:38 am, William Hughes <wpihug...(a)hotmail.com> wrote: > On Feb 9, 6:01 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > There are many similarities between Yessenin-Volpin and WM > > that lead me to believe that each deserves to be a "crank" if > > the other so deserves. These similarities include: > > 1. Both are ultrafinitists. > Except for WM who actively denies the existence of a largest > integer. ....which I acknowledge in the very next sentence: > > 2. Neither actually has a fixed upper bound on the magnitude > > of a permissible natural number. Thus Y-V can't answer the > > question "What is the largest number?" > > WM has also stated > > that although some Peano natural numbers "don't exist," there > > is no largest permissible natural number. The original purpose of my post is to acknowledge that there are some posters so "crank"-y that even I shouldn't defend them. (To the standard theorists, of course, that describes _every_ "crank," but still, I'm trying to be somewhat more selective in which of the "cranks" I will defend.) I was trying to propose that WM is so "crank"-y that I shouldn't defend him, except that I mentioned Y-V in the same sentence, setting the standard theorists off. I've noticed that of the three criteria that I mentioned to identify the "cranks," the third one has received the least hostility from the standard theorists. Let me repeat this one for emphasis: > > III. Standard theorists prefer the use of symbolic object > > language to natural metalanguage. To standard theorists, > > natural languages such as English lack the mathematical > > precision of symbolic language, and so all axioms must be > > stated using symbolic language only. An axiom written in > > metalanguage isn't truly an axiom to standard theorists. > > "Cranks," on the other hand, prefer the use of metalanguage > > to object language. To "cranks," purely symbolic language > > lacks the direct applicability to real-world phenomena that > > can be described in metalanguage, and that working with > > symbolic language is merely playing around with symbols. An > > axiom written in object language isn't truly an axiom to > > the "cranks." It appears that of the three criteria that I listed, this one appears to be the most effective in distinguishing "cranks." After all, William Hughes writes: > In my opinion WM is a crank because of the way > he argues. E.g. > - he mostly refuses to give definitions, > and many of the definitions he does use > (generally implicitly) are idiosyncratic in > the extreme. Giving definitions, especially rigorous definitions that can be reduced to primitives, is obviously something that "cranks" are much less likely to do than standard theorists. It goes with what I'm saying above in III, namely that "cranks" feel that giving rigorous (especially symbolic) definitions is just playing around with symbols, not doing mathematics. This is likely one of the reasons WM calls such rigor "matheology" and not truly mathematics. Keeping this in mind, this means that the "crank"-iest posters are therefore those posters whose writing is as far from being a rigorous definition of anything as possible. Such posters include, for example, Ross Finlayson in this thread, and T.H. Ray in another active thread. Indeed, MoeBlee has quipped that it's easier to convince "a pastrami sandwhich to do the backstroke" than it is to convince RF to make his arguments more comprehensible. Standard theorists in the past have compared RF's writing to the Postmodern Generator, a website that chooses words at random, including many buzzwords, and grammatically combines them to form what appears on the surface to be an academic essay but in reality contains little content. Thus, if we judge "cranks" on a continuum from the most rigorous (i.e., formal symbolic language) to the least (i.e., writing resembling the PoMo generator), then RF and THR would be the "crank"-iest posters of all -- and therefore the posters whom even I shouldn't defend. But between the formalist rigor of the standard theorists and the PoMo-like writing of RF and THR, there must be some sort of middle ground. For example, we might have: 1. (Formalist) Axyz (xRx & (xRy <-> yRx) & ((xRy & yRz) -> xRz)) 2. (Middle) R is an equivalence relation. 3. (Po-Mo) "The equivalency function [...] works in general models of the reals [...]" (direct quote from RF) Option 3 is incomprehensible to read. Option 1, while comprehensible, requires some thought to process and may appear to be more "playing with symbols" with no connection to anything mathematical. To the standard theorists, those who write like Option 3 aren't really doing mathematics, and to the "cranks," those who write like Option 1 aren't really doing mathematics. My desire is that Option 2 will be a happy compromise. And if it isn't, then I want to change it so that it will be a happy compromise. I think about the pre-formalist writing of Euclid, who wrote (in Greek) axioms such as "equals added to equals are equal," which isn't Po-Mo but is less formal than Aabcd ((a=b & c=d) -> a+c=b+d). So, the standard theorists have convinced me of the following: 1. I no longer consider Y-V to be any sort of "crank." 2. RF and THR are cranks (without scare quotes), since their writing is too far removed from mathematical concepts. 3. WM is still up in the air. The "Axiom of Potential Infinity," written by WM himself and quoted elsewhere, appears to be written more rigorously than Po-Mo, but not symbolically, of course. This would put WM in the "crank" (with scare quotes) range -- someone who doesn't write as formally as the standard theorists and writes about concepts that contradict ZFC, but still wrote at least one axiom that can be the basic of a new non-ZFC theory. > [Note that your categorization of crank only applies > to construction of mathematics. E.g. your stuff does > not apply to JSH.] I admit that JSH doesn't fit in the categories I mentioned so far. But still, I usually avoid defending JSH (since to me, his latest claims of having proved the Twin Primes and Goldbach Conjectures in PA using only elementary math is indefensible). Thus, JSH will be considered a crank (no scare quotes).
From: Transfer Principle on 12 Feb 2010 18:48 On Feb 10, 4:56 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Feb 9, 2:01 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > <snip /> > > the standard theorist Jeffries > That you consider Fred Jeffries a standard theorist does not say much > for your method of classification. In the past few days, there have been several more objections to my use of the phrase "standard theorist." I came up with the name "standard theorist" when many posters objected to the name "Cantorian" (which was invented by several so-called "cranks" who opposed Cantor or ZFC). But the name "standard theorist" is still apparently objectionable. Unfortunately, I still have yet to come up with a better name for those who defend standard theories such as ZFC, ZF, NBG, and FOL from "crank" attacks. So I'll continue to use the name, but perhaps sometimes interspersing it with the labels non-"crank" for those who don't exhibit "crank" tendencies, and anti-"crank" for those who actively oppose the "cranks." No matter how the standard theorists/non-"cranks"/anti-"cranks" object, I will never replace those labels with any label that acknowledges their opinion that only they are actually doing mathematics or that what the "cranks" are doing isn't actually mathematics at all. So labels like "actual/true mathematicians" vs. "non-mathematicians" (for the "cranks") are out. The standard theorists would love it if I started using those labels, but I definitely will not.
From: Jesse F. Hughes on 12 Feb 2010 19:12 Transfer Principle <lwalke3(a)lausd.net> writes: > Unfortunately, I still have yet to come up with a better name for > those who defend standard theories such as ZFC, ZF, NBG, and > FOL from "crank" attacks. Have you ever considered the possibility that the issue is not "crank" attacks, but crank "attacks"? -- Jesse F. Hughes "Being wrong is easy, knowing when you're right can be hard, but actually being right and knowing it, is the hardest thing of all." -- James S. Harris
From: Jesse F. Hughes on 12 Feb 2010 19:07 Transfer Principle <lwalke3(a)lausd.net> writes: > I admit that JSH doesn't fit in the categories I mentioned so > far. But still, I usually avoid defending JSH (since to me, his > latest claims of having proved the Twin Primes and Goldbach > Conjectures in PA using only elementary math is indefensible). Thus, > JSH will be considered a crank (no scare quotes). As usual, you are incapable of reporting on the claims of others. JSH does *not* claim to have proved anything in PA. As far as I recall, JSH has never worked in Peano Arithmetic nor given any hint that he knows what PA is -- even in an addled sense. Moreover, his current claims involve probabilistic reasoning, which perhaps one may represent in PA in some way, but he surely isn't bothering to do so. So, back off my favorite crank (indeed, my favorite sci.math poster). You don't get him. -- Jesse F. Hughes "What does soap kill? Germs or Germans?" -- Quincy P. Hughes (age 3 1/2) asks for clarification
From: Frederick Williams on 12 Feb 2010 19:21
Transfer Principle wrote: > > [...] Let me repeat this one for emphasis: > > > > III. Standard theorists prefer the use of symbolic object > > > language to natural metalanguage. To standard theorists, > > > natural languages such as English lack the mathematical > > > precision of symbolic language, and so all axioms must be > > > stated using symbolic language only. An axiom written in > > > metalanguage isn't truly an axiom to standard theorists. > > > "Cranks," on the other hand, prefer the use of metalanguage > > > to object language. Symbolic object languages alone won't do. As well as axioms one needs rules of inference and for those a metalanguage must be used. -- .... A lamprophyre containing small phenocrysts of olivine and augite, and usually also biotite or an amphibole, in a glassy groundmass containing analcime. |