Cantor's Diagonal?
in [Math]
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From: FredJeffries on 9 Feb 2010 11:14 On Feb 7, 10:55 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > Therefore, _ultra_finitists such as WM and > Yessenin-Volpin will now be considered cranks (and that's > _without_ the scare quotes, since the label is deserved). That you consider Alexander Yessenin-Volpin a crank does not say much for your method of classification.
From: Transfer Principle on 9 Feb 2010 17:01 On Feb 9, 8:14 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Feb 7, 10:55 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > Therefore, _ultra_finitists such as WM and > > Yessenin-Volpin will now be considered cranks (and that's > > _without_ the scare quotes, since the label is deserved). > That you consider Alexander Yessenin-Volpin a crank does not say much > for your method of classification. I believe that Yessenin-Volpin is a "crank" if and only if WM is a "crank." (For this post, I return to scare quotes until the "crank"-hood of WM and Y-V is settled.) Recall that Yessenin-Volpin is the mathematician famous for, upon being asked "Is n a natural number?" (for the standard or Peano natural n), waiting until a time proportional to the magnitude of n has passed before answering "yes." In other words, he waits until time nt has passed (for some fixed constant time t) before answering the question. 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. 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. 3. Both rely on the physical universe to determine whether a natural number is permissible. Y-V's definition depends on the physical property of _time_ -- in particular, whether there is enough time for him to answer the question. WM also relies on a temporal notion -- namely, whether there is sufficient time for its digits to be calculated. 4. For completeness, Y-V was affiliated with a university (Boston U). WM is also affiliated with a university. (One key difference between Y-V and WM is that for the former, if n is a permissible natural number and m<n is a Peano natural number, then m is also permissible. This is because in order for Y-V to answer "Is n a natural number?" a period of time nt must pass, while for him to answer "Is m a natural number?" a period of mt must pass. Since m<n, mt<nt, and so if time nt is available, then so is the shorter time interval mt. On the other hand, WM's natural numbers contain gaps, a fact admitted by WM himself.) Notice that Y-V is currently 85 years old. Thus, he'd just now be answering yes to the question (if posed at his birth) "Is 2.7 billion a natural number?" for t=1 second. He'd never be able to answer the question "Is the size of President Obama's latest stimulus package in dollars, rounded to the nearest dollar, a natural number?" in his lifetime, unless we were to reduce the constant t to about one millisecond. Even if we were to reduce t to a very short interval, such as the Planck time, Y-V would only be able to reach about 10^52. Even if Y-V were eternal, and we posed the question right after the Big Bang, and Y-V could count until the Heat Death, he'd still be hundreds of orders of magnitude shy of the largest RSA numbers. So Y-V can never answer the question "Is RSA-2048 a natural number?" despite its usefulness in computing and cryptography. (On the other hand, note that to WM, RSA-2048 is a natural number since we know all of its digits!) The main point here is not to attack ultrafinitism, but merely to point out that if I hold Y-V and WM to my same standards, then either both are "cranks," or neither are. But Fred Jeffries wrote, to repeat: > That you consider Alexander Yessenin-Volpin a crank does not say much > for your method of classification. strongly implying that in Jeffries's own method of classification Y-V isn't a "crank," even though WM is. The ball is now in Jeffries's court to defend his classification and give the differences between Y-V and WM such that the latter is a "crank," but not the former. Since the standard theorists like Jeffries are the ones who came up with the term "crank" in the first place, if Jefferies says that Y-V isn't a "crank," then Y-V indeed isn't a "crank." If Jeffries would please explain why Y-V isn't a "crank" despite making several WM-like statements... (Note: "Y-V is a mathematician and WM isn't" is insufficient. To me, that's little different from saying "WM is a 'crank' because he's a 'crank,' and Y-V isn't because he isn't.") (Recently there is a debate in the AP threads in which AP, while not truly a finitist, is trying to argue that 10^500 is the largest number useful to physics, and that any natural number larger than 10^500 is "incognitum," not finite. Later on today, I might find and propose in one of the AP threads that a finite natural number be defined as a number to which Y-V will answer "Yes, that is a natural number," and "incognitum" as one to which Y-V cannot so answer. Then "finite" and "incognitum" will be reduced to the whims of Y-V -- a mathematician considered to be a non-"crank" by the standard theorist Jeffries.)
From: Transfer Principle on 9 Feb 2010 23:52 On Feb 8, 8:05 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Transfer Principle wrote: > > I. Standard theorists think of mathematics in terms of the > > objects that they want to _exist_ in their own conception > > of the mathematical universe. > > "Cranks," on the other hand, think of mathematics in terms > > of the objects that they _don't_ want to exist. > What about those posters who subscribe to none of those "thinking"? > Are they "crank", or "standard"? The two options that I gave were not intended to be either mutually exclusive or mutually exhaustive. They are intended merely as generalizations of the two sides of the issue. To illustrate what I mean, let's look at an old thread from last year, one in which Nam Nguyen himself was a participant (via Google archive): http://groups.google.com/group/sci.logic/browse_thread/thread/7634dadf33044697/dba25c60398020dd?lnk=gst&q=cantor+crank# In this thread, the standard theorist MoeBlee wrote: "Why would the point of such exercises merely be to fiddle with axioms irrespective of what those axioms PROVE, especially when we're talking about comparisons with a foundational theory such as Z that axiomatizes calculus?" Typical of the standard theorists, MoeBlee desires a theory in which calculus can be axiomatized. It matters little to such standard theorists what "extra" can be axiomatized in theory, however "useless" such objects may be (objects such as the set P(P(P(R)))). As long as calculus is axiomatized, that's what mainly matters in a foundational theory that's adequate for standard theorists. An axiomatization for the calculus is MoeBlee's priority in this post. On the other hand, the following WM quote appears in the thread to which I linked above: > "Axiom Of Potential Infinity: For every natural number there is a set > that contains this number together with all smaller natural numbers, > and for every set of natural numbers there is a natural number that > is larger than every number of the set." In this post, WM has expressed that he wants potentially infinite sets to exist, but for no _actually_ infinite sets to exist. It matters little to "cranks" like WM that without actually infinite sets, calculus is most likely impossible. Avoiding actually infinite sets is WM's priority in this post. The priorities of MoeBlee and WM in these quotes are typical of the priorities of the standard theorists and the "cranks," respectively. The standard theorist MoeBlee says to include calculus in the foundational theory (no matter what else gets included). The "crank" WM says to exclude actually infinite sets in the foundational theory (no matter what what else gets excluded). Standard theorists think in terms of what to _in_clude. So-called "cranks" think in terms of what to _ex_clude. Hence my generalization above. > What about those posters who subscribe to none of those "thinking"? > Are they "crank", or "standard"? It's possible (indeed probable) that Nguyen intends to include himself as one of "those posters who subscribe to none of those" generalizations. In this case, I would use a different criterion in order to determine whether Nam Nguyen is "crank" or standard. Consider the thread in the link above. In this thread, Nguyen used mostly symbolic language, which according to criterion III above, suggests that he is standard. But then again, the argument that most of the standard theorists was making is that Nguyen actually was thinking in terms of _excluding_ something from standard logic, in this case, excluding the proof of "Axy (x+y=0)" from a theory with axiom "Axy (x=y)" yet no axioms containing the symbol "+" from the language of the theory, was enough to consider Nguyen to be a "crank" -- hence the existence of that long discussion with him in the link. Of course, Nguyen hasn't argued in a long time that "Axy (x+y=0)" isn't derivable from the lone axiom "Axy (x=y)" in a theory with "+" in the language. So I have no idea whether the standard theorists with whom he argued in that thread still consider him to be a "crank" anymore or not. Remember that theose standard theorists are the ones who came up with the "crank" label in the first place, so they are the ultimate judges as to whether Nguyen is a "crank" or not.
From: Transfer Principle on 10 Feb 2010 00:03 On Feb 8, 1:33 am, "Peter Webb" <webbfam...(a)DIESPAMDIEoptusnet.com.au> wrote: > I don't think zuhair is a "crank" in the more general Usenet sense. He does > appear to eventually agree he was wrong, which cranks don't - they either > change the topic, make an ad-hominem attack, or do both. Standard theorists make ad-hominem attacks all the time. But I do concede that zuhair does admit that he's wrong -- indeed he did so in this very thread. > And of course your views on crankdom are very specific to logic. It's hard for me to generalize beyond mathematics and logic what traits make a "crank." Of course, Webb does mention how "cranks" rarely admit their errors, which does generalize to fields outside mathematics and logic. > One thing I disagreed on: > > II. Standard theorists believe that physics favors their > > side of the debate. Physical processes can be modeled by > > differential equations, which require the existence of a > > complete ordered field to solve. Thus, they believe that > > standard analysis is necessary for the sciences, and so > > science is impossible in the "crank" theories. > All physical science can be done using only the countable (but uncomputable) > set of computable Reals. Calculus also works fine over this subset of R. > There is no hard evidence to me that uncountable sets need exist in nature > at all. I grant that it would be somewhat arbitrary for magnetic field > strengths to only take only computable values, but for all we know they may > do so. You might call this UltraCountabilism in comparison to UltraFinitism > Indeed, if you accept that you can make precise measurements of anything at > all in nature, and you believe in QM, then in this Universe I can well order > the Reals. I do so as follows. I place an electron inside an energy well > such that its location between 0 and 1 is randomly determined by QM. The > location at which I precisely measure the electron is the lowest (first) > Real. Then I continue doing so for the second and subsequent measurements, > making sure each time I haven't listed that Real already. In this manner, I > can select a random Real from any subset of Reals, which in ZFC requires a > limitted form of choice. The random outcomes in QM provides a mechanism not > directly available inside ZF for choosing elements, and may provide a choice > function in this Universe. Interesting idea! > The point of all this being it is very dangerous to try and use the physical > world to validate set theory. I agree with Webb here -- in the end, it's a wash as to which side physics really favors, so perhaps it's best just to avoid physics in any discussion of set theory. The point that I was making is that members of both sides _do_ make physical-based arguments, not that such arguments are actually valid.
From: Herman Jurjus on 10 Feb 2010 04:30
Transfer Principle wrote: > > The main point here is not to attack ultrafinitism, but merely > to point out that if I hold Y-V and WM to my same standards, > then either both are "cranks," or neither are. So much for your standards, then. The problem with WM is not so much that he's 'wrong', but that he's a lousy mathematician. He's confused about lots of things: He confuses rejection of actual infinity with ultra-finitism. He claims to have proved ZFC inconsistent with a proof that's not formalizable in ZFC (and doesn't show any signs of being aware of the latter). He claims he agrees with Robinson where he doesn't. Etc, etc. But worst of all: he doesn't even attempt a formalization or precise elucidation of what is and is not allowed in his mathematics. A rather big contrast with Yessenin-Volpin, wouldn't you say? -- Cheers, Herman Jurjus |