How Can ZFC/PA do much of Math - it Can't Even Prove PA is
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From: Daryl McCullough on 9 Jul 2010 14:43 Marshall says... > >On Jul 8, 8:00=A0pm, Transfer Principle <lwal...(a)lausd.net> wrote: >> On Jul 8, 6:15=A0pm, Marshall <marshall.spi...(a)gmail.com> wrote:. >> >> > And I [don't] trust your ability to faithfully represent >> > others' views >> >> No need to. The link is right here: >> >> http://www.math.princeton.edu/~nelson/papers/hm.pdf > >An interesting read! And far less stupid than I had feared. >But ultimately I found it unconvincing. I think that "Transfer Principle" should pay attention to the difference in respect towards someone like Nelson and someone like WM or |-|erc or those people who are called crackpots. Nelson has some very unorthodox beliefs about mathematics, but he is a *competent* mathematician. He is able to explain what he means in a language that mainstream mathematicians can follow. He understands what a valid proof is, and is able to construct them. In contrast, |-|erc and WM are *not* competent mathematicians. They are unable to construct a valid proof, they are unable to recognize a correct proof when they see one. Since proof is the language that mathematicians use to communicate, that means that it is basically impossible for such people to communicate to mainstream mathematicians. It is actually for them to communicate anything of substance to *ANYONE* (not just a mainstream mathematician), which is why their attempts at developing an alternative foundation for mathematics are dead ends. They don't have the skills to develop a rigorous theory, and without rigor, what you have is not mathematics but a word that rhymes with "celebration" (thanks to Barb Knox). -- Daryl McCullough Ithaca, NY
From: George Greene on 10 Jul 2010 12:40 On Jul 9, 2:43 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > I think that "Transfer Principle" should pay attention to > the difference in respect towards someone like Nelson and > someone like WM or |-|erc or those people who are called > crackpots. Nelson has some very unorthodox beliefs about > mathematics, but he is a *competent* mathematician. Perhaps, but at the beginning of this paper, he is not looking like a competent philosopher. In addition to completely misunderstanding what "formal" (or abstract) means in contradistinction to being "genetic" (which he defines as "relating to origins") and to "human activity" (which he takes as *pro*-"formal", when it is in fact anti-), he next attacks infinite sets for not being concretely realizable -- AS IF *ANY*FUNCTION* were EITHER! AS IF ANY FINITE SET WERE, EITHER! Math is ABOUT abstractions! THE WHOLE FIELD is abstract! In other words, dissing something for not being concretely realizable (have you ever seen or TOUCHED zero?? What about the letter 'a'??) IS JUST *STUPID*. This could be introductory whimsey as an audience-hook but I for one was appalled by the next heading: "Try to imagine N as if it were real." HOW PEJORATIVE (and question-BEGGING) is THAT?!? N *is* real! NO "imagination" REQUIRED! The fact that you cannot (as this section continues) > Buy a copy of N! > Contains zerocontains the successor of everything > it containscontains only these. > Just $100. is HARDLY any sort of PROBLEM, at least not to a mathematician (it's the unwashed OTHER masses who would have THAT problem)! But the continuation makes it clear that he should have said "model", not "copy". That would make the next part valid: > I bought a model [copy] from another dealer and am > quite pleased with it. My friend maintains that > it contains an ineffable number, although zero is > effable and the successor of every effable number is > effable, but I dont know what he is talking about. I think he is just jealous. > The point of this conceit is that it is impossible > to characterize N unambiguously, as we shall > argue in detail. THOU SHALT NOT *BORE* me! This HAS ALREADY BEEN argued in detail, in Godel's 1st incompleteness theorem! You cannot do this in FIRST-order logic with a recursive axiom-set! But to state this as baldly as Nelson has stated it here, in natural language, IS SELF-contradictory!! Student (asking him a question abuot this): "It is impossible to characterize *WHAT* unambiguously, Prof.Nelson?" Prof: Why, N, of course. Student: "N? what is that?" My point is, the question, "can we characterize N unambiguously?" CANNOT even be intelligible UNLESS N *HAS ALREADY BEEN* characterized unambiguously during its mention IN THE QUESTION!! OF COURSE WE CAN AND HAVE characterized N unambiguously IN MANY ways, and IN MANY contexts, INCLUDING THIS ONE!! The POINT, though, is that howEVER we may have managed to pull off this feat, we did NOT do it with a recursive axiom-set under the usual paradigm of a first-order language. Do Mathematics became a faith-based initiative. A
From: George Greene on 10 Jul 2010 13:39
Nelson contradicting himself AGAIN: > Despite all the accumulated evidence to the > contrary, mathematicians persist in believing in N > as a real object existing independently of any formal > human construction. All the evidence that Nelson has been presenting here IS NOT "to the contrary"! All of Nelson's presented evidence is evidence that N *canNOT* be properly described ("unambiguously characterized") by any formal construction! If it exists at all, then OF COURSE it's independent of human construction! And that is equally true of finite numbers as well: 2 existed long before the axioms of PA were formally constructed by P himself. This paper also has a serious problem in choosing to try to define "number". What is ACTUALLY going on here involves numERALS inSTEAD of numbers, and what is actually being defined here is FINITE and NOT "number": to use "number" as Nelson is using it here is to pretend (wrongly) that negative numbers, rational numbers, real numbers, and complex numbers (indeed, all numbers other than vonNeumann ordinals) do not exist. > As a genetic concept, the notion of numeral is clear. That may be true, but to the extent that genetic was supposed to be an opposite of formal, it is clearly RIDICULOUS: numerals are among the MOST formal things there are! Formal/genetic is a false dichotomy in any case: there are some things -- math especially -- whose genetic origins really are in and about THE USE OF SIGNS and shapes, i.e., PURELY FORMAL AND ABSTRACT things. Letters and numerals, despite the fact that people had to think them up and concretely write them down with actual sticks, ARE ABSTRACT, NOT physical! They are what they are in virtue OF THEIR FORM, which MAKES them FORMAL!! > The attempt to formalize the concept usually proceeds as follows: Grotesque antecedent failure alert: the antecedent of "the concept" is "the notion of numeral" in the above, yet in what follows, we are ALL about NUMBER. Numbers and numerals ARE DIFFERENT, Prof. Nelson! Too bad a distinguished professor at Princeton is not going to allow one of his most famous papers to be proofread by somebody who failed to finish a dissertation. (For those arriving late, this is reaction to http://www.math.princeton.edu/~nelson/papers/hm.pdf ) > (i) zero is a number; > (ii) the successor of a number is a number; > (iii) zero is not the successor of any number; > (iv) different numbers have different successors; > (v) something is a number only if it is so by virtue of (i) and (ii). > We shall refer to this as the usual definition. > Sometimes (iii) and (iv) are not stated explicitly, > but it is the extremal clause (v) that is unclear. > What is the meaning of by virtue of? It is > obviously circular to define a number as something > constructible by applying (i) and (ii) any number of times. People think I overuse emphasis, but there is some emphasis missing in the original here. The word "number" should have been italicized in both occurrences in the last two lines. But the more important point here is that the Wrong Concept is being defined. What Nelson actually NEEDS to be defining here IS NOT the noun "number" BUT RATHER the ADJECTIVE "finite". The Actual Problem is NOT that we can't "describe numbers", but rather that PA doesn't know the difference between what's finite and what's not (and neither, it turns out, does anybody else). |