Guess I must be really tired. Its all making sense to me in this past few
messages!!
What is the magic spell he weaves
Who Goedel's proof he thus conceives?
mike
On 1/4/07, Ed Wall <ewall@umich.edu> wrote:
>
> Paul
>
> Godel's Undecidability Theorem is, in a sense, about truth. He
> showed that within a particular system of first order logic that you
> could not with, more or less, the tools of first order logic decide
> the truth or falsity of all statements. However, one need only extend
> these logics. A sort of example is that you cannot, with a ruler and
> a compass, trisect an angle.
>
> Chaitin (and, of course, Turing) has taken all this a bit further
> in a somewhat readable book titled "The Limits of Mathematics"
>
> Thinking about mathematicians as weavers makes a sort of sense if
> somehow reproducibility is factored in. Also mathematics seems to be
> both out there and in there re Debaene.
>
> Ed Wall
>
> Ed
> >Mike,
> > You wrote,
> >
> > "Unless you are going up against Goedel, there IS presumably an
> >answer, a way to figure "it" out."
> >
> > which made me think I might have created a false idea about Godel,
> >maybe not. I don't think Godel was trying to show "there is no
> >answer".
> >
> > My understanding of Godel comes from Newman's book: Godel's Proof,
> >which is written so that anyone who understands arithmetic can get
> >an idea what Godel did. He was basically refuting Russell and
> >Whitehead's attempt to reduce mathematics to logic, that is, to
> >refute the idea that given certain axioms (Peano's basically) that
> >all of math would follow thereon according to the laws of
> >first-order logic--it must be admitted that along the way Russell
> >did develop his theory of types from which Bateson got so much
> >mileage. Godel simply showed that you can't do this. I can't say I
> >have understood Godel's proof, but I did understand the book of the
> >same name. At least I think I did.
> >
> > It really comes down to saying that numbers are "out there", which
> >should be OK for CHAT folk since the mind itself is "out there", not
> >inside the head, from a CHAT perspective, isn't it. Math, as
> >practiced by mathematicians, is really the "theory of math" not the
> >math itself, e.g., weavers perform incredibly complex arithmetical
> >operations in order to produce their designs without having the
> >slightest knowledge of that theory or any formal training in
> >arithmetic
> >
> > Paul Dillon
> >
> > Paul
> >
> >
> >
> >Mike Cole <lchcmike@gmail.com> wrote:
> > Ed-- Sorry-- I needed a quiet enough moment to read your example,
> re-read
> >your
> >initial inquiry, and then come back to your initial example.
> >
> >There are two things you need to know in interpreting my response. First,
> I
> >am TERRIBLE
> >at proofs such as that you give for the infinite number of prime numbers.
> I
> >would have failed
> >at several points, but the essential starting point (suppose the opposite
> is
> >true and show it can't be)
> >would not occur to me never mind I would screw up at some other
> bifurcation
> >point!
> >
> >Second, I am married to a fiction writer.
> >
> >I sometimes get the feeling that maybe what you are saying could be true
> of
> >Dickens, but to describe
> >the experiences and events that constitute one of my wife's books as "the
> >writing writing itself" would
> >simply go against all I have witnessed and been party to. The only way I
> >could get there is if all of the
> >incredibly uneven backs and forths, and getting stuck in blind alleys,
> and
> >then getting distracted by
> >other life exigencies and then returning, reaching "an" end only to have
> a
> >completely different end
> >emerge, if THAT is writing itself, then ok. But what an "itself"!!
> >
> >We sometimes discuss how what I do is different. I, presumably, write
> about
> >"things as they are" such as,
> >for example, the role of culture and biology in ontogenesis. There is a
> >putative, as if reality out there (I
> >naively assume) and I set out to write about it, to describe it, to
> >speculate about, to (ha ha!!) explain it. What
> >I write seems a whole lot easier to do than what my wife does...... and
> what
> >she does seems, in some ways
> >to be "halfway" between what I do and a mathematical proof. Unless you
> are
> >going up against Goedel, there IS
> >presumably an answer, a way to figure "it" out. But what if there is no
> if,
> >if if has to be created from...........
> >
> >So I write fictions that pass as descritions of reality and my wife
> creates
> >descriptions of a reality that pass as
> >fictions.
> >
> >I doubt if that helps you, but it helped me. thanks
> >mike
> >
> >On 1/2/07, Ed Wall wrote:
> >>
> >> Mike
> >>
> >> Here is a sort of expansion (and I am by no means sure about the
> >> authoring business, but the process sounded somehow similar) and
> >> perhaps the best place to begin is with a story. A number of year ago
> >> I was teach a graduate course in mathematics and had, for most of the
> >> period, been working on one or two proofs. At the end of class, a
> >> young woman approached me (she was, my impression, one of the more
> >> knowledgeable students) and said something like "I understood
> >> everything you did, but I didn't understand why you did it. I don't
> >> think I'll ever be able to do proofs." I said the usual dumb thing
> >> something like "It is just a matter of writing down what you were
> >> doing and you'll catch on after doing it for awhile and I've just
> >> been doing it for awhile" and left it at that.
> >> Latter that day and yet still I've been thinking about this. I
> >> tend, I suspect like a lot of others who teach some content, to have
> >> an idea of the direction and a 'feel' for the terrain and then,
> >> depending on where people are at, tend to somewhat improvise. What
> >> makes it difficult is that the young woman was asking me for a
> >> 'formula' for proof and there, in a sense, isn't one. One's beginning
> >> constrains one somewhat, one pulls out of experience some likely
> >> scenarios which have their own affordances and limitations, and one
> >> sort of keeps one's end in sight (sort of what Dewey talks about in
> >> the Theory of Inquiry).
> >>
> >> Perhaps another way to say it is that in a moderately complex
> >> proof there seems to before the 'novice' a huge amount of leeway as
> >> almost every time you write a line you come to a bifurcation point.
> >> However, that is misleading as what has gone before both supports and
> >> simultaneously constrains where you can 'reasonably' go next (holding
> >> that end in sight).
> >>
> >> Let me be more specific and give a very simple example (there is a
> >> lot missing form this so this isn't exactly what I had in mind, but
> >> it perhaps illustrates). Okay, I want to prove there are an infinite
> >> number of prime numbers. The wrong way to do this is write some
> >> formula which gives you an infinite number of primes. There isn't
> >> one. [bifurcation] So a scenario would be to assume the converse -
> >> i.e. there are only a finite number of primes and show this leads to
> >> a contradiction (hence, showing 'logically' that there is indeed an
> >> infinite number of primes). [bifurcation] Now you have an finite
> >> number of something so you write them down (skipping 1 just in case
> >> you want that to be a prime) p1, p2, p3, out to pN and, of course as
> >> you are working with primes (and they are mucked up with
> >> multiplication and division), you write the product p1*p2* out to pN
> >> and set that equal to K. [bifurcation] Then you look at K+1.
> >> [bifurcation] Well, K+1 certainly isn't divisible by p1 or p2 out to
> > > pN so either K+1 is a prime or there is a prime pm less than K+1 that
> >> was not in the original list. Hence a contradiction as was hoped for.
> >>
> >> Okay, I've used some basic knowledge about primes to begin and
> >> that with some arithmetic has both constrained and enabled the proof
> >> at each step. However, there is a sense in which I know that the
> >> appropriate thing to do is multiply the primes and then, of course,
> >> adding 1 is the elegant thing to do (smile).
> >>
> >> Does this help?
> >>
> >> Ed
> >>
> >> >Ed--
> >> >Never mind off topic. We are always shifting topics. And I would be
> happy
> >> to
> >> >respond usefully to your query if I knew how!! The problem is that I
> do
> >> not
> >> >understand
> >> >what you wrote! I am GUESSING that what you are talking about has to
> do
> >> with
> >> >origins and
> >> >change. ("the mathematics one does is both circumscribed and
> supported by
> >> >the math one is
> >> >doing" coupled with expertise-- which I think of as a developmental
> >> >process). But I cannot get
> >> >from that to authoring a novel. And I am not even sure what the math
> > > example
> >> >is about. Can you
> >> >expand?
> >> >
> >> >I am not sure, either, what David is after. My suggestions were
> intended
> >> to
> >> >focus on the origins
> >> >or graphic representations of ....... things.... ideas...... language
> >> (all
> >> >big issues in the history of writing).I picked my
> >> >suggestions for David thinking that what he was interested in the
> origins
> >> of
> >> >scripts of various kinds. Others have gone
> >> >to goody and watt on the consequences of writing, ong, etc. Havelock
> is
> >> an
> >> >interesting "half way" point because he makes
> >> >a big deal of the special properties of the alphabet and hits on
> Chinese
> >> >ideographic writing.
> >> >
> >> >Perhaps you can expand? (And be ready for someone to comment on the
> >> article
> >> >of the month-for-discussion, although who knows!!)
> >> >
> >> >mike
> >> >On 1/2/07, Ed Wall wrote:
> >> >>
> >> >>Mike and all
> >> >>
> >> >> This is not quite on the topic (and, thus, I have held back a
> >> >>bit), but given the amount of expertise that people are bringin I
> ask
> >> >>a question I have asked elsewhere (I apologize for how it is
> phrased,
> >> >>but something like this was appropriate in that particular
> community):
> >> >>
> >> >>> I had a question and wonder if you might point me in a useful
> >> >>>direction(s). The situation is such: It has been argued of late
> that
> >> >>>the work mathematicians do - proof and the such - proceeds within
> the
> >> >>>mathematics being created. That is, without going into a lot of
> >> >>>detail, the mathematics one does is both circumscribed and
> supported
> >> >>>by the mathematics one is doing. This is not exactly a matter of
> >> >>>prior knowledge or the hermeneutic circle per se although it might
> >> >>>have something to do with being an 'expert.'
> >> >>> The reason why I am asking is that, the other day in a somewhat
> >> >>>philosophic discussion around a novel, a participant noted that
> some
> >> >>>authors describe the authoring process as open-ended in the sense
> >> >>>that what finally takes place may differ from what was originally
> >> >>>intended. That is, in a certain sense, the writing writes itself.
> As
> >> >>>this sounded somewhat parallel to the phenomenon I mentioned in
> >> >>>mathematics, I was wondering if you knew of someone(s) who makes
> >> >>>remarks about a similar phenomenon re writing.
> >> >>
> >> >>Ed Wall
> >> >>
> >> >>>Hi David--
> >> >>>
> >> >>>There is a LOT of material on the topic of writing systems.
> >> >>>Two interesting places to start are:
> >> >>>
> >> >>>D. Schmandt-Besserat, Before Writing:. U of Texas Press. 1992 (two
> >> >>volumes)
> >> >>>
> >> >>>R. Harris. The origin of writing. Open Court. 1986.
> >> >>>
> >> >>>David Olson has written extensively on this topic, primarily from
> >> >>secondary
> >> >>>sources.
> >> >>>
> >> >>>I am unsure of best sources that delve into origins of writing in
> China
> >> >>>which were more or less co-incident with
> >> >>>events in Euphrates area.
> >> >>>mike
> >> >>>_______________________________________________
> >> >>>xmca mailing list
> >> >>>xmca@weber.ucsd.edu
> >> >>>http://dss.ucsd.edu/mailman/listinfo/xmca
> >> >>
> >> >>_______________________________________________
> >> >>xmca mailing list
> > > >>xmca@weber.ucsd.edu
> >> >>http://dss.ucsd.edu/mailman/listinfo/xmca
> >> >>
> >> >_______________________________________________
> >> >xmca mailing list
> >> >xmca@weber.ucsd.edu
> >> >http://dss.ucsd.edu/mailman/listinfo/xmca
> >>
> >>
> >_______________________________________________
> >xmca mailing list
> >xmca@weber.ucsd.edu
> >http://dss.ucsd.edu/mailman/listinfo/xmca
> >
> >
> > __________________________________________________
> >Do You Yahoo!?
> >Tired of spam? Yahoo! Mail has the best spam protection around
> >http://mail.yahoo.com
> >_______________________________________________
> >xmca mailing list
> >xmca@weber.ucsd.edu
> >http://dss.ucsd.edu/mailman/listinfo/xmca
>
> _______________________________________________
> xmca mailing list
> xmca@weber.ucsd.edu
> http://dss.ucsd.edu/mailman/listinfo/xmca
>
_______________________________________________
xmca mailing list
xmca@weber.ucsd.edu
http://dss.ucsd.edu/mailman/listinfo/xmca
This archive was generated by hypermail 2b29 : Thu Feb 01 2007 - 10:11:31 PST