**Next message:**Mike Cole: "Re: [xmca] Writing systems (Godel)"**Previous message:**Ed Wall: "Re: [xmca] Writing systems (Godel)"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

That all makes sense to me, Ed. Which probably means you have either

risen to a concrete I can recognize or have decenced to the lowest common

denominator!

Thanks

mike

On 1/4/07, Ed Wall <ewall@umich.edu> wrote:

*>
*

*> Mike
*

*>
*

*> The type of example I gave causes problems for those 'new' to
*

*> proof and choosing the converse is not, for most, an at all obvious
*

*> option (and doing something like this was hotly debated in the early
*

*> 1900s). However, it is a standard technique in proof (although some
*

*> topic are probably more amenable than others). I remember talking
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*> with one mathematician who, as a grad student, was given a problem by
*

*> an advisor and one week would try to prove it was true by deductive
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*> means and the other by assuming the converse and trying to get a
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*> contradiction. This went on for some time. I can't remember if he was
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*> ever able to come to a solution.
*

*> Perhaps saying something like the "the writing writes itself" was
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*> misleading as what I am wondering if one can build up enough momentum
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*> (and this is hard work) so that things sort of unfold (I know that
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*> sometimes happens in my writing or, at least at moments, it seems to)
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*> in a way such that resources, technique, and constraints sort of
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*> blend (perhaps the image of a flower). This is not necessarily a
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*> continuous process and is brought about, in part, through having
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*> built up some resources and some knowing about how one navigates a
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*> genre/topic. I know that doesn't happen for many in proof (and, of
*

*> course, this might be more usual for somebody in algebra rather than
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*> geometry) and I know it doesn't happen on a regular basis for me in
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*> writing (I don't do fiction writing although I sometimes wonder).
*

*> I do see things like this happening in the elementary mathematics
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*> classroom (certainly those that are a bit more like the one Joe
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*> mentioned in another thread). Somehow a child builds up enough
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*> momentum and it all unfolds. I'm mixing threads a bit, I guess, but
*

*> Davydov Has an intriguing subchapter in Types of Generalization in
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*> Instruction titled "The Method of Ascent from the Abstract to the
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*> Concrete." Here he is speaking of, it seems, the intellectually
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*> concrete. There is a way that this sort of captures what I'm talking
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*> about, but I admit to wondering whether I am reading into it.
*

*>
*

*> Ed
*

*>
*

*> >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.
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*> >
*

*> >There are two things you need to know in interpreting my response.
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*> >First, I am TERRIBLE
*

*> >at proofs such as that you give for the infinite number of prime
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*> >numbers. I would have failed
*

*> >at several points, but the essential starting point (suppose the
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*> >opposite is true and show it can't be)
*

*> >would not occur to me never mind I would screw up at some other
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*> >bifurcation point!
*

*> >
*

*> >Second, I am married to a fiction writer.
*

*> >
*

*> >I sometimes get the feeling that maybe what you are saying could be
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*> >true of Dickens, but to describe
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*> >the experiences and events that constitute one of my wife's books as
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*> >"the writing writing itself" would
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*> >simply go against all I have witnessed and been party to. The only
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*> >way I could get there is if all of the
*

*> >incredibly uneven backs and forths, and getting stuck in blind
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*> >alleys, and then getting distracted by
*

*> >other life exigencies and then returning, reaching "an" end only to
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*> >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,
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*> >write about "things as they are" such as,
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*> >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
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*> >speculate about, to (ha ha!!) explain it. What
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*> >I write seems a whole lot easier to do than what my wife does......
*

*> >and what she does seems, in some ways
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*> >to be "halfway" between what I do and a mathematical proof. Unless
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*> >you are going up against Goedel, there IS
*

*> >presumably an answer, a way to figure "it" out. But what if there is
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*> >no if, if if has to be created from...........
*

*> >
*

*> >So I write fictions that pass as descritions of reality and my wife
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*> >creates descriptions of a reality that pass as
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*> >fictions.
*

*> >
*

*> >I doubt if that helps you, but it helped me. thanks
*

*> >mike
*

*> >
*

*> >On 1/2/07, Ed Wall <<mailto:ewall@umich.edu>ewall@umich.edu> wrote:
*

*> >
*

*> >Mike
*

*> >
*

*> > Here is a sort of expansion (and I am by no means sure about the
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*> >authoring business, but the process sounded somehow similar) and
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*> >perhaps the best place to begin is with a story. A number of year ago
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*> >I was teach a graduate course in mathematics and had, for most of the
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*> >period, been working on one or two proofs. At the end of class, a
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*> >young woman approached me (she was, my impression, one of the more
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*> >knowledgeable students) and said something like "I understood
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*> >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
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*> >doing and you'll catch on after doing it for awhile and I've just
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*> >been doing it for awhile" and left it at that.
*

*> > Latter that day and yet still I've been thinking about this. I
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*> >tend, I suspect like a lot of others who teach some content, to have
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*> >an idea of the direction and a 'feel' for the terrain and then,
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*> >depending on where people are at, tend to somewhat improvise. What
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*> >makes it difficult is that the young woman was asking me for a
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*> >'formula' for proof and there, in a sense, isn't one. One's beginning
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*> >constrains one somewhat, one pulls out of experience some likely
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*> >scenarios which have their own affordances and limitations, and one
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*> >sort of keeps one's end in sight (sort of what Dewey talks about in
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*> >the Theory of Inquiry).
*

*> >
*

*> > Perhaps another way to say it is that in a moderately complex
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*> >proof there seems to before the 'novice' a huge amount of leeway as
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*> >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
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*> >simultaneously constrains where you can 'reasonably' go next (holding
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*> >that end in sight).
*

*> >
*

*> > Let me be more specific and give a very simple example (there is a
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*> >lot missing form this so this isn't exactly what I had in mind, but
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*> >it perhaps illustrates). Okay, I want to prove there are an infinite
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*> >number of prime numbers. The wrong way to do this is write some
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*> >formula which gives you an infinite number of primes. There isn't
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*> >one. [bifurcation] So a scenario would be to assume the converse -
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*> >i.e. there are only a finite number of primes and show this leads to
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*> >a contradiction (hence, showing 'logically' that there is indeed an
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*> >infinite number of primes). [bifurcation] Now you have an finite
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*> >number of something so you write them down (skipping 1 just in case
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*> >you want that to be a prime) p1, p2, p3, out to pN and, of course as
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*> >you are working with primes (and they are mucked up with
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*> >multiplication and division), you write the product p1*p2* out to pN
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*> >and set that equal to K. [bifurcation] Then you look at K+1.
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*> >[bifurcation] Well, K+1 certainly isn't divisible by p1 or p2 out to
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*> >pN so either K+1 is a prime or there is a prime pm less than K+1 that
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*> >was not in the original list. Hence a contradiction as was hoped for.
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*> >
*

*> > Okay, I've used some basic knowledge about primes to begin and
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*> >that with some arithmetic has both constrained and enabled the proof
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*> >at each step. However, there is a sense in which I know that the
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*> >appropriate thing to do is multiply the primes and then, of course,
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*> >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
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*> by
*

*> >>the math one is
*

*> >>doing" coupled with expertise-- which I think of as a developmental
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*> >>process). But I cannot get
*

*> >>from that to authoring a novel. And I am not even sure what the math
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*> example
*

*> >>is about. Can you
*

*> >>expand?
*

*> >>
*

*> >>I am not sure, either, what David is after. My suggestions were intended
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*> to
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*> >>focus on the origins
*

*> >>or graphic representations of ....... things.... ideas...... language
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*> (all
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*> > >big issues in the history of writing).I picked my
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*> >>suggestions for David thinking that what he was interested in the
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*> origins of
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*> >>scripts of various kinds. Others have gone
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*> >>to goody and watt on the consequences of writing, ong, etc. Havelock is
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*> an
*

*> >>interesting "half way" point because he makes
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*> >>a big deal of the special properties of the alphabet and hits on Chinese
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*> >>ideographic writing.
*

*> >>
*

*> >>Perhaps you can expand? (And be ready for someone to comment on the
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*> article
*

*> >>of the month-for-discussion, although who knows!!)
*

*> >>
*

*> >>mike
*

*> >>On 1/2/07, Ed Wall < <mailto:ewall@umich.edu>ewall@umich.edu> 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
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*> >>>>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
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*> >>>>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
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*> >>>>mathematics, I was wondering if you knew of someone(s) who makes
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*> >>>>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:
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*> >>>>
*

*> >>>>D. Schmandt-Besserat, Before Writing:. U of Texas Press. 1992 (two
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*> >>>volumes)
*

*> >>>>
*

*> >>>>R. Harris. The origin of writing. Open Court. 1986.
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*> >>>>
*

*> >>>>David Olson has written extensively on this topic, primarily from
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*> >>>secondary
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*> >>>>sources.
*

*> >>>>
*

*> >>>>I am unsure of best sources that delve into origins of writing in
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*> China
*

*> >>>>which were more or less co-incident with
*

*> >>>>events in Euphrates area.
*

*> >>>>mike
*

*> >>>>_______________________________________________
*

*> >>>>xmca mailing list
*

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*

*> >>>><http://dss.ucsd.edu/mailman/listinfo/xmca>
*

*> >>>>http://dss.ucsd.edu/mailman/listinfo/xmca
*

*> >>>
*

*> >>>_______________________________________________
*

*> >>>xmca mailing list
*

*> >>><mailto:xmca@weber.ucsd.edu>xmca@weber.ucsd.edu
*

*> >>><http://dss.ucsd.edu/mailman/listinfo/xmca>
*

*> http://dss.ucsd.edu/mailman/listinfo/xmca
*

*> >>>
*

*> >>_______________________________________________
*

*> >>xmca mailing list
*

*> >><mailto:xmca@weber.ucsd.edu> xmca@weber.ucsd.edu
*

*> >><http://dss.ucsd.edu/mailman/listinfo/xmca>
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*>
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*>
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**Next message:**Mike Cole: "Re: [xmca] Writing systems (Godel)"**Previous message:**Ed Wall: "Re: [xmca] Writing systems (Godel)"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

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