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
On 1/4/07, Ed Wall <email@example.com> wrote:
> 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
> 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
> means and the other by assuming the converse and trying to get a
> contradiction. This went on for some time. I can't remember if he was
> ever able to come to a solution.
> Perhaps saying something like the "the writing writes itself" was
> misleading as what I am wondering if one can build up enough momentum
> (and this is hard work) so that things sort of unfold (I know that
> sometimes happens in my writing or, at least at moments, it seems to)
> in a way such that resources, technique, and constraints sort of
> blend (perhaps the image of a flower). This is not necessarily a
> continuous process and is brought about, in part, through having
> built up some resources and some knowing about how one navigates a
> 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
> geometry) and I know it doesn't happen on a regular basis for me in
> writing (I don't do fiction writing although I sometimes wonder).
> I do see things like this happening in the elementary mathematics
> classroom (certainly those that are a bit more like the one Joe
> mentioned in another thread). Somehow a child builds up enough
> momentum and it all unfolds. I'm mixing threads a bit, I guess, but
> Davydov Has an intriguing subchapter in Types of Generalization in
> Instruction titled "The Method of Ascent from the Abstract to the
> Concrete." Here he is speaking of, it seems, the intellectually
> concrete. There is a way that this sort of captures what I'm talking
> about, but I admit to wondering whether I am reading into it.
> >Ed-- Sorry-- I needed a quiet enough moment to read your example, re-read
> >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
> >I doubt if that helps you, but it helped me. thanks
> >On 1/2/07, Ed Wall <<mailto:firstname.lastname@example.org>email@example.com> wrote:
> > 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?
> >>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
> >>what you wrote! I am GUESSING that what you are talking about has to do
> >>origins and
> >>change. ("the mathematics one does is both circumscribed and supported
> >>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
> >>is about. Can you
> >>I am not sure, either, what David is after. My suggestions were intended
> >>focus on the origins
> >>or graphic representations of ....... things.... ideas...... language
> > >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
> >>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
> >>of the month-for-discussion, although who knows!!)
> >>On 1/2/07, Ed Wall < <mailto:firstname.lastname@example.org>email@example.com> 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
> >>>>R. Harris. The origin of writing. Open Court. 1986.
> >>>>David Olson has written extensively on this topic, primarily from
> >>>>I am unsure of best sources that delve into origins of writing in
> >>>>which were more or less co-incident with
> >>>>events in Euphrates area.
> >>>>xmca mailing list
> >>>xmca mailing list
> >>xmca mailing list
> >><mailto:firstname.lastname@example.org> email@example.com
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