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
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
>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 <<mailto:ewall@umich.edu>ewall@umich.edu> 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 < <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
>>>>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
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