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"Integral Mathematics: An AQAL Approach"

Last post 08-06-2007, 10:14 PM by ralphweidner. 19 replies.
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  •  04-13-2007, 4:42 PM 21739 in reply to 21579

    Re: "Integral Mathematics: An AQAL Approach"

         Yes--I completely agree that the level of abstraction and development in mathematics has grown by leaps and bounds since Galois' conception of group theory applied specifically to polynomials.  In algebraic number theory, groups and fields are studied initially as tools for solving problems in number theory, but very soon in the course of study these mathematical tools take on a meaning of their own and the abstract entities become their own worlds, abstractions building upon abstractions, with hypotheses and theorems and open-ended questions to explore.  The level of thinking is staggering in its complexity, and for me it is an eternal source of intellectual beauty.  If I were going to try to assign any kind of AQAL level to this, I think something like Turquoise would be appropriate.  For the reference levels of these abstract fields encompass all the lower levels ranging from practical usefulness to concrete operations to relative abstractness along the lines of Galois, but develop into amazingly abstract worlds for its own intrinsic artistic intellectual pleasure.
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  •  04-14-2007, 7:33 AM 21753 in reply to 21739

    Re: "Integral Mathematics: An AQAL Approach"


    elliot,

    i feel about the same as you. i was just listening to the latest concall segment in which ken talks alot about eastern spiritual practices. then, seeing your message, i began wondering if western 'mathematical practices' might share some similarities, although not all, of course. i still need to see what piaget had to say about development, and another reference you gave, dehaene.

    i've ordered harold edwards' 'fermat's last theorem'. he takes a construvist approach, which has the advantage that it is closer to the way the mathematics actually developed--without all the structure we now ensconce earlier mathematics in. hopefully, i'll find the time to study it carefully, as is necessary for such a book. i feel i need to do this, because it's important that we don't just talk about mathematics in general, but concretely, which presents a problem for the two of us, you being a number theorist and myself a topologist, i guess, in this age of specialization.

    incidentally, ken would like to know what we integral enthusiasts think of howard gardner's new book 'five minds for the future', so of course i want to read that, too--looks like it will be a quick read.

    and only a couple of months till summer!

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  •  04-22-2007, 4:53 PM 21942 in reply to 21739

    Re: "Integral Mathematics: An AQAL Approach"


    the book you recommend by latoff and nun~ez is better than 'philosophy in the flesh', by latoff and johnson in that it at least recognizes other methodologies such as zone 2 (piaget). nun~ez evidently lived in europe for several years and hasn't yet, at least, succumbed to the zone 5, subtly materialistic absolutism of some of his north american colleagues.

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  •  08-06-2007, 7:03 PM 26995 in reply to 21942

    Re: "Integral Mathematics: An AQAL Approach"'

    it looks like this is going to remain a monologue or, at best, a silent dialogue between me and any mathematically oriented book i happen to read, in this case, 'the artist and the mathematician, the story of nicolas bourbaki ....', by amir aczel, who evidently was an undergraduate in mathematics in berkeley about the same time i was a graduate student there. aczel has become a prolific author, coming out with a new book almost every year.

    in this book aczel, in effect, traces to some extent the development mathematics went through in the 20th century, particularly in france. he focuses on structuralism, which in fact was the primary focus of the bourbaki group. with reference more to philosophy, i guess, he lists existentialism as what came before and postmodernism (?) as what came after.

    what is clear, in this regard, is that he fails to achieve an integral perspective, but, then, who has? his description of how the bourbaki group functioned is, nevertheless, very intriguing:

    (p.91) In September 1936, Bourbaki convened for the "Escorial congress" in an alternative location, in Chanc()ay, in the French region of Touraine. Here, Claude Chevalley"s family owned a beautiful house, which Claude's mother gladly offered as the location for the meetings. Two Bourbaki congresses would take place at the Chevalley family property.
    Reminiscing about these meetings at his parents' estate, Chevalley asid, "The Bourbakis arrived at the station at Amboise, and those who were already there let out a frightful howl: 'Bourbaki! Bourbaki! You would have taken us for a band of madmen. There, that was the Bourbaki style!"
    Chevalley continued: "Strong bonds of friendship existed between us; and when the problem of recruiting new members was raised, we were all in agreement that members should be chosen as much for their social manner as for their mathematical ability. This allowed our work to submit to a rule of unanimity: anyone had the right to impose a veto. As a general rule, unanimity over a text only appeared at the end of seven or eight successsive drafts [writtten by distinct members]. When a draft was rejected, there was a procedure forseen for its improvement. 'The Tribe', a report of the congress, related the discussions and decisions on the subject."
    At this congress and the following ones, the Bourbaki group changed its working methods. For each topic, a general discussion took place. Bourbaki always argued among themselves [often shouting]...This, in fact, became the group's hallmark. And one of their innovations was to do mathematics in the open air--on lawns and in nature. Traditionally, mathematics was a dry discipline, taught with chalk on a blackboard in a stuffy room. The members wanted to enjoy life, and to enliven mathematics.
    Oncer a discussion was over, a writer was designated. This writer produced a preliminary draft of the topic, and then the group discussed the draft and argued about its various points. Another writer was then disignated to write the second draft, after which a new discussion would take place, and so on, until the final draft was produced. Given this work system, it was impossible to ever attribute any given text to any particular person in the group. All decisions had to be unanimous, and any decision could be challenged at any time. This made for a very cumbersome process of writing, but it did ensure that the final product was always created by a group rather than by any particular individual.
    Describing this process, Andre Weil [close brother of Simone Weil] wrote in his memoirs: "No doubt it required a major act of faith to think that this process would produce results, but we had faith in Bourbaki." In reality, it so happened that Jean Dieudonne' ended up writing most of the final drafts... He was an excellent writer, and he had a very good style.
    The final consensus necessary to produce a text surprised even the members of the group. It would happen as if by magic after many hours of arguing and discussing every small detail.

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  •  08-06-2007, 10:14 PM 27001 in reply to 26995

    Re: "Integral Mathematics: An AQAL Approach"'

    continuing, the criterion of 'social manner' may possibly have taken into account c.o.g., with mathematical ability doing the same for altitude. the veto power sounds something like one of the principles of holacracy, so it's conceivable that this was a 2nd tier organization to a significant extent. however, their focus was on a non-developmental (1st tier) form of structuralism, which they never appear to have gotten beyond. aczel lauds grothendieck's push to make category theory foundational, replacing set theory, but by itself it over emphasizes lumping, i.e. is not at all integral. for example, 57 came to be known as grothendieck's prime, because when he was pushed to give an actual example of what he was talking about in such great generality, to wit a particular prime, that was his exasperated reply.

    their aim was truly to demonstrate 'unity in diversity' (p.99), a 2nd tier view, and they actually floundered eventually in their over emphasis on unity, namely, the unity that an over adherence to structure (not dynamically conceived, unfortunately) could provide.

    mathematics itself, of course, focuses almost exclusively on the objective, so it's refreshing to see the lives of this group of mathematicians included in the story, being in fact the substance of the story. the dilemma of european jews in the early 20th century is an essential part of the story. but aczel's training appears to be primarily in the RQs, so while he does as good a job as could be expected, it lacks the expertise of someone specifically trained in the left hand quadrants.

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