AndyThe same way you explain the darkening blue twilight I saw yesterday framed within an arch of the campus building. You don't. You do them.
As far as practice and mathematical symbolism goes, there certainly seems to be a need for some sort of practicing (how does one, for example, draw an integral sign or meander through a geometric proof). The problem with practice often is, as Goffman puts it I think, the play often never goes on and, perhaps one might say, the abstract never becomes concrete.
Ed On Apr 24, 2009, at 9:24 PM, Andy Blunden wrote:
I've never heard this discussed before Ed. Very interesting. How can you explain to someone the beauty of Goedel's proof of his Undecidability Theorem? Like with Euler's equation and so on, it is one thing to give the "easy" explanation which appeals to intuition, but the mathematical formalism is something else again. How do you communicate the beauty of these symbolic practices, other than taking people through years of routinized practice exercises?Andy Ed Wall wrote:Martin and AndyThis is interesting (and an experience I had also with Shrodinger's wave equation for a hydrogen atom although I was lured away by education) as I am trying with some of my students (who, so they believe, have neither strong interests or abilities in mathematics - elementary school teachers presently or on the way) to develop a sense of beauty within mathematics. Part of this is because their students do have sort of a sense and part of this is because I have been wondering if I can crank it up a notch (not all the way to tensor algebra - smile) and also, partially negate, their, these teachers, abominable mathematical experiences. I have begun to have a little success as without coaching (and an hour or so thinking, talking, and working) they seem to be able to distinguish on their own between two or so acceptable proofs - ones they, for the most part, understand and generate - as to the one that somehow is elegant (whatever that means although I happen to agree with their choice). Assuming that taste is cultural (although there are ways in which mathematics, one might say, isn't. I don't mean by this Platonic), I've been sort of bemused by the response. Anyway, it seems that Vygotsky would have been interested in 'intellectual' taste.Ed On Apr 24, 2009, at 5:20 PM, Martin Packer wrote:As an undergraduate I was in a class in which we solved Shrodinger's wave equation for a hydrogen atom (the simplest case, and I think the only salvable one) using tensor notation. I can confirm that it is beautiful mathematics, and it almost prevented me from becoming a psychologist.Martin On 4/23/09 10:32 PM, "Andy Blunden" <ablunden@mira.net> wrote::) Yes, Ed, I found tensor calculus a genuine thing of beauty. After learning about e^ip=-1 a couple of years earlier only ijGlm=0 could top it (excuse lack of sub- and superscripts and Greek letters). But it is not so much the mathematics that is at issue I think, when someone says "relativity is simple" but just how the mathematics is related to experience. Einstein himself wrote an introduction to the Special Theory which does the whole thing up to the variation of length with relative speed, without using mathematics. But tensors are a mathematics whose object is not physical relations, but differential equations. That's tricky! Any way, it's a long time ago for me too! Andy e^ip means the base of natural logatrithms raised to the power of the square root of minus one times the ratio of the diameter to the circumference of a circle, and it = -1 Beautiful. In ijGlm , G is a tensor of space-time, ij are subscripts and lm are superscripts. But I may have that wrong! Ed Wall wrote:Andy It has been quite awhile since I have taught a course in special/general relativity (about 20 years); however, the tensorcalculus is, I thought then, a nice way to go about it and brings some things to light that are important on the way to general relativity. Tensor algebra is actually somewhat straightforward by the way, but that is a matter of opinion. However, all of this has now become perhaps a bit off topic (smile) and you are correct that special relativity does not, at a certain level of understanding, require manipulation of tensors.Ed On Apr 22, 2009, at 10:40 PM, Andy Blunden wrote:Yes and No. I was using the word "metaphysics" in the way Pragmatistsuse it. Strictly speaking, of course, *all* thinking containsmetaphysical assumptions. So in that you and Kuhn are right and I waswrong. Perhaps I could stop using the word Metaphysics to mean thereification of thought forms into independently existing substances, and others stop using the word Ontology to refer to personal identityformation? :)But I disagree with you about your Kantian conclusion that "science is a purely logical". It was this Kantian belief (along with Euclid) thatwas overthrown by Einstein. The Logical positivists were wrong ofcourse, because they interpreted the subject in Kantian terms, as an individual person and their private psyche having direct access toeternal reason. Interstingly Einstein disagreed with Bridgman. Einstein said thatwithin the context of a consistent theory, not every entity in the theory has to be subject to an operational definition. Einstein right, Bridgman wrong. But I think Bridgman got the right idea nonetheless.Where Hegel and you are wrong, I believe, is the presumption that we are at the end of history (neither of you claim that of course, but it is a valid implication in both cases.) If the nature of time and space can be deduced completely from a critique of the cultural practices at any given time, e.g. in 1807 before the Michaelson-Morley experiment was possible, then obviously the practices whose critique will allowthe Special Theory of Relativity to be deduced "by logic" i.e.,critique of practice, are impossible. If "science is a purely logical" then that presumes that no further significant developments in socialpractices (such as the Michelson-Morlet experiment) can be made.BTW Ed, I think we have to treat the Special Theory and the GeneralTheory differently. There is absolutely nothing simple about the general theory and its tensor calculus! Andy Martin Packer wrote:Oh Andy, I'm going to have to disagree with you once again!At least, I'm going to disagree if by your statement here you mean tosaythat Einstein was avoiding metaphysics. That was the interpretation the logical positivists made, arguing that Einstein had exposed the fact Newtonian physics had hidden metaphysical assumptions, but that, withhisoperational definitions (Bridgman's term, but his ilustrations were fromEinstein), Einstein had finally showed that science was a purely logical (orif you prefer practical) activity, free from metaphysics. What a messthat has led us into! I'm on Kuhn's side on this issue: every scientific paradigm has metaphysicalassumptions embedded in its practices. So we don't have metaphysicson the one hand and practice on the other. We have alternative kinds of scientificpractice, each with their metaphysical assumptions. (The metaphysics of Einsteinian physics include the assumption that space is somethingthat can be curved by a mass, for example.) The merits of each of the alternatives is what scientists spend their careers hotly debating. Even what *counts* as metaphysics is different from one paradigm to another. But that's probably what you meant! :) Martin On 4/22/09 8:17 PM, "Andy Blunden" <ablunden@mira.net> wrote:All Einstein did was, instead of regarding time and space as metaphysical entities existing independently of human practice, he closely examined the practice of measuring time and distance. That's all._______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca-- ------------------------------------------------------------------------ Andy Blunden http://home.mira.net/~andy/ Hegel's Logic with a Foreword by Andy Blunden: From Erythrós Press and Media <http://www.erythrospress.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-- ------------------------------------------------------------------------ Andy Blunden http://home.mira.net/~andy/ Hegel's Logic with a Foreword by Andy Blunden: From Erythrós Press and Media <http://www.erythrospress.com/>. _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca
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