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MEMORANDUM April 10, 1998 TO: Paul Ernest, School of Education, University of Exeter, UK FR: Kirby Urner, 4D Solutions, Portland, Oregon, USA RE: Your Social Constructivism as a Philosophy of Mathematics etc. CC: Synergetics-L (e-list and archives) Dear Sir, I was just now perusing your book, while sipping coffee, at our local Powell's Books on Hawthorne Blvd (while getting the oil changed in the Subaru at Jiffy Lube). As I tend to treat the local Powell's as an extension of my own study, or living room, I reshelved the book for further consultation, and so don't have it in front of me -- so I'm writing this now while the memories are still fresh. [Editor's Note: I also buy plenty of tomes from Powell's, and fork out for refreshments]. The timing was right, for me, to find your opus, as just last night I was replying to a college math teacher on one of the e-lists to which I'm subscribed and she had written "We're working under the model of experiential learning and social constructivism". I was scratching my head as to what was meant by the latter. Having your book fall into my hands was most helpful (Powell's is good in that way). I liked what I saw for several reasons. You navigate the convoluted twists and turns in recent philosophy with some skill and brevity it seems to me, cluing the reader about Rorty's linguistic turn away from positivism, closer rapport with a Continental brew, the short shrifting of Wittgenstein's philosophy of mathematics over the years (I wasn't aware that Benacerraf and Putnum dropped that whole section in the later edition -- Powell's had a copy of the earlier). I especially enjoyed your chronicling of your own personal journey re LW's work, seeing it at first as 'irrelevant and obscurantist' but coming to a new appreciation commensurate with the evolution of your own philosophical position. I see from your website that you edit an online journal and are maybe looking for submissions. I'm writing in part to give you a preview of where I've been socially constructive regarding mathematics, to find out if maybe you'd like an essay -- or better yet you could just web-publish this memo, typos fixed if you catch any (I'd then link to it from my website, where many memoranda are already archived). From Wittgenstein's focus on usage patterns as meaningful (which patterns comprise more than just 'context' in some temporally immediate sense -- context over time we might say, which would include many instructive special-case events) I move to P.W. Bridgman's "operational mathematics" as incorporated into the writings of R. Buckminster Fuller. To render a term "operational" is to use it within "language games" and if these latter are highly precise (and meet other criteria) then perhaps we're operating within the domain of mathematics, or at least in the company of chess-playing computers (like IBM's Big Blue). In particular, Fuller was enthused and infused by the early century hype surrounding the "dimension" concept, with writers going everywhichway in their attempts to give a creative -- perhaps breakthrough -- spin to this signifier, now in "free fall" (almost) owing to its dislodging from Euclidean and Newtonian moorings. It was a heady time for all, not just Fuller. We are familiar with two of the resulting vectors (operationally defined) for "dimension": Einstein's and "time as the fourth dimension" talk, plus extrapolated Euclideanism, where 3-tuples become n-tuples, with Pythagorean distance remaining intact, giving us 'dim 24' sphere packings and the like (Conway et al). We are less familiar with a third trajectory for 'dimension' (and '4D' in specific), which was Fuller's. I trace the trajectory of "dimension" through his writings (beginning with 4D Timelock) in my paper On Redefining "Dimension" in Synergetics -- showing where 4D ends up as a signifier for the tetrahedron (primitive conceptuality) in a "geometry of lumps". Euclidean --> Cartesianism --> Non-XYZ Geometries | | |--- Einstein's Relativity (xyzt) | Gaussian manifolds, hyperbolic spaces etc. | "time" as 4th dimension | |--- Hypercubes etc. | n-tuple polytope geometry | |--- Fuller's 4D geometry unit volume tetrahedron Fig. 1 Synergetics as a Non-XYZ Geometry In Fuller, we even find polemics against ordinary 'dimension talk', the commonsense vernacular (as informed by academese) which would make everyday space '3D'. In Synergetics, heighth, width and depth do not comprise building-block seperables, such that one or the other might be sensibly subtracted from the others. Conceptual volume marks a primitive conceptual beginning. Locations (points) exist in the same dimensional space (have the same dimensional characteristics) as cubes (or tetrahedra) if for no other reason that we observe them from multiple angles. You might say that Fuller's geometry is volumetric because it always reminds us of the observer -- of a plane, of a line. His thinking anticipates computer graphics, wherein we must pay as much attention to the camera and its angle (the viewpoint) as to whatever objects (e.g. points). Because the tetrahedron is a primitive signifier of volume, being the "box" or "room" with the fewest walls, is the topologically minimal inside- versus-outside containment, considering edges, vertices and windows our only constituents (E, V, F: V+F=E+2 -- Euler), it makes some sense that we consider volume to be 4D. The tetrahedron has 4 vertices, 4 windows. 0D, 1D, 2D and 3D remain undefined as such (but remain meaningful in other contexts of course). Of course we needn't waste ammo trying to dislodge conventional dimension talk from its deeply embedded position -- my goal has been more to clarify Fuller's meaning in order to pave the way for those with a sincere interest in penetrating more deeply into his magnum opus, Synergetics, now on the web (linked from my http://www.teleport.com/~pdx4d/links.html). If Fuller's "dimension talk" is internally consistent enough to be interesting and worthy of study, it doesn't follow that everything we've invested in up until now must be jettisoned -- I confess that I tire of the defensiveness I encounter in people who think this must be my point of view. I'm more a live and let live type, in the true liberal arts tradition. Synergetics has a reputation for being closer to 'Finnegans Wake' than any kind of principia of mathematical relevance, and in part that's because we don't have this social constructivist context or willingness to take an operational stance. Instead of allowing Fuller's usage patterns to build and self-reinforce, giving them space to revector key terms into orbits of internal consistency, we destructively interfere with our own prejudices, believing all key words already spoken for in the King's English (see my memo re Aldersey-Williams' chapter on Fuller at my http://www.teleport.com/~pdx4d/virus.html). Positivist name->object nominalism makes it harder for us to let go of the 'objective anchor' we suppose gives words their weight in the final analysis. But in Synergetics, a key word like 'gravity' also has subjective significance and defines a metaphysics with permission to reflect this, even while remaining faithful to Newton when visiting his neighborhood. We need to understand that Synergetics has a soul, is a work in the humanities, meaning metaphor and hyperlinks are a built in aspect of its primitives, because operationally a part of our form of life. Elsewhere I've dubbed Fuller "the Lacan of mathematics" and linked his polymorphic (but he says never scientifically perverse) language to Norman O. Brown's (see http://www.teleport.com/~pdx4d/psych.html). I think you might be getting the picture here: I'm using Wittgenstein, and by extension your social constructivist school, as a potential bridge into Synergetics within the philosophy department. Synergetics is a philosophy. What you'll find at my website is some further interpreting of Wittgenstein's philosophy of mathematics and a link to an "object lesson" in how usage patterns revector key terms: the dimension stuff, polemics against "hypercross dogmatics", and a fully elaborated '4D coordinate system' designed to further pry loose the prejudicial assumption that ordinary '3D talk' is the only self-consistent chatter. I invite you to explore these exhibits and share any feedback: On Ludwig Wittgenstein's Contribution to a Pragmatic Philosophy http://www.teleport.com/~pdx4d/lw.html Investigations into the Linear Algebra Concepts used in the XYZ and Quadray Language Games http://www.teleport.com/~pdx4d/quadphil.html (see attached memo to Dr. Paul Benacerraf) Four Dimensional in Synergetics http://www.teleport.com/~pdx4d/terms.html#4d Synergetics Versus HyperCross Dogmatics http://www.teleport.com/~pdx4d/hypercross.html On Redefining 'Dimension' in Synergetics is linked from the bottom of this essay. |
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