Typos fixed, italics added, embedded hyperlinks activated. Original e-mail archived to Synergetics-L

Relevant links:

                     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 

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".
    --> 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 

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 

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

Investigations into the Linear Algebra Concepts used in the XYZ 
and Quadray Language Games
(see attached memo to Dr. Paul Benacerraf)

Four Dimensional in Synergetics

Synergetics Versus HyperCross Dogmatics
On Redefining 'Dimension' in Synergetics is linked from the bottom of this essay.

Synergetics on the Web
maintained by Kirby Urner