"Mathematics hasn't been philosophy for 1000's of years Kirby,
and in preparation to my reponse to Richard, Physics hasn't
been philosophy for 100's of years." -- Robert Hansen

[ http://mathforum.org/kb/message.jspa?messageID=7123390&tstart=0 ]

The context of the above remark is my ongoing thread
about exploring alternative "foundations" in the realm
of mathematics, drawing on philosophical work of the
late 20th century.

Long timers on this list may recognize this as my pet topic.
I am interested in language games that "revector" (change
the meaning of) such basic words as "dimension" and "volume"
-- not for all time for all people, but within one more sandbox
or sandcastle on the beaches of possibility, another way
to think and compute, design, get results, with a place in
the sun.

We're talking about "forms of life" in other words, or call
them "ethnicities" (the anthropological dimension is apropos
when yakking about Wittgenstein).

The underlying plot line (you need one of those, to sustain
interest and commitment) is that civilization got off on the
wrong foot to some extent, in becoming so enamored with
the cube as its favorite space-filling polyhedron.

We consider volume to be the product of three mutually
perpendicular edge lengths multiplied together and assign
a cube the role of "unit volume" as a result of this mindset.
To turn a 90 degree corner is to access a whole new
dimension, making 90 degrees a somewhat mystical angle.
Right angles are "normal" and to be orthogonal is to
be orthodox (to believe all the right things). The cube is
an ultimate bastion of conservatism, and to question its
primacy is indeed to engage in a radical operation (or
philosophical investigation).

This rectilinear beginning is now thoroughly taken for
granted of course, gets passed along essentially
unquestioned by one generation after another. It would
take a rather willful and obstreperous youth with
privileged access to education (e.g. Harvard) and a
commitment to make a name for himself, to ever buck
this trend. One in a million or billion might try this.
Most would be quickly overwhelmed by the seeming
hopelessness of their calling.

So yeah, along came R. Buckminster Fuller, born in the
late 1800s, lived until 1983. He developed a philosophy
which gave primacy *not* to the cube, but to the tetrahedron
instead.

The latter is topologically simpler in having fewer edges,
vertexes and windows (thinking of it more like a network
than a "solid"). It's known as a "simplex" for this
reason.

You may likewise use it to anchor a notion of 3rd powering
i.e. volumetric growing and shrinking relative to linear
growing and shrinking of its edges. Triangles may be
used for 2nd powering the same way.

There's no logical reason that 3rd powering *has to be*
called "cubing" and modeled as such. You need to
get back to your mathematical foundations to "see"
this -- as I've endeavored to do on several occasions
on this list, in the spirit of Remarks on the Foundations
of Mathematics and Philosophical Investigations.

Assign the role of "unit volume" to a tetrahedron
(a regular one) and some magical things start to
happen. This shape plays well with others and
although it does not fill space alone, it does
in complement with an octahedron of precisely
four times the volume.

Another tetrahedron, known as the Mite (volume 1/8)
*does* fill space without complements (Aristotle was
right, remember the mite -- a new slogan).

The cube is reintroduced in this language game,
but with a volume of three this time.

The rhombic dodecahedron, which embraces the
octahedron of volume 4, the cube of volume 3,
has a volume of 6.

Simple whole number beginnings, not shared with
any students in US elementary schools because
the Way of the Cube is considered the only way,
best way, and it's "my way or the highway" when
it comes teaching math's foundations (the life
form in question is totalitarian in that respect,
Borg-like ("resistance is futile")) -- partly why math
is often such a turn-off to those who think freely
and creatively (e.g. artists).

It needn't stay this way, were "philosophy for
children" to open more doors, challenge the
dogmatists.

Yes, the current foundations are primarily
dogmatic in their delivery, like a catechism.
Philosophers could be chipping away here,
restoring some mental flexibility, freeing us
from overly straitjacketed thinking, a kind of
paralysis that keeps us stuck, awkwardly
trapped.

A few of us are doing that work -- a real uphill
slog given how people glaze over at the slightest
mention of anything "mathy". We obviously could
use some more help. Consider me a recruiter
for the cause then, looking for allies.

A favorite way to keep "tetrahedral mensuration"
from making any headway in the current "devo"
context (ultra dumbed down, ethically in the toilet)
is to dismiss it as "trivially true" i.e. the mathematics
is well above the threshold of "false" (cannot be
falsified) but then it's just too easy and simple
to merit the attention of high level guru-geniuses,
the caliphate as it were.

Elementary school kids who might benefit from
earlier exposure to spatial geometry with this
newfangled approach, never get the opportunity.
They don't even have a clue what they're missing.

I call this "verboten math" therefore, because
people such as myself, Amy Edmondson (Harvard
Business School), Ed Applewhite (CIA),
David Koski etc., who put many years of work
into this project, encounter mostly resistance
and put downs, transparent delay tactics.

Fuller is still a frequent target of character
assassinations, even though he's dead (especially
because he's dead?). Once on the cover of TIME,
he was more recently ridiculed in the same
magazine for his ugly "lemon" of a car (so how
many philosophers do you know who invented
a car? -- another reason he can't be a *real*
philosopher, like Hegel or Marx: he had patents
and inventions (a huge dis-qualifier, by today's
academic standards)).

Amy wrote a book ('A Fuller Explanation') which
Branko Grunbaum nastily panned, Ed collaborated
on Fuller's magnum opus (wrote 'Cosmic Fishing'
about the experience (talk about uphill slog!)
and David Koski has mapped all of the Archimedean
honeycomb duals to Fuller's more simply named
and volumed "modules" or "cells" (among many
other achievements).

I've posted a table of David's results to the
Math Forum thread above, fingers crossed it
gets through. My previous response appears
not to have made it past the censor, was perhaps
too vituperative in tone (par for the course on
that list, but I'm held to a higher standard
perhaps).

So where does Wittgenstein fit in again?

I think one meaning of "show" stemming from
Tractatus days, relates to what in psychology
we might call a "gestalt switch" -- except
sometimes that gets too narrowly interpreted
as a merely visual phenomenon, such as in
the case of the duckrabbit, Necker Cube and
such.

When LW talks about the world waxing and
waning, from the perspective of the subjective
viewpoint (in some sense synonymous with
the world itself because that perspective or
angle colors everything), he's talking about
how meaning is "orthogonal" to facticity i.e.
to the world of facts (of true and false). A
gestalt switch or new way of seeing (feeling,
being) may leave everything as it as, factually
speaking, and yet the world has changed its
meaning in some way. This relates to what
we mean by Zeitgeist, as many people seem
to come to similar realizations, or call them
"currents in the collective unconscious" (lots
of ways to talk).

I realize talk of "many people" may not sound
solipsistic enough to fit the mood of the TLP,
but by the time of the PI, I think we're looking
for people who "breathe a different air" (to
understand what's presaged).

We could connect to William James at this
point, or any of a myriad number of writers
more cogent than I on this topic. My core
thesis about Philosophical Investigations and
LW's philo more generally, was that it aimed
to catalyze or induce precisely such gestalt
switches (aspect changes). Hence: philosophy
leaves everything as it is, contains no theses
(except of a tautological nature), is about
liberation from reflexive, unexamined habits
of thought. It's an ethical work in other words
(ethics = aesthetics, per TLP) and therefore
religious in some dimension.

In the Fullerian world view, human beings have
crossed a threshold in their ability to leverage
eternal principles (nature's "rules of the road")
such that they have the option to take care of
themselves at a pretty high living standard,
though that doesn't mean simply amplifying
the wasteful and resource-intensive lifestyles
of North Americans etc.

This ability to mitigate human suffering on a vast
scale is actually within reach, from an engineering
perspective, but at the level of perception and
conditioned reflex, our language is keeping us
imprisoned (would be the view -- a tough one to
stomach, as so much unnecessary evil appears
humanly contrived -- not so easy to blame the
gods then, nor even "politicians").

We're still enslaved by the thought patterns of
darker ages, and cling to older dogmas out
of habit and a need for security. People don't
like having their cages rattled. The idea that we
actually could eliminate death by starvation from
the planet is a huge threat to business as usual,
which is entirely premised on 'never enough to
go around' or 'enough is never enough' as they
say in 'Over the Hedge' (a fun cartoon).

Again in the Fullerian world view, getting more
air time for tetrahedral mensuration was a
"foot in the door" that would get the sciences
and the humanities to open more of a dialog.
He aimed to bridge that C.P. Snow chasm,
seeking a common language for both sides
to invest in. Literary critics want to read texts
on many levels, not get too mired in "the one
literal truth" (per Norman O. Brown). Fuller's
text does not disappoint in this regard, yet
the ability to grab literal meanings from his
fish tank is still very much there.

Fluency with sciences and maths on the
humanities side has the potential to skyrocket,
given a philosophical language well stocked with
core memes from those disciplines, organized
according to some broad heuristics centered
around syntropy and entropy as the countervailing
tendencies.

Yes, you could see this as just one more metaphysics,
a Neoplatonism we could say, but then doesn't
every age need to keep upgrading and updating?
Are such language games entirely dispensable,
now that we've gone through a linguistic turn?

I would argue that "new ways of thinking" remain
as relevant as ever, and that a lot hinges on our
ability to remain flexible and non-dogmatic, not
overly reliant on inherited mental habits.

Kirby
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