Ben, list,
this thread on "The New Elements of Mathematics" started with Charles
Peirce writing:
"None of them approved of my book, because it put perspective before
metrical geometry, and topical
geometry before either."
Even today if one would consider to engage in the project of writing such
a book, one should really think
twice. Nobody has a really good idea how to write it and if it were
written, nobody would understand it,
and if one would understand it, one would have to unlearn lots of things
one already knows and that only
for a curiosity.
One criterion for scientific progress is, that a new theory should explain
everything that the preceding
ones explained and something else besides (ha!).
Charles was, together with his father Benjamin Peirce, part of a movement
in 19th Century mathematics
called "Universal Algebra". Others were e.g. William Rowan Hamilton and
Hermann Grassmann. All of them or
their followers erected a "philosophy" on their mathematical ideas, by the
way.
What Felix Klein has written about Hermann Grassman's Ausdehnungslehre
("Theory of Extension") in his
"Lectures on the Development of Mathematics in 19th Century" (1926)
applies to Charles Peirce too and is
still considered relevant today. The main point is on page 178 in my
Springer Reprint of Klein's book (I
believe there exists an English translation too).
It is this:
The grand project in mathematics for much more than a century now has been
"arithmetization, i.e. to
reduce mathematical structures to the abstract structure of the natural
numbers. If you put the
continuous before the discrete, then you are not alone in history, but
nobody has as yet really succeeded
with such a project. The problem is, simplistically speaking, that,
starting with a continuum, you will
have great difficulties to introduce discrete entities, except by way of
an arbitrary addition. So the
relevant book today is David Hilbert's "Grundlagen der Geometrie
(Foundations of Geometry). There are
today followers of the other approach, especially in Grassmann's
footsteps, e.g. David Hestenes with his
"Geometric Calculus and "Geometric Algebra, but their success, despite
some very striking
simplifications and insights, till today is quite limited. It is more or
less regarded as a curiosity,
some "flashes of brilliant light relieved against Cimmerian darkness��� ...
On the other hand there is in Sir Roger Penrose's "Road to Reality (now
we come to the noble celebrities)
an introductory chapter on "The roots of science and especially "Three
worlds and three deep mysteries
(chap 1.5) with the usual Popperian sermon preached (sorry, Sir Karl
Raimund).
But one "deep puzzle for Sir Roger is "why mathematical laws should apply
to the world with such
phenomenal precision. Moreover, it is not just the precision but also the
subtle sophistication and
mathematical beauty of the successful theories that is profoundly
mysterious(p.21).
Finally Roger Penrose writes in this context: "There is, finally, a
further mystery concerning figure 1.3,
which I have left to the last. I have deliberately drawn the figure so as
to illustrate a paradox. How
can it be that, in accordance with my own prejudices, each world appears
to encompass the next one in its
entirety? I do not regard this issue as a reason for abandoning my
prejudices, but merely for
demonstrating the presence of an even deeper mystery that transcends those
that I have been pointing to
above. There may be a sense in which the three worlds are not separate at
all, but merely reflect,
individually, aspects of a deeper truth about the world as a whole of
which we have little conception at
the present time. We have a long way to go before such matters can be
properly illuminated.(pp. 22/23)
Noble words to be considered well! But don't tell Sir Roger about the sign
and it's interpretants. That
will not do for him. There are a lot of philosophical soap shops out
there. You had better understand fully
what his problems are in the next 980 or so pages of mathematics and
physics that come then, before you
tell him about "The New Elements of Mathematics���.
So what we do with Peirce's work appears to the outside world either as a
more or less philatelistic
pastime with historical curiosities. It's all good and fine and edifying
and very logical except for a
few paradoxes here and there, perhaps. Or else you start getting your
hands really dirty and do whatever
it takes to find out what is going on behind the scenes. We had better
find out and make our mistakes as
quickly as possible in order not to flog a dead horse, I believe.
Enough name dropping for now.
Ben, you write:
<begin citation>
1. The idealized system of motions & forces -- classical Newtonian or
pure-quantum-system -- is time-
symmetric, completely deterministic in the given relevant sense,
unmuddled, pure OBJECT to us, information about which object we can only
approach indefinitely, as to a limit.
2. The material system is time-nonsymmetric, stochastic-processual, in
which the system at a given stage
is only ALMOST the system at another given stage, i.e., a SIGN to us of
the system at other stage.
3. The vegetable-level biological system is time-nonsymmetric but LOCALLY
pointed thermodynamically in
the opposite direction from that of its material world, from which it
filters order and is an INTERPRETANT to us.
4. The intelligent living system is time-nonsymmetric but INDIVIDUALLY
pointed variously in both
directions thermodynamically -- as living thing, it filters for order --
as intelligent, it is a sink, retaining
sign-rich disorder as recorded -- I don't know how it pulls
double-direction "trick" off --
anyway it is a RECOGNITION which we are.
The sign defined by its relationship to recogition is a proxy.
<end citation>
Peirce distinguishes equiparants and disquiparants ("Classification of
Simple Relatives; CP 3.136):
"Classification of Simple Relatives (cont.)
"Third, relatives are divisible into those for which every
element of the form (A:B) have another of the form (B:A),
and those which want this symmetry. This is the old
division into 'equiparants' and 'disquiparants',
or in Professor De Morgan's language, convertible
and inconvertible relatives. Equiparants are
their own correlatives. All copulatives are
equiparant.���
That was in 1870 ("Description of a notation for the logic of relatives���).
In MS 293 (1906) under the title: "The Logical Form of Identity, he says
(I have to retranslate things
into English, since I only have Helmut Pape's German translation here
before me), considering the question
whether it is disquiparancy or equiparancy that is the more fundamental,
important, elementary, simple:
"I hold that it is disquiparancy or, rather, it is the opposition or the
relation, of which disquiparancy
can be a specialization���.
Sounds ugly, but maybe someone out there can give us the original English
text. (My command of English is limited.)
We need something fundamentally asymmetric in physics as you implicitly
remark.
We need a fundamental asymmetry in logic, since there are such things as
memory,
history, time.
I guess we should discuss this "how it pulls double-direction "trick" off"
further. No mercy!-) This is
very important and something that seems to me to have been neglected as
yet!
You write: "anyway it is a RECOGNITION which we are"
This "RECOGNITION effect, this is tremendously important. You've got it!
That's it!
We'll get that! We'll get that damned thing out. Be sure.
Ben, you write:
"ERGO: As sign, man is most of all a proxy. At intelligent life's best,
only indefinitely approached,
intelligent life is a genuine, legitimate proxy acting & deciding on
behalf of the ideal, in being
determined _by_ the ideal. Intelligent life shouldn't let it go to his/her
head, though. Hard it is to be
good; harder still to confirm & solidify it by entelechy = by staying good
=> continual renovation and
occasional rearchitecting (entelechy is not necessarily a freeze) amid
changing & evolvable conditions.
I fully agree.
Later more,
Thomas.
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