v Macrobe on Axiomatics

Originally Posted by Macrobius
Returning to the discussion of Ayer (though the side trip was not irrelevant) -- axiomatic systems have several different purposes. There is one purpose in Greek philosophy, where reasoning is seen itself as a sort of triad of origin, process, and reversion. 'Axiom' is supposed to denote a propositional truth so obvious that no one (who is not being perverse) contests it, as, 2+2=4. It has come to mean something different today, with axiomatic set theory, in which the 'axioms' are really speculative hypotheses, and the deductive results are in fact more certain than they, and amount to what evidence there is for them. This is practically a reversal of what an axiom is.





OK. I'll read Albanius in a minute.

But about the formal status of "axiomatics": Russell began from Peano's 5 axioms for the reconstruction of arithmetic. These included as primitive (undefined) terms:
"0 is a number" ("assertion*: of an "S is P*" type proposition) ..."successor of" ... properties of P* --inductive properties of P* (to get an axiom of infinity, which other systems that take number signs as primitive must assume) I credit this as a contribution to metaphysics gained by the axiomatic method.

This reconstruction, whatever one says about the 5 primitive propositions with the undefined primitive terms, demonstrates that signs for numbers are dispensible, in principle, i.e., can be introduced on a definitional basis into any statements of fact (Ex. "The Temple of the Ark was 300 meters long, 50 wide", etc.) in a syntactical system correlating truths in it (logical) with truths of arithmetic. Now, this is what Russell and Whitehead did in the Principia Mathematica, much of which I actually read, in the olden days, trying to figure how the Symboli/Glyph story played out to get Quine ... This is the only sense in which I need acknowledge use of the term for what I do: 1. a small set of proposition taken as necessary (for mathematics: the denial of which is self-contradiction: from Aristotle, nevermind 0 is not a number!); 2. a formal (token-checkable; digitalized) apparatus for formal deduction: transfering predicate "LT" of P* to Q* when every instance of P* is necessarily an instance of Q*. 3. An array of tokens available for use as undefined descriptive constants for interpreting* the formal syntacticic apparatus -- with application arranged in orders of logical type, barring predication over illegitimate totalities* (at the theological level: the sin of idolotry) in the formation rules of basic sentences. (stratifying the 'x's' that there are -- a real constructivist theory of individuals).

There is a "Y" -- fork in the road -- here. To the left goes set theory, Cantor -- and metaphysical deficit (judg.). To the right goes --new territory here: it is a proposal, a posited 'model', like the Mercator projection of earth's globe onto a plane of two dimensions. This path seeks to predicate on an original order: The Quadrivium ?


I have used the lines of a four-space formed by folding a square sheet of paper once over, left-to-right; once up, bottom-to-top ... then straightened out again, corner squares labelled NW NE

SW SE

for reference. Let these correlate with four Quadrants

QI = SW
QII = NE
QIII = NW
QIV = SE

This QI -QII - QIII - QIV sequence deffines a line running incounter-clockwise direction from lower-left, looping up through the four planes, and exiting lower right.
Along this loop is posited four disrtinct in-out domains in a single in-out processing unit from perception (QI = input) to act-ing (QIV = output). QII is the domain of "historical echo" attaching to an inner inervation originating in QI, on the way to brain-consciousness, QIII. This, the upper left square, matched point-for point on/by the other three, is where content of consciousness under the 7 tokens of Sign-use are displayed -- in TokenSpace*, which, through the textual content of signs, maps any alternate communication system on the same page. QIV is the "-I -N - G" Quadrant: the reality-box of what is presently passing into history, frozen in time foreverafter, amen.

It is onto this as a formal schema I propose to project the individual consciousness; BUT (and this is big, metaphysically) also in a way that spans the collective psyche which Mousavieff brilliantly predictes on (reading his remarks on the global/international judicial situation of anarchy, heeding his call for a New Assembly, to do what Richileu did, and try to stave off catasrophe.
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Everything is two