February 23, 1977; revised June 1999

Plex is a scientific philosophy -- not a philosophy of science, but a philosophy which itself is scientific in its character and practice.

Plex has no assumptions or axioms -- only definitions -- expressed in ordinary language -- but used with extraordinary purpose and precision. No formal logic or mathematics appears. Even counting is not allowed, at first.

The entire **Epistemology of Plex** is the Formal Saying:

Only that which is known-by-definition is known -- by definition.

where the **Definition of Definition** follows from

If D defines X, then Y satisfies D

if and only if <Y> is <X>.

where "is" is the ontological copula, and to define
"satisfies" needs a longer derivation than is
suitable for presentation here. Ordinary-speech understanding of
<<to satisfy> a definition> is satisfactory, however. The
self-definition of X, is X itself; "X is X" *expresses*
definition of X. {The < >s are called Meaning Quotes; the " "s
are ordinary Naming Quotes.}

The **Possibility Definition **is:

A possibility is that which may, but need not, BE.and <may not BE> and <may BE> are the two States of Possibility

while ISN'T and IS are the two States of BEingso <may not BE>/ISN'T and <may BE>/IS are the two States of Reality

Notice that BEing itself is a possibility (for it satisfies the Possibility Definition). IS is its realization.

To complete these few Plex foundation definitions:

Given that "Nothing" is our proper name for <that which does not exist> --

The **First Definition of Plex** is the Formal Saying:

Nothing doesn’t exist.

where "existence" is BEing and is that which is defined by the First Definition -- as pure non-Nothing-ness -- i.e. IS itself. [Notice that the First Definition does not define Nothing, which has no definition at all -- for <it> ISN’T -- having no self-definition.]

By the Plex Epistemology -- without ** any** definition,
Nothing is known.

so Nothing is <The Ultimate Knowledge>

by means of which all else is known.

Finally, the **Definition of Meanin**g arises in the Formal
Saying:

-- a most marvelously recursive discovery. Yes, this is our ordinary every-day meaning of "meaning" although it is pure Plex and is not to be found in any dictionary. To understand it, it is merely necessary to appreciate the Possibility Definition, for by this definition, here, that which there is defined -- (that <may BE>|<may not BE> dichotomy of the States of Possibility) -- IS the meaning (of that possibility).

What difference does it make whether that possibility is realized or not?

-- i.e. in the overall scheme of things,

What is the <<before>-to-<after>>effectof its birth?What

roledoes it fulfill in the scheme of things?

Surely that is what matters about <it>. That is what gives it meaning.

__Let__ <point> = <word>, for any word. Then prove
the following Propositions:

P1)Letpoints be such that, except for identity, they all are indistinguishable.P2)

Letthere be only points.P3)

Letthe world be the collection of all points.P4)

Thenthe identity of a point is the collection of all other points,P5)

Andevery point is the (whole) world.

I (n = 1) A world of one point is the world.II Assume the theorem true for n-1 points. (n>l)

I.e. for any collection of n-1 points, every point is the world.III To prove the theorem for n points given its truth for n-l (n>1)

a) The identity of any one point, p, in the collection of n is a collection of n-1 points, each of which is the world, by II. [Please skip to b).]In the case of n = 2, so n - 1 = 1, that one point will be q, so q is the world by I = II.

No equivocation is involved or allowed. The theorem is by I;

there is no assumption.In the case of n > 2, so n - 1 = 2, 3, … -- the inductive assumption of II

(just provedwithoutassumption) applies; hence for q and

<(the/all) other point(/s)> of p’s identity -- each is the world.b) The identity of any other point, q, i.e. a point of the identity of p, is a collection of n - 1 points, each of which is the world, by II. [Skip to c).]

In the case of n = 2, so n - 1 = 1, this one point is p, so p is the world by I = II.

No equivocation is involved or allowed. The theorem is by I;

there is no assumption.In the case of n > 2, so n - 1 = 2, 3,... -- the inductive assumption of II

(just provedwithoutassumption) applies; hence for p and

<(again, still) the other point(s)of q’s identity> -- each is the world.

Here the proof by induction already is complete, fully explicated for case of n = 1, 2, 3,… The P4identityof a P1 point is not a property of the individual point; it is whatprovidesthat point with itsunique individuality, by definition -- for the definition is the same for each one, but each result is different. (A property could be distinguishable; hence none are relevant, should they exist. This also is why P1-P5 is cited as "the meaning of any word" forThe sole

qualityof P1 points is their Pl through P5equality;they, themselves,theirareinequality(as the mere P3 collection of P1 individuals).The

possibility of inequality(undoing the P5 equality in favor of a renewed inequality) is what ensures that each <new P1 point> (as it is realized asrelevant to the proof) instantly joins the P3 world of relevance to the on-going proof --<P3 <<world of relevance> to the on-going proof> (before) =<world of <relevance to the on-going proof>> (after)>--

BY DEFINITION-- for what is to delay or prevent it? Already it is "in there", when it’s time comes!Definitions must be accepted without reservation or equivocation in mathematics.

Proposition P3's <"all"> really <all>means-- just as Proposition P2's <"only"> = <only> to specify that relevance. (There is no consideration of any "space" for P1 points; they

mustsuperimpose, in their non-Nothing indistinguishability.)

c) The identity of p and the identity of q are identical except that where the identity of p has q, the identity of q has p. In any case, p is the world by b) and q is the world by a).d) Hence both p and q are the world, as are all the other points (if any) in their respective identities (and shared between them).

e) Hence all n points are the world.

IV For n = 2, I is used (via II) in IIIa and IIIb, q.e.d. [Please skip to V.]

This <q.e.d> actually makes redundant the interjected commentary

[--except for the explanatory remarks about the consequences of P4’s definition of <identity of a P1 point> and the permissiveness of thelimbotransition (the shared "may" of the <<BEing>overlap>>linking the stable states of reality) -- accessible for each <new>’s birth in their superposition].V Q.E.D. by natural induction on the integers. [Now please do a full re-read.]

Natural induction on the integers has either to be believed or not. There is no flaw in this presentation of it, with or without the explanatory commentary. In fact:

P6)Letaninstantbe a world with one distinguished point.

With this additional proposition, the inductive proof’s steps
become not merely logical, but ontological -- and the result is not
merely mathematical, but physical. The P3 world becomes not just the
world of the proof, but the real world of <all possibility>,
for P6 completes the basis definitions for Plex’s
__thime__**-- **the factual substratum of Actual
Reality and the precursor (both conceptually and ontologically) to
the spacetime of Physics. In thime, **time**(here) and
**place**(there) __coincide__ in
**<now>** **= **<all that
exists> **-- **by the First Definition of Plex.** **It is
the pure non-Nothingness of The Sole Possibility, which is the
<possibility of **<verity>> =** the <possibility of
<possibility of <possibility>>> which is assured by
the <impossibility of <impossibility>>. [Also the
<meaning of <meaning of<meaning>>> cannot be
Nothing. These can be proved, when the relevant definitions are
supplied. Although it sounds like a possibility, <verity> =
<the <possibility of possibility>> is not a possibility
at all; for it lacks the <may not be> state of
possibility.]

Verity **can’t not BE**; it is the ontological equivalent
of the logical tautology.

The true import of the meaning definition and of existence itself is embodied in the Propositions, P1 through P5, which resolve Plato’s "ideals" ("general elements" in modern parlance) -- Is there an ideal cat through which (by some relationship) I can know

Once the possibility of P6 is recognized, Time going forward in an
expanding universe of <now> **= **<all that exists>
(both immaterial and material) results. Thime becomes SpaceTime only
when thinks/things matter. [Because of P40): "Change is not
caused, it is *allowed *by change of perception." -- what this
mattering means in Plex is not yet fully established. There is no
cause and effect in Plex’s thime fundament. It's all a matter of
definition.]

The PLM proof is much simpler to see.

**[This 1975 proof was first presented in public in an abortive Graduate Seminar on Plex in the Fall of 1984 in the MIT EECS Department. It has since then appeared in several publications, including:

D.T.Ross: From Scientific Practice to Epistemological Discovery

In: C.Floyd, H.Zullighoven, R.Budde, R.KeiI-Slawik, (Eds.):
*Software Development and Reality Construction *Springer-Verlag,
1992 pp. 60-70.

D.T.Ross: Understanding /:\The Key to Software

In: *Scaling Up: A Research Agenda for Software Engineering*,
National Research Council: NRC0041317, 1989, pp. 66-70.

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