the unity of opposites and differential ontology

[Mitchell Smith]

For reference, I include my original attempt at axiom writing. The illustrations on page 2 reflect my assessment that the expressions,

"a set is a collection taken as an object"

"a theory of pure sets suffices for mathematics"

describe a "universe of discourse" incompatible with the view typically found in the literature from its historical development.

About two years ago, I learned of "differential ontology."

Let me immediately introduce some terminology. As a contrast to "differential ontology," let me use the expression "objectual ontology." Say, then, that a correspondence theory of truth presupposes an objectual ontology.

The site where I first read about differential ontology had been the Internet Encyclopedia of Philosophy,

https://iep.utm.edu/differential-ontology/

There is little of mathematical use here. However, there is a useful terminological distinction made in this description.

Consider the difference between "intrinsic curvature" and "extrinsic curvature" from differential geometry. The author of that web page associates the expression "immanence" with differential ontology and "transcendence" with objectual ontology.

Does the "unfolding" of models of set theory or a hierarchy of metalanguages to explain "truth" not sound like "transcendence"?

On my reading, "immanence" may be compared with intrinsic curvature whereas "transcendence" may be compared with extrinsic curvature.

Among places where one can learn about differential ontology a specific principle is often mentioned: the unity of opposites. This appears to have some useful relatiin with foundational mathematics.

William Lawvere ran across this idea -- apparently, because Karl Marx had written something on the mathematics of the differential calculus -- and wrote a paper on the topic. Lawvere's paper is available at the link,

https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1996-unity-and-identity-of-opposites-in-calculus-and-physics.pdf

It seems to have gained some ground in the category-theoretic community. In looking up the links for this post, I found the MSE question,

https://math.stackexchange.com/questions/2357569/can-you-explain-lawveres-work-on-hegel-to-someone-who-knows-basic-category-theo

with an accepted answer providing links to ncatlab.

My particular interest, however, lies with a paper by Mcgill and Parry,

https://www.semanticscholar.org/paper/THE-UNITY-OF-OPPOSITES-%3A-A-DIALECTICAL-Mcgill-Parry/68d76061e72ccbb08f62d59e509ddfd49f2cbf4f

leading to the immediate download link,

http://thetempleofnature.org/_dox/unity-of-opposites-dialectic.pdf

The introduction of this paper discusses the fact that Russell had concerned himself with this problem. Viewing it as a contradiction, Russell developed a theory whereby classes are fictional. This had become his no-classes theory.

While some will hold to a dogma of progress, the fact of the matter is that the status of mathematcal objects is not universally agreed upon. That mathematical objects be treated as fictions has its advocates.

Personally, I could care less about such beliefs. What I find significnt here -- because of where my own work had led -- is that this deliberation by Russell is intimately bound to description theory.

It would seem that the view attached to my "intuitiin" about domains of discourse had led, quite naturally, to consideration of descriptions. And, using refined logical constructs not available to Russell, my work led to a logical calculus that implements a form of description compatible with a development bsed upon relations.

Certainly, "difference" is a relation.

In more typical foundational studies, "difference" is studied as "apartness." It is significant for intuitionism. And, although it is behind a paywall, the paper "Equality in the Presence of Apartness" by Statman and van Dalen relates the intuitionistic paradigm to modal model theory.

When Mcgill and Parry list forms of the principle several pages into their paper, they include:

2) Polar opposites are identical.

As is made clear in the expository which follows, this statement is actually referring to indiscernibility relative to sufficient abstraction.

But, that is exactly what is considered by Max Black in his paper on the identity of indiscernibles,

http://home.sandiego.edu/~baber/analytic/blacksballs.pdf

Black's argument corresponds to a justification for excluding the principle of the identity of indiscernibles from the first-order paradigm as a logical principle. This is why it does not appear among the inference rules of first-order logic.

There are those who would like to say that the foundations of mathematics reduces to the first-order paradigm. I do not agree with this. However, I fully reject programs which would try to replace the methods of the first-order paradigm.

Algebra is not a foundation for mathematics because mathematics includes analysis. Differential ontology and the unity of opposites is more directly related to analysis in its usual form (as opposed to "third-order" arithmetic).

Just an opinion.

mitch

[Ross Finlayson]

When we were talking about order theory in foundations and Cohen's proof of the
independence of the Continuum Hypothesis from ZF, and forcing, I still feel like
or get the impression or would gather where I don't much "care" per se, that most
people don't really get that he added an axiom of sorts to make it so.

Or rather, he added an axiom of "not sorts".

Now, I'm for that, because, it's total or about how ordering theory and collection
are so opposite, that, then when we get back to tautology and identity or various
forms of equality, then here this idea of "opposites" yet reflects a space.

Anyways there's though that Cohen's forcing as axiomatic, is as well included
from otherwise a non-well-founded approach, what would be the space where such
things would exist, then as to whether that otherwise "inconsistent multiplicities" are exclude.d

This is where though I've already seated such notions in ubiquitous ordinals where
"successor is order type is powerset", then also since some time it's been
"there's no standard model of integers, only fragments and extensions".

[Mostowski Collapse]

There is a nice principle of difference already built-in
into first order logic (and some other logics as well),

namely we find this inference rule:

P(a), ~P(b) => ~a=b

Interestingly you could reject LEM (P v ~P) and/or reject
LNC (~(P & ~P)), it would be still derivable, via

minimal logic principles from this principle:

P(a), a=b => P(b)

Mitchell Smith schrieb:

[Mitchell Smith]

Ross,

For what this is worth, Tom Leinstner posted on order theory and set theory in 2012 on n-category cafe,

https://golem.ph.utexas.edu/category/2012/10/the_curious_dependence_of_set.html

Because of my own work, I view this as a puff piece that does little except to identify something that is obvious from READING THE PROOFS (an activity rarely practiced in mathematics and science).

Since last I posted here (miss Peter) I have come to the conclusion that set theory in the philosophical sense gets its dependence upon order from the fact that urelements must constitute an antichain.

To be a set is to relate through the denial of antichains. Urelements cannot relate to one another in this way.

In turn, Cohen's forcing (not in the syntactic sense through the finiteness of proofs) exposes how the "width" of a given model cannot be bound.

By contrast, the topological sense I began with leads to the introduction of a transitive, irreflexive order as a "self-defining" primitive (I no longer like to use tgat word). In turn, membrrship is introduced with a syntactic dependence on the order relation syntax.

This actually fails in the sense of my first attempt (relative to objectives).

At some point I read Czarsar's book, "The Foundations of Topology." Cohen's outer models seem to be comparable to orders within orders that can never fill their "ALL container." But, this is precisely the description of Czarsar's syntopogenous orders relative to "subset" in the usual formulations.

Although I posted the link in the other thread, the file,

https://drive.google.com/file/d/15goGcZyQuG7fAq70eBb6mYcRsJFZ_lMp/view?usp=drivesdk

reflects the change to an intensional proper part ordering "filled, but not filled up" by an extensional subset order.


Yes. The dependency of set theory is significant and has ramifications.

mitch

[Mostowski Collapse]

For identity of indiscernibles, a solution to:

∃P(~x=y & P(x) => ~P(y))

In the form of P(z) <=> x=z, might be considered impure.
I could have also written P={x}, but then I might be subject
to the accusation of using some sort of set theory?

Although the Peano unit was possibly originally conceived
as a class. But to solve this identity of indiscernibles,
no order was needed! The order is set theory is quite

an illusion. Already Prolog cyclic terms are hard to be
ordered without breaking some assumptions about an order,
that the ordering of acylic terms easily satisfy.

[Mitchell Smith]

I have no problem with how other logics implement "difference principles," Jan. The documents in my post on "potential infinity" have a fully specified set of non-propositional inference rules to implement a logic whose descriptions behave in the way Skolem explained in his paper on arithmetic.

No one reads. Skolem makes specific statements about descriptions ignored in the literature because of the influence of Hilbert and Goedel.

And, the same is true of Aristotle.

And, the same is true of Abraham Robinson.

And, the same is true of Karel Lambert.

And, the same is true of Dana Scott.

And, the same is true of David Lewis.

You are good at logic. Look at the derivations. Don't worry about "meanings."
The syntax presents opportunity for specific discussion. Rhetorical ninsense about what is or what is not mathematics is useless.

[Mostowski Collapse]

But identity of indiscernibles has some funny
solutions if ordering is available. Because ~x=y
implies x < y or y < x for a total order.

So if the order is dense we might further have
some t such that x < t < y or y < t < x. And then
a discernible P might be a for example

P(z) <=> z<t. Unlike the Peano unit, the construction
depends on both x and y. Is it pure?

[Mitchell Smith]

On Monday, March 27, 2023 at 8:42:46 PM UTC-5, Mostowski Collapse wrote:
> But identity of indiscernibles has some funny
> solutions if ordering is available. Because ~x=y
> implies x < y or y < x for a total order.
>
> So if the order is dense we might further have
> some t such that x < t < y or y < t < x. And then
> a discernible P might be a for example
>

> P(z) <=> z<t. Unlike the Peano unit, the con2struction

> depends on both x and y. Is it pure?
> Mostowski Collapse schrieb am Dienstag, 28. März 2023 um 03:37:46 UTC+2:
> > For identity of indiscernibles, a solution to:
> >
> > ∃P(~x=y & P(x) => ~P(y))
> >
> > In the form of P(z) <=> x=z, might be considered impure.
> > I could have also written P={x}, but then I might be subject
> > to the accusation of using some sort of set theory?
> >
> > Although the Peano unit was possibly originally conceived
> > as a class. But to solve this identity of indiscernibles,
> > no order was needed! The order is set theory is quite
> >
> > an illusion. Already Prolog cyclic terms are hard to be
> > ordered without breaking some assumptions about an order,
> > that the ordering of acylic terms easily satisfy.

So, the logic which I have attempted to develop bears relation to negative free logic. That logic includes a principle of indiscernibility of nonexistents. Relative to that, the expression,

~x=y

could mean 2 or nonexistence for one of the terms. However, the inference rules are structured to never admit a description for a nonexistent (I think). So, while that expression might be discussed informally with respect to a nonexistent argument, it should actually never appear in the formalism.

Negative free logic is generally discussed as if it ought to be able to "substantiate" reference to nonexistent objects. But, my logic is better understood with respect to Markov's constructivism. His language over 1 symbol generates WM's monotone inclusive sequences,

|
||
|||
etc.

To retain classical bivalence, Markov introduced a strengthened implication interpreted with respect to givenness. This is one way how my universal quantifier may be read: if x is given.... But, reflexive equalities are used in this reading: if x=x is given....

The point is that comparing what I have tried to do with the vast array of logics that take the existence of terms for granted is comparing apples with oranges.

It is not that I am uninterested in other logics. Nor is it that I am discounting them. Rather, their presuppositions are inappropriate for what I have been trying to do.

For the record, my computer has a number of your postings from several years ago because of your knowledge on these matters.

[Ross Finlayson]

We've had some talks about quantifier disambiguation in the "impredicative", about
for all
for any
for every
for each
reflecting universal quantifications for various types of domains that admit predicativity
for various predicates for the universal quantifier(s).

The "urelements" reflect that there's a one theory that has all the elements if there is
going to be any fundamental theory, whether sets, ordinals, parts, fractions, numbers,
objects of geometry, for function theory, for operator theory, ....

The "primary ur-element" of a sort still arises from a monadology.

A proof is only its context, its definitions, its derivations, that surely the innards usually
reflect an opinion and a course, about that when there are things like "at least three
replete models of completeness for real numbers: line-reals field-reals signal-reals",
that then it results a space of "real-valued" formulas.

For what's really "fundamental" in the logical objects and the mathematical objects besides,
is one theory for all, ....

[Ross Finlayson]

There are lots of schemes what admit being topologies. I'm mostly interested in the
most simple and natural topologies. Then, for the "quasi-invariant" which is a term from
measure theory and Ramsey theory and the "symmetry-flex" which is term I coined to
reflect angling in a sense, is for a natural topological setting as a natural geometric setting,
where in terms of the algebraic and geometric, they often start with different contexts
where geometry has a more plain context and because there are non-Cartesian functions
and the Cartesian space of free analysis for algebra doesn't always result existing, that,
my line-reals and field-reals and signal-reals are three sorts of "geometry's real numbers".



About tautologies and intensionality and extensionality and discirnibles and equatables
and indiscirnibles the impredicative, I like the idea of opposites with the respect to
"the opposite is a particular different that the opposite of the opposite is not different",
with respect to always having an OBJECTIVE space where both systems are always so,
because, it reflects that otherwise they would alternate and neither be so.


Then it just results that the void and universal work out "Janus' introspection", where Janus
is a two-faced person who looks two different ways, whether his introspection results self-knowledge.

[Ross Finlayson]

Here's some more reading from Maugin, and looking at Leitch and Da Vinci.


https://www.youtube.com/watch?v=xIcZim7Y53U


Longest complete run-on sentence ever: "... we are here".

[Mostowski Collapse]

Here is a formal proof of Matt Carlsons counter example.
Wiithout loss of generality its enough to handle the case
Lab, although Matt Carlson mentioned both proofs:

/* Lexical Ordering */
∀x∀y∀z∀t(Ls(x,y)s(z,t) ↔ (Lxz ∨ (x=z ∧ Lyt))),

/* Example A,B satisfies A @< B and ~(B @< A) */
Lab, ¬Lba,

/* Example A,B is A=s(B,c) and B=s(A,d), proof works for any c,d */
a=s(b,c), b=s(a,d)

/* We can derive a Contradiction */
entails p∧¬p.
https://www.umsu.de/trees/#~6x~6y~6z~6t(Ls(x,y)s(z,t)~4Lxz~2(x=z~1Lyt)),Lab,~3Lba,a=s(b,c),b=s(a,d)%7C=p~1~3p

The inconsistency means unless hell freezes over,
we can never find a less than relationship L.

[Mitchell Smith]

Jan,

Without specifics of what you are formalizing, I am uncertain of what you are conveying here.

I looked up Matt Carlson. He seems to be a somewhat junior faculty member at Wabash College. So, I am not certain of the counterexample to which you are referring.

[Mitchell Smith]

So, you are doing youtube readings now?

I listened to the first one. I'll try this one when I get more time.

[Mostowski Collapse]

The context is here:

Mats Carlsson schrieb am Dienstag, 16. Juli 1996 um 09:00:00 UTC+2:

comparing infinite terms

In the '80s, Alain Comerauer, Joxan Jaffar and others published
algorithms for unification without occurs check that could work
with infinite (cyclic) terms with termination guaranteed.
Until yesterday, I always believed that the standard total order for
finite Prolog terms readily extended to include infinite terms as
well. However, the following example shows two terms that can't be
ordered.

Consider the terms A and B defined by the equations

[1] A = s(B,0).
[2] B = s(A,1).

Clearly, A and B are not identical, so is A @< B or A @> B?

Assume A @< B. But then s(A,1) @< s(B,0) i.e. B @< A. Contradiction.

Assume A @> B. But then s(A,1) @> s(B,0) i.e. B @> A. Contradiction.

So the standard order of Prolog terms cannot include (some) infinite
terms. On reflection, the same is true for the integers---infinity
minus infinity is not defined.

Perhaps this is all obvious, but I was a bit jarred at first.

Mats Carlsson, PhD
Computing Science Department tel: +46 18 187691
Uppsala University fax: +46 18 511925
P.O. Box 311 Email: m...@csd.uu.se
S-751 05 Uppsala, Sweden

https://groups.google.com/g/comp.lang.prolog/c/Om8bTZ_Mom4

[Ross Finlayson]

It's an outlet, basically as "simulacrums" keep getting more sophisticated,
it's just adding a few bars to mocking the content.

Today's was setting up Weyl's "The Continuum", basically establishing some
of the principles of quantum theory and an overview of standard real analysis
and "foundations today", setting up to enumerate properties of "sweep"
thus introducing line-reals, for field-reals, and signal-reals, "foundations,
less and more".

Then if you'll excuse me, that you get back to
"unity of opposites" and about making equals to opposites
thus establishing in the space "between" them the continuum
of their context, is that I interpret your posts in this thread as
establishing space terms as it were for usual "arithmetizations"
and usual "geometrizations".

[Ross Finlayson]

(
"

Here I read a few passages from Weyl's "The Continuum", talk about it an hour,
explain standard real analysis.

It starts with a description of quantum theory.

https://www.youtube.com/watch?v=sRr1gBLEmo0


You know, you might rather wonder how Zermelo-Franekel made it
so that von Neumann's ordinals don't make Russell make them contain
themself/themselves.

I.e., where the "sets that don't contain themselves" are only the integers,
then, don't they? Contain themselves not containing themselves?

This is where "Axiom of Infinity" is the only axiom both
"expansion of comprehension" and "restriction of comprehension".

People think that a Cartesian space of ordered pairs or "the
most many-many function modeled as a set" is "the" definition
of function, each subset a function, and those all functions,
is without restriction in function theory in set theory, but, it's not so.

Here then "ubiquitous ordinals": is what happens when numbers are in sets.

"
)

[Mostowski Collapse]

Here is a proposal to get out of the dilemma of a compare
for cyclic terms. But we would drop the lexical ordering
requirement. But otherwise we could retain all other laws,

if we find an injective representation to acyclic terms,
and would define:

Here is a proof that connectedness is preserved:

S @<' T :<=> rep(S) @< rep(T)

/* I am using r for the function rep,
R for the relation @<' and L for the relation @< */
/* Injectivity */
∀x∀y(r(x)=r(y) → x=y),

/* Bootstrapping */
∀x∀y(Rxy ↔ Lr(x)r(y)),

/* Connectedness */
∀x∀y(¬x=y → (Lyx ∨ Lxy))

/* Connectedness */
entails ∀x∀y(¬x=y → Ryx ∨ Rxy)

https://www.umsu.de/trees/#~6x~6y(r(x)=r(y)~5x=y),~6x~6y(Rxy~4Lr(x)r(y)),~6x~6y(~3x=y~5Lyx~2Lxy)%7C=~6x~6y(~3x=y~5Ryx~2Rxy)

[Mitchell Smith]

You prefer posting on top.

Thanks for this. I do know that questions involving equality and substitutivity for logic programming fall under the term "unification." And, I did spend some time looking at that after reading some of the thread to which you linked.

I am not entirely certain about what you may have read from my post or work, but the syntax of the cyclic terms does reflect my usage.

I do not use the expression "definition" because it has been appropriated by a community who believes their view of logic to have been divinely received at a burning bush.

What I refer to as "symbol introductions" do include statements formulated with a circular syntax when the introduced symbol is a "non-eliminable" relation.

When only a single relation is involved -- for example,

AxAy( x=y <-> Ez( x=z /\ z=y ))

-- what you really have in relation to substitutions in the right hand side is an infinite scheme and a free algebra with infinitely many terms. Unless one fools one's self into making this infinity semantically important, it is harmless -- deflationary.

A slightly more interesting case is when there is a syntactic dependency:

AxAy( x ppart y <-> ( Az( y ppart z -> x ppart z ) /\ Ez( x ppart z /\ ~( y ppart z ) ) ))

AxAy( x in y <-> ( Az( y ppart z -> x in z ) /\ Ez( x in z /\ ~( y ppart z ) ) ))

One can imagine greater complexity, but, it still appears to be harmlesss (or deflationary) Whatever interpretation is given to the relations is unchanged by expanding the right hand sides with substitutions.

The issue here is that relations are a matter of "form" as opposed to "substance." This is why the analysis cannot lead to a classical calculus.

And, in so far as the first statement above is from Tarski's cylindric algebra, departure from a classical calculus is related to algebraization. Appendix C from Blok and Pigozzi is clear about how algebraization necessitates a change to inference rules.

There is a sequent in homotopy type theory,

C; a
----------
C; a=a

My work divides this into two paradigms. Above the line is "objectual." Below the line is "relational." The latter, then, becomes an aspect of "differential ontology."

Related to this is the truth preservation of universals for submodels and the truth preservation of existentials for extensions from first-order model theory.

For the existentials, the symbol introduction for equality is accomplished with Tarski's axiom from above.

For the universals, one has,

AxAy( x=y <-> ( Az~( ~x=z <-> ( x=z /\ z=y ) ) /\ Az~( ~y=z <-> ( y=z /\ x=z ) ) ))

Although I have not checked this with derivations, this ought to be compatible with the "objectual ontology" of first-order inference. This merely ensures that substitutivity is compatible with transitivity for an otherwise unspecified domain of discourse.

Relating equality to a universal in this way admits the possibility of distinguishing between formalism and logicism.

Let a discernibility relation be introduced with

AxAy( x!=y <-> ( y!=y /\ (( y!=y -> y!=x ) /\ ( y!=x -> x!= y )) ))

Because their are no nested quantifiers, the exclusive disjunction,

AxAy~( x=y <-> x!=y )

yields an equality that is interpretable for a first-order formalist ontology. The "self-identity" is derived from the syntax of the discernibility relation.

Let a distinctness relation be introduced with

AxAy( x=/=y <-> ( Az~( ~x=z <-> ( x=z /\ z=/=y ) ) \/ Az~( ~y=z <-> ( y=z /\ x=/=z ) ) ))

Notice that a denied existence will convey existential import. So, the exclusive disjunction,

AxAy~( x=y <-> x=/=y )

yields an equality interpretable as a metaphysical law of identity.

In any case, there are many places in ordinary mathematics where transitivity serves to "glue" structures together. This informal term -- namely, "gluing" -- is found in topology.

I think, however, that it also reflects much of what I read about "unification."

The thread to which you linked also made a comparison with "undefined" situations.

Mitchell Spector's contribution to the October 2015 thread on free logic,

https://cs.nyu.edu/pipermail/fom/2015-October/019285.html

is a simple statement suggesting that the sign of equality can provide the means for sorting through these issues.

Again, though, thanks for the link to the thread. I know very little about Prolog and its methods.

[Mostowski Collapse]

You need to read past the reference to Colmerauer und Jaffar.
The post by Matt Carlsson is about a precendence relation
not about unification. Prolog has things like (==)/2 and (@<)/2.

Whereas (==)/2 is relatively tame for cyclic terms, the (@<)/2
gets a little bit more nasty for cyclic terms. In particular
Matt Carlsson shows a counter example

to lexical ordering of cyclic terms, this doesn't mean they
cannot have a total order, but it can be a little annoying.

[Ross Finlayson]

I enjoy Mitch, he details some
much-involved questions about the
wider relevance, of logic and such.

Also he noticed interesting points of others,
that generally fly right over their head.

[Ross Finlayson]

The stronger Mitch's stronger mathematical developments are very much
appreciating on sci.logic.

His results make "collected coat-tailers of the TL/DR set" lose their
weight, the results have a nice heft.