a geometric metamathematics

[Mitchell Smith]

It was around 2003 --- maybe even 2002 --- when I first posted to sci.math and sci.logic. I had questions for which I sought answers or guidance. Instead, I recieved useless lectures and criticism.

Shortly after I started posting, I found a paper showing that propositional logic is not categorical. The paper is at the link,

https://arxiv.org/abs/quant-ph/9906101#

The result is algebraic, whence symbolic logicians might take issue with it. But, in 2006, Eric Schechter formulated a syntactic interpretation called the "hexagon interpretation."

Now, orthomodular lattice theory had arisen through the work of Birkhoff and von Neumann when attempting to formulate a quantum logic. Quantum mechanics can be characterized as a synthesis of group theory and orthogonality of dimension for the present context.

It turns out that every Boolean lattice is an orthomodular lattice. The group-theoretic correlation lies with reflection groups. Geometrically, admitting high-dimension Euclidean geometry, the geometric correlation lies with hypercubes.

The free Boolean lattice on two generators is order-isomorphic with a 4-dimensional orthomodular lattice. Its hypercube correlate is the tesseract.

Whereas the term "metamathematics" has come to have a vague meaning, its origin (with respect to the Hilbert school) lies with the visual impression of symbols on a page.

A more modern text, namely Kreisel's, affirms this view with respect to what he calls combinatorial foundations (to be contrasted with semantic foundations).

Relative to the Hilbert school, this perspective leads to formation rules based upon structural recursion and proof theory.

My concerns about claims for logic as a foundation for mathematics or claims that mathematics is merely logic led me to study the forms of truth tables from an extremely abstract point of view.

The file at the link,

https://drive.google.com/file/d/1BiMoZd9cdEWwFR5PP8F7JF_94UMLJk-h/view?usp=drivesdk

is a scan of a handwritten note organizing column vectors of horizontal and vertical lines into classes based upon the operations of "symbol exchange" and "read-order reversal."

As this clearly involves the sensible impression of symbols on a page, I view the analysis arising from this starting point as "metamathematics" even though it is not directed at the spellings of expressions in a formal language or recursions over the syntactic structure of formulas.

As it turns out, these "sensible operations" from my metamathematics are arithmetized in a non-standard logic called "trilattice logic." As can be found in the paper at the link,

https://philpapers.org/archive/MUSIIV.pdf

trilattice logic is characterized by three order relations simultaneously imposed upon tesseract projections. There is a truth order, a falsity order, and an information order. It is the information order described with Boolean-valued components that bears relationship to my metamathematical operations.

Although I had to suspend my writing at a certain point because of combinatorial complexity which I had hoped to avoid, the file at the link,

https://drive.google.com/file/d/15087p1Io6K8xOZO2NUFzv_m7UAks1rTq/view?usp=drivesdk

is a development which portrays dichotomy in the sense of oppositely directed tetrahedra as quasigroups over four symbols (there are only two non-isomorphic representations of such.

By representing the borders within each cell of the quasigroups, the two quasigroups become arrays of ordered triples. This transformation can be found at the bottom of page 9 and the top of page 10.

In turn, these ordered triples can be put into relation with one another as "squares of opposition." This gets discussed in section 2.4 on about page 38.

A comprehensive comparison of notations (arrows, mappings, and 4x4 arrays) can be found on pages 47-49.

Naturally, trying to explain everything in this document is beyond the scope of this post. But, the appendices begin on page 96.

Scrolling through the appendices would provide some idea of what it might take to obtain truth table semantics from a general notion of dichotomy. The last appendix uses the XOR connective to make assignments of Boolean vectors because this is the only table from Appendix A that satisfies the definition for affine spaces or Euclidean point spaces stated on page 85.

For what this is worth, I am not responsible for any of this complexity. You can attribute it to the two centuries of jackasses at universities that most of you have idolized as intellectual heroes.

In any case, while the second document is too onerous to actually read in full, I hope some participants in this forum might find interest in excerpts they might examine.

For my part, I have come to believe that this complexity is related to Hilbert's proof of uniqueness for the Cartesian plane from "Foundations of Geometry." My reasons for this lie, in part, with 16 configurations from modern physics,

https://www.researchgate.net/figure/Sixteen-vertex-configurations-and-their-statistical-weights-oi-exp-bi-The-first_fig1_235633908

and from the fact that Hilbert's proof is a proof by contradiction. Such proofs, of course, are always questioned with regard to the fact that they are less informative than constructive methods.

Section 2.2 contains the deliberation on the use of "canonical symbols" reflecting spatial orientation. This is what I believe may be compared 16 vertex configurations --- not the physics, of course.

mitch

[Julio Di Egidio]

On Thursday, 6 April 2023 at 02:13:01 UTC+2, Mitchell Smith wrote:

> Whereas the term "metamathematics" has come to have
> a vague meaning, its origin (with respect to the Hilbert
> school) lies with the visual impression of symbols on a page.

What a pile of absolute crap. You may not realize it, but you are
just yet another shameless liar and polluter of ponds. Indeed,
just shit on that page, that's gonna be the apotheosis of your
meaningfulness.

Julio

[Mitchell Smith]

Fortunately, there are translations of Hilbert and books by Kreisel that verify the actuality of my remark as a matter of what is available in print independent of my existence and your opinion.

mitch

[Julio Di Egidio]

On Thursday, 6 April 2023 at 12:59:24 UTC+2, Mitchell Smith wrote:
> On Thursday, April 6, 2023 at 5:29:21 AM UTC-5, Julio Di Egidio wrote:
> > On Thursday, 6 April 2023 at 02:13:01 UTC+2, Mitchell Smith wrote:
> >
> > > Whereas the term "metamathematics" has come to have
> > > a vague meaning, its origin (with respect to the Hilbert
> > > school) lies with the visual impression of symbols on a page.
> > What a pile of absolute crap. You may not realize it, but you are
> > just yet another shameless liar and polluter of ponds. Indeed,
> > just shit on that page, that's gonna be the apotheosis of your
> > meaningfulness.
>

> Fortunately, there are translations of Hilbert and books by Kreisel
> that verify the actuality of my remark as a matter of what is available
> in print independent of my existence and your opinion.

But you are the one buying into it and only into it. Liar.

Julio

[Graham Cooper]

hi Mitch

great to see you back!

i hope you get a better go with the mediocre minds in sci.logic this time!

I had a quick glance at your 187 pages of logic lattices.

I will only post a comment because I doubt the usenet logic community will grasp much themselves.

I think your only seeing HALF the picture.

How does a lattice framework of logic primitives SOLVE a general logic enquiry ..?

POINT 1 - only 2 logic primitives are needed - which simplifies logic lattice to a binary tree.

POINT 2 - logic uses terms and VARIABLES

usually VARS are upper case and terms lower case - DC Proof has TERMS upper case and VARS lower case.


Logic is actually depicted with HORN CLAUSES

a standard definition uses equivalence

DEFININGTERM ARG ARG
<->
CLAUSE 1 ARG ARG
CLAUSE 2 ARG ARG

==================

However this is too strong for logical solving - so HORN CLAUSES use

DEFININGTERM ARG ARG
<-
CLAUSE 1 ARG ARG
CLAUSE 2 ARG ARG

===================

You dont appear to have a lattice for HORN CLAUSES

Using horn clauses multiple "definitions" are possible

or A B
<-
A

or A B
<-
B

==================

UNIFY ( f A B ) ( f C D )

is simply a method of comparing 2 strings and making them compatible

UNIFY ( cat X mat ) ( HIGH on LOW )

X = on

==================

When given a GENERAL FORMULA

PROLOG solves it by applying ALL POSSIBLE FUNCTIONS

-----------------------------------

So using a simple recursive UNIFY FUNCTION
with just *terms* and *VARIABLES* in STRINGS

a DATABASE of HORN CLAUSES is used to MATCH
to any given query
(or returns NO)



a simple PROLOG function

niece P N
<-
brother P B
daughter B N



This is ALL THERE IS TO LOGIC

All that is needed is
IF
ELSE
ASSERT
REMOVE
PLUS
TIMES
EQUAL

and a dozen other functions to form a computer language.

THEN you have to PROGRAM the LOGIC SOLVER to do pure logic.

I think PROLOG will make a resurgence when LOGIC SOLVERS do general maths
and you type in ANYTHING mathematical and they do peoples homework (maybe 10 years away)

BTW what do you think of CHAT-GPT! should sort out the morons for you pretty quick!

hope you hang around this time and just ignore the few bad eggs plenty of good folk still

[Mitchell Smith]

Hi Graham,

Your observations are entirely correct with respect to POINT 1 and POINT 2.Nothing in that note translates into using logic to solve problems.

I am not responsible for the views of others who opine that mathematics reduces to logic. So, for me, "logic" is THE PROBLEM rather than a means to solving problems.

Thirty-five years ago I thought of "logic" and "mathematics" as univocal and well-construed topics of study. So, when the dot-com debacle left me with no information technology career and a plentitude of free time, I found the NNTP Usenet groups. It had been here --- on these groups --- where I learned that many people learn mathematics in philosophy departments. And, the view of mathematics as taught in philosophy departments is very different from the standard curriculum of mathematics departments training students for applications.

Problematically, others --- most notably, Mr. Greene --- perceived my work as a rejection of "received views." What had, in fact, been the case is that my training had been inadequate. However, the fact that the literature is replete with challenges to classical methods does not mean that learning the rationale behind the first-order paradigm resolves my questions. All that had been accomplished through that effort had been the realization that mathematics is studied in different ways.

As I have often stated, my interest in logic arose because of the continuum hypothesis. That question may only be meaningful with respect to one particular paradigm. And, it may be resolved for that paradigm. But, Goedel assumes consistency in order to formulate the constructible universe. Is this consistency assumption a "truth" which must be accepted? And, Cohen's forcing only works if a given model is *assumed* to be partial. Is Cohen's assumption a "truth" which must be accepted?

I have been time-constrained since posting that research note. There are a few more pieces of my puzzle yet to come. The particular issue I attempt in the posted has to do with the fact that all of this debate in the foundations of mathematics has largely made "truth" and "not truth" meaningless. It is merely a dichotomy within language that is not intrinsically different from any other dichotomy.

The connection to "mathematical logic" in so far as that expression refers to truth table semantics appears to involve the representation of oppositely directed tetrahedra encoded into a pair of 4-symbol quasigroups described with 4x4 arrays of cells.

In saying this, note that my concerns over the last few years has shifted from the continuum hypothesis to "the unity of mathematics."

When I started 35 years ago, information technology had not been ubiquitous, category theory had not become a mainstream method for teaching algebra, and, there simply was no homotopy type theory implementing Martin-Lof's work. In one form or another I can relate most of these developments to what I have been doing because I began with the idea that "individuation" could be correlated with Cantor's nested set theorem for closed sets of vanishing diameter. This, of course, relates to the principle of the identity of indiscernibles which is denied as a logical principle in the first-order paradigm. So, I inadvertently stumbled into the very questions upon which these competing paradigms cannot agree.

Whoops! Signs of equality are ubiquitous!

Anyway, Graham, its good to see you are still at it. Stay tuned for a little more. As for ChatGPT, I applaud it, although I question those who believe that artificial intelligence is equated with human intelligence. Such individuals will use rhetoric to demand that I show how their "in principle" arguments are in error. The issue is simply that "in principle" arguments are not fully specified models. The burden of proof will always (and eternally) be on them to produce a fully specified model. Not only is neuroscience nowhere near such a definite state, it is likely that a fully specified model will involve non-eliminable infinities or circularities (which is why I used the expression "eternally.")

I suppose that reflects my current opinions on "truth" and "not truth." For logic to be useful, debate must begin with respect to a common belief. Debate over artificial intelligence introduces irreconcilable beliefs.

mitch

[Julio Di Egidio]

On Sunday, 9 April 2023 at 17:07:22 UTC+2, Mitchell Smith wrote:

> So, for me, "logic" is THE PROBLEM rather than a means to solving problems.

LOL.

> I suppose that reflects my current opinions on "truth" and "not truth."
> For logic to be useful, debate must begin with respect to a common
> belief. Debate over artificial intelligence introduces irreconcilable beliefs.

False, debate starts from agreed stipulations;
and false, OpenAI is open source (isn't it?) and so
is most of the underlying research (at least so far),
indeed it's based on known mathematical models
and methods.

Julio

[Julio Di Egidio]

On Sunday, 9 April 2023 at 17:07:22 UTC+2, Mitchell Smith wrote:

> On Thursday, April 6, 2023 at 10:57:39 AM UTC-5, Graham Cooper wrote:

> > I think your only seeing HALF the picture.

> Whoops! Signs of equality are ubiquitous!

The other half of the picture is the role of
"(function) application" ("arrows"), aka the
ubiquity of Modus Ponens.

Julio

[Ross Finlayson]

Irreconcilable beliefs, or faiths? Here is for a distinction between "belief" and "faith",
where faiths are conscientious suspensions of what would be dis-belief.

About the geometric, I scanned the paper about TRILATTICE logics, then tried
to fit that into my understanding or codified belief, "universal logic" then for
that I'm wondering how to explain what are the commonalities of calculi,
that "the logic" is a "an universal logic" and "a logic" is a calculus or notation.

Then, where logics are just systems for tableau mostly and connectives and
rules, making for a sequent (sequenced) operation or for example the higher
order where properties arises in the parallels or what may be detached or embedded,
"parallel", then I'm wondering what are the commonalities of logics the calculi,
for having a "universal calculus of logics" in terms.

Then thanks for pointing out Hilbert's treatise, I'll be reading it.

I'm a bit more interested in the "table of 16" or about running out conjunctives,
and wondering if there's a brief note that defines for all practical purposes,
what are conjunctives and connectives, and rules or judgments and strokes,
what are then for spaces with forms like "a graph has an adjacency matrix" and
"a graph has other matrices that represent it" and "a graph has other forms in
geometry that represent it" and "a graph has graphs that represent it", about
the Cartesian I suppose, to get De Morgan and Des Cartes together, figuring
that such a "table of common logic" should be about as standard as the old
etched bones of diviners about the fire-light.

I have "logic" the entire apparatus rather separate from calculi.

[Mitchell Smith]

Exactly, Julio.

But, when Russell "grounds" logicism on consequence via the material implication, the law of excluded middle is described using a negation and an inclusuve disjunction. One may, however, understand mutual exclusion relative to the exclusive disjunction.

Relative to the kind of geometric sensibility behind artificial intelligence, exclusive disjunction is fundamentally different --- it is not a linearly separable switching function.

This distinction is obscured when negation is combined with a linearly separable switching function like inclusive disjunction.

Gotta go... work is starting.

mitch

[Ross Finlayson]

The other day I was reminded of the Moebius strip, that
the cross product is closed only in 3 and 7 dimensions,
and there are no knots in 4+ dimensions.

Whether the hairy ball theorem has a pigtail instead
of a cowlick, is an example of the de-generate for the
de-tailed, that the de-constructed makes for the assembly
of "false primitives", what results for example an arithmetic
defined as addition and division, that decompose variously,
then for "line-continuity sweep, field-continuity spray,
and the signal-continuity slide".

These days I'm looking at a 1-dimension case of Jordan measure
as line-continuity of iota-values, and I have a sigma algebra for it,
and for signal-continuity there is a usual milieu of Dirichlet function.

However, in this context, when there's "algebra versus geometry",
where that for example was detailed differences in trhe Vitali-Hausdorff
"geometry's view" of the Banach-Tarski versus the "algebraist's view",
is for getting them back together where for example there exist functions
that aren't Cartesian, what are functions.

Then, AND and OR and XOR and NOR trees, or "the exhaustion of Boolean
operators", is for both when a) out-of-bounds inputs and truthiness/falsiness,
then also about b) out-of-bounds predicates and the transfer principle.
The, inner and outer products, for where products are closed and their
inner and outer products are variously not closed, and where the inner and outer
products reflect wider or narrower and closure in the de-generate or closure in
the sum or products the space terms, where they go or what they fill or
how they're closed, is for a geometric view, and an algebraist's view,
where the "geometric" is not the "geometer's" as it's more objective.

[Julio Di Egidio]

On Monday, 10 April 2023 at 14:08:41 UTC+2, Mitchell Smith wrote:
> On Sunday, April 9, 2023 at 10:53:49 AM UTC-5, Julio Di Egidio wrote:
> > On Sunday, 9 April 2023 at 17:07:22 UTC+2, Mitchell Smith wrote:
> > > On Thursday, April 6, 2023 at 10:57:39 AM UTC-5, Graham Cooper wrote:
> >
> > > > I think your only seeing HALF the picture.
> > >
> > > Whoops! Signs of equality are ubiquitous!
> >
> > The other half of the picture is the role of
> > "(function) application" ("arrows"), aka the
> > ubiquity of Modus Ponens.
>

> Exactly, Julio.

Great, at least one answer we have managed to give.

> But, when Russell "grounds" logicism on
> consequence via the material implication,

Who cares: as Burse said, it's Easter 2023... so, first you
should state clearly what you are talking about, the limits
of it, otherwise it becomes a lie; secondly, I still cannot
understand why you cannot be interested in anything but
that. Have you really exhausted any interest for further
research/any research?

> Relative to the kind of geometric sensibility behind
> artificial intelligence, exclusive disjunction is fundamentally
> different --- it is not a linearly separable switching function.

Utter nonsense, honestly. Here is a very quick intro: the AI
you are seeing is actually (almost only) plain ML (Machine
Learning), which in turn is application of statistical methods
together with the magic (still to be really understood, AFAIK)
of non linear transformations... Period, pretty much (and I
am not saying it is little thing), then they have realized that
that still doesn't cut it (attention is not enough), so now they
are combining into it rules for logical reasoning, mainly at
the input/output stages AFAIK, which does improve the
effectiveness dramatically, though overall it isn't then much
different from an expert system at that point, is it, just on
steroids and with the whole Internet (hopefully, almost, so
far) at hand...

Nothing substantial to do with logic proper except for what
is *not* ML and is indeed logic proper... and, indeed, as for
what I am doing with Coq (thank you) and with the game
theoretic ramifications (as a way to say it and for comparison)
is, in my dreams at least (but I have been working on this
problem for 30+ years), an implementation of the Characteristica
Universalis of Leibniz...

And if that doesn't sound interesting to you, a least "theoretically",
I really don't see why you are here as opposed to sci.math (with
all due respect where is due).

Julio

[Mitchell Smith]

So, I could not get to this before.

Hilbert's "On the infinite" is translated in van Heijenoort. What Tarski refers to as Hilbert's metamathematical numerals are equivalent to WM's monotone inclusive sequences. Hilbert's paper contains a paragraph bracketed with the statements'

"Let us call to mind the nature and methods of ordinary finitary number theory. [...] Let us now consider number theory in detail."

The next paragraph begins with:

"In number theory we have the numerals

|, ||, |||, ||||, ...

each numeral being perceptually recognizable by the fact that in it | is always again followed by | (if it is followed by anything). These numerals, which are the object of our consideration, have no meaning at all in themselves. In elementary number theory, however, we already require, besides these signs, others that mean something and convey information, for example, the sign '2' as an abbreviation for the numeral ||, ..."

That passage only makes reference to "perceptually recognizable."

Ewald has translated Hilbert's "The new grounding of mathematics." In this paper he writes:

".... If logical inference is to be certain, then these objects must be capable of being completely surveyed in all their parts, and their presentation, their difference, their succession (like the objects themselves) must exist for us immediately, intuitively, as something that cannot be reduced to something else. Because I take this standpoint, the objects of number theory are for me --- in direct contrast to Dedekind and Frege --- the signs themselves, whose shape can be generally and certainly recognized by us --- independently of space and time, of the special conditions of the production of the sign, and of insignificant differences in the finished product.The solid philosophical attitude that I think iz required for the grounding of pure mathematics --- as well as for all scientific thought, understanding, and communication --- is this: In the beginning was the sign."

So, this passage speaks of recognizanle shapes.

In Kreisel's book "Elements of Mathematical Logic," there is an appendix on "combinatorial foundations" which reflects the modern form of Hilbert's metamathematics. His section 0 begins with the statement:

"The objects with which this kind of reasoning is concerned, and whence it takes its name, are finite combinations of concrete objects such as letters of an alphabet, numerals, symbols of a formal language, etc."

A few paragraphs later he writes:

"In (combinatorial) mathematical practice, the act of recognizing that two expressions are identical is accepted as part of the data without gurther analysis.Such an analysis is needed here since the importance of combinatorial reasoning for foundations depends precisely on the particular nature of these acts, which are on par with the simplest sense perceptions, the objects being conceived as finite spatio-temporal confugurations."

It is true that Kriesel goes on to say that the only objects with a place in his theory are proofs. This begs the question of "Why?" Thanks to the advent of information technology, a great deal of mathematics relegated to "recreational mathematics" has become important.

All I have done is to "encode" certain arrangements into partitions based on symnol exchange ane read-order reversal. These are att least as "apparent" as succession between Hilbert's metamatjematical numerals.

In any case, my statement is supported in the literature.

mitch

[Julio Di Egidio]

On Tuesday, 11 April 2023 at 02:05:06 UTC+2, Mitchell Smith wrote:
> On Thursday, April 6, 2023 at 5:29:21 AM UTC-5, Julio Di Egidio wrote:
> > On Thursday, 6 April 2023 at 02:13:01 UTC+2, Mitchell Smith wrote:
> >
> > > Whereas the term "metamathematics" has come to have
> > > a vague meaning, its origin (with respect to the Hilbert
> > > school) lies with the visual impression of symbols on a page.
> >
> > What a pile of absolute crap. You may not realize it, but you are
> > just yet another shameless liar and polluter of ponds. Indeed,
> > just shit on that page, that's gonna be the apotheosis of your
> > meaningfulness.
> >

> So, I could not get to this before.
>
> Hilbert's "On the infinite" is translated in van Heijenoort.

<snip>

> In any case, my statement is supported in the literature.

Back to square one: whomever the translator, you have amply
demonstrated that you don't and won't understand a single
word of it. Indeed, not only you (at best) know only half of the
picture, you also (at best) know only HALF OF THE LITERATURE.

So, you'll have to excuse me if from this point on, thanks for the
feedback, whenever you start pontificating, or, to be more bland,
casting spells, I'll simply call you names.

TL;DR Please just get the fuck out of sci.logic.

*Plonk*

Julio

[Mitchell Smith]

On Monday, April 10, 2023 at 3:35:08 PM UTC-5, Julio Di Egidio wrote:
> On Monday, 10 April 2023 at 14:08:41 UTC+2, Mitchell Smith wrote:
> > On Sunday, April 9, 2023 at 10:53:49 AM UTC-5, Julio Di Egidio wrote:
> > > On Sunday, 9 April 2023 at 17:07:22 UTC+2, Mitchell Smith wrote:
> > > > On Thursday, April 6, 2023 at 10:57:39 AM UTC-5, Graham Cooper wrote:
> > >
> > > > > I think your only seeing HALF the picture.
> > > >
> > > > Whoops! Signs of equality are ubiquitous!
> > >
> > > The other half of the picture is the role of
> > > "(function) application" ("arrows"), aka the
> > > ubiquity of Modus Ponens.
> >
> > Exactly, Julio.
>
> Great, at least one answer we have managed to give.
> > But, when Russell "grounds" logicism on
> > consequence via the material implication,
> Who cares: as Burse said, it's Easter 2023...

And, Mr. Greene used to proclaim that all mathematics prior to 1933 is irrelevant.

Am I to assume that all knowledge prior to the present is irrelevant? Using the date to declare "progress" is simply nonsense. The only rational conclusion -- other than attributing laziness and a personal preference for ignorance to certain individuals -- is that an all-knowing diety substitutes whatever store of knowledge you believe at a current moment with updated knowledge at the next current moment.

In case you do not understand this, you are human beings who have learned statements believed to be factual from other human beings. Some aspects of using natural language may be evolutionarily innate. There are philosophical views which deny this. Certain aspects of mathematics may also be innate. And, there are philosophical views that deny this. Regardless, whatever human beings discuss the most likely explanation is that it is the product of human agency and not voodoo substituting truths in human minds from moment to moment in support of nonsensical rhetoric.

If you want to beat your chest like a great ape, you are certainly entitled. But do not expect intelligent human beings to be taken in by it.

> so, first you
> should state clearly what you are talking about

Really? Section 74 of Kleene's "Introduction to Metamathematics" clearly states that the formalization of ordinary mathematics will take the forms of a sequence of statements extending signatures as is exhibited by my presentations.

You people (used with the contempt commonly attributed to the most reprehensible racism of dialogue in the United States) choose to be ignorant and lazy. That is entirely your own problem.

It is ridiculous to pretend that "mathematics is formal" if you are incapable of parsing formal statements for yourselves.

The hypocrisy is astounding. We declare pretend languages to be important, and, then make "great claims" on the basis of pretend languages.

> the limits
> of it, otherwise it becomes a lie;

And what incompetent philosophy class did you learn this accusation from? It had also been a favorite of Mr. Greene. Everyone is a liar.

You are the one throwing out unsupported "facts" which, apparently have been placed in your mind supdrnaturally.

secondly, I still cannot
> understand why you cannot be interested in anything but
> that. Have you really exhausted any interest for further
> research/any research?

You mean the fantasies and dead ends taught as mathematics in philosophy departments?

> > Relative to the kind of geometric sensibility behind
> > artificial intelligence, exclusive disjunction is fundamentally
> > different --- it is not a linearly separable switching function.
> Utter nonsense, honestly.

You are an idiot. What you go on to say below is why I refrain from using "pop science" terms like AI.

To create pretend intelligences that can "classify," the very first thing that must be studied is the difference between linearly separable and linearly inseparable switching fumctions relative to Boolean polynomials. They even talk about "solving" this problem with "hidden layers,"

https://towardsdatascience.com/how-neural-networks-solve-the-xor-problem-59763136bdd7

And, let me point out that one of the books with which I shall always have on my shelf is "Threshold Logic" by Hu,

https://archive.org/details/thresholdlogic0000szet/page/n10/mode/1up

Open a free account and learn something about the mathematics underlying neural networks.

Yes, there are also other aspects. I read books on genetic programming and "perceptrons" over 20 years ago. I've read books on information theory, including Shannon's original monograph. I understand the use of Bayesian probabilities and information-theoretic statistics.

Pretend languages, pretend realities, pretend intelligences, pretend mathematicians? Do you seriously believe that intelligent human beings should drool over these things?

I worked in information technology for a decade. Setting aside the hard problems of computation theory it was trivial mathematics employed to put food on the table and nothing more. The people who dressed up in Star Trek uniforms and fuzzy outfits to have sex did not care for me very much.

Not surprising.

> Here is a very quick intro: the AI
> you are seeing is actually (almost only) plain ML (Machine
> Learning), which in turn is application of statistical methods
> together with the magic (still to be really understood, AFAIK)
> of non linear transformations... Period, pretty much (and I
> am not saying it is little thing), then they have realized that
> that still doesn't cut it (attention is not enough), so now they
> are combining into it rules for logical reasoning, mainly at
> the input/output stages AFAIK, which does improve the
> effectiveness dramatically, though overall it isn't then much
> different from an expert system at that point, is it, just on
> steroids and with the whole Internet (hopefully, almost, so
> far) at hand...
>
> Nothing substantial to do with logic proper except for what
> is *not* ML and is indeed logic proper... and, indeed, as for
> what I am doing with Coq (thank you) and with the game
> theoretic ramifications (as a way to say it and for comparison)
> is, in my dreams at least (but I have been working on this
> problem for 30+ years), an implementation of the Characteristica
> Universalis of Leibniz...
>
> And if that doesn't sound interesting to you, a least "theoretically",
> I really don't see why you are here as opposed to sci.math (with
> all due respect where is due).

Because logicians have claimed that mathematics is merely logic.

Prove it!

mitch

[Julio Di Egidio]

On Tuesday, 11 April 2023 at 03:11:17 UTC+2, Mitchell Smith wrote:
> On Monday, April 10, 2023 at 3:35:08 PM UTC-5, Julio Di Egidio wrote:

> If you want to beat your chest like a great ape, you are certainly entitled.

Fuck you, that's simply unfair and yet another lie. OTOH,
NO-body is entitled to shit all over the place, period.
Debunk that.

> Not surprising.

That you are so utterly incompetent on the whole
line? Every day less.

> > is, in my dreams at least (but I have been working on this
> > problem for 30+ years), an implementation of the Characteristica
> > Universalis of Leibniz...
> >
> > And if that doesn't sound interesting to you, a least "theoretically",
> > I really don't see why you are here as opposed to sci.math (with
> > all due respect where is due).
>
> Because logicians have claimed that mathematics is merely logic.

Logicisians maybe, except that I have even stopped
criticizing Hilbert after learning that you are even
worse at history than you are at mathematics.

> Prove it!

What?? That you do not understand trivial facts?!
*You* are a prove of it, and you won't get it.

Julio

[Ross Finlayson]

Hey, you said perceptron, I was just thinking about perceptrons the other day.

Consider this sort of thought experiment, what if we're all and each just
automatons, i.e., what if pretty much all the accounts that post to usenet
really are sock-puppets and bots of various sorts.

Then, what is the IQ test, that establishes a natural intelligence?


I don't know much about J.d.E. beyond what's posted here, I mostly
ignore his vituperation, because, it's ubiquitous. Then, sometimes he
will make a good point, then other times his complex of skepticism
makes then that until his nose stops bleeding it's rather tender.


I'm a logicist but when I say "strong" platonism it's a monist's, establishing
that there _is_ a system of pure ideals then that in an overall true theory of truth,
with for example "conservation of truth, it's also a natural dichotomy", that
it's a _strong_ logical positivism, not the _hypocrite's_ essential _lack thereof_.

(I don't believe in atheists, and have no dis-proof of G-d. There is though that
among the burdens divinity might assign to Man, is founding science. This
then I'm happy to arrive at via "axiomless natural deduction" or reflections on
dichotomy. Then I can say it's not my fault nor my function nor my faith,
though that I can fully believe it and you can, too, which justifies a faith
in its function without fault. Then, insofar as it's perfect, it's not wrong.)


Anyways Mitch you're warmly appreciated because when there's talk about
ontological commitment, and, the briefest figleaf thereof, is for this sort
of "theory of truth" that happens to result a "mathematical universe hypothesis"
and a science of it, and a natural philosophy, the theory does all the work then
that what remains for ontological commitment can be taken or left.

So, I suppose what I'm most reminded about is the talk about ontological commitment,
then that with regards to the objects of reason the objects of discourse, then is for
how that arrives at some "holgram of 3-D space from one continuum" and "a ray of
time from 1 continuum" and "a dichotomous projection of space and a ray of time,
together, resulting from 1 continuum", a continuum of numerical resources and
logical resources what are "the metal and concrete numbers" of a mathematical universe
hypothesis' the field occupation numbers which envalue the field of the space, and just
so how easily that simple principles result the entire thing at least as an exercise in
the utter fruiting of a richness of the cosmology, a cosmological clockworks including
a clock hypothesis, from the "utterly principled humility of reflections on principles
as universal gift".

Then, I've arrived at that, "foundations", can be pretty simple, then of course for
that things like complexity theory and emergence and convergence in phenomena
are still utterly more rich than my tools make tractable, there's still means or as
of "the lightning bolt from the center of the universe, straight to the head".

Mostly there's that continuum mechanics is enriched with _at least three definitions
of continuity_, the line-continuity field-continuity signal-continuity, mathematics owes
physics these for lots of things in real effect.

Then, the models of small machines the cellular automata, here the boolean and the
lattices and the crystallography of the lattices and the order and disorder, these most
interest me in their domain, the true and false where 0 and 1 are indicators of the ergodic
in a Dirac positronic sea or the example of the restless universe, then the true and false of
the deterministic in large systems of predicates, what result for the applied the implementation
of large solvers, and, the implementations of solvers in the large.

I'm not much of an electrical engineer and my experience in the fabless semiconductor world
was mostly implementing bit- and clock-accurate simulators of small working units involved
in a usual data pipeline of image compression, but study around the field also exposed me to
the, "free-form 3-D IC", that beyond "standard logic the semiconductor templates that fill space
in the design of fab-etched gate arrays", there's also custom logic, then about what software
makes for the adaptive logic, and about how flow models advise, or don't, changes, what reflect
the most judgments matching ontological commitment, for the least burnt bits, in guarantees.


So, Mitch you quite impress me because I read from your opinion, which I esteem, in logical calculi,
the concerns which so relevant to usual adaptive systems, provide context where then if I work
up various organizations in free logic in the VLSI very-large-scale-integration, that there is the
adaptation of it also to the free logic of the analog computing, and the free-form 3-D IC, and besides,
it also works up rules that according to scratches with a stick in the sand, break down tableau to strokes.

[Julio Di Egidio]

GET THE FUCK OUT OF SCI LOGIC.

you shameless frauds and pieces of systematic insane lying shit.

Polluters of ponds should be shot in the face!

Julio

[Mitchell Smith]

To be honest, Ross, I have not thought much about differentiating belief and faith beyond the fact that we all live through faith, in my opinion. By faith, I do not mean belief in the tenets of any particular religion. Rather, there are simply questions we cannot answer, and, we are undoubtedly overinfluenced by the indoctrination we receive as children. Having received a Catholic catechism, I had been taught that "there are no atheists in foxholes" (Veterans from WWII and Korea had been a large proportion of the adult population in my childhood). However, as an adult, it is clear to me that children of atheist households will never be taught to turn to prayer in crisis. So, whatever their response to their possible demise in a war, it will not be prayer to a diety.

On the other hand, thanks to a cross-posting from an evangelist named Dale about a decade ago, I spent a month on alt.atheist (criticizing evangelists in defense of atheists). Among the things I learned during that month is that a proportion of atheists seem to have been taught that science and mathematics prove "truths." This had been surprising for me since I had always been of the opinion that scientific assertions must always be tentative, whence an agnostic posture would seem necessary.

In this regard, I see little difference between faith in religious tenets and faith that the scientific method produces "truths."

About two years ago an Internet interlocutor recommended the book, "Retreat to Committment" available at the link,

https://archive.org/details/retreattocommitm00bart

Bartley had been a student of Popper. The book is largely about how science and liberal Protestantism parted ways at the end of the 19th century and the early 20th century.

It does have bearing on modern logic in that it explains the failure of foundationalism in the sense of the Munchhausen trilemma,

https://en.m.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma

Once everything is reduced to dogmatic stipulation, both science and religion are on equal footing -- dogmatism defended with rhetoric.

So, can anyone finish the sentence, "Mathematics is ..." without an empty foundationalism?

mitch

[Mitchell Smith]

On Sunday, April 9, 2023 at 12:06:23 PM UTC-5, Ross Finlayson wrote:

With regard to the 16 set, Ross, I had first noticed that one could apply negation and de Morgan conjugation across a system of definite truth tables as an involution governed by symbol exchanges. Initially, I referred to the composition of the two as "conversion" in the sense of order converses. Later, I realized that "contraposition" better suited the situation.

When reading "Projective Geometry" by Veblen and Young, a theorem caught my eye which made me realize that what I had been looking at was very similar to collineations in a geometry.

Then, I learned of finite geometries and their relationship to group theory.

The long research note actually began with trying to arrange named truth tables into a representation for a finite geometry. The simplest document I have with such a labeling is in the link,

https://drive.google.com/file/d/1J4udRS-mWiT__ia4PnBU80gxdqPt4aeV/view?usp=drivesdk

I take a difference set presentation from an elementary projective geometry text and rearrange it so thst its parallel classes are evident. Then I rename the numeric parameters with the "logic word inscriptions" such that the involutions mentioned above are exhibited as collineations.

While the 21-point projective plane has 21 lines, the affine subplane of 16 points has 20 lines. To perform this labeling, I needed additional mnemonics. As it turned out, I could differentiate "absurdity" from "falsity" and add four "quantifier mnemonics." In the planar geometry, the quantifier mnemonics have no meaningful relation to one another. They are, in fact, their own negations.

But, I did find a 20-element order into which I could arrange the line names so that the "quantifier mnemonics" related to one another in terms of order-theoretic negation.

Thanks to Cullinane's site on finite geometry, I learned that these arrangements could be placed into 4x4 arrays,

http://finitegeometry.org/sc/16/geometry.html

Next, I learned of Assmus' work on Kummer configurations. There is a relationship between the comllete graph on 6 points and the 4x4 arrays. It can be found in the paper,

https://www.semanticscholar.org/paper/The-(16%2C16%2C2)-designs-Assmus-Salwach/0a33b7e1ea3e8cf80218379d5ac38eda449c38ec

Although I do not currently have a document with collineations exchanging "the point at infinity" in the style of the document above, I used such collineations to formulate explicit labelings of the hexagonal graphs in the document,

https://drive.google.com/file/d/1JAegr0qqHpn3SM3hallaKpHirKHnMgHz/view?usp=drivesdk

The Kummer configuration is encoded into the Rook's graph on each page. Pick a label. It is connected to six others. These six would be a 6-element block of the Kummer configuration.

These particular labelings are not "canonical." The complicated document from the top post is intended to constrain admissible labelings in relation to other structures.

Gotta go... work.

mitch

[Ross Finlayson]

That sounds kind of usual. I always felt lucky to grow up in an era
when there wasn't conscription and the Berlin Wall fell and that kind of thing.
(Still I reserved a pride that going "in" was always an option.)

Then, about "higher powers", if "atheism is conscious disbelief in the ultimately potent",
then it's for example along the lines of that I believe in truth, and there's a monist's
absolute truth, then that it's a matter of objectivity that "not being an atheist" doesn't
necessarily reject that, as, not allow confirming its negation. So, there's always room
for omnipotence, but, it's exactly because it's utterly vanishing. It results that's also
a mono-theism. It squares very well with universalism, which many have as their professed ideology.
It's inclusive.

It's easy to understand otherwise then why religions, and I esteem religion because
it provides a philosophical setting then besides social aspects of religion, and there's
that one can be a faithful adherent while detached from ontological commitment,
from an ontological commitment only to its ultimate means and ends. It's the role
of apologists and apologetics like this, to install the high points of higher-order
symbology and meaning in religion, to corresponding higher-order tenets of the
conscious commitment to ultimate truth, for example, for a gentle commonality,
then for the benefits of religion that encourage the higher-order thinking,
for example selflessness, or, the personal connection to the higher-order.

It would be foolish to not accept that people are driven by their own exercises in will,
then though it's not necessary to get bound up in Mammon while getting along and
getting ahead, or confuse defense of ideals with making and destroying perceived enemies,
with having a model of psychology that is their beliefs and their faiths, in terms of
the working theory of their beliefs and essentially, the "will to submit", of their faiths.

So, this way I have been able to enjoy lots of the good things in most religions, though
in most cases it's partially because of inherited faith in self-righteousness and few
natural enemies, and not resulting rejection from resentment or disappointment.

So anyways then the "ontological commitment", then, is for a philosophy of being or
philosophy of mind, here for these fundamental elements in the conjuctives , inclusive,
and conjunctivity, of connectives, about the grain of the smooth and that a table or tablet,
is of tableau, is of all the tableau or the great tableau, the algebraists', what is the world, the geometers'.

Then, the idea of application after organization, is that "line-drawing" result a parallel act,
on the distinct and discrete tableau. This is where the idea is to build machines of the connectives
the conjunctives that applications of the strokes, the "action", that what results the laminar stoke,
operates in parallel, then that the overall organization of truth tables admits a constant "gradient of truth",
that in step with time more-or-less, continuously curries these "mono-tops of truth", how it's so
that this action, basically points to then the "table of 16" or "above the binary and Boolean table of 2",
has that then this is for a sort of unified action, then that "connectives" are as about projections to
reflect off these, how it results that an entire apparatus, is always carrying on this currying of truth,
for the applied.

This gets into both the multi-valent as composite and the indeterminate as nullified,
the outer and inner products of a sort, but what I'm imagining is that there's enough
of a machine or an automaton, that results a sort of "logical computing atom",
sits or floats in a realm of gradients of flows, of the "wind of truth", what results
that more-or-less an optical model, constantly computes all things.

Then, that's an ontological commitment, where, it helps to already have a 'theory of truth'
some pure theory and also a model of physics for relating that to the phenomenological
or sensory, it's not different this "sensory evaluation of inputs" and some "pure-sensory
model of the computation of truth by state, of logical computing atoms".

Then the idea is that also makes for a science of an engineering of the applied, why all
sorts computers can also be implemented in this sort of framework, and that it's naturally
geometrical and natural operations like line-drawing reflect how some "logical computing atom"
can be modeled in a higher-order value and structure and the surrounds similarly simulated,
at a very low level and a very grand scale.

So, TRILATTICE, as an example, or, parallel-stroke, pretty much from combinations of
"table of 16" the conjunctives inclusive, is what I have in mind for analog computing devices
implemented in circuits, with basically the photonic and electronic as models of propagation
and flow, and flux, this sort of thing is for "an atomic theory of a universe of logic".


It's a continuum mechanics, ..., also any computing thing is a model of a fragment of it.
Then, the tableau their entries are like the super-strings of this theory, fundamentally
small to the atoms of this theory, working all the way up to classical logic as a view,
while describing the mechanics of a "wind of truth" or a truth-gradient, that always
cleaves, draws, and strokes, in the present.

So, it's a combined world-view of an algebraist and a geometer an "applied" the "pure".



Then, for your question, the idea is that "the 'empty' and 'full' are both inverse and same".
This way the simple reflection on chance or uncertainty flips all the bits, reverses the wind,
and it's the same. Such extensions are available from most religions that are philosophical
grounds, or one such that it is.

I address this variously as "null axiom theory" just like anybody else like Leibniz and Heidegger
and since antiquity of course and so on. This reading from Nozick about Philosophical Explanations
frames it kind of in this manner.

It's a continuum mechanics.

[Ross Finlayson]

One good thing about the mind is that it remembers -
another good thing about the mind is that it forgets.

These days I've been framing the "tendencies and propensities"
in theories of physics
or "the action" for "the principle of least action" as about the
"restitutive and oscillating" and the
"dissipative and attenuating" as the
common terms about the "continuity laws",
it's a continuum mechanics.

Mathematics needs one and physics does, also.

It's rather neat and kempt to frame theory this way.

Then, the idea that this "empty" theory is the same as the oscillating "full" theory,
it's utter "reversibility" because of some "existence of inverse", makes for what
Nozick, Heidegger, and Kant call "the fundamental question of metaphysics", that,
yes, indeed, a cognitive agent or thinking thing can arrive at this.

"Where it's at."

What it is - where it's at.


The contrapositive is about the strongest tool, because, when weighing alternatives,
which is any matter of judgment and not just repetitation or carved form an instance
of an inference scheme, in the symmetric and reflexive frame, it remains a scheme,
so that "ontological commitment" to "lack thereof, weighing judgments" doesn't result
"baggage".

I recall before when you mentioned Kummer and considering it profound, there's much
to be said for that thinking is an act, in terms of memory and the cast of memory,
the learning.


Much obliged - que sera sera.

[Ross Finlayson]

Eight bits is a dollar.

(Mathematics is a meta-geometry.)

[Graham Cooper]

there are NO general problem solvers at this time
perhaps in 10years PROLOG will have a GOOD THEOREM PROVER
that solves all engineering mathematics will change your mind

>
> Thirty-five years ago I thought of "logic" and "mathematics" as univocal and well-construed topics of study. So, when the dot-com debacle left me with no information technology career and a plentitude of free time,

Yes I was in and out of IT jobs then I switched to WEB TECHNOLOGY and make $200,000+ a year the last 7 years.

> I found the NNTP Usenet groups. It had been here --- on these groups --- where I learned that many people learn mathematics in philosophy departments. And, the view of mathematics as taught in philosophy departments is very different from the standard curriculum of mathematics departments training students for applications.

Exactly right! Its a CURRICULUM argument to denounce IT with (impossible problems) that computers cant solve....
YET!

>
> Problematically, others --- most notably, Mr. Greene --- perceived my work as a rejection of "received views." What had, in fact, been the case is that my training had been inadequate. However, the fact that the literature is replete with challenges to classical methods does not mean that learning the rationale behind the first-order paradigm resolves my questions.

Right again! FOL is riddled with paradoxes that the MATHS PHILOSOPHERS use as gospel to invent (HIGHER MATHEMATICS)

theorem (R)
<-
if( L R )
^
theorem( L )


applies to ALL FUNCTIONS R
so theorem proving itself is not in scope of FOL

> All that had been accomplished through that effort had been the realization that mathematics is studied in different ways.
>
> As I have often stated, my interest in logic arose because of the continuum hypothesis. That question may only be meaningful with respect to one particular paradigm. And, it may be resolved for that paradigm. But, Goedel assumes consistency in order to formulate the constructible universe. Is this consistency assumption a "truth" which must be accepted? And, Cohen's forcing only works if a given model is *assumed* to be partial. Is Cohen's assumption a "truth" which must be accepted?

Godel has a simple weak flaw

G has 2 truth values
1 is the INCOMPLETENESS THEOREM

proof( !proof(G) )


The other is BY SUBSTITUTION which is not even mentioned in Godels Proof

G <-> !proof(G)
G <-> !proof(!proof(G))

SPOT THE CONTRADICTION with godels proof.


Godels' proof fails to check ALL MODELS (derivations from axioms) AGREE

GODELS PROOF PREDICATE

proof(C)
<->
A^B->C
^
proof(A)
^
proof(B)

--------------

COMPLETE PROOF PREDICATE

proof(C)
<->
A^B->C
^
proof(A)
^
proof(B)
^
!proof(!C)


That COMPLETES the INCOMPLETENESS THEOREM

>
> I have been time-constrained since posting that research note. There are a few more pieces of my puzzle yet to come. The particular issue I attempt in the posted has to do with the fact that all of this debate in the foundations of mathematics has largely made "truth" and "not truth" meaningless. It is merely a dichotomy within language that is not intrinsically different from any other dichotomy.
>
> The connection to "mathematical logic" in so far as that expression refers to truth table semantics appears to involve the representation of oppositely directed tetrahedra encoded into a pair of 4-symbol quasigroups described with 4x4 arrays of cells.
>
> In saying this, note that my concerns over the last few years has shifted from the continuum hypothesis to "the unity of mathematics."
>
> When I started 35 years ago, information technology had not been ubiquitous, category theory had not become a mainstream method for teaching algebra, and, there simply was no homotopy type theory implementing Martin-Lof's work. In one form or another I can relate most of these developments to what I have been doing because I began with the idea that "individuation" could be correlated with Cantor's nested set theorem for closed sets of vanishing diameter. This, of course, relates to the principle of the identity of indiscernibles which is denied as a logical principle in the first-order paradigm. So, I inadvertently stumbled into the very questions upon which these competing paradigms cannot agree.

Mathematics IS algebra

AXIOMS
+
INFERENCE RULES

form a FINITE platonic set


The geometric extrapolation of ALL AXIOMS , DERIVATIONS RULES and VALUES
is INFINITE and I dont think it exists platonically

Like

N = { 1 2 3 4 AND SO ON ... }

doesn't EXIST even platonically - as WM puts it you cannot see the end of a infinite expansion.


Minds are LIMITED to finite axioms and rules - extrapolation from those FORMS THE GEOMETRIC MATHS finite bit by bit.


ACTUAL INFINITY is just

0 in N
X in N -> s(X) in N

Thats why VARIABLES ( algebra ) from the basis of mathematics
not Geometry


Geometry of mathematics is too large a set (infinite)

>
> Whoops! Signs of equality are ubiquitous!
>
> Anyway, Graham, its good to see you are still at it. Stay tuned for a little more. As for ChatGPT, I applaud it, although I question those who believe that artificial intelligence is equated with human intelligence. Such individuals will use rhetoric to demand that I show how their "in principle" arguments are in error. The issue is simply that "in principle" arguments are not fully specified models. The burden of proof will always (and eternally) be on them to produce a fully specified model. Not only is neuroscience nowhere near such a definite state, it is likely that a fully specified model will involve non-eliminable infinities or circularities (which is why I used the expression "eternally.")
>
> I suppose that reflects my current opinions on "truth" and "not truth." For logic to be useful, debate must begin with respect to a common belief. Debate over artificial intelligence introduces irreconcilable beliefs.
>
> mitch

As GENERAL PURPOSE theorem provers evolve we will know the truth. (by mutual agreement)
This may occur when AI masters computer languages - universal theorem provers as well as it mastered natural language

We have a theorem prover for this (in 1OL)
and a theorem prover for that

but none of them even do general mathematics so I wouldn't underestimate ALL LOGIC and MATHS unified with 1 method

COMPARE 2 STRINGS (with variables)

[Julio Di Egidio]

On Wednesday, 12 April 2023 at 09:43:00 UTC+2, Graham Cooper wrote:

> Godel has a simple weak flaw

Goedel has no flaws, indeed he says it all,
as there is no such thing as an infinite proof.

(And your formalization is as wrong as your derivations.)

> Mathematics IS algebra
> AXIOMS + INFERENCE RULES
> form a FINITE platonic set

Wrong, there are two sides to that coin and you too
keep conflating them. There is a sense in which
mathematics reduces to algebra, i.e. "equations"
or reflexivity or syntactic equality/substitutivity, and
there is a sense in which logic reduces to "application",
i.e. modus ponens.

And still then the devil is in the detail, and there is
alpha, beta, eta, zeta, and what-not equivalences
and reductions (lambda calculus docet)...

Julio

[Julio Di Egidio]

On Wednesday, 12 April 2023 at 10:18:48 UTC+2, Julio Di Egidio wrote:
> On Wednesday, 12 April 2023 at 09:43:00 UTC+2, Graham Cooper wrote:
>
> > Godel has a simple weak flaw
>
> Goedel has no flaws, indeed he says it all,
> as there is no such thing as an infinite proof.
>
> (And your formalization is as wrong as your derivations.)
>
> > Mathematics IS algebra
> > AXIOMS + INFERENCE RULES
> > form a FINITE platonic set
>
> Wrong, there are two sides to that coin and you too
> keep conflating them. There is a sense in which
> mathematics reduces to algebra, i.e. "equations"
> or reflexivity or syntactic equality/substitutivity, and
> there is a sense in which logic reduces to "application",

> i.e. modus ponens [aka inference rules].

>
> And still then the devil is in the detail, and there is
> alpha, beta, eta, zeta, and what-not equivalences
> and reductions (lambda calculus docet)...

P.S. Up to Univalence, where the two paradigms
appear to meet, under the scope of Logic (mathematics
must be logical) and in the language of (full-fledged)
type theory.

Something along that line...

Julio

[Graham Cooper]

On Wednesday, April 12, 2023 at 6:18:48 PM UTC+10, Julio Di Egidio wrote:
> On Wednesday, 12 April 2023 at 09:43:00 UTC+2, Graham Cooper wrote:
>
> > Godel has a simple weak flaw
> Goedel has no flaws, indeed he says it all,
> as there is no such thing as an infinite proof.
>

he doesn't examine SUBSTITUTION of G into G

G <-> !proof(G)
G <-> !proof(!proof(G))

G oscillates between T and F

and doesn't check ALL MODELS AGREE with derivation of G

but anyway... no convincing you
European philosophers are the worst

[Julio Di Egidio]

You need convincing? Firstly, that should be "provable",
not "proof", then, anyway, that substitution is INVALID.
But also a big hint should be that G is "identically true
hence unprovable" while you keep conflating it with the liar.

Julio

[Graham Cooper]

The question is can a single total theorem prover work with G

So I'm not disputing the meta-theory method and its conclusions

G <-> !proof(G)
G <-> !proof( [[[ !proof(G) ]]] )

but its not the only way to attack the problem - single total TP works too!

Since G oscillates between T and F

'this is not provable' - F
'this is not provable' is not provable - T
'this is not provable' is not provable is not provable - F

? proof(G)
> NO


There is no GODEL ARGUMENT that 'AHA it says its not provable' so its TRUE!

SIMPLY BECAUSE G HAS 2 TRUTH VALUES it is INCONSISTENT
(in a single total TP)


So although GODELS METHOD is sound too... it is superfluous and amounts to nothing

[Julio Di Egidio]

On Wednesday, 12 April 2023 at 14:51:11 UTC+2, Graham Cooper wrote:
> On Wednesday, April 12, 2023 at 8:22:56 PM UTC+10, Julio Di Egidio wrote:
> > On Wednesday, 12 April 2023 at 11:03:59 UTC+2, Graham Cooper wrote:
> > > On Wednesday, April 12, 2023 at 6:18:48 PM UTC+10, Julio Di Egidio wrote:
> > > > On Wednesday, 12 April 2023 at 09:43:00 UTC+2, Graham Cooper wrote:
> > > >
> > > > > Godel has a simple weak flaw
> > > >
> > > > Goedel has no flaws, indeed he says it all,
> > > > as there is no such thing as an infinite proof.
> > >
> > > he doesn't examine SUBSTITUTION of G into G
> > > G <-> !proof(G)
> > > G <-> !proof(!proof(G))
> > > G oscillates between T and F
> > >
> > > and doesn't check ALL MODELS AGREE with derivation of G
> > >
> > > but anyway... no convincing you
> > > European philosophers are the worst
> > You need convincing? Firstly, that should be "provable",
> > not "proof", then, anyway, that substitution is INVALID.
> > But also a big hint should be that G is "identically true
> > hence unprovable" while you keep conflating it with the liar.
>
> The question is can a single total theorem prover work with G

A meaningless question with an answer already:
the theory of all is not the theory of everything.

> So I'm not disputing the meta-theory method and its conclusions
> G <-> !proof(G)
> G <-> !proof( [[[ !proof(G) ]]] )
> but its not the only way to attack the problem

That is bullshit, not any way at all.

> - single total TP works too!
> Since G oscillates between T and F

Yet, since it doesn't, you are provably talking nonsense.

> There is no GODEL ARGUMENT that 'AHA it says its not provable' so its TRUE!

Indeed, that is NOT the argument... but I won't
repeat that argument (and that then model theory
is a lie) for the UMPTEENTH time!

STFU and start doing your own home work for a change.

(EOD.)

Julio

[Graham Cooper]

thats actually fair treatment coming from Julio

Julio, like GODEL forgot to substitute G into G


If G is SELF-REFERENTIAL
then it has MULTIPLE INSTANCES of the formula

Godel doesn't check all models (derivations of G) match

Yeh I'll pass of the META-THEORY WAFFLE

EOD

[Julio Di Egidio]

On Wednesday, 12 April 2023 at 15:28:55 UTC+2, Graham Cooper wrote:

> thats actually fair treatment coming from Julio
> Julio, like GODEL forgot to substitute G into G

Idiot, moron, spammer, and eventually liar.

Get the fuck out of sci.logic.

*Plonk*

Julio

[Graham Cooper]

Ah thats more like it!

JUILIO is unable to FATHOM

G <-> !proof(G)
G <-> !proof( !proof(G) )

wot a moron

[Ross Finlayson]

Mitch or Smith is really pretty great, here he presents a strong account
of the necessity of structuralism, which is naturally arithmetical.