Re: The unified theory of all we can do with it (seriously)
amy666
> any of the fields I am going to touch, I am rather
> some kind of abstract logician plus a professional
> programmer, so please mind the step(s). What I'm
> after is for (dis-)proofs to the following chain of
> assertions. As to the "discussion", that might very
> well start later, and I do mean it. OTOH, questions
> are always welcome. Again, please mind the step(s),
> there are none.]
>
> Subject: The unified theory of all we can do with it
> (seriously).
>
> Freely mentioning Russell, Cantor, Goedel, Turing,
> Complexity, Walster, Golden, and Di Egidio.
>
> Below, "dis-x" stands for "x and only x", where the
> connotations are thought to be even more
> interesting.
>
> >> Walster shows[*] that a number is a set, and that
> the empty set is a number.
>
> Walster extends interval arithmetic with the empty
> interval and intervals with one or both infinite
> end-points. He first defines operations on the empty
> interval; from there, he closes arithmetic to any
> operator and function combinations on the entire
> domain. In his system, the empty interval is dominant
> to any other, including the entire interval, i.e.
> {}.rel.X = {} and X.rel.{} = {}, for all X in IR*,
> for all interval relation. (More from Walster
> later.)
>
> (1) We now can say: a number dis-is a set, and the
> empty set is dominant to any number, dis-including
> itself.
>
> >> Russell asks what is the set of all sets not
> having themselves as elements.
>
> (2) We now can say: the set of all sets not having
> themselves as elements dis-is the empty set, i.e. the
> paradoxical set of paradoxical sets not having any
> elements, dis-including themselves.
>
> >> Cantor wanders how the diagonal argument leads to
> entities outside the domain.
>
> (3) We now can say: the diagonal argument leads to
> the all-encompassing void of the empty set, i.e. the
> paradoxical number of paradoxical numbers (or, more
> simply, the "without" (outside) of the domain, as
> seen from "within" (inside) the domain).
>
> >> Goedel proves that self-referential entities must
> exist, yet undecidably.
>
> (4) We now can say: the purely self-referential set
> dis-is the empty set, by foundation; in fact, a
> progression up the chain of provability systems can
> be seen from "I can't be proved", to "I am not true",
> up to "You fail", where this last sentence dis-is the
> empty set and expresses the limit ad infinitum of
> goedelization.
>
> Incidentally, "You fail" might express more than the
> abstract sentence ~Bf ("consis"; can read
> "not-believe-that"), where belief is the foundational
> meaning behind logical negation and falsehood. I
> cannot say (for lack of specific knowledge) what this
> strictly entails on the "incompleteness arguments in
> a general setting" (Smullyan, 1992), but the
> introduction of the empty set as a full fledged and
> foundational entity, and the closedness of algebraic
> systems it brings, seem to suggest a profound impact.
> For instance, it seems quite evident that belief
> should itself be founded on "us", the subjects, and
> aren't we, with respect to the system, just its very
> external domain? In a sense, ultimately, "we" are the
> empty set and the self-referential dis-proof of all
> proofs.
>
> Indeed, OTOH, I must note that, as far as "real"
> systems are concerned (i.e., in any form of
> "engineering"), the introduction of the empty set is
> in itself enough to formally found our very daily
> "practices", where we are used to discard apparently
> incorrect answers (i.e., dis-answers within our
> accepted domain), and where we, ultimately, improve
> throw failure (i.e., breaking out of the boundaries
> of the accepted). This is in the lights of what
> straight follows.
>
> >> Turing, in shades of Hilbert's tenth problem,
> restates the question in terms of the "halting
> problem".
>
> (5) We now can say: All machines indeed stop, sooner
> or later; in the worst case, it is "us" (see my
> preceding note) stopping them, and, in any meaningful
> sense, the machines "we" stop have failed, and
> dis-belong to the void of the empty set, which
> happens to be the void "we" are.
>
> In simpler terms, the set of failing machines is the
> set of machines that are "not machines" at all, with
> respect to the bounds "we" impose on the accepted
> domain.
>
> Incidentally, for all this to be (believed) true, we
> must accept that classical mathematics (pardon my
> lack of "sharpness" on that, but, strictly speaking,
> we could go back to Aristotle), from its very logical
> foundations onward, is rooted on a fallacy. I wish to
> stress here that correcting that fallacy doesn't mean
> throwing to the bin all of the great accomplishments
> so far, it rather means we could enlarge our
> perspectives beyond what is today believed to be out
> of our reach.
>
> As an anticipation, Walster talks about an "exception
> free system", which is a way to show that this "new"
> system (in the sense just given) must be simpler, not
> more complex than our current systems. This "new"
> system must still show some kind of instrinsic limit,
> though, and that is what I am (at a naive level)
> dis-expressing within the very last sentence in this
> document. However, let's finish the tour first.
>
> >> Within Complexity, NP problems are said to be
> "intractable", and it is yet an open question whether
> or not P = NP.
>
> Going back to Walster: "The use of interval methods
> provides computational proofs of existence and
> location of global optima. Computer software
> implementations use outwardly-rounded interval (cset)
> arithmetic to guarantee that even rounding errors and
> bounded in the computations. The results are
> mathematically rigorous." For example, the Kepler
> conjecture has been proved after 300 years by means
> of computers and outwardly-rounded cset arithmetic.
>
> (6) We now can say: NP is too in the tractable
> domain, though NP is not P in that they represent the
> two opposite approaches to problem solving, the
> latter finding the correct solutions, the first
> discarding the incorrect ones.
>
> Incidentally, if NP happened to be intractable, in
> real life we would forever be stuck in undecidedness,
> which is apparently not the case (and here I mean,
> apart from any heuristics: human beings don't need
> heuristics in every-day life, though every-day life
> is full of undecidable questions, rooted into the
> very structure of natural language).
>
> >> Di Egidio asks what then is a "number" (or a
> "set") for the sake?
>
> (7) We now can say (extensionally): a number (or a
> set) dis-is all we can do with-in and with-out it,
> i.e. the closed number system exhausts the whole
> domain of tractability, by
> tautological-within-self-referential foundation.
>
> >> Golden shows[**] a family of number systems called
> "polysign numbers", having a natural number of
> signs.
>
> (8) We now can say: Walster shows the "natural"
> closure of real numbers (avoiding undefined numbers
> and bounding computational errors); Golden shows the
> "natural" interplay between natural and real numbers
> (avoiding the asymmetries introduced with the
> imaginary numbers[***]); finally, Di Egidio shows
> (yet to be dis-proved), the general "natural" meaning
> of "number", as rooted into the concept of a subject
> which is the empty set (avoiding undefined
> reasoning).
>
> >> Di Egidio screams, what's the outcomes then?
>
> (9) We now can say: the outcome is the unified theory
> of all we can do with it (seriously).
>
> >> Di Egidio cries, come on, you're saying
> "seriously"!?
>
> (10) The outcomes are still to be inspected and
> worked out, but... even if only half of what is
> stated here has some reasonable foundation, then its
> incidence on everybody's lives could be dramatic, and
> to the better(!!). For instance, we have here the
> foundation for a final convergence of natural and
> logical languages, and, if you cannot imagine how
> that could only and only only change our lives for
> the better, then "You" fail.
>
> If you managed to get to this point, I'll be looking
> forward to your knowledgeable dis-proofs.
>
> (As you may guess, I do really need your feed-back.)
>
> Thank you very much,
>
> Julio
>
> --------------------------
> Julio Di Egidio
> Analyst/Programmer
> http://julio.diegidio.name
>
> [*] More on Walster's Closed Sets in this thread
> (it's the sci.math group):
> http://mathforum.org/kb/message.jspa?messageID=6126425
> &tstart=0
>
> [**] More on Golden's Polysigned Numbers on his web
> site:
> http://www.bandtechnology.com/PolySigned/index.html
>
> [***] The asymmetries are not completely avoided, but
> I guess Mr Golden might not have heard of closed
> number systems, yet.
? whats that last remark suppose to mean ?
polysign numbers are algebraicly closed.
their only "weakness" is that there are still zerodivisors...
but that is unavoidable in higher dimensions with commutative addition and multiplication.
do you think you can avoid that ??
do you think you can avoid zero-divisors or " improve " the polysign numbers in a certain way ?
as for a unified theory , i rather accept tommy1729 ideas:
set theory :
x = [x]
aleph_1 = aleph_2
2^c = c
logic :
all paradoxes are just illusions , slightly counterintuitive facts or simply assuming a badly defined question can be answered.
abstract algebra :
all integer domains can be expressed in a set of polysigned numbers by isomorphism.
iterations :
infinite interations with fixed parameters are just the solution to a zero of a related equation.
( like continued fractions or infinitely nested roots )
calculus :
fourrier series
productrule
contourintegration
tommy1729 analytic foundation
algebra :
galois theory
knot polynomials
minimum polynomial
game theory :
statistics
k-level thinking
nim sums
chess
go
regression analysis
cellular automaton :
equivalent to number theory , but not always easy to see.
***
according to tommy1729 , all undecidable problems reduce to the most intresting disciplines of mathematics being :
number theory
enumerative combinatorics
calculus
critical lines and its variants
tetration and operations beyond that
(and maybe minkowsky products or other minkowsky operators)
he believes P = NP can be decided.
he dislikes infinite ordinals.
his examples are riemann and matheyasevich mainly.
( if golden is your friend he should now him. )
amy666
Julio Di Egidio
> as for a unified theory , i rather accept tommy1729 ideas:
Please keep in mind my lack of specific mathematical knowledge.
> set theory :
> x = [x]
Yes, I'd write: x = [x] = [x,x]
> aleph_1 = aleph_2
Have no idea what that means.
> 2^c = c
Have no idea what that entails.
> logic :
> all paradoxes are just illusions , slightly counterintuitive facts or simply assuming a badly defined question can be answered.
I'd say I agree.
> abstract algebra :
> all integer domains can be expressed in a set of polysigned numbers by isomorphism.
I'd say I agree.
> iterations :
> infinite interations with fixed parameters are just the solution to a zero of a related equation.
> ( like continued fractions or infinitely nested roots )
I'd say I strongly agree, though just on an intuitive basis.
> calculus :
> fourrier series
> productrule
> contourintegration
> tommy1729 analytic foundation
> algebra :
> galois theory
> knot polynomials
> minimum polynomial
> game theory :
> statistics
> k-level thinking
> nim sums
> chess
> go
> regression analysis
Have no idea what all that means and/or entails.
> cellular automaton :
> equivalent to number theory , but not always easy to see.
I'd say I completely agree, as should be evident with TOA.
> according to tommy1729 , all undecidable problems reduce to the most intresting disciplines of mathematics being :
> number theory
> enumerative combinatorics
> calculus
> critical lines and its variants
> tetration and operations beyond that
> (and maybe minkowsky products or other minkowsky operators)
Have no idea what all that means and/or entails, I am not a mathematician.
> he believes P = NP can be decided.
I believe TOA has solved that.
> he dislikes infinite ordinals.
Don't know what that means.
> his examples are riemann and matheyasevich mainly.
> ( if golden is your friend he should now him. )
Have no idea what you mean or entail, but I might have answered this already.
> amy666
Thanks a lot Amy. IMVHO, and until dis-proved, TOA and tommy1729 are in perfect accordance.
Julio
Neilist
> Please keep in mind my lack of specific mathematical knowledge.
...
> Have no idea what that means.
...
> Have no idea what that entails.
...
> I'd say I strongly agree, though just on an intuitive basis.
...
> Have no idea what all that means and/or entails.
...
> Have no idea what all that means and/or entails, I am not a mathematician.
...
> Don't know what that means.
...
> Have no idea what you mean or entail, but I might have answered this already.
So, you haven't a clue, but you agree with "amy"? Brilliant!
>
> > amy666
>
> Thanks a lotAmy. IMVHO, and until dis-proved, TOA and tommy1729 are in perfect accordance.
You apparently don't realize that "amy666" is tommy1729 in
schizophrenic mode, talking and (here's the funny part) agreeing with
himself.
Pathetic.
Tonico
******************************************************
I most heartedly recommend Amy-Tommy-Aedipus-Whoever to Dedanoe, and
the other way around. I think they all have a lot to share with each
other, and the other way around...or up...or something.
Regards
Tonio