Re: Few words on theory in logic, mathematics, and science

[Graham Cooper]

On Nov 9, 4:13 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:
> Hello the group,
>
> Well it's been some month or so and I return to find some regular discussion on basically the import of Cantor's theorems and then with regards to that, for some, they find intuitionistic, at least, reasoning that there would be some alternative to no well-ordered progression of dense real numbers.
>
> Constructively, I have simply ten years of output well establishing a framework of real numbers, i.e., suitable for analysis, where it is not the end result that bijections don't exist between the natural integers and reals, instead that as a very concrete form, a particular function, unique among functions, actually serves as a basis (in the mathematically strong sense of the word), here in what was established as a theory, and set theory, and a foundation for geometry, and a foundation for real analysis.
>
> I understand pretty much nobody gets that.  That may well be expected among the general public, but among those interested and familiar with mathematical foundations, for example Goedel's results that there are true statements about these objects of the theory not in the theory, that for something like the universe existing and to some, paradoxically, thus being its own powerset and establishing via very existence a prima facie counterexample, I'm very impressed.  Here, with that I discovered these simple features, and brought forward much of a development, searchable on these forae.
>
> Then, and you will find a ready group of folks willing to discuss the foundations of mathematics, any one of them would be proud to discover a feature of mathematics advancing knowledge, collectively, beyond what it was.  That takes not just creativity, and a penchant for mathematical thought and understanding of the modern, standard foundations of mathematics.  That takes not just a will to complete the bridge of the constructive divide between the discrete and continuous, while basically all vocal around you are against you.  That takes not just luck or some merited authority in bringing forward truth to the people and rabble.
>
> It takes all that.
>


or just a bit of LOGIC!

All these centuries of insurmountable problems must be hard right?

What problems are hard that you know the solution to?

We just all took a wrong turn, or a few. Picture back to 1971, the
sun was shining, University computers were doing logic, computers were
chatting about building steeples out of blocks of arches and querying
why they were asked to do so according to the stack of instructions in
their microPLANNER memory..

Then in 1972, UNIFY( f(a,C) , f(Z,g) ) was S.T.R.I.P.'ed (Stanford
Research Institute Planner) out of microPLANNER, PROgramming in LOGic
was born and LISP was tacked on and it solved 1000s of different
problems in this new age of 4th Generation languages...

There was only one problem... PROLOG was a single logic value
language, 1 RECORD FOUND using backward chaining and the Theorem
Provers of the time were Resolution Based 2 valued logic where
Theorems were TRUE and Assumptions were reversed otherwise! Set
Theory adopted a construction system of Predicates, true or false, as
long as they were well formed, functions were formulas, no sets, more
than infinity of them at any rate, don't try listing a function now-a-
days... axioms were replaced as the theorem provider with |= missing
theorems from somewhere else, and every thing started with first order
logic!

Herc
--


f(0).
t(1).
t(X) :- f(f(X)).


wff(X) :- t(X).
wff(X) :- f(X).
what(X,true) :- t(X).
what(X,false) :- f(X).



t(if(X,Y)) :- t(X), t(Y).
t(if(X,Y)) :- f(X), f(Y).
t(if(X,Y)) :- f(X), t(Y).
t(or(X,Y)) :- t(X).
t(or(X,Y)) :- t(Y).
t(and(X,Y)) :- t(X),t(Y).
t(iff(X,Y)) :- t(X),t(Y).
t(iff(X,Y)) :- f(X),f(Y).
t(xor(X,Y)) :- t(X),f(Y).
t(xor(X,Y)) :- f(X),t(Y).

f(if(X,Y)) :- t(X),f(Y).
f(or(X,Y)) :- f(X),f(Y).
f(and(X,Y)) :- f(X).
f(and(X,Y)) :- f(Y).
f(iff(X,Y)) :- t(X),f(Y).
f(iff(X,Y)) :- f(X),t(Y).
f(xor(X,Y)) :- t(X),t(Y).
f(xor(X,Y)) :- f(X),f(Y).

RESOLUTION
or(R,Q) :- if(L,R), or(L,Q).
or(R,Q) :- if(L,R), or(Q,L).
or(Q,R) :- if(L,R), or(L,Q).
or(Q,R) :- if(L,R), or(Q,L).

MODUS PONENS
t(R) :- if(L,R), t(L).
t(R) :- or(f(L),R), t(L).
t(R) :- or(R,f(L)), t(L).


INFERENCE RULE
if( if(t(S),f(R)) , if(t(R),f(S)) ).
if it's sunny then it's not raining
ergo
if it's raining then it's not sunny