Date: Mar 26, 2013 4:43 PM
Author: fom
Subject: Re: Mathematics and the Roots of Postmodern Thought

On 3/26/2013 1:18 PM, david petry wrote:
> On Tuesday, March 26, 2013 4:05:38 AM UTC-7, Dan wrote:
>

>> Gödel's theorem is here to stay .
>
> As I have argued previously, if we treat mathematics as
> a science and accept falsifiability as the cornerstone of
> mathematical reasoning, then Godel's theorem is utterly utterly
> trivial, while at the same time, his proof of the theorem is
> not a valid proof.
>
> https://groups.google.com/group/sci.math/msg/25be708362cb7e?
>


In your argument you write:

"When we reason about how our thought processes work, we come
to the conclusion that every thought process we have can be modelled
on a digital computer (that's not to say that our brains are digital
computers, but there's an equivalence between what a computer can
do and what we can do)."

At an earlier time, you asked about the difficulties
I had with established mathematics. You did not, however,
ask me about my mathematics.

The following remarks should impress upon you that how you reason
about your thought process is very different from how other people
reason about their thought processes.

I believe, for example, that because the propositional logic has
two complete connectives that unary negation is eliminable.

I believe, for example, that the eliminability of unary negation
suggests the possibility that the objective of foundational research
is thwarted.

I believe, for example, that the existence of two complete
connectives dictates a determination of whether or not there
exists a fundamental asymmetry whereby the formulas of logic --
representable in two distinct forms by virtue of those complete
connectives -- have a canonical form.

I believe, for example, that these questions were not possible
until the historical developments whereby truth-functionality
became a decision procedure for propositional formulas.

My investigation into these matters led to my understanding
logic in terms of a system of symbols rather than random symbols
manipulated by rules.

That system forms a geometry described in

news://news.giganews.com:119/Jr2dnbdYvtfPdlrNnZ2dnUVZ_t-dnZ2d@giganews.com

It is a projective geometry because when the involutions
discussed in

news://news.giganews.com:119/EsqdnX0_NvwwKlzNnZ2dnUVZ_o-dnZ2d@giganews.com

are taken with the identity map, they exchange the truth
functions according to a general system of permutations
characteristic of projective geometries.

Because a set of names on points does not make a truth
function, the functionality is expressed by combining
the ideas of Church and Birkhoff. Namely, functionality
is described with an equational theory of intensional
functions. The axioms of that theory can be found
in

news://news.giganews.com:119/IqudndogJ8-VB1zNnZ2dnUVZ_qydnZ2d@giganews.com

To establish an intrinsic notion of negation, Curry's
notion of a logistic system based on Schonfinkel's applicative
structures motivated the treatment of the complete
connectives as algebraic products. This is in

news://news.giganews.com:119/Jr2dnbZYvtc2dlrNnZ2dnUVZ_t-dnZ2d@giganews.com

Truth functionality is described in terms of a free
DeMorgan algebra on one generator mapping into the
ortholattice O_6 in

news://news.giganews.com:119/Jr2dnbNYvtf9cFrNnZ2dnUVZ_t-dnZ2d@giganews.com

It is convenient for the received paradigm that the DeMorgan
lattice on 16 symbols is order-isomorphic to the free Boolean
lattice on 2 generators. The DeMorgan lattice on 16 symbols
is the Cartesian product of the maximal subdirectly irreducible
DeMorgan algebra. Thus, the construction in

news://news.giganews.com:119/Jr2dnbBYvtedcFrNnZ2dnUVZ_t-dnZ2d@giganews.com

is essentially a claim that DeMorgan algebra rather than Boolean
algebra is the foundational structure in mathematics.

In other work that is simply too difficult to post, the
argument proceeds from the different orientations of
tetrahedral simplexes and partition lattices on four
symbols.

I had been unsatisfied with what I had been taught.

I formulated alternatives.

Those alternatives had been based on the presumption
that mathematics was fine (by virtue of its explanatory
force). So, the problem had come from the application
of logic. Investigating the eliminability of unary
negation ensued.

Computers do not respond to the world the way that
vital, living things do. You may attribute it to
God's love, Darwin's genius, or both. But your statement
concerning equivalences conflates acts with the motivation
for acts.