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:

>

>> GĂ¶del'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.