fom
Posts:
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12/4/12


Re: Mathematics and the Roots of Postmodern Thought
Posted:
Mar 26, 2013 4:43 PM


On 3/26/2013 1:18 PM, david petry wrote: > On Tuesday, March 26, 2013 4:05:38 AM UTC7, 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 truthfunctionality 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_tdnZ2d@giganews.com
It is a projective geometry because when the involutions discussed in
news://news.giganews.com:119/EsqdnX0_NvwwKlzNnZ2dnUVZ_odnZ2d@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/IqudndogJ8VB1zNnZ2dnUVZ_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_tdnZ2d@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_tdnZ2d@giganews.com
It is convenient for the received paradigm that the DeMorgan lattice on 16 symbols is orderisomorphic 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_tdnZ2d@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.

