Showing posts with label Completeness. Show all posts
Showing posts with label Completeness. Show all posts

Friday, June 26, 2015

The Enormous Theorem

In the early 1960s, Group Theory was a way of describing several phenomena in a concise manner.

Morton Hamermesh's book is great.  Chapter 1 is "Elements of Group Theory".

The applications of Group Theory have moved to "Representation Theory".  The theory side seems to have evolved to Category Theory.

Category theory can be used to formalize concepts of other high-level abstractions such as set theoryring theory, and group theory. Several terms used in category theory, including the term "morphism", differ from their uses within mathematics itself. In category theory, a "morphism" obeys a set of conditions specific to category theory itself. Thus, care must be taken to understand the context in which statements are made.

Morphism In mathematical category theory, a generalization or abstraction of the concept of a structure-preserving function.

In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebralinear transformations; in group theorygroup homomorphisms; in topologycontinuous functions, and so on.

In category theory, morphism is a broadly similar idea, but somewhat more abstract:  the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map.

The July 2015 issue of Scientific American on page 72 summarizes "Four Enormous Families" that contain all the finite simple groups.  An exemplar for completeness. 



Saturday, July 20, 2013

The Adolescent Brain


Teenagers

Trayvon Benjamin Martin was a teen ager when he was shot through the heart and died at the age of 17 years and 21 days.

Ye Mengyuan was a 16-year-old teenager when she died.

Dzhokhar Tsarnaev is 19 years old.

Two of these teenagers are dead.  Without doubt, Ye Mengyuan is the most “innocent.”

Henry Kissinger supposedly said, “Just because you are paranoid doesn’t mean people aren’t out to get you.”  Just because you are a teenager doesn’t mean you were misbehaving.  Ye Mengyuan was not.  Where is justice for her?

“Beautiful Brains” by David Dobbs in the Best American Science and Nature Writings in 2012 available as an e-book from Amazon explains the baffling phenomena of teen brain development.

The first full series of scans of the developing adolescent brain – a National Institutes of Health (NIH) project that studied over a hundred young people as they grew up in the 1990s—showed that our brains undergo a massive reorganization between our twelfth and twenty-fifth years.  The brain doesn’t actually grow very much during this period.  It has already reached 90 percent of its full size by the time a person is six, and a thickening skull accounts for most head growth afterward.  But as we move through adolescence, the brain undergoes extensive remodeling, into the resembling a network and wiring upgrade.

Dobbs says that the teens desire for thrills peaks at 16.  Dobbs differs with the “work in progress” view of adolescent and writes:

The resulting account of the adolescent brain –call it the adaptive-adolescent story—casts the teen less as a rough draft than as an exquisitely sensitive, highly adaptable creature wired almost perfectly for the job of moving from the safety of home into the complicated world outside.

Presented with the circumstances of February 2012, Trayvon may have yielded to teen thrill seeking tendencies.

Dzhokhar's behavior is consistent with thrill seeking.  Ye Mengyuan’s thrill seeking was to come to the United States of America as a 16 year old girl.

Thursday, September 27, 2012

The Internet - The End of Localized Completeness

 The Supreme Court of Florida has adopted rules that require “the allegations” be signed and under oath.  A copy of the rule imposing this obligation (3-7.3(c)) may (or may not) be found at the Florida Bar’s web site at www.floridabar.org.

The oath consists of “Under penalty of perjury, I declare that the facts contained in … are, true, correct and complete.”  I wonder what definitions they are using.  My definitions of “true” and “correct” overlap considerably.  As for “complete,” the universe is complete but even it is evolving.  The Courts seem to be stuck in the days of William Blackstone:

“Blackstone had developed a great interest in common law, and in 1753 he began to lecture on that subject. These were the first lectures on English law ever delivered in a university. His listeners were captivated by the lucidity and charm of his style and by the simplicity with which he presented the subject. The latter virtue, however, was attained in part because Blackstone blurred the difficulties and contradictions of English law. He gave the whole subject an air of completeness and mutual interdependence as if it were a uniform logical system, and he suppressed or ignored its archaic aspects and instead acclaimed English law as the embodiment of 18th-century wisdom. He stated his aims in a notice of his lectures dated June 23, 1753:

“It is proposed to lay down a general and comprehensive Plan of the Laws of England; to deduce their History; to enforce and illustrate their leading Rules and fundamental Principles; and to compare them with the Laws of Nature and of other Nations.

“In 1754 Blackstone published Analysis of the Laws of England, a synopsis of his lectures for the guidance of his pupils. In October 1758 he was elected the first holder of a chair (the Vinerian professorship) of common law. His lectures formed the basis of his Commentaries, which were published in four successive volumes between 1765 and 1769.”

For a continuation of the article in Britannica Online see http://www.britannica.com/EBchecked/topic/68589/Sir-William-Blackstone/729/Assessment

The terminology is breathtaking:  an air of completeness and mutual interdependence as if it were a uniform logical system” and he “suppressed or ignored its archaic aspects”.

The 200+-year history of the United States of America hadn’t even gotten started when Blackstone published his four volumes.  The internet is introducing an interdependence that was unimaginable in Blackstone’s day.  Only the web has potential for "completeness."  No single node can be "complete."

In the Florida judicial system “complete” means that no other factors or evidence need be considered.  Such completeness would allow the Judiciary decision making agent/node to act as if they were a closed system.  For applicable definitions of Open System and Closed System, see http://en.wikipedia.org/wiki/Open_and_closed_systems_in_social_science.

The Florida Judicial Systems needs to look closely at feedback loops.  Circuit Court judges in Family Law matters definitely need their findings of fact to be subject to review/correction.  If my personal experience is “typical,” we are in deep trouble.

My personal situation is an example of “The Road to Hell is paved with Good Intentions.”  My ex-spouse is an entitled victim in the 47%.  She has two (2) Master’s degrees.  One is basically in getting free help to help her do good/well.  She has a Director of Christian Education (DCE) degree from Emory University.

She has utilized student loans, victim’s support group(s), and extended unemployment.  (Her highest paying job was with Volunteers of America - Florida where she helped Vets with homelessness, alcohol, and drug problems.)  She wants everything to which she is entitled.  She got her second Master’s in Mental Health counseling in 2003.  Doesn’t she owe the current veterans her services at a reasonable price?  She is/was a licensed mental health counselor in Florida.  Would Florida license a person inappropriate for our Vets?

Should I pay alimony to keep her in the extremely comfortable life style to which she wanted to become accustomed?  Her two (2) real doctor brothers could afford her desired lifestyle.  With only a PhD in experimental particle physics, I could not.  My ex-spouse was enabled by various support groups.  If not the Road to Hell, then at least the Road to the Poor House for me is paved with the Good Intentions of many (good intentioned) people.  Judge Charles J. Roberts, 18th Circuit, may be the only one to suffer any negative consequences for his poorly directed good intentions.