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. 



Sunday, June 14, 2015

The Road to Hell is paved with Good Intentions

1.  Recent standards for teaching science say that students should be taught "The Law of Unintended Consequences" before they graduate from high school.  The title combined with "Fools rush in where angels fear to tread" suggest my views on the subject.

Systems Dynamics is an approach that applies to dynamic problems arising in complex social, managerial, economic, or ecological systems — literally any dynamic systems characterized by interdependence, mutual interaction, information feedback, and circular causality.
The field developed initially from the work of Jay W. Forrester at MIT.  His seminal book Industrial Dynamics (Forrester 1961) is still a significant statement of philosophy and methodology in the field.  Within ten years of its publication, the span of applications grew from corporate and industrial problems to include the management of research and development, urban stagnation and decay, commodity cycles, and the dynamics of growth in a finite world.   It is now applied in economics, public policy, environmental studies, defense, theory-building in social science, and other areas, as well as its home field, management.  The name industrial dynamics no longer does justice to the breadth of the field, so it has become generalized to system dynamics.    The modern name suggests links to other systems methodologies, but the links are weak and misleading.  System dynamics emerges out of servomechanisms engineering, not general systems theory or cybernetics.
System Dynamics has a chance of usefully handling the complexities of the Modern world.  However, Auerbach reports stated about that time that only 15% of Analysts could use PSL/PSA (which Tom Demarco described as the best Automated Data Dictionary System available at the time).  Systems Dynamics is inherently non-linear.
Newton developed methods to linearize things.  Human thinking does rather well in a linear world.  R. A. Fisher did some outstanding work concerning the Design of Experiments and tools for the analysis of experimental data. See http://dr2htay.blogspot.com/2015/01/the-design-of-experiments.html
Social scientists did a very poor job of understanding the system before the tried "to improve it".  Several problems were presented in Jay W. Forrester, "Counterintuitive Behavior of Social Systems", 
Technology Review, Vol. 73, No. 3, Jan. 1971, pp. 52-68.  See http://www.constitution.org/ps/cbss.htm

2.  "Gall's Law"

Although dubbed Gall's Law by some, the actual quote is not labeled as such in the original work. The work cites Murphy's Law and the Peter Principle and is filled with similar sayings.

John Gall (born 1925) is an American author and retired pediatrician. Gall is known for his 1975 book General systemantics : an essay on how systems work, and especially how they fail..., a critique of systems theory. One of the statements from this book has become known as Gall's law.
Gall's Law is a rule of thumb for systems design from Gall's book Systemantics: How Systems Really Work and How They Fail. It states:

A complex system that works is invariably found to have evolved from a simple system that worked. A complex system designed from scratch never works and cannot be patched up to make it work. You have to start over with a working simple system. – John Gall (1975, p.71)

This law is essentially an argument in favour of underspecification: it can be used to explain the success of systems like the World Wide Web and Blogosphere, which grew from simple to complex systems incrementally, and the failure of systems like CORBA, which began with complex specifications. Gall's Law is said to have strong affinities to the practice of agile software development.

Although the quote may seem to validate the merits of simple systems, it is preceded by the qualifier "A simple system may or may not work." (p. 70). 

One of the first systems designers to quote Gall's law was Ken Orr in 1981. Notable were the quotations of Gall's Law by Grady Booch since 1991.


Systemantics (a change in typography and underlining indicate that the title is better rendered as "SystemANTICS") is a commentary on systems theory and general semantics publications by such thinkers as Ludwig von Bertalanffy and Alfred Korzybski.

3.  Alfred Korzyski was the only one above unknown to me previously. Following the link, I find William Burroughs went to a Korzybski workshop in the Autumn of 1939. He was 25 years old, and paid $40. His fellow students—there were 38 in all—included young Samuel I. Hayakawa (later to become a Republican member of the U.S. Senate), Ralph Moriarty deBit (later to become the spiritual teacher Vitvan) and Wendell Johnson (founder of the Monster Study). I used S. I. Hayakawa's book Language in Thought and Action as an undergraduate.

4.  Wiki's 1st para on Korzybski:

"Alfred Habdank Skarbek Korzybski ([kɔˈʐɨpski]; July 3, 1879 – March 1, 1950) was a Polish-American independent scholar who developed a field called general semantics, which he viewed as both distinct from, and more encompassing than, the field of semantics. He argued that human knowledge of the world is limited both by the human nervous system and the languages humans have developed, and thus no one can have direct access to reality, given that the most we can know is that which is filtered through the brain's responses to reality. His best known dictum is "The map is not the territory""

5.Willard Van Orman Quine

Quine's Ph.D. thesis and early publications were on formal logic and set theory. Only after World War II did he, by virtue of seminal papers on ontologyepistemology and language, emerge as a major philosopher. By the 1960s, he had worked out his "naturalized epistemology" whose aim was to answer all substantive questions of knowledge and meaning using the methods and tools of the natural sciences. Quine roundly rejected the notion that there should be a "first philosophy", a theoretical standpoint somehow prior to natural science and capable of justifying it. These views are intrinsic to his naturalism.

6.  Karl Popper versus Thomas Kuhn

I rejected Thomas Kuhn's stuff that was contrary to German Idealism.  I was in graduate school in the 1960s.  The Encyclopedia Britannica definition of "Physics" was basically the Hegelian dialectic applied to theory and experiment.  Seemed great to me.  I thought my rejection of Kuhn meant that I agreed with Karl Popper.  I'm revisiting this now.  What I thought  agreed with Karl Popper on was that assertions should be backed by (independently) verifiable evidence.  I have always been more interested in Metaphysics than "physics."  I  never shared the Logical Positivist's desire to do away with Metaphysics.  I am learning that Popper's focus was on distinguishing science from pseudo science.  That has not been a problem for me.  Physics, systems, and/or economics is the real world science and mathematics is the logical science.

7.  Charles S. Peirce

September 10, 1839 – April 19, 1914) was an American philosopherlogician,mathematician, and scientist who is sometimes known as "the father of pragmatism".  He was educated as a chemist and employed as a scientist for 30 years. Today he is appreciated largely for his contributions to logic, mathematics, philosophy, scientific methodology, and semiotics, and for his founding of pragmatism

Semiotics is closely related to the field of linguistics, which, for its part, studies the structure and meaning of language more specifically.  A language-game (GermanSprachspiel) is a philosophical concept developed by Ludwig Wittgenstein and Friedrich Waismann, referring to simple examples of language use and the actions into which the language is woven.

Jean-François Lyotard explicitly drew upon Wittgenstein's concept of language-games in developing his own notion of metanarratives in The Postmodern Condition. However, Wittgenstein's concept is, from its inception, of a plurality of language games; their plurality is not taken to be a feature solely of contemporary discourse. Lyotard's discussion is primarily applied in the contexts of authority, power and legitimation, where Wittgenstein's is concerned to mark distinctions between a wide range of activities in which language users engage.

Michel Foucault  frequently lectured outside France, particularly in the United States, and in 1983 agreed to teach annually at the University of California at Berkeley. An early victim of AIDS, Foucault died in Paris on June 25, 1984. 

Since its beginnings with Socrates, philosophy has typically involved the project of questioning the accepted knowledge of the day. Later, Locke, Hume, and especially, Kant developed a distinctively modern idea of philosophy as the critique of knowledge. Kant's great epistemological innovation was to maintain that the same critique that revealed the limits of our knowing powers could also reveal necessary conditions for their exercise. What might have seemed just contingent features of human cognition (for example, the spatial and temporal character of its objects) turn out to be necessary truths. Foucault, however, suggests the need to invert this Kantian move. Rather than asking what, in the apparently contingent, is actually necessary, he suggests asking what, in the apparently necessary, might be contingent. The focus of his questioning is the modern human sciences (biological, psychological, social). These purport to offer universal scientific truths about human nature that are, in fact, often mere expressions of ethical and political commitments of a particular society. Foucault's “critical philosophy” undermines such claims by exhibiting how they are just the outcome of contingent historical forces, and are not scientifically grounded truths.

Foucault left clear instructions that there should be no posthumous publication of his writings that he had not published in his lifetime. His estate has obeyed, with one major qualification. Foucault is deemed to have published lectures that he allowed to be taped. This has, in particular, allowed print editions of the annual courses of lectures that he delivered at the Collège de France from 1970–71 through 1983–84 (except for a sabbatical year in 1980–81). This has made an enormous body of important material available. Some of it covers (although in a different way) material later published, but some presents ideas (for example, on ancient philosophy) that appear nowhere else.

9.  Jurgen Habermas

Jürgen Habermas is a German sociologist and philosopher in the tradition of critical theory and pragmatism. He is perhaps best known for his theories on communicative rationality and the public sphere.


The Theory of Communicative Action (GermanTheorie des kommunikativen Handelns) is a 1981 book by Jürgen Habermas, in which he continues his project set out in On the Logic of the Social Sciences[1][2] of finding a way to ground "the social sciences in a theory of language."[1] The two volumes are Reason and the Rationalization of Society (Handlungsrationalität und gesellschaftliche Rationalisierung)[3] in which Habermas establishes a concept of communicative rationality,[4] and Lifeworld and System: A Critique of Functionalist Reason (Zur Kritik der funktionalistischen Vernunft),[5] in which Habermas creates the two level concept of society and lays out the critical theory for modernity.[4] After writing The Theory of Communicative Action, Habermas expanded upon the theory of communicative action by using it as the basis of his theory of morality, democracy, and law.[6] The work was the subject of a collection of critical essays published in 1986,[7] has inspired many responses by social theorists and philosophers, and in 1998 was listed by the International Sociological Association as the eighth most important sociological book of the 20th century.[8]