Archive for November, 2015

Four Transformations of Chu Spaces

November 29, 2015

sq_four_transformationsCan mathematics help us reformulate Cartesian Dualism? I have previously tried to diagram some of computer scientist Vaughn Pratt’s notions, such as a Duality of Time and Information and the Stone Gamut. Another recent attempt is the diagram above of four transformations that issue out of his analysis of Chu Spaces. Pratt’s conceptualization of these generalized topological spaces led him to propose a mathematization of mind and body dualism.

The duality of time and information was actually an interplay of several dualities, such as the aforementioned time and information, plus states and events, and changing and bearing (or dynamic and static). The philosophical mathematization in his paper “Rational Mechanics and Natural Mathematics” leads to additional but somewhat different dualities, shown in the following table:

Mind Body
Mental Physical
States Events
Anti-functions Functions
Anti-sets Sets
Operational Denotational
Infers Impresses
Logical Causal
Against time With time
Menu Object
Contingent Necessary

Pratt reveals two transformations that are “mental”: delete and copy, and two that are “physical”: adjoin and identify.

These four transformations are functions and their converses which:

  • Identify when the function is not injective.
  • Adjoin when the function is not surjective.
  • Copy when the converse is not injective.
  • Delete when the converse is not surjective.

Ordinarily we think of mind and body as being radically different in kind, but perhaps they are the same but merely viewed from a different perspective or direction. Recall what Heraclitus says, “the road up and the road down are the same thing”.


[*6.74, *6.75, *9.76]


The Day the Earth Stood Still

November 20, 2015

sq_day_the_earth_stood_still3“I am leaving soon, and you will forgive me if I speak bluntly. The universe grows smaller every day, and the threat of aggression by any group, anywhere, can no longer be tolerated. There must be security for all, or no one is secure. Now, this does not mean giving up any freedom, except the freedom to act irresponsibly. Your ancestors knew this when they made laws to govern themselves and hired policemen to enforce them. We, of the other planets, have long accepted this principle. We have an organization for the mutual protection of all planets and for the complete elimination of aggression. The test of any such higher authority is, of course, the police force that supports it. For our policemen, we created a race of robots. Their function is to patrol the planets in spaceships like this one and preserve the peace. In matters of aggression, we have given them absolute power over us. This power cannot be revoked. At the first sign of violence, they act automatically against the aggressor. The penalty for provoking their action is too terrible to risk. The result is, we live in peace, without arms or armies, secure in the knowledge that we are free from aggression and war. Free to pursue more… profitable enterprises. Now, we do not pretend to have achieved perfection, but we do have a system, and it works. I came here to give you these facts. It is no concern of ours how you run your own planet, but if you threaten to extend your violence, this Earth of yours will be reduced to a burned-out cinder. Your choice is simple: join us and live in peace, or pursue your present course and face obliteration. We shall be waiting for your answer. The decision rests with you.”

— Klaatu’s closing words

“Gort! Klaatu barada nikto!”

— Helen Benson

I, for one, welcome our new X overlords!




The Four Requisites of Randall Collins

November 16, 2015

sq_four_requisites2“Everything in the human world has four aspects” states sociologist Randall Collins, and I couldn’t agree more. For his view of sociology, these four aspects are the Social, the Political, the Cultural, and the Economic.

These four requisites are adapted from Talcott Parsons, who was Collins’ undergraduate teacher. Parsons’ four requisites were named differently, and together they are known as the “AGIL” model. “A” stood for Adaptation, “G” for Goal-attainment, “I” for Integration, and “L” for Latency. It was also called the Structural-Functional model of society. Besides the change in names, Collins also says that the functionalism inherent in Parsons’ model has been downplayed in his because a biological, functional approach cannot model conflict, which is pervasive in human interaction.

Collins wrote a book on the historical “sociology” of philosophies, “The Sociology of Philosophies”. This book was the reason I first noted Collins, but I haven’t studied the book in any detail to note if any fourness falls out of the analysis. This kind of historical and organizational model of philosophy seems to be popular, and several others have attempted to compile something similar.

Collins also wrote “Four Sociological Traditions”, a history of sociology organized around the development of four classic schools of thought: the conflict tradition of Marx and Weber, the ritual solidarity of Durkheim, the microinteractionist tradition of Mead, Blumer, and Garfinkel, and the utilitarian/rational choice tradition. This book was the second reason I noted Collins, but not having read the book, I wasn’t sure how to interpret these four schools as a four-fold.


Books by Randall Collins:

Four Sociological Traditions

The Sociology of Philosophies : a global theory of intellectual change

Philosophical Family Trees:

The social networks theory of philosophy

[*3.79, *4.6, *4.7, *9.7, *9.71]


The Four Cardinal Virtues

November 3, 2015

sq_four_virtuesI shall begin with the assumption that a perfect state is one that is rightly ordered, and is therefore wise, brave, temperate and just.

— Plato, from The Republic

From Wikipedia:

The Cardinal virtues are a quartette set of virtues recognized in the writings of Classical Antiquity and, along with the theological virtues, also in Christian tradition. They are comprised of the following qualities:

  • Prudence: also described as wisdom, the ability to judge between actions with regard to appropriate actions at a given time.
  • Justice: also considered as fairness, the most extensive and most important virtue.
  • Restraint: also known as temperance, the practice of self-control, abstention, and moderation tempering the appetition.
  • Courage: also named fortitude, forbearance, strength, endurance, and the ability to confront fear, uncertainty, and intimidation.


Images of Four Cardinal Virtues

Also see: