Computation, fourfolds, logic, Mathematics

Propositions as Types

February 14, 2016 Martin K. Jones 3 Comments

For almost 100 years, there have been linkages forged between certain notions of logic and of computation. As more associations have been discovered, the bonds between the two have grown stronger and richer.

Propositions in logic can be considered equivalent to types in programming languages.
Proofs of propositions in logic can be considered equivalent to programs of given type in computation.
The simplification of proofs of propositions in logic can be considered equivalent to the evaluation of programs of types in computation.

The separate work of various logicians and computer scientists (and their precursors) can be paired:

Gerhard Gertzen’s work on proofs in intuitionistic natural deduction and Alonzo Church’s work on the simply typed lambda calculus.
J. Roger Hindley and Robin Milner’s work on type systems for combinatory logic and programming languages, respectively.
J. Y. Girard and John Reynold’s work on the second order lambda calculus and parametric polymorphic programs, respectively.
Haskell Curry’s and W. A. Howard’s work on the overall correspondence between these notions of proofs as programs or positions as types.

Logic and computation are the sequential chains of efficient causation and actions. Propositions and types are the abstract grids of formal causation and structures. Proofs and programs are the normative cycles of final causation and functions. Simplification and evaluation are the reductive solids of material causation and parts.

References:

Philip Wadler / Propositions as Types, in Communications of the ACM, Vol. 58 No. 12 (Dec 2015) Pages 75-85.

http://cacm.acm.org/magazines/2015/12/194626-propositions-as-types/fulltext

Preprint at

Click to access propositions-as-types.pdf

Also see:

http://www.drdobbs.com/old-ideas-form-the-basis-of-advancements/184404384

https://en.wikipedia.org/wiki/Hindley%E2%80%93Milner_type_system

https://en.wikipedia.org/wiki/System_F

https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence

[*9.92-9.94]

fourfolds

Means and Ends

December 30, 2015 Martin K. Jones 2 Comments

An interesting fourfold I saw while browsing through “The Power of the 2×2 Matrix” that I mentioned previously was the Means and Ends matrix of Russell Ackoff, known as a pioneer in the fields of systems and management sciences.

Composed of the relationships between two purposeful agents, where the means and the ends of each are separately considered to be compatible or incompatible.

Conflict: Incompatible means, incompatible ends
Competition: Incompatible means, compatible ends
Coalition: Compatible means, incompatible ends
Cooperation: Compatible means, compatible ends

Ackoff is also known for the “Hierarchy of Understanding” of Data, Information, Knowledge, and Wisdom, which probably begs for its own entry.

References:

https://en.wikipedia.org/wiki/Russell_L._Ackoff

Russell Ackoff and Fred Emery / On Purposeful Systems: an interdisciplinary analysis of individual and social behavior as a system of purposeful events

Jamshid Gharajedaghi / Systems Thinking: managing chaos and complexity: a platform for designing business architecture

Images of Data Information Knowledge Wisdom

[*9.88, *9.89]

fourfolds

The Four Hats of Creativity

December 29, 2015 Martin K. Jones 4 Comments

These four livelihoods: artist, designer, scientist, and engineer, make a nice fourfold. They are called the “four hats of creativity” by Rich Gold. They are also called the “four winds of making” by computer scientist Richard P. Gabriel.

Some say the artist and scientist are “inward” looking, and the designer and engineer are “outward” looking. Some say the artist and the designer “move minds”, and the scientist and engineer “move matter”. One can observe that the artist sorts the important from the boring, the scientist separates the true from the false, the designer discerns the cool from the uncool, and the engineer divides the good from the bad.

References:

Rich Gold / The Plenitude: creativity, innovation, and making stuff

https://www.dreamsongs.com/

https://www.linkedin.com/pulse/20130727173842-1391-the-creativity-compass

Images of Artist Scientist Designer Engineer.

[*9.20]

fourfolds

A Solstice Message

December 22, 2015 Martin K. Jones Leave a comment

“As far as we can discern, the sole purpose of human existence is to kindle a light of meaning in the darkness of mere being.”

— Carl Jung, from Memories, Dreams, Reflections

[*9.80]

fourfolds, sociology

The Four Cultures Model of Fons Trompenaars

December 18, 2015 Martin K. Jones 2 Comments

Here is a model of cultural differences, with two major axes:

Egalitarian (Decentralized) vs. Hierarchical (Centralized)
Person (Informal) vs. Task (Formal)

Leading to the following types (and orientations):

Incubator (Fulfilment) [Egalitarian/Person]
Family (Power) [Hierarchical/Person]
Guided Missile (Project) [Egalitarian/Task]
Eiffel Tower (Role) [Hierarchical/Task]

Trompenaars’ research later expanded these into seven cultural differences (universalism vs. particularism, individualism vs. communitarianism, neutral vs. emotional, specific vs. diffuse, achievement vs. ascription, sequential vs. synchronic, and internal vs. external control)! I’m not clear on how the four map into the seven.

Another model of cultural dimensions was developed by Geert Hofstede, who first found four dimensions (power distance index, individualism vs. collectivism, uncertainty avoidance index, and masculinity vs. femininity), and later increased these to six (adding long-term vs. short-term, and indulgence vs. restraint). Again, I’m unsure what the differences are between Trompenaars’ and Hofstede’s models.

Trompenaars’ model of four cultures is somewhat similar to another fourfold I found in the article “How to Build Scenarios”. It consists of two axes: individual vs. community and fragmentation vs. coherence.

Ectopia [Community/Fragmented]
I Will [Individual/Fragmented]
Consumerland [Individual/Coherent]
New Civics [Community/Coherent]

This fourfold is also mentioned in the book “The Power of the 2×2 Matrix”, which looks quite interesting. I think it is generally geared towards business decision applications, but has a compendium of various 2×2 matrices that appear to be broadly useful.

Also, the ChangingMinds.org website looks like it has a wealth of models and introductory information about them (and not only those with four aspects).

References:

http://changingminds.org/explanations/culture/trompenaars_four_cultures.htm

https://en.wikipedia.org/wiki/Fons_Trompenaars

http://en.wikipedia.org/wiki/Trompenaars’_model_of_national_culture_differences

http://en.wikipedia.org/wiki/Hofstede’s_cultural_dimensions_theory

Books / Articles:

Fons Trompenaars, Peter Woolliams / Business Across Cultures

Lawrence Wilkinson / How to Build Scenarios (in Wired Scenarios 1.01)

Alex Lowy, Phil Hood / The Power of the 2×2 Matrix: using 2×2 thinking to solve business problems and make better decisions

[*4.130, *9.64, *9.82]

fourfolds

Childhood’s End

December 14, 2015 Martin K. Jones 1 Comment

“No utopia can ever give satisfaction to everyone, all the time. As their material conditions improve, men raise their sights and become discontented with power and possessions that once would have seemed beyond their wildest dreams. And even when the external world has granted all it can, there still remain the searchings of the mind and the longings of the heart.”

― Arthur C. Clarke, Childhood’s End

References:

https://www.goodreads.com/work/quotes/209414-childhood-s-end

[*9.86]

fourfolds, Science

The Four Idols of Francis Bacon

December 11, 2015 Martin K. Jones 3 Comments

As a counterpoint to my previous post (and perhaps most of my previous posts) I present the following from Francis Bacon’s Novum Organum:

Idols of the Tribe
Errors in the mind of the group
Beliefs because most people have them

Idols of the Cave
Errors in the mind of the individual
Beliefs due to limited experience

Idols of the Marketplace
Errors in the use of words
Beliefs due to misuse of words

Idols of the Theater
Errors in false learning
Beliefs colored by religion and personal philosophy

References:

http://www.sirbacon.org/links/4idols.htm

http://www.iep.utm.edu/bacon/#SH2j

http://plato.stanford.edu/entries/francis-bacon/#Ido

[*9.84]

Consciousness, fourfolds

The Four Transcendental Imperatives of Bernard Lonergan

December 9, 2015 Martin K. Jones 3 Comments

Jesuit and theologian Bernard Lonergan had a worthy goal: to generalize the successful methods of science to all facets of human inquiry. Most particularly, he sought to consider not only exterior data from sensation but interior data from consciousness. He presented four epistemological precepts of ‘being’ that transcended cultural norms, to inform all domains of human knowing and knowledge:

Being Attentive
in Experience

Being Intelligent
in Understanding

Being Reasonable
in Judgment

Being Responsible
in Deciding

In addition, his Generalized Empirical Method (GEM) had four facets, and four methodological questions:

Cognitional
What do I do when I know?

Epistemological
Why is doing that knowing?

Metaphysical
What do I know when I do it?

Methodological
What therefore should we do?

These are certainly worthy precepts, domains, facets, and questions. How well have his techniques worked? At one time, Lonergan was said to be considered by many to be one of the finest thinkers of the 20th century.

The summary of the article on Lonergan at the IEP states:

A generalized empirical method in ethics clarifies the subject’s operations regarding values. The effort relies on a personal appropriation of what occurs when making value judgments, on a discovery of innate moral norms, and on a grasp of the meaning of moral objectivity. These innate methods of moral consciousness are expressed in explanatory categories, to be used both for conceptualizing for oneself what occurs regarding value judgments and for expressing to others the actual grounds for one’s value positions.

GEM is based on a gamble that the odds of genuine moral development are best when the players lay these intellectual, moral and affective cards on the table. Concretely, this implies a duty to acknowledge the historicity of one’s moral views as well as a readiness to admit oversights in one’s self-knowledge. Moreover, given the proliferation of moral issues that affect confronting cultures with different histories today, it also implies a duty to meet the stranger in a place where this openness can occur.

Lonergan’s imperatives are also somewhat similar to Kolb’s Learning Cycle and the Scientific Method. In both of those fourfolds, observation (sensation) occurs both before (but also cyclically after) experimentation (action). Could this be because these are methods for inquiry as opposed to one of making? Some think there is no distinction, that discovery is always socially constructed anyway, but I disagree.

References:

https://en.wikipedia.org/wiki/Bernard_Lonergan

Lonergan, Bernard

http://tipphilosophy.weebly.com/heaps-essay.html

http://www.metanexus.net/essay/learning-be-reflections-bernard-lonergans-transcendental-philosophy-education-towards-integral

https://en.wikipedia.org/wiki/Being

http://www.bernardlonergan.com/

Notes:

A good discussion and comparison of learning and inquiry at the Tetrast:

http://tetrast2.blogspot.com/2013/04/methods-of-learning.html

Books:

Bernard Lonergan / Insight: A Study of Human Understanding

Bernard Lonergen / Method in Theology

[*3.122, *3.124, *9.70, *9.72]

Cartesianism, fourfolds, Mathematics, Philosophy

Four Transformations of Chu Spaces

November 29, 2015 Martin K. Jones 1 Comment

Can mathematics help us reformulate Cartesian Dualism? I have previously tried to diagram some of computer scientist Vaughan 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”.

References:

https://en.wikipedia.org/wiki/Dualism_%28philosophy_of_mind%29

Click to access ratmech.pdf

http://chu.stanford.edu/

http://en.wikipedia.org/wiki/Chu_space

[*6.74, *6.75, *9.76]

fourfolds

The Day the Earth Stood Still

November 20, 2015 Martin K. Jones Leave a comment

“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.”