Fourier Analysis, V2

December 8, 2017

Here is another example of a fourfold, in the mathematics of Fourier Analysis. Here the four elements of our investigation resolve into Discrete Time, Continuous Time, the Fourier Series, and the Fourier Transform.

From the three dualities of Time – Frequency, Periodic – Aperiodic, and Discrete – Continuous, we obtain the four combinations Discrete Time/Periodic Frequency, Continuous Time/Aperiodic Frequency, the Fourier Series (Periodic Time/Discrete Frequency), and the Fourier Transform (Aperiodic Time/Continuous Frequency).

In the table below, T stands for Time and f for Frequency. The subscripts denote the attributes of each: D for Discrete, C for Continuous, P for Periodic, and A for Aperiodic. So T subscript C, f subscript A means that when Time is Continuous, Frequency is Aperiodic, etc. Please see Steve Tjoa’s web site for the equations for the Fourier Series and the Fourier Transform in Continuous and Discrete Time.References:

[*7.74, *7.108]



A Rosetta Stone

December 6, 2017

Abstract of Physics, Topology, Logic and Computation: A Rosetta Stone by John Baez and Michael Stay:

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much like a “cobordism”. Similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of “closed symmetric monoidal category”. We assume no prior knowledge of category theory, proof theory or computer science.

  • Physics
  • Logic
  • Topology
  • Computation

Perhaps Category Theory is a “Fifth Essence”?

Further Reading:

[*9.168, *10.50]


Speak, Listen, Write, and Read

December 3, 2017

Here’s another simple fourfold and maybe sixfold.

Speaking, Listening, Writing, and Reading are commonly presented together in elementary education as interrelated language skills. One is speaking for a listener and one is writing for a reader. One is listening to a speaker and one is reading a writer.

In the computer age another pair needs to be mentioned, that of programming for computers and the execution or running of that code by the computer. In a way, this new pair doesn’t fit, since one is writing code for computers, not people. And the execution of the program is not performed by a person, but by a computer.

(That’s not entirely true. Programs are also written for other human programmers in mind so that they can debug or maintain or modify the code if the original programmer isn’t available. Structured programming is one method to simplify the logical organization of the program so that others can comprehend it more readily. Object oriented programming is another method to allow multiple programmers to work independently without conflict.)

  • Speak – Listen
  • Write – Read
  • Program – Execute

But perhaps there is a different way to understand these duals. A speaker understands that a listener is following their speech by their response. A writer understands that a reader is comprehending their writing by their response. A programmer understands that a computer is ‘understanding’ the code by its response or output when the program is run.

Also, one can consider speech and writing to be encodings of thoughts into physical representations, and listening and reading to be decoding of the representations back into thoughts. Running or executing a program is not really decoding, or is it? But it is something like processing the speech or writing, like a computer is processing the program.

One might say that listening and reading are like processing the speech and text as programs on the computer of our brains. They are normally thought to be processed as data, as in Natural Language Processing, but it is an interesting twist if one considers them as programs. (Actually, I just recalled that the 1992 science-fiction novel “Snow Crash” by Neal Stephenson used this notion.)

To program effectively the programmer must execute their code in their mind, at least piecemeal and partially, just as a speaker or a writer must listen as they speak and read over what they have written. They can’t understand the full effect of the program’s execution, especially once the program becomes larger than a few statements, just as the full effect of speech or writing that is being processed by another person cannot be completely understood.

Not considered are computers themselves writing programs for other computers to “read” or execute. As the science of artificial intelligence becomes mature, computers writing and reading among themselves may become a common thing. Wasn’t there a news article about that recently? They pulled the plug on that pretty fast.

As digital assistants become more ubiquitous, they are fully participating in our language games of speaking, listening, writing, and reading. Will these so-called virtual assistants program for us next, as in Automatic Programming? That day may already be here.

Further Reading:

N. Katherine Hayles / My Mother was a Computer: digital subjects and literary texts

Neal Stephenson / Snow Crash


[*9.50, *10.36, *10.46]


The Who, What, Where, When, Why, and How of Everything

November 6, 2017

I’ve mentioned several times the fourfold of Who, What, Why, and How, mostly because I perceive an association between it and Aristotle’s Four Causes: Who for Efficient Cause, What for Material Cause, Why for Final Cause, and How for Formal Cause.

Two other “W”’s, Where and When, weren’t mentioned because they add two more to the four, and don’t fit into the Four Causes anyway.

But evidently they are mentioned often with the first four, as the several links below indicate. These six questions are used in elementary education and even beginning journalism to elicit six answers. They also form three pairs as follows:

  • Who – What
  • Where – When
  • How – Why

Further Reading:

Other Images:



Richard McKeon’s Aspects of Knowing, Part 2, V3

October 10, 2017

The duals in Richard McKeon’s system of Philosophical Semantics can also be arranged in a three-dimensional tetrahedron, where the dual pairs are on opposing edges. The universal and particular methods, the phenomenal and ontic interpretations, and the meroscopic and holoscopic principles are shown above.

Universal methods, between knower and knowledge, are applicable to all problems and all subject matters. Particular methods, between the knowable and the known, require distinct methodological procedures for different problems or subject matters.

Holoscopic principles, looking at the parts from the perspective of the whole, provide a coincidence of knowledge and known. Meroscopic principles, looking at the whole from the perspective of the parts, separate the knower and the knowable from each other and from influence between each other.

Ontic interpretations, between the knowable and knowledge, derive their character from a reality assumed to transcend or to underlie phenomena and statements. Phenomenal interpretations, between knower and the known, may reduce reality and values to aspects or consequences of phenomena.

Alternatively, the four vertices of  knower, knowledge, known, and knowable can be labeled by their method, principle, and interpretation as shown at right.

Further Reading:


As McKeon said in his lecture of  October 23, 1972 (the “Topics course”, unpublished): “You can either look at things from the point of view of the whole — then the principles are holoscopic (holos means whole, skopein means to look). Or, you can take the same set of facts, and view them from the part: then you have meroscopic principles. (Meros means part, skopein still means to look at).”


The Duality of Time and Information, V3

October 1, 2017


The states of a computing system bear information and change time, while its events bear time and change information.

from The Duality of Time and Information by Vaughn Pratt

The most promising transformational logic seems to us to be Girard’s linear logic.

— from Rational Mechanics and Natural Mathematics by Vaughn Pratt


Here we have three duals:

  • Information – Time
  • States – Events
  • Bear – Change

Further Reading:

Vaughan Pratt / The Duality of Time and Information

Vaughan Pratt / Time and Information in Sequential and Concurrent Computation

Vaughan Pratt / Rational Mechanics and Natural Mathematics



The Philosopher’s Wheel

September 24, 2017

Eric A. Meece’s web site has been in existence for a while, and it claims there is a forthcoming book called “The Philosopher’s Wheel”. This seems like an interesting project and it’s a shame that the book is still “in process”.

Meece has certainly been working on these ideas for few years. Starting with a Master’s thesis in 1979, and a presentation in 2001, he also has a collection of articles available to the reader that are related to his theme.

The philosopher’s wheel is essentially composed of three polarities:

  • Materialism vs. Spiritualism
  • Rationalism vs. Empiricism
  • Essentialism vs. Existentialism

Two polarities are related to Jung’s Psychological Types:

  • Rationalism (Thinking)
  • Empiricism (Sensing)
  • Existentialism (Feeling)
  • Essentialism (Intuiting)

I’ve tried to represent these dualities a little differently than Meece. Note in the above diagram that Materialism mediates Rationalism and Empiricism, and Spiritualism mediates Essentialism and Existentialism, similar to the wheel representation.

At right is an attempt at eliminating the “isms”. Perhaps I should have read some more of his writings before making these efforts.

Further Reading:


The more I think about it, the more I like to compare this with

Images of the Philosopher’s Wheel:



September 15, 2017

“Thanks, and farewell, faithful explorer.”

— Cassini program manager Earl Maize

Today, Cassini’s impetus succumbed to Saturn’s gravity.

Further Reading:


The Tetramorph

September 15, 2017

I’ve felt a fondness for the Tetramorph for a long time. Four beings considered simultaneously: a man, an ox, a lion, an eagle.

They are the Babylonian symbols of the four fixed signs of the zodiac: ox or bull for Taurus (and earth), lion for Leo (and fire), eagle for Scorpio (and water), man for Aquarius (and air).

They can be thought of as representing the ancient four elements: earth, fire, water, and air.

They are mentioned in the Judeo-Christian Bible in the books of Ezekiel and Revelation. They have also been paired with the four evangelists and their books: Matthew, Mark, Luke, and John (although the pairing is not fixed).

They are shown on the Tarot card for the “Wheel of Fortune” in the Rider-Waite deck, and the card for the “World” in the Tarot of Marseilles.

Why these four creatures, and not others?

Further Reading:

The Tetramorph; The Sumerian Origins of a Christian Symbol

[*8.38, *8.96]


Simon Magus and the Six Roots of Boundless Power

September 9, 2017

Through the darkness of future’s past,
The magician longs to see.
One chants out between two worlds…
“Fire… walk with me.”

— Mike from Twin Peaks

As we had a taste of Christian Eschatology last time we now nibble on Christian Gnosticism. Simon was mentioned in the bible and in apocryphal works was purported to be a sorcerer of some renown. He developed a philosophically idealistic system wherein six roots of mental aspects issue from a fundamental principle of “Fire”:

  • Mind (Heaven)
  • Voice (Sun)
  • Reason (Air)
  • Reflection (Water)
  • Name (Moon)
  • Thought (Earth)

These six roots form three pairs:

  • Mind [Nous] – Thought [Epinoia]
  • Voice [Phone] – Name [Onoma]
  • Reason [Logismos] – Reflection [Enthumesis]

Above is a representation of these six roots forming their three pairs or syzygies. Most of Simon’s teachings have been either been lost or perhaps even degraded from their original meaning. Do I waste both my time as well as yours to perpetuate this esoteric nonsense? Or is there some merit in it, if only for idle amusement?

Further Reading:

G. R. S. (George Robert Stow) Mead / Simon Magus: His Philosophy and Teachings


Sometimes Mind is called Reason, and then Reason is called Judgment.

Greek philosopher Heraclitus also thought that fire was the fundamental element.

Simon was thought to have written the book “The Four Quarters of the World”, now lost to us. Perhaps it was in the Library of Alexandria?

Perhaps Mike’s spell from “Twin Peaks” is inspired by Simon Magus? See

Compare the six roots to de Bono’s six thinking hats:

[*9.222, *9.223]