How to Solve It

“How to Solve It” (first published in 1945) is a small volume by mathematician George Pólya describing methods of problem solving.  The book suggests the following steps when solving a mathematical problem:

1. Understand the problem.
2. Devise a plan to solve it.
3. Carry out the plan.
4. Revise and extend: look back on your work.

Further Reading:

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

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

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Yonkoma

From Wikipedia:

Yonkoma manga (4コマ漫画, “four cell manga” or 4-koma for short), a comic strip format, generally consists of gag comic strips within four panels of equal size ordered from top to bottom.

And also:

Traditionally, yonkoma follow a structure known as kishōtenketsu. This word is a compound formed from the following Japanese kanji characters:

    • Ki (起): The first panel forms the basis of the story; it sets the scene.
    • Shō (承): The second panel develops upon the foundation of the story laid down in the first panel.
    • Ten (転): The third panel is the climax, in which an unforeseen development occurs.
    • Ketsu (結): The fourth panel is the conclusion, in which the effects of the third panel are seen.

Further Reading:

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

https://en.wikipedia.org/wiki/Kish%C5%8Dtenketsu

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The Golden Ratio

Here are shown various ways to write the Golden Ratio, a favorite theme of mathematical recreation and investigation:

  • The equality of two ratios of simple algebraic quantities in two unknowns
  • The infinite limit of the ratio of successive Fibonacci numbers
  • In an infinite continued fraction representation
  • As a specific irrational number involving the square root of five (but cross multiply and 4 = 4 is explicit)

These instances remind me a bit of the Four Causes in their representations:

  • Relative and relational, but indefinite (Efficient)
  • Idealized as an infinite sequence and infinite completion (Formal)
  • Infinitely nested and recursively reductive (Material)
  • Definite but with embedded irrationality (Final)

Further Reading:

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

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

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

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Four Dimensions of Knowledge

What is the best way to educate, to teach and learn? Ideally, students shouldn’t merely memorize facts and recall them on demand, although retaining well accepted knowledge is important. Certainly, students need mental structures to organize these facts, so that they form associated groups of categories and classification. Additionally, methods are needed for accepting and rejecting facts, and procedures for organizing facts, although facts may often be revisited for truth, or to reorganize them, and so on.

Benjamin Bloom et al. developed a taxonomy for educators of six mental aspects for the acquisition of knowledge, which was revised later into six cognitive actions or processes by L. Anderson, B. S. Krathwohl and others. These six actions form a sort of food pyramid for knowledge, so that lower actions form a broader base for higher ones. Both taxonomies sort a dimensional hierarchy of knowledge operated by these aspects or actions from concrete to abstract: Factual, Conceptual, Procedural, and Metacognitive (the last added when revised).

In the revised taxonomy, “Remember” is the lowest cognitive action, described by “remember facts and basic concepts,” which deals with the Factual and Conceptual. Above it is “Understand,” described by “explain ideas or concepts,” and I imagine an idea can be a fact. Next is a procedural action “Apply: use information in new situations,” and in fact all six actions are procedural by being actions. At the top of the pyramid is “Analyze,” “Evaluate,” and “Create”. The Metacognitive dimension (“thinking about thinking”) is for thinking about these six actions and these four dimensions, as to how they are related and differ.

These taxonomies are well considered and there are many resources to investigate.

Further Reading:

http://edtheory.blogspot.com/2016/03/knowledge-dimensions-factual-conceptual.html

Bloom’s Taxonomy

Revised Bloom’s Taxonomy

https://www.schoolnet.org.za/teach10/resources/dep/thinking_frameworks/bloom_taxonomy_4.htm

Bloom’s Taxonomy Revised

https://www.goconqr.com/en/mindmap/12973371/four-types-of-knowledge

Pedagogy of book and chapter organization

Click to access Anderson-and-Krathwohl_Revised-Blooms-Taxonomy.pdf

Click to access krathwohl.pdf

Anderson, L., Bloom, B. S., Krathwohl, D., & Airasian, P. (2000). Taxonomy for learning, teaching and assessing: A revision of Bloom’s Taxonomy of Educational Objectives (2nd ed.). New York: Allyn & Bacon, Inc.

Anderson, L. W., Krathwohl, D. R., Airasian, P. W., Cruikshank, K. A., Mayer, R. E., Pin- trich, P. R., … & Wittrock, M. C. (2001). A taxonomy for learning, teaching, and assessing: A revision of Bloom’s taxonomy of educational objectives, abridged edition. White Plains, NY: Longman.

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

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

https://plato.stanford.edu/entries/concepts/

Concepts, Theory-Theory of

 

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

https://plato.stanford.edu/entries/innateness-cognition/

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Yojijukugo

A Yojijukugo (四字熟語) is a unit of four kanji characters that usually represents an idiomatic saying in Japanese. It is itself a yojijukugo, even though it isn’t idiomatic, since the term can also broadly refer to a non-idiomatic phrase of four characters.

Shunkashūtō (春夏秋冬) is a nice one that means the four seasons of the year, and so is a fourfold written in four kanji characters.

Eshajōri (会者定離) is hopefully appropriate, meaning “every meeting must involve a parting”.

Further Reading:

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

https://en.wikipedia.org/wiki/Shunkash%C5%ABt%C5%8D

https://www.tofugu.com/japanese/yojijukugo-idioms/

https://www.nippon.com/en/nipponblog/m00111/

https://www.kanjiclinic.com/kc37final.htm

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

And these are just interesting:

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

https://gogonihon.com/en/blog/japanese-characters

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Mitama

  • Ara-mitama (荒魂): wild soul
  • Nigi-mitana (和魂): tranquil soul
  • Saki-mitama (幸魂): love soul
  • Kushi-mitama (奇魂): wise soul

Further Reading:

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

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

[*12.74]

Every Which Way

  • Top-down
  • Bottom-up
  • Inside-out
  • Outside-in

Further Reading:

https://en.wikipedia.org/wiki/Inside_Out_(2015_film)

https://en.wikipedia.org/wiki/Top-down_and_bottom-up_design

https://en.wikipedia.org/wiki/Outside_In_(film)

https://www.merriam-webster.com/dictionary/every%20which%20way

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

https://www.thersa.org/blog/2014/04/top-down-bottom-up-side-to-side-inside-out-4-types-of-social-change-and-why-we-need-them-all

What’s your approach to innovation?

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Our Demarcation Problem

I have a foreboding of an America in my children’s or grandchildren’s time — when the United States is a service and information economy; when nearly all the manufacturing industries have slipped away to other countries; when awesome technological powers are in the hands of a very few, and no one representing the public interest can even grasp the issues; when the people have lost the ability to set their own agendas or knowledgeably question those in authority; when, clutching our crystals and nervously consulting our horoscopes, our critical faculties in decline, unable to distinguish between what feels good and what’s true, we slide, almost without noticing, back into superstition and darkness.

— Carl Sagan from The Demon-haunted World

As science is confused with pseudo-science, as real news is conflated with fake, we need much better ways to judge the truth of the information we require to be good citizens. Unfortunately, in this age of nontraditional television networks, kitchen-sink cable, and internet news sources, our information sources can be subverted by entities that wish to bend our mindset to their agenda, rather than giving us measured and reasonable knowledge. When these entities wish to fracture and divide our polity, our social fabric strains and unravels.

Here are four (or five minus one) distinctions for information or knowledge claims, based upon their type of warrant, or context of truthfulness. Three of them are modalities from Kant’s doctrine of judgments, and I suggest that Dialectic could reasonably be added to them, but I do not know if they form a complete set or not. I would suppose they can be ordered by their level of assurance, from low to high. Another more scientific option might be Probablistic instead of Dialectic, based upon measurements or even theoretical arguments. Certainly there must be something between a bald assertion or the questionable and the certain.

  • Assertoric: assert to be true or false without (inherent) proof
  • Problematic: assert as possibly true (or false)
  • Dialectic: philosophically reasoned as true or false (qualified?)
  • Probabilistic: quantified or theoretically argued as mostly true or false
  • Apodictic: clearly provable as true (or false) or logically certain

From Wikipedia:

Apodictic propositions contrast with assertoric propositions, which merely assert that something is (or is not) true, and with problematic propositions, which assert only the possibility of something being true. Apodictic judgments are clearly provable or logically certain. For instance, “Two plus two equals four” is apodictic. “Chicago is larger than Omaha” is assertoric. “A corporation could be wealthier than a country” is problematic. In Aristotelian logic, “apodictic” is opposed to “dialectic,” as scientific proof is opposed to philosophical reasoning.

For example, the president’s language (“many say”, “everyone knows”, “we’ll see”) is full of assertoric and problematic claims (to be extremely generous), and perhaps that’s the limit of his ability. I don’t think he could manage part of a measured dialectical argument if pressed, and if he manages an apodictic statement it would be like a clock that tells the time correctly twice a day. To have the head of the executive branch of our government to be so untrustworthy in providing information and knowledge hurts us all, and misleads those that takes his words at face value.

And then there are the news sources that cater to the president and his followers. Perhaps they present some warranted information, but mix plenty of misleading punditry in to tickle the fancy of unquestioning minds. As a result we have citizens who only digest information from sources that appeal to their sensibilities. Some of these news sources disseminate their fabrications via a flood in social media and the internet, because our ability to stifle them is almost nonexistent. And when these news sources originate from foreign countries wanting to influence us for their own purposes, how is it that they are allowed to continue?

In truth, people can be misled on scientific topics like the coronavirus and COVID-19, vaccinations, face masks, climate change or global warming, environmentalism and pollution, pseudoscience, and political topics like mail-in voting, Russian meddling with the 2016 and 2020 elections, conspiracy theories such as QAnon, etc. The lists seem almost endless.

Further Reading:

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

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

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

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

https://plato.stanford.edu/entries/aristotle-logic/

https://plato.stanford.edu/entries/kant-judgment/

Immanuel Kant: Logic

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

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

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

https://en.wikipedia.org/wiki/Russian_interference_in_the_2016_United_States_elections

https://en.wikipedia.org/wiki/Russian_interference_in_the_2020_United_States_elections

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

[*11.84]

The Four Tendencies

In a new spin on the four temperaments, here is a book that describes four “tendencies” for personality profiles, based on meeting or resisting inner and outer expectations.

  • Upholders: Meets inner expectations, meets outer expectations
  • Obligers: Resists inner expectations, meets outer expectations
  • Inquirers: Meets inner expectations, resists outer expectations (Questioners)
  • Dissenters: Resists inner expectations, resists outer expectations (Rebels)

Further Reading:

Gretchen Rubin / The Four Tendencies: the indispensable personality profiles that reveal how to make your life better

The Four Tendencies Quiz

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

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

Two Factor Models of Personality

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The Logic of Universalization Guides Moral Judgment

From the abstract:

To explain why an action is wrong, we sometimes say, “What if everybody did that?” In other words, even if a single person’s behavior is harmless, that behavior may be wrong if it would be harmful once universalized. We formalize the process of universalization in a computational model, test its quantitative predictions in studies of human moral judgment, and distinguish it from alternative models. We show that adults spontaneously make moral judgments consistent with the logic of universalization, and report comparable patterns of judgment in children. We conclude that, alongside other well-characterized mechanisms of moral judgment, such as outcome-based and rule-based thinking, the logic of universalizing holds an important place in our moral minds.

  • Universalization: What if everyone did this?
  • Rule-based: An implicit or explicit rule says I may not do this.
  • Norm-based: I shouldn’t do this because many say not to.
  • Outcome-based: How would it be bad if I did this?

Further Reading:

Sydney Levine, Max Kleiman-Weiner, Laura Schulz, Joshua Tenenbaum, and Fiery Cushman / The logic of universalization guides moral judgment

PNAS first published October 2, 2020; https://doi.org/10.1073/pnas.2014505117

https://www.pnas.org/content/early/2020/10/01/2014505117

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