why/how explain?

[ajschumacher]

A proof can be correct but incomplete in the sense of leaving out the intuition that led to it. (https://twitter.com/planarrowspace/status/1285951583415664643)

As also e.g. code that isn't commented might work but be unclear/hard to understand/update.

"We live in an age of exponential growth in knowledge, and it is increasingly futile to teach only polished theorems and proofs. We must abandon the guided tour through the art gallery of mathematics, and instead teach how to create the mathematics we need. In my opinion, there is no long-term practical alternative." (page v, epigraph quoting Hamming, in Art of Doing Science and Engineering)

"It is another example of why you need to know the fundamentals very well; the fancy parts then follow easily and you can do things that they never told you about." (page 186)

"Now this fact, once understood, impacts design! Good design protects you from the need for too many highly accurate components in the system. But such design principles are still, to this date, ill understood and need to be researched extensively. Not that good designers do not understand this intuitively, merely it is not easily incorporated into the design methods you were taught in school. Good minds are still needed in spite of all the computing tools we have developed. But the best mind will be the one who gets the principle into the design methods taught so it will be automatically available for lesser minds!" (page 269)

"Probably the most important tool in creativity is the use of an analogy. Something seems like something else which we knew in the past. Wide acquaintance with various fields of knowledge is thus a help—provided you have the knowledge filed away so it is available when needed, rather than to be found only when led directly to it. This flexible access to pieces of knowledge seems to come from looking at knowledge while you are acquiring it from many different angles, turning over any new idea to see its many sides before filing it away. This implies effort on your part not to take the easy, immediately useful "memorizing the material" path, but to prepare your mind for the future. It is for this reason I have urged you in many of the chapters to get down to the fundamentals of a field, since it implies you must examine things in many ways before you can decide what is fundamental and what is frills. In fact, for one person they may be in one order, and for another in the opposite order. What is fundamental partly depends on the individual and their mental makeup. It is obvious you need many "hooks" on the knowledge if you are to use it in new situations.

"... I can only advise you to do what I do—when you learn something new, think of other applications of it, ones which have not arisen in your past but which might in your future. How easy to say, but how hard to do! Yet what else can I say about how to organize your mind so useful things will be recalled readily at the right time?" (pages 328-329)

[ajschumacher]

Is intense specialization really necessary, or is it an artifact of communities that become exclusive and defend their territory rather than sharing and communicating sufficiently well to lose their mystique? https://twitter.com/planarrowspace/status/1287459383199367168

[ajschumacher]

related thought mentioned in https://planspace.org/20200816-e_the_story_of_a_number_and_more/ re: Feynman explanations

[ajschumacher]

A Mathematician's Lament:

"By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the "truth" but in the explanation, the argument. It is the argument itself that gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation." (page 29)

[ajschumacher]

https://www.lesswrong.com/posts/6t4EjTfNzC6jN69ad/great-explanations

[ajschumacher]

"(In mathematics, you very seldom get the clearest account of an idea from the person who invented it.)" (page 49, Ellenberg)

I think this is tied in with #197 (whether it should be or not)

[ajschumacher]

"Cauchy was not interested in the needs of engineers. Cauchy was interested in the truth. It's hard to defend Cauchy's stance on pedagogical grounds." (page 49)

[ajschumacher]

"There is a large literature on the subject, but as the field is relatively recent, there does not yet appear to be a good non-technical introduction to the subject."

https://terrytao.wordpress.com/2007/04/13/compressed-sensing-and-single-pixel-cameras/

[ajschumacher]

"Mathematicians ever since Descartes have enjoyed the wonderful freedom to flip back and forth between algebraic and geometric descriptions of the world. The advantage of algebra is that it's easier to formalize and to type into a computer. The advantage of geometry is that it allows us to bring our physical intuition to bear on the situation, particularly when you can draw a picture. I seldom feel I really understand a piece of mathematics until I know what it's about in geometric language." (Ellenberg page 336)

This relates to "The Rule of Three" in the McCallum et al. Multivariable Calculus:

"Every topic should be presented geometrically, numerically and algebraically." (page v)

[ajschumacher]

There’s often an implicit assumption in education that to reach higher-level concepts one has to somehow recapitulate the history of how those concepts were historically arrived at. But usually—and perhaps always—this doesn’t seem to be true. In an extreme case, one might imagine that to teach about computers, one would have to recapitulate the history of mathematical logic. But actually we know that people can go straight to modern concepts of computing, without recapitulating any of the history.

But what is ultimately the understandability network of concepts? Are there concepts that can only be understood if one already understands other concepts? Given a particular ambient experience for a human (or particular background training for a neural network) there is presumably some ordering.

https://writings.stephenwolfram.com/2018/11/logic-explainability-and-the-future-of-understanding/

[ajschumacher]

"conceptual foundations" (as opposed to axiomatic foundations)

[ajschumacher]

on quantum mechanics:

Jaynes' CLEARING UP MYSTERIES { THE ORIGINAL GOAL } (1989) https://bayes.wustl.edu/etj/articles/cmystery.pdf (https://en.wikipedia.org/wiki/Mind_projection_fallacy)

The current literature of quantum theory is saturated with the Mind Projection Fallacy.

A standard of logic that would be considered a psychiatric disorder in other fields, is the accepted norm in quantum theory. But this is really a form of arrogance, as if one were claiming to control Nature by psychokinesis.

In our more humble view of things, the probability distributions that we use for inference do not describe any property of the world, only a certain state of information about the world.

But QM has two difficulties; firstly, like all empirical equations, the process by which it was found gives no clue as to its meaning. QM has the additional difficulty that its predictions are incomplete, since in general it gives only probabilities instead of definite predictions, and it does not indicate what extra information would be required to make definite predictions.

Einstein and Schrodinger saw this incompleteness as a defect calling for correction in some future more complete theory. Niels Bohr tried instead to turn it into a merit by fitting it into his philosophy of complementarity, according to which one can have two different sets of concepts, mutually incompatible, one set meaningful in one situation, the complementary set in another. As several of his early acquaintances have testified (Rozental, 1964), the idea of complementarity had taken control of his mind years before he started to study quantum physics.

... QM is not a physical theory at all, only an empty mathematical shell in which a future theory may, perhaps, be built.

For 60 years, acceptance of the Copenhagen interpretation has prevented any further progress in basic understanding of physical law.

Quantum Mechanics can be understood through stochastic optimization on spacetimes https://www.nature.com/articles/s41598-019-56357-3 representing some new attack on explaining quantum (pop coverage: https://phys.org/news/2020-10-quantum-mechanics-reality-person.html)

[ajschumacher]

Julia Evans is an example of a good explainer: https://twitter.com/b0rk

[ajschumacher]

"In my experience, any learning environment that encourages students to be fast learners is setting its students up for failure." (page xi, Bad Choices by Almossawi)

[ajschumacher]

"We can't really explain a heap by analogy." (page 120, Bad Choices)

[ajschumacher]

https://math.ucsd.edu/~harel/What%20Is%20Mathematics.pdf Guershon Harel

"DNR" - Duality, Necessity, Repeated reasoning

ways of thinking lead to ways of understanding

And the premises, in: https://math.ucsd.edu/~harel/pdf/DNRII.pdf

1. Mathematics • Mathematics: Knowledge of mathematics consists of all ways of understanding and ways of thinking that have been institutionalized throughout history.
2. Learning • Epistemophilia: Humans—all humans—possess the capacity to develop a desire to be puzzled and to learn to carry out mental acts to solve the puzzles they create. Individual differences in this capacity, though present, do not reflect innate capacities that cannot be modified through adequate experience. • Knowing: Knowing is a developmental process that proceeds through a continual tension between assimilation and accommodation, directed toward a (temporary) equilibrium. • Knowing-Knowledge Linkage: Any piece of knowledge humans know is an outcome of their resolution of a problematic situation. • Context Dependency: Learning is context dependent.
3. Teaching • Teaching: Learning mathematics is not spontaneous. There will always be a difference between what one can do under expert guidance or in collaboration with more capable peers and what he or she can do without guidance.
4. Ontology • Subjectivity: Any observations humans claim to have made is due to what their mental structure attributes to their environment. • Interdependency: Humans’ actions are induced and governed by their views of the world, and, conversely, their views of the world are formed by their actions.

[ajschumacher]

"Since breaking out of bad habits, rather than acquiring new ones, is the toughest part of learning, we must expect from that system permanent mental damage for most students exposed to it."

On the cruelty of really teaching computing science (Dijkstra) https://www.cs.utexas.edu/users/EWD/transcriptions/EWD10xx/EWD1036.html

also he's pretty formalist; prefers not to appeal to prior knowledge, etc.

and he seems to agree with others that quantum can't be explained in terms of other things

[ajschumacher]

Terry Tao has an interesting 3-stage analysis of mathematical understanding, which goes:

1. pre-formal
2. formal
3. post-formal

(and most people don't get to 2)

https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

Very interesting to me; I hadn't thought of it this way before...

Update 2020-12-30: It strikes me now that this is sort of a reduced Dreyfus model for expertise in formal methods. Roughly.

[ajschumacher]

"two trivialities omitted can add up to an impasse"

John Littlewood, “A Mathematician’s Miscellany,” quoted in https://terrytao.wordpress.com/advice-on-writing-papers/give-appropriate-amounts-of-detail/

[ajschumacher]

https://distill.pub/2017/research-debt/

[ajschumacher]

https://betterexplained.com/ has some math stuff and a philosophy of explanation called ADEPT:

• Analogy
• Diagram
• Example
• Plain English
• Technical definition

[ajschumacher]

Patterns in confusing explanations (Julia Evans) https://jvns.ca/blog/confusing-explanations/

pattern 1: making outdated assumptions about the audience’s knowledge
pattern 2: having inconsistent expectations of the reader’s knowledge
pattern 3: strained analogies
pattern 4: pretty pictures on confusing explanations
pattern 5: unrealistic examples
pattern 6: jargon that doesn’t mean anything
pattern 7: missing key information
pattern 8: introducing too many concepts at a time
pattern 9: starting out abstract
pattern 10: unsupported statements
pattern 11: no examples
pattern 12: explaining the “wrong” way to do something without saying it’s wrong

[ajschumacher]

"Anyway, back to chemistry: laboratory work seemed to me an uninspiring and messy waste of time. I was never told and I never caught on that a laboratory could teach you new facts and insights; I regarded it as just one of those chores (like irregular verbs in German and finger exercises for the piano) that the world assigns to apprentices before allowing them to become journeymen. I knew in advance what each experiment was intended to prove, and I proved it; I cooked the books mercilessly. Before the year was over I knew that chemistry was not for me, and I arranged to be transferred to the general liberal arts curriculum." (pages 23-24)

See also #276 (skills versus understanding...)

"The [calculus] text was the infamous Granville, Smith, and Longley that, according to rumor, brought each of its authors a royalty income of many thousands of dollars for at least 20 years. It was very bad. The explanations were not explanations—they were neither clear nor correct—they were cookbook instructions, no more. The selling virtue of the book was that it had many exercises, amost all of the the routine mechanical kind." (pages 26-27, Halmos, I want to be a mathematician)

[ajschumacher]

"You will see the spirit right away. The emphasis is on understanding—I try to explain rather than to deduce. This is a book about real mathematics, not endless drill. In class, I am constantly working with examples to teach what students need."

from Strang's Linear Algebra and its Applications

[ajschumacher]

New Math Book Rescues Landmark Topology Proof https://www.quantamagazine.org/new-math-book-rescues-landmark-topology-proof-20210909/

people went back and explained an unreadable original proof - pretty neat!

I saw this tweet: https://twitter.com/meaningness/status/1436048614569103362

“Ghost knowledge” in mathematics: a major result in topology was almost lost because the handful of people who could understand the proof are retirement age. Rescued by a team of unreasonably motivated young mathematicians!

[ajschumacher]

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

[ajschumacher]

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

[ajschumacher]

"For too long, educators have followed blindly the pleasure principle. This oversimplified approach is rejected here."

Carl E. Linderholm, Mathematics Made Difficult, page 10

desirable difficulties etc.

[ajschumacher]

copying in from #279:

---

"I cannot believe that anything so ugly as multiplication of matrices is an essential part of the scheme of nature."

Eddington https://en.wikipedia.org/wiki/Arthur_Eddington page 39 Relativity Theory of Electrons and Protons, Cambridge University Press, 1936.

as included in: Typing linear algebra: A biproduct-oriented approach https://arxiv.org/abs/1312.4818

---

"Determinants are difficult, non-intuitive, and often defined without motivation."

Sheldon Axler, "Down With Determinants!" https://www.maa.org/sites/default/files/pdf/awards/Axler-Ford-1996.pdf https://www.axler.net/DwD.html

---

There are lots of things that have been poorly understood in the past that now have better explanations and are better and more widely understood.

[ajschumacher]

https://byorgey.wordpress.com/2009/01/12/abstraction-intuition-and-the-monad-tutorial-fallacy/

[ajschumacher]

you don’t always need to convey how you arrived at an idea

https://www.johndcook.com/blog/2016/06/27/category-theory-and-homiletics/