Short musings

2021-Jun-20

There’s a meta-contrarian idea that the mechanisms of academia exclude some really good science that’s just too unconventional. This is not true to the extent claimed.

Computer algebra is useful but discovering new algorithms to automate mathematical work is hard.

As Robin Hanson and Steve Levitt say, life is long. There’s lots of time to do lots of different things.

Juergen Schmidhuber is right: China will surpass the US in dominance this century.

Here Robin Hanson proposes a much more efficient method of small claims resolution. The Enlightenment was about such ideas: approaching economic problems rationally where previously no one realized there was a problem.

The rapid decision-making abilities of basketball and soccer players impress me as much as their physical skills.

“Up to 40%” of travelers from developed to developing countries get travelers’ diarrhea; “in the normal population 1% to 2% of persons per year will develop irritable bowel syndrome (IBS), while 5% to 6% of travelers after traveler’s diarrhea will develop IBS”; and “the prevalence of depression and anxiety in IBS patients is 37.1 and 31.4% respectively”.

The Princeton Companion to Mathematics says “algebraists like to work with exact formulas and analysts use estimates. Or, to put it even more succinctly, algebraists like equalities and analysts like inequalities”. In computer science, algebraists like programming languages and analysts like algorithms and complexity. Or, to put it even more succinctly, algebraists like lambda calculus and analysts like Turing machines.

During retirement, write a memoir to be read by your descendants if no one else.

Mathematics, to a first approximation, is a 20th century phenomenon.

Problem solving in mathematics and beyond

2021-Jan-27

One doesn’t discover new lands without consenting to lose sight, for a very long time, of the shore.

-André Gide

One of the things that I often faced in my clinical practice and with the students that I mentored was this confusion about acting. “I don’t know what to do, so what should I do? Well nothing, I’ll wait around until I figure out what to do.” No you should put together a bad plan and you should implement it because even if you fail in the implementation you’ll gather information and then you can rectify the plan.

-Jordan Peterson

Math

Math problems are some of the hardest problems humans solve. Besides the obvious things like practicing and learning more theory, are there concrete techniques that can make us better problem solvers?

If you open a book on math problem solving, it will probably talk about heuristics. Heuristics are relevant because they sit in between generating candidate steps automatically (which requires no special teaching or memory aid) and generating them uniformly at random or exhaustively (which is useless). Heuristics may be general (see below) or branch-specific (e.g. major counting techniques, inequalities cheat sheet). They also may be more for information gathering or more for directly taking a step towards a solution.

General heuristics:

• perform change of variables
• How to Solve It
• Schoenfeld, A.H. “Teaching problem-solving skills.”
• Ch. 1 of Larson, L. C. Problem-solving through problems.
• if you’re working from definitions try leveraging theory or vice versa
• name and conquer

Claude Shannon suggests “try to restate [the problem] in just as many different forms as you can”. T. Tao even says “The human brain has got many different modes of thinking. So we have visual modes, we have symbolic modes, we have modes where we are trying to fight some sort of adversary. And by changing the language of your problem, you are activating different areas of your brain.”

Comedian Lee Mack on generating ideas:

I was suddenly in a position where I could perform as many sketches as I could think of in front of millions of people. But I didn’t have enough. So I went off with my old college mate Neil Webster, and we locked ourselves away in a cottage for three days. After a day of achieving nothing I decided that I’d had enough of staring at the fireplace not knowing what to write about, so out of frustration I picked up a magazine and told him to pick a random page number. There were loads of different little articles on the page, so I asked him to narrow it down to a corner. He said top left. It was an article about fishing. I told him that we would both sit and write a sketch about fishing. So we did. […] Then we did it again, a new page, a new random corner. We found that about every one in four mini-sketches had just about a funny enough premise, or key joke, that it was worth exploring a bit more.

Similar techniques are described here. However, in math if we’re solving a particular problem we want to generate ideas in a more constrained manner. Two possible methods are computer tools or bootstrapping by riffing off of your own discoveries while making attempts towards a solution.

Computer tools for generating ideas:

• https://github.com/andrew222651/mathtext/tree/master/info_retrieval
• AI

With regards to bootstrapping, Richard Rusczyk’s top tip is “Do something … At some point you have to stop staring and start trying stuff”. Rusczyk elsewhere: “Notice that we didn’t just sit and stare at the problem and wait for it to solve itself. We have to add lines and variables so we can build equations. Don’t expect to just memorize formulas and bash geometry problems.”

• You can’t write piano music all in your head, you have to press some keys to help prompt ideas. in math, paper is the piano: write stuff down. this can be thought of as freeing mental RAM.
• When deciding whether to try something or not, factor in not just how likely it is to solve the problem in one step but also the fact that the process of trying it may help you generate further ideas

Beyond

The “Build-Measure-Learn” loop in startup strategy is a lot like math problem solving in that it says you don’t have to start with the complete solution, and data generated by exploring the implications of an idea can help find further ideas. See also https://longform.asmartbear.com/posts/extreme-questions/ for some heuristics

Does problem solving training in one domain improve problem solving ability in another domain? Cognitive psychology finds that cross-domain transfer is not automatic. Trinchero, R., “Chess Training and Mathematical Problem-Solving” (2016) says

The results suggest that chess practice can enhance problem-solving abilities in children, but only if chess training conveys problem-solving heuristics to pupils.

The Confucian virtue of learning

2021-Jan-18

The Three Character Classic is a 13th century Chinese text with three characters per line which is traditionally read by children. Below is an excerpt from the 1812 translation by Robert Morrison, Presbyterian missionary and author of the first Chinese-English dictionary.

Chung-ni [another name for Confucius] once called a boy of ten years of age his instructor; for, of old, even perfect and wise men learned diligently.

Chao, when he held the office of Chung-ling, read Sun-yu. Though filling so high a situation, he yet learned diligently – so much so, that he never laid the book out of his hand.

In the time of the emperor Sung, Lu-wen-shu was constantly looking over the books engraven on leaves.

Wu-yao made leaves of the reed bamboo, by paring it thin. Though he did not possess books [as we do], he exerted himself in the pursuit of knowledge.

Sun-king suspended his head by its hair to the beam of his house, to prevent his sleeping over his books.

Su-tsin pricked his thigh with an awl, to prevent his sleeping.

Those persons, though not taught, of themselves rigorously pursued their studies.

Che-yin, when a boy, being poor, read his book by the light of a glow-worm which he confined. And Sun-kang, in winter, read his book by the light reflected from snow. Though their families were poor they studied incessantly.

Chu-mai-chin, though he subsisted by carrying fire-wood round the town to sell, yet carefully read his book. At last he became capable of, and filled a public office.

Li-mie, while watching his cattle in the field, always had his book at hand, suspended to the horn of a cow. These two persons, though their bodies were wearied by labor yet studied hard.

Su-lao-tsiuen, at the age of twenty-seven years began to exert himself, and read a great many books. He, when at that age, repented of his delay: you, a little boy, should early consider.

Leang-hao, at the age of eighty-two, was permitted to answer the emperor in his palace, and was placed at the head of all the literati. In the evening of life his wishes were fulfilled, and all spoke of his extraordinary learning. You, a little boy, ought to determine to pursue your studies.

Yung, at eight hears of age could recite the Odes. Li-pi, at seven years of age could play chess. These clever and studious boys were called by everyone wonderful. You, youths, ought to imitate them.

Tsai-wen-ki could play a stringed instrument. Sie-tao-wen could sing well. These ladies were clever. You, who are a gentleman, ought at an early time of life, to perfect that which is suitable.

Chin-tung, a remarkable lad, was raised by the emperor to fill the office of Ching-tsi. He, though a youth, was made a public officer. Do you, youths, exert yourselves to learn, and you may arrive at the same. Let all who make learning their pursuit be as those persons whom we have mentioned.

It is natural for a dog to watch at night, and for a cock to crow in the morning; if anyone does not learn, how can he be called a man?

(Above: In the Temple of Literature in Hanoi.)

Highlights of Canadian geography

2020-Apr-12

Mainland Canada extends south to a latitude found in California

There’s a piece of France in between Nova Scotia and Newfoundland

Victoria, BC has a “warm-summer Mediterranean climate” like Porto, Portugal and Cape Town, South Africa.

Canada’s most picturesque spot is Lake Louise

Mathematics as a service

2020-Apr-1

What would a market for mathematics look like?

Formal verification might allow an elegant mechanism: Someone posts a proposition in a formal language like Coq and the first to submit a proof that passes verification wins the bounty. Everything can be automated and maybe even trustless. This has been tried, at proofmarket.org, which was shut down due to consistency bugs in the verifier. Even without bugs, proof assistants are still difficult to use; mathematician Thomas Hales says “It is very hard to learn to use Lean proficiently. Are you a graduate student at Stanford, CMU, or Pitt writing a thesis on Lean? Are you a student at Imperial being guided by Kevin Buzzard? If not, Lean might not be for you.”

If we stick to natural language to avoid the learning curve, things get messy. How does the market decide what a complete proof is, which proof is first, and who did it? Perhaps the only tenable solution is to leave these decisions up to the individuals who post the bounties. How would we know that bounties would ever get paid? Stack Exchange forces bounties to be put in escrow and if they’re not awarded to someone there’s no refund. Another option is to rely on reputation by using certified identities (e.g. users’ email addresses are verified and public so they can be checked against personal webpages).

Something along these lines might be doable (name: proofbounty.io?) but what’s the use case? Monetary rewards for mathematical problems are rare and mathematicians generally already earn a salary, so the interest would likely be modest. Students (anywhere in the world) are plausible suppliers though, perhaps even high school students, while consumers could be anyone with a research grant usable for paying “research assistants”, or industry and non-profit research groups. A market that brings these two sides together could be of some value.

Paid question answering has been tried before, e.g. Google Answers which wasn’t very popular. Did it fail due to lack of network effects, lack of innovative mechanisms, or an essential flaw in the concept? I don’t know. Bounties on GitHub issues seem to be a bit more successful.

In addition to bounties, there could be a prediction market. The time of resolution may have to be indefinite, though, since resolving “proposition X will be publicly proved by date Y” would in general require determining the nonexistence of a public proof, which is at least somewhat error-prone. However, prediction markets are basically illegal so it’s a moot point.

March 2020 links

2020-Mar-24

James I’s 1597 book Daemonologie, “a philosophical dissertation on contemporary necromancy … touches on topics such as werewolves and vampires”.

96.5% of 19-year-old males in Seoul have myopia.

List of Scottish Canadians.

Free ebook of classic novel plot summaries.

“Kime”: complex-valued time.