Difficult math is about recognizing patterns

Some people will recognize patterns pretty fast, while others must solve literally hundreds of different problems, before becoming comfortable with the concepts.

People always seem amazed and baffled that some candidates can practically walk into white-board interviews unprepared, other than what they learned / did in their DS&A classes in college, and nail the interviews, while others have to basically prep 6-12 months before passing the same interview.

"Both these properties, predictability and stability, are special to integrable systems... Since classical mechanics has dealt exclusively with integrable systems for so many years, we have been left with wrong ideas about causality. The mathematical truth, coming from non-integrable systems, is that everything is the cause of everything else: to predict what will happen tomorrow, we must take into account everything that is happening today.

Except in very special cases, there is no clear-cut "causality chain," relating successive events, where each one is the (only) cause of the next in line. Integrable systems are such special cases, and they have led to a view of the world as a juxtaposition of causal chains, running parallel to each other with little or no interference."

- Ivar Ekeland

Functional/Relational programming models are just a trivial layer on top of math. Everything is pattern recognition at the end of the day.

Domain modeling is the logical extension of building standardized "patterns" that can be leveraged for rapidly building & replicating similar ideas.

Using good modeling techniques is the most important thing for managing complex systems. If you aren't sure, you can always start modeling at 6th normal form, then walk it back to 3NF as the various pieces start to make sense together. If you have your domain in 6NF and are using purely functional/relational programming, there are mountains of mathematical guarantees you can make about the correctness of your software. For instance, 6NF gets rid of null. It forces you to deal with the notion of optional facts using 0..1-1 relations and applicable query constraints.

> When my students encounter a math problem they can’t answer, I have them put it in the error log with an explanation of how they did and how they knew how to do it.

If they can't answer it, where does the "how they knew how to do it" come from? Their teacher/tutor?

Programming is significantly easier than math (for something equivalently complex) because of things like syntax checking and compiler/interpreter errors. This speeds up the pattern recognition process in the human brain.

People who are identified as being skilled at math or programming at a relatively early age are usually those who understood it in spite of the teacher/curriculum, so the ability comes as a surprise.

But many such people do not go on to distinguish themselves in either field in any way. There are always things that come easily to one person vs another, but in math and programming, the early birds are typically the only ones whose interest in the subject isn't destroyed by the teaching methods (because the learning happened in spite of them).

Consequently during math classes I used to sit at the back of the class and play counter strike all day on my laptop. Nobody seemed to care since I'd ace all the tests and still compete for my school in math competitions and stuff. However I completely wrecked my math education, and come university (I skipped last year of high school for uni, there's a standard program for it in my country) I had completely forgotten how to prepare for a math exam and was systematically left further behind with every year Lol.

Looking back I still kind of regret my perspective on doing practice problems. In hindsight it was kind of stupid but it was mostly because I thought it was kind of lame that I did well sometimes because I practiced more than other people, whereas some other students seemed to do pretty good without (seemingly) having practiced at all. On the plus side I do feel I learn things a lot faster than much people and am pretty descent at a wider variety of things

That's why an expert can charge so much for 1 hour of time. It is more valuable than days or weeks or months of a non-expert's time who doesn't have the library and can't recognise the pattern.

I'm gonna assume a step where they learned how to do it?

TFA's method is for incremental discovery expertise. Feynman talks about an inverse, where he maintained a list of interesting problems, and when he learnt a new technique, tried it on each one.

But Feynman's actual breakthroughs came from playfully looking at phenomena.

I think the incremental skills are basics like reading, writing and arithmetic - it's harder to really get to grips with something you've noticed without them.

I mean, Einstein famously didn't have adequate math for special relativity and sought help. He was however the one to notice something.

A library of techniques is a poor substitute for actual thought.

I remember during my first year buying a book called something like 1000 limit problems. I "just" did about 300. It was definitely pattern matching and nothing Mathematica wouldn't do better than me.

https://github.com/adamnemecek/adjoint/

I think instilling this optomism in students --- following their curiosity won't lead deeper in a bottomless pit, if something doesn't make sense it's might be them but a lack of information, etc. --- is the essentially hard part, and requires undoing a lot alienation people experience.

Conversely, I think messing around with block boxes like machine learning we don't understand is giving into the alienation. (Studying it to understand it rather than do things is fine.) I worry more use of machine learning like things will be a another nail in liberalism's coffin as do the equivalent of regressing back to alchemy from chemistry.

Now, looking for patterns is what machine learning does, but while Rorschach-test-style grappling in the dark might be the basal "reptiling" instinct that lead to more high level theory-based pattern renegotiation, they should not be conflated.