More generally, mathematics is experimental not just in the sense that it can be used to make physical predictions, but also (probably more importantly) in that definitions are "experiments" whose outcome is judged by their usefulness.

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The most important object in modern geometry is the manifold. This is a space that looks locally like n-dimensional Euclidean space -- 1-dimensional manifolds are curves, 2-dimensional manifolds are surfaces, and higher dimensional manifolds are simply called n-manifolds. All of physics takes place on manifolds. Differential equations correspond to vector fields on manifolds. The manifold hypothesis says that much of the high-dimensional data we see actually lives on much lower-dimensional manifolds (partially explaining the unreasonable effectiveness of deep learning on very high-dimensional datasets).

The most important object in algebra is the group. The collection of symmetries of any object (e.g. a Rubick's cube, a piece of paper, or three-dimensional space) forms a group under composition. A group that is also a manifold is called a Lie group. These are everywhere -- n-dimensional rotations form groups, fundamental particles correspond to representations of Lie groups, invertible matrices form a group. Spherical harmonics and Fourier series are both naturally viewed in terms of representations of Lie groups.

The most important object in analysis is the limit. Limits first appear in the construction of the real line by adjoining limits of Cauchy sequences to the rational numbers. Using the real line, one can measure volumes, probabilities, and distances in geometric spaces such as manifolds, but also in spaces of functions, sequences, and more abstract objects. The proof of the fundamental theorem of calculus (that derivatives and integrals are roughly inverse operations) requires rigorous analysis of the definitions of derivative and integral as limits.

To learn math, you should begin by understanding what a proof is. All of mathematics is based on proving theorems. A mathematical proof is a sequence of statements that explains the logical steps required to use the assumptions of the theorem to verify the result. Just as a computer program cannot "almost output" the correct answer, there is no such thing as an "almost correct" proof. A proof either describes a correct chain of logic to reach the conclusion, or it does not. The reason math is based on proofs is because more advanced math and science builds upon more basic math. An error in a mathematical theorem or an imprecise definition will lead to bigger problems down the line, so every step must be carefully validated. For an individual student as well, only through proving theorems can one deeply understand a mathematical subject, and a solid understanding of basic subjects is required to understand more advanced topics.

Fortunately, you can learn to prove theorems at the same time as learning the foundations of math. The first books you should work through are "Principles of Mathematical Analysis" by Walter Rudin, and "Linear Algebra" by Georgi Shilov. This will be hard, not for an arbitrary reason, but because assimilating new math into your brain is intrinsically difficult, especially at the beginning. If possible, try to find a teacher.

Re: goals —- the goal is to let the kid learn as fast as they want assuming they have solid foundations. If they like proofs let them do proofs, if they like applications let them do that. Just don’t force them to sit in a classroom doing busywork for the most formative years of their lives.

Kids intuitively understand this -- they like doing what they're good at. Unfortunately, most schools are not good at serving this need. A very important part of being a parent is to encourage kids to start compounding positive habits/learning early, and to prevent the schools from dragging them back to the median.

It's not clear to me that prediction markets usually have natural hedging participants (maybe political operatives, but the tx costs are probably too high relative to the value at stake).

Walkable cities: NYC, Chicago, SF, Boston, Seattle

Urban hellscapes: LA, Houston, Atlanta, Phoenix

Simple explanation -- people don't want to walk around in the heat.

The great thing about life is that it's so multidimensional. If you want to be the richest person in the world, of course you're setting yourself up for failure. But if you want to be the best version of yourself, you can easily be the best father-husband-son-coder-blogger-walker-painter to your children+wife+colleagues in your city in July 2021.

More than that, you can do things no one else has done. If you like research, the frontier is endless and extremely high dimensional. Find some niche that you enjoy and crush it. If you like helping people, there will never be an end of people you can help. You don't have to be average -- you can be in the 1% of what you're passionate about, easily, because there are so many possible choices of passion.

90% more advanced type systems

50% IDEs, github, and slack will merge into one monolithic social developer experience

10% end user programming using NLP

Those are the basic axioms to say you're working with a nice operation that turns paths into edges in a reasonable way. It gets interesting when you add more identifications (commutative diagrams) and operations (functors). Prototypical example: vertices are sets, edges are functions, you get identification of paths (sequential compositions of functions) whenever the underlying composite functions are pointwise equal.

“No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years.”

GH Hardy 1941