Gunnar_Zarncke comments on What math is essential to the art of rationality?

[Gunnar_Zarncke]

My document of life-lessons spits out this (it has a focus on teaching children, but it aims high):

Key math insights of general value:

• What is a number really—Peano’s sentence

• Equality (do the same to both sides, equivalence classes)

• Negation and inversion (reversing any relationship in general)

• Variables, functions, domains

• Continuous functions

• Limits, infinities (leads e.g. to real analysis)

• Postponing operations (fractions, ‘primitive functions’, lazy evaluation)

• Probability (enumerating paths that can are taken fractionally, Bayes rule)

• Tracking errors (dealing with two or more functions/​results at the same time)

• Induction, proofs

• Transformation into another space (Fourier, dual spaces, radix sort)

• Representations of sequences and trees and graphs

• Decomposition of plans and algorithms (O-notation)

• Encoding of plans as numbers (Turing, Curry, Gödel)

The idea is to see the patterns behind the patterns (link in Einsteins Speed).

[othercriteria]

This is really good and impressive. Do you have such a list for statistics?

[Gunnar_Zarncke]

My main aha-moment in statistics occurred when I encountered the lebesgue integral. Integrals suddenly generalized a lot. Lebesgue also allows a lot more nifty but intuitive integral transformations. And of course it is needed for dealing cleanly with probability densities.

Causal networks despite needing tricky rules follow from the other points on my list (trees and probability measures)