On Pure Mathematics

I was always good at math in school, but when I reached the end of the standard Calculus curriculum at the college level and kept going, I realized that everything I had seen for the past decade was child's play.

Studying pure math is like a phase change in the state of matter: what you thought was a solid can now be rearranged in infinite combinations. Every assumption can be disassembled down to the plus, minus, and equals and rebuilt in a new way. The tools of pure math enable the creation of new universes. Many of these universes are merely intellectual curiosities, but sometimes they become breakthrough tools for thinking.

Pure math teaches you how to think better. It is a neuro-programming upgrade for your brain. After grinding through enough proofs at a deep level and understanding the structure of numbers, the learning tools are sharpened. One can learn any other subject quickly because it's an easy lift. 

The first exposure to pure math concepts in grade school usually arises in geometry. The grade school curriculum would benefit from more emphasis on discrete math, logic, and number theory and less on calculus. These will better prepare kids for the successful deployment of mathematical thinking in the digital age.

On Probability and Statistics

Mathematicians have a bias toward certainty, which may result from natural selection for success in the field. For example, little attention is given to inconsistent mathematical systems, even though the dominant, consistent, systems are known to be incomplete according to the Godel Incompleteness Theorem. There is some intuitive discomfort with accepting that mathematical systems may be incomplete.

Probability theory and statistics open up a mathematical window to reasoning with unknowns. I only studied Bayesian probability theory several years after my math degree. But when I did, it provided a thinking framework upgrade that enabled better decision-making in countless life situations.

In business, the Pareto principle refers to the 20% of inputs that produce 80% of the result (e.g., effort to productivity, customers to revenue, etc.). I have adopted an internal Pareto principle for decision-making: I can't be more than 80% certain of anything; therefore, making executive decisions at the 80% certainty level should be considered equivalent to 100%. Thresholds between 60% to 80% are acceptable operating parameters for most situations.