# Levels of Mathematics

by Justin Skycak on

Research mathematicians are like professional athletes.

Most people have no idea how deep the levels of mathematics can go, and how cognitively taxing it is to learn the deepest levels.

Arithmetic is a completely different ballpark from graduate-level math (and beyond).

Most people consider calculus to be “really advanced math,” but calculus is not even halfway to the level at which expert mathematicians operate.

For reference, here’s a loose formulation of the levels of mathematics below:

1. Arithmetic – seldom considered “hard math”
2. Algebra – often considered “hard math” by people who disliked math in school
3. Calculus – considered “hard math” by the general public
4. Real Analysis, Abstract Algebra, Partial Differential Equations, etc. – considered “hard math” by most college students majoring in math
5. Algebraic Topology, Differential Geometry, etc. – considered “hard math” by most graduate students doing PhDs in math, as well as many research professors in math
6. The math underlying solutions to the most famous problems in modern mathematics, e.g. Ricci Flow with Surgery which underlies the proof of Poincaré Conjecture – considered “hard math” by the world’s top mathematicians

To put these levels in perspective, it can be helpful to draw an analogy to athletics:

• Learning arithmetic is like basic ambulatory movement: almost everyone can do it.
• Learning high school calculus is like being able to run ten miles without stopping: it takes time and effort to work up to it, but by training effectively and consistently, many people can accomplish it.
• Learning research-level mathematics is like qualifying for the 100-meter dash at the Olympics: it requires a certain biological predisposition coupled with the commitment of thousands of hours to the most grueling forms of training.

Nobody has a problem accepting this in athletics.

However, it’s harder to accept in the context of math because we cannot observe the makeup and functioning of our brains as clearly as we can our bodies.

But individual differences in brains do exist (e.g., working memory capacity) and are relevant to key mathematical skills (e.g., abstraction ability).

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