Is there a math subject harder than calculus?

Answer by Jay Verkuilen:

The only people who think that calculus is the pinnacle of mathematics are people who have no knowledge past K12 mathematics.

Very roughly speaking there are two branches of mathematics: Algebra and analysis.

Algebra starts with theory of equations and polynomials (aka what you encounter in high school) but areas like combinatorics, graph theory, linear algebra, number theory, algebraic geometry, etc., depart quickly and develop this further. Calculus is in the analysis branch, which also includes much of probability theory, topology, functional analysis, differential equations, differential geometry, etc., and is focused on the notion of continuity. These are all developments from calculus.

(I fully acknowledge that these groupings are at best approximate and arguable.)

The thing is that different areas of mathematics borrow from each other routinely. So for instance, graph theory makes use of linear algebra heavily (see Spectral graph theory) but actual application of linear algebra involves quite a bit of analysis, so if you want to do any computation on graphs, you'll be using both. Probability theory started in a very combinatoric way (i.e., in algebra) but very rapidly ran into the need to use continuous approximations to complicated combinatoric functions (via Stirling's approximation) and thus makes heavy use of analysis.

In applied areas (e.g., statistics) this continues. Most theorems in statistics require applications of both linear algebra and analysis. Here's a great example: Cochran's theorem, which makes elegant use of orthogonality, spectral decomposition of projectors, and Fourier transforms. If you combine it with the Central limit theorem you'll end up making use of Taylor series approximation as well.

Is there a math subject harder than calculus?

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