Areas of Pure Maths: Less Abstract & More Concrete

[tgt]

Pure maths is obviously abstract compared to the applied areas but which areas of pure maths are considered not as abstract or relatively less abstract?

 

[HallsofIvy]

I'm afraid you would have to say what you mean by "pure math"! Would you consider Calculus or Differential Equations to be pure math or applied math? I would consider them both "applicable math" but you don't give that category. I would suggest linear algebra as being as "concrete" as pure math gets.

 

[Tac-Tics]

Pure maths is obviously abstract compared to the applied areas but which areas of pure maths are considered not as abstract or relatively less abstract?

Much of mathematics is founded on models of the physical world. We invented counting numbers because we had to keep track of our sheep. We invented rationals to measure lengths of string and volumes of water. We invented reals to make sense of peculiarities in the lengths of triangles and circles. We invented calculus to predict the movement of planets. We invented predicate calculus to make sense of logical statements. We invented turing machines to make sense of precise procedures.

The purity of a branch of mathematics is just a measure of what is more important to you: the physical world or the model.

In set theory, no one ever flinches at the thought of an infinite collection of infinite sets. The axioms are in place, and the mathematician doesn't really care how it is reflected in the real world. Differential equations are just the opposite. There, the mathematician will forgo precise definitions and try to find a set of axioms which works pretty close to the real world phenomenon.

But it's a fuzzy, social line. Being too extreme on either end can make it difficult to apply (for pure math) or prove (for applied math). Something like Category Theory, we understand pretty perfectly, but it's a stretch to find anything useful or meaningful within it. On the other end, AI is a great use of applied math, but building a great AI system doesn't really teach us about the core principles of intelligence.

 

[hilbert250]

Probability. It begins with coin tosses, and goes into Von Mises–Fisher distributions.

 

[Werg22]

Discrete math/combinatorics.