Areas of Pure Maths: Less Abstract & More Concrete

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Discussion Overview

The discussion revolves around identifying areas of pure mathematics that are perceived as less abstract or more concrete compared to applied mathematics. Participants explore the boundaries and definitions of pure versus applied mathematics, along with examples and perspectives on various mathematical fields.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions the definition of "pure math" and suggests that areas like Calculus and Differential Equations could be seen as applicable rather than purely abstract.
  • Another participant argues that linear algebra might represent the most concrete aspect of pure mathematics.
  • A different viewpoint emphasizes that much of mathematics is rooted in physical models, suggesting that the perceived purity of a mathematical branch depends on its relation to the physical world versus its abstract model.
  • It is noted that set theory allows for infinite collections without concern for real-world reflection, while differential equations prioritize real-world applicability over precise definitions.
  • One participant mentions probability as a field that starts with simple concepts like coin tosses and evolves into more complex distributions.
  • Discrete mathematics and combinatorics are also proposed as areas of pure mathematics that may be considered less abstract.

Areas of Agreement / Disagreement

Participants express differing views on what constitutes pure versus applied mathematics, and there is no consensus on which areas are definitively less abstract. The discussion remains unresolved regarding the definitions and boundaries of these categories.

Contextual Notes

The discussion reflects varying interpretations of mathematical purity and its relationship to real-world applications, highlighting the subjective nature of these classifications.

tgt
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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?
 
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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.
 
tgt said:
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.
 
Probability. It begins with coin tosses, and goes into Von Mises–Fisher distributions.
 
Discrete math/combinatorics.
 

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