Mathematics as isolation of logical difficulties of learning
Mathematics (especially pure mathematics at the non-research level) provides a kind of "lab environment" for analyzing the logical difficulties that arise during learning. Many other subjects besides mathematics have other difficulties that make learning difficult, including:
- Grappling with counterintuitive empirical (natural and social) phenomena (physics, economics, evolutionary biology (see section 2, "Anthropomorphic Bias"))
- Dealing with subtleties of measurement, the scientific method, etc. (the sciences)
- Dealing with models of phenomena that might be wrong or simplistic (economics, social sciences)
- Dealing with nonsense (post-modernism, literary analysis, pseudosciences)
Pure mathematics (at the undergraduate and beginning graduate levels) generally does not deal with the difficulties mentioned above. Instead, it is all about the "logical difficulties" of dealing with things like abstractions, rigorous argumentation, and the surprising connections one can find in a purely deductive environment.
By looking at the difficulties that arise when humans try to learn mathematics, we can answer questions like:
- Why is it sometimes so difficult to follow a completely deductive proof?
- How do humans learn to deal with abstract concepts?
- Why is learning mathematics so difficult when in principle "all the answers are there"?