80-110: The Nature of Mathematical Reasoning

Carnegie Mellon University

Spring 2015

Homework 4

Due Monday, Feb. 9

Respond to the following prompt, in 3-4 paragraphs.

We've discussed how mathematical theories are based upon definitions, axioms, etc. But any subject -- biology, fiction writing, philosophy -- has some foundational vocabulary, and some rules (or general principles) that guide its practice.

Pick a subject that you know something about. (Your major would be a good choice, but you may interpret "subject" broadly: a hobby could work too, for instance.) Explain to me the foundational vocabulary of your subject. What are the terms that are commonly used when discussing it? Are these terms defined by other terms (like "triangle" in geometry), or are they atomic (like "point")? What are some of the "axioms" that govern the subject? (For instance, perhaps poetry has the axiom "all poems must conform to some fixed meter.")

Examine the objectivity of your chosen subject. Does it make claims to truth? (Are biological statements true or false? Are poems?) Are the "axioms" subjective, or objective? Do people universally follow the axioms, or can they be broken?