Self-Evident, but not Simple

“We hold these truths to be self-evident…” So reads the Declaration of Independence, perhaps the most famous document in the history of our United States. Read it again, maybe say it out loud and you can almost see one of our bewigged founders standing at a podium, declaiming to the agitated crowd. It rings of democracy, it rings of revolution, and by golly it rings of math!

Wait, math? Yup, math. That’s at least what I was thinking as I stared at these words and those that followed during a recent family trip to Philadelphia, the first leg of an end of summer family summer vacation wherein like many others we engaged in a little bit of “civics tourism” and a last moment for summer-inspired mathematical reflections.

“Self-evident truths” are the generally the currency (oh yes, we visited the National Mint too…) of the mathematician rather than the politician. These are statements that are meant to be taken on faith as obvious, which might then be used as starting material for a mechanical kind of reasoning process of “proof” that is an application of the rules of basic logic.

A first experience with mathematical self-evident truths that many of us have comes in a Euclidean geometry class. That’s how it was for me, more years ago than I care to count, in my first year of high school. Guided by the ironic but friendly Mr. Bulman, my friends and I spent a year exploring the power of the basic postulates of Euclidean geometry. Instead of referencing the aspirational beliefs of a nascent nation (e.g., the existence of certain “inalienable rights”, or the assertion that “all men are created equal”) Euclid’s postulates, definitions and axioms frame a stripped down set of seemingly irrefutable observations —> Read More