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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. It rings of democracy, it rings of revolution and by golly it rings of math!
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"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 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 about points and lines and right angles, sometimes in language almost as flowery as that used by our nation's founders ("a point is that which has no part" and "a line is breadthless length"). We spent the year building a veritable city of increasingly complicated truths from these poetic first principles.

This felt more like art than math. I was playing with ideas here, like a mental game of Tinkertoys, using concepts instead of dowels and connectors and I really liked it. I liked the mixture of orderliness, thoughtfulness and creativity that was the process of proof construction. I enjoyed the introduction to a whole new kind of math that was very different from calculation. I found myself continually surprised (ok, delighted!) that from a small and terse set of truths you could create some pretty complex statements. It was a harbinger of things to come when later in college I found my first real mathematical "love" in the study of mathematical logic.

I heard math in the Declaration, but I don't think it was just an artifact of my own mathematical orientation and upbringing (although that helps!). A little poking around on the web turns up that late in his life, Thomas Jefferson, the chief author of the Declaration wrote, "When I was young, mathematics was the passion of my life". An early math teacher, Dr. William Small was the man who "probably fixed the destinies of [his] life". These quotes (and much other interesting material) can be found at the fascinating "Jefferson and Mathematics" website.

The yearnings of mathematical clarity for a nascent nation are understandable, but hardly realistic. The thing about mathematical reasoning is that if you start with a truth, then it only produces more truth. Mathematics and pure logic don't support simultaneous contradiction, but somehow in life, this isn't the case. Our founders were able to hold in both their heads and their behavior the "self-evident" fact of all men being created equal with the persistence of slavery, a systemic contradiction that had to be addressed not by simply pointing it out, but by a Civil War, and whose effects continued after that through to today.

Our trip to Philadelphia was followed by a few days in Washington, DC. Our civics theme required a trip to a Nationals game (baseball is the national pastime after all) as well as trips to the monuments and a museum or two. A morning walk through the necklace of beautiful and thoughtful memorials, Jefferson to Roosevelt to King made a particularly huge impression on my kids. We were all struck by the way in which each of these great leaders in their own times seemed to be working on the many of the same kinds of great problems: the achievement of social justice, equal rights, economic opportunity and freedom and peace. This is perhaps why the most affecting part of the trip to these cities was the witnessing of widespread homelessness in the shadows of these tributes to the efforts to solve the problem of enabling "the pursuit of happiness" for all.

My son said, "It's right there in the Declaration, so why don't we do it?" Well, just because a problem is easy to state, it doesn't mean that it's easy to solve. Maybe it defies solution. It's true in life and even in math. In the early twentieth century the logician Kurt Gödel stunned the mathematical world by discovering that if you had a logical system capable of producing basic arithmetic (not unlike the logical circuitry that enables the operation of your computer), then as long as you didn't embed any contradictions in the system (like including some sequence of commands able to access a hidden self-destruct instruction in your computer!) then it was possible to write down statements about mathematics using the logical language that were true, but couldn't actually be proved with the logic machine. I.e., not all math problems could be solved. Gödel's famous "Incompleteness Theorem" dashed the hopes and plans of many mathematicians who were sure that all true facts about mathematics could be derived through simple logic.

Gödel's Theorem is a fascinating and supremely important result in mathematics, but surprisingly, it doesn't seem to affect the way most mathematicians work. We all generally proceed as if all the problems we pose are solvable, even the ones that have resisted solution for hundreds of years. Sometimes it actually takes hundreds of years to find the answer (cue "Fermat's Last Theorem"). We do this, hacking away, thinking hard, enlisting our colleagues, looking back at the work of our predecessors, hoping to make progress even if there's a little voice in the back of your mind telling you that maybe it can't be done. This kind of dogged pursuit of truth is something that we should all hope our greatest leaders will continue to possess. Not working on math, but perhaps working just a little bit like mathematicians.

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