Declaration of Independence of the Thirteen Colonies, Paragraph 2, first sentence We hold these truths to be self-evident: that all men are created equal; that they are endowed by their Creator with certain unalienable Rights; that among these are Life, Liberty and the pursuit of Happiness. Morning Prayers talk, Friday, 21 April 2006 Thank you for inviting me back to speak at Morning Prayers. I know that it is a tradition of long standing that a Morning Prayers talk open with a reading from religious Scripture, either the Hebrew Bible [displaying the front of the pew Bible, then the back side for...] or the Christian New Testament. For today's service, coming four days after Patriots Day, I chose instead a celebrated passage from what might called the secular scripture of American patriotism: the noble foundational myth that is our Declaration of Independence. This text is secular, too, in another way: though it invokes the Creator, it is modeled not after Genesis but after the opening of another work, a work said to be second only to the Bible as the book most translated, published and studied in the world. I speak of [display] Euclid's Elements of geometry. The Elements begin with a short list of "common notions" and "postulates" regarded as self-evident: things that equal the same thing are equal to each other; the whole is greater than the part; any two points are joined by a straight line; all right angles are congruent; and a handful of others. On these few but well-chosen foundations, Euclid then proceeds to build, with precise and ingenious logical argument, the whole edifice of what we now call Euclidean geometry, including classics that we still admire, teach and use today -- such as the Pythagorean theorem, the Platonic solids, the area of the circle, and the Euclidean algorithm. Over the millennia, Euclid's Elements have been an inspiration not just within mathematics but to all educated people. Every educated reader in 1776 would have recognized that the writers of the Declaration of Independence allude to Euclid by beginning their document with a short list of "self-evident truths" and framing the rest of the Declaration as a logical argument based on those truths. The allusion is intentional, as is the choice to start not from the revealed truth of the Bible but from "self-evident" truths. People argue and go to war over religious truth; but self-evident truths compel agreement, as do their logical consequences, however unexpected or even uncomfortable. If you grant Euclid's postulates, you must accept the regular dodecahedron [show picture] and unique factorization into primes. If you grant that all are created equal, and entitled to life and liberty, then this entitlement too must be equal, and a government of colonial rule must be replaced by a system where every person has equal rights and equal opportunities. OK, that's not quite what they had in mind in 1776 -- for instance, at first only property-owning males had the vote. But, as Euclid's postulates logically lead us to new geometrical insights beyond what even Euclid could see, the postulate of equality implies the gradual widening of the circle of American democracy across barriers of class, race, and gender. But there's a problem here: is it really true that we are all created equal? The assertion certainly has strong moral appeal, especially when we compare modern Western democracy with states built on a foundation of basic inequality, such as Nazi Germany and, to our shame, our own United States during slavery. Another appeal is that equality simplifies policy: if we're all equal, then clearly the only fair and reasonable thing to do is to treat everybody the same, with equal rights and opportunities. And yet: if it is true that we are all equal, it sure isn't self-evident. No two of us look, think, or act the same. It takes quite a leap of vision, or a leap of faith, to look past our self-evident inequality and conceive that all of us could in fact be created equal. Indeed it is a veritable miracle if we really are created equal by every measure that counts, now that it is known that we, individually as well as in groups, are created unequal by so many other measures, measures as trivial as hair color and as complex as temperament. Remember Euclid, our paradigm of logical reasoning from self-evident truth? That paradigm worked great in 1776. But since then it has become something of a cautionary tale. In the 1800's mathematicians constructed and studied non-Euclidean geometries: alternative mathematical universes that reject one or more of Euclid's postulates but remain valid logical systems. At first these were regarded as mere curiosities, the pathological perversions of theory divorced from common sense. Then it gradually became clear that these geometries could be appreciated for their own exotic beauty [display Escher's _Angels and Devils_, based on a tiling of the non-Euclidean Poincare disc], and that the inner structure of mathematics absolutely demands study of non-Euclidean as well as Euclidean geometry. But surely the real world is still self-evidently Euclidean? Well, in the 20th century we learned that not just theoretical math but real-world physics is fundamentally non-Euclidean: the world we find ourselves in contradicts not just that troublesome parallel postulate, but even the apparently unimpeachable fact that all right angles are created equal -- that is, that any two points in space are equivalent. Euclid's geometry is a really, really good approximation; but it doesn't quite hold up if you go and measure carefully and accurately enough; and it breaks down completely if you look in the right place, as near a black hole or -- if string theory is to be believed -- anywhere at all at a small enough scale. When even Euclid's hallowed postulates can fail us, we should be wary of taking as self-evident the truths asserted in our Declaration of Independence. This is not a welcome thought for an American. If the Declaration of Independence is our scripture of patriotism, then doubting it makes us feel both unpatriotic and heretical. From early on we train ourselves to believe that the counterintuitive notions of equality and inalieanble rights for all are in fact self-evident. To be sure, we have the best of motives. We reflexively recoil from any idea of innate inequality, because we know how often such ideas are used to support and incite bigotry, hatred, and persecution. Leonard Bernstein relied on this reflex when writing _Trouble in Tahiti_: he had Sam proclaim at the top of his voice that "men are created unequal!" -- a sure way for a character to lose the sympathy of an American audience. Closer to home, the tempest that led to the resignation of President Summers began to boil over when he suggested that differences in the standard deviation of men's and women's aptitude -- not the average, just the variability -- might partly explain the persistent gender disparity at the top of many professions. For all the qualifiers and hedges, this was close enough to the taboo to make local waves and national headlines. Now inalienable rights, self-evident or not, assert a moral ideal, an ideal whose truth or falsity are beyond the realm or science. But the claim that we are created equal, that we can test, to an extent -- not as precisely as we can test Euclid's postulates, and taking great care against many confounding cultural factors on one side and our liberal, patriotic wishes on the other, but we can measure it. Indeed it is precisely to support our moral goals of rights and fairness that we must measure it. And then, in those areas where we learn that we are in fact equal, we shall dedicate ourselves with new confidence to the simple and compelling aim of equal treatment throughout our cultural and legal system; and if and where we are different, our knowledge of the difference will inform our much more complicated judgements of how best to balance everybody's inalienable rights when simple equal treatment might not be enough. To conclude: beware of self-evident truths. In mathematics we learn to watch out for arguments that, instead of logically justifying claims, assert them as obvious or self-evident, implicitly daring the reader to disagree. Such a failure of logic is called "proof by intimidation". We also learn how easy it can be to intimidate ourselves in this way. So, even when it comes to our patriotic foundations, let us fearlessly test them where we can. And in those areas where we find that, for all our evident differences, we are in fact created equal, let us celebrate that miracle and praise the Creator. Thank you.