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OUR AXIOMATIC FOUNDING FATHER

“WE HOLD THESE TRUTHS to be self-evident” begins the second paragraph of the United States Declaration of Independence. Its author Thomas Jefferson then enumerates these truths: “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.”

Daniele Struppa is a mathematician, newly appointed as the president of southern California’s Chapman University. What’s more, he recognizes Jefferson’s declaration of self-evident truths as an example of the axiomatic method, one of the foundations of mathematics. This method of thought was a significant achievement in The Elements of Euclid, this Greek mathematician’s treatise on geometry composed more than 2300 years ago.

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Daniele Struppa, Milan-born American academic, in-coming president of southern California’s Chapman University. Image from The Orange County Register, September 29, 2016.

That Professor Struppa would link ancient geometer Euclid and Founding Father Thomas Jefferson gives an indication of what a gifted educator he must be. His analogy is straightforward, not a fanciful stretch.

The power of mathematics lies in its certainty in modeling reality. Once basic assumptions about the real world are made, other things can be proved, not simply guessed at.

That is, the axiomatic method begins by stating a subject’s basic assumptions, its axioms. Then their consequences follow with an unassailable “if-then” certainty. There’s elegance in this approach because it deduces so much from so little.

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Euclid of Alexandra, c. 300 B.C. Author of The Elements. Image from Monticello College.

The Elements of Euclid begins with definitions of point, line, circle, angle and the like. Then follow self-evident truths, also known as axioms or postulates, about how these concepts relate to each other.

Euclid’s First Postulate says two points determine a line segment. The Second Postulate, that any line segment can be extended indefinitely in a straight line. The Third, that a circle can be drawn having a given line segment as radius and one of its endpoints as center. The Fourth, that all right angles are congruent. The Fifth Postulate, also known as the Parallel Postulate, is more subtle: It’s equivalent to the self-evident notion that parallel lines never meet.

The incredibly rich subject of Euclidean geometry followed with logical deductions from these few definitions and five axioms.

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Thomas Jefferson, 1743–1826, American Founding Father, principal author of the U.S. Declaration of Independence, third President. Portrait by Rembrandt Peale, 1800.

And so it was with Thomas Jefferson’s wording in the Declaration of Independence. The basics of American freedom were self-evident to Jefferson, assumptions from which the declaration’s subsequent assertions follow with unassailable logic.

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The Elements of Euclid viz The First Six Books Together with the Eleventh and Twelfth, Robert Simson, Glasgow, 1756. This edition would have been contemporary with Thomas Jefferson. Image from William & Mary Law Library.

The Elements of Euclid was a well-read book in Jefferson’s extensive library at Monticello. In ”An “Old-Fashioned” Nationalism: Lincoln, Jefferson, and the Classical Tradition, Drew R. McCoy writes, “Specifically, Euclid’s geometry had become, by Jefferson’s time, a testament to the power of human reason to deduce truth.” Jefferson recognized this in writing the Declaration of Independence.

The Euclid-Jefferson link doesn’t end there either. Another beauty of the axiomatic method is that it doesn’t lose its deductive power with modified axioms, with reinterpreted self-evidence.

As an example, Euclid’s geometry flourished for 2200 years and, of course, still has validity today. However, mathematicians in the 19th century posited different possible truths of Euclid’s Fifth Postulate: What if parallel lines meet? What world might be deduced from changing this single axiom?

They didn’t realize it at the time, but these 19th century mathematicians were constructing a geometry that would be an appropriate model of reality in 20th century physics.

Similarly, when Jefferson penned “all men are created equal,” he was a man of his time. Four score and seven years later, Abraham Lincoln and others felt that the words “all men” have a different meaning.

The rest of the Declaration of Independence still rings with unassailable axiomatic certainty. ds

© Dennis Simanaitis, SimanaitisSays.com, 2016

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