Topsy-turvy Maths: Proving Axioms from Theorems

Mathematics is distinguished from the sciences by the freedom it enjoys in choosing basic assumptions from which consequences can be deduced by applying the laws of logic. We call the basic assumptions axioms and the consequent results theorems. But can things be done the other way around, using theorems to prove axioms? This is a central question of reverse mathematics  [TM222 or search for “thatsmaths” at irishtimes.com].

What axioms are the “right”, or the “best”, axioms? Ideally, they should be intuitive and instinctive, simple and undoubtedly true. Reverse mathematics seeks the best system of axioms required to prove a set of theorems. Its essence is to “work backwards”, using theorems as a starting point and proving that the axioms follow from them. This implies a logical equivalence. The programme was initiated by Harvey Friedman of Ohio State University, who was once listed in the Guinness Book of World Records as the world’s youngest professor.

The basic axioms for modern mathematics are the Zermelo-Fraenkel axioms and the axiom of choice (the ZFC system), but the solution of some of the great unsolved problems of maths may require something over and above ZFC. Most mathematicians never go beyond these, but Harvey Friedman has done so.

Harvey Friedman [image Wikimedia Commons]

The foundations of mathematics are concerned with the consistency, unity, and structure of mathematics. But for nearly a century, these foundations have been known to be less than rock-solid. Friedman has been searching for decades for a new theory, one that will introduce “natural” ways for determining the best axioms. His dictum is “When a theorem is proved from the right axioms, the axioms can be proved from the theorem.”

The starting point for reverse maths is a base theory – a core axiom system – that is strong enough to state the theorems of interest, but not strong enough to prove them. The goal is then to determine the best axiom system needed to prove these theorems. As an example, we could take Euclid’s first four axioms as the base, omitting the fifth, the parallel postulate.

Euclidean Geometry

The word geometry means measurement of the Earth and the Greeks strove to understand the world in which they lived. Mathematics was a means to do this, and Euclid formulated basic assumptions or axioms that seemed self-evidently true: there is a straight line between any two points; a centre and radius determine a circle uniquely; and so on. But the fifth axiom, the parallel postulate, was more awkward and cumbersome and less obviously true than the first four axioms. It assumed that, given a line and a point, there is always another line through the point and parallel to the given line.

Thales of Miletus

Thales of Miletus is generally credited with introducing rigorous proof into mathematics centuries before Euclid. One of his most famous results states that a triangle drawn in a semi-circle has a right angle. Although this theorem was known in Babylon and India, Thales is believed to have been the first to prove it in a systematic fashion. His proof depended on assuming that the angles of any triangle add up to 180 degrees.

But how did Thales prove that last result, that three angles in a triangle sum to two right angles? Later, Euclid proved this, but only by using his fifth axiom. The parallel postulate was essential to Euclid in many of his proofs. But he could have chosen another axiom. For example, he could have started, as Thales did, with the sum of the angles of a triangle. That’s reverse maths!

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by Peter Lynch

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