Opinion 44: Two Lessons I learned from Shalosh B. Ekhad XIV's 2050 Webbook

By Doron Zeilberger

Written: Nov. 1, 2001

I just posted Shalosh B. Ekhad's PLANE GEOMETRY TEXTBOOK

that is going to be written in 2050, and that I was able, luckily, to download from the future.

Lesson 1: The Statement of the Theorem IS the Proof

Jacques Derrida taught us that Modern philosophy is based on tacit metaphysical assumptions and binary dichotomies, like cause and effect, signifier and signified, etc., that are assumed to be distinct and opposite, but really overlap. One of the main dichotomies in modern math is that of theorem/proof. The proof is distinct from the theorem, and is usually much longer. In this webbook, every statement is also the proof! Also the statements/proofs are written in a formal programming language (Maple), hence are much more reliable than any textbook written in English (or any other natural language). Also the proof is completely hyper-texed, so that nothing is left unsaid, and the proof can be performed by Maple right away.

The idea is very old, it is Rene Descartes's brilliant way of transforming Geometry to high-school algebra, and with modern compute-algebra, it is all implementable. But the statements are also a pleasure to read, since they are English-based, and with the diagram, self-explanatory.

So it is no longer necessary to have 1)Statement in English 2)Translation into Maple 3) Proof in English. Each program-theorem-statement are self-explanatory and ready to be run.

Lesson 2: Theorems are not Miracles, but Incestuous Relationships between Overdetermined Inbred Mathematical Objects

In a fascinating invited MAA talk (for students) in Jan. 1998, Roger Howe said: ``Everybody knows that mathematics is about miracles, but mathematicians have a special name for them: theorems''. While this is a lovely thought, it is unfortunately not quite right. Why are there so many ``miracles'' in, say, triangle geometry, and analogously elsewhere in mathematics? Take a triangle ABC. W.l.o.g A=[0,0], B=[1,0], and C=[c1,c2]; Hence a triangle is determined by only TWO degrees of freedom! Now one defines lots of new points and lines and circles, but all of them descended from the original ABC , and hence with coordinates that are rational functions of just c1,c2. Add to it that these rational functions are often symmetric, and that there are not too many symmetric polynomials of low degree. Hence lots of such ``miracles'' are bound to happen, for the same reason that many of my ancestors are the same, or else there would be at least 2^1000 different people 30000 years ago.

Added May 27, 2021: I just found this gem written by Caleb Ji
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