FOM: Sum of the first n positive integers

[Mark Steiner]

About twenty years ago, I published a paper entitled "Mathematical
Explanation" where I tried to explore the characteristics of an
explanatory proof (a proof that shows why, and not merely that).  My
idea was that an explanatory proof is a generalizable proof, meaning
that it is a proof P(a) about a mathematical object in a family F such
that by substituting b, another object in F, we get a proof P(b), which
preserves the same proof idea, modulo the necessary deformation of the
proof needed to treat b.  Proofs in elementary number theory by
induction usually do not have this characteristic, because one usually
has to have discovered the theorem before one can prove it by
induction.  On the other hand, the proof by counting lattice points, or
the "Gaussian" proof by summing forward and backward, rely on a specific
symmetry of a mathematical object (e.g. the sequences 1,2,...,n) such
that we could substitute a different object and get another theorem,
such as the general sum of an arithmetic progression.  Or we could
change the geometric objects from triangles to squares, etc.
As a result of this idea, I actually made a "discovery," the only
mathematical idea I have ever had.  Namely, most of the usual proofs of
the Pythagorean theorem are not explanatory in the above sense.  But
there is one given by Polya (he states that it's the one Euclid really
had in mind), which is that if we draw the altitude of the triangle to
the hypotenuse as base, this decomposes the triangle into two triangles
(I and II) which are similar to each other and to the entire triangle
(call it III).  This property uniquely characterizes the right
triangle.  Now consider triangles I, II, III as constructed on the three
sides of the triangle; I+II=III.  By considerations of similarity, three
squares constructed on the sides of the triangle will also sum, qed.
Now if we "deform" the triangle by increasing or decreasing the right
angle and "try" to do the decomposition, we can decompose the triangle
into triangles I and II, but there is an area left over.  Calculating
this area we get the law of cosines, which is usually proved FROM the
Pythagorean theorem, where in fact it is a generalization.
I would be interested to see whether this idea is formalizable.