Wednesday, July 9, 2014

Pasch's Theorem/Axiom

Learn something new everyday….

I recently discovered a math misconception I was unaware of… and will pass along for any others not aware of it. "Pasch's axiom" is a little-discussed "axiom," discovered by Moritz Pasch in 1882, that is needed for, but missing from, Euclid's axioms of plane geometry. I'd heard of it previously but now learn that what I'd always considered Pasch's axiom is actually Pasch's "theorem," and that his actual "axiom" is slightly different.

Here is what Davis/Hersh write of Pasch's "axiom" in their classic, "The Mathematical Experience," where I think I may have first learned of it (you can use the line depiction below for visual reference):

"Just what constitutes the 'straightness' of the straight line? There is undoubtedly more in this notion than we know and more than we can state in words or formulas. Here is an instance of this 'more.' Suppose a, b, c, d are four points on a line. Suppose b is between a and c, and c is between b and d. Then what can we conclude about a, b, and d? It will not take you long to conclude that b must lie between a and d.
This fact, surprisingly, cannot be proved from Euclid's axioms: it has to be added as an additional axiom in geometry. This omission of Euclid was first noticed 2000 years after Euclid, by M. Pasch in 1882! Moreover, there are important theorems in Euclid whose complete proof requires Pasch's axiom; without it, the proofs are not valid."
But it turns out that this almost trivial-sounding statement of Pasch's is actually his "theorem," while his "axiom," is stated rather more interestingly and differently:

"In a plane, if a line intersects one side of a triangle internally then it intersects precisely one other side internally and the third side externally, if it does not pass through a vertex of the triangle." (from Wikipedia)

This is interesting in part, because the "theorem" sounds so much simpler or more basic than the "axiom"… reminiscent of Euclid's 5th postulate sounding so much more involved than his other axioms (also interesting that it took until 1882 to discover this missing component of Euclid's axioms). In fact, bopping around the Web, the two Pasch outcomes are often confused, and it is not even clear to me what makes one an "axiom" and the other a "theorem" (and so the confusion is understandable, and not limited to Davis and Hersh).

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