Sunday, November 29

Principia Mathematica for Children

You may have already heard of Logicomix; visit the website and click on the preview link to read some of it; better yet, buy the book. Wonderful stuff.

It no doubt will sound odd, a bizarre sort of bragging, but from the time I was old enough to read my Dad would talk to me about the theory of sets and types developed in The Principles of Mathematics. Russell and Whitehead's heroic quest for certainty in the face of paradox made for great drama in my mind. Dad was right that all of these stories could be grasped by (even) a child, and he enjoyed hearing me expounding on them to my mother and sister.

I can repeat what he taught me, a half-century ago, verbatim.
The class of all classes that are not members of themselves: is it a member of itself? Put another way: the Barber of Seville only shaves those who do not shave themselves. Who shaves the Barber of Seville? He can't shave himself — he shaves only those who don't shave themselves! But then he must shave himself — he shaves all who don't shave themselves!
We arrive at Cantor's Infinite Hotel. Unfortunately, the desk clerk tells us, the Hotel is full! Cleverly, he makes a room available for us by having the occupants of room 1 move to room 2, those in room 2 move to room 3, and so on, leaving room 1 empty. In the same way, when 100 guests arrive at once, the clerk can free the first 100 rooms by moving the occupant of room N to room N+100. Then, disaster strikes. An infinite number of new guests arrive! What to do? Simple: the occupant of room N moves to room N*2 — an infinite number of rooms are now available.
Dad's history of the class-of-all-classes paradox has Russell realizing the problem just after Principles was sent to the publishers, completely demolishing years of work and requiring the material be rewritten. Logicomix emphasizes the impact on Frege's work (also at the printers!) instead. Wikipedia has it that Russell's book was indeed at the printers, but he simply added the theory of types as an appendix. I like my Dad's version better. What heroic intellectual honesty: years of work creating this perfect logical structure and destroying it yourself in a single flash of insight.

I miss my Dad, but I haven't lost him: he made me.