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Whitehead and Russell famously took several hundred pages in Principia Mathematica to prove the validity of the proposition 1+1=2. What is it that kids are understanding that W&R took much pain over? Understanding equates to learning the rules and when and where they can be applied. That's where drilling comes in.

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> Whitehead and Russell famously took several hundred pages in Principia Mathematica to prove the validity of the proposition 1+1=2. What is it that kids are understanding that W&R took much pain over?

W&R proved 1+1=2 in their axiomatic system. It wasn't done to prove once and for all that 1+1 is in fact 2, but to show that their system produced mathematical truths. The truth of 1+1=2 was already assumed and understood since they were kids, and that's why it was necessary that their system also produce it.

Of course PM was obliterated by Godel shortly after.

Godel did not destroy formalization. He showed that there would always be true things that cannot be proved. It's true that the axioms used by Whitehead and Russell are not the axioms usually used today, but PM is still an important book that has influenced much work through today and Beyond. You might find this website interesting, which formalizes axioms and then proves many things from them: http://us.metamath.org/index.html

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