Peano surface area mean value theorem Dirichlet function Peano's counterexamples Weierstrass maximum theorem foundation reduction of mathematics to sets mathematical definitions definitions by abstraction axiom of choice Peano's axioms of arithmetic Peano's axioms of geometry affine exterior algebra reduction formulae differentiability lower and upper limits of sets tangency optimality conditions linear differential equations nonlinear differential equations Peano's filling curve interior closure distributive families anti-distributive families compactness planar measure measure theory abstract measures differentiation of measures sweeping-tangent theorem 1886 existence theorem proof subtangents