David Hilbert and Paul du Bois-Reymond: Limits and ideals

Paul du Bois-Reymond (1831-1889), younger brother of physiologist Emil du Bois-Reymond, was among the most accomplished and influential mathematicians in the second half of the 19th Century. He made substantial contributions to the areas (we would today call) differential equations, calculus of variations, functional analysis, topology and nonstandard analysis, not to mention the foundations of mathematics. Paul du Bois-Reymond may have been the first to employ an explicitly diagonal argument and is credited with introducing into analysis the notions of dense set and choice sequence. In his "General Function Theory" (1882), du Bois-Reymond presented and defended a skeptical philosophy of mathematics and argued informally, on the basis of that philosophy, that mathematics contains absolutely undecidable statements. In the first decades of the 20th Century, mathematician David Hilbert explicitly attacked Emil du Bois-Reymond's famous "Ignorabimus" in several articles and lectures. It is here argued that Paul du Bois-Reymond's ideas, although not so famous, afforded a natural target for the positions Hilbert took up in his proof-theoretic Program, perhaps a more natural target than Brouwer's intuitionism.