Applied Foundations: Proof Mining in Mathematics

A central theme in the foundations of mathematics, dating back to D. Hilbert, can be paraphrased by the following question ‘How is it that abstract methods (‘ideal elements’) can be used to prove ‘real’ statements e.g. about the natural numbers and is this use necessary in principle?’ Hilbert’s aim was to show that the use of such ideal elements can be shown to be consistent by finitistic means (‘Hilbert’s program’). Hilbert’s program turned out to be impossible in the original form by the seminal results of K. Gödel. However, more recent developments show it can be carried out in a partial form in that one can design formal systems A which are sufficient to formalize substantial parts of mathematics and yet can be reduced proof-theoretically to primitive recursive arithmetic PRA, a formal system usually associated with ‘finitism’. These systems