In Search of the Roots of Formal Computation

It became clear by the 1930s that steps of formal computation are also steps of formal deduction as defined by recursion equations and other principles of arithmetic. This development began with attempts at giving a foundation to the laws of commutativity and associativity of sum: Followers of Kant's doctrine of the synthetic a priori in arithmetic, a certain Johann Schultz as foremost, missed by a hair's breadth the proper recursive definition of addition that appeared instead first in a book of Hermann Grassmann of 1861. Schultz in his book Anfangsgründe der reinen Mathesis (Basics of pure mathesis, 1790) missed the in-ductive proofs of commutativity and associa-tivity, and could not do better than claim that any attempted proof of the commutativity of addition would be circular. He gives instead an inductive proof of the commutativity of product in which the right recursion equations appear as " corollaries " , a reverse of the conceptual order recursive definition – inductive proof. This order was found by Grassmann and a line can be followed from it to Hankel, Schröder, Dedekind, Peano, and Skolem, the last mentioned marking the birth of recursive arithmetic. Schultz in his two-volume Prüfung der Kantischen Critik der reinen Vernunft (Examination of the Kantian critique of pure reason, 1789 and 1792) asks (vol. 1, p. 220): How do I know, then, that this willful procedure [of changing the order of a sum] that lies in no way in the concept of addition itself, provokes no difference in the sum 7+5? Schultz gives a proof of the equation 7+5= 1 – perhaps the only formula in Kant's Kritik, a " smoking gun " of sorts in later writers. He uses the axioms of commutativity and as-sociativity of addition, then states that they are " indispensable for arithmetic " (p. 219), for without them, one would get only that 7+5 = 7+(4+1) = 7+(1+1+1+1+1). I take this to be Schultz' reading and interpretation of Kant passage. With commutativity and associativ-ity taken into use, one gets the following proof of the Kantian formula (Prüfung, p. 220): Instead of all of 5, or its units taken together and added at once to 7, I must take them instead successively one by one, and instead of 7+(4+1) first set 7+(1+4), and in place of it (7+1)+4, so then I get thanks to the concept of the number 8, namely that it is 7+1, first 8+4 …