Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Proof structures

The proof shown is in a sense based on very low-level steps, each consisting of applying a single axiom from the original axiom system. But in practical mathematics it is usual for proofs to be built up in a more hierarchical fashion using intermediate results or lemmas. In the way I set things up lemmas can in effect be introduced as new axioms which can be applied repeatedly during a proof. And in the case shown here if one first proves the lemma

a \[Nand] (a \[Nand] (b \[Nand] ((a \[Nand] a) \[Nand] c))) == b \[Nand] a

and treats it as rule 6, then the main proof can be shortened:

When one just applies axioms from the original axiom system one is in effect following a single line of steps. But when one proves a lemma one is in effect on a separate branch, which only merges with the main proof when one uses the lemma. And if one has nested lemmas one can end up with a proof that is in effect like a tree. (Repeated use of a single lemma can also lead to cycles.) Allowing lemmas can in extreme cases probably make proofs as much as exponentially shorter. (Note that lemmas can also be used in multiway systems.)

In the way I have set things up one always gets from one step in a proof to the next by taking an expression and applying some transformation rule to it. But while this is familiar from algebraic mathematics and from the operation of Mathematica it is not the model of proofs that has traditionally been used in mainstream mathematical logic. For there one tends to think not so much about transforming expressions as about taking collections of true statements (such as equations u == v), and using so-called rules of inference to deduce other ones. Most often there are two basic rules of inference: modus ponens or detachment which uses the logic result Implies[x && Implies[x, y], y] to deduce the statement y from statements x and Implies[x, y], and substitution, which takes statements x and y and deduces x /. p -> y, where p is a logical variable in x (see page 1151). And with this approach axioms enter merely as initial true statements, leaving rules of inference to generate successive steps in proofs. And instead of being mainly linear sequences of results, proofs instead become networks in which pairs of results are always combined when modus ponens is used. But it is still always in principle possible to convert any proof to a purely sequential one—though perhaps at the cost of having exponentially many more steps.

From Stephen Wolfram: A New Kind of Science [citation]