[Mathematical] proofs in practice
At some level the purpose of a proof is to establish that something is true. But in the practice of modern mathematics proofs have taken on a broader role; indeed they have become the primary framework for the vast majority of mathematical thinking and discourse. And from this perspective the kinds of proofs given on pages 810 and 811—or typically generated by automated theorem proving—are quite unsatisfactory. For while they make it easy at a formal level to check that certain statements are true, they do little at a more conceptual level to illuminate why this might be so. And indeed the kinds of proofs normally considered most mathematically valuable are ones that get built up in terms of concepts and constructs that are somehow expected to be as generally applicable as possible. But such proofs are inevitably difficult to study in a uniform and systematic way (though see page 1176). And as I argue in the main text, it is in fact only for the rather limited kinds of mathematics that have historically been pursued that such proofs can be expected to be sufficient. For in general proofs can be arbitrarily long, and can be quite devoid of what might be considered meaningful structure.
Among practical proofs that show signs of this (and whose mathematical value is thus often considered controversial) most have been done with aid of computers. Examples include the Four-Color Theorem (coloring of maps), the optimality of the Kepler packing (see page 986), the completeness of the Robbins axiom system (see page 1151) and the universality of rule 110 (see page 678).
In the past it was sometimes claimed that using computers is somehow fundamentally incompatible with developing mathematical understanding. But particularly as the use of Mathematica has become more widespread there has been increasing recognition that computers can provide crucial raw material for mathematical intuition—a point made rather forcefully by the discoveries in this book. Less well recognized is the fact that formulating mathematical ideas in a Mathematica program is at least as effective a way to produce clarity of thinking and understanding as formulating a traditional proof.