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This section is taken with very little change from my article on ``The Sources of Certainty in Computation and Formal Systems.''
The idea of a schematic derivation is worth some attention, as it illustrates the highly reflexive way in which formal systems provide reasoning power. Most of the intuitively important observations about formal systems are schematic---they are observations of patterns in the derivations of the formal system, rather than individual derivations. But, there is another formal system containing the derivations of the Combinator Calculus, and also derivations with formal variable symbols. Individual derivations in the Combinator Calculus with variables correspond to schematic patterns of derivations in the Combinator Calculus, in a rigorous way. There is yet another formal system that models the correspondence between schematic derivations in the Combinator Calculus and derivations in the Combinator Calculus with variables.
But, the trickiest twists are yet to come. The Combinator Calculus was designed specifically to be able to simulate the behavior of systems with variables, in a variable-free style. So, the Combinator Calculus contains a precise model of the behavior of the Combinator Calculus with variables, and therefore single derivations in the Combinator Calculus can demonstrate the behaviors of schematic derivations in the Combinator Calculus. And, there's a formal system that models the correspondence between the Combinator Calculi with and without variables, and the Combinator Calculus contains a model of that system, and .... Figure 1 below suggests the systems and relations described above, but of course the real picture is infinitely large, and infinitely more complicated.
Formal systems can be used to study one another in highly tangled and reflexive, but also powerful and productive, ways. This tangled reflexivity has a lot to do with the effectiveness of formal methods, and it is no doubt the source of a lot of the confusion about the precise relationship between formal systems, mathematics, and the rest of the world. Saunders Mac Lane traces the reflexivity of formal systems within mathematics rather thoroughly in Mathematics, form and function.
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