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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.''
Not absolutely certain, since they depend on consensus regarding formal distinctions, and the correct perception of those formal distinctions through whatever senses we choose for their presentation. I find this not very disturbing, and doubt the possibility of absolute certainty about anything. The correctness of formal derivations is at least as certain as primitive sensual observations, such as the sky is blue and the sun rose this morning. They are more robust, since we are at liberty to repeat the verification of a derivation to the limits of our attention and tolerance for tedium, and to recast the presentation of symbols whenever we notice a potential for error or ambiguity. Although not absolute, formal derivations arguably enjoy the highest degree of objective certainty that is attainable by rational intelligence.
The strength of certainty is not purely quantitative---there are different sorts of certainty. When we stand on Gibraltar, we feel an a priori sort of certainty in the independent quality of that rock as a support for our feet. Certainty in the derivation of formal systems is a more social sort of certainty. It shares some of the qualities of our certainty in an automobile that is warranted by a reliable firm. We are confident, but not that the auto will always function perfectly. Rather, we are confident that we can recognize deviations, and adjust the machine, with occasional appeal to the maker, so that it eventually gets us where we want to go. Similarly, we are certain about derivations in formal systems because we can detect errors, and we can refine our physical presentations of arrays of symbols to overcome momentary confusion and ambiguity. Because of the extreme efficiency and malleability of the basic symbolic materials underlying formal systems, the degree of certainty in final success is much stronger than the degree of certainty in even the best engineered automobiles.
Strictly, a formal system only gives us strong certainty that certain derivations do or do not follow the rules of the system. They do not and cannot provide certainty that particular natural phenomena, such as the configuration of paths followed by particles of light, follow precisely the rules of a formal system, such as Euclidean or non-Euclidean geometry. But, the scope of formally derived certainty is much more valuable practically than this mere certainty relative to the rules suggests to pessimists. Reasoning about formal systems can give us extremely high certainty that the presence of one formal pattern entails the presence of another. Since our observations of the universe abound in, and arguably consist entirely of, recognitions of formal patterns, the actual effectiveness of formal systems is substantial, and not at all unreasonable.
Gödel's famous incompleteness theorem shows the first step in Hilbert's program to be inherently impossible to achieve. No single formal system can derive all of the truths of integer number theory. If we accept that Descartes' program contains the first step in Hilbert's, then his program is also inherently impossible. The second step in Hilbert's program depends on the first. But, if we choose a single formal system that is sufficient for some part of mathematical practice, there is still value in proving the consistency of that limited system. Gödel also showed that consistency of one formal system requires reasoning that is in some technical sense too powerful to be carried out in the system under study. It is natural to conclude that Hilbert's second step is impossible, even accepting a limited accomplishment of the first step. Takeuti pointed out that the technical power of a system involved in Gödel's theorem is not necessarily connected to the ontological level of our confidence in the system. So, it makes sense to prove the consistency of a formal system using rules of reasoning that are technically more powerful, but intuitively more secure, than those of the system under investigation. The practical impact of this approach to Hilbert's second step has only been partially explored.
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