By Niels Henrik Gregersen
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Chaitin, G. J. 1987. Algorithmic Information Theory. Cambridge: Cambridge University Press. , and T. Toffoli. 1982. ” International Journal of Theoretical Physics 21: 219–253. Gacs, P. 1986. ” Journal of Computer and Systems Sciences 32: 15–78. , and J. Reif. 1985. ” Proceedings of the Seventeenth Association of Comput ing Machinery Symposium on the Theory of Computing: 388–395. , and D. Johnson. 1979. Computers and Intractability: A Guide to NP Completeness. San Fransisco: Freeman. Levin, L. A.
Such a language, with its rules, is called a formal system. A formal system is defined so precisely that a proof can be evaluated by a recursive procedure involving only simple logical and arithmetical manipulations. In other words, in the formal system there is an algo rithm for testing the validity of proofs. Today, although not in Hilbert’s time, the algorithm could be executed on a digital computer and the machine could be asked to “judge” the merits of the proof. Because of Hilbert’s requirement that a formal system have a proofchecking algorithm, it is possible in theory to list one by one all the theorems that can be proved in a particular system.
We seem to be constructed from an overwhelmingly vast tapestry of biological possibilities, yet our lives also consist of singularities. The hard mysteries of existence are no longer placed only in the very small (in quantum physics) and in the vastness of the universe but also in the realm of the exceedingly complex. A high degree of complexity always implies a high degree of specificity: a “thisness,” or haecceity, as the medieval philosopher Duns Scotus called it. At a personal level, we may tend to regard our lives as a gift or as a burden.
