On Computability

concepts that cut across different areas of mathematics. Logical formality per se does not facilitate the finding of arguments from given assumptions to a particular conclusion. However, strategic considerations can be formulated (for natural deduction calculi) and help to bridge the gap between assumptions and conclusion, suggesting at least a very rough structure of arguments. These logical structures depend solely on the syntactic form of assumptions and conclusion; they provide a seemingly modest, but in fact very important starting-point for strategies that promote automated proof search in mathematics. Here is a pregnant general statement that appeals primarily to the first feature of mathematical practice mentioned above: Proofs provide explanations of what they prove by putting their conclusion in a context that shows them to be correct. The deductive organization of parts of mathematics is the classical methodology for specifying such contexts. “Leading mathematical ideas” have to be found, proofs have to be planned: I take this to be the axiomatic method turned dynamic and local. This requires undoubtedly the introduction of heuristics that reflect a deep understanding of the underlying mathematical subject matter. The broad and operationally significant claim is, that we have succeeded in isolating the leading ideas for a part of mathematics, if that part can be developed by machine — automatically, efficiently, and in a way that is furthermore easily accessible to human mathematicians. This feature can undoubtedly serve as a springboard for the second feature I mentioned earlier, one that is so characteristic of the developments in modern mathematics, beginning in the second half of the 19 century: the introduction of abstract notions that do not have an intended interpretation, but rather are applicable in many different contexts. (Cf. section 5.5.) The above general statement concerning mathematical explanation can now be directly extended to incorporate also the second feature of actual mathematical experience. Turing might ask, whether machines can be educated to make such reflective moves on their own. It remains a deep challenge to understand better the very nature of reasoning. A marvelous place to start is mathematics; where else do we find such a rich body of systematically and rigorously organized knowledge that is structured for intelligibility and discovery? The appropriate logical framework should undoubtedly include a structure theory of (mathematical) proofs. Such an extension of mathematical logic and in particular of proof theory interacts directly with a That is, it seems to me, still far removed from the introduction of “abstract terms” in Gödel’s discussions. They are also, if not mainly, concerned with the introduction of new mathematical objects. Cf. note 10. That is a classical observation; just recall the dual experiences of Hobbes and Newton with the Pythagorean Theorem, when reading Book 1 of Euclid’s Elements. Saunders MacLane articulated such a perspective and pursued matters to a certain extent in his Göttingen dissertation. See his papers [1935] and [1979]. To mention one example: in an abstract setting, where representability and derivability conditions, but also instances of the diagonal lemma are taken for granted as axioms, Gödel’s proofs can be found fully automatically; see [Sieg and Field]. The leading ideas used to extend the basic logical strategies are very natural; they allow moving between object and meta-theoretic considerations via provability elimination and introduction rules. On Computability 613 sophisticated automated search for humanly intelligible proofs. How far can this be pushed? What kind of broader leading ideas will emerge? What deeper understanding of basic operations of the mind will be gained? —We’ll hopefully find out and, thus, uncover with strategic ingenuity part of Turing’s residue and capture also part of what Gödel considered as “humanly effective”, but not mechanical — “by asking the right questions on the basis of a mechanical procedure”. 6.4 (Supra-) Mechanical devices Turing machines codify directly the most basic operations of a human computor and can be realized as physical devices, up to a point. Gödel took for granted that finite machines just are (computationally equivalent to) Turing machines. Similarly, Church claimed that Turing machines are obtained by natural restrictions from machines occupying a finite space and with working parts of finite size; he viewed the restrictions “of such a nature as obviously to cause no loss of generality”. (Cf. section 4.5.) In contrast to Gödel and Church, Gandy did not take this equivalence for granted and certainly not as being supported by Turing’s analysis. He characterized machines informally as discrete mechanical devices that can carry out massively parallel operations. Mathematically Gandy machines are discrete dynamical systems satisfying boundedness and locality conditions that are physically motivated; they are provably not more powerful than Turing machines. (Cf. section 5.2.) Clearly one may ask: Are there plausible broader concepts of computations for physical systems? If there are systems that carry out supra-Turing processes they cannot satisfy the physical restrictions motivating the boundedness and locality conditions for Gandy machines. I.e., such systems must violate either the upper bound on signal propagation or the lower bound on the size of distinguishable atomic components. In Paper machines, Mundici and I diagnosed matters concerning physical processes in the following way. Every mathematical model of physical processes comes with at least two problems, “How accurately does the model capture physical reality, and how efficiently can the model be used to make predictions?” What is distinctive about modern developments is the fact that, on the one hand, computer simulations have led to an emphasis on algorithmic aspects of scientific laws and, on the other hand, physical systems are being considered as computational devices that process information much as computers do. It seems, ironically, that the mathematical inquiry into paper machines has led to the point where (effective) mathematical descriptions of nature and (natural) computations for mathematical