Philosophical aspects of proof theory DAG PRAWITZ University of Stockholm I. INTRODUCTION. DIFFERENT DIRECTIONS OF THE SUBJECT 1. The subject and its name The term proof theory, in Gennan Beweistheorie, was introduced by Hilbert [1] in 1922 as the name of a subject which he had already described in a lecture [2] in 1917 as 'an important, new field of research' where 'we must make the concept of specific mathematical proof itself object of investigation, just as also the astronomer pays attention to his place of observation, the physicist must care about the theory of his instrument, and the philo sopher criticizes reason itsdf'. Hilbert had great expectations of this new discipline. In [2] he mentions a number of questions that belong to it: 'the problem of the solvability of every mathematical question, the problem of checking the result of a mathematical investigation, further the question of a criterion for the simplicity of a mathematical proof, the question about the relation between matter and form (lnhalt­ lichkeit und Formalismus) in mathematics and logic, and finally, the problem of the decidability of a mathematical question by a finite number of operations.' He even thought he had found a solution of the continuum hypothesis by the help of his proof theory [3 J. But first of all, proof theory was meant to be the essential ingredient in Hilbert's program. The aim of this program was to obtain a finitary foundation of mathematics, and the way in which this was to be done has been variously formulated as showing that the use of transfinite principles leads to correct finitary results or that all transfinite elements can be eliminated in Contemporary philosophy. A new survey. Vol. 1, pp. 235-277. ©1981. Martinus NiihoffPublishers. Tile Hague/Boston/London. 236 D. Prawitz principle from proofs of finitary results or, more simply, as demonstrating the consistency of mathematics, which problems are all eq uivalent under certain conditions. In spite of Hilbert's greater ambitions, to which also Bernays [4] has recently called attention, proof theory has to a large extent been dominated by Hilbert's program and has even come to be identified with work on it. Therefore, with growing interest in other problems concerning proofs, there has seemed to be a need to distinguish between a narrower and a broader use of the term proof theory. Prawitz [5, 6] suggests the name reductive proof theory for a study of proofs aimed at a reduction of trans­ finite or in general abstract principles to more elementary ones along the lines suggested by Hilbert and the name general proof Theory for a broader investigation when proofs are taken as objects of study for their own intrinsic interest and not for reductive aims. This terminology is also followed e.g. in the recent monograph on proof theory by Cellucci [7]. Proof theory taken in a very broad sense is certainly not a creation of Hilbert. The subject can fairly be said to have started with Aristotle's works in logic. Indeed, logic as it was cultivated until quite recently was to a great extent identical with proof theory: it studied arguments, distinguished between valid and invalid forms of arguments, and tried to give a systematic presen­ tation of the valid ones. In short, its general aim was to give a sys­ tematic account of correct, strict reasoning or, in other words, of what constitutes a proof. In Hilbert's vision of the subject, however, proof theory was to be distinguished from the older tradition of logic in especially two ways: firstly, instead of proofs in a more absolute sense, he wanted to study formal proofs, i.e. derivations in given formal sys­ tems, and, secondly, he thought of the subject as a strictly mathe­ matical one. The first feature was made possible by Frege's sharpening of the axiomatic ideal by the creation of logistic or formal systems. The second feature was essentially Hilbert's own idea. Frege and his immediate successors such as Russell and Whitehead may be said to have studied proofs in the sense of having isolated different rules of inference and having investigated by what principles various mathematical proofs can be carried out, but the proofs themselves did not then occur as objects of a mathematical theory. , Philosophical aspects o/proof theory 237 Proof theory in Hilbert's sense thus makes its appearance at about the same time as model theory begins in the shape of Skolem's investigations and Tarski's semantics where also formal languages are objects of mathematical study. But in contrast to proof theory, model theory tries to study a notion of truth and logical consequence independently of the question of how we come to know that a sentence is true or a logical consequence of given premisses - it is true that most presentations of model theory contain proof-theoretical elements, but pure model theory may be presented without mentioning a deductive apparatus for the language studied (as most consistently done by Kreisel and Krivine [8]). If proof theory is to pay attention to the epistemological nature of proofs, i.e. to their raison d'etre as the means of getting to know truths and logical consequences, then, however, it is highly doubtful whether the two special features which charac­ terize proof theory in Hilbert's sense are appropriate. Firstly, we know from G6del's first incompleteness proof that the notion of proof cannot in general be characterized in terms of formal systems: given any formal notion of proof, there is e.g. a universal numerical proposition which has no proof of the given kind but which nevertheless is not only true but can also be proved to be true:'by Godel's method. Secondly, one would expect that a study of proofs in their capacity as means of getting to know truths must take into account also the meanings of the sentences involved, and that it thus has to merge with a meaning theory as argued by e.g. Prawitz [9, 10]. It may be only a terminological question whether such a theory is called mathematical or not, but in any case, it has to attend to concepts quite different from the ones Hilbert had in mind. It is also to be noted (as in Prawitz [10]) that the meaning theory in question must be quite different from Tarskian seman­ tics, and that there is no direct route from the notion of logical consequence to the notion of proof: a proof may correctly be said to consist of a number of valid inferences, but the condition that the conclusion is a logical consequence of the premisses is clearly only a necessary and not a sufficient condition for the appearance of an inference in a proof; nobody counts just some axioms immediately followed by a non-trivial theorem as a proof 238 D. Prawitz even if the theorem happens to follow logically from the axioms. Obviously, some kind of evidence is also required of the inferences in the proof; one must in some way take seriously that the con­ clusion is seen to be true. In conclusion, three kinds of proof theory emerge: (1) reductive proof theory directed towards questions of consistency or, more generally, towards reductions of abstract principles to more elementary ones, (2) a more embracing, mathematical proof theory as envisaged by Hilbert (to be distinguished from his program), and (3) a more general proof theory not restricted by the features Hilbert had in mind but attending to the general epistemological nature of proofs and questions of validity of argu­ ments. Kreisel has in some papers, e.g. [11, 12J , suggested the name theory of proofs for a study that concentrates on such things as the structural complexity of proofs but disregards questions about what makes a proof a proof, or in other words, what makes an argument valid, which has been the focus of papers such as Prawitz [10, 13J. What Kreisel has in mind here seems to be proof theory in the sense of (2) except that it is to be explicitly separated from the aims in (1) and (3). Kreisel [14] argues that there are reasonably adequate data for a 'natural history' of proofs (comparable to early mineralogy or biology) but that there is no hope at present for a rewarding, systematic science of proofs (by which he means a theoretical science that is rewarding in the same way as "success­ ful" natural sciences like crystallography or genetics, to continue his examples above). We may here see a fourth suggested direction of proof theory if we want, viz. in the form of case studies of striking phenomena of proofs, exemplified by Kreisel [14 J (but in the terminology of that paper, also proof theory in the sense of (2) and (3) is to be classified as 'natural history'). There is thus no complete agreement in the literature about what direction the subject should or could take nor about the question what it should be called. As to the last question, the idea to use 'proof theory' and 'theory of proofs' as two non-synony­ mous terms denoting different aims in the study of proofs does not strike one as very suggestive (and Kreisel need not be under­ stood as proposing that but only as wanting to change the name of the subject to avoid associations with directions that according Philosophical aspects of proof theory 239 to him are better disregarded). One may ask whether it is proper to widen the term proof theory to cover also studies of kind (3), whose aims more or less coincide with that of traditional logic as noted above.