Lógos and Máthma 2. Roman Murawski

-inductive is exactly the same as what can be proved in PA plus transfinite induction for ordinals (for all k ∈N) or in PA plus appropriate consistency statements. Similarly for PA plus full inductive satisfaction (truth) on the one hand and PA plus transfinite induction for ordinals (for all k ∈N) or PA plus appropriate consistency statements on the other. They show also that by adding to PA the notion of satisfaction (truth) and assuming that it is full and makes all theorems of PA true, one obtains a theory with exactly the same theorems about natural numbers as by taking PA augmented with a concept of a full and inductive satisfaction (truth) or PA plus appropriate consistency statements. In the above considerations, we restricted ourselves to formal proofs and to the semantical notion of truth in mathematics. We tried to show how the awareness of differences between them has been developed – from the hopes that formal proofs provide sufficient means to exhaust the mathematical truth to the discovery of various limitations of them. Let us finish with some general remarks.

      Concepts of proof and truth are (even in mathematics) ambiguous. One should distinguish between working proofs of everyday mathematics and idealized formal proofs used by logicians. On the other hand, a proof in mathematics has various aspects and can be studied from various points of view. One can distinguish psychological, social, cultural and logical aspects of proofs. a proof can be studied as a mathematical or as an epistemological object. The former is precisely defined on the basis of mathematical logic, and the latter is a vague concept. The former is an idealization of proofs occurring in a research practice of mathematicians, is a reconstruction of them. Recently, one can observe in the philosophy of mathematics a tendency to concentrate on the actual research practice of mathematicians rather than on idealized foundational reconstructions of it and consequently to study the methods actually used by mathematicians.

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Similar distinctions can be made with respect to the concept of truth. The semantical concept of truth precisely defined by Tarski is in fact a mathematical notion. It provides a definition of truth in mathematics;20 it is the concept of truth for a model in a formal language (its essential feature is to define truth in terms of reference or satisfaction on the basis of a particular kind of syntactico-semantical analysis of the language). But one can also speak about epistemic truth – cf., e.g. Isaacson (1987, 1992) where it is argued that Peano arithmetic is complete with respect to an epistemic notion of arithmetical truth.

The distinction between proof and truth in mathematics presupposes of course some philosophical assumptions. In fact for pure formalists and for intuitionists there exists no truth/proof problem. For them a mathematical statement is true just in case it is provable, and proofs are syntactic or mental constructions of our own making. In the case of a Platonist (realist) philosophy of mathematics, the situation is different. One can say that Platonist approach to mathematics enabled Gödel to state the problem and to be able to distinguish between proof and truth, between syntax and semantics.21

14 ,,Wenn wir die Tatsachen eines bestimmtenmehr oder minder umfassenden Wissensgebiete zusammenstellen, so bemerken wir bald, daß diese Tatsachen einer Ordnung fähig sind. Diese Ordnung erfolgt jedesmal mit Hilfe eines gewissen Fachwerkes von Begriffen in der Weise, daß dem einzelnen Gegenstande des Wissensgebietes ein Begriff dieses Fachwerkes und jeder Tatsache innerhalb des Wissensgebietes eine logische Beziehung zwischen den Begriffen entspricht. Das Fachwerk der Begriffe ist nicht Anderes als die Theorie des Wissensgebietes”.

15 ,,Ja, es ist doch selbsverständlich eine jede Theorie nur ein Fachwerk oder Schema von Begriffen nebst ihren nothwendigen Beziehungen zu einander, und die Grundelemente können in beliebiger Weise gedacht werden. Wenn ich unter meinen Punkten irgendwelche Systeme von Dingen, z.B. das System: Liebe, Gesetz, Schornsteinfeger [...] denke und dann nur meine sämtlichen Axiome als Beziehungen zwischen diesen Dingen annehme, so gelten meine Sätze, z.B. der Pythagoras auch von diesen Dingen.Mit anderen Worten: eine jede Theorie kann stets auf unendliche viele Systeme von Grundelementen angewandt werden”.

16 Cf. Gödel’s letter to Hao Wang dated 7th December 1967 – see Wang (1974), p. 8.

17 For the problem of the priority of proving the undefinability of truth, see Woleński (1991) and Murawski (1998).

18 Construction of SatΣn and SatΠn can be found in Kaye (1991) and Murawski (1999).

19 Note that many of those theorems hold not only for Peano arithmetic PA but also for a broad class of theories – cf.Niebergall (1996).

20 One should distinguish the truth in mathematics and the truth of mathematics.

21 Note that, as indicated above, Hilbert was not interested in philosophical questions and did not consider them.

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      Some Historical, Philosophical and Methodological Remarks on Proof in Mathematics

      Introduction

      Proofs play an important role in mathematics and its methodology (in the context of justification).They form the main method of justifying mathematical statements. Only statements that have been proved can be treated as belonging to the corpus of mathematical knowledge. Proofs are used to convince the readers of the truth of presented theorems. But what is in fact a proof? In mathematical research practice, proof is a sequence of arguments that should show the truth of the claim. Of course, the particular arguments used in a proof depend on the situation, on the audience, on the type of a claim, etc. Hence a concept of a proof has in fact a cultural, psychological and historical character. In practice, mathematicians generally agree whether a given argumentation is or is not a proof. More difficult is the task to define a proof as such. Beside proofs used in the research practice there is a concept of a formal proof developed by logic. What are the relations between them? What roles do they play in mathematics?

      Problems of that type will be considered in the paper. We start by some historical remarks showing in what circumstance the idea of a proof (informal and formal) appeared. Next, the features and role played by informal proofs will be considered. The subject of the next section will be formal proofs and their relation to the concept of truth. In the closing section, some conclusions will be made and a thesis (similar to Church-Turing Thesis in the computation theory) formulated.

      Historical remarks

      The model of mathematics as a science, its paradigm functioning in fact till today, was formulated and developed in the ancient Greece about 4th century B.C. Earlier, e.g. in ancient Egypt or Babylon, mathematics consisted of practical procedures that should help to solve everyday problems such as measuring surface area or the amount of cereal in a granary or oil in a jug. In those both pre- Greek mathematics – though they were advanced and sophisticated (especially the Babylonian mathematics) – there was no need to prove statements. In fact, there were no general statements and no attempts were undertaken to deduce the results or to explain their validity. One was satisfied by instructions what should be done in order to receive the result or