Gerhard Heinzmann

Department of Philosophy, University of Nancy 2, Laboratoire de Philosophie et d'Histoire des Sciences – Archives Henri Poincaré, UMR 7117 du CNRS

23, Bd. Albert-Ier

F-54015 Nancy Cedex

Tel and Fax: 0033/383967083

E-mail: Gerhard.Heinzmann@univ-nancy2.fr

in D. Gabbay,/S.Rahman/J.M. Torres/J.-P. van Bendegem (eds.), Logic, Epistemology and the Unity of Science, Dordrecht/Boston/London : Kluwer, à paraître

I. TOWARDS A FOUNDATIONAL ECHEC

1.1–The best known studies on the foundations of mathematics in the end of the 19th century were less written by philosophers as by philosophically minded mathematicians: the titles I have in mind concern the foundations of arithmetic and geometry, and were written by Frege (Grundlagen der Arithmetik, 1884), Dedekind (Was sind und was sollen die Zahlen? 1888) and Peano (Arithmetices principia, 1888; I principii di geometria, 1889), by Poincaré (On the Foundations of Geometry, 1898) and by Hilbert (Grundlagen der Geometrie, 1899). The genesis of the questions underlying these studies is well known. They studies themselves are already partially concerned with what I call the principal tools of foundational studies in the 20th century, i. e. unification and presentation concerns of the whole field (such as generality, minimum number of primitive terms and axioms, clear language), epistemological concerns (consistency, existence, relation between mathematical and physical objects, what is a proof?) and historical aspects (How does mathematical knowledge grow? How are methods of proofs modified?). Historical aspects are present in Frege's booklet, but absent in the leading movement of foundational studies in the first half of the century: I allude naturally to Logical Empiricism. Now, if the concept underlying the historical interest is not reduced to a strictly expository accumulation of the details of past science (cf. Shea 1983, 3), and if, nevertheless, the history is written from some point of view and with some unifying theme without to be reduced to a simple confirmation theory (cf. Grattan-Guinness 1996), then historical studies fall under the domain of foundations. I think we all can agree with the observation that the equilibrium of historical and philosophical aspects and the addition of a sociological component separates more recent views about the foundations of mathematics from the studies in the first half of this century (cf. Echeverria et al. 1992, XI).

More generally, the mentioned foundational concerns relate – like mathematical sciences themselves – to questions of content or of method. When we examine content we find natural numbers, special sets, relations, operations; in the case of method, basic elements are definition, proof and construction (cf. Henkin 1967, 116). In each case, the analysis my proceed either along unificational, historical or epistemological concerns. The first one emphasises more mathematical, the last one more philosophical interests and the historical concerns is of a mixed form. The line is, of course, difficult to draw on all sides, but the attempt may be reasonable in order to keep a clear idea.

It is not surprising, but perhaps not worthless to mention, that the birth of symbolic logic and set theory together with the new elucidation role of the axiomatic method is accompanied by an unificational aspect which is already manifest by the choice of titles: Foundations of Mathematics replaces since Peano's Formulaire de mathématique (1895ff) the Foundations of Geometry and Arithmetic and remains current during the whole century. Beginning with Russell's The Principles of Mathematics (1903) and ending with Hintikka's The Principles of Mathematics Revisited (1996) one finds a great number of prominent essays entitled Foundations, Philosophy, Principles or Elements of Mathematics.

Let's not exaggerate, but the large majority of these studies by philosophers and mathematicians concern Set theory and Logic. Non-historical books on the foundations including other fields than set theory or Logic and perhaps elementary arithmetic or the birth of non-Euclidean geometry are rather seldom and have often the form of readings. Nevertheless, in the last thirty years we find a crowing number of conference-proceedings often structured around a special topic, concerned with a larger part of mathematics and mixed in character, i. e. systematic and historic.

Naturally, there are a lot of textbooks in special mathematical domains containing a chapter on foundations and there are a large number of historically or systematically oriented studies and readings in special mathematical domains.

1.2 – But let us return to Set Theory and Logic. There are at least three different approaches in these fields. On can lead the research

a) with a foundation or justification interest

b) with a unification and clarification interest

c) in a technical view.

From a classical point of view, the two first approaches were pursued by both, philosophers and mathematicians, until about 1930 to respect the question of completeness, until about 1933/35 to respect the vision of a single universal language, until about 1948 to respect the question of existence, until 1963 to respect the question of axiomatization. Concerning consistency, there are two dates, one, 1931, for philosophers, the other, 1940, for mathematicians. Concerning the works of Church and Curry, the original goal was first to encompass with their calculi all of mathematics but, after the discovery of the Kleene-Rosser paradoxes, the research on the lambda calculus was restricted on the notion of computability that gives the technical foundation of computer science and concerns mostly interest c).

Apart from the questions of semantic completeness, universal language and computability, all other questions have their roots in the nineteenth-century discussion about monsters and the formalisation-tendency, itself motivated by the critical assessment of classical rationalism pretending the possibility of a clear and distinct apperception of concepts by rational evidence (cf. Vuillemin 1979).

Studies on consistency are directly motivated by Hilbert's Program to eliminate intuition from mathematics and the logical antinomies discovered by Zermelo and Russell (the "paradoxes" of Cantor and Burali-Forti were originally perhaps only interpreted as a step in a reductio ad absurdum procedure). Now, the consistency of a theory is made philosophically credible only to the extent that the principles used in the consistency proof are trustworthy. So, after Gödel's 1931 proof, everybody, except philosophers and constructivists, have desert the epistemological reductive proof theory in favour of what Prawitz calls general proof theory analysing our understanding of proof-complexity, derivability in a given calculus and relative consistency (cf. Prawitz 1974). Most prominent results are here Gentzen's 1936 consistency-proof of elementary arithmetic and, above all, Gödel's 1940 proof of the consistency of the axiom of choice and of the continuum hypothesis with the axioms of set theory. Herewith we have reached the technical view. In this approach, modern formal logic has turned into a mathematical field and has essentially cut the bounds to its original task of serving as a means to mathematical inference and also lost its relevance for investigations into the foundations of mathematics. – I will later return to the relation of logic and mathematics.

Concerning the completeness of the first-order calculus, proved in 1930 by Gödel, it compensates, so to speak, the undecidability of the syntactical deduction concept, proved only in 1936 by Church. Now all logical truth is deductible. But is this a result providing more clarification? I think, Dummett is right in emphasising (cf. Dummett 1973, 204), that Gödel's result which presupposes set theoretical means in its proof, should not be called "semantic" but "algebraic" completeness in order to indicate its defect of foundational force but its technical usefulness. Contrary to its first order analogue, the pure second order calculus is "algebraically" incomplete and cannot be made complete by the addition of any finite number of axioms (cf. Henkin 1967, 124).

The Logical Empiricists' vision of a single universal language was negatively resolved by Tarski's theorem (1933/35) of the indefinability of truth (given a language rich enough, truth for this language is inexpressible in it), or more generally, by the ineffability of semantic concepts such as truth, designation, satisfaction etc.

Concerning the axiomatisation of set theory, there were proposed various manner to axiomatize the original idea of Cantor: the systems of ZF, of von Neumann, Bernays and Gödel (NBC) or of Quine (NF) are not equivalent but all seem adequate for providing a basis for the traditional mathematical theories of analysis, algebra and geometry (cf. Henkin, 127/128). Nevertheless, it is well known that the axiomatisation of set theory in a first order language is not only characterised by the not intended existence of non-standard models (Gödel 1931) and by the general fact that deductive first order theories cannot provide an adequate description of mathematical structure (cf. Beth 1965, 643) – a consequence of the Löwenheim-Skolem-Tarski theorem (1915-1920) – but the axiomatisation of set theory has even an other defect: its originates from Cohen's 1963 independence proof of the axiom of choice (ZF+ÿAC is consistent). Clearly, NF seems not be concerned because AC is not compatible with it. Nevertheless, chapter 10, entitled Mathematics without choice, of Thomas Jech's well known book The axiom of Choice (Jech 1973) illustrates very well the consequences for, say, an algebra without choice: there exists a vector space which has no basis, a field which has noalgebraic closure, a free group whose subgroup is not free etc. On the other han d, the addition of AC to ZF leads to paradoxical consequences, too: one is, for example, Hausdorff's discovery that a half of a surface is congruent to a third of it (1914), an other is the Banach-Tarski paradox that a sphere with a given radius may be decomposed in such a way that, after rotations and translations, one can obtain two spheres of the same radius (1924). So, Cohen's work created a to-day undecided question: should one accept the existence of two incompatible set theories, each one being useful concerning certain interests (although this view leaves the success unexplained), should one find new axioms, which eliminate one of the incompatible theories (cf. Mostowski 1966, 149), should one search after an other framework for mathematics (e. g. category theory) or, finally, should one argue that the difficulties are inherent in the very nature of mathematics (cf. Cohen 1966, 1)?

Concerning existence, Quine's celebrated ontological criterion To be is to be the value of a variable (Quine 1953, 15), installing classical first order logic as ontological measure, specifies technically the philosophical open-end discussion about Platonism (Logicism), Intuitionism and Formalism, standardised 1931 by Carnap's, Heyting's and von Neumann's articles published in Erkenntnis: naturally, Quine's criterion doesn't resolve the philosophical question what there is but gives "to know what a given remark or doctrine [...] says there is" (ibid.). "Platonism condones the use of bound variables to refer to abstract entities [...] specifiable and unspecifiable, indiscriminately. [...] Intuitionism [...] countenances the use of bound variables to refer to abstract entities only when those entities are capable of being cooked up individually from ingredients specified in advance". Nominalism "object to admitting abstract entities at all. [... According to this view], an adequate basis for agreement among mathematicians can be found simply in the rules which govern the manipulation of the notations" (ibid. 14). Significant are not the notations themselves, but only the syntactical rules. The philosophical literature in this domain is immense.

Whereas the axiomatic method was in the both discussed perspectives a) and b) used for the purpose of elucidating the foundations on which mathematicians build (Hilbert's position), it has become, according to the technical view c), a tool for concrete mathematical research; while formerly Axiomatics was concerned with axioms which determine the structure of the system, axiomatic systems are now the common basis for the investigation of individual entities arising by specified constructions (cf. Weyl 1985,13), such as the study of definable sets of real numbers (descriptive set theory), and differentiation or by the variety of models of a given system. The first orientation stand close to the inheritance of Cantor's and Hausdorff's studies on ordinal and cardinal numbers on the basis of set theoretical axioms supplemented by various extensions, the second line is the birth of model theory. Both domains are themselves considered as part of mathematics, and must also have their proper foundations although they may have consequences for the foundations: the theorem of Löwenheim-Skolem is such an example.

In the 19th century, the relation of mathematics to general formal logic was impressed on by "Boole's fundamental idea that the language of mathematics is the most perfect form of the universal language of thought, and that general logic, therefore, is mathematics with all conceptions of quantity struck out" (Bryant 1902). In the 20th century, Quine's idea is pre-eminent: a constant is called logic iff it is invariant in all uses. The language of logic contains only brackets, logical constants and schema-letters for propositions and relations, the language of mathematics in addition at least one special binary predicate,the Œ-relation. Hence, if one would qualify today logic (in a strong sense) as mathematics, it should signify something others. It would signify that mathematics is an auxiliary science in order to formulate or to prove –with the induction principle or structures like group theory – logical facts. One sees that in this interpretation logic could not have a foundational function in an epistemological sense.

It is also not surprising, that there exist works, many works, about the foundations of Logic and Set Theory, written either in a systematic or historical perspective. Here we are sent back to parts a) and b). Nevertheless, we have already seen that technically motivated investigations can have interesting consequences for foundational questions.

On the other hand, it is right that mathematical Logic (in a large sense) as mathematical discipline whose branches set theory, recursion theory, proof theory and model theory are described in Barwise's Handbook of Mathematical Logic (1977), has no properly intrinsic interest for the foundations of mathematics.

Mathematical Logic is a very marginal mathematical discipline – Dieudonné used to prove this fact by brandishing the Zentralblatt as witness for the small percentage of logical studies in respect to the totality of mathematical publications. Indeed, although the historical question at issue here is not the foundational one, the history of Mathematical Logic is probably the best documented of 20th century's mathematics. From Lewis' A Survey of Symbolic Logic (1918) to now there is a great number of competent surveys and histories.

In recapitulating now the main stream of Set Theory and Logic, one can see that there is a tendency to distinguish most important positive results which are over all interesting from a technical point of view from most important negative results which are even of first importance from a philosophical point of view. More precisely, the situation amounts to the following table:

Surely, technical results, as for example the consistency proof for elementary arithmetic, can by their of certain technical tools again generate most interesting philosophical questions, in the given example the differentiation of the infinite domain. A corresponding transformation can even observed for philosophical results: first negatively considered, the existence of non-standard models can even work as an aspect for technical problems, in the given example the development of the theory of category. But these are, so to speak, not intended "second" order results.

So, in reading this table, which is of course only selective, on can perhaps not suppress the following question: is the translatability in the language of set theory and logic really the exclusive form of justification and rigour in mathematics? In fact, the first order axiomatic set-theoretical universe is deductively incomplete, inevitably non-standard, and that we have no unproblematically clear idea of what the intended models of set theory are (cf. Hintikka 1996, 166ff). In other terms, we cannot have confidence in a formal theory without having confidence in our intuition of a conceptual content.

One possible answer could be: "Mathematics without foundations," and it could be evidenced by the fact that the existence of formally undecidable propositions (within a given arithmetical system) or of problems unsettled by standard axioms (within set theory) does not obstruct the development of a viable and, in fact, powerful science. Accordingly, the foundational view of mathematics itself might then be suspect. Mathematics can to be understood from mathematical praxis alone. Indeed, there is even an other possibility.

II. THE POÏETIC REVISION

Since Poincaré there have always been some outsiders who rejected the standard view about the foundations of mathematics. Formulated in modern terms, Poincaré held that the varieties of formal logical theories–which he thought to be considerably attached to set operations–don't express the structure which is essential for a genuine understanding of mathematical proof (cf. Poincaré 1908, 149). Surely, his critic cannot mean that we should fully give up formal languages as the corresponding well justified critic of rational evidence did not mean to give up all evidence in favour of logical reasoning. What happens is that the new formal categories of deductive mathematical reasoning are no longer specific enough for the idea of a comprehensive formalised language. For this reason, one should perhaps make a step from a first "intuitive" epistemological level to a "theoretical" epistemological level. On this latter, one should proceed by a revision of the criterion of deductive rigour. This revision affords a way of pragmatically circumscribing the correlation of the concepts involved. Such consequences seem to be drawn from the core of Poincaré's critic on logic. Voices in this sense were more and more audible at the end of the century. Poincaré was fighting with the logicists in a large sense, Gonseth with the logical empiricists, Beth with Church, Steiner argued in its Mathematical Knowledge (1975) for the importance of non deductive arguments in mathematical justification (cf. Aspray/Kitcher 1988, 18) and more generally, philosophers of mathematics argued against mathematical logicians (cf. Detlefsen's collection Proof, Logic and Formalization, 1992).

But nevertheless, one will reduce Arithmetic and Analysis to set theory! Why? According to Hintikka, one relevant reason may be the important limitation theorem of Tarski (1933): the theoretical truth predicate of a first-order language containing elementary arithmetic is not definable in that language itself. Now, we would like to formalise truth: all model theory depends on truth definitions. As long as these definitions can only be given on second order level or in set theory, then model theory depends on second order logic or set theory (Hintikka 1996, VIII).

It is even very known that first order logic without individual variables including higher-order entities cannot dealing with the most characteristic modes of concept and inference in mathematics such us complete induction, well-ordering or power-set formation. So, mathematical induction is not generated but only represented by indefinite repetition of different levels (cf. Heinzmann 1987, 72), that means that the separation between object and symbol is not yet accomplished: it postulates a survey of a potentially reiterated stroke-concatenation or something analogous and a survey of a potentially reiterated modus ponens. An analogues principle consist in the fact that mathematical symbols can be read with respect to different contents without the possibility to check this ambiguity on the level of the given notation. Herewith we have a typical case of partial information. Poincaré suggests that this difficult situation including semantic ambiguity should be overcome by the introduction of aesthetic feeling in mathematics. The mastering of simultaneous reasoning about different contents, provoked by the lack of perfect information in one field, requires the acquisition of a practice and can be made only explicit by model theoretical means. What are requisite, are new deductive methods corresponding to factual mathematical reasoning.

Is there any formal way-out?

In one sense, yes: if the "correspondence" is interpreted large enough. The solution is suggested in the last fight of the XXth century: Hintikka IF-Logic against Frege's first order logic. Hintikka proved the surprising and revolutionary result that the game theoretical truth predicate of a so called Independence-friendly first-order language containing elementary arithmetic is definable in that language itself in the sense "that there is a complex predicate (of the Gödel numbers of the sentences of the given first-order language) which applies to a number if and only if it is the Gödel number of a true sentence" (cf. Hintikka 1996, 118f). We just have to be carefully with respect to the informational independence of quantifiers in their role of lying numbers as numbers and numbers as Gödel codifications. IF-Logic is a conservative extension of ordinary first-order logic, but it does not admit of a complete axiomatization (cf. Hintikka 1996, 65, 88): "There is no finite (or recursive) set of axioms from which all valid sentences of this logic can be derived as theorems by means of completely formal (recursive) rules of inference." So, perhaps the efforts made during all the century along the axiomatisation-line was from the justificational point of view doomed to failure as a foundational enterprise.

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