Synopsis of Jean Ladrière, ‘Le théorème de Löwenheim-Skolem’
[‘The Löwenheim-Skolem Theorem’]
Born in Belgium in 1921, where he spent most of his career until his death in 2007, Jean Ladrière was one of the twentieth century’s major philosophers working at the crossroads of phenomenology and Catholic theology. Departing from his early engagements with mathematical logic, Ladrière retained a lifelong concern for the relation between science and culture, broadly conceived. Whether dealing with mathematics or linguistics, the incapacity of a formalized science to include the ‘endlessly deferred horizon’ which served as the condition of possibility for the sense it expressed was a guiding concern in Ladrière’s work. Along with Bouveresse’s article on Wittgenstein (CpA 10.9), the inclusion of Ladrière’s contribution to this final issue of the Cahiers pour l’Analyse makes clear, with its robust defence of intuitive thought, that ‘formalisation’ was not simply a matter of uncritical affirmation for the editors of the journal.
In this article, written in September 1967, Ladrière produces a critique of formalism that was first developed in his doctoral thesis, Les limitations internes des formalismes: Étude sur la signification du théorème de Gödel et des théorèmes apparentés dans la théorie des fondements des mathématiques (1957). In an essay titled ‘Mathematics and Formalism’ (1955), Ladrière had conceived of the relationship between ‘formalist’ and ‘realist’ mathematics (or, between Hilbertian Platonists and Brouwerian intuitionists) in terms of a dialectic at the heart of mathematics itself: ‘The inadequation which always separates being from its manifestation and prevents it from ever being reduced to an objectified datum expresses the temporality of the movement which strives to complete it, or again, that internal dialectic of the concept in which the aspect of perpetual incompleteness is united with that of an ever efficacious plenitude which prevents the movement from being halted’.1 In addition to lauding ‘expression’ as the optimal mode of mathematical thought, Ladrière’s essay on the Löwenheim-Skolem theorem also follows the basic movement of this passage, reading in the tendency to formalise itself, i.e., to produce an ‘objectified datum’, the very movement that transgresses objectification, ensuring mathematical thought’s unfolding in an ‘efficacious plenitude’.
Before attending to the details of the theorem itself, Ladrière begins his article with a general discussion of the relation between semantics and syntax as methods of formalisation. A semantic method is one that formalises a theory extrinsically, that is, by describing its construction with reference to a domain of objects to which it might apply. By contrast, a syntactical method of formalisation is intrinsic, in that it departs from axioms internal to the theory itself and obtains all the propositions of the theory from this point of departure. The challenge, Ladrière observes, is how to coordinate these two approaches, that is, to verify that a theory semantically ‘confirmed’ can be syntactically formalized through axioms, or vice versa to determine if a thoroughly axiomatised theory might still semantically ‘apply’ to something extrinsic to it.
This relation is made problematic by the Löwenheim-Skolem theorem, a theorem of logic which describes first-order predicates and their relation to individual variables. ‘First-order logic’ is a logic that essentially deals with the function of ‘true’ or ‘false’, that is, whether a given predicate can (true) or cannot (false) be ascribed to a given variable. The Löwenheim-Skolem theorem is a generalisation of a property in one of Löwenheim’s previous theorems, which states that if a proposition of a given language (L) is valid throughout an infinite denumerable domain, that is, the ‘first-order predicate’ can be affirmed unanimously throughout the domain, then it is also valid in any non-empty domain. But if this is true, then its corollary is as well, namely: if a proposition of L is exemplifiable (that is, the proposition can be attributed to individuals as a predicate) in a given non-empty domain, then it can apply to a denumerable domain as well. Skolem generalised this property so that if all the propositions of a given class within L are simultaneously exemplifiable in a given non-empty domain, then they are all simultaneously exemplifiable in a denumerable domain. Whence the Löwenheim-Skolem theorem: if a class of propositions of L admits a model, then it admits a denumerable model (CpA 10.6:118-19).
Despite being written mainly in plain language, Ladrière’s text pays considerable technical attention to the syntactical, or axiomatic, generation of the theorem (which he abbreviates LP). It is clear, however, that his main concern is the coordination introduced at the outset, namely, the tension that can arise between syntactic formalisation and semantic coherence or adequation. In effect, LP introduces a paradox into the heart of set theory, showing how its syntactic application produces semantic contradictions and, consequently, revealing the limits of axiomatic formalization. To be compelling as a theory of anything (for instance, numbers or other mathematical objects), set theory must be ‘exemplifiable’. What this means is that set theory itself – as a system of axioms and operations – must be able to be affirmed, or verified, as a predicate. Stated most simply: it must ‘describe’ something. ‘Set theory’ must apply to some set. Here we see the slide from a syntactically established theory to the crucible of its semantic application. Ladrière explains the predicament as follows:
If set theory is exemplifiable, then it is exemplifiable in a denumerable domain. This property is paradoxical because it is possible to represent, within axiomatic set theory itself, Cantor’s celebrated reasoning that proves the existence of non-denumerable sets. This reasoning shows that, in order to obtain a non-denumerable set, it is sufficient, for example, to take the set of all the subsets (finite or infinite) of the set of whole numbers. As one can give a denumerable model for set theory, there must therefore exist, in this model, an object ND which has the properties of a non-denumerable set. […] In other words, if the object ND exists (and it must exist for the model to effectively offer an exemplification of the theory), there exists a subset of the model that is non-denumerable. But the model itself constitutes a denumerable set. […] The sets that represent, in the model, the non-denumerable sets of the theory are themselves denumerable sets (CpA 10.6:122-3).
In Ladrière’s view this paradox shows a ‘systematic discrepancy [décalage] between the formalised theory and the metatheory’, in this case axiomatic set theory being the ‘formalised theory’ with its ‘model’ functioning as the ‘metatheory’. In Skolem’s view, the effect of his theorem, and the paradox he drew from it, was to relativise all of mathematics, to show that ‘mathematical concepts…no longer have an absolute sense, given in itself, independent of all representation, but that they take on different meanings [sens] according to the domain in which they are implemented [ils se réalisent]’ (CpA 10.6:124).
For Ladrière, the signal innovation of LP is to show the inadequacy of axiomatic formalisation by making clear the ‘regulative role’ that ‘intuitive mathematical notions continue to play with regard to axiomatised concepts’ (124). This is not to say that axiomatic formalisation is not still useful and in fact necessary. Indeed, the establishment of a metatheory – or model – to describe a theory always involves an element of formalisation. Cavaillès called this process ‘thematisation’, a concept which Ladrière invokes as well. Like Cavaillès, Ladrière thinks of thematisation primarily in terms of a process of reflexivity which links expression and act. ‘There is reflection of the intuitive theory in itself, in the sense that in it there is the passage from the plane of expression to the plane of the act, and reciprocally: the aim [visée] is an act, not an expression, but it presupposes expressions, and thematisation, objectivising the aim, recuperates [reprend] the act which constituted it in an expression of a higher level’ (CpA 10.6:128).
And yet, Ladrière maintains that the expressions one finds in an axiomatised formal system are not properly expressive. Rather, these processes ‘represent’ or ‘furnish a tangible equivalent of the operations of intuitive thought’ (CpA 10.6:129). In its drive toward autosufficiency, formalisation has the tendency to convert form itself into an object, a move which constitutes a kind of reification of intuitive thought. The virtue of intuitive thought for Ladrière, the reason for its superiority to purely formalised thought, is that it is intrinsically transgressive and thus constitutive of progress itself, even as its results must always be formalised in order to be transgressed again: ‘the reflection that [intuitive thought] involves is not a simple recapitulation, but a transgression’ (CpA 10.6:130). At the outset, Ladrière began by noting the problems the concept of the infinite creates for formalisation. Intuitive thought’s value in this regard is affirmed in the conclusion. ‘When intuitive thought represents the infinite to itself, it grasps it as the always open horizon of acts [demarches] that can be reiterated and superimposed indefinitely, as the field that is always available through an incessant transgression’ (CpA 10.6:130). Again, Ladrière notes that formalism is necessary for the establishment and advance of any science. Nonetheless, ‘formalism’s fecundity is retrospective, while intuitive thought’s is anticipatory. The force of the formal is to make present; but presentation [présentification] of the infinite finitises it. The force of intuition is to announce; but the pure announcement always risks being lost in the unsayable’ (CpA 10.6:130).
References to this text in other articles in the Cahiers pour l’Analyse:
- Ladrière, Jean. Les Limitations Internes des Formalismes: Étude sur la signification du théorème de Gödel et des theorems apparentés dans la théorie des fondements des mathématiques. Louvain: E. Nauwalaerts/Paris: Gauthier-Villars, 1957 [reprint, same pagination, with Ladrière’s corrections, Sceaux: Editions Jacques Gabay, 1992].
- ---.. L’Articulation du sens: Discours scientifique et parole de la foi. Paris: Aubier Montaigne-Editions du Cerf, 1970.
- ---. ‘Mathematics in the Philosophy of Science’ (1959), trans. Theodore J. Kiesel, in Joseph J. Kockelmans and Theodore J. Kisiel. Phenomenology and the Natural Sciences: Essays and Translations. Evanston, IL: Northwestern University Press, 1970, pp. 443-65.
- ---. ‘Mathematics and Formalism’ (1955), trans. Theodore J. Kiesel, in Phenomenology and the Natural Sciences: Essays and Translations, pp. 466-99.
Selected secondary literature
- Badiou, Alain. The Concept of Model: An Introduction to the Materialist Epistemology of Mathematics, trans. Zachary Luke Fraser and Tzuchien Tho, Melbourne: Re.press, 2007.
- Malherbe, Jean-Francois. Le Langage théologique à l’âge de la science: Lecture de Jean Ladrière, Paris: Les Editions du Cerf, 1985.
- Shapiro, Steven, ed.. The Limits of Logic: Higher-Order Logic and the Löwenheim-Skolem Theorem, Aldershot, UK: Dartmouth, 1996.
- Van Heijenoort, Jean. From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879-1931. Cambridge, MA: Harvard University Press, 1957.
- Wang, Hao. ‘On Formalization’. Mind, vol. 64, 1955.
1. Ladrière, ‘Mathematics in the Philosophy of Science’, 496 ↵