7th International Meeting of the Association for the Philosophy of Mathematical Practice (APMP)School for Advanced Studies IUSS Pavia, Italy

Activity: Participating in or organising an eventParticipation in conference


Title of Talk: Mal'tsev early contributions to model theory

Abstract: Although the history of early model-theoretic ideas, between Loewenheim
and Tarski, has been widely studied, comparable attention has not been
paid to the model-theoretic work of A.I. Mal'tsev.
This imbalance is perhaps not entirely surprising, since Mal'tsev's earlier
contributions are marked by a discontinuity with dominant concerns
and approaches that had characterised the best known model-theoretic work predating his. What is distinctive in Mal'tsev is a general neglect of foundational or
metamathematical issues: in its place stands a newly gained awareness that
logical results can be used to tackle algebraic problems, in the same way
in which, e.g. analytical results could be used to tackle number-theoretic problems. As a consequence, standard logical equipment acquires a novel
significance and function. First-order languages are essentially problem-solving instruments, not means of rigorous codification; cardinality constraints on the language solely depend on the character of the algebraic problem involved, not on
foundational scruple; logical results are regarded as a general framework
presiding over the controlled reconstruction of algebraic particulars, rather than evincing a self-contained significance.
I illustrate and examine these features of Malt'sev's work with special reference to three key articles on local theorems (Mal'tsev (1941, 1956, 1959), relating them to similar, contemporaneous investigations (notably Neumann (1954) and
Robinson (1955, 1960)).
The central idea I extract from this analysis is that Mal'tsev's use of logical ideas and results to transform algebraic subject matter endows those ideas and results with a novel significance. They are now regarded as constituents of a formal working
environment in which the treatment and understanding of algebraic
problems can be evolved in a new direction. New epistemic activities are
consequently enabled: foremost among them are the detection of
fundamental unifying features, the reconstruction of algebraic concepts
and results, the expansion of logical work directed towards further
refinements of interactions with algebraic content. This outcome
displays a mathematical attitude of lasting significance in subsequent
model-theoretic work.

Mal'tsev, A.I. (1971) The Metamathematics of Algebraic
Systems. Amsterdam: North-Holland. Mal'tsev, A.I. (1936) `Untersuchungen
aus dem Gebiete der mathematischen Logik', Matematicheskii Sbornik 1 (43): 323-
-336 [translated in Mal'tsev (1971), pp.1--14]
Mal'tsev, A.I. (1941) `Ob odnom obščem metode polučenija lokal'nyh
teorem teorii grupp', Ucenye Zapiski Ivanov. Ped. Inst. (Fiz-mat. Fakul'tet)
1 (1): 3--9 [translated in Mal'tsev (1971), pp. 15--21]
Mal'tsev, A.I. (1956) `O predstavlenijah modeleĭ', Doklady
Akademii Nauk (SSSR) 108: 27--29 [translated in Mal'tsev (1971), pp. 22-
Mal'tsev, A.I. (1957) `Model'nye sootvetstvija', Izvestiya Akademii
Nauk (SSSR), Seriya Matematicheskaya 23: 313--336
[translated in Mal'tsev (1971), pp. 66--94]
Neumann, B.H. (1954) `An embedding theorem for algebraic
systems', \textit{Proceedings of the London Mathematical Society 3: 138-
Robinson, A. (1955) `Note on an embedding theorem for algebraic
systems', Journal of the London Mathematical Society 30: 249--252
Robinson, A. (1960) `Recent Developments in Model Theory', in
Logic, Methodology and Philosophy of Science: Proceedings of the 1960
International Congress, E. Nagel, P. Suppes and A. Tarski (eds), Stanford
University Press, pp. 60--79
PeriodJun 2024
Event typeConference
Degree of RecognitionInternational