## Description

Title of Talk: Mal'tsev early contributions to model theoryAbstract: 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.

REFERENCES

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-

-26]

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-

-153

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

Period | Jul 2024 |
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Event type | Conference |

Degree of Recognition | International |