**Arbitrary Objects and Structures**

15 July - 12 August 2017

Stanislav O. Speranski is an Associate Professor in Mathematical Logic at St Petersburg State University. He received his B.Sc. (2008) and M.Sc. (2010) from Novosibirsk State University. After completing his Ph.D. (2013), he accepted a research position at Sobolev Institute of Mathematics, one of the leading institutes of the Russian Academy of Sciences. Then he spent one year as a Humboldt Research Fellow in Logic at the Munich Center for Mathematical Philosophy, which is a part of the Ludwig-Maximilian University of Munich. In 2016, he accepted his current position in Saint Petersburg.

His work addresses topics in: logic (mathematical, philosophical and computational); the foundations of mathematics (broadly construed to include statistics) and those of computing (broadly construed to include artificial intelligence); the philosophy of mathematics and that of computing.

Most recently, he has been writing on the computational aspects of Kripke's theory of truth, and developing effective versions of the game-theoretic semantics for independence-friendly first-order logic (which is an extension of first-order logic advocated by Hintikka in his influential book `The Principles of Mathematics Revisited'). Among the other topics S.O. Speranski has been working on are modal logic, defeasible reasoning, probability logic, second-order arithmetic and elementary theories.

He has over a dozen articles in peer-reviewed international journals, and is a co-author of a textbook on non-classical logics (in Russian, written jointly with S.P. Odinstov and S.A. Drobyshevich). He gave over two dozens of talks at international conferences, workshops, colloquia and seminars (in philosophy, mathematics, computer science and linguistics), and taught lecture courses in logic, theoretical computer science and mathematical linguistics.

Following the philosopher Kit Fine, imagine that in addition to the particular natural numbers 0,1,2,..., there are also arbitrary natural numbers. These are numbers that are not any particular numbers, but can take the value of any particular number. It is the kind of number that a mathematician refers to when she says "let x be an arbitrary natural number..." This view would have seemed natural in the 19th century, but since the late 19th century it has come under heavy attack, particularly by Gottlob Frege and Bertrand Russell. However, Fine claims that a form of the theory of arbitrary objects is not only defensible, but also valuable in applications to the logic of generality, the use of variables in mathematics, the role of pronouns in natural language, and other topics. Moreover, Leon Horsten has recently suggested that the theory of arbitrary objects has applications in the debate on structuralism in the philosophy of mathematics.

Although Fine gave some good arguments in defence of his theory of arbitrary objects, many researchers still reject the concept of arbitrary object as incoherent, and its role in philosophy seems underestimated or even neglected. We think that Fine's theory and its applications are worth developing further, in accord with modern standards of rigor.

The situation here is somewhat analogous to what happened earlier with infinitesimals in analysis, which had been heavily attacked at least since Bishop Berkeley, but which have made a remarkable comeback in what is known as non-standard analysis, through the work of Abraham Robinson and others. In effect, the way Fine avoids certain paradoxes concerning arbitrary objects can be compared with Edward Nelson's axiomatic approach to non-standard analysis (inspired in part by Abraham Robinson's pioneering model-theoretic approach). Our proposed IAS research project aims at giving a completely formal treatment of arbitrary objects. We will develop a rigorous framework for the theory of arbitrary objects along the lines of Nelson's Internal Set Theory, and use it to study basic meta-mathematical properties of the theory of arbitrary objects (such as conservativeness properties, and consistency). We will also apply the resulting framework to philosophical questions about mathematical structuralism.

Dr Speranski's visit was be hosted by Professor Leon Horsten (Philosophy) and featured the following events:

- Graduate Student Seminar: Measuring complexity of Kripkean truth predicates 1 August, 16:00
- Public Lecture: On the discovery of incomputability and unsolvability 3 August, 16:00
- Departmental Lecture: A constructive realizability interpretation for Hintikka's independence-friendly first-order logic (work with S.P. Odintsov and I.Yu. Shevchenko) 8 August, 16:00