•  643
    Textbook for students in mathematical logic. Part 1. Total formalization is possible! Formal theories. First order languages. Axioms of constructive and classical logic. Proving formulas in propositional and predicate logic. Glivenko's theorem and constructive embedding. Axiom independence. Interpretations, models and completeness theorems. Normal forms. Tableaux method. Resolution method. Herbrand's theorem.
  •  620
    Introduction to mathematical logic. Part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loe…Read more
  •  596
    Philosophy of Modeling: Neglected Pages of History
    Baltic Journal of Modern Computing 6 (3). 2018.
    The work done in the philosophy of modeling by Vaihinger (1876), Craik (1943), Rosenblueth and Wiener (1945), Apostel (1960), Minsky (1965), Klaus (1966) and Stachowiak (1973) is still almost completely neglected in the mainstream literature. However, this work seems to contain original ideas worth to be discussed. For example, the idea that diverse functions of models can be better structured as follows: in fact, models perform only a single function – they are replacing their target systems, b…Read more
  •  459
    Fourteen Arguments in Favour of a Formalist Philosophy of Real Mathematics
    Baltic Journal of Modern Computing 3 (1): 1-15. 2015.
    The formalist philosophy of mathematics (in its purest, most extreme version) is widely regarded as a “discredited position”. This pure and extreme version of formalism is called by some authors “game formalism”, because it is alleged to represent mathematics as a meaningless game with strings of symbols. Nevertheless, I would like to draw attention to some arguments in favour of game formalism as an appropriate philosophy of real mathematics. For the most part, these arguments have not yet been…Read more
  •  306
    Philosophy of Modeling in the 1870s: A Tribute to Hans Vaihinger
    Baltic Journal of Modern Computing 9 (1): 67-110. 2021.
    This paper contains a detailed exposition and analysis of The Philosophy of “As If“ proposed by Hans Vaihinger in his book published in 1911. However, the principal chapters of the book (Part I) reproduce Vaihinger’s Habilitationsschrift, which was written during the autumn and winter of 1876. Part I is extended by Part II based on texts written during 1877–1878, when Vaihinger began preparing the book. The project was interrupted, resuming only in the 1900s. My conclusion is based exclusively…Read more
  •  262
    The Dappled World Perspective Refined
    The Reasoner 8 (1): 3--4. 2014.
    The concept of the Dappled World Perspective was first proposed by Nancy Cartwright. I propose a new argument in favour of the Dappled World Perspective, and show how this Perspective can be refined in the model-based model of cognition. Limitations to modeling are not caused by limitations of human cognition, but are limitations built into the very structure of the Universe. At the level of models, we will always have only a patchwork of models, each very restricted in its application scope.
  •  238
    Is Scientific Modeling an Indirect Methodology?
    The Reasoner 3 (1): 4-5. 2009.
    If we consider modeling not as a heap of contingent structures, but (where possible) as evolving coordinated systems of models, then we can reasonably explain as "direct representations" even some very complicated model-based cognitive situations. Scientific modeling is not as indirect as it may seem. "Direct theorizing" comes later, as the result of a successful model evolution.
  •  216
    This article is an experiment. Consider a minimalist model of cognition (models, means of model-building and history of their evolution). In this model, explanation could be defined as a means allowing to advance: production of models and means of model-building (thus, yielding 1st class understanding), exploration and use of them (2nd class), and/or teaching (3rd class). At minimum, 3rd class understanding is necessary for an explanation to be respected.
  •  212
    Freges Puzzle from a Model-Based Point of View
    The Reasoner 6 (1): 5--6. 2012.
    Frege's puzzle about propositional attitude reports is considered. Proposed solution: Every utterance comes from the world model of the speaker, and sometimes it may contain references to (speaker's models of) other world models. More generally, every sentence comes from some kind of world model. It may be the world model of a (real or imagined) person, the world model represented in a novel, movie, scientific book, virtual reality, etc. In principle, even smaller informational units (stories, p…Read more
  •  197
    What is a model? Surprisingly, in philosophical texts, this question is asked (sometimes), but almost never – answered. Instead of a general answer, usually, some classification of models is considered. The broadest possible definition of modeling could sound as follows: a model is anything that is (or could be) used, for some purpose, in place of something else. If the purpose is “answering questions”, then one has a cognitive model. Could such a broad definition be useful? Isn't it empty? Can …Read more
  •  187
    Indispensability Argument and Set Theory
    The Reasoner 2 (11): 8--9. 2008.
    Most set theorists accept AC, and reject AD, i.e. for them, AC is true in the "world of sets", and AD is false. Applying to set theory the above-mentioned formalistic explanation of the existence of quarks, we could say: if, for a long time in the future, set theorists will continue their believing in AC, then one may think of a unique "world of sets" as existing in the same sense as quarks are believed to exist.
  •  177
    Towards Model-Based Model of Cognition
    The Reasoner 3 (6): 5--6. 2009.
    Models are the ultimate results of all (scientific, non-scientific, and anti-scientific) kinds of cognition. Therefore, philosophy of cognition should start with the following fundamental distinction: there are models, and there are means of model-building. Laws of nature and theories are useful only as a means of model-building. If it's true that models are the ultimate results of cognition, then shouldn't we try reordering the field, starting with the notion of model? In this way, couldn't we …Read more
  •  167
    This paper represents a philosophical experiment inspired by the formalist philosophy of mathematics. In the formalist picture of cognition, the principal act of knowledge generation is represented as tentative postulation – as introduction of a new knowledge construct followed by exploration of the consequences that can be derived from it. Depending on the result, the new construct may be accepted as normative, rejected, modified etc. Languages and means of reasoning are generated and selected …Read more
  •  150
    First, I propose a new argument in favor of the Dappled World perspective introduced by Nancy Cartwright. There are systems, for which detailed models can't exist in the natural world. And this has nothing to do with the limitations of human minds or technical resources. The limitation is built into the very principle of modeling: we are trying to replace some system by another one. In full detail, this may be impossible. Secondly, I'm trying to refine the Dappled World perspective by applying t…Read more
  •  23
    Many people think that mathematical models are built using well-known “mathematical things” such as numbers and geometry. But since the 19th century, mathematicians have investigated various non-numerical and non-geometrical structures: groups, fields, sets, graphs, algorithms, categories etc. What could be the most general distinguishing feature that would separate mathematical models from non-mathematical ones? I would describe this feature by using such terms as autonomous, isolated, stable, …Read more