Karlis Podnieks

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 on the texts written in 1876-1878: Vaihinger was, decades ahead of the time, a philosopher of modeling in the modern sense – a brilliant achievement for the 1870s! And, in the demystification of such principal aspects of cognition as truth, understanding and causality, is he not still ahead of many of us? According to Vaihinger, what we set beyond sensations is our invention (fiction), the correspondence of which with reality cannot (and need not) be verified in the mystical, absolute sense many people expect.

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, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.

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, but for different purposes. Another example: the idea that all of cognition is cognition in models or by means of models. Even perception, reflexes and instincts (animal and human) can be best analyzed as modeling. The paper presents an analysis of the above-mentioned work.

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, self-contained, and – as a summary – formal. Autonomous and isolated – because mathematical models can be investigated “on their own” in isolation from the modeled objects. And one can do this for many years without any external information flow. Stable – because any modification of a mathematical model is qualified explicitly as defining a new model. No implicit modifications are allowed. Self- contained – because all properties of a mathematical model must be formulated explicitly. The term “formal model” can be used to summarize all these features.

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 the correct distinction between models and theories. At the level of models, because of the above-mentioned limitations, we will always have only a patchwork of models each very restricted in its application scope. And at the level of theories, we will never have a single complete Theory of Everything allowing, without additional postulates, to generate all the models we may need for surviving in this world.

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 one derive useful consequences from it? I'm trying to show that there is a lot of them.

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 used or were neglected in past discussions.

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 obtain a unified and more productive picture - a model-based model of cognition?

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, poems, newspaper articles, jokes, mathematical proofs, video-clips, dreams, halucinations, etc.) may introduce their own “partial world models” – as small additions to “bigger” world models (regarded as background knowledge). Sometimes, sentences contain references to other world models. Trying to understand such sentences, we should identify and keep separated the world models involved.

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 in a similar process. In the formalist picture, all kinds of “truth” are detected intra-theoretically. Some knowledge construct may be considered as “true”, if it is accepted in a particular normative knowledge system. Some knowledge construct may be considered as persistently true, if it remains invariant during the evolution of some knowledge system for a sufficiently long time. And, if you wish, you may consider some knowledge construct as absolutely true, if you do not intend abandoning it in your knowledge system. And finally, in the formalist picture, all kinds of ontologies generated by humans can be demystified by reconstructing them within the basic solipsist ontology simply as hypothetical branches of it.