In the contemporary philosophy of mathematics there is an ongoing discussion concerning the issue of the justification of mathematical axioms and independent sentences. The works of P. Maddy in which the author focuses on the set theory are extensive studies devoted to that problem. In one of the monographs she deals with the problem of the justification of set theory's sentences in a new way, assuming a different metaphilosophical standpoint. In the paper (which is the first part of the two-part (...) whole) Maddy's conception is faithfully presented. The author makes no comments of his own, except for the first - introductory - chapter and footnotes, in which some additional explanations and comments can be found. (shrink)

We define a semantics for conditionals in terms of stochastic graphs which gives a straightforward and simple method of evaluating the probabilities of conditionals. It seems to be a good and useful method in the cases already discussed in the literature, and it can easily be extended to cover more complex situations. In particular, it allows us to describe several possible interpretations of the conditional and to formalize some intuitively valid but formally incorrect considerations concerning the probabilities of conditionals under (...) these two interpretations. It also yields a powerful method of handling more complex issues. The stochastic graph semantics provides a satisfactory answer to Lewis’s arguments against the PC = CP principle, and defends important intuitions which connect the notion of probability of a conditional with the notion of conditional probability. It also illustrates the general problem of finding formal explications of philosophically important notions and applying mathematical methods in analyzing philosophical issues. (shrink)

We define a semantics for conditionals in terms of stochastic graphs which gives a straightforward and simple method of evaluating the probabilities of conditionals. It seems to be a good and useful method in the cases already discussed in the literature, and it can easily be extended to cover more complex situations. In particular, it allows us to describe several possible interpretations of the conditional and to formalize some intuitively valid but formally incorrect considerations concerning the probabilities of conditionals under (...) these two interpretations. It also yields a powerful method of handling more complex issues. The stochastic graph semantics provides a satisfactory answer to Lewis’s arguments against the PC = CP principle, and defends important intuitions which connect the notion of probability of a conditional with the notion of conditional probability. It also illustrates the general problem of finding formal explications of philosophically important notions and applying mathematical methods in analyzing philosophical issues. (shrink)

We define a semantics for conditionals in terms of stochastic graphs which gives a straightforward and simple method of evaluating the probabilities of conditionals. It seems to be a good and useful method in the cases already discussed in the literature, and it can easily be extended to cover more complex situations. In particular, it allows us to describe several possible interpretations of the conditional and to formalize some intuitively valid but formally incorrect considerations concerning the probabilities of conditionals under (...) these two interpretations. It also yields a powerful method of handling more complex issues. The stochastic graph semantics provides a satisfactory answer to Lewis’s arguments against the PC = CP principle, and defends important intuitions which connect the notion of probability of a conditional with the notion of conditional probability. It also illustrates the general problem of finding formal explications of philosophically important notions and applying mathematical methods in analyzing philosophical issues. (shrink)

It is a well-known fact that mathematics plays a crucial role in physics; in fact, it is virtually impossible to imagine contemporary physics without it. But it is questionable whether mathematical concepts could ever play such a role in psychology or philosophy. In this paper, we set out to examine a rather unobvious example of the application of topology, in the form of the theory of persons proposed by Kurt Lewin in his Principles of Topological Psychology. Our aim is to (...) show that this branch of mathematics can furnish a natural conceptual system for Gestalt psychology, in that it provides effective tools for describing global qualitative aspects of the latter’s object of investigation. We distinguish three possible ways in which mathematics can contribute to this: explanation, explication and metaphor. We hold that all three of these can be usefully characterized as throwing light on their subject matter, and argue that in each case this contrasts with the role of explanations in physics. Mathematics itself, we argue, provides something different from such explanations when applied in the field of psychology, and this is nevertheless still cognitively fruitful. (shrink)

It is a well-known fact that mathematics plays a crucial role in physics; in fact, it is virtually impossible to imagine contemporary physics without it. But it is questionable whether mathematical concepts could ever play such a role in psychology or philosophy. In this paper, we set out to examine a rather unobvious example of the application of topology, in the form of the theory of persons proposed by Kurt Lewin in his Principles of Topological Psychology. Our aim is to (...) show that this branch of mathematics can furnish a natural conceptual system for Gestalt psychology, in that it provides effective tools for describing global qualitative aspects of the latter’s object of investigation. We distinguish three possible ways in which mathematics can contribute to this: explanation, explication and metaphor. We hold that all three of these can be usefully characterized as throwing light on their subject matter, and argue that in each case this contrasts with the role of explanations in physics. Mathematics itself, we argue, provides something different from such explanations when applied in the field of psychology, and this is nevertheless still cognitively fruitful. (shrink)

It is widely believed that mathematics carries a substantial part of the explanatory burden in science. However, mathematics can also play important heuristic roles of a different kind, being a source of new ideas and approaches, allowing us to build toy models, enhancing expressive power and providing fruitful conceptualizations. In this paper, we focus on the application of dynamical systems theory (DST) within the extended cognition (EC) field of cognitive science, considering this case study to be a good illustration of (...) a general phenomenon. In the paper, we justify both a negative and a positive claim. The negative claim is that dynamical systems theory hardly plays any explanatory role in EC research. We justify our claim by analyzing several accounts of the explanatory role of mathematics and stressing the way mathematical arguments are used in explanations. Our positive claim is that even though, for now, DST has no explanatory power in many of the EC approaches, it still plays an important heuristic role there. In particular, using mathematical notions improves the expressive power of the language and gives a sense of understanding of the phenomena under investigation. The case study of EC allows us to identify and analyze this important role of mathematics, which seems to be neglected in contemporary discussions. (shrink)

This article examines the use of mathematical concepts in philosophy, focusing on topology, which may be viewed as a modern supplement to geometry. We show that Plato and Parmenides were already employing geometric ideas in their research, and discuss three examples of the application of topology to philosophical problems: the first concerns the analysis of the Cartesian distinction between res extensa and res cogitans, the second the ontology of possible worlds of Wittgenstein's Tractatus, and the third Leibniz's monadology. We also (...) consider the role of topology in mathematical explanations of the sort found in science, arguing that it can perform a role in philosophy that is of comparable importance. (shrink)

PCCP is the much discussed claim that the probability of a conditional A → B is conditional probability. Triviality results purport to show that PCCP – as a general claim – is false. A particularly interesting proof has been presented in (Hájek, 2011), who shows that – even if a probability distribution P initially satisfied PCCP – a rational update can produce a non-PCCP probability distribution. We argue that the notion of rational update in this argumentation is construed in much (...) too broad a way. In order to make the argumentation precise, we discuss the general rules for modeling conditionals in probability spaces and present formalized version(s) of PCCP and of minimal assumptions concerning the appropriate spaces. Using the introduced apparatus we give a detailed analysis of Hájek’s (2011) triviality proof and show that it is based on an application of revision rules which allow one to construct probability distributions violating not only PCCP, but also fundamental properties of conditionals. This means that they do not really provide arguments against PCCP, properly formalized. We also discuss a Dutch Book argument which shows that the updated belief system is not coherent. This gives an additional, strong argument against accepting the update rules. We also discuss the Converse Dutch Book theorem and argue, that even if the produced probability measure seems to violate it, it cannot serve as the counterexample, as it is not an appropriate model for conditionals. Ultimately, we show that important arguments against PCCP fail. (shrink)

In the recent years we can observe a sort of renaissance of the philosophy of mathematics. More and more papers and books are published. A few years ago a new journal (Philosophia Mathematica) devoted exclusively to the philosophy of mathematics started appearing. In the contemporary discussions - especially in the context of the question of the applicability of mathematics to the description of the physical world - the issue of the existence and the ontological status of mathematical objects plays a (...) particular role. Many new conceptions have appeared - both realistic and antirealistic. In the paper one of the antirealistic conceptions formulated in the recent years is presented - namely a conception by C. Chihara [1990]. The author rejects both Gödel's and Quine's arguments for the realistic standpoint. First, these standpoints will be briefly summarised, since they play a significant role in the contemporary philosophy of mathematics - and are particularly important in the context of the discussion upon Chihara's conception. Next, Chihara's arguments will be analysed in detail. As a result it will turn out that these arguments are not conclusive. Chihara's system relies on certain unjustified assumptions. Moreover, the philosophical difficulties that it encounters are not sufficiently discussed. (shrink)

This article consists of the two parts: the first on presents Hartry Field's nominalistic theory of science contained in his „Science Without Numbers”. The second part points to certain difficulties, which the realization of Field's program is faced with. The problem of an exact translation of nominalistic theories to mathematical theories, the connection between the incompleteness of the nominalistic theory and the conservativeness of its mathematical extension and an example of a theorem about finite sets, which needs some strong assumptions (...) about infinite sets in its proof are presented. (shrink)

In the article the problem of independence in mathematics is discussed. The status of the continuum hypothesis, large cardinal axioms and the axiom of constructablility is presented in some detail. The problem whether incompleteness is really relevant for ordinary mathematics and for empirical science is investigated. Another aim of the article is to give some arguments for the thesis that the problem of reliability and justification of new axioms is well-posed and worthy of attention. In my opinion, investigations concerning the (...) status of independent sentences give insight into our understanding of mathematical concepts, of mathematical knowledge and of the role of mathematics in empirical science. (shrink)

Artykuł dotyczy zagadnienia, w jakim sensie można stosować kategorię wyjaśnienia do interpretacji filozofii matematyki Kurta Gödla. Gödel – jako realista matematyczny – twierdzi bowiem, że w wypadku matematyki mamy do czynienia z niezależnymi od nas faktami. Jednym z owych faktów jest właśnie rozwiązywalność wszystkich dobrze postawionych problemów matematycznych – i ten fakt domaga się wyjaśnienia. Kluczem do zrozumienia stanowiska Gödla jest identyfikacja założeń, na których się opiera: metafizyczny realizm: istnieje uniwersum matematyczne, ma ono charakter obiektywny, niezależny od nas; optymizm epistemologiczny: (...) jesteśmy wyposażeni w wystarczająco dobre środki poznawcze, aby uzyskać wgląd w owo uniwersum. Pojęcie rozwiązania problemu matematycznego Gödel rozumie znacznie szerzej niż jako podanie matematycznego dowodu – chodzi raczej o znalezienie wiarogodnych aksjomatów, prowadzących do rozwiązania. Stawiany w artykule problem analizuję na przykładzie hipotezy kontinuum. (shrink)

This is the second part of the study concerning the problem of ontological reductions in mathematics. In this part, some major strategies of argumentation in favor of mathematical realism are presented. The versions to be considered are: Gödel’s realism, Quine’s quasi-empiricism and Balaguer’s Full-Blooded Platonism. Some introductory remarks considering the problem of ontological reductions in the context of these three stances are also presented.

This is the third part of the study concerning the problem of ontological reductions in mathematics. In this part, the problem of reconstruction of (parts of) mathematics in theories weaker than full ZFC is discussed. The tools from reverse mathematics are used, and the results are discussed from the point of view of various versions of realism (Gödel’s realism, Quine’s quasi-empiricism and Balaguer’s Full-Blooded Platonism). Some problems concerning the possibility of discussing these problem outside the conceptual system of set theory (...) are also addressed. (shrink)

The article is the first part of a series of papers devoted to the problem of ontological reductions in mathematics – in particular, of choosing the basic category of mathematical entities. The received view is that such a category is provided by set theory, which serves as the ontological framework for the whole of mathematics (as all mathematical entities can be represented as sets). However, from the point of view of "naive mathematical realism" we should rather think of the mathematical (...) universe as populated by a variety of diverse mathematical objects, and the set-theoretic reduction seems to be rather unnatural. In the first (introductory) part I discuss the general problem of providing an ontological foundation for mathematics. (shrink)

In the article, I discuss the differences between mathematical structuralism and object realism. I argue that they are partly only a matter of formulation and that some basic theses of both standpoints can be translated into the opponent’s language.

In the article, Woodin’s program (for setting up axioms, which decide the continuum hypothesis) is presented, and some philosophical aspects of it are discussed. In particular, the general problem of justifying axioms of set theory is discussed in the context of the relation between set theory and mainstream mathematics.