The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the role of diagrams in mathematics, and the (...) effectiveness of mathematics in natural science. (shrink)

The work done so far on the understanding of mathematical (proof) texts focuses mostly on logical and heuristical aspects; a proof text is considered to be understood when the reader is able to justify inferential steps occurring in it, to defend it against objections, to give an account of the “main ideas”, to transfer the proof idea to other contexts etc. (see, e.g., Avigad in The philosophy of mathematical practice, Oxford University Press, Oxford, 2008). In contrast, there is a rich (...) philosophical tradition dealing with the concept of understanding and interpreting texts, namely philosophical hermeneutics, represented, e.g., by Schleiermacher, Dilthey, Heidegger or Gadamer. In this tradition, “understanding” generally refers to the integration in a comprehensive (historical, existential, life-worldly,...) context. In this article, we take some first steps towards exploring the question how the ideas from philosophical hermeneutics presented in Gadamer’s “Truth and Method” apply to mathematical texts and what (if anything) can be learned from these for the didactics and presentation of mathematics. (shrink)

B a c k g r o u n d. The ordinary language philosophy should be considered as a set of different but interconnected research projects within the Anglo-American analytical philosophy of the first half and middle of the 20th century. A common factor for these studies is the application of the method of linguistic analysis of natural language expressions to solve many classical problems for philosophy. This method replaced the prevailing idealistic concepts, and was picked up and developed in (...) Great Britain by the works and scientific explorations of J. Moore and B. Russell. It is a generally accepted opinion that the sphere of interests of representatives of the ordinary language philosophy is divided between two poles, represented by the works of L. Wittgenstein and D. Austin. M e t h o d s. In the article, the author used the method of historical-philosophical reconstruction to clarify the role, place and genesis of the ordinary language philosophy as a dominant methodology within the Anglo-American analytical philosophy of the middle of the 20th century, general scientific methods of cognition: analysis, synthesis, generalization to clarify key ideas and concepts, which is the basis of the methodology of the ordinary language philosophy, as well as methods of systematization and comparison. R e s u l t s. The article clarifies the genesis of the main ideas and concepts of the philosophy of everyday language, as well as gives an example of the modern strategy of criticism of this methodological direction. The key postulates of the ordinary language philosophy age are clarified, namely: (i) the thesis that philosophical problems arise due to the incorrect use of language and must be solved by clarifying the meanings of words, (ii) philosophical analysis should not be based on the discovery of hidden logical forms of expression, and on the clarification of the necessary and sufficient conditions for the use of the expression, including the analysis of the situation of use, (iii) meaning as use, (iv) the philosophical analysis of the meaning should be carried out informally, taking into account the analysis of each specific case of the use of the expression, (v) criticism systematic theories of meaning. C o n c l u s i o n s. At the current stage of the development of analytical philosophy, there is a noticeable increase in attention to the achievements of the ordinary language philosophy. Among modern researchers and critics of this direction, the names of A. Baz, D. Nagel, S. Somes, and T. Williamson can be singled out. The presence of the interest of modern researchers indicates premature statements about the completion of the ordinary language philosophy. This concept should be rehabilitated, and its methodological apparatus should enrich the arsenal of scientific methods of modern analytical philosophers. (shrink)

We reformulate slightly Russell's notion of typicality, so as to eliminate its circularity and make it applicable to elements of any first‐order structure. We argue that the notion parallels Martin‐Löf (ML) randomness, in the sense that it uses definable sets in place of computable ones and sets of “small” cardinality (i.e., strictly smaller than that of the structure domain) in place of measure zero sets. It is shown that if the domain M satisfies, then there exist typical elements and only (...) non‐typical ones. In particular this is true for the standard model of second‐order arithmetic. By allowing parameters in the defining formulas, we are led to relative typicality, which satisfies most of van Lambalgen's axioms for relative randomness. However van Lambalgen's theorem is false for relative typicality. The class of typical reals is incomparable (with respect to ⊆) with the classes of ML‐random, Schnorr random and computably random reals. Also the class of typical reals is closed under Turing degrees and under the jump operation (both ways). (shrink)

This article argues that philosophers tend to employ a particular method in constructing their theories and critiquing their opponents. To substantiate this claim, the article examines the work of Nietzsche and Locke, the Empiricists and Rationalists, Heidegger, Levinas, and Derrida, and Russell and Wittgenstein, showing how each relies on a method the article labels “revolution-through-return.” The method consists in identifying the authority behind your opponent's theory, then appealing to something “prior to” that authority, from which you then proceed to derive (...) your own theory. The article distinguishes between several senses of priority, argues that modern philosophers tend to rely on temporal priority, and discusses the questions in priority theory that need to be addressed in order evaluate and construct revolution-through-return arguments. (shrink)

A central problem for any truthmaker theory is the problem of negative truths. In this paper, I develop a novel, piecemeal strategy for solving this problem. The strategy puts central focus on a truth-relevant notion of aboutness within a metaphysically modest version of truthmaker theory and uses key conceptual tools gained by taking a deeper look at the best attempts to solve the problem of intentionality. I begin this task by critically discussing past proposed solutions to P-NEG in light of (...) Russell’s debate with Demos. This reveals a central difficulty with addressing the problem, specifically that one cannot be committed to incompatibility facts in one’s account of negation and of the truth of negative truths. I then present an aboutness-based version of truthmaker theory. Utilising what I call the strict and full account of aboutness, I extract aboutness-based theories of truth and falsity. I use this machinery to present a promising new strategy for solving P-NEG which does not have the problems of alternative approaches. Finally, I present and respond to some potential objections. (shrink)

In this paper, I will show to what extent we can use our modern understanding of the Square of Opposition in order to make sense of Kant 's double standard solution to the cosmological antinomies. Notoriously, for Kant, both theses and antitheses of the mathematical antinomies are false, while both theses and antitheses of the dynamical antinomies are true. Kantian philosophers and interpreters have criticized Kant 's solution as artificial and prejudicial. In the paper, I do not dispute such claims, (...) but I show that our modern understanding of the Square of Opposition enables us to more naturally deliver the result Kant was aiming at. Accordingly, the paper does not pretend to be exegetically accurate. It is an attempt to revise the antinomies with the help of standard classical logic. And although such a revision entails some re-interpretation, in the end, it will actually help to unveil some of Kant 's thoughts. (shrink)

Review of Russell’s Philosophy of Logical Atomism 1897–1905, by Jolen Galaugher (Palgrave Macmillan 2013).

Every scientific or intellectual movement rests on central premises and assumptions that shape its philosophy. The purpose of this study is to review a brief account of the main philosophical bases of grey systems theory (GST) and the paradigm governing its principles. So, the recent studies on the philosophical foundations of GST have been reviewed and tried to pay attention to some key ambiguities in the previous studies and give more and clearer explanations in this paper. Also, this paper tries (...) to fill the gap among the previous studies and to provide a purposeful connection between them by expressing two key concepts: complete information and imbalance knowledge. Primarily, the study addresses the theoretical foundations of uncertainty and the concept of greyness. Next, it focuses on the notion of “complete information” and challenges to it. Then, it reviews such processes as perception, cognition, and understanding, as well as their dynamic nature. It explains how knowledge is produced through understanding and interpreting information/data and the dynamics governing the whole process. Also, the study describes any dataset, no matter how large it may be, will remain incomplete, imperfect, and grey, so humans only rely on incomplete datasets to interpret the world. As such, information and knowledge are always grey and uncertain because they are basically contingent on subjective understandings and interpretations and imperfect inputs and data. Finally, as a key development, this study also demonstrates that human grey knowledge remains imbalanced across different disciplines and spheres. In the end, a brief overview of the philosophical paradigm of GST is also provided. GST is depicted as an anti-realistic, anti-positivistic, and non-deterministic approach, which is inherently pluralistic and ideographic. According to GST principles, dynamicity and change are essential parts of human narratives of the world and systems, and human knowledge is constantly reproduced through collecting new information. As a result, knowledge, theories, narratives, and scientific laws dynamically change. Given this premise, one could argue that GST is considerably compatible with the postulates of post-modern thinking. (shrink)

This article first surveys the established views on Wittgenstein's relation to analytic philosophy. Next it distinguishes among different ways of defining analytic philosophy—topical, doctrinal, methodological, stylistic, historical, and the idea that it is a family‐resemblance concept. It argues that while certain stylistic features are important, the historical and the family‐resemblance conceptions are the most auspicious, especially in combination. The answer to the title question is given in section 3. Contrary to currently popular “irrationalist” interpretations, Wittgenstein was an analytic philosopher in (...) all phases of his career, albeit an exceedingly exotic one whose style transcends the limits of academic philosophy in general. On the historical understanding he qualifies because he was influenced by and in turn influenced mainly analytic philosophers. On the family‐resemblance conception he qualifies both because he developed and employed logico‐linguistic analysis and because he initiated the linguistic turn and the distinction between philosophy and science that characterizes one important strand in analytic philosophy. (shrink)

This essay argues that non-linguistic animals qualify not just for externalist notions of rationality (maximizing biological fitness or utility), but also for internal ones. They can act for reasons in several senses: their behaviour is subject to intentional explanations, they can act in the light of reasons - provided that the latter are conceived as objective facts rather than subjective mental states - and they can deliberate. Finally, even if they could not, it would still be misguided to maintain that (...) animals are capable only of (mechanical) behaviour, not of (intentional) action. (shrink)

This paper deploys a Cantor-style diagonal argument which indicates that there is more possible mathematical content than there are propositional functions in Russell and Whitehead's Principia Mathematica and similar formal systems. This technical result raises a historical question: "How did Russell, who was himself an expert in diagonal arguments, not see this coming?" It turns out that answering this question requires an appreciation of Russell's understanding of what logic is, and how he construed the relationship between logic and Principia Mathematica.

William Lane Craig has revived interest in the medieval kalām argument to the point where it is now one of the most discussed arguments for God’s existence in the secondary literature. Still, the reception of Craig’s argument among philosophers of religion has been mostly critical. In the interest of developing an argument that more philosophers of religion would be inclined to support, I will lay the philosophical groundwork for a new kalām cosmological argument that, in contrast with Craig’s argument, does (...) not adopt such controversial positions as the dynamic theory of time and the metaphysical impossibility of an actual infinite. (shrink)