The present book is an introduction to the philosophy of mathematics. It asks philosophical questions concerning fundamental concepts, constructions and methods - this is done from the standpoint of mathematical research and teaching. It looks for answers both in mathematics and in the philosophy of mathematics from their beginnings till today. The reference point of the considerations is the introducing of the reals in the 19th century that marked an epochal turn in the foundations of mathematics. In the book problems (...) connected with the concept of a number, with the infinity, the continuum and the infinitely small, with the applicability of mathematics as well as with sets, logic, provability and truth and with the axiomatic approach to mathematics are considered. In Chapter 6 the meaning of infinitesimals to mathematics and to the elements of analysis is presented. The authors of the present book are mathematicians. Their aim is to introduce mathematicians and teachers of mathematics as well as students into the philosophy of mathematics. The book is suitable also for professional philosophers as well as for students of philosophy, just because it approaches philosophy from the side of mathematics. The knowledge of mathematics needed to understand the text is elementary. Reports on historical conceptions. Thinking about today‘s mathematical doing and thinking. Recent developments. Based on the third, revised German edition. For mathematicians - students, teachers, researchers and lecturers - and readersinterested in mathematics and philosophy. Contents On the way to the reals On the history of the philosophy of mathematics On fundamental questions of the philosophy of mathematics Sets and set theories Axiomatic approach and logic Thinking and calculating infinitesimally – First nonstandard steps Retrospection. (shrink)

Simple Heuristics That Make Us Smart invites readers to embark on a new journey into a land of rationality that differs from the familiar territory of cognitive science and economics. Traditional views of rationality tend to see decision makers as possessing superhuman powers of reason, limitless knowledge, and all of eternity in which to ponder choices. To understand decisions in the real world, we need a different, more psychologically plausible notion of rationality, and this book provides it. It is about (...) fast and frugal heuristics--simple rules for making decisions when time is pressing and deep thought an unaffordable luxury. These heuristics can enable both living organisms and artificial systems to make smart choices, classifications, and predictions by employing bounded rationality. But when and how can such fast and frugal heuristics work? Can judgments based simply on one good reason be as accurate as those based on many reasons? Could less knowledge even lead to systematically better predictions than more knowledge? Simple Heuristics explores these questions, developing computational models of heuristics and testing them through experiments and analyses. It shows how fast and frugal heuristics can produce adaptive decisions in situations as varied as choosing a mate, dividing resources among offspring, predicting high school drop out rates, and playing the stock market. As an interdisciplinary work that is both useful and engaging, this book will appeal to a wide audience. It is ideal for researchers in cognitive psychology, evolutionary psychology, and cognitive science, as well as in economics and artificial intelligence. It will also inspire anyone interested in simply making good decisions. (shrink)

Two crucial concepts of the methodology and philosophy of mathematics are considered: proof and truth. We distinguish between informal proofs constructed by mathematicians in their research practice and formal proofs as defined in the foundations of mathematics (in metamathematics). Their role, features and interconnections are discussed. They are confronted with the concept of truth in mathematics. Relations between proofs and truth are analysed.

This volume collects Gigerenzer's recent articles on the psychology of rationality. This volume should appeal, like the earlier volumes, to a broad mixture of cognitive psychologists, philosophers, economists, and others who study decision making.

The essays in this collection have been written for Gerd Buchdahl, by colleagues, students and friends, and are self-standing pieces of original research which have as their main concern the metaphysics and philosophy of science of the seventeenth and eighteenth centuries. They focus on issues about the development of philosophical and scientific thought which are raised by or in the work of such as Bernoulli, Descartes, Galileo, Kant, Leibniz, Maclaurin, Priestly, Schelling, Vico. Apart from the initial bio-bibliographical piece and (...) those by Robert Butts and Michael Power, they do not discuss Buchdahl or his ideas in any systematic, lengthy, or detailed way. But they are collected under a title which alludes to the book, Metaphysics and the Philosophy of Science: The Classical Origins, Descartes to Kant (1969), which is central in the corpus of his work, and deal with the period and some of the topics with which that book deals. (shrink)

Nicholas of Cusa was first of all a theologian but he was interested also in mathematic and natural sciences. In fact philosophico-theological and mathematical ideas were intertwined by him, theological and philosophical ideas influenced his mathematical considerations, in particular when he considered philosophical problems connected with mathematics and vice versa, mathematical ideas and examples were used by him to explain some ideas from theology. In this paper we attempt to indicate this mutual influence. We shall concentrate on the following problems: (...) the role and place of mathematics and mathematical knowledge in knowledge in general and in particular in theological knowledge, ontology of mathematical objects and their origin, in particular their relations to God and their meaning for the description of the world and physical reality, infinity in mathematics versus infinity in theology and their mutual relations and connections. It will be shown that—according to Nicholas—mathematics and mathematical thinking are tools of rationalization of theology and liberating it in a certain sense from the trap of apophatic theology. (shrink)

The paper is devoted to the description of life and scientific achievements as well as the influence of Seweryna Łuszczewska-Romahnowa.

The paper is devoted to the discussion of some philosophical and historical problems connected with the theorem on the undefinability of the notion of truth. In particular the problem of the priority of proving this theorem will be considered. It is claimed that Tarski obtained this theorem independently though he made clear his indebtedness to Gödel’s methods. On the other hand, Gödel was aware of the formal undefinability of truth in 1931, but he did not publish this result. Reasons for (...) that are also considered. (shrink)

Lvov-Warsaw School in Philosophy – as the very name suggests – was connected mainly with two academic centers: universities in Lvov and Warsaw. However, it had a broader impact. The members of this school were active also at other universities, in particular in Cracow, Vilnius and Poznań. The aim of the paper is to present and analyze the connections of Lvov-Warsaw School with the University in Poznań.

Axiomatic rationality is defined in terms of conformity to abstract axioms. Savage limited axiomatic rationality to small worlds, that is, situations in which the exhaustive and mutually exclusive set of future states S and their consequences C are known. Others have interpreted axiomatic rationality as a categorical norm for how human beings should reason, arguing in addition that violations would lead to real costs such as money pumps. Yet a review of the literature shows little evidence that violations are actually (...) associated with any measurable costs. Limiting axiomatic rationality to small worlds, I propose a naturalized version of rationality for situations of intractability and uncertainty, all of which are not in. In these situations, humans can achieve their goals by relying on heuristics that may violate axiomatic rationality. The study of ecological rationality requires formal models of heuristics and an analysis of the structures of environments these can exploit. It lays the foundation of a moderate naturalism in epistemology, providing statements about heuristics we should use in a given situation. Unlike axiomatic rationality, ecological rationality can explain less-is-more effects, formalize when one should move from ‘is’ to ‘ought,’ and be evaluated by goals beyond coherence, such as predictive accuracy, frugality, and efficiency. Ecological rationality can be seen as a formalization of means–end instrumentalist rationality, based on Herbert Simon’s insight that rational behavior is a function of the mind and its environment. (shrink)

Praca poświęcona jest prezentacji i krytycznej ocenie poglądów filozoficznych Jana Franciszka Drewnowskiego na matematykę i logikę. Podstawą rozważań są cztery źródła. Podstawowym źródłem jest Drewnowskiego "Zarys programu filozoficznego", dalej dwa artykuły, a mianowicie "Stosowanie logiki symbolicznej w filozofii" oraz "Uwagi o stosowaniu logiki symbolicznej", w końcu fragmenty dziennika. Pokazuje się w niej, jak Drewnowski rozumiał matematykę i teorie matematyczne, jak widział logikę i jej rolę w nauce oraz na ile był zaznajomiony ze współczesnymi osiągnięciami w zakresie logiki matematycznej i podstaw (...) matematyki oraz z ich konsekwencjami dla filozofii matematyki. (shrink)

The volume consists of thirteen papers devoted to various problems of the philosophy of logic and mathematics. They can be divided into two groups. The first group contains papers devoted to some general problems of the philosophy of mathematics whereas the second group - papers devoted to the history of logic in Poland and to the work of Polish logicians and math-ematicians in the philosophy of mathematics and logic. Among considered problems are: meaning of reverse mathematics, proof in mathematics, the (...) status of Church's Thesis, phenomenology in the philosophy of mathematics, mathematics vs. theology, the problem of truth, philosophy of logic and mathematics in the interwar Poland. (shrink)

The aim of the paper is to consider ontological views connected with mathematics and logic of main representatives of Lvov-Warsaw School of Philosophy. In particular views of the following scholars will be presented and discussed: Jan Łukasiewicz, Stanisław Leśniewski, Alfred Tarski, Tadeusz Kotarbiński and Kazimierz Ajdukiewicz. We shall consider also views of Andrzej Mostowski who belonged to the second generation of the school as well as of Leon Chwistek who was not directly the member of this group but whose conceptions (...) are of interest. (shrink)

The paper is devoted to two Polish philosophers and logicians Henryk Struve and Władysław Biegański. They represented the traditional pre-mathematical approach to logic and stood on the threshold of the new paradigm of logic as formal mathematical logic. Their views concerning logic and its philosophy as well as its relations to philosophy and mathematics will be analyzed. Those views will be compared with views of their contemporaries and with views of Jan Łukasiewicz—one of the main leaders of Warsaw School of (...) Logic who represented the new paradigm. (shrink)

The aim of this paper is to present and analyse the views on logic of the members of the so-called Cracow Circle, namely the Dominican Father Józef M. Bocheński, Rev. Jan Salamucha, and Jan Franciszek Drewnowski. They tried to apply the methods of modern formal/mathematical logic to philosophical and theological problems. In particular, they attempted to modernise contemporary Thomism by employing logical tools. The influence of Jan Łukasiewicz, the co-founder of the Warsaw School of Logic will be also discussed. --------------- (...) Received: 16/01/2021. Reviewed: 24/03/2021. Accepted: 13/05/2021. (shrink)

Gerd Buchdahl; XIV—The Relation Between ‘Understanding’ and ‘Reason’ in the Architectonic of Kant's Philosophy1, Proceedings of the Aristotelian Society, Volume.

The aim of this book is to present and analyze philosophical conceptions concerning mathematics and logic as formulated by Polish logicians, mathematicians and philosophers in the 1920s and 1930s. It was a remarkable period in the history of Polish science, in particular in the history of Polish logic and mathematics. Therefore, it is justified to ask whether and to what extent the development of logic and mathematics was accompanied by a philosophical reflection. We try to answer those questions by analyzing (...) both works of Polish logicians and mathematicians who have a philosophical temperament as well as their research practice. Works and philosophical views of the following Polish scientists will be analyzed: Wacław Sierpiński, Zygmunt Janiszewski, Stefan Mazurkiewicz, Stefan Banach Hugo Steinhaus, Eustachy Żylińsk and Leon Chwistek, Jan Łukasiewicz, Zygmunt Zawirski, Stanisław Leśniewski, Tadeusz Kotarbiński, Kazimierz Ajdukiewicz, Alfred Tarski, Andrzej Mostowski and Henryk Mehlberg, Jan Sleszyński, Stanisław Zaremba and Witold Wilkosz. To indicate the background of scientists being active in the 1920s and 1930s we consider in Chapter 1 some predecessors, in particular: Jan Śniadecki, Józef Maria Hoene-Wroński, Samuel Dickstein and Edward Stamm. (shrink)

The paper is devoted to the presentation and analysis of the philosophical views concerning logic and mathematics of the leading members of Cracow Circle, i.e., of Jan Salamucha, Jan Franciszek Drewnowski and Józef Maria Bocheński. Their views on the problem of possible applicability of logical tools in metaphysical and theological researches is also discussed.

The degree of success in creating quality care for people suffering from dementia is limited despite extensive research. This article describes Healthcare providers’ experience with the ethical challenges and possibilities in the relationship with patients suffering from dementia and its impact on quality care. The material is based on qualitative, in-depth individual narrative interviews with 12 professional Healthcare providers from two different nursing homes. The transcribed interview texts were subjected to a phenomenological–hermeneutical interpretation. To provide quality care to patients with (...) dementia, the Healthcare providers emphasized the importance of sensing and understanding the patients’ emotional and bodily expressions through sentient attentiveness and recognition of the patient as a person. They also described reciprocity of expressions in the relationship where the patient recognized them both as persons and Healthcare providers. The analyses of the findings are, inter alia, discussed in light of Løgstrup’s relational philosophy of ethics. (shrink)

What is the nature of moral behavior? According to the study of bounded rationality, it results not from character traits or rational deliberation alone, but from the interplay between mind and environment. In this view, moral behavior is based on pragmatic social heuristics rather than moral rules or maximization principles. These social heuristics are not good or bad per se, but solely in relation to the environments in which they are used. This has methodological implications for the study of morality: (...) Behavior needs to be studied in social groups as well as in isolation, in natural environments as well as in labs. It also has implications for moral policy: Only by accepting the fact that behavior is a function of both mind and environmental structures can realistic prescriptive means of achieving moral goals be developed. (shrink)

The contrast between the two approaches alluded to in the title has gained a certain prominence in our own day. With the knowledge of hindsight it will be of interest therefore to study its incidence in an earlier period, in the writings of Whewell and Mill, Which may thus yield added significance for a later generation. Right at the start there is a difficulty. Not all inductivists agree on their principles, or their interpretation of the logic of scientific reasoning, and (...) the same is true of deductivism, and the differences can therefore be discussed only in connection with individual writers. When this is done, we do find moreover some considerable variations in the treatment of their respective doctrines by the members of each of our two schools. To define these differences, we need a finer structure of elements of classification, indicative of criteria for the acceptance of scientific hypotheses. (shrink)

The aim of this paper is to describe the history of logical investigations at the University of Poznań. The organisational structures within the discipline as well as the outstanding logicians and their achievements are presented. Connections with the Lviv–Warsaw School are indicated.

We consider expansions of models of Peano arithmetic to models ofA 2 s -¦Δ 1 1 +Σ 1 1 −AC which consist of families of sets definable by nonstandard formulas.

A demarcation between kant's general metaphysics (transcendental principles) and his special metaphysics is attempted, through a discussion of kant's three accounts of lawlikeness, 'transcendental', 'empirical' and 'metaphysical'. the distinctions are defended via a number of 'indicators' in kant's writings, and the 'looseness of fit' between the different types of lawlikeness is discussed.

Research practice is too often shaped by routines rather than reflection. The routine of sampling subjects, but not stimuli, is a case in point, leading to unwarranted generalizations. It likely originated out of administrative rather than scientific concerns. The routine of sampling subjects and testing their averages for significance is reinforced by delusions about its meaningfulness, including the replicability delusion.

In the debate regarding the different possibilities for gene therapy, it is presupposed that the manipulations are limited to the nuclear genome (nDNA). Given recent advances in genetics, mitochondrial genome (mtDNA) and diseases must be considered as well. In this paper, we propose a three dimensional framework for the ethical debate of gene therapy where we add the genomic type (nDNA vs. mtDNA) as a third dimension to be considered beside the paradigmatic dimensions of target cell (somatic vs. germ-line) and (...) purpose (therapeutic vs. enhancement). Somatic gene therapy can be viewed today as generally accepted, and we review t he contemporary arguments surrounding it on the basis of bioethical-pragmatic, socio-political and deontological classifications. Many of the supposed ethical questions of somatic gene therapy today are not new; they are well-known issues of research ethics. We also critically summarize the different international perspectives and the German ethical discussion regarding manipulations of germ-line cells. (shrink)