In 1900 David Hilbert announced his famous list of then-opened mathematical problems; the problem number 6 in this list is axiomatization of physical theories. Since then a lot of systematic efforts have been invested into solving this problem. However the results of these efforts turned to be less successful than the early enthusiasts of axiomatic method expected. The existing axiomatizations of physical and biological theories provide a valuable logical analysis of these theories but they do not constitute anything (...) like their standard presentation,which can be used for transmission, evaluation, and justification of physical and biological knowledge. This state of theart in the axiomatization of physics is strong evidence that the standard notion of axiomatic theory stemming from Hilbert and Tarski is not appropriate for thet ask. However in the recent years in mathematics there emerged a new axiomatic approach best represented by the Homotopy Type theory (HoTT). We argue that the constructive axiomatic architecture used in HoTT has better chances to be successfully applied in physics as well as in computer science and engineering. (shrink)

In 1900 David Hilbert announced his famous list of then-opened mathematical problems; the problem number 6 in this list is axiomatization of physical theories. Since then a lot of systematic efforts have been invested into solving this problem. However the results of these efforts turned to be less successful than the early enthusiasts of axiomatic method expected. The existing axiomatizations of physical and biological theories provide a valuable logical analysis of these theories but they do not constitute anything (...) like their standard presentation,which can be used for transmission, evaluation, and justification of physical and biological knowledge. This state of theart in the axiomatization of physics is strong evidence that the standard notion of axiomatic theory stemming from Hilbert and Tarski is not appropriate for thet ask. However in the recent years in mathematics there emerged a new axiomatic approach best represented by the Homotopy Type theory (HoTT). We argue that the constructive axiomatic architecture used in HoTT has better chances to be successfully applied in physics as well as in computer science and engineering. (shrink)

This volume draws together Allan Gotthelf's pioneering work on Aristotle's biology. He examines Aristotle's natural teleology, the axiomatic structure of biological explanation, and the reliance on scientifically organized data in the three great works with which Aristotle laid the foundations of biological science.

The Cell Ontology (CL) is designed to provide a standardized representation of cell types for data annotation. Currently, the CL employs multiple is_a relations, defining cell types in terms of histological, functional, and lineage properties, and the majority of definitions are written with sufficient generality to hold across multiple species. This approach limits the CL’s utility for cross-species data integration. To address this problem, we developed a method for the ontological representation of cells and applied this method to (...) develop a dendritic cell ontology (DC-CL). DC-CL subtypes are delineated on the basis of surface protein expression, systematically including both species-general and species-specific types and optimizing DC-CL for the analysis of flow cytometry data. This approach brings benefits in the form of increased accuracy, support for reasoning, and interoperability with other ontology resources. 104. Barry Smith, “Toward a Realistic Science of Environments”, Ecological Psychology, 2009, 21 (2), April-June, 121-130. Abstract: The perceptual psychologist J. J. Gibson embraces a radically externalistic view of mind and action. We have, for Gibson, not a Cartesian mind or soul, with its interior theater of contents and the consequent problem of explaining how this mind or soul and its psychological environment can succeed in grasping physical objects external to itself. Rather, we have a perceiving, acting organism, whose perceptions and actions are always already tuned to the parts and moments, the things and surfaces, of its external environment. We describe how on this basis Gibson sought to develop a realist science of environments which will be ‘consistent with physics, mechanics, optics, acoustics, and chemistry’. (shrink)

This article is a systematic critical survey of work done in the philosophy of biology within the logical empiricist tradition, beginning in the 1930s and until the end of the 1950s. It challenges a popular view that the logical empiricists either ignored biology altogether or produced analyses of little value. The earliest work on the philosophy of biology within the logical empiricist corpus was that of Philipp Frank, Ludwig von Bertalanffy, and Felix Mainx. Mainx, in particular, provided (...) a detailed analysis of biology in the 1930s and 1940s in his contribution to the logical empiricists’ Encyclopedia of Unified Science. However, the most important contributions to the philosophy of biology were those of Joseph Henry Woodger and Ernest Nagel. Woodger is primarily remembered for deploying the axiomatic method in biology but he also used semiformal methods for the analysis of many biological problems. While Woodger’s axiomatic work was often derided by some later philosophers of biology (e.g., David Hull and Michael Ruse), this article defends both the biological and the philosophical significance of some of that work, for instance, those aspects that led to the recognition of the conceptual complexity of mereology and temporal identity in biological systems. Woodger’s semiformal analyses were even more important, for instance, his explication of the concepts of the Bauplan and of innateness. Nagel’s importance lies in his analyses of reduction and emergence in the context of all empirical sciences and his use of these analyses in a careful exploration of biological problems. While Nagel’s model of reduction was generally rejected by philosophers of science in the 1970s and 1980s, particularly for biological contexts, it has recently been sympathetically reconstructed by many commentators; this article defends its continued relevance for the philosophy of biology. (shrink)

Although the introduction of the modern axiomatic method in physics is attributed to Hilbert, it is only recently that physicists and mathematicians have applied it significantly, i.e. on a basis extensive enough to promise fruitful results. Carnap, for one, stresses the importance of the axiomatic method, yet he considers its application in physics as a task for the future.

Dr Batóg is Lecturer in Logic at the University of Poznan. In Poland mathematical logic has been as characteristic an object of study and development as that of the atom elsewhere. The names of Fr Bochenski and Tarski are of as much importance in the logical field as are those of Bohr and Rutherford in the physical. The Polish approach seems to have been based chiefly on Whitehead & Russell’s Principia Mathematica, that Homeric work whose effect abroad was far greater (...) than at home. The bearing of this fact on the present book will be discussed in due course, for it has determined our author’s treatment of his subject. (shrink)

We characterize access to empirical objects in biology from a theoretical perspective. Unlike objects in current physical theories, biological objects are the result of a history and their variations continue to generate a history. This property is the starting point of our concept of measurement. We argue that biological measurement is relative to a natural history which is shared by the different objects subjected to the measurement and is more or less constrained by biologists. We call symmetrization the theoretical (...) and often concrete operation which leads to considering biological objects as equivalent in a measurement. Last, we use our notion of measurement to analyze research strategies. Some strategies aim to bring biology closer to the epistemology of physical theories, by studying objects as similar as possible, while others build on biological diversity. (shrink)

The aim of the paper is this: Instead of presenting a provisional and necessarily insufficient characterization of what mathematical physics is, I will ask the reader to take it just as that, what he or she thinks or believes it is, yet to be prepared to revise his opinion in the light of what I am going to tell. Because this is precisely, what I intend to do. I will challenge some of the received or standard views about mathematical physics (...) and replace them by a more sophisticated picture, which takes into account the methodological and philosophical roots of mathematical physics in Göttingen. (shrink)

The modern notion of the axiomatic method developed as a part of the conceptualization of mathematics starting in the nineteenth century. The basic idea of the method is the capture of a class of structures as the models of an axiomatic system. The mathematical study of such classes of structures is not exhausted by the derivation of theorems from the axioms but includes normally the metatheory of the axiom system. This conception of axiomatization satisfies the crucial (...) requirement that the derivation of theorems from axioms does not produce new information in the usual sense of the term called depth information. It can produce new information in a different sense of information called surface information. It is argued in this paper that the derivation should be based on a model-theoretical relation of logical consequence rather than derivability by means of mechanical (recursive) rules. Likewise completeness must be understood by reference to a model-theoretical consequence relation. A correctly understood notion of axiomatization does not apply to purely logical theories. In the latter the only relevant kind of axiomatization amounts to recursive enumeration of logical truths. First-order “axiomatic” set theories are not genuine axiomatizations. The main reason is that their models are structures of particulars, not of sets. Axiomatization cannot usually be motivated epistemologically, but it is related to the idea of explanation. (shrink)

This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their (...) ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method. (shrink)

The dramatic development of physics in the twentieth century has thrown philosophy into a crisis regarding its self-image, from which today's philosophy has still not fully recovered.1 The crisis had two consequences or complementary manifestations: First, there was a gradual retreat of philosophy from the natural sciences. Because physics turned out to be an autonomous discipline, which aims at cognition of nature apparently totally independent of any kind of philosophy, "natural philosophy" as a particular branch of philosophy seemed superfluous. Second, (...) in order to prevent its own abdication, natural philosophy mutated into a kind of meta-science, called philosophy of science, that avoids any serious .. (shrink)

Robert Brandon is one of the most important and influential of contemporary philosophers of biology. This collection of his recent essays covers all the traditional topics in the philosophy of evolutionary biology and as such could serve as an introduction to the field. There are essays on the nature of fitness, teleology, the structure of the theory of natural selection, and the levels of selection. The book also deals with newer topics that are less frequently discussed but are (...) of growing interest, for example the evolution of human language and the role of experimentation in evolutionary biology. A special feature of the collection is that it avoids jargon and is written in a style that will appeal to working evolutionary biologists as well as philosophers. (shrink)

In the last part of the Timaeus, where Plato presents his ideas about human physiology but also about biology in general, we find the combined activity of Intellect and Necessity. In this essay I investigate whether Plato, apart from his general statement about the combined activity of Reason and Necessity, proposes a more specific method of biological research. For this purpose I am going to examine some methodological passages as well as the way in which he exposes and (...) develops his views about the human body and its organs. (shrink)

This collection of essays by Robert Brandon spans two decades and most of the important problems in the philosophy of biology. Four of his five most important contributions to the philosophy of biology can be found here: the concept of relative adaptedness and its role in the propensity interpretation of fitness; the principle of natural selection; the use of the screening-off relation in defense of organismic selection; and the distinction between units of selection and levels of selection. The (...) fifth major contribution, an analysis of the concept of "environment," mentioned briefly in an essay on the co-evolution of organisms and environment, is given an extended treatment in his 1990 book, Adaptation and Environment. (shrink)

Feferman argues that category theory cannot stand on its own as a structuralist foundation for mathematics: he claims that, because the notions of operation and collection are both epistemically and logically prior, we require a background theory of operations and collections. Recently [2011], I have argued that in rationally reconstructing Hilbert’s organizational use of the axiomatic method, we can construct an algebraic version of category-theoretic structuralism. That is, in reply to Shapiro, we can be structuralists all the way (...) down ; we do not have to appeal to some background theory to guarantee the truth of our axioms. In this paper, I again turn to Hilbert; I borrow his distinction between the genetic method and the axiomatic method to argue that even if the genetic method requires the notions of operation and collection, the axiomatic method does not. Even if the genetic method is in some sense epistemically or logically prior, the axiomatic method stands alone. Thus, if the claim that category theory can act as a structuralist foundation for mathematics arises from the organizational use of the axiomatic method, then it does not depend on the prior notions of operation or collection, and so we can be structuralists all the way up. (shrink)

The general principles of the scientific method, which are applicable across all of the sciences, are essential for perspective, productivity, and innovation. These principles include deductive and inductive logic, probability, parsimony, and hypothesis testing, as well as science's presuppositions, limitations, ethics, and bold claims of rationality and truth. The implicit contrast is with specialized techniques confined to a given discipline, such as DNA sequencing in biology. Neither general principles nor specialized techniques can substitute for one another, but rather (...) the winning combination for scientists is mastery of both. The purposes of this book are to enhance perspective on science by drawing insights from the humanities, and to increase productivity by fostering a deep understanding of the general principles of scientific method. The examples and case studies span the physical, biological, and social sciences; include applications in agriculture, engineering, and medicine; and also explore science's interrelationships with disciplines in the humanities such as philosophy and law. This book engages a great diversity of viewpoints on science, both historical and contemporary, and responds by affirming science's rationality. Informed by position papers on science from the American Association for the Advancement of Science, National Academy of Sciences, and National Science Foundation, this book aligns with a distinctively mainstream vision of science. It is an ideal resource for anyone undertaking a systematic study of scientific method for the first time, from undergraduates to professionals in both the sciences and the humanities. (shrink)

Reductionism encompasses a set of ontological, epistemological, and methodological claims about the relation of different scientific domains. The basic question of reduction is whether the properties, concepts, explanations, or methods from one scientific domain (typically at higher levels of organization) can be deduced from or explained by the properties, concepts, explanations, or methods from another domain of science (typically one about lower levels of organization). Reduction is germane to a variety of issues in philosophy of science, including the structure of (...) scientific theories, the relations between different scientific disciplines, the nature of explanation, the diversity of methodology, and the very idea of theoretical progress, as well as to numerous topics in metaphysics and philosophy of mind, such as emergence, mereology, and supervenience. (shrink)

Although this excellent introductory and intermediate level text is intended for students of mathematics, it could serve well in any course for philosophers on that level. The first two chapters present the propositional and predicate calculi, along with an informal discussion of some of the set-theoretic concepts needed to study logic. The third chapter discusses what exactly an axiomatic system is, and examples of various mathematical systems cast in axiomatic form are provided; the discussion here, as elsewhere in (...) the book, is generally self-contained and presupposes only an ability to deal with abstract ideas. The study of the propositional calculus is resumed in chapter four, but its formulation is now more rigorous and thoroughly axiomatic; in a similar manner the predicate calculus is treated in the next chapter, and the usual metatheorems are proved about each, but the specific techniques differ from the usual. The last chapter concentrates on the completeness proof of quantification theory, relating it to some of the axiomatic algebraic systems considered earlier. An appendix sketches some sophisticated results of Vaught and Robinson on model theory and completeness. In sum, this is one of the very best of a plethora of logic texts on the market, and it should serve well for years to come.—P. J. M. (shrink)

In the paper we examine the method of axiomatic rejection used to describe the set of nonvalid formulae of Aristotle's syllogistic. First we show that the condition which the system of syllogistic has to fulfil to be ompletely axiomatised, is identical to the condition for any first order theory to be used as a logic program. Than we study the connection between models used or refutation in a first order theory and rejected axioms for that theory. We show (...) that any formula of syllogistic enriched with classical connectives is decidable using models in the domain with three members. (shrink)

The criticism of Hilbert's axiomatic system of geometry by Mate Meršić (Merchich, 1850-1928), presented in his work "Organistik der Geometrie" (1914, also in "Modernes und Modriges", 1914), is analyzed and discussed. According to Meršić, geometry cannot be based on its own axioms, as a logical analysis of spatial intuition, but must be derived as a "spatial concretion" using "higher" axioms of arithmetic, logic, and "rational algorithmics." Geometry can only be one, because space is also only one. It cannot be (...) reduced to manipulation with symbols but has to deal with concepts and provide real knowledge. According to Meršić, aiming at some particular proof of the consistency of the whole axiomatic system is meaningless because the consistency follows from the truth of individual axioms based on spatial intuition. It is impossible, according to Meršić, for the geometric axioms to be mutually independent because they must be interconnected as essential parts of a whole. Although Meršić did not recognize all the advantages of Hilbert's formalization of geometry, he clearly pointed out some crucial limitations of the axiomatic method. (shrink)

A nonstandard set theory ∗ZFC is proposed that axiomatizes the nonstandard embedding ∗. Besides the usual principles of nonstandard analysis, all axioms of ZFC except regularity are assumed. A strong form of saturation is also postulated. ∗ZFC is a conservative extension of ZFC.

In this article the predominant, purely theoretical perspectives on animal ethics are questioned and two important sources for empirical data in the context of animal ethics are discussed: methods of the social and methods of the natural sciences. Including these methods can lead to an empirical animal ethics approach that is far more adapted to the needs of humans and nonhuman animals and more appropriate in different circumstances than a purely theoretical concept solely premised on rational arguments. However, the potential (...) tension between lay people’s moral judgements and ethical theory must be handled with care. The thorough analysis of qualitative data can lead to a deep insight into e.g. ethical problems with the application of laws and guidelines, practicality issues with ethical theories, personal ambivalence, and cognitive biases. The interaction between animal ethics theory and empirical findings can lead to both a more context-sensitive and applicable ethical theory and a less arbitrary folk moral system. Findings from the natural sciences can also contribute valuable information to animal ethics theory—the more we know about the properties and preferences of nonhuman animals the better we can respect them. Here, however, it is vital not to justify invasive procedures for the sake of “ethical progress”. It might be ethically required to forego some scientific findings about nonhuman animals if it is not clear how much a procedure would harm them. Only with robust empirical methods will light ultimately be shed on the nature of our moral relationship with animals. (shrink)

This book publishes 31 of the author's selected papers which have appeared, with one exception, since 1970. The papers cover a wide range of topics in the philosophy of science. Part I is concerned with general methodology, including formal and axiomatic methods in science. Part II is concerned with causality and explanation. The papers extend the author's earlier work on a probabilistic theory of causality. The papers in Part III are concerned with probability and measurement, especially foundational questions about (...) probability. Part IV consists of several papers, including two historical ones, on the foundations of physics, with the main emphasis being on quantum mechanics. Part V, the longest part, is on the foundations of psychology and includes papers mainly on learning and perception. The book is aimed at philosophers of science, scientists concerned with the methodology of the social sciences, and mathematical psychologists interested in theories of learning, perception and measurement. (shrink)

There is, of course, one clear sense in which Spinoza’s axiomatic method is a method of presentation: this is the sense which contrasts a method of presentation with a method of discovery. In the Ethics, Spinoza is stating and explaining his views, not describing how he arrived at them or telling us how to make discoveries for ourselves. Nor does he elsewhere present the axiomatic method as a method of discovery. In the (...) seventeenth century, the distinction between a method of discovery and a method of presentation was often described as a distinction between "analytic" and "synthetic" method; Arnauld and Nicole write. (shrink)

Plato began it. After thinking about the nature of argument he concluded that the correct way of reasoning was the axiomatic way, and formulated the programme of axiomatization that Eudoxus and Euclid subsequently carried out. Since then the axiomatic method has been firmly established, not only as the method for mathematics, but as a paradigm to which all other disciplines should strive to be assimilated; and in this present century not only has axiomatization been carried through (...) as completely as it can be, but the most determined efforts have been made to wish hypotheticodeductive schemata on to biology, economics, and even history. (shrink)