Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses the linguistic (...) pitfalls and misperceptions philosophers in this field are often prone to, and explores the misapplications of epistemic principles from the empirical sciences to the exact sciences. What emerges is a picture of mathematics both sensitive to mathematical practice, and to the ontological and epistemological issues that concern philosophers. (shrink)

(2000). CASSIRER, SCHLICK AND ‘STRUCTURAL’ REALISM: THE PHILOSOPHY OF THE EXACT SCIENCES IN THE BACKGROUND TO EARLY LOGICAL EMPIRICISM. British Journal for the History of Philosophy: Vol. 8, No. 1, pp. 71-106.

In this paper I argue that there is a difficulty for Fine's exact semantics for counterfactuals. The difficulty undermines Fine's reasons for preferring exact semantics to possible worlds semantics.

The role of mathematics in scientific practice is too readily relegated to that of formulating equations that model or describe what is being investigated, and then finding solutions to those equations. I survey the role of mathematics in: 1. Exact solutions of differential equations, especially conformal mapping; and 2. Simulations of solutions to differential equations via numerical methods and via agent-based models; and 3. The use of experimental models to solve equations (a) via physical analogies based on similarity of (...) the form of the equations, such as Prandtl’s soap-film method, and (b) the method of physically similar systems. Two major themes emerge: First, the role of mathematics in science is not well described by deduction from axioms, although it generally involves deductive reasoning. Creative leaps, the integration of experimental or observational evidence, synthesis of ideas from different areas of mathematics, and insight regarding analogous forms are required to find solutions to equations. Second, methods that involve mappings or transformations are in use in disparate contexts, from the purely mathematical context of conformal mapping where it is mathematical objects that are mapped, to the use of concrete physical experimental models, where one concrete thing is shown to correspond to another. (shrink)

There is a fundamental subsets–partitions duality that runs through the exact sciences. In more concrete terms, it is the duality between elements of a subset and the distinctions of a partition. In more abstract terms, it is the reverse-the-arrows of category theory that provides a major architectonic of mathematics. The paper first develops the duality between the Boolean logic of subsets and the logic of partitions. Then, probability theory and information theory (as based on logical entropy) are shown to (...) start with the quantitative versions of subsets and partitions. Some basic universal mapping properties in the category of Sets are developed that precede the abstract duality of category theory. But by far the main application is to the clarification and interpretation of quantum mechanics. Since classical mechanics illustrates the Boolean worldview of full distinctness, it is natural that quantum mechanics would be based on the indefiniteness of its characteristic superposition states, which is modeled at the set level by partitions (or equivalence relations). This approach to interpreting quantum mechanics is not a jury-rigged or ad hoc attempt at the interpretation of quantum mechanics but is a natural application of the fundamental duality running throughout the exact sciences. (shrink)

This is the first in a series of papers on Predicative Algebraic Set Theory, where we lay the necessary groundwork for the subsequent parts, one on realizability [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory II: Realizability, Theoret. Comput. Sci. . Available from: arXiv:0801.2305, 2008], and the other on sheaves [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory III: Sheaf models, 2008 ]. We introduce the notion of a predicative category with small (...) maps and show that it provides a sound and complete semantics for constructive set theories like IZF and CZF. The main technical contribution of this paper is that it shows in detail that such categories can always be conservatively embedded in categories that are exact. These exactness properties play a crucial rôle in showing that predicative categories with small maps contain models of set theory and that they are closed under sheaves and realizability. We will prove the former statement in this paper as well, leaving a proof of the closure properties to the papers on realizability and sheaves as mentioned above. (shrink)

This paper analyses the attitude of the contemporary philosophy to the problems associated with increasingly radical diagnoses concerning anthropogenic climate changes that may lead the human civilization on Earth to a global catastrophe. One can identify three approaches to this issue in contemporary philosophy: involvement in the breakthrough taking place; evaluation of the change process from an axiological perspective or ignoring the evolving phenomena on the grounds that it is not possible to define them meaningfully from the perspective of theoretical (...) orientations that currently dominate the contemporary philosophy. (shrink)

We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and (...) uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chaos; we also show how efficient algorithms have been obtained for computing elementary functions in exact real arithmetic. (shrink)

The theoretical framework adopted in the exact sciences, for constructing and testing deterministic theories on the one hand, and modelling and analysis of observed phenomena on the other, is often implicitly assumed to be that of structural stability. In view of recent developments in nonlinear dynamics, it is argued here that in general it may not be possible to assume strict determinism and structural stability simultaneously; either strict determinism holds, in which case the fragility framework may turn out to (...) be the appropriate framework for the study of certain phenomena in the exact sciences, or ‘structural stability’ is restored at the expense of introducing stochasticity. In this sense a certain degree of indeterminacy may be unavoidable even at the classical level. (shrink)

This essay traces the history of early molecular dynamics simulations, specifically exploring the development of SHAKE, a constraint-based technique devised in 1976 by Jean-Paul Ryckaert, Giovanni Ciccotti and the late Herman Berendsen at CECAM (Centre Européen de Calcul Atomique et Moléculaire). The work of the three scientists proved to be instrumental in giving impetus to the MD simulation of complex polymer systems and it currently underpins the work of thousands of researchers worldwide who are engaged in computational physics, chemistry and (...) biology. Despite its impact and its role in bringing different scientific fields together, accurate historical studies on the birth of SHAKE are virtually absent. By collecting and elaborating on the accounts of Ryckaert and Ciccotti, this essay aims to fill this gap, while also commenting on the conceptual and computational difficulties faced by its developers. (shrink)

The essays in this volume concern the points of intersection between analytic philosophy and the philosophy of the exact sciences. More precisely, it concern connections between knowledge in mathematics and the exact sciences, on the one hand, and the conceptual foundations of knowledge in general. Its guiding idea is that, in contemporary philosophy of science, there are profound problems of theoretical interpretation-- problems that transcend both the methodological concerns of general philosophy of science, and the technical concerns of (...) philosophers of particular sciences. A fruitful approach to these problems combines the study of scientific detail with the kind of conceptual analysis that is characteristic of the modern analytic tradition. Such an approach is shared by these contributors: some primarily known as analytic philosophers, some as philosophers of science, but all deeply aware that the problems of analysis and interpretation link these fields together. (shrink)

The Scientific Revolution 1500/1600–1700.

Richard Swinburne’s argument in The Existence of God discusses many probabilities, ultimately concluding that God probably exists. Swinburne gives exact values to almost none of these probabilities. I attempted to assign values to the probabilities that met that weak condition that they could be correct. In this paper, I first present a brief outline of Swinburne’s argument in The Existence of God. I then present the problems I encountered in Swinburne’s argument, specifically problems that interfered with my attempt to (...) arrive at values for the probabilities discussed by Swinburne. Finally, I suggest that Swinburne’s argument would be more persuasive if he assigned exact values to his probabilities. (shrink)

The present paper examines the role of exact results in the theory of many‐body physics, and specifically the example of the Mermin‐Wagner theorem, a rigorous result concerning the absence of phase transitions in low‐dimensional systems. While the theorem has been shown to hold for a wide range of many‐body models, it is frequently ‘violated’ by results derived from the same models using numerical techniques. This raises the question of how scientists regulate their theoretical commitments in such cases, given that (...) the models, too, are often described as approximations to the underlying ‘full’ many‐body problem. (shrink)

This work studies the systematic role of the deduction of categories in Paul Natorp’s Logical Foundations of the Exact Sciences. Through an analysis of the deduction of the categories of quantity and quality, we contend that this deduction is not merely a historiographical exercise, but it is the core of Natorp’s system. It is argued that Natorp follows a synthetic method, rather than an analytic, similar to that employed by Kant in the Critique of Pure Reason. We argue that (...) the core of Natorp’s deduction is rooted in the principle of correlation. Unlike Kant, who derives categories from the table of judgments, Natorp constructs his deduction by examining the structure of thought itself. We demonstrate that this approach allows for a systematic deduction of the properties of number and the fundamental series. The study shows how this deduction of categories is the ground for the construction of the fundamental series and how it is aligned with Natorp’s methodological prescriptions. (shrink)

The study reviews particular conceptions of natural and exact sciences which had a significant impact on social and humanitarian knowledge primarily on philosophy: the theories of sets, heterarchy, cybernetic approach and others. G. Bateson’s cybernetic or systems theory allows more comprehensive understanding of religious phenomena and the Pentecostal expansion in Brasilia among others in the context of all relevant areas taking into account their interconnections and mutual influence. The heterarchy conception allowed to I.V. Krasavin to solve the problem in (...) general intellect theory of the Italian post-operaisms (M. Lazzarato, A. Negri, P. Virno, M. Pasquinelli, C. Vercellone, T. Terranova and ect) and conclude that it’s equally able as to liberate the society of the exploitation as to be appropriated by the society of control. The reverse of conceptions, usage of connectionism although criticized for the excessive simplifications, in the philosophy of brain and pedagogy gives the possibility to improve the methods of teaching increasing its efficiency. The natural sciences’ conceptions, the quantum physics for example, are applied in the philosophy of mind. Following W.V.O. Quine the representatives of the analytic philosophy more often propose to naturalize the discipline due to the natural sciences achievements. Although the undue tendency for formalization can divert from meaning as in F. Salis’ dealing with the problem of the literature characters’ narcissism. The application of a priory developed mathematical models trying to fit phenomena into them often harm philosophy. The mathematical formalization of the «second law of criminal behavior» suggested by the Norwegian forensic psychiatrist and psychoanalyst D. Abrahamsen seems to be not only meaningless but also ethically unacceptable. Bringing methodology of other disciplines designed for practical results and really “working” to science allows take a fresh look at the problem, covering it wider and from new angles. (shrink)

In 2015, the FDA revised its indefinite deferral policy towards MSM blood donors. I develop an empirical case for the revised policy and show why rights-based objections fail. I introduce and defend a distinction between “ordinary” and “exact” justification that accounts for the worries in the empirical case. I conclude that the FDA is committed to further revising the policy to let MSM donors with exact justification donate.

In a previous paper we have shown that in quantum chromodynamics the gluon propagator vanishes in the infrared limit, while the ghost propagator is more singular than a simple pole. These results were obtained after angular averaging, but in the current paper we go beyond this approximation and perform an exact calculation of the angular integrals. The powers of the infrared behaviour of the propagators are changed substantially. We find the very intriguing result that the gluon propagator vanishes in (...) the infrared exactly like p2, whilst the ghost propagator is exactly as singular as 1/p4. We also find that the value of the infrared fixed point of the QCD coupling is much decreased from the y-max estimate: it is now equal to 4π/3. Following a recent study by von Smekal et al.[1], we analyzed in Ref.[2] the coupled Dyson- Schwinger equations for the gluon and ghost form factors F and G. The approximations were two-fold: firstly the vertices were taken bare, and secondly angular averaging was introduced (the so-called y-max approximation). Deferring to later work an improvement of the vertices. (shrink)

This collection of essays reflects the wide range of David Pingree's expertise in the scientific texts of Ancient Mesopotamia, Greece, India, Persia, and the medieval Arabic, Hebrew and Latin traditions. Both theoretical aspects and the practical applications of the exact sciences-in time keeping, prediction of the future, and the operation of magic-are dealt with. The book includes several critical editions and translations of hitherto unknown or understudied texts, and a particular emphasis is on the diffusion of scientific learning from (...) one culture to another, and through time. Above all, the essays show the variety and sophistication of the exact sciences in non-Western societies in pre-modern times. (shrink)

In the second chapter of his book Kant and the Exact Sciences Michael Friedman deals with two different interpretations of the relation or the difference between algebra and arithmetic in Kant's thought. According to the first interpretation algebra can be described as general arithmetic because it generalizes over all numbers by the use of variables, whereas arithmetic only deals with particular numbers. The alternative suggestion is that algebra is more general than arithmetic because it considers a more general class (...) of magnitudes. This means that arithmetic is concerned only with rational magnitudes, whereas algebra is also concerned with irrational magnitudes. In this article, I will discuss which of the two aforementioned approaches is to be considered the most plausible interpretation of Kant's theory of algebra and arithmetic. According to Friedman, the first interpretation cannot be reconciled with certain statements made by Kant on various occasions. The second interpretation is developed by Friedman himself. It is meant to be an attempt to avoid such inconsistencies. By a detailed analysis of the texts Friedman himself cites I shall examine the soundness of his arguments against the first interpretation and the compatibility of his own interpretation of the difference between algebra and arithmetic with the relevant passages in Kant's texts. It will turn out that the reasons that make Friedman reject the first interpretation are invalid as they are based on misunderstandings and that his own interpretation does not expound Kant's notions on that subject correctly, whereas the first interpretation is compatible with these passages. Thus, I conclude that the interpretation rejected by Friedman, unlike his own approach, is actually the more adequate interpretation of Kant. (shrink)

Summary The Sanskrit term ‘Yavana’, originally a transliteration of ‘Ionian (Greek)’ but later applied to other foreigners as well, was used throughout the common era to designate various foreign importations in the exact sciences. Likewise, the name ‘Indian’ was attached to several mathematical concepts and techniques in the Islamic world (as well as Europe) from about the seventh century onward. However, not all innovations adopted from or into the Indian tradition were labeled ‘Indian’ or ‘Yavana’ respectively. This paper examines (...) the question of what characteristics marked some borrowed techniques and concepts as ‘foreign’ and stamped them with their outlandish origin, while others were quietly assimilated into ‘indigenous’ learning. (shrink)

A social history of Andalusi exact sciences should deal with scientific schools, teaching, travels to the East, patronage and scientific professions. I claim that such a social history of Andalusi science is almost impossible because of the lack of sources.

Young children initially learn to ‘count’ without understanding either what counting means, or what numerical quantities the individual number words pick out. Over a period of many months, children assign progressively more sophisticated meanings to the number words, linking them to discrete objects, to quantification, to numerosity, and so on. Eventually, children come to understand the logic of counting. Along with this knowledge comes an implicit understanding of the successor function, as well as of the principle of equinumerosity, or (...) class='Hi'>exact equality between sets. Thus, when children arrive at a mature understanding of counting, they have a way of mentally representing exact, large numbers. (shrink)

The Vienna Circle's philosophical views were ludicrously simplistic and their perception of their place in history inflated, but like the Bloomsbury Circle with which they had connections, they managed to be interesting.

Without addressing the measurement problem (i. e., what causes the wave function to “collapse,” or to ”branch,” or a history to become realized, or a property to actualize), I discuss the problem of the timing of the quantum measurement: Assuming that in an appropriate sense a measurement happens, when precisely does it happen? This question can be posed within most interpretations of quantum mechanics. By introducing the operator M, which measures whether or not the quantum measurement has happened, I suggest (...) that, contrary to what is often claimed, quantum mechanics does provide a precise answer to this question, although a somewhat surprising one. (shrink)

This essay considers the nature of conceptual frameworks in science, and suggests a reconsideration of the role played by philosophy in radical conceptual change. On Kuhn's view of conceptual conflict, the scientist's appeal to philosophical principles is an obvious symptom of incommensurability; philosophical preferences are merely “subjective factors” that play a part in the “necessarily circular” arguments that scientists offer for their own conceptual commitments. Recent work by Friedman has persuasively challenged this view, revealing the roles that philosophical concerns have (...) played in preparing the way for conceptual change, creating an enlarged conceptual space in which alternatives to the prevailing framework become intelligible and can be rationally discussed. If we shift our focus from philosophical themes or preferences to the process of philosophical analysis, however, we can see philosophy in a different and much more significant historic role: not merely as an external source of general heuristic principles and new conceptual possibilities, but, at least in the most important revolutionary developments, as an objective tool of scientific inquiry. I suggest that this approach offers some insight into the philosophical significance of Newton's and Einstein's revolutionary work in physics, and of the interpretation of their work by (respectively) Kant and the logical positivists. It also offers insight into the connections between modern philosophy of science and some traditional philosophical concerns about the nature of a priori knowledge. (shrink)

In the latter half of the 19th century, economic thought in the Germanspeaking world was dominated, both intellectually and academically, by the so-called historical school, from Wilhelm Roscher to Gustav Schmoller and others. In 1871, the Austrian Carl Menger published his Grun&tze der Volkswirtschaftslehre (Menger, 1976 (1871)), customarily referred to as one of the three simultaneous discoveries of marginalist economics-the other two marginalist ‘revolutionaries’ being Jevons in England and Walras in France. Twelve years later, in 1883, Menger published a major (...) methodological treatise entitled Untersuchungen iiber die Methode der Socialwissenschaften und der Politischen Oekonomie insbesondere (Menger, 1963 (1883)). This book included criticisms of some of the historicist principles of doing economics. In the same year, Schmoller, leader of the German historicists, wrote a critical review of Menger’s book (Schmoller, 1883). Menger reacted forcefully with a more straightforwardly polemical small book, Die Irrthiimer des Historismus in der deutschen National6konomie (Menger, 1884). Commentaries by others appeared in later years, but this brief episode amounted to what has thereafter been called the Methodenstreit between Menger and Schmoller. It has been established as perhaps the most famous methodological controversy in the history of the social sciences. (shrink)

This paper describes the local context of Kant’s scientific education. It provides an informed sense of what Kant’s scientific training was like by presenting each relevant member of the philosophy faculty at the university in Königsberg where Kant was a student, and the scientific activities each one was engaged in. On the basis of this picture, it is argued that Kant’s relationship with one of his teachers, Martin Knutzen, may have been much more negative or critical than is typically supposed. (...) This evaluation has important implications especially for understanding Kant’s early works in natural philosophy. (shrink)

The precise 1:1 segregation of Mendelian heredity is ordinarily taken for granted, yet there are numerous examples of ‘cheating’ genes that perpetuate themselves in the population by biasing the Mendelian process in their favor. One example is the Segregation Distortion system of Drosophila melanogaster, in which the distorting gene causes its homologous chromosome to produce a nonfunctional sperm. This system depends on three closely linked components, whose molecular basis is beginning to be understood.The system is characterized by numerous modifiers changing (...) the degree of distortion. Mathematical theory shows that unlinked modifiers that change the degree of distortion in the direction of Mendelism always increase in the population. This provides a mechanism for removing cheaters and preserving the honesty of the Mendelian gene‐shuffle. (shrink)

ONE of the most celebrated mathematical physicists, Pierre-Simon Laplace is often remembered as the mathematician who showed that despite appearances, the Solar System does conform to Newton’s theories. Together with distinguished scholars Robert Fox and Ivor Grattan-Guinness, Charles Gillispie gives us a new perspective, showing that Laplace did not merely vindicate Newton’s system, but had a uniquely creative and independent mind.

When ancient mathematical treatises lack expositions of numerical techniques, what purposes could ancient mathematical theories be expected to serve? Ancient writers only rarely address questions of this sort directly. Possible answers are suggested by surveying geometry, mechanics, optics, and spherics to discover how the mathematical treatments imply positions on this issue. This survey shows the ways in which these ancient theoretical inquiries reflect practical activity in their fields. This account, in turn, suggests that the authors may have intended their theorems (...) not to predict, but to explain phenomena. We may then consider what kind of explanations they were seeking. (shrink)