In 1998 a long-lost proposal for an election law by Gottlob Frege (1848–1925) was rediscovered in the _Thüringer Universitäts- und Landesbibliothek_ in Jena, Germany. The method that Frege proposed for the election of representatives of a constituency features a remarkable concern for the representation of minorities. Its core idea is that votes cast for unelected candidates are carried over to the next election, while elected candidates incur a cost of winning. We prove that this sensitivity to past elections guarantees a (...) proportional representation of political opinions in the long run. We find that through a slight modification of Frege’s original method even stronger proportionality guarantees can be achieved. This modified version of Frege’s method also provides a novel solution to the apportionment problem, which is distinct from all of the best-known apportionment methods, while still possessing noteworthy proportionality properties. (shrink)

A two strain HIV/AIDS model with treatment which allows AIDS patients with sensitive HIV-strain to undergo amelioration is presented as a system of non-linear ordinary differential equations. The disease-free equilibrium is shown to be globally asymptotically stable when the associated epidemic threshold known as the basic reproduction number for the model is less than unity. The centre manifold theory is used to show that the sensitive HIV-strain only and resistant HIV-strain only endemic equilibria are locally asymptotically stable when the associated (...) reproduction numbers are greater than unity. Qualitative analysis of the model including positivity, boundedness and persistence of solutions are presented. The model is numerically analysed to assess the effects of treatment with amelioration on the dynamics of a two strain HIV/AIDS model. Numerical simulations of the model show that the two strains co-exist whenever the reproduction numbers exceed unity. Further, treatment with amelioration may result in an increase in the total number of infective individuals (asymptomatic) but results in a decrease in the number of AIDS patients. Further, analysis of the reproduction numbers show that antiretroviral resistance increases with increase in antiretroviral use. (shrink)

A nonlinear differential equation model is proposed to study the dynamics of HIV/AIDS and its effects on workforce productivity. The disease-free equilibrium point of the model is shown to be locally asymptotically stable when the associated basic reproduction number $$\mathcal{{R}}_{0}$$ is less than unity. The model is also shown to exhibit multiple endemic states for some parameter values when $$\mathcal{{R}}_{0} 1$$. Global asymptotic stability of the disease-free equilibrium is guaranteed only when the fractions of the Susceptible subclass populations are within (...) some bounds. Optimal control analysis of the model revealed that the most cost effective strategy that should be adopted in the fight against HIV/AIDS spread within the workforce is one that seeks to prevent infections and the treatment of infected individuals. (shrink)

In his article ‘The Structure of Emptiness’, 467–80. doi: 10.1353/pew.0.0069[Crossref], [Web of Science ®] [Google Scholar]) Graham Priest examines the concept of emptiness in the Mādhyamaka school of Nāgārjuna and his commentators Candrakīırti and Tsongkhapa from a mathematical point of view. The approach attempted in this article does not involve any commitment to Priest's more controversial dialethic Mādhyamaka interpretation. The purpose of the present paper is to explain Priest's sketchy but very insightful interpretation of objects as non-well-founded sets in (...) greater detail. Some problems concerning his idea to model the Mādhyamaka claim of the emptiness of emptiness by means of this kind of framework will be noted. Moreover, we will also discuss the possibility to represent the Mādhyamika's denial of the existence of irreducible constituents of empirical reality within a well-founded system of set theory. Finally, some slight mistakes in Priest's mathematical construction need to be pointed out. (shrink)

George Boole published the pamphlet The Mathematical Analysis of Logic in 1847. He believed that logic should belong to a universal mathematics that would cover both quantitative and nonquantitative research. With his pamphlet, Boole signalled an important change in symbolic logic: in contrast with his predecessors, his thinking was exclusively extensional. Notwithstanding the innovations introduced he accepted all traditional Aristotelean syllogisms. Nevertheless, some criticisms have been raised concerning Boole’s view of Aristotelean logic as the solution of algebraic equations. (...) In order to circumvent such criticisms, we show here how Boole’s conclusions may be deduced from the axioms and inference rules proposed in his pamphlet, from which Aristotle’s deductions in Prior Analytics (including those that are completed through an impossibility) can be stated as theorems. We clarify his method for dealing with both universal and particular premises by means of his symbol v and demystify some criticisms concerning the way he expressed particular premises. Boole’s conclusions for some of those premises for which according to Aristotle there is no deduction are also discussed. The deductive system presented here has the potential to provide a new perspective on Boole’s approach to logic in The Mathematical Analysis of Logic. (shrink)

Frisch’s 1933 macroeconomic model for business cycles has been extensively studied. The present study is the first comprehensive mathematical analysis of Frisch’s model. It provides a detailed reconstruction of how the model was built. We demonstrate the workability of Frisch’s PPIP model without adding hypotheses or changing the value of Frisch’s parameters. We prove that (1) the propagation model oscillates; (2) the PPIP model is mathematically incomplete; (3) the latter could have been calibrated by Frisch; (4) Frisch’s (...) class='Hi'>analysis and demonstration are based on Poincaré’s methodology. (shrink)

The meaning of the term “analysis” in Leibniz’s work is multifarious and it is doubtful that one could ever succeed in gathering this variety of meanings into a unified whole. However it has long been remarked that a landmass seems to detach itself from these moving waters – an island sometimes called by its inventor “The Most General Analytics of Human Thoughts”. Already sketched in the De Arte Combinatoria (1666) as a reform of the “analytical part” of Logic (pars (...) logices analytica), it rested on a very simple intuition: the possibility of developing, in parallel, three analyses, namely an analysis of thoughts (analysis notionum et veritatum), an analysis of the structure of language by which we express these thoughts (analysis linguarum or analysis grammatica) and an analysis of the symbols by which we represent them (analysis characterum). This “General Analytics” achieved, any human thought could be related to a catalog of simple notions expressed by basic characters (the so called alphabetum cogitationum humanarum) and, as a direct consequence, any truth could be expressed as a combination of these primitive symbols. (shrink)

George Boole collected ideas for the improvement of his Mathematical analysis of logic(1847) on interleaved copies of that work. Some of the notes on the interleaves are merely minor changes in explanation. Others amount to considerable extension of method in his mathematical approach to logic. In particular, he developed his technique in solving simultaneous elective equations and handling hypotheticals and elective functions. These notes and extensions provided a source for his later book Laws of thought(1854).

This article is devoted to some of the dominant positions in the philosophy of mathematics, in Descartes and in Leibniz, and to their consequences drawn by these authors in mathematical analysis. I shall treat of Descartes’ epistemological conceptions of analysis and of the primacy of order, both of which are excellently exposited in the _Rules for the direction of the mind_ and of various mathematical devices which he developed later, from the time of the _Cogitationes (...) privatae_ until the _Géométrie_. (shrink)

Mathematics is often positioned as either neutral or non-neutral by mathematicians. However, in practice, issues of neutrality arise in situated contexts, and the positioning of mathematics as either neutral or non-neutral is done for many purposes. We interpret positioning of mathematical work, with different degrees of neutrality, as a response to conflicts of interest and power dynamics. Using a framework from ecofeminist critical theory, we examine the ways that mathematical neutrality is positioned and communicated to different audiences in (...) ways that can appear contradictory. Our goal is to demonstrate that the neutral/non-neutral dualism is insufficient for the analysis of neutrality in mathematics, which requires instead a robust analytical lens. We situate our discussion of these issues in the context of communication regarding the mathematical analysis of partisan gerrymandering in the United States. Through a study of communication to different audiences by mathematicians regarding mathematical techniques used to study partisan gerrymandering, we illustrate various ways in which dualistic views of neutrality are insufficient to describe and understand the complicated role of neutrality in this context. (shrink)

Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number (...) of themes in recent philosophy of mathematics (concerning a priority and fictionalism, for example) in revealing new light. (shrink)

Trust and monitoring are traditionally antithetical concepts. Describing trust as a property of a relationship of reliance, we introduce a theory of trust and monitoring, which uses mathematical models based on two classes of functions, including _q_-exponentials, and relates the levels of trust to the costs of monitoring. As opposed to several accounts of trust that attempt to identify the special ingredient of reliance and trust relationships, our theory characterizes trust as a quantitative property of certain relations of reliance (...) that can be quantified and expressed as a scalar quantity. Our theory is applicable to both human–human and human–artificial agent interactions, as it is agnostic with respect to the concrete realization of trustworthiness properties, and is compatible with many views differing on which properties contribute to trust and trustworthiness. Finally, as our mathematical models make the quantitative features of trust measurable, they provide empirical studies on trust with a rigorous methodology for its measurement. (shrink)

The concept of similarity has had a rather mixed reputation in philosophy and the sciences. On the one hand, philosophers such as Goodman and Quine emphasized the „logically repugnant“ and „insidious“ character of the concept of similarity that allegedly renders it inaccessible for a proper logical analysis. On the other hand, a philosopher such as Carnap assigned a central role to similarity in his constitutional theory. Moreover, the importance and perhaps even indispensibility of the concept of similarity for (...) many empirical sciences can hardly be denied. The aim of this paper is to show that Quine’s and Goodman’s harsh verdicts about this notion are mistaken. The concept of similarity is susceptible to a precise logico-mathematical analysis through which its place in the conceptual landscape of modern mathematical theories such as order theory, topology, and graph theory becomes visible. Thereby it can be shown that a quasi-analysis of a similarity structure S can be conceived of as a sheaf (etale space) over S. (shrink)

This paper traces August Comte’s attempts to get clear about the concept of mathematical analysis at various stages in his intellectual development. Comte was especially concerned with distinguishing a method of analysis for the resolution of complex prolems from analysis in the sense of a method of drawing inferences. Geometrical analysis serves as his model for the former. In his attempt to get clear about this notion, he discovers an historical succession of different methods all (...) of which may be labeled “analytic.” In modern terms, Comte reveals how each of these methods of analysis characterizes a research program in mathematics, even showing us how more powerful methods came to supplant less powerful methods of analysis. (shrink)

This paper traces August Comte’s attempts to get clear about the concept of mathematical analysis at various stages in his intellectual development. Comte was especially concerned with distinguishing a method of analysis for the resolution of complex prolems from analysis in the sense of a method of drawing inferences. Geometrical analysis serves as his model for the former. In his attempt to get clear about this notion, he discovers an historical succession of different methods all (...) of which may be labeled “analytic.” In modern terms, Comte reveals how each of these methods of analysis characterizes a research program in mathematics, even showing us how more powerful methods came to supplant less powerful methods of analysis. (shrink)

Semantic analysis in early analytic philosophy belongs to a long tradition of adopting geometrical methodologies to the solution of philosophical problems. In particular, it adapts Descartes’ development of formalization as a mechanism of analytic representation, for its application in natural language semantics. This article aims to trace the mathematical roots of Frege, Russel and Carnap’s analytic method. Special attention is paid to the formal character of modern analysis introduced by Descartes. The goal is to identify the (...) particular conception of “form” developed by the analytic tradition, from Descartes to early analytic philosophy, and to determine its relation to similar notions, like ‘function’ and ‘syntax’. Finally, I focus on how Frege, Russell and Carnap’s methods of semantic analysis fit the general characterization of formal analysis previously developed. (shrink)

Alan Baker argues that mathematical objects play an indispensable explanatory role in science. There are several examples cited in the literature as solid candidates for such a role. We discuss two such examples and show that they are very different in their strength and (im)perfection, although both are recognized by the scientific community as examples of the best scientific explanations of particular phenomena. More specifically, it will be shown that the explanation of the cicada case has serious shortcomings compared (...) with the explanation of the case of Königsberg’s bridges. We will argue that the latter is a perfectly reliable scientific explanation that employs mathematical reasoning whereas the former is not. (shrink)

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)

This book provides a reading of Kant's theory of the construction of mathematical concepts through a fully contextualised analysis. In this work the author argues that it is only through an understanding of the relevant eighteenth century mathematics textbooks, and the related mathematical practice, that the material and context necessary for a successful interpretation of Kant's philosophy can be provided.

In this book, which is both a philosophical and historiographical study, the author investigates the fallibility and the rationality of mathematics by means of rational reconstructions of developments in mathematics. The initial chapters are devoted to a critical discussion of Lakatos' philosophy of mathematics. In the remaining chapters several episodes in the history of mathematics are discussed, such as the appearance of deduction in Greek mathematics and the transition from Eighteenth-Century to Nineteenth-Century analysis. The author aims at developing (...) a notion of mathematical rationality that agrees with the historical facts. A modified version of Lakatos' methodology is proposed. The resulting constructions show that mathematical knowledge is fallible, but that its fallibility is remarkably weak. (shrink)

The unified theory of dose and effect, as indicated by the median-effect equation for single and multiple entities and for the first and higher order kinetic/dynamic, has been established by T.C. Chou and it is based on the physical/chemical principle of the massaction law (J. Theor. Biol. 59: 253-276, 1976 (質量作用中效定理) and Pharmacological Rev. 58: 621-681, 2006) (普世中效指數定理). The theory was developed by the principle of mathematical induction and deduction (數學演繹歸納法). Rearrangements of the median-effect equation lead to Michaelis-Menten, Hill, (...) Scatchard, and Henderson-Hasselbalch equations. The “median” serves as the universal reference point and the “common link” for the relationship of all entities and is also the “harmonic mean” of kinetic dissociation constants. Over 300 mechanism-specific equations have been derived and published using the mathematical induction-deduction process. These equations can be deduced into several general equations, including the median-mediated whole/part equation, combination index theorem, isobologram equation, and polygonogram. It is proven that “dose” and “effect” are interchangeable, thus, “substance” and “function” are interchangeable, which leads to “the unity theory” (劑效、心物、知行一元論) in quantitative mathematical philosophy (數學的定量哲學) in functional context. Therefore, a general theory centered on the “median” and based on equilibrium dynamics has evolved. In other words: [「中」的宇宙觀： 以「中」爲基凖的動力學生態平衡]. Based on the median-effect equation of the mass-action law, the fundamental claim is that we can draw “a specific cure” for only two data points, if they are determined accurately. This claim has far reaching consequences since it defies the general held belief that two points can dray only a straight line. Remarkably, the unity theory (一元論) providesscientific/mathematical interpretation in equations and in graphics of Chinese ancient philosophy, including Fu-Si Ba Gua (伏羲八卦), Dao’s Harmony (和諧), the Confucian doctrine of the mean (儒家中庸之道), Chou Dun-Yi’s (周敦頤, 1017-1073) From Wu-ji to Tai-ji and Taiji Tu Sho (無極而太極及太極圖說). The moderntopological analysis for trinity yields an exact correspondence to the Ba-Gua, which was introduced over 4,000 years ago. Furthermore, the median-centered algorithm, promotes modern ecological content (生態學) in the equilibral dynamic state of harmony. It is concluded that Western science and Eastern philosophy are directly linked and complementary to each other. Since the truth in mathematical quantitative philosophy (數學的定量哲學) has no boundaries, East and West philosophies can flourish together for the common goal and ideal in science and in humanity (世界大同). (shrink)

Interest in the computational aspects of modeling has been steadily growing in philosophy of science. This paper aims to advance the discussion by articulating the way in which modeling and computational errors are related and by explaining the significance of error management strategies for the rational reconstruction of scientific practice. To this end, we first characterize the role and nature of modeling error in relation to a recipe for model construction known as Euler’s recipe. We then describe a general (...) model that allows us to assess the quality of numerical solutions in terms of measures of computational errors that are completely interpretable in terms of modeling error. Finally, we emphasize that this type of error analysis involves forms of perturbation analysis that go beyond the basic model-theoretical and statistical/probabilistic tools typically used to characterize the scientific method; this demands that we revise and complement our reconstructive toolbox in a way that can affect our normative image of science. (shrink)

Some concepts that are now part and parcel of mathematics used to be, at least until the beginning of the twentieth century, a central preoccupation of mathematicians and philosophers. The concept of continuity, or the continuous, is one of them. Nowadays, many philosophers of mathematics take it for granted that mathematicians of the last quarter of the nineteenth century found an adequate conceptual analysis of the continuous in terms of limits and that serious philosophical thinking is no longer required, (...) except perhaps when the question of the continuum is transferred to the arena of set theory where it takes the form of the infamous continuum hypothesis. As Philip Ehrlich has recently shown, this conviction goes back to the early writings of Russell who, in 1903 and then again in later writings, forcefully and eloquently pushed the view that mathematicians had given the final answer to immemorial conundrums arising from the continuous and infinitesimals . This proclamation of victory came with what was announced as the necessary defeat of the notion of the infinitesimal, despite the fact that mathematicians like Thomae, Du Bois-Reymond, Stolz, Bettazi, Veronese, Levi-Civita, and Hahn were investigating mathematical structures containing infinitesimals in a mathematically rigorous and logically consistent manner. In this respect Russell was merely walking in the footsteps of Cantor, and many of Russell's contemporaries were only too keen to keep infinitesimals out of Cantor's paradise. However, although Cantor certainly wanted to send the notion of infinitesimal to Hell, the notion kept a low profile in the mathematical purgatory, making its way in the study of non-Archimedean ordered algebraic systems. Of course, nowadays everyone has heard of Robinson's attempt at resurrecting infinitesimals in analysis in the form of non-standard analysis, but Robinson's work, despite the fact that it had, in …. (shrink)

This text is organized around the distinction between finite and infinite. It includes a brief overview of what different philosophers have said about infinity, and looks at some of the arguments to the effect that one should adopt a pro-infinity attitude. Other chapters contain an exposition of the ontological schools; interactions among these schools and various theories of truth; the relationship between mathematics and values; a history of mathematics; an analysis of mathematical knowledge; the role of mathematics in (...) eduction; the implications of religion for the philosophy of mathematics; and referenc eto mathematical objects. (shrink)

This article contributes to the revision of the procedure of robustness analysis of mathematical models in epistemic democracy using the systematic review method. It identifies the drawbacks of robustness analysis in epistemic democracy in terms of sample universality and inference from samples with the same results. To exemplify the effectiveness of systematic review, this article conducted a pilot review of diversity trumps ability theorem models, which are mathematical models of deliberation often cited by epistemic democrats. A (...) review of nine models extracted from 352 papers exemplifies the effectiveness of robustness analysis supplemented by systematic review in epistemic democracy. (shrink)

Contents: Acknowledgements. Conventions. Preface. Biographical sketch. 1 Introduction. 2 The Contributions. 3 Early work in analysis. 4 The Theory of Science . 5. Later mathematical studies. A On Kantian Intuitions. B The Bolzano-Cauchy Theorem.

We give an overview of recent results in ordinal analysis. Therefore, we discuss the different frameworks used in mathematical proof-theory, namely "subsystem of analysis" including "reverse mathematics", "Kripke-Platek set theory", "explicit mathematics", "theories of inductive definitions", "constructive set theory", and "Martin-Löf's type theory".

This dissertation looks to make sense of the role 'grammar' plays in Wittgenstein's philosophy of mathematics during the middle period of his career. It constructs a formal model of Wittgenstein's notion of grammar as expressed in his writings of the early thirties, addresses the appropriateness of that model and then employs it to test Wittgenstein's claim that mathematical propositions are ultimately grammatical. ;In order to test Wittgenstein's claim that mathematical propositions are grammatical, the dissertation provides a formalized (...) theory of grammatical analysis and applies it to a portion of language big enough to contain numerical expressions. It attempts to prove that, if the object language contains the appropriate numerical expressions, the resulting grammar will include at least some rules with a natural mathematical interpretation. In particular, it tries to show that Wittgenstein's grammatical analysis of the ordinary use of numerical expressions yields familiar theorems of arithmetic. ;The dissertation also endeavors to extend these results to a coherent picture of Wittgenstein's peculiar view of mathematics as grammar. It fits the formal results into the larger picture of Wittgenstein's philosophy of mathematics during this period. A large portion of the dissertation explains how, by combining the notions of grammar and mathematics, Wittgenstein allowed himself to offer original answers to some central questions in the philosophy of mathematics. In this regard, the dissertation pays special attention to four main issues: Wittgenstein's use of the term 'grammar' in his philosophical writings of this period, in comparison with traditional understandings of 'grammar', mathematics as part of the syntax of language, his explanation of mathematical application, and Wittgenstein's account of mathematical necessity. Taking these points in consideration situates the dissertation's results in the larger philosophical discussion of these themes. (shrink)