The article deals with historical dynamics of implicit and intuitive elements of mathematical knowledge. The author describes historical dynamics of implicit and intuitive elements and discloses a historical and evolutionary mechanism of building up mathematical knowledge. Each requirement to increase the level of theoretical rigor in mathematics is historically realized as a three-stage process. The first stage considers some general conditions of valid mathematical knowledge recognized by the mathematical community. The second (...) one reveals the level of theoretical rigor increasing, while the third one is characterized by explication of the hidden lemmas. A detailed discussion of historical substantiation of the basic algebra theorem is conducted according to the proposed technique. (shrink)

Evolutionary systems biology (ESB) is a rapidly growing integrative approach that has the core aim of generating mechanistic and evolutionary understanding of genotype‐phenotype relationships at multiple levels. ESB's more specific objectives include extending knowledge gained from model organisms to non‐model organisms, predicting the effects of mutations, and defining the core network structures and dynamics that have evolved to cause particular intracellular and intercellular responses. By combining mathematical, molecular, and cellular approaches to evolution, ESB adds new insights (...) and methods to the modern evolutionary synthesis, and offers ways in which to enhance its explanatory and predictive capacities. This combination of prediction and explanation marks ESB out as a research manifesto that goes further than its two contributing fields. Here, we summarize ESB via an analysis of characteristic research examples and exploratory questions, while also making a case for why these integrative efforts are worth pursuing. (shrink)

In the first part of this article we survey general similarities and differences between biological and social macroevolution. In the second (and main) part, we consider a concrete mathematical model capable of describing important features of both biological and social macroevolution. In mathematical models of historical macrodynamics, a hyperbolic pattern of world population growth arises from non-linear, second-order positive feedback between demographic growth and technological development. This is more or less identical with the working of the (...) collective learning mechanism. Based on diverse paleontological data and an analogy with macrosociological models, we suggest that the hyperbolic character of biodiversity growth can be similarly accounted for by non-linear, second-order positive feedback between diversity growth and the complexity of community structure, suggesting the presence within the biosphere of a certain analogue of the collective learning mechanism. We discuss how such positive feedback mechanisms can be modelled mathematically. (shrink)

An interaction between the ways morality and society develop in the face of the challenges they are exposed to seems consistent with empirical evidence. However, within this frame man's instinct of self preservation and his individualism seem to play an important part. The former by its influence on the evolution of what I refer to as 'our social morality'. The latter by finding its expression in conservative and radical ideologies whose political confrontation appears to give rise to an evolutionary (...) develop ment. The paper draws attention to the fact that in the face of contemporary challenges the evolutionary development of our morals may determine whether the corresponding evolutionary development of our societies will approach the patterns of those of insects, or whether our highly developed individual istic traits will result in new moral trends creating a society which is better suited to the present characteristics of humankind. It is also argued that the present growth of international organizations for cooperation and peaceful co-existence should make it possible to build a society worthy of the human race, but that it could be done only if we are able to develop a sufficient sense of responsibility to deliberately keep over-consumption and human population within a natural balance. (shrink)

The questions and methods of molecular biology and evolutionary biology are clearly distinct, yet a unified approach can lead to deep insights. Unfortunately, attempts to unify these approaches are fraught with pitfalls. In this informal series of questions and answers, we offer the mechanistically oriented biologist a set of steps to come up with evolutionarily reasonable and meaningful hypotheses. We emphasize the critical power and importance of carefully constructed null hypotheses, and we illustrate our ideas with examples representing a (...) range of topics, from the biology of aging, to protein structure, to speciation, and more. We also stress the importance of mathematics as the lingua franca for biologists of all stripes, and encourage mechanistic biologists to seek out quantitative collaborators to build explicit mathematical models, making their assumptions explicit, and their logic clear and testable. Biologists in all realms of inquiry stand to gain from strong bridges between our disciplines. (shrink)

Active participation closely associates with the sustainable operation of co-creation communities. Different from recent studies on the promotion of sustainable operation by identifying the internal and external motivations of user participation, this paper aims to analyze the mechanism regarding how different motivations affect the decision of user participation from group-level perspective. To better understand the mechanism, internal and external motivations are, respectively, captured by return-cost analysis and user interactive network. Afterwards, a network evolutionary game model was formulated (...) to analyze the dynamic strategy selection of all users. In addition, the stable equilibrium and evolutionary path of strategies are analyzed through computational experiments. Results indicate the following: Rewards have an influence on the promotion of active participation. However, with the continued growth of rewards, this promotion does not make sense sustainably. The promotional effect of information noise on the selection of active participation can be found when passive participation is the dominant strategy. However, the inhibitory effect can be seen in populations that mainly adopt active participation. The scale-free feature of user interactive network inhibits the selection of active participation when active participation is the dominant strategy in populations. Results found here is beneficial for managers to implement the specified policies and thus to achieve the sustainability of co-creation community. (shrink)

The high-tech industry is the main force promoting the development of China’s national economy. As its industrial economic strength grows, China’s high-tech industry is increasingly using cross-border mergers and acquisitions as an important way to “go out.” To explore the rules governing the process and operation mechanism of reverse knowledge transfer through the CBM&A of China’s high-tech industry under government intervention, a tripartite evolutionary game model of the government, the parent company, and the subsidiary as the main (...) subjects is constructed in this paper. The strategies adopted by the three subjects in the RKT game process are analysed, and the factors influencing RKT through CBM&A under government intervention are simulated and analysed using Python 3.7 software. The results show that, under government intervention, the parent company and subsidiary have different degrees of influence on each other. Subsidiaries are highly sensitive to the compensation rate of RKT. Positive intervention by the government tends to foster stable cooperation between the parent company and the subsidiary. However, over time, the government gradually relaxes its intervention in the RKT and innovation of multinational companies. (shrink)

This book draws its inspiration from Hilbert, Wittgenstein, Cavaillès and Lakatos and is designed to reconfigure contemporary philosophy of mathematics by making the growth of knowledge rather than its foundations central to the study of mathematical rationality, and by analyzing the notion of growth in historical as well as logical terms. Not a mere compendium of opinions, it is organised in dialogical forms, with each philosophical thesis answered by one or more historical case studies designed to (...) support, complicate or question it. The first part of the book examines the role of scientific theory and empirical fact in the growth of mathematical knowledge. The second examines the role of abstraction, analysis and axiomatization. The third raises the question of whether the growth of mathematical knowledge constitutes progress, and how progress may be understood. Readership: Students and scholars concerned with the history and philosophy of mathematics and the formal sciences. (shrink)

In the context of life sciences, we are constantly confronted with information that possesses precise semantic values and appears essentially immersed in a specific evolutionary trend. In such a framework, Nature appears, in Monod’s words, as a tinkerer characterized by the presence of precise principles of self-organization. However, while Monod was obliged to incorporate his brilliant intuitions into the framework of first-order cybernetics and a theory of information with an exclusively syntactic character such as that defined by Shannon, research (...) advances in recent decades have led not only to the definition of a second-order cybernetics but also to an exploration of the boundaries of semantic information. As H. Atlan states, on a biological level "the function self-organizes together with its meaning". Hence the need to refer to a conceptual theory of complexity and to a theory of self-organization characterized in an intentional sense. There is also a need to introduce, at the genetic level, a distinction between coder and ruler as well as the opportunity to define a real software space for natural evolution. The recourse to non-standard model theory, the opening to a new general semantics, and the innovative definition of the relationship between coder and ruler can be considered, today, among the most powerful theoretical tools at our disposal in order to correctly define the contours of that new conceptual revolution increasingly referred to as metabiology. This book focuses on identifying and investigating the role played by these particular theoretical tools in the development of this new scientific paradigm. Nature "speaks" by means of mathematical forms: we can observe these forms, but they are, at the same time, inside us as they populate our organs of cognition. In this context, the volume highlights how metabiology appears primarily to refer to the growth itself of our instruments of participatory knowledge of the world. (shrink)

Purpose:On the background of innovation-driven growth strategy of the Chinese government, this study aims to explore the impact of the knowledge base on innovation-driven growth of a firm, which is moderated by organizational character.Design/methodology/approach:Based on the data of 965 Chinese listed companies, some hypotheses were tested using the method of hierarchical regression analysis.Findings:Organizational growth relies on both technological and business model innovations and their interactive effect. Knowledge base, both breadth and depth, makes a positive impact (...) on the innovation-driven growth of an enterprise. In the impacting mechanism, an explicit organizational character not only has direct positive effects on business model innovation, it also strengthens the effect of knowledge breadth on business model innovation. On the contrary, an implicit organizational character is not significantly related to innovation.Research limitations/implications:In order to achieve growth, enterprises are suggested to adopt such dual innovation strategy, led by technological innovation and supplemented with business model innovation, which is supported by the integrated management of intangible resources, deep and broad knowledge, and explicit organizational character.Originality/value:A new theoretical framework of organizational innovation-driven growth was proposed. The realization paths of innovation-driven growth were explored. The idea of collaborative governance between the knowledge base and organizational character was raised. (shrink)

This paper attempts to show that mathematical knowledge does not grow by a simple process of accumulation and that it is possible to provide a quasi-empirical (in Lakatos's sense) account of mathematical theories. Arguments supporting the first thesis are based on the study of the changes occurred within Eudidean geometry from the time of Euclid to that of Hilbert; whereas those in favour of the second arise from reflections on the criteria for refutation of mathematical theories.

In his book The Value of Science Poincaré criticizes a certain view on the growth of mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the (...) centuries past” (Poincaré 1958, p. 14). The view criticized by Poincaré corresponds to Frege’s idea that the development of mathematics can be described as an activity of system building, where each system is supposed to provide a complete representation for a certain mathematical field and must be pitilessly torn down whenever it fails to achieve such an aim. All facts concerning any mathematical field must be fully organized in a given system because “in mathematics we must always strive after a system that is complete in itself” (Frege 1979, p. 279). Frege is aware that systems introduce rigidity and are in conflict with the actual development of mathematics because “in history we have development; a system is static”, but he sticks to the view that “science only comes to fruition in a system” because “only through a system can we achieve complete clarity and order” (Frege 1979, p. 242). He even goes so far as saying that “no science can be so enveloped in obscurity as mathematics, if it fails to construct a system” (Frege 1979, p. 242). By ‘system’ Frege means ‘axiomatic system’. In his view, in mathematics we cannot rest content with the fact that “we are convinced of something, but we must strive to obtain a clear insight into the network of inferences that support our conviction”, that is, to find “what the primitive truths are”, because “only in this way can a system be constructed” (Frege 1979, p. 205). The primitive truths are the principles of the axiomatic system. Frege’s stress on the role of systems also determines his views on the growth of mathematical knowledge.. (shrink)

I begin by arguing that a consistent general naturalism must be understood in terms of methodological maxims rather than metaphysical doctrines. Some specific maxims are proposed. I then defend a generalized naturalism from the common objection that it is incapable of accounting for the normative aspects of human life, including those of scientific practice itself. Evolutionary naturalism, however, is criticized as being incapable of providing a sufficient explanation of categorical moral norms. Turning to the epistemological norms of science itself, (...) particularly those governing the empirical testing of specific models, I argue that these should be regarded as conditional rather than categorical and that, as such, can be given a naturalistic justification. The justification, however, is more cognitive than evolutionary. The historical development of science is found to be a better place for applying evolutionary ideas. After briefly considering the possibility of a naturalistic understanding of mathematics and logic, I turn to the problem of reconciling scientific realism with an evolutionary picture of scientific development. The solution, I suggest, is to understand scientific knowledge as being “perspectival” rather than absolutely objective. I first argue that scientific observation, whether by humans or instruments, is perspectival. This argument is extended to scientific theorizing which is regarded not as the formulation of universal laws of nature but as the construction of principles to be used in the construction of models to be applied to specific natural systems. The application of models, however, is argued to be not merely opportunistic but constrained by the methodological presumption that we live in a world with a definite causal structure even though we can understand it only from various perspectives. (shrink)

Knowledge generation can be naturalized by adopting computational model of cognition and evolutionary approach. In this framework knowledge is seen as a result of the structuring of input data (data → information → knowledge) by an interactive computational process going on in the agent during the adaptive interplay with the environment, which clearly presents developmental advantage by increasing agent’s ability to cope with the situation dynamics. This paper addresses the mechanism of knowledge generation, a (...) process that may be modeled as natural computation in order to be better understood and improved. (shrink)

This paper aims to reassess the philosophical puzzle of the “applicability of mathematics to physical sciences” as a misunderstood disciplinary interplay. If the border isolating mathematics from the empirical world is based on appropriate criteria, how does one explain the fruitfulness of its systematic crossings in recent centuries? An analysis of the evolution of the criteria used to separate mathematics from experimental sciences will shed some light on this question. In this respect, we will highlight the historical influence of three (...) major disciplinary paradigms. According to the Aristotelian classification of the sciences, the separation of mathematics from physics is based on their respective objects of study. The Baconian system distinguishes these sciences by the type of knowledge involved in each field. Finally, the Whewellian disciplinary layout categorises these disciplines by their respective methods. In this paper, we argue that the cascading effect of such disciplinary categorisations—based successively on ontological, epistemological, and heuristic criteria—resulted in a profound redefinition of the border between mathematics and physics, with the consequence of obscuring the foundations of the interplay between these fields. Our approach intends to put forward such a mechanism as a constitutive piece of the puzzle of the applicability of mathematics. (shrink)

In the last decades two different and apparently unrelated lines of research have increasingly connected mathematics and evolutionism. Indeed, on the one hand different attempts to formalize darwinism have been made, while, on the other hand, different attempts to naturalize logic and mathematics have been put forward. Those researches may appear either to be completely distinct or at least in some way convergent. They may in fact both be seen as supporting a naturalistic stance. Evolutionism is indeed crucial for a (...) naturalistic perspective, and formalizing it seems to be a way to strengthen its scientificity. The paper shows that, on the contrary, those directions of research may be seen as conflicting, since the conception of knowledge on which they rest may be undermined by the consequences of accepting an evolutionary perspective. (shrink)

This paper is a critique of Darwinian models of the growth of scientific knowledge. Donald Campbell, Karl Popper, Stephen Toulmin, and others have discussed analogies between the development of biological species and the development of scientific knowledge: in both kinds of development, we find variation, selection, and transmission. It is argued that these similarities are superficial, and that closer examination of biological evolution and of the history of science shows that a non-Darwinian approach to historical epistemology is (...) needed. An adequate model of the growth of knowledge will have to go beyond evolutionary epistemology in discussing the role of intentional, abductive theory construction in scientific discovery, the selection of theories according to general criteria, the achievement of progress by sustained application of criteria, and the transmission of selected theories in highly organized scientific communities. (shrink)

Kuhn's Structure of Scientific Revolutions has been enduringly influential in philosophy of science, challenging many common presuppositions about the nature of science and the growth of scientific knowledge. However, philosophers have misunderstood Kuhn's view, treating him as a relativist or social constructionist. In this book, Brad Wray argues that Kuhn provides a useful framework for developing an epistemology of science that takes account of the constructive role that social factors play in scientific inquiry. He examines the core concepts (...) of Structure and explains the main characteristics of both Kuhn's evolutionary epistemology and his social epistemology, relating Structure to Kuhn's developed view presented in his later writings. The discussion includes analyses of the Copernican revolution in astronomy and the plate tectonics revolution in geology. The book will be useful for scholars working in science studies, sociologists and historians of science as well as philosophers of science. (shrink)

This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors (...) tried to naturalize mathematics by relying on evolutionism. But several difficulties arise when we try to do this. This chapter suggests that, in order to naturalize mathematics, it is better to take the method of mathematics to be the analytic method, rather than the axiomatic method, and thus conceive of mathematical knowledge as plausible knowledge. (shrink)

In 1908, Henri Poincar? claimed that: ...the mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a mathematical law, just as experimental facts lead us to the knowledge of a physical law. They are those which reveal to us unsuspected kinship between other facts, long known, but wrongly believed to be strangers to one another. Towards the end of the twentieth century, (...) with many more mathematical facts since discovered, several mathematicians proposed overarching schemes to organise the facts they considered most significant. In this paper, I shall briefly discuss three of these schemes (those of Arnold, Atiyah, and Baez and Dolan), before drawing some philosophical consequences from their attempts. Rather than the kind of claim made by Frege that with the entry of imaginary numbers we reach the ?natural end of the domain of numbers?, we are dealing here with a more open-ended sense of conceptual growth. I shall illustrate this theme by discussing the elaboration of algebraic structures designed to measure symmetry. What emerges is that at any one time mathematicians are operating with a notion very similar to that discussed in the philosophy of science under the heading of ?natural kinds?. We find ?quasi-causal? talk of properties being ?responsible? for a phenomenon, projectability, the transfer of robust mechanisms between domains, and reference to entities not yet fully determined. Debates in philosophy of science prompt further questioning as to the ?naturalness? of these mathematical kinds, whether one should expect them to be dependent on varying human interests, whether there is a distinction between artefactual and real kinds, and whether there is convergence of kinds. I believe that these questions present a wonderful opportunity for a philosophy of mathematics to treat ?real? mathematics, while making powerful points of contact with philosophy of science, and philosophy in general. (shrink)

In his philosophical history of nineteenth-century mathematics, Proofs and Persuasions: The Logic of Mathematical Discovery, Imre Lakatos asserts that mathematical criticism was the driving force in the growth of mathematical knowledge during the nineteenth century, and provided the impetus for some of the deepest conceptual reformulations of the century. The philosophy of mathematics represented by Proofs and Refutations also presents a rich analysis of how mathematics can be thought of as an essentially historical discipline. Despite (...) protestations by Lakatos that he completely discarded Hegel when he discovered the work of Karl Popper, his philosophy of error closely resembles Hegel's in the Phenomenology of Spirit. According to Lakatos, the impact of proofs and refutations on naive concepts is to erase them completely and replace them by proof-generated concepts. His historical point about proofs and refutations is that this pattern in the growth of mathematical knowledge is a relatively recent innovation. Hegel's and Lakatos' shared vision of theoretical knowledge is that rather than being the inspired work of timeless, totally subjective, intellectual intuition , it is, more often than not, mediated by a lengthy history of speculation and failure. (shrink)

In this paper, I introduce and examine the notion of “mathematical engineering” and its impact on mathematical change. Mathematical engineering is an important part of contemporary mathematics and it roughly consists of the “construction” and development of various machines, probes and instruments used in numerous mathematical fields. As an example of such constructions, I briefly present the basic steps and properties of homology theory. I then try to show that this aspect of contemporary mathematics has important (...) consequences on our conception of mathematical knowledge, in particular mathematical growth. (shrink)

Kitcher's philosophical approach has moved from the reflection on the nature of mathematical knowledge to an explicit social concern about science, because he considers seriously the relevance of democratic values to scientific activity. Focal issues in this trajectory - from the internal perspective to the external - have been naturalism and scientific progress, which includes studies of the uses of scientific findings in the social milieu. Within this intellectual context, the chapter pays particular attention to his epistemological and (...) methodological evolution. The analysis of Philip Kitcher's contents on progress begins with mathematics, a conception that follows a naturalist perspective. Thereafter, the growth of science comes to the front line, an advancement that he views according to realism and cognitive naturalism. Later, the social concern about science receives a visible consideration, when his vision of scientific undertaking is characterized following modest realism and social naturalism. After these four steps (philosophical context, progress in mathematics, the growth in science, and the social concern about science), there is an analysis of his philosophical-methodological framework in retrospective. This is continued by the presentation of the origins of this book and the bibliography related to this thinker. (shrink)

Chapters 7–9 offer a general account of the growth of mathematics. Introduce the notion of a mathematical practice, a multidimensional entity consisting of a language, accepted statements, accepted questions, accepted means of inference, and methodological maxims. Mathematics grows by modifying one or more components in response to the problems posed by others. So new language, language that is not initially well understood, may be introduced in order to answer questions taken to be important but resisting solution by available (...) methods; in consequence, there may be a new task of clarifying the language or taming the methods that the extended language allows. The chapters attempt to show how such processes have occurred in the history of mathematics, and how they link the rich state of present mathematics to the crude beginnings of the subject. In Chapter 7, in particular, Kitcher compares mathematical change with scientific change, attempting to show that the growth of mathematical knowledge is far more similar to the growth of scientific knowledge than is usually appreciated. (shrink)

What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions.

Table of Contents: Preface vii Introduction: Philosophy and Operationalism 1 1. "What was Worth Knowing" in 1500 10 2. Humanism and Ancient Wisdom: How to Learn Things in the Sixteenth Century 30 3. The Scholar and the Craftsman: Paracelsus, Gilbert, Bacon 49 4. Mathematics Challenges Philosphy: Galileo, Kepler, and the Surveyors 65 5. Mechanism: Descartes Builds a Universe 80 6. Extra-Curricular Activities: New Homes for Natural Knowledge 101 7. Experiment: How to Learn Things about Nature in the Seventeenth (...) Century 131 8. Cartesians and Newtonians 149 Conclusion: What was Worth Knowing by the Eighteenth Century 168 Notes and References 171 Documentation and Further Reading 181 Dramatis Personae 193 Glossary of Major Terms 197 Index 201. (shrink)

As familiar as the form of the mathematical equation is to us, the ostensibly simple act of equating unlike things was an achievement many centuries in the making, and one that would ultimately redefine European mathematical enquiry such that its bias toward geometry and the concrete would be displaced by a bias toward algebraic abstraction. The moment of that displacement was the nineteenth century, and its broader significance is on particularly striking display in the British context, where the (...) implications of algebraic abstraction were the object of sustained enquiry among mathematicians, logicians, and economists. This article argues that the ascendance of the algebraic equation, and the transformation in the conception of number on which it was premised, were not simply the product of evolutionary pressures internal to mathematics; the Victorian embrace of algebra was also a response to the practical and cognitive demands of Victorian economic life, which was increasingly reliant on attenuated exchange relations and merely nominal forms of ownership. This was an economy organized around the global extension of trade and characterized by the exponential growth of financial intermediation, of what Walter Bagehot called “number abstracted from reference.” Victorian economic practices thus modeled an abstraction that helped to justify the abstractions of mathematics, and that mathematics in turn was used by economic theorists to argue for the necessity and objectivity of their models. This mutually sustaining dialogue is particularly visible in the writings of William Stanley Jevons, who applied the principles of algebra to philosophy and economic theory so as to reconceive the logic of cognitive and social life in terms of equations. This logic, for which he was merely a spokesman, continues to shape our faith in the special value of abstract, theoretical knowledge. (shrink)

Explanation in terms of gene regulatory networks has become standard practice in evolutionary developmental biology. In this paper, we argue that GRNs fail to provide a robust, mechanistic, and dynamic understanding of the developmental processes underlying the genotype–phenotype map. Explanations based on GRNs are limited by three main problems: the problem of genetic determinism, the problem of correspondence between network structure and function, and the problem of diachronicity, as in the unfolding of causal interactions over time. Overcoming these problems (...) requires dynamic mechanistic explanations, which rely not only on mechanistic decomposition, but also on dynamic modeling to reconstitute the causal chain of events underlying the process of development. We illustrate the power and potential of this type of explanation with a number of biological case studies that integrate empirical investigations with mathematical modeling and analysis. We conclude with general considerations on the relation between mechanism and process in evo-devo. (shrink)

This paper is concerned with the genesis of mathematical knowledge. While some philosophers might argue that mathematics has no real subject matter and thus is not a body of knowledge, I will not try to dissuade them directly. I shall not attempt such a refutation because it seems clear to me that mathematicians do know such things as the Mean Value Theorem, The Fundamental Theorem of Arithmetic, Godel's Theorems, etc. Moreover, this is much more evident to me (...) than any philosophical view of mathematics I know of — including my own. So I am going to take mathematics as my starting point. (shrink)

Universal Basic Income (UBI) has been proposed as a potential way in which welfare states could be made more responsive to the ever-shifting evolutionary challenges of institutional adaptation in a dynamic environment. It has been proposed as a tool of “real freedom” (Van Parijs) and as a tool of making the welfare state more efficient. (Friedman) From the point of view of complexity theory and evolutionary economics, I argue that only a welfare state model that is “polycentrically” (Polanyi, (...) Hayek) organized as an evolving network of distributed decision-making can respond to the challenges of complexity in an adaptively efficient manner. UBI has the theoretical possibility to facilitate flexible, bottom-up innovation, since a) it embodies the “rule of law” principles of generality, nondiscrimination, simplicity, and transparency, and b) it grants people widespread freedom to experiment, innovate, and deviate from established practices and norms; but whether it can be made to work as intended depends on a number of variables in its design. I situate my research in the contemporary UBI debate around automation and technological development, which I interpret creatively through the lens of contemporary evolutionary political economy. I advance a syncretic model of institutional design based upon the shared insights of the Santa Fe, Neo-Hayekian, and Neo-Schumpeterian schools of evolutionary economics that, following Adam Thierer and Michael Munger, I call the framework of Permissionless Innovation (PI). Under the PI framework, which is a development of the tradition of evolutionary liberalism (Hume, Smith, Mandeville, Mill, Hayek, Polanyi, Schumpeter, Hodgson), individuals are granted the default right to innovate without having to ask for permission, and the right to basic income, which they can use to support their right to innovate, mutate, and experiment. This framework is a liberal framework to the extent it relies on widespread human freedom as a means of fostering innovation and social learning. And it is an evolutionary framework to the extent that it seeks to facilitate evolutionary socioeconomic processes for the sake of social progress and welfare advancement. I proceed to show, through an analysis of how innovation operates in both the economy and the cultural domain, that UBI can be used as an important cornerstone of an evolutionary liberal model of “ecostructural” governance that treats the socioeconomic order as an ecological garden of spontaneous growths, adaptations, and innovations. I call this the Permissionless Innovation Universal Basic Income (PIUBI) framework. In it, the right to innovate, deviate, and mutate is upheld as the “gold standard” of institutional design whereby individuals are granted widespread freedoms combined with the provision of UBI and other innovation-fostering (carefully bound) services and regulations. In the coming decades, UBI may even enable poor people to take better advantage of Human Enhancement Technologies (HETs) and other experimental innovations. Such a framework is compatible with a fast pace of evolutionary development, but the regulatory model accompanying such a framework must be capable of responding to evolutionary lock-ins, maladaptive innovations, self-destructive behaviour on the part of UBI recipients, and the management of catastrophic and existential risks. In the end, I will have shown that the PIUBI framework can be justified as a plausible default institutional mechanism for responding to radical uncertainty, but it needs to be experimentally adapted to changing circumstances with the help of various situation-specific, polycentric, multi-level, approaches. (shrink)

We argue that mathematical knowledge is context dependent. Our main argument is that on pain of distorting mathematical practice, one must analyse the notion of having available a proof, which supplies justification in mathematics, in a context dependent way.

The notion of indivisibles and atoms arose in ancient Greece. The continuum—that is, the collection of points in a straight line segment, appeared to have paradoxical properties, arising from the ‘indivisibles’ that remain after a process of division has been carried out throughout the continuum. In the seventeenth century, Italian mathematicians were using new methods involving the notion of indivisibles, and the paradoxes of the continuum appeared in a new context. This cast doubt on the validity of the methods and (...) the reliability of mathematical knowledge which had been regarded as established by the axiomatic method in geometry expounded by Aristotle’s younger contemporary Euclid. The teaching of indivisibles was banned within the Society of Jesus, the Jesuits. In England, indivisibles were used by the mathematician John Wallis, and there was an acrimonious and extended feud between Wallis and the philosopher Thomas Hobbes over legitimate methods of argument in mathematics. Notions of the infinitesimal were used by Isaac Newton and Gottfried Leibniz, and were attacked by Bishop Berkeley for the vagueness of the concept and the illegitimate reasoning applied to it. This article discusses aspects of these events with reference to the book Infinitesimal by Amir Alexander and to other sources. Also discussed are wider issues arising from Alexander’s book including: the changes in cultural sensibility associated with the growth of new mathematical and scientific knowledge in the seventeenth century, the changes in language concomitant with these changes, what constitutes valid methods of enquiry in various contexts, and the question of authoritarianism in knowledge. More general aims of this article are to widen the immediate mathematical and historical contexts in Alexander’s book, to bridge a gap in conversations between mathematics and the humanities, and to relate mathematical ideas to wider human and contemporary issues. (shrink)

A new diagnostic modeling system for automatically synthesizing a deep‐structure model of a student's misconceptions or bugs in his basic mathematical skills provides a mechanism for explaining why a student is making a mistake as opposed to simply identifying the mistake. This report is divided into four sections: The first provides examples of the problems that must be handled by a diagnostic model. It then introduces procedural networks as a general framework for representing the knowledge underlying a (...) skill. The challenge in designing this representation is to find one that facilitates the discovery of misconceptions or bugs existing in a particular student's encoding of this knowledge. The second section discusses some of the pedagogical issues that have emerged from the use of diagnostic models within an instructional system. This discussion is framed in the context of a computer‐based tutoring/gaming system developed to teach students and student teachers how to diagnose bugs strategically as well as how to provide a better understanding of the underlying structure of arithmetic skills. The third section describes our uses of an executable network as a tool for automatically diagnosing student behavior, for automatically generating “diagnostic” tests, and for judging the diagnostic quality of a given exam. Included in this section is a discussion of the success of this system in diagnosing 1300 school students from a data base of 20.000 test items. The last section discusses future research directions. (shrink)

How should one conceive of the method of mathematics, if one takes a naturalist stance? Mathematical knowledge is regarded as the paradigm of certain knowledge, since mathematics is based on the axiomatic method. Natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some naturalists try to naturalize mathematics relying on Darwinism. But several difficulties arise when one (...) tries to naturalize in this way the traditional view of mathematics, according to which mathematical knowledge is certain and the method of mathematics is the axiomatic method. This paper suggests that, in order to naturalize mathematics through Darwinism, it is better to take the method of mathematics not to be the axiomatic method. (shrink)

I present a hypothetical account of how the ancients might have come to introduce mathematical objects in order to describe patterns, and I explain how working with patterns can generate information about the mathematical realm. The ancients might have started using what I call templates, i.e. concrete devices, like blueprints or drawings, to represent how things are shaped or structured, and this could have evolved into representing the abstract patterns that concrete things might fit. In this way, they (...) might have come to believe that written constructions and computations could provide information about the mathematical realm, for by their very nature, patterns should be structurally analogous to their templates and in positing that they are, one simply projects onto structures and features already attributed to templates. By reflecting on systems of dots representing cardinalities, the ancients could generate a body of results that then evolved into a systematic theory of numbers, but the approach fails when there is no direct connection between the computations and the patterns they are supposed to concern, e.g. those concerning trigonometric functions or transfinite ordinal numbers. In these cases, we forge a connection between proofs and patterns by positing that the premises of the proofs state uncontroversial features of the patterns, i.e. the premises constitute implicit definitions of the patterns. I explain how this position does not entail that mathematics is analytic or a priori. (shrink)

Philippe Huneman has recently questioned the widespread application of mechanistic models of scientific explanation based on the existence of structural explanations, i.e. explanations that account for the phenomenon to be explained in virtue of the mathematical properties of the system where the phenomenon obtains, rather than in terms of the mechanisms that causally produce the phenomenon. Structural explanations are very diverse, including cases like explanations in terms of bowtie structures, in terms of the topological properties of the system, or (...) in terms of equilibrium. The role of mathematics in bowtie structured systems and in topologically constrained systems has recently been examined in different papers. However, the specific role that mathematical properties play in equilibrium explanations requires further examination, as different authors defend different interpretations, some of them closer to the new-mechanistic approach than to the structural model advocated by Huneman. In this paper, we cover this gap by investigating the explanatory role that mathematics play in Blaser and Kirschner’s nested equilibrium model of the stability of persistent long-term human-microbe associations. We argue that their model is explanatory because: i) it provides a mathematical structure in the form of a set of differential equations that together satisfy an ESS; ii) that the nested nature of the ESSs makes the explanation of host-microbe persistent associations robust to any perturbation; iii) that this is so because the properties of the ESS directly mirror the properties of the biological system in a non-causal way. The combination of these three theses make equilibrium explanations look more similar to structural explanations than to causal-mechanistic explanation. (shrink)

The goal of this paper is to raise a novel objection to Lewis’s modal realist epistemology. After reformulating his modal epistemology, I shall argue that his view that we have necessary knowledge of the existence of counterparts ends up with an absurdity. Specifically, his analogy between mathematical knowledge and modal knowledge leads to an unpleasant conclusion that one’s counterpart exists in all possible worlds. My argument shows that if Lewis’s modal realism is true, we cannot know (...) what is possible. Conversely, if we can know what is possible, his modal realism is false. In the remainder of the paper, I shall consider and block possible objections to my argument. (shrink)

Logic and Knowledge -/- Editor: Carlo Cellucci, Emily Grosholz and Emiliano Ippoliti Date Of Publication: Aug 2011 Isbn13: 978-1-4438-3008-9 Isbn: 1-4438-3008-9 -/- The problematic relation between logic and knowledge has given rise to some of the most important works in the history of philosophy, from Books VI–VII of Plato’s Republic and Aristotle’s Prior and Posterior Analytics, to Kant’s Critique of Pure Reason and Mill’s A System of Logic, Ratiocinative and Inductive. It provides the title of an important collection (...) of papers by Bertrand Russell (Logic and Knowledge. Essays, 1901–1950). However, it has remained an underdeveloped theme in the last century, because logic has been treated as separate from knowledge. -/- This book does not hope to make up for a century-long absence of discussion. Rather, its ambition is to call attention to the theme and stimulating renewed reflection upon it. The book collects essays of leading figures in the field and it addresses the theme as a topic of current debate, or as a historical case study, or when appropriate as both. Each essay is followed by the comments of a younger discussant, in an attempt to transform what might otherwise appear as a monologue into an ongoing dialogue; each section begins with an historical essay and ends with an essay by one of the editors. -/- Carlo Cellucci is Emeritus Professor of Philosophy at the University of Rome ‘La Sapienza,’ Italy. He is currently completing a book entitled, Remaking Logic: What is Logic Really? -/- Emily Grosholz is Professor of Philosophy at the Pennsylvania State University, USA. She is the author of Representation and Productive Ambiguity in Mathematics and the Sciences (Oxford University Press, 2007). -/- Emiliano Ippoliti is a Research Fellow at the University of Rome ‘La Sapienza,’ Italy. His main interests are heuristics, the logic of discovery, and problem-solving. He is currently working on a book, Ampliating Knowledge: Data, Hypotheses and Novelty. -/- TABLE OF CONTENTS -/- Foreword .................................................................................................... ix Acknowledgements ................................................................................. xxv -/- Section I: Logic and Knowledge -/- Chapter One................................................................................................. 3 The Cognitive Importance of Sight and Hearing in Seventeenthand Eighteenth-Century Logic (Mirella Capozzi) Discussion .............................................................................................. 26 (Chiara Fabbrizi) Chapter Two .............................................................................................. 33 Nominalistic Content (Jody Azzouni) Discussion ............................................................................................... 52 (Silvia De Bianchi) Chapter Three ............................................................................................ 57 A Garden of Grounding Trees (Göran Sundholm) Discussion.......................................................................................... .. 75 (Luca Incurvati) Chapter Four .............................................................................................. 81 Logics and Metalogics (Timothy Williamson) Discussion.......................................................................................... 101 (Cesare Cozzo) Chapter Five ............................................................................................ 109 Is Knowledge the Most General Factive Stative Attitude? (Cesare Cozzo) Discussion.......................................................................................... 117 (Timothy Williamson) Chapter Six .............................................................................................. 123 Classifying and Justifying Inference Rules (Carlo Cellucci) Discussion.......................................................................................... 143 (Norma B. Goethe) -/- Section II: Logic and Science -/- Chapter Seven.......................................................................................... 151 The Universal Generalization Problem and the Epistemic Status of Ancient Medicine: Aristotle and Galen (Riccardo Chiaradonna) Discussion.......................................................................................... 168 (Diana Quarantotto) Chapter Eight........................................................................................... 175 The Empiricist View of Logic (Donald Gillies) Discussion.......................................................................................... 191 (Paolo Pecere) Chapter Nine............................................................................................ 197 Artificial Intelligence and Evolutionary Theory: Herbert Simon’s Unifying Framework (Roberto Cordeschi) Discussion.......................................................................................... 216 (Francesca Ervas) Chapter Ten ............................................................................................. 221 Evolutionary Psychology and Morality: The Renaissance of Emotivism? (Mario De Caro) Discussion.......................................................................................... 232 (Annalisa Paese) Chapter Eleven ........................................................................................ 237 Between Data and Hypotheses (Emiliano Ippoliti) Discussion.......................................................................................... 262 (Fabio Sterpetti) -/- Section III: Logic and Mathematics -/- Chapter Twelve ....................................................................................... 273 Dedekind Against Intuition: Rigor, Scope and the Motives of his Logicism (Michael Detlefsen) Discussion.......................................................................................... 290 (Marianna Antonutti) Chapter Thirteen...................................................................................... 297 Mathematical Intuition: Poincaré, Polya, Dewey (Reuben Hersh) Discussion.......................................................................................... 324 (Claudio Bernardi) Chapter Fourteen ..................................................................................... 329 On the Finite: Kant and the Paradoxes of Knowledge (Carl Posy) Discussion.......................................................................................... 358 (Silvia Di Paolo) Chapter Fifteen ........................................................................................ 363 Assimilation: Not Only Indiscernibles are Identified (Robert Thomas) Discussion.......................................................................................... 380 (Diego De Simone) Chapter Sixteen ....................................................................................... 385 Proofs and Perfect Syllogisms (Dag Prawitz) Discussion.......................................................................................... 403 (Julien Murzi) Chapter Seventeen ................................................................................... 411 Logic, Mathematics, Heterogeneity (Emily Grosholz) Discussion.......................................................................................... 427 (Valeria Giardino) -/- Contributors........................................................................................ ..... 433 Index............................................................................................... ......... 437 -/- Price Uk Gbp: 49.99 Price Us Usd: 74.99. 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The original proof of the four-color theorem by Appel and Haken sparked a controversy when Tymoczko used it to argue that the justification provided by unsurveyable proofs carried out by computers cannot be a priori. It also created a lingering impression to the effect that such proofs depend heavily for their soundness on large amounts of computation-intensive custom-built software. Contra Tymoczko, we argue that the justification provided by certain computerized mathematical proofs is not fundamentally different from that provided by (...) surveyable proofs, and can be sensibly regarded as a priori. We also show that the aforementioned impression is mistaken because it fails to distinguish between proof search (the context of discovery) and proof checking (the context of justification). By using mechanized proof assistants capable of producing certificates that can be independently checked, it is possible to carry out complex proofs without the need to trust arbitrary custom-written code. We only need to trust one fixed, small, and simple piece of software: the proof checker. This is not only possible in principle, but is in fact becoming a viable methodology for performing complicated mathematical reasoning. This is evinced by a new proof of the four-color theorem that appeared in 2005, and which was developed and checked in its entirety by a mechanical proof system. (shrink)