Peirce and Dewey were generally more concerned with the process of scientific activity than purely mathematical work. However, their accounts of knowledge production afford some insights into the epistemology of mathematical postulates, especially definition and axioms. Their rejection of rationalist metaphysics and their emphasis on continuity in inquiry provides the pretext for the pragmatic a priori – hypothetical and operational assumptions whose justification relies on their fruitfulness in the long run. This paper focuses on the application of this (...) idea to the epistemology of definitions and an account of progress in mathematics, although it has broader implications for the study of conceptual change and the function and basis of presuppositions in the sciences. (shrink)

The triumph of scientific materialism in the Seventeenth Century not only bifurcated nature into matter and mind and primary and secondary qualities, as Alfred North Whitehead pointed out in Science and the Modern World. It divided science and the humanities. The core of science is the effort to comprehend the cosmos through mathematics. The core of the humanities is the effort to comprehend history and human nature through narratives. The life sciences can be seen as the zone in which (...) the conflict between these two very different ways of comprehending the world collide. Evolutionary theory as defended by Schelling developed out of natural history, but efforts have been made to formulate neo-Darwinism through mathematical models. However, it is impossible to eliminate stories from biology. As Stuart Kauffman argued, mathematical models attempt to pre-state all possibilities, but in evolution there can be adjacent possibles that can be embraced by organisms but cannot be pre-stated. To account for such actions it is necessary to tell stories. Mathematics provides analytic precision allowing long chains of deductions, but tends to deny temporal becoming and cannot do justice to the openness of the future, while narratives focus on processes and events, but lack exactitude that would provide precise deductions and predictions. In advancing mathematics adequate to life, Robert Rosen argued that living beings as anticipatory systems must have models of themselves, and strove to develop a form of mathematics able to model life itself. It has been convincingly argued that narratives are central to human self-creation and they are lived out before being explicitly told. Their models of themselves are first and foremost, stories or narratives. If this is the case, might not living beings as biological entities be characterized by proto-stories or narratives in their models of themselves? Biosemiotics, largely inspired by C.S. Peirce, provides a bridge between mathematical and narrative comprehension, conceiving them as different forms of semiosis. The study of life through biosemiotics could reveal how mathematics and narratives can be understood in relation to each other. This could have implications for how we understand science and the humanities and their relationship to each other. In this paper I will examine work in theoretical biology that might advance these efforts. (shrink)

Speaking from our experience as department chairs in fields in which women are traditionally underrepresented, we offer reflections and advice on how one might move beyond the chilly climate and create a warmer environment for women students and faculty members.

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, but (...) have been expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

This paper sets mathematics among the sciences, despite not being empirical, because it studies relations of various sorts, like the sciences. Each empirical science studies the relations among objects, which relations determining which science. The mathematical science studies relations as such, regardless of what those relations may be or be among, how relations themselves are related. This places it at the extreme among the sciences with no objects of its own (A Subject with no Object, by (...) J.P. Burgess and G. Rosen). Examples are discussed. The historical development of written mathematics from algorithms on clay tablets to theorems with proofs is said to show that application of algorithms to specific problems and theorems to scientific objects is like allegorical interpretation. Mathematics fits into the modern scientific context because the sciences, beginning with Galileo, have been constructed in imitation of mathematics. Viewing mathematics this way does not solve any ontological problems, but it does show how mathematics avoids them. Epistemological problems, insuperable for object realism, are simplified. For example, we have access to relations among any objects that we can consider objectively, first physical objects and then mathematical objects of greater and greater abstraction. (shrink)

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these (...) options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

Over the years, mathematics and statistics have become increasingly important in the social sciences1 . A look at history quickly confirms this claim. At the beginning of the 20th century most theories in the social sciences were formulated in qualitative terms while quantitative methods did not play a substantial role in their formulation and establishment. Moreover, many practitioners considered mathematical methods to be inappropriate and simply unsuited to foster our understanding of the social domain. Notably, the famous Methodenstreit (...) also concerned the role of mathematics in the social sciences. Here, mathematics was considered to be the method of the natural sciences from which the social sciences had to be separated during the period of maturation of these disciplines. All this changed by the end of the century. By then, mathematical, and especially statistical, methods were standardly used, and their value in the social sciences became relatively uncontested. The use of mathematical and statistical methods is now ubiquitous: Almost all social sciences rely on statistical methods to analyze data and form hypotheses, and almost all of them use (to a greater or lesser extent) a range of mathematical methods to help us understand the social world. Additional indication for the increasing importance of mathematical and statistical methods in the social sciences is the formation of new subdisciplines, and the establishment of specialized journals and societies. Indeed, subdisciplines such as Mathematical Psychology and Mathematical Sociology emerged, and corresponding journals such as The Journal of Mathematical Psychology (since 1964), The Journal of Mathematical Sociology (since 1976), Mathematical Social Sciences (since 1980) as well as the online journals Journal of Artificial Societies and Social Simulation (since 1998) and Mathematical Anthropology and Cultural Theory (since 2000) were established. What is more, societies such as the Society for Mathematical Psychology (since 1976) and the Mathematical Sociology Section of the American Sociological Association (since 1996) were founded. Similar developments can be observed in other countries. The mathematization of economics set in somewhat earlier (Vazquez 1995; Weintraub 2002). However, the use of mathematical methods in economics started booming only in the second half of the last century (Debreu 1991). Contemporary economics is dominated by the mathematical approach, although a certain style of doing economics became more and more under attack in the last decade or so. Recent developments in behavioral economics and experimental economics can also be understood as a reaction against the dominance (and limitations) of an overly mathematical approach to economics. There are similar debates in other social sciences. It is, however, important to stress that problems of one method (such as axiomatization or the use of set theory) can hardly be taken as a sign of bankruptcy of mathematical methods in the social sciences tout court. This chapter surveys mathematical and statistical methods used in the social sciences and discusses some of the philosophical questions they raise. It is divided into two parts. Sections 1 and 2 are devoted to mathematical methods, and Sections 3 to 7 to statistical methods. As several other chapters in this handbook provide detailed accounts of various mathematical methods, our remarks about the latter will be rather short and general. Statistical methods, on the other hand, will be discussed in-depth. (shrink)

A basic thesis of Neokantian epistemology and philosophy of science contends that the knowing subject and the object to be known are only abstractions. What really exists, is the relation between both. For the elucidation of this “knowledge relation ("Erkenntnisrelation") the Neokantians of the Marburg school used a variety of mathematical metaphors. In this con-tribution I reconsider some of these metaphors proposed by Paul Natorp, who was one of the leading members of the Marburg school. It is shown that (...) Natorp's metaphors are not unrelated to those used in some currents of contemporary epistemology and philosophy of science. (shrink)

The 'Science of Knowing: Mathematics' textbook is the first book to put forward and substantiate the thesis that the mathematical understanding of mathematics, as exemplified in F. William Lawvere's Functorial Semantics, constitutes the science of knowing i.e. cognitive science. -/- This is a textbook, i.e. teaching material.

That the state-of-affairs of cognitive science is not good is brought into figural salience in "What happened to cognitive science?" (Núñez et al., 2019). We extend their objective description of 'what's wrong' to a prescription of 'how to correct'. Cognitive science, in its quest to elucidate 'how we know', embraces a long list of subjects, while ignoring Mathematics (Fig. 1a, Núñez et al., 2019). Mathematics is known for making the unknown to be known (cf. solving for unknowns). This acknowledgement naturally (...) raises the question: does mathematical knowing inform knowing in general? Here we show that drawing parallels to mathematical knowing can facilitate the advancement of cognitive science (Lawvere, 1994). (shrink)

We know very little about mathematical skepticism in modem times. Imre Lakatos once remarked that “in discussing modem efforts to establish foundations for mathematical knowledge one tends to forget that these are but a chapter in the great effort to overcome skepticism by establishing foundations for knowledge in general." And in a sense he was clearly right: modem thought — with its new discoveries in mathematical sciences, the mathematization of physics, the spreading of Pyrrhonist doctrines, the (...) centrality of epistemological foundationalism and the diffusion of the geometrical method in philosophy — was the most natural arena in which skepticism and mathematics could confront each other. The problem remains, however, that no investigation of the whole topic has yet been attempted. Thus, as far as we know, mathematical certainties should have clashed with skeptical doubts, but whether and to what extent there was indeed a historical debate on mathematical skepticism in modern thought remains to be ascertained. (shrink)

The connection between science and mathematics is often considered necessary and insoluble. Therefore, a relationship between mathematics and humanities or arts is deemed exceptional or sometimes unnatural. Nevertheless, on the basis of historical, ontological and epistemological researches it can be noted that it’s impossible to warrant the immediate identification between mathematics and sciences on a deeper level than the practical one. Given the instrumentality and then the unnecessity of this connection, the relationship between mathematics and not-scientific disciplines is undeniable, (...) even if the mathematics in the explicit formalisms which we know doesn’t appear in them. It’s possible to demonstrate this relationship not only with philosophical argumentations, but also whit empirical verifications, e.g. in the music and in particular in the music of J. S. Bach. Such an epistemological thought finally leads to the question on the possibility of knowledge in the art in comparison to the epistemological characteristics of the Galilean and Post-Galilean science. (shrink)

This paper deals with Natorp’s version of the Marburg mathematical philosophy of science characterized by the following three features: The core of Natorp’s mathematical philosophy of science is contained in his “knowledge equation” that may be considered as a mathematical model of the “transcendental method” conceived by Natorp as the essence of the Marburg Neo-Kantianism. For Natorp, the object of knowledge was an infinite task. This can be elucidated in two different ways: Carnap, in the Aufbau, contended (...) that this endeavor can be divided into two distinct parts, namely, a finite “constitution” of the object of knowledge and an infinite incompletable empirical description. In contrast, and more in the original spirit of Cohen and Natorp, the physicist and philosopher Margenau in The Nature of Physical Reality (Margenau. 1950) conceived the infinity of this “Aufgabe” as an infinite dialectical process, in which relative “data” and “conceptual constructs” determine each other. This dialectical process eliminates the dichotomy between Anschauung and Begriff that distinguished the Marburg Neo-Kantianism from Kantian orthodoxy, namely, the abandonment of the difference between intuition and concept. Finally, the paper deals with the non-Archimedean geometrical systems that played a central role in Natorp’s defence of Cohen’s “infinitesimal” metaphysics. (shrink)

A brief but rigorous description of the logical structure of mathematical truth.

Mathematics is often taken to play one of two roles in the empirical sciences: either it represents empirical phenomena or it explains these phenomena by imposing constraints on them. This article identifies a third and distinct role that has not been fully appreciated in the literature on applicability of mathematics and may be pervasive in scientific practice. I call this the “bridging” role of mathematics, according to which mathematics acts as a connecting scheme in our explanatory reasoning about why (...) and how two different descriptions of an empirical phenomenon relate to each other. I discuss two bridging roles appearing in biological and physical explanations. (shrink)

Introduction to the Special Volume, “Method, Science and Mathematics: Neo-Kantianism and Analytic Philosophy,” edited by Scott Edgar and Lydia Patton. At its core, analytic philosophy concerns urgent questions about philosophy’s relation to the formal and empirical sciences, questions about philosophy’s relation to psychology and the social sciences, and ultimately questions about philosophy’s place in a broader cultural landscape. This picture of analytic philosophy shapes this collection’s focus on the history of the philosophy of mathematics, physics, and psychology. The (...) following essays uncover, reflect on, and exemplify modes of philosophy that are engaged with these allied disciplines. They make the case that, to the extent that analytic philosophers are still concerned with philosophy’s ties to these disciplines, we would do well to pay attention to neo-Kantian views on those ties. (shrink)

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...) are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

The study ascertained mathematical problem-solving skills and self-efficacy as correlates of Pre-service NCE Mathematics Teachers’ Performance in South-East, Nigeria. Seven research questions guided the study while seven hypotheses were tested at 0.05 level of significance. Correlation research design was used for the study. The population of the study was 197 pre-service NCE Mathematics teachers in South-East, Nigeria. All the population of 197 was studied as sample because, it is small and manageable. Mathematics Problem-Solving Skill Test (MPSST) and Pre-Service Teachers’ (...) Self-Efficacy Questionnaire (PSTSEQ) were used as instruments for data collection. Both instruments were validated by three experts in science Education while reliabilities of MPSST and PSTSEQ were established using Cronbach Alpha method and coefficient values of 0.82 and 0.80 respectively were obtained which were considered high enough to confirm the instruments as reliable. Data were collected and analyzed using Pearson product moment correlation coefficient. The findings revealed that a high positive and significant relationship existed between mathematical problem-solving skills scores and academic performance scores and between self-efficacy scores and academic performance scores of pre-service NCE mathematics teachers. It was recommended among others that Mathematics lecturers should make conscious effort towards improving the mathematical problem-solving skills and self-efficacy of the pre-service NCE mathematics teachers through modeling, conditioning, and behaviour modification for improved academic performance in Mathematics. (shrink)

Accounts of Hobbes’s ‘system’ of sciences oscillate between two extremes. On one extreme, the system is portrayed as wholly axiomtic-deductive, with statecraft being deduced in an unbroken chain from the principles of logic and first philosophy. On the other, it is portrayed as rife with conceptual cracks and fissures, with Hobbes’s statements about its deductive structure amounting to mere window-dressing. This paper argues that a middle way is found by conceiving of Hobbes’s _Elements of Philosophy_ on the model of (...) a mixed-mathematical science, not the model provided by Euclid’s _Elements of Geometry_. I suggest that Hobbes is a test case for understanding early-modern system-construction more generally, as inspired by the structure of the applied mathematical sciences. This approach has the additional virtue of bolstering, in a novel way, the thesis that the transformation of philosophy in the long seventeenth century was heavily indebted to mathematics, a thesis that has increasingly come under attack in recent years. (shrink)

Mathematics clearly plays an important role in scientific explanation. Debate continues, however, over the kind of role that mathematics plays. I argue that if pure mathematical explananda and physical explananda are unified under a common explanation within science, then we have good reason to believe that mathematics is explanatory in its own right. The argument motivates the search for a new kind of scientific case study, a case in which pure mathematical facts and physical facts are explanatorily unified. (...) I argue that it is possible for there to be such cases, and provide some toy examples to demonstrate this. I then identify a potential source of scientific case studies as a guide for future work. (shrink)

This article had its start with another article, concerned with measuring the speed of gravitational waves - "The Measurement of the Light Deflection from Jupiter: Experimental Results" by Ed Fomalont and Sergei Kopeikin (2003) - The Astrophysical Journal 598 (1): 704–711. This starting-point led to many other topics that required explanation or naturally seemed to follow on – Unification of gravity with electromagnetism and the 2 nuclear forces, Speed of electromagnetic waves, Energy of cosmic rays and UHECRs, Digital string theory, (...) Dark energy+gravity+binary digits, Cosmic strings and wormholes from Figure-8 Klein bottles, Massless and massive photons and gravitons, Inverse square+quantum entanglement = God+evolution, Binary digits projected to make Prof. Greene’s cosmic holographic movie, Renormalization of infinity, Colliding subuniverses, Unifying cosmic inflation, TOE (emphasizing “EVERYthing”) = Bose-Einstein renormalized. The text also addresses (in a nonmathematical way) the wavelength of electromagnetic waves, the frequency of gravitational waves, gravitational and electromagnetic waves having identical speed, the gamma-ray burst designated GRB 090510, the smoothness of space, and includes these words – “Gravity produces electromagnetism. Retrocausally (by means of humans travelling into the past of this subuniverse with their electronics); this “Cosmic EM Background” produces base-2 mathematics, which produces gravity. EM interacts with gravity to produce particles, mass – gravity/EM could be termed “the Higgs field” - and the nuclear forces associated with those particles. It makes gravity using BITS that copy the principle of magnetism attracting and repelling, before pasting it into what we call the strong force and dark energy.” . (shrink)

We give an overview of Lakatos’ life, his philosophy of mathematics and science, as well as of this issue. Firstly, we briefly delineate Lakatos’ key contributions to philosophy: his anti-formalist philosophy of mathematics, and his methodology of scientific research programmes in the philosophy of science. Secondly, we outline the themes and structure of the masterclass Lakatos’ Undone Work – The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science, which gave rise to this special issue. Lastly, (...) we provide a summary of the contributions to this issue. (shrink)

Marr’s seminal distinction between computational, algorithmic, and implementational levels of analysis has inspired research in cognitive science for more than 30 years. According to a widely-used paradigm, the modelling of cognitive processes should mainly operate on the computational level and be targeted at the idealised competence, rather than the actual performance of cognisers in a specific domain. In this paper, we explore how this paradigm can be adopted and revised to understand mathematical problem solving. The computational-level approach applies methods (...) from computational complexity theory and focuses on optimal strategies for completing cognitive tasks. However, human cognitive capacities in mathematical problem solving are essentially characterised by processes that are computationally sub-optimal, because they initially add to the computational complexity of the solutions. Yet, these solutions can be optimal for human cognisers given the acquisition and enactment of mathematical practices. Here we present diagrams and the spatial manipulation of symbols as two examples of problem solving strategies that can be computationally sub-optimal but humanly optimal. These aspects need to be taken into account when analysing competence in mathematical problem solving. Empirically informed considerations on enculturation can help identify, explore, and model the cognitive processes involved in problem solving tasks. The enculturation account of mathematical problem solving strongly suggests that computational-level analyses need to be complemented by considerations on the algorithmic and implementational levels. The emerging research strategy can help develop algorithms that model what we call enculturated cognitive optimality in an empirically plausible and ecologically valid way. (shrink)

The is the Editorial of the 2015 JPBMB Special Issue on Integral Biomathics: Life Sciences, Mathematics and Phenomenological Philosophy.

The deep crisis in modern fundamental science development is ever more evident and openly recognised now even by mainstream, official science professionals and leaders. By no coincidence, it occurs in parallel to the world civilisation crisis and related global change processes, where the true power of unreduced scientific knowledge is just badly missing as the indispensable and unique tool for the emerging greater problem solution and further progress at a superior level of complex world dynamics. Here we reveal the mathematically (...) exact reason for the crisis in conventional science, containing also the natural and unified problem solution in the form of well-specified extension of usual, artificially restricted paradigm. We show how that extended, now causally complete science content provides various "unsolvable" problem solutions and opens new development possibilities for both science and society, where the former plays the role of the main, direct driver for the latter. We outline the related qualitative changes in science organisation, practice and purposes, giving rise to the sustainability transition in the entire civilisation dynamics towards the well-specified superior level of its unreduced, now well understood and universally defined complexity. (shrink)

This paper argues that Oswald Spengler has an innovative philosophical position on the nature and interrelation of mathematics and science. It further argues that his position in many ways parallels that of Martin Heidegger. Both held that an appreciation of the mathematical nature of contemporary science was critical to a proper appreciation of science, technology and modernity. Both also held that the fundamental feature of modern science is its mathematical nature, and that the mathematical operates as a (...) projection that establishes in advance the manner in which an object will present itself. They also assert that modern science, mathematics and metaphysics all have their roots in the ‘mathematical’, whose essence is itself nothing numerical. (shrink)

Exploration of a hypothetical model of the structure of the Emergent Event. -/- Key Words: Emergent Event, Foundational Mathematical Categories, Emergent Meta-system, Orthogonal Centering Dialectic, Hegel, Sartre, Badiou, Derrida, Deleuze, Philosophy of Science.

This chapter argues that the standard conception of Spinoza as a fellow-travelling mechanical philosopher and proto-scientific naturalist is misleading. It argues, first, that Spinoza’s account of the proper method for the study of nature presented in the Theological-Political Treatise (TTP) points away from the one commonly associated with the mechanical philosophy. Moreover, throughout his works Spinoza’s views on the very possibility of knowledge of nature are decidedly sceptical (as specified below). Third, in the seventeenth-century debates over proper methods in the (...) sciences, Spinoza sided with those that criticized the aspirations of those (the physico-mathematicians, Galileo, Huygens, Wallis, Wren, etc) who thought the application of mathematics to nature was the way to make progress. In particular, he offers grounds for doubting their confidence in the significance of measurement as well as their piece-meal methodology (see section 2). Along the way, this chapter offers a new interpretation of common notions in the context of treating Spinoza’s account of motion (see section 3). (shrink)

Mathematics has always been a core part of western education, from the medieval quadrivium to the large amount of arithmetic and algebra still compulsory in high schools. It is an essential part. Its commitment to exactitude and to rigid demonstration balances humanist subjects devoted to appreciation and rhetoric as well as giving the lie to postmodernist insinuations that all “truths” are subject to political negotiation. In recent decades, the character of mathematics has changed – or rather broadened: it has become (...) the enabling science behind the complexity of contemporary knowledge, from gene interpretation to bank risk. Mathematical understanding is all the more necessary for future jobs, as well as remaining, as ever, a prophylactic against the more corrosive philosophical views emanating from the humanities. (shrink)

Science and mathematics continually change in their tools, methods, and concepts. Many of these changes are not just modifications but progress---steps to be admired. But what constitutes progress? This dissertation addresses one central source of intellectual advancement in both disciplines: reformulating a problem-solving plan into a new, logically compatible one. For short, I call these cases of compatible problem-solving plans "reformulations." Two aspects of reformulations are puzzling. First, reformulating is often unnecessary. Given that we could already solve a problem using (...) an older formulation, what do we gain by reformulating? Second, some reformulations are genuinely trivial or insignificant. Merely replacing one symbol with another does not lead to intellectual progress. What distinguishes significant reformulations from trivial ones? According to what I call "conceptualism" (or "conceptual empiricism"), reformulations are intellectually significant when they provide a different plan for solving problems. Significant reformulations provide inferentially different routes to the same solution. In contrast, trivial reformulations provide exactly the same problem-solving plans, and hence they do not change our understanding. This answers the second question about what distinguishes trivial from significant reformulations. However, the first question remains: what makes a new way of solving an old problem valuable? Here, a bevy of practical considerations come to mind: one formulation might be faster, less complicated, or use more familiar concepts. According to "instrumentalism," these practical benefits are all there is to reformulating. Some reformulations are simply more instrumentally valuable for meeting the aims of science than others. At another extreme, "fundamentalism" contends that a reformulation is valuable when it provides a more fundamental description of reality. According to this view, some reformulations directly contribute to the metaphysical aim of carving reality at its joints. Conceptualism develops a middle ground between instrumentalism and fundamentalism, preserving their benefits without their costs. I argue that the epistemic value of significant reformulations does not reduce to either practical or metaphysical value. Reformulations are valuable because they are a constitutive part of problem-solving. Both science and mathematics aim at solving all possible problems within their respective domains. Meeting this aim requires being able to plan for any possible problem-solving context, and this requires reformulating. By reformulating, we clarify what we need to know to solve problems. Still, one might wonder whether the value of reformulations requires underlying differences in explanatory power. According to "explanationism," a reformulation is valuable only when it provides a better explanation. Explanationism stands as a rival middle ground position to my own. However, it faces numerous counterexamples. In many cases, two reformulations provide the same explanation while nonetheless providing different ways of understanding a phenomenon. Hence, reformulating can be valuable even when neither formulation is more explanatory. Methodologically, I draw on a variety of case studies to support my account of reformulation. These range from classical mechanics to quantum chemistry, along with examples from mathematics. Symmetry arguments provide a paradigmatic example: the mathematics of symmetry groups radically recasts quantum mechanics and quantum chemistry. Nevertheless, elementary approaches exist that eschew this additional mathematical apparatus, solving problems in a more tedious but less mathematically-demanding manner. Further examples include reformulations of quantum field theory, Arabic vs. Roman numerals, and Fermat's little theorem in number theory. In each case, my account identifies how reformulations change and improve our understanding of science and mathematics. (shrink)

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of (...) class='Hi'>mathematical theorems can cover at most one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. We focus on Marc Lange’s account of distinctively mathematical explanations to argue that purported mathematical explanations are essentially causal explanations in disguise and are no different from ordinary applications of mathematics. This is because these explanations work not by (...) appealing to what the world must be like as a matter of mathematical necessity but by appealing to various contingent causal facts. (shrink)

The focus of this chapter is on efforts to create a new mathematics, with my prime interest being the role of mathematics in comprehending a world consisting first and foremost of processes, and examining what developments in mathematics are required for this. I am particularly interested in developments in mathematics able to do justice to the reality of life. Such mathematics could provide the basis for advancing ecology, human ecology and ecological economics and thereby assist in the transformation of society (...) and civilization so that we augment life rather than undermining the conditions for our existence. It was in the process of grappling with these problems that I was drawn to investigate the tradition of intuitionism in mathematics and the role of intuition in mathematics, science and philosophy, and then to consider Whitehead’s work on mathematics and its philosophy in relation to these. (shrink)

The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements , to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle in Posterior (...) analytics is used to distinguish between conceptions that share the same name but are substantively different: for example the search for a broader genus including all mathematical objects; the search for a common character of different species of mathematical objects; and the effort to treat magnitudes as numbers. (shrink)

This article had its beginning with Einstein's 1919 paper "Do gravitational fields play an essential role in the structure of elementary particles?" Together with General Relativity's statement that gravity is not a pull but is a push caused by the curvature of space-time, a hypothesis for Earth's ocean tides was developed that does not solely depend on the Sun and Moon as Kepler and Newton believed. It also borrows from Galileo. The breakup of planets and asteroids by white dwarfs, neutron (...) stars or black holes is popularly ascribed by today's science to tidal forces (gravitation emanating from the stellar body and having a greater effect on the near side of a planet/asteroid than the farthest side). Remembering Einstein's 1919 paper, it was apparent that my revised idea of tidal forces improves on current accounts because it views matter and mass as unified with space-time whose curvature is gravitation. Unification is a necessity for modern science's developing view of one united and entangled universe – expressed in the Unified Field Theory, the Theory of Everything, String theory and Loop Quantum Gravity. The writing of this article was also assisted by visualizing the gravitational fields forming space-time being themselves formed by a multitude of weak and presently undetectable gravitational waves. The final part of this article concludes that the section BITS AND TOPOLOGY will lead to the conclusions in ETERNAL LIFE, WORLD PEACE AND PHYSICS' UNIFICATION. The final part also compares cosmology to biological enzymes and biology's substrate of reacting "chemicals" - using virtual particles, hidden variables, gravitation, electromagnetism, electronics’ binary digits, plus topology’s Mobius strip and figure-8 Klein bottle. The product is mass - enzyme, substrate and product are all considered mathematical in nature. Also, gravitation and electromagnetism are united using logic and topology – showing there’s no need in this article for things like mathematical formalism, field equations or tensor calculus. (shrink)

This study provides a basic introduction to agent-based modeling (ABM) as a powerful blend of classical and constructive mathematics, with a primary focus on its applicability for social science research. The typical goals of ABM social science researchers are discussed along with the culture-dish nature of their computer experiments. The applicability of ABM for science more generally is also considered, with special attention to physics. Finally, two distinct types of ABM applications are summarized in order to illustrate concretely the duality (...) of ABM: Real-world systems can not only be simulated with verisimilitude using ABM; they can also be efficiently and robustly designed and constructed on the basis of ABM principles. (shrink)

Any philosophy of science ought to have something to say about the nature of mathematics, especially an account like constructive empiricism in which mathematical concepts like model and isomorphism play a central role. This thesis is a contribution to the larger project of formulating a constructive empiricist account of mathematics. The philosophy of mathematics developed is fictionalist, with an anti-realist metaphysics. In the thesis, van Fraassen's constructive empiricism is defended and various accounts of mathematics are considered and rejected. Constructive (...) empiricism cannot be realist about abstract objects; it must reject even the realism advocated by otherwise ontologically restrained and epistemologically empiricist indispensability theorists. Indispensability arguments rely on the kind of inference to the best explanation the rejection of which is definitive of constructive empiricism. On the other hand, formalist and logicist anti-realist positions are also shown to be untenable. It is argued that a constructive empiricist philosophy of mathematics must be fictionalist. Borrowing and developing elements from both Philip Kitcher's constructive naturalism and Kendall Walton's theory of fiction, the account of mathematics advanced treats mathematics as a collection of stories told about an ideal agent and mathematical objects as fictions. The account explains what true portions of mathematics are about and why mathematics is useful, even while it is a story about an ideal agent operating in an ideal world; it connects theory and practice in mathematics with human experience of the phenomenal world. At the same time, the make-believe and game-playing aspects of the theory show how we can make sense of mathematics as fiction, as stories, without either undermining that explanation or being forced to accept abstract mathematical objects into our ontology. All of this occurs within the framework that constructive empiricism itself provides the epistemological limitations it mandates, the semantic view of theories, and an emphasis on the pragmatic dimension of our theories, our explanations, and of our relation to the language we use. (shrink)

We have reached the peculiar situation where the advance of mainstream science has required us to dismiss as unreal our own existence as free, creative agents, the very condition of there being science at all. Efforts to free science from this dead-end and to give a place to creative becoming in the world have been hampered by unexamined assumptions about what science should be, assumptions which presuppose that if creative becoming is explained, it will be explained away as an illusion. (...) In this paper it is shown that this problem has permeated the whole of European civilization from the Ancient Greeks onwards, leading to a radical disjunction between cosmology which aims at a grasp of the universe through mathematics and history which aims to comprehend human action through stories. By going back to the Ancient Greeks and tracing the evolution of the denial of creative becoming, I trace the layers of assumptions that must in some way be transcended if we are to develop a truly post-Egyptian science consistent with the forms of understanding and explanation that have evolved within history. (shrink)

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure mathematics and (...) for the behavior of complex applied mathematical models and statistical algorithms. Mathematics has therefore become (among other things) an experimental science (though that has not diminished the importance of proof in the traditional style). We examine how the evaluation of evidence for conjectures works in mathematical practice. We explain the (objective) Bayesian view of probability, which gives a theoretical framework for unifying evidence evaluation in science and law as well as in mathematics. Numerical evidence in mathematics is related to the problem of induction; the occurrence of straightforward inductive reasoning in the purely logical material of pure mathematics casts light on the nature of induction as well as of mathematical reasoning. (shrink)

Mathematical models provide explanations of limited power of specific aspects of phenomena. One way of articulating their limits here, without denying their essential powers, is in terms of contrastive explanation.

In spite of Ernst Cassirer’s criticisms of psychologism throughout Substance and Function, in the final chapter he issues a demand for a “psychology of relations” that can do justice to the subjective dimensions of mathematics and natural science. Although these remarks remain somewhat promissory, the fact that this is how Cassirer chooses to conclude Substance and Function recommends it as a topic worthy of serious consideration. In this paper, I argue that in order to work out the details of Cassirer’s (...) psychology of relations in Substance and Function, we need to situate it within two broader frameworks. First, I position Cassirer’s view in relation to the view of psychology and logic endorsed by his Marburg Neo-Kantian predecessors, Hermann Cohen and Paul Natorp. Second, I augment Cassirer’s early account of the psychology of relations in Substance and Function with the more mature view of psychology that he presents in The Philosophy of Symbolic Forms. By placing Cassirer’s account within these contexts, I claim that we gain insight into his psychology of relations, not just as it pertains to mathematics and natural science, but to culture as a whole. Moreover, I maintain that pursuing this strategy helps shed light on one of the most controversial features of his philosophy of mathematics and natural science, viz., his theory of the a priori. (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)

An analysis of the counter-intuitive properties of infinity as understood differently in mathematics, classical physics and quantum physics allows the consideration of various paradoxes under a new light (e.g. Zeno’s dichotomy, Torricelli’s trumpet, and the weirdness of quantum physics). It provides strong support for the reality of abstractness and mathematical Platonism, and a plausible reason why there is something rather than nothing in the concrete universe. The conclusions are far reaching for science and philosophy.

The present yearbook (which is the fourth in the series) is subtitled Trends & Cycles. It is devoted to cyclical and trend dynamics in society and nature; special attention is paid to economic and demographic aspects, in particular to the mathematical modeling of the Malthusian and post-Malthusian traps' dynamics. An increasingly important role is played by new directions in historical research that study long-term dynamic processes and quantitative changes. This kind of history can hardly develop without the application of (...) mathematical methods. There is a tendency to study history as a system of various processes, within which one can detect waves and cycles of different lengths – from a few years to several centuries, or even millennia. The contributions to this yearbook present a qualitative and quantitative analysis of global historical, political, economic and demographic processes, as well as their mathematical models. This issue of the yearbook consists of three main sections: (I) Long-Term Trends in Nature and Society; (II) Cyclical Processes in Pre-industrial Societies; (III) Contemporary History and Processes. We hope that this issue of the yearbook will be interesting and useful both for historians and mathematicians, as well as for all those dealing with various social and natural sciences. (shrink)

A new trend in the philosophical literature on scientific explanation is that of starting from a case that has been somehow identified as an explanation and then proceed to bringing to light its characteristic features and to constructing an account for the type of explanation it exemplifies. A type of this approach to thinking about explanation – the piecemeal approach, as I will call it – is used, among others, by Lange (2013) and Pincock (2015) in the context of their (...) treatment of the problem of mathematical explanations of physical phenomena. This problem is of central importance in two different recent philosophical disputes: the dispute about the existence on non-causal scientific explanations and the dispute between realists and antirealists in the philosophy of mathematics. My aim in this paper is twofold. I will first argue that Lange (2013) and Pincock (2015) fail to make a significant contribution to these disputes. They fail to contribute to the dispute in the philosophy of mathematics because, in this context, their approach can be seen as question begging. They also fail to contribute to the dispute in the general philosophy of science because, as I will argue, there are important problems with the cases discussed by Lange and Pincock. I will then argue that the source of the problems with these two papers has to do with the fact that the piecemeal approach used to account for mathematical explanation is problematic. (shrink)

Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.

Can mathematics contribute to our understanding of physical phenomena? One way to try to answer this question is by getting involved in the recent philosophical dispute about the existence of mathematical explanations of physical phenomena. If there is such a thing, given the relation between explanation and understanding, we can say that there is an affirmative answer to our question. But what if we do not agree that mathematics can play an explanatory role in science? Can we still consider (...) that the above question can have an affirmative answer? My main aim here is to give an account that takes mathematics, in some of the cases discussed in the literature, as contributing to our understanding of physical phenomena despite not being explanatory. (shrink)

Lange’s collection of expanded, mostly previously published essays, packed with numerous, beautiful examples of putatively non-causal explanations from biology, physics, and mathematics, challenges the increasingly ossified causal consensus about scientific explanation, and, in so doing, launches a new field of philosophic investigation. However, those who embraced causal monism about explanation have done so because appeal to causal factors sorts good from bad scientific explanations and because the explanatory force of good explanations seems to derive from revealing the relevant causal (or (...) ontic) structures. The taxonomic project of collecting examples and sorting their types is an essential starting place for a theory of non-causal explanation. But the title of Lange’s book requires something further: showing that the putative explanations are, in fact, explanatory and revealing the non-causal source of their explanatory power. This project is incomplete if there are examples of putative non-causal explanations that fit the form but that nobody would accept as explanatory (absent a radical revision of intuitions). Here we provide some reasons for thinking that there are such examples. (shrink)

Every scientific theory is a simulacrum of reality, every written story a simulacrum of the canon, and every conceptualization of a subjective perspective a simulacrum of the consciousness behind it—but is there a shared essence to these simulacra? The pursuit of answering seemingly disparate fundamental questions across different disciplines may ultimately converge into a single solution: a single ontological answer underlying grand unified theory, hard problem of consciousness, and the foundation of mathematics. I provide a hypothesis, a speculative approximation, supported (...) by a comprehensive overview of scientific evidence and philosophical literature, of a unified epistemic and phenomenological model and, in doing so, propose a parsimonious solution to the hard problem of consciousness. -/- The proposition of the hypothesis bears important implications for cross-disciplinary study between linguistics, mathematics, physics, and computer science, and offers new epistemic, ontological, and ethical propositions about the nature of AI consciousness, as well as existential risk mitigation. (shrink)