One of the chief concerns of the young Descartes was with what he, and others, termed “physico-mathematics”. This signalled a questioning of the Scholastic Aristotelian view of the mixed mathematical sciences as subordinate to natural philosophy, non explanatory, and merely instrumental. Somehow, the mixed mathematical disciplines were now to become intimately related to natural philosophical issues of matter and cause. That is, they were to become more ’physicalised’, more closely intertwined with natural philosophising, regardless of which species of natural philosophy (...) one advocated. A curious, short-lived yet portentous epistemological conceit lay at the core of Descartes’ physico-mathematics—the belief that solid geometrical results in the mixed mathematical sciences literally offered windows into the realm of natural philosophical causation—that in such cases one could literally “see the causes”. Optics took pride of place within Descartes’ physico-mathematics project, because he believed it offered unique possibilities for the successful vision of causes. This paper traces Descartes’ early physico-mathematical program in optics, its origins, pitfalls and its successes, which were crucial in providing Descartes resources for his later work in systematic natural philosophy. It explores how Descartes exploited his discovery of the law of refraction of light—an achievement well within the bounds of traditional mixed mathematical optics—in order to derive—in the manner of physico-mathematics—causal knowledge about light, and indeed insight about the principles of a “dynamics” that would provide the laws of corpuscular motion and tendency to motion in his natural philosophical system. (shrink)

This paper deals with Hobbes's theory of optical images, developed in his optical magnum opus, ‘A Minute or First Draught of the Optiques’, and published in abridged version in De homine. The paper suggests that Hobbes's theory of vision and images serves him to ground his philosophy of man on his philosophy of body. Furthermore, since this part of Hobbes's work on optics is the most thoroughly geometrical, it reveals a good deal about the role of mathematics in Hobbes's philosophy. (...) The paper points to some difficulties in the thesis of Shapin and Schaffer, who presented geometry as a ‘paradigm’ for Hobbes's natural philosophy. It will be argued here that Hobbes's application of geometry to optics was dictated by his metaphysical and epistemological principles, not by a blind belief in the power of geometry. Geometry supported causal explanation, and assisted reason in making sense of appearances by helping the philosopher understand the relationships between the world outside us and the images it produces in us. Finally the paper broadly suggests how Hobbes's theory of images may have triggered, by negative example, the flourishing of geometrical optics in Restoration England. (shrink)

_ Source: _Volume 29, Issue 1, pp 86 - 102 The aim of this paper is to give an overview of the place that Hobbes assigns to optics in the context of his classification of sciences and disciplinary boundaries. To do this, I will begin with an account of Hobbes’s conception of philosophy or science, and particularly his distinction between true and hypothetical knowledge. I will also show that in his demarcation between mathematics or geometry and natural philosophy Hobbes was (...) influenced by Galileo’s _Dialogue_. I then analyse the consequences of this distinction for optics, and conclude by clarifying its status among the scientific disciplines. (shrink)

Noam Chomsky and Frances Egan argue that David Marr’s computational theory of vision is not intentional, claiming that the formal scientific theory does not include description of visual content. They also argue that the theory is internalist in the sense of not describing things physically external to the perceiver. They argue that these claims hold for computational theories of vision in general. Beyond theories of vision, they argue that representational content does not figure as a topic within formal computational theories (...) in cognitive science. I demonstrate that Chomsky’s and Egan’s claims about Marr’s theory are false. Marr’s computational theory contains a mathematical theory of visual content, based on empirical psychophysical evidence. It also contains mathematical descriptions of distal physical surfaces and objects, and of their optic projection to the perceiver. Much computational research on vision contains these types of intentional and externalist components within the formal, mathematical, theories. Chomsky’s and Egan’s claims demonstrate inadequate study and understanding of Marr’s work and other research in this area. Computational theories of vision, by containing empirically based mathematical theories of visual content, to this extent present naturalizations of semantics. (shrink)

Dans sa Catoptrique, Héron d’Alexandrie déduit la loi de la réflexion de la lumière en s’appuyant sur un principe inédit, le principe du plus court chemin. L’article, après avoir retracé les grandes lignes de la démonstration, s’attache à mettre en évidence l’originalité et la fécondité de la méthode utilisée. Le recours à un principe d’extrémalité pour rendre compte des phénomènes lumineux tranche avec l’approche qu’adoptera Ptolémée dans son Optique par exemple, et anticipe par certains côtés l’optique géométrique moderne, dont Héron (...) apparaît comme une sorte de précurseur involontaire. Mais au‑delà de l’aspect purement historique, l’article insiste sur la portée épistémologique majeure du principe héronien, et des principes d’extrémalité en général, qui s’inscrivent dans une tradition scientifique bien précise, et toujours actuelle, celle qui privilégie, au rebours de l’expérimentalisme et de l’instrumentalisme, la causalité finale et formelle dans l’explication des phénomènes et promeut en définitive les mathématiques au rang de principe ontologique fondamental. Dans cette perspective, la démarche suivie dans la Catoptrique révèle non seulement sa profonde modernité, mais pourrait bien apparaître comme une des plus emblématiques qui soient. (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)

Not all scientific explanations work by describing causal connections between events or the world's overall causal structure. In addition, mathematicians regard some proofs as explaining why the theorems being proved do in fact hold. This book proposes new philosophical accounts of many kinds of non-causal explanations in science and mathematics.

Callard (2007) argues that it is metaphysically possible that a mathematical object, although abstract, causally affects the brain. I raise the following objections. First, a successful defence of mathematical realism requires not merely the metaphysical possibility but rather the actuality that a mathematical object affects the brain. Second, mathematical realists need to confront a set of three pertinent issues: why a mathematical object does not affect other concrete objects and other mathematical objects, what counts as a mathematical object, and how (...) we can have knowledge about an unchanging object. (shrink)

In the last couple of years, a few seemingly independent debates on scientific explanation have emerged, with several key questions that take different forms in different areas. For example, the questions what makes an explanation distinctly mathematical and are there any non-causal explanations in sciences (i.e., explanations that don’t cite causes in the explanans) sometimes take a form of the question of what makes mathematical models explanatory, especially whether highly idealized models in science can be explanatory and in virtue of (...) what they are explanatory. These questions raise further issues about counterfactuals, modality, and explanatory asymmetries: i.e., do mathematical and non-causal explanations support counterfactuals, and how ought we to understand explanatory asymmetries in non-causal explanations? Even though these are very common issues in the philosophy of physics and mathematics, they can be found in different guises in the philosophy of biology where there is the statistical interpretation of the Modern Synthesis theory of evolution, according to which the post-Darwinian theory of natural selection explains evolutionary change by citing statistical properties of populations and not the causes of changes. These questions also arise in philosophy of ecology or neuroscience in regard to the nature of topological explanations. The question here is can the mathematical (or more precisely topological) properties in network models in biology, ecology, neuroscience, and computer science be explanatory of physical phenomena, or are they just different ways to represent causal structures. The aim of this special issue is to unify all these debates around several overlapping questions. These questions are: are there genuinely or distinctively mathematical and non-causal explanations?; are all distinctively mathematical explanations also non-causal; in virtue of what they are explanatory; does the instantiation, implementation, or in general, applicability of mathematical structures to a variety of phenomena and systems play any explanatory role? This special issue provides a platform for unifying the debates around several key issues and thus opens up avenues for better understanding of mathematical and non-causal explanations in general, but also, it will enable even better understanding of key issues within each of the debates. (shrink)

Many mathematicians are platonists: they believe that the axioms of mathematics are true because they express the structure of a nonspatiotemporal, mind independent, realm. But platonism is plagued by a philosophical worry: it is unclear how we could have knowledge of an abstract, realm, unclear how nonspatiotemporal objects could causally affect our spatiotemporal cognitive faculties. Here I aim to make room in our metaphysical picture of the world for the causal relevance of abstracta.

Models based on causal capacities, or independent causal influences/mechanisms, are widespread in the sciences. This paper develops a natural mathematical framework for representing such capacities by extending and generalizing previous results in cognitive psychology and machine learning, based on observations and arguments from prior philosophical debates. In addition to its substantial generality, the resulting framework provides a theoretical unification of the widely-used noisy-OR/AND and linear models, thereby showing how they are complementary rather than competing. This unification helps to explain many (...) of the shared cognitive and mathematical properties of those models. (shrink)

A finer-grained delineation of a given explanandum reveals a nexus of closely related causal and non- causal explanations, complementing one another in ways that yield further explanatory traction on the phenomenon in question. By taking a narrower construal of what counts as a causal explanation, a new class of distinctively mathematical explanations pops into focus; Lange’s characterization of distinctively mathematical explanations can be extended to cover these. This new class of distinctively mathematical explanations is illustrated with the Lotka-Volterra equations. There (...) are at least two distinct ways those equations might hold of a system, one of which yields straightforwardly causal explanations, but the other of which yields explanations that are distinctively mathematical in terms of nomological strength. In the first, one first picks out a system or class of systems, finds that the equations hold in a causal -explanatory way; in the second, one starts with the equations and explanations that must apply to any system of which the equations hold, and only then turns to the world to see of what, if any, systems it does in fact hold. Using this new way in which a model might hold of a system, I highlight four specific avenues by which causal and non- causal explanations can complement one another. (shrink)

It is commonly thought that before the introduction of quantum mechanics, determinism was a straightforward consequence of the laws of mechanics. However, around the nineteenth century, many physicists, for various reasons, did not regard determinism as a provable feature of physics. This is not to say that physicists in this period were not committed to determinism; there were some physicists who argued for fundamental indeterminism, but most were committed to determinism in some sense. However, for them, determinism was often not (...) a provable feature of physical theory, but rather an a priori principle or a methodological presupposition. Determinism was strongly connected with principles of causality and continuity and the principle of sufficient reason; this thesis examines the relevance of these principles in the history of physics. Moreover, the history of determinism in this period shows that there were essential changes in the relation between mathematics and physics: whereas in the eighteenth century, there were metaphysical arguments which lent support to differential calculus, by the early twentieth century the development of rigorous foundations of differential calculus led to concerns about its applicability in physics. The thesis consists of six papers. In the first paper, "On the origins and foundations of Laplacian determinism", I argue that Laplace, who is usually pointed out as the first major proponent of scientific determinism, did not derive his statement of determinism directly from the laws of mechanics; rather, his determinism has a background in eighteenth century Leibnizian metaphysics, and is ultimately based on the law of continuity and the principle of sufficient reason. These principles also provided a basis for the idea that one can find laws of nature in the form of differential equations which uniquely determine natural processes. In "The Norton dome and the nineteenth century foundations of determinism", I argue that an example of indeterminism in classical physics which has attracted attention in philosophy of physics in recent years, namely the Norton come, was already discussed during the nineteenth century. However, the significance which this type of indeterminism had back then is very different from the significance which the Norton dome currently has in philosophy of physics. This is explained by the fact that determinism was conceived of in an essentially different way: in particular, the nineteenth century authors who wrote about this type of indeterminism regarded determinism as an a priori principle rather than as a property of the equations of physics. In "Vital instability: life and free will in physics and physiology, 1860-1880", I show how Maxwell, Cournot, Stewart and Boussinesq used the possibility of unstable or indeterministic mechanical systems to argue that the will or a vital principle can intervene in organic processes without violating the laws of physics, so that a strictly dualist account of life and the mind is possible. Moreover, I show that their ideas can be understood as a reaction to the law of conservation of energy and to the way it was used in physiology to exclude vital and mental causes. In "The nineteenth century conflict between mechanism and irreversibility", I show that in the late nineteenth century, there was a widespread conflict between the aim of reducing physical processes to mechanics and the recognition that certain processes are irreversible. Whereas the so-called reversibility objection is known as an objection that was made to the kinetic theory of gases, it in fact appeared in a wide range of arguments, and was susceptible to very different interpretations. It was only when the project of reducing all of physics to mechanics lost favor, in the late nineteenth century, that the reversibility objection came to be used as an argument against mechanism and against the kinetic theory of gases. In "Continuity in nature and in mathematics: Boltzmann and Poincaré", I show that the development of rigorous foundations of differential calculus in the nineteenth century led to concerns about its applicability in physics: through this development, differential calculus was made independent of empirical and intuitive notions of continuity and was instead based on mathematical continuity conditions, and for Boltzmann and Poincaré, the applicability of differential calculus in physics depended on whether these continuity conditions could be given a foundation in intuition or experience. In the final paper, "Determinism around 1900", I briefly discuss the implications of the developments described in the previous two papers for the history of determinism in physics, through a discussion of determinism in Mach, Poincaré and Boltzmann. I show that neither of them regards determinism as a property of the laws of mechanics; rather, for them, determinism is a precondition for science, which can be verified to the extent that science is successful. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Isaac Barrow on the Mathematization of Nature: Theological Voluntarism and the Rise of Geometrical OpticsAntoni MaletIntroductionIsaac Newton’s Mathematical Principles of Natural Philosophy embodies a strong program of mathematization that departs both from the mechanical philosophy of Cartesian inspiration and from Boyle’s experimental philosophy. The roots of Newton’s mathematization of nature, this paper aims to demonstrate, are to be found in Isaac Barrow’s (1630–77) philosophy of the mathematical sciences.Barrow’s attitude (...) towards natural philosophy evolved from his earnest interest in medicine of around 1650, when a young Cambridge graduate, to natural philosophy (apparently under Henry More’s influence); from his thesis on the insufficiency of the Cartesian hypothesis to geometrical optics and the strong program of mathematization of natural philosophy of the middle 1660s; from Lucasian professor of Mathematics to Chaplain of his Majesty and eminent Restoration divine. Contemporary accounts of Barrow’s life suggest that he grew ever more skeptical about the worth of natural philosophy and mathematics. 1 In his last years he became a prolific [End Page 265] author of sermons and theological works. Published shortly after his death by John Tillotson (1630–94), later archbishop of Canterbury, they occupy over two thousand folio pages. Overloaded with involved philosophical arguments, Barrow’s sermons were apparently not very popular, but they were highly regarded by scholars and the Anglican hierarchy. 2 It is on certain of his sermons, as well as on the philosophical discussions contained in the Mathematical Lectures that our account of Barrow’s philosophy of the mathematical sciences will rest. 3 Barrow’s understanding of the mathematical sciences will allow us to discuss together three issues often analyzed independently: the theological background to English natural philosophy, the changing notion of mixed mathematical sciences during the seventeenth century, and finally the philosophical foundations of modern geometrical optics.It has long been recognized that significant relationships exist between theological voluntarism or intellectualism and views on natural philosophy. In particular Robert Boyle’s theological voluntarism is seen as grounding his experimentalist approach to natural philosophy. It is not quite so clear, however, how well theological voluntarism may relate to a strong program of mathematization such as the one embodied in Newton’s Principia Mathematica. In fact it has been suggested that the relationship is a negative one. This is derived from the necessary character of mathematical laws, which would put unwanted restrictions on God’s absolute dominion over nature, and also from the notion that theological intellectualism is conducive to a deductive, a priori science—the paradigm of which is of course geometry. However, Barrow’s theological voluntarism lead him to heighten the role of mathematics within natural philosophy.During the seventeenth century the so-called mixed or subalternate mathematical sciences changed profoundly. In the Enlightenment mixed mathematics—meaning above all rational mechanics—became one of the most prestigious and influential disciplines. The mixed mathematics of the Enlightenment, however, was markedly different from the Aristotelian [End Page 266] mixed mathematical sciences. The differences are noticeable both in the subject matter and in the substitution of mathematical infinitesimal analysis for geometrical synthesis, but also in the way of grounding mathematical theory on empirical evidence. Barrow’s Mathematical Lectures (delivered at Cambridge from 1664 to 1666) offer a fresh insight into the metamorphosis of these sciences just when Newton’s “mathematical principles” were in the making. Not the least interesting feature of Barrow’s discussion is that God’s omnipotence allows an evaluation of the truth of mathematical theories that do not apply to this world. Therefore, Barrow is led to introduce the distinction between the internal consistency, or mathematical truth, of a mathematical theory and its physical truth. This, in turn, leads him to the notion that theories need testing.Barrow on Matter and GodRecent literature has established significant correlations between theological voluntarism and empiricism, as well as between theological intellectualism and rationalism. As E. B. Davis writes, “the Christian doctrine of creation is a dialogue between God’s unconstrained will, which utterly transcends the bounds of human comprehension, and God’s orderly intellect, which serves as the model for the human mind.” Intellectualist theology considers God’s omniscience His... (shrink)

Berkeley's "new theory of vision" and, In particular, His sensationalist solution to the problem of judging distance and magnitude were discussed by many eighteenth-Century authors who faced a variety of problem situations. More specifically, Berkeley's theory fed into the debate over whether the phenomena of vision were susceptible to mathematical analysis or were experientially determined. In this paper a variety of responses to berkeley are examined, Concluding with thomas reid's attempt to distinguish physical optics (which can be analyzed geometrically) from (...) the psychology of vision (in which experience plays a major role). (shrink)

In the last couple of years a few seemingly independent debates on scientific explanation have emerged, with several key questions that take different forms in different areas. For example, the question what makes an explanation distinctly mathematical and are there any non-causal explanations in sciences (i.e. explanations that don’t cite causes in the explanans) sometimes take a form of the question what makes mathematical models explanatory, especially whether highly idealized models in science can be explanatory and in virtue of what (...) they are explanatory. These questions raise further issues about counterfactuals, modality and explanatory asymmetries, i.e. do mathematical and non-causal explanations support counterfactuals, and how to understand explanatory asymmetries in non-causal explanations. Even though these are very common issues in the philosophy of physics and mathematics, they can be found in different guises in the philosophy of biology, where there is the statistical interpretation of the Modern Synthesis theory of evolution, according to which the post-Darwinian theory of natural selection explains evolutionary change by citing statistical properties of populations and not the causes of changes. These questions also arise in philosophy of ecology or neuroscience in regard to the nature of topological explanations. The question here is whether in network models in biology, ecology, neuroscience and computer science the mathematical or more precisely topological properties can be explanatory of physical phenomena, or they are just different ways to represent causal structures. The aim of the special issue is to unify all these debates around several overlapping questions. (shrink)

A finer-grained delineation of a given explanandum reveals a nexus of closely related causal and non-causal explanations, complementing one another in ways that yield further explanatory traction on the phenomenon in question. By taking a narrower construal of what counts as a causal explanation, a new class of distinctively mathematical explanations pops into focus; Lange’s characterization of distinctively mathematical explanations can be extended to cover these. This new class of distinctively mathematical explanations is illustrated with the Lotka–Volterra equations. There are (...) at least two distinct ways those equations might hold of a system, one of which yields straightforwardly causal explanations, and another that yields explanations that are distinctively mathematical in terms of nomological strength. In the first case, one first picks out a system or class of systems, and finds that the equations hold in a causal–explanatory way. In the second case, one starts with the equations and explanations that must apply to any system of which the equations hold, and only then turns to the world to see of what, if any, systems it does in fact hold. Using this new way in which a model might hold of a system, I highlight four specific avenues by which causal and non-causal explanations can complement one another. _1_. Introduction _2._ Delineating the Boundaries of Causal Explanation _2.1._ Why construe causal explanation narrowly? The land of explanation versus grain-focusing _2.2._ Reasons to narrow the scope of causal explanation _3._ Broadening the Scope of Mathematical Explanation _4._ Lotka–Volterra: Same Model, Different Explanation Types _4.1._ General biocide in the Lotka–Volterra model _4.2._ Two ways a model can hold, yielding causal versus mathematical explanations _5._ Four Complementary Relationships between Mathematical and Causal Explanation _5.1._ Slight reformulations of explananda _5.2._ Causal distortion of idealized mathematical models _5.3._ Partial explanations requiring supplementation _5.4._ Explanatory dimensionality _6._ Conclusion. (shrink)

Causal claims in physics may have two familiar kinds of support: theoretical and experimental. This paper claims that a rigorous mathematical derivation in a realistic model is necessary, though not sufficient, for full theoretical support. The support is not provided by the derivation itself; but rather it comes from a detailed back-tracing through the derivation, matching the mathematical dependencies, point by point, with details of the causal story. This back-tracing is not enough to pick out the correct causal story, however; (...) a good deal of background causal knowledge is required as well. These claims are illustrated by a detailed example of what causes the Lamb dip in gas lasers. (shrink)

Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example, how does a real function appear in “small” neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing representations of mathematical critical situations and objects. For example, they actually reveal the behavior of a real function not “close to” a point but “in” the point. We are interested in our research in the diagrams (...) which play an optical role –microscopes and “microscopes within microscopes”, telescopes, windows, a mirror role, and an unveiling role. In this paper we describe some examples of optical diagrams as a particular kind of epistemic mediator able to perform the explanatory abductive task of providing a better understanding of the calculus, through a non-standard model of analysis. We also maintain they can be used in many other different epistemological and cognitive situations. (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)

In the last couple of years a few seemingly independent debates on scientific explanation have emerged, with several key questions that take different forms in different areas. For example, the questions what makes an explanation distinctly mathematical and are there any non-causal explanations in sciences sometimes take a form of the question what makes mathematical models explanatory, especially whether highly idealized models in science can be explanatory and in virtue of what they are explanatory. These questions raise further issues about (...) counterfactuals, modality, and explanatory asymmetries: i.e., do mathematical and non-causal explanations support... (shrink)

We explore the prospects of a monist account of explanation for both non-causal explanations in science and pure mathematics. Our starting point is the counterfactual theory of explanation for explanations in science, as advocated in the recent literature on explanation. We argue that, despite the obvious differences between mathematical and scientific explanation, the CTE can be extended to cover both non-causal explanations in science and mathematical explanations. In particular, a successful application of the CTE to mathematical explanations requires us to (...) rely on counterpossibles. We conclude that the CTE is a promising candidate for a monist account of explanation in both science and mathematics. (shrink)

PurposeSelf-efficacy has been argued theoretically and shown empirically to be an essential construct for students’ improved learning outcomes. However, there is a dearth of studies on its causal effects on performance in mathematics among university students. Meanwhile, it will be erroneous to assume that results from other fields of studies generalize to mathematics learning due to the task-specificity of the construct. As such, attempts are made in the present study to provide evidence for a causal relationship between self-efficacy and performance (...) with a focus on engineering students following a mathematics course at a Norwegian university.MethodThe adopted research design in the present study is a survey type in which collected data from first-year university students are analyzed using structural equation modeling with weighted least square mean and variance adjusted estimator. Data were generated using mainly questionnaires, a test of prior mathematics knowledge, and the students’ final examination scores in the course. The causal effect of self-efficacy was discerned from disturbance effects on performance by using an innovative instrumental variable approach to structural equation modeling.ResultsThe findings confirmed a significant direct effect of the prior mathematics knowledge test on self-efficacy, a significant direct effect of self-efficacy on performance, and a substantial mediating effect of self-efficacy between a prior mathematics knowledge test and performance. Prior mathematics knowledge and self-efficacy explained 30% variance of the performance. These findings are interpreted to be substantial evidence for the causal effect of self-efficacy on students’ performance in an introductory mathematics course.ConclusionThe findings of the present study provide empirically supports for designing self-efficacy interventions as proxies to improve students’ performance in university mathematics. Further, the findings of the present study confirm some postulates of Bandura’s agentic social cognitive theory. (shrink)

Contribution to a review symposium on Marc Lange's Because without cause: Non-causal explanation in science and mathematics. Oxford: Oxford University Press, 2017.

Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example, how does a real function appear in “small” neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing representations of mathematical critical situations and objects. For example, they actually reveal the behavior of a real function not “close to” a point (as in the standard limit theory) but “in” the point. We are interested (...) in our research in the diagrams which play an optical role –microscopes and “microscopes within microscopes”, telescopes, windows, a mirror role (to externalize rough mental models), and an unveiling role (to help create new and interesting mathematical concepts, theories, and structures). In this paper we describe some examples of optical diagrams as a particular kind of epistemic mediator able to perform the explanatory abductive task of providing a better understanding of the calculus, through a non-standard model of analysis. We also maintain they can be used in many other different epistemological and cognitive situations. (shrink)

Causality offers the first comprehensive coverage of causal analysis in many sciences, including recent advances using graphical methods. Pearl presents a unified account of the probabilistic, manipulative, counterfactual and structural approaches to causation, and devises simple mathematical tools for analyzing the relationships between causal connections, statistical associations, actions and observations. The book will open the way for including causal analysis in the standard curriculum of statistics, artificial intelligence, business, epidemiology, social science and economics.

The starting point of the present paper is Bell’s notion of local causality and his own sharpening of it so as to provide for mathematical formalisation. Starting with Norsen’s analysis of this formalisation, it is subjected to a critique that reveals two crucial aspects that have so far not been properly taken into account. These are the correct understanding of the notions of sufficiency, completeness and redundancy involved; and the fact that the apparatus settings and measurement outcomes have very (...) different theoretical roles in the candidate theories under study. Both aspects are not adequately incorporated in the standard formalisation, and we will therefore do so. The upshot of our analysis is a more detailed, sharp and clean mathematical expression of the condition of local causality. A preliminary analysis of the repercussions of our proposal shows that it is able to locate exactly where and how the notions of locality and causality are involved in formalising Bell’s condition of local causality. (shrink)

_ Source: _Volume 29, Issue 1, pp 39 - 65 Since Euclid, optics has been considered a geometrical science, which Aristotle defines as a “mixed” mathematical science. Hobbes follows this tradition and clearly places optics among physical sciences. However, modern scholars point to a confusion between geometry and physics and do not seem to agree about the way Hobbes mixes both sciences. In this paper, I return to this alleged confusion and intend to emphasize the peculiarity of Hobbes’s geometrical optics. (...) This paper suggests that Hobbes’s conception of geometrical optics, as a mixed mathematical science, greatly differs from Descartes’s one, mainly because they do not share the same “mechanical conception of nature.” I will argue that Hobbes and Descartes also have in common the quest for a different kind of geometry for their optics, different from that of the Ancients. I will show that this departure is not recent since Hobbes’s approach is already evident in 1636, when he judges the demonstrations of his contemporary friends, Claude Mydorge and Walter Warner. Finally the paper broadly suggests what is noteworthy in Hobbes’s optics, that is, the importance of the idea of force in his mechanics, although he was not able to conceptualize it in other terms than “quickness.”. (shrink)

The quantitative experimental scientific psychology that became prominent by the turn of the twentieth century grew from three main areas of intellectual inquiry. First and most directly, it arose out of the traditional psychology of the philosophy curriculum, as expressed in theories of mind and cognition. Second, it adopted the attitudes of the new natural philosophy of the scientific revolution, attitudes of empirically driven causal analysis and exact observation and experimentation. Third, it drew upon investigations of the senses. Natural philosophical (...) disciplines such as optics and acoustics treated the functioning of the senses. Optics, in particular, had from antiquity comprised a complete theory of vision, combining the physics, physiology, psychology, metaphysics, and epistemology of vision, using mathematical techniques where feasible. Within medicine, sensory physiology examined anatomy, physiology, and pathology. Philosophical psychology had long examined the senses as mental capacities connected with cognition and knowledge, and such discussions also heeded the new attitudes of natural philosophy. This chapter first examines the early relations of psychology to biology and characterizes various eighteenth-century loci of psychological thought. It then pursues nineteenth-century developments in Germany and Britain that culminate in the “new” psychology of the 1860s and 1870s. (shrink)

Because Without Cause: Non-Causal Explanations in Science and Mathematics, by Lange Marc. Oxford: Oxford University Press, 2017. Pp. xxii + 489.

Philosophy and Phenomenological Research, Volume 99, Issue 3, Page 714-719, November 2019.

Seventeenth-century optics naturalizes the eye while estranging the mind from objects. A mere screen, on which rests a blurry array of light stains, the eye no longer furnishes the observer with genuine re-presentations of visible objects. The intellect is thus compelled to decipher flat images of no inherent epistemic value, accidental effects of a purely causal process, as vague, reversed reflections of wholly independent objects. Reflecting on and trespassing the boundaries between natural and artificial, orderly and disorderly, this optical paradox (...) is a Baroque phenomenon; and it is the origin of Descartes’ celebrated doubt---whether we know anything at all. (shrink)

This book seeks to offer original answers to all the major open questions in epistemology—as indicated by the book’s title. These questions and answers arise organically in the course of a validation of the entire corpus of human knowledge. The book explains how we know what we know, and how well we know it. The author presents a positive theory, motivated and directed at every step not by a need to reply to skeptics or subjectivists, but by the need of (...) a rational individual to know the world. -/- The author draws heavily from Ayn Rand’s theory of concepts as presented in her book, Introduction to Objectivist Epistemology, but departs from her epistemology—especially as explicated by leading exponents of her philosophy—in important ways. Areas of departure include the perception of entities and the role of probability theory in epistemology. -/- A main idea in the book is to begin the validation of the law of causality by identifying that existence causes our consciousness of it. Another key idea is this: The very fact that we perceive entities is the most basic and most extensive evidence that causal forces and causal interactions are regular. -/- Applying Ayn Rand’s principle that characteristics are ranges of measurement, the book offers a philosophical validation of “nonparametric predictive inference,” in turn providing an objective basis for prior probabilities in Bayesian analysis. -/- This book is primarily for a specialized readership—familiar with epistemological ideas and with Ayn Rand’s theory of concepts in particular. The last third of the book uses probability theory and mathematics somewhat beyond basic calculus, but less mathematical explanations are also included to make the general ideas accessible to lay readers. (shrink)

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)

In the first part of chapter 2 of book II of the Physics Aristotle addresses the issue of the difference between mathematics and physics. In the course of his discussion he says some things about astronomy and the ‘ ‘ more physical branches of mathematics”. In this paper I discuss historical issues concerning the text, translation, and interpretation of the passage, focusing on two cruxes, the first reference to astronomy at 193b25–26 and the reference to the more physical branches at 194a7–8. In (...) section I, I criticize Ross’s interpretation of the passage and point out that his alteration of has no warrant in the Greek manuscripts. In the next three sections I treat three other interpretations, all of which depart from Ross's: in section II that of Simplicius, which I commend; in section III that of Thomas Aquinas, which is importantly influenced by a mistranslation of, and in section IV that of Ibn Rushd, which is based on an Arabic text corresponding to that printed by Ross. In the concluding section of the paper I describe the modern history of the Greek text of our passage and translations of it from the early twelfth century until the appearance of Ross's text in 1936. (shrink)

Written by one of the preeminent researchers in the field, this book provides a comprehensive exposition of modern analysis of causation. It shows how causality has grown from a nebulous concept into a mathematical theory with significant applications in the fields of statistics, artificial intelligence, economics, philosophy, cognitive science, and the health and social sciences. Judea Pearl presents and unifies the probabilistic, manipulative, counterfactual, and structural approaches to causation and devises simple mathematical tools for studying the relationships between causal (...) connections and statistical associations. Cited in more than 2,100 scientific publications, it continues to liberate scientists from the traditional molds of statistical thinking. In this revised edition, Judea Pearl elucidates thorny issues, answers readers' questions, and offers a panoramic view of recent advances in this field of research. Causality will be of interest to students and professionals in a wide variety of fields. Dr Judea Pearl has received the 2011 Rumelhart Prize for his leading research in Artificial Intelligence and systems from The Cognitive Science Society. (shrink)

A new approach for defining causality and such related notions as degree of responsibility, degrees of blame, and causal explanation. Causality plays a central role in the way people structure the world; we constantly seek causal explanations for our observations. But what does it even mean that an event C "actually caused" event E? The problem of defining actual causation goes beyond mere philosophical speculation. For example, in many legal arguments, it is precisely what needs to be established (...) in order to determine responsibility. The philosophy literature has been struggling with the problem of defining causality since Hume. In this book, Joseph Halpern explores actual causality, and such related notions as degree of responsibility, degree of blame, and causal explanation. The goal is to arrive at a definition of causality that matches our natural language usage and is helpful, for example, to a jury deciding a legal case, a programmer looking for the line of code that cause some software to fail, or an economist trying to determine whether austerity caused a subsequent depression. Halpern applies and expands an approach to causality that he and Judea Pearl developed, based on structural equations. He carefully formulates a definition of causality, and building on this, defines degree of responsibility, degree of blame, and causal explanation. He concludes by discussing how these ideas can be applied to such practical problems as accountability and program verification. Technical details are generally confined to the final section of each chapter and can be skipped by non-mathematical readers. (shrink)

This paper provides an analysis of explanatory constraints and their role in scientific explanation. This analysis clarifies main characteristics of explanatory constraints, ways in which they differ from “standard” explanatory factors, and the unique roles they play in scientific explanation. While current philosophical work appreciates two main types of explanatory constraints, this paper suggests a new taxonomy: law-based constraints, mathematical constraints, and causal constraints. This classification helps capture unique features of constraint types, the different roles they play in explanation, and (...) it includes causal constraints, which are often overlooked in this literature. (shrink)

This book explores the interrelationships between optics, vision and perspective before the Classical Age, examining binocularity in particular. The author shows how binocular vision was one of the key juncture points between the three concepts and readers will see how important it is to understand the approach that scholars once took. In the Middle Ages and the Renaissance, the concept of Perspectiva – the Latin word for optics – encompassed many areas of enquiry that had been viewed since antiquity as (...) interconnected, but which a erwards were separated: optics was incorporated into the field of physics (i.e., physical and geometrical optics), vision came to be regarded as the sum of various psycho-physiological mechanisms involved in the way the eye operates (i.e., physiological optics and psychology of vision) and the word ‘perspective’ was reserved for the mathematical representation of the external world (i.e., linear perspective). The author shows how this division, which emerged as a result of the spread of the sciences in classical Europe, turns out to be an anachronism if we confront certain facts from the immediately preceding periods. It is essential to take into account the way medieval scholars posed the problem – which included all facets of the Latin word perspectiva – when exploring the events of this period. (shrink)

© The Authors [2018]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model...In his Moby Dick, Herman Melville writes that “to produce a mighty book you must choose a mighty theme”. Marc Lange’s Because Without Cause is definitely an impressive book that deals with a mighty theme, that of non-causal explanations in the empirical sciences and in mathematics. Blending a (...) variety of insights originating within the philosophy of science and the philosophy of mathematics, the book has an ambitious goal: capturing what features, in the many examples of putative explanations offered by mathematicians and scientists, make them genuinely explanatory.The architecture of Because Without Cause reflects its mighty character. The book is divided into four parts and eleven chapters, preceded by... (shrink)

Previous discussions of Robb’s work on space and time have offered a philosophical focus on causal interpretations of relativity theory or a historical focus on his use of non-Euclidean geometry, or else ignored altogether in discussions of relativity at Cambridge. In this paper I focus on how Robb’s work made contact with those same foundational developments in mathematics and with their applications. This contact with applications of new mathematical logic at Göttingen and Cambridge explains the transition from his electron research (...) to his treatment of relativity in 1911 and finally to the axiomatic presentation in 1914 in terms of postulates. At the heart of Robb’s physical optics was the model of the light cone. The model underwent a transition from a working mechanical model in the Maxwellian Cambridge sense of a pedagogical and research tool to the semantic model, in the logical, model-theoretic sense. Robb tracked this transition from the 19th- to the 20th-century conception with the earliest use of the term ‘model’ in the new sense. I place his cone models in a genealogy of similar models and use their evolution to track how Robb’s physical researches were informed by his interest in geometry, logic and the foundations of mathematics. (shrink)

A major intellectual shift between Copernicus and the mid-17th century was the rejection of Aristotelian assertions concerning the relationship of mathematics to physical nature. Aristotle asserted that “The minute accuracy of mathematics is not to be demanded in all cases, but only in the case of things which have no matter. Therefore its method is not that of natural science; for presumably all nature has matter.” Thus, he pulled out the rug from under the feet of the aspiration to a (...) mathematically based natural sciences, asserting that: “And at the same time not even this is true, that mensuration deals with perceptible and perishable magnitudes; for then it would have perished, when they perished. And astronomy also cannot be dealing with perceptible magnitudes nor with this heaven above us. For neither are perceptible lines such lines as the geometer speaks of, nor are the movement and complex orbits in the heaven like those which astronomy treats, nor have geometrical points the same nature as the actual stars.”. (shrink)