This paper examines explanations that turn on non-local geometrical facts about the space of possible configurations a system can occupy. I argue that it makes sense to contrast such explanations from "geometry of motion" with causal explanations. I also explore how my analysis of these explanations cuts across the distinction between kinematics and dynamics.

At some point in the Incessu Animalium, Aristotle appeals to some geometrical claims in order to explain why animal progression necessarily involves the bending (of the limbs), and this appeal to geometrical claims might be taking as violating the recommendation to avoid “kind-crossing” (as found in the Posterior Analytic). But a very unclear notion of kind-crossing has been assumed in most debates. I will argue that kind-crossing in the Posterior Analytics does not mean any employment of premises from (...) a discipline other than that to which the explanandum belongs. Kind-crossing was meant to cover a specific sort of employment of premises from a different discipline, namely, the case in which premises from a discipline X are taken as the most important explanatory factor that delivers the fullest appropriate explanation of an explanandum within discipline Y. If this is so, the employment of geometrical premises in the Incessu Animalium is not an instance of the prohibited kind-crossing, but something that is in line with the theory of the Posterior Analytics. (shrink)

This study analyses the predictions of the General Theory of Relativity (GTR) against a slightly modified version of the standard central mass solution (Schwarzschild solution). It is applied to central gravity in the solar system, the Pioneer spacecraft anomalies (which GTR fails to predict correctly), and planetary orbit distances and times, etc (where GTR is thought consistent.) -/- The modified gravity equation was motivated by a theory originally called ‘TFP’ (Time Flow Physics, 2004). This is now replaced by the ‘Geometric (...) Model’, 2014 [20], which retains the same theory of gravity. This analysis is offered partially as supporting detail for the claim in [20] that the theory is realistic in the solar system and explains the Pioneer anomalies. The overall conclusion is that the model can claim to explain the Pioneer anomalies, contingent on the analysis being independently verified and duplicated of course. -/- However the interest lies beyond testing this theory. To start with, it gives us a realistic scale on which gravity might vary from the accepted theory, remain consistent with most solar-scale astronomical observations. It is found here that the modified gravity equation would appear consistent with GTR for most phenomena, but it would retard the Pioneer spacecraft by about the observed amount (15 seconds or so at time). Hence it is a possible explanation of this anomaly, which as far as I know remains unexplained now for 20 years. -/- It also shows what many philosophers of science have emphasized: the pivotal role of counterfactual reasoning. By putting forward an exact alternative solution, and working through the full explanation, we discover a surprising ‘counterfactual paradox’: the modified theory slightly weakens GTR gravity – and yet the effect is to slow down the Pioneer trajectory, making it appear as if gravity is stronger than GTR. The inference that “there must be some tiny extra force…” (Musser, 1998 [1]) is wrong: there is a second option: “…or there may be a slightly weaker form of gravity than GTR.” . (shrink)

The causal closure of the physical poses a familiar causal exclusion problem for the special sciences that stems from the idea that if closure is true, then fundamental physical properties do all the causal work involved in bringing about physical effects. In this paper I aim to show that the strongest causal closure principle that is not ruled out by some simple physics in fact allows for a certain kind of downward causation, which in turn makes room for robust special (...) science autonomy. I focus on the case of vector composition, arguing that it involves irreducibly multilevel causation. To give a complete causal explanation of what goes on in cases that involve the effects of multiple forces, we need to appeal not only to the fundamental properties that generate those forces, but also to geometric properties that determine how multiple vectors compose. However, if all the causal work involved in such cases were due to fundamental physical properties, there would be a one-level explanation available. Hence, I argue, it cannot be the case that fundamental physical properties do all the causal work. The extra causal work, I suggest, comes from the fact that the causal powers bestowed by fundamental physical properties have irreducibly geometric manifestation conditions. I defend the resulting form of downward causation as a solution to the problem of special science autonomy, and discuss two potential accounts of its source, in terms of the debate between kinematic and dynamic theories of the origins of spacetime symmetries. (shrink)

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to (...) Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation. (shrink)

For an Aristotelian observer, the halo is a puzzling phenomenon since it is apparently sublunary, and yet perfectly circular. This paper studies Aristotle's explanation of the halo in Meteorology III 2-3 as an optical illusion, as opposed to a substantial thing (like a cloud), as was thought by his predecessors and even many successors. Aristotle's explanation follows the method of explanation of the Posterior Analytics for "subordinate" or "mixed" mathematical-physical sciences. The accompanying diagram described by Aristotle is (...) one of the earliest lettered geometrical diagrams, in particular of a terrestrial phenomenon, and versions of it can still be found in modern textbooks on meteorological optics. (shrink)

This paper presents a new approach to resolving an apparent tension in Descartes’ discussion of scientific theories and explanations in the Principles of Philosophy. On the one hand, Descartes repeatedly claims that any theories presented in science must be certain and indubitable. On the other hand, Descartes himself presents an astonishing number of speculative explanations of various scientific phenomena. In response to this tension, commentators have suggested that Descartes changed his mind about scientific theories having to be certain and indubitable, (...) that he lacked the conceptual resources to describe the appropriate epistemic attitude towards speculative theories, or that the presence of geometrical principles in these explanations guarantee their certainty. I argue that none of these responses is satisfactory and suggest a different resolution to the tension by examining Descartes’ notion of explanation. On Descartes’ view, providing an adequate explanation does not require being certain of the theories that constitute the explanans. Relatedly, the purpose of Cartesian explanations is not to discover the truth about the various underlying mechanisms that such explanations appeal to, but to support his general philosophical thesis that all natural phenomena can be explained by appealing to the extension of matter. (shrink)

Robert Batterman’s ontological insights (2002, 2004, 2005) are apt: Nature abhors singularities. “So should we,” responds the physicist. However, the epistemic assessments of Batterman concerning the matter prove to be less clear, for in the same vein he write that singularities play an essential role in certain classes of physical theories referring to certain types of critical phenomena. I devise a procedure (“methodological fundamentalism”) which exhibits how singularities, at least in principle, may be avoided within the same classes of formalisms (...) discussed by Batterman. I show that we need not accept some divergence between explanation and reduction (Batterman 2002), or between epistemological and ontological fundamentalism (Batterman 2004, 2005). Though I remain sympathetic to the ‘principle of charity’ (Frisch (2005)), which appears to favor a pluralist outlook, I nevertheless call into question some of the forms such pluralist implications take in Robert Batterman’s conclusions. It is difficult to reconcile some of the pluralist assessments that he and some of his contemporaries advocate with what appears to be a countervailing trend in a burgeoning research tradition known as Clifford (or geometric) algebra. In my critical chapters (2 and 3) I use some of the demonstrated formal unity of Clifford algebra to argue that Batterman (2002) equivocates a physical theory’s ontology with its purely mathematical content. Carefully distinguishing the two, and employing Clifford algebraic methods reveals a symmetry between reduction and explanation that Batterman overlooks. I refine this point by indicating that geometric algebraic methods are an active area of research in computational fluid dynamics, and applied in modeling the behavior of droplet-formation appear to instantiate a “methodologically fundamental” approach. I argue in my introductory and concluding chapters that the model of inter-theoretic reduction and explanation offered by Fritz Rohrlich (1988, 1994) provides the best framework for accommodating the burgeoning pluralism in philosophical studies of physics, with the presumed claims of formal unification demonstrated by physicists choices of mathematical formalisms such as Clifford algebra. I show how Batterman’s insights can be reconstructed in Rohrlich’s framework, preserving Batterman’s important philosophical work, minus what I consider are his incorrect conclusions. (shrink)

The author presents the history of gravitational waves according to Einstein, linking it to his biography and his time in order to understand it in his connection with the history of the Semites, the personality of Einstein in the handling of his conflict-generating circumstances in his relationships competition with his colleagues and in the formulation of the so-called general theory of relativity. We will fall back on the vicissitudes that Einstein experienced in the transition from his scientific work to normal (...) science as a pillar of theoretical physics. We will deal with how Einstein introduced the relativistic ether, conferring an "odor of materiality" to his geometric explanation of gravity, where undoubtedly it does not fit, but that he had to give in to the pressure that was justified by his most renowned colleagues, led by Lorentz. Einstein had to do it to stay in the queue that would lead him to the Nobel. It was thus, as developing the relativistic ether thread, in June 1916, he introduced the gravitational waves of which, in an act of personal liberation and scientific honesty, when he could, in 1938, he demonstrated how they could not exist, within the scenario of his relativity, to immediately also put an end to the relativistic ether. (shrink)

A semantics of pictorial representation should provide an account of how pictorial signs are associated with the contents they express. Unlike the familiar semantics of spoken languages, this problem has a distinctively spatial cast for depiction. Pictures themselves are two-dimensional artifacts, and their contents take the form of pictorial spaces, perspectival arrangements of objects and properties in three dimensions. A basic challenge is to explain how pictures are associated with the particular pictorial spaces they express. Inspiration here comes from recent (...) proposals that analyze depiction in terms of geometrical projection. In this essay, I will argue that, for a central class of pictures, the projection-based theory of depiction provides the best explanation for how pictures express pictorial spaces, while rival perceptual and resemblance theories fall short. Since the composition of pictorial space is itself the basis for all other aspects of pictorial content, the proposal provides a natural foundation for further pictorial semantics. (shrink)

I offer an alternative account of the relationship of Hobbesian geometry to natural philosophy by arguing that mixed mathematics provided Hobbes with a model for thinking about it. In mixed mathematics, one may borrow causal principles from one science and use them in another science without there being a deductive relationship between those two sciences. Natural philosophy for Hobbes is mixed because an explanation may combine observations from experience (the ‘that’) with causal principles from geometry (the ‘why’). My argument (...) shows that Hobbesian natural philosophy relies upon suppositions that bodies plausibly behave according to these borrowed causal principles from geometry, acknowledging that bodies in the world may not actually behave this way. First, I consider Hobbes's relation to Aristotelian mixed mathematics and to Isaac Barrow's broadening of mixed mathematics in Mathematical Lectures (1683). I show that for Hobbes maker's knowledge from geometry provides the ‘why’ in mixed-mathematical explanations. Next, I examine two explanations from De corpore Part IV: (1) the explanation of sense in De corpore 25.1-2; and (2) the explanation of the swelling of parts of the body when they become warm in De corpore 27.3. In both explanations, I show Hobbes borrowing and citing geometrical principles and mixing these principles with appeals to experience. (shrink)

Although logical consistency is desirable in scientific research, standard statistical hypothesis tests are typically logically inconsistent. To address this issue, previous work introduced agnostic hypothesis tests and proved that they can be logically consistent while retaining statistical optimality properties. This article characterizes the credal modalities in agnostic hypothesis tests and uses the hexagon of oppositions to explain the logical relations between these modalities. Geometric solids that are composed of hexagons of oppositions illustrate the conditions for these modalities to be logically (...) consistent. Prisms composed of hexagons of oppositions show how the credal modalities obtained from two agnostic tests vary according to their threshold values. Nested hexagons of oppositions summarize logical relations between the credal modalities in these tests and prove new relations. (shrink)

In demonstration, speakers use real-world activity both for its practical effects and to help make their points. The demonstrations of origami mathematics, for example, reconfigure pieces of paper by folding, while simultaneously allowing their author to signal geometric inferences. Demonstration challenges us to explain how practical actions can get such precise significance and how this meaning compares with that of other representations. In this paper, we propose an explanation inspired by David Lewis’s characterizations of coordination and scorekeeping in conversation. (...) In particular, we argue that words, gestures, diagrams and demonstrations can function together as integrated ensembles that contribute to conversation, because interlocutors use them in parallel ways to coordinate updates to the conversational record. (shrink)

Cross-linguistic strategies for mapping lexical and spatial relations from body partonym systems to external object meronymies (as in English ‘table leg’, ‘mountain face’) have attracted substantial research and debate over the past three decades. Due to the systematic mappings, lexical productivity and geometric complexities of body-based meronymies found in many Mesoamerican languages, the region has become focal for these discussions, prominently including contrastive accounts of the phenomenon in Zapotec and Tzeltal, leading researchers to question whether such systems should be explained (...) as global metaphorical mappings from bodily source to target holonym or as vector mappings of shape and axis generated “algorithmically”. I propose a synthesis of these accounts in this paper by drawing on the species-specific cognitive affordances of human upright posture grounded in the reorganization of the anatomical planes, with a special emphasis on antisymmetrical relations that emerge between arm-leg and face-groin antinomies cross-culturally. Whereas Levinson argues that the internal geometry of objects “stripped of their bodily associations” (1994: 821) is sufficient to account for Tzeltal meronymy, making metaphorical explanations entirely unnecessary, I propose a more powerful, elegant explanation of Tzeltal meronymic mapping that affirms both the geometric-analytic and the global-metaphorical nature of Tzeltal meaning construal. I do this by demonstrating that the “algorithm” in question arises from the phenomenology of movement and correlative body memories—an experiential ground which generates a culturally selected pair of inverse contrastive paradigm sets with marked and unmarked membership emerging antithetically relative to the transverse anatomical plane. These relations are then selected diagrammatically for the classification of object orientations according to systematic geometric iconicities. Results not only serve to clarify the case in question but also point to the relatively untapped potential that upright posture holds for theorizing the emergence of human cognition, highlighting in the process the nature, origins and theoretical validity of markedness and double scope conceptual integration. (shrink)

Jakob Friedrich Fries (1773-1843): A Philosophy of the Exact Sciences -/- Shortened version of the article of the same name in: Tabula Rasa. Jenenser magazine for critical thinking. 6th of November 1994 edition -/- 1. Biography -/- Jakob Friedrich Fries was born on the 23rd of August, 1773 in Barby on the Elbe. Because Fries' father had little time, on account of his journeying, he gave up both his sons, of whom Jakob Friedrich was the elder, to the Herrnhut Teaching (...) Institution in Niesky in 1778. Fries attended the theological seminar in Niesky in autumn 1792, which lasted for three years. There he (secretly) began to study Kant. The reading of Kant's works led Fries, for the first time, to a deep philosophical satisfaction. His enthusiasm for Kant is to be understood against the background that a considerable measure of Kant's philosophy is based on a firm foundation of what happens in an analogous and similar manner in mathematics. -/- During this period he also read Heinrich Jacobi's novels, as well as works of the awakening classic German literature; in particular Friedrich Schiller's works. In 1795, Fries arrived at Leipzig University to study law. During his time in Leipzig he became acquainted with Fichte's philosophy. In autumn of the same year he moved to Jena to hear Fichte at first hand, but was soon disappointed. -/- During his first sojourn in Jenaer (1796), Fries got to know the chemist A. N. Scherer who was very influenced by the work of the chemist A. L. Lavoisier. Fries discovered, at Scherer's suggestion, the law of stoichiometric composition. Because he felt that his work still need some time before completion, he withdrew as a private tutor to Zofingen (in Switzerland). There Fries worked on his main critical work, and studied Newton's "Philosophiae naturalis principia mathematica". He remained a lifelong admirer of Newton, whom he praised as a perfectionist of astronomy. Fries saw the final aim of his mathematical natural philosophy in the union of Newton's Principia with Kant's philosophy. -/- With the aim of qualifying as a lecturer, he returned to Jena in 1800. Now Fries was known from his independent writings, such as "Reinhold, Fichte and Schelling" (1st edition in 1803), and "Systems of Philosophy as an Evident Science" (1804). The relationship between G. W. F. Hegel and Fries did not develop favourably. Hegel speaks of "the leader of the superficial army", and at other places he expresses: "he is an extremely narrow-minded bragger". On the other hand, Fries also has an unfavourable take on Hegel. He writes of the "Redundancy of the Hegelistic dialectic" (1828). In his History of Philosophy (1837/40) he writes of Hegel, amongst other things: "Your way of philosophising seems just to give expression to nonsense in the shortest possible way". In this work, Fries appears to argue with Hegel in an objective manner, and expresses a positive attitude to his work. -/- In 1805, Fries was appointed professor for philosophy in Heidelberg. In his time spent in Heidelberg, he married Caroline Erdmann. He also sealed his friendships with W. M. L. de Wette and F. H. Jacobi. Jacobi was amongst the contemporaries who most impressed Fries during this period. In Heidelberg, Fries wrote, amongst other things, his three-volume main work New Critique of Reason (1807). -/- In 1816 Fries returned to Jena. When in 1817 the Wartburg festival took place, Fries was among the guests, and made a small speech. 1819 was the so-called "Great Year" for Fries: His wife Caroline died, and Karl Sand, a member of a student fraternity, and one of Fries' former students stabbed the author August von Kotzebue to death. Fries was punished with a philosophy teaching ban but still received a professorship for physics and mathematics. Only after a period of years, and under restrictions, he was again allowed to read philosophy. From now on, Fries was excluded from political influence. The rest of his life he devoted himself once again to philosophical and natural studies. During this period, he wrote "Mathematical Natural Philosophy" (1822) and the "History of Philosophy" (1837/40). -/- Fries suffered from a stroke on New Year's Day 1843, and a second stroke, on the 10th of August 1843 ended his life. -/- 2. Fries' Work Fries left an extensive body of work. A look at the subject areas he worked on makes us aware of the universality of his thinking. Amongst these subjects are: Psychic anthropology, psychology, pure philosophy, logic, metaphysics, ethics, politics, religious philosophy, aesthetics, natural philosophy, mathematics, physics and medical subjects, to which, e.g., the text "Regarding the optical centre in the eye together with general remarks about the theory of seeing" (1839) bear witness. With popular philosophical writings like the novel "Julius and Evagoras" (1822), or the arabesque "Longing, and a Trip to the Middle of Nowhere" (1820), he tried to make his philosophy accessible to a broader public. Anthropological considerations are shown in the methodical basis of his philosophy, and to this end, he provides the following didactic instruction for the study of his work: "If somebody wishes to study philosophy on the basis of this guide, I would recommend that after studying natural philosophy, a strict study of logic should follow in order to peruse metaphysics and its applied teachings more rapidly, followed by a strict study of criticism, followed once again by a return to an even closer study of metaphysics and its applied teachings." -/- 3. Continuation of Fries' work through the Friesian School -/- Fries' ideas found general acceptance amongst scientists and mathematicians. A large part of the followers of the "Fries School of Thought" had a scientific or mathematical background. Amongst them were biologist Matthias Jakob Schleiden, mathematics and science specialist philosopher Ernst Friedrich Apelt, the zoologist Oscar Schmidt, and the mathematician Oscar Xavier Schlömilch. Between the years 1847 and 1849, the treatises of the "Fries School of Thought", with which the publishers aimed to pursue philosophy according to the model of the natural sciences appeared. In the Kant-Fries philosophy, they saw the realisation of this ideal. The history of the "New Fries School of Thought" began in 1903. It was in this year that the philosopher Leonard Nelson gathered together a small discussion circle in Goettingen. Amongst the founding members of this circle were: A. Rüstow, C. Brinkmann and H. Goesch. In 1904 L. Nelson, A. Rüstow, H. Goesch and the student W. Mecklenburg travelled to Thuringia to find the missing Fries writings. In the same year, G. Hessenberg, K. Kaiser and Nelson published the first pamphlet from their first volume of the "Treatises of the Fries School of Thought, New Edition". -/- The school set out with the aim of searching for the missing Fries' texts, and re-publishing them with a view to re-opening discussion of Fries' brand of philosophy. The members of the circle met regularly for discussions. Additionally, larger conferences took place, mostly during the holidays. Featuring as speakers were: Otto Apelt, Otto Berg, Paul Bernays, G. Fraenkel, K. Grelling, G. Hessenberg, A. Kronfeld, O. Meyerhof, L. Nelson and R. Otto. On the 1st of March 1913, the Jakob-Friedrich-Fries society was founded. Whilst the Fries' school of thought dealt in continuum with the advancement of the Kant-Fries philosophy, the members of the Jakob-Friedrich-Fries society's main task was the dissemination of the Fries' school publications. In May/June, 1914, the organisations took part in their last common conference before the gulf created by the outbreak of the First World War. Several members died during the war. Others returned disabled. The next conference took place in 1919. A second conference followed in 1921. Nevertheless, such intensive work as had been undertaken between 1903 and 1914 was no longer possible. -/- Leonard Nelson died in October 1927. In the 1930's, the 6th and final volume of "Treatises of the Fries School of Thought, New Edition" was published. Franz Oppenheimer, Otto Meyerhof, Minna Specht and Grete Hermann were involved in their publication. -/- 4. About Mathematical Natural Philosophy -/- In 1822, Fries' "Mathematical Natural Philosophy" appeared. Fries rejects the speculative natural philosophy of his time - above all Schelling's natural philosophy. A natural study, founded on speculative philosophy, ceases with its collection, arrangement and order of well-known facts. Only a mathematical natural philosophy can deliver the necessary explanatory reasoning. The basic dictum of his mathematical natural philosophy is: "All natural theories must be definable using purely mathematically determinable reasons of explanation." Fries is of the opinion that science can attain completeness only by the subordination of the empirical facts to the metaphysical categories and mathematical laws. -/- The crux of Fries' natural philosophy is the thought that mathematics must be made fertile for use by the natural sciences. However, pure mathematics displays solely empty abstraction. To be able to apply them to the sensory world, an intermediatory connection is required. Mathematics must be connected to metaphysics. The pure mechanics, consisting of three parts are these: a) A study of geometrical movement, which considers solely the direction of the movement, b) A study of kinematics, which considers velocity in Addition, c) A study of dynamic movement, which also incorporates mass and power, as well as direction and velocity. -/- Of great interest is Fries' natural philosophy in view of its methodology, particularly with regard to the doctrine "leading maxims". Fries calls these "leading maxims" "heuristic", "because they are principal rules for scientific invention". -/- Fries' philosophy found great recognition with Carl Friedrich Gauss, amongst others. Fries asked for Gauss's opinion on his work "An Attempt at a Criticism based on the Principles of the Probability Calculus" (1842). Gauss also provided his opinions on "Mathematical Natural Philosophy" (1822) and on Fries' "History of Philosophy". Gauss acknowledged Fries' philosophy and wrote in a letter to Fries: "I have always had a great predilection for philosophical speculation, and now I am all the more happy to have a reliable teacher in you in the study of the destinies of science, from the most ancient up to the latest times, as I have not always found the desired satisfaction in my own reading of the writings of some of the philosophers. In particular, the writings of several famous (maybe better, so-called famous) philosophers who have appeared since Kant have reminded me of the sieve of a goat-milker, or to use a modern image instead of an old-fashioned one, of Münchhausen's plait, with which he pulled himself from out of the water. These amateurs would not dare make such a confession before their Masters; it would not happen were they were to consider the case upon its merits. I have often regretted not living in your locality, so as to be able to glean much pleasurable entertainment from philosophical verbal discourse." -/- The starting point of the new adoption of Fries was Nelson's article "The critical method and the relation of psychology to philosophy" (1904). Nelson dedicates special attention to Fries' re-interpretation of Kant's deduction concept. Fries awards Kant's criticism the rationale of anthropological idiom, in that he is guided by the idea that one can examine in a psychological way which knowledge we have "a priori", and how this is created, so that we can therefore recognise our own knowledge "a priori" in an empirical way. Fries understands deduction to mean an "awareness residing darkly in us is, and only open to basic metaphysical principles through conscious reflection.". -/- Nelson has pointed to an analogy between Fries' deduction and modern metamathematics. In the same manner, as with the anthropological deduction of the content of the critical investigation into the metaphysical object show, the content of mathematics become, in David Hilbert's view, the object of metamathematics. -/-. (shrink)

Wolfgang Pauli was influenced by Carl Jung and the Platonism of Arnold Sommerfeld, who introduced the fine-structure constant. Pauli’s vision of a World Clock is related to the symbolic form of the Emerald Tablet of Hermes and Plato’s geometric allegory otherwise known as the Cosmological Circle attributed to ancient tradition. With this vision Pauli revealed geometric clues to the mystery of the fine-structure constant that determines the strength of the electromagnetic interaction. A Platonic interpretation of the World Clock and the (...) Cosmological Circle provides an explanation that includes the geometric structure of the pineal gland described by the golden ratio. In his experience of archetypal images Pauli encounters the synchronicity of events that contribute to his quest for physical symmetry relevant to the development of quantum electrodynamics. (shrink)

What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the transfinite deadlock of (...) higher set theory the jewel of mathematical Continuum -- this genuine, even if mostly forgotten today raison d'etre of all set-theoretical enterprises to Infinity and beyond, from Georg Cantor to W. Hugh Woodin to Buzz Lightyear, by simultaneously exhibiting the limits and pitfalls of all old and new reductionist foundational approaches to mathematical truth: be it Cantor's or post-Cantorian Idealism, Brouwer's or post-Brouwerian Constructivism, Hilbert's or post-Hilbertian Formalism, Goedel's or post-Goedelian Platonism. -/- In the spirit of Zeno's paradoxes, but with the enormous historical advantage of hindsight, we claim that Cantor's set-theoretical methodology, powerful and reach in proof-theoretic and similar applications as it might be, is inherently limited by its epistemological framework of transfinite local causality, and neither can be held accountable for the properties of the Continuum already acquired through geometrical, analytical, and arithmetical studies, nor can it be used for an adequate, conceptually sensible, operationally workable, and axiomatically sustainable re-creation of the Continuum. -/- From a strictly mathematical point of view, this intrinsic limitation of the constative and explicative power of higher set theory finds its explanation in the identified in this study ultimate phenomenological obstacle to Cantor's transfinite construction, similar to topological obstacles in homotopy theory and theoretical physics: the entanglement capacity of the mathematical Continuum. (shrink)

This article draws on several crucial and unpublished manuscripts from the Scholem Archive in exploration of Gershom Scholem's youthful statements on mathematics and its relation to extra-mathematical facts and, more broadly, to a concept of history that would prove to be consequential for Walter Benjamin's own thinking on "messianism" and a "futuristic politics." In context of critiquing the German Youth Movement's subsumption of active life to the nationalistic conditions of the "earth" during the First World War, Scholem turns to mathematics (...) for a genuine and self-consistent theory of action. In the concept of actual infinity (in Cantor and Bolzano) he finds an explanation of how mathematics relates to "the physical" without reducing the former to an "image" of the latter, and without relying on the concept of geometric intuition. This explanation, insofar as it relies on the notion of actual infinity, provides Scholem with a conception of mathematics (and the history of mathematics) that reconciles freedom and necessity—remarks on which he outlines in his diaries and communicates to Benjamin in early March 1916. (shrink)

After sketching the historical development of “emergence” and noting several recent problems relating to “emergent properties”, this essay proposes that properties may be either “emergent” or “mergent” and either “intrinsic” or “extrinsic”. These two distinctions define four basic types of change: stagnation, permanence, flux, and evolution. To illustrate how emergence can operate in a purely logical system, the Geometry of Logic is introduced. This new method of analyzing conceptual systems involves the mapping of logical relations onto geometrical figures, following (...) either an analytic or a synthetic pattern (or both together). Evolution is portrayed as a form of discontinuous change characterized by emergent properties that take on an intrinsic quality with respect to the object(s) or proposition(s) involved. Causal leaps, not continuous development, characterize the evolution of human life in a developing foetus, of a thought out of certain brain states, of a new idea (or insight) out of ordinary thoughts, and of a great person out of a set of historical experiences. The tendency to assume that understanding evolutionary change requires a step-by-step explanation of the historical development that led to the appearance of a certain emergent property is thereby discredited. (shrink)

When faced with uncertainty, consequentialists often advocate choosing the option with the largest expected utility, as calculated using the arithmetic average. I provide some arguments to suggest that instead, one should consider choosing the option with the largest geometric average of utility. I explore the difference between these two approaches in a variety of ethical dilemmas and argue that geometric averaging has some appealing properties as a normative decision-making tool.

The ontic conception of scientific explanation has been constructed and motivated on the basis of a putative lexical ambiguity in the term explanation. I raise a puzzle for this ambiguity claim, and then give a deflationary solution under which all ontically-rendered talk of explanation is merely elliptical; what it is elliptical for is a view of scientific explanation that altogether avoids the ontic conception. This result has revisionary consequences for New Mechanists and other philosophers of science, (...) many of whom have assimilated their conception of explanation to the ontic conception. (shrink)

"Explanation and Understanding" (1971) by Georg Henrik von Wright is a modern classic in analytic hermeneutics, and in the philosophy of the social sciences and humanities in general. In this work, von Wright argues against naturalism, or methodological monism, i.e. the idea that both the natural sciences and the social sciences follow broadly the same general scientific approach and aim to achieve causal explanations. Against this view, von Wright contends that the social sciences are qualitatively different from the natural (...) sciences: according to his view, the natural sciences aim at causal explanations, whereas the purpose of the social sciences is to understand their subjects. In support of this conviction, von Wright also puts forward a version of the so-called logical connection argument. -/- Von Wright views scientific explanation along the lines of the traditional covering law model. He suggests that the social sciences, in contrast, utilize what he calls “practical syllogism” in understanding human actions. In addition, von Wright presents in this work an original picture on causation: a version of the manipulability theory of causation. -/- In the four decades following von Wright’s classic work, the overall picture in in the philosophy of science has changed significantly, and much progress has been made in various fronts. The aim of the contribution is to revisit the central ideas of "Explanation and Understanding" and evaluate them from this perspective. The covering law model of explanation and the regularity theory of causation behind it have since then fallen into disfavor, and virtually no one believes that causal explanations even in the natural sciences comply with the covering law model. No wonder then that covering law explanations are not found in the social sciences either. Ironically, the most popular theory of causal explanation in the philosophy of science nowadays is the interventionist theory, which is a descendant of the manipulability theory of von Wright and others. However, this theory can be applied with no special difficulties in both the natural sciences and the social sciences. -/- Von Wright’s logical connection argument and his ideas concerning practical syllogisms are also critically assessed. It is argued that in closer scrutiny, they do not pose serious problems for the view that the social sciences too provide causal explanations. In sum, von Wright’s arguments against naturalism do not appear, in today’s perspective, particularly convincing. (shrink)

We argue that the trend toward providing users with feasible and actionable explanations of AI decisions—known as recourse explanations—comes with ethical downsides. Specifically, we argue that recourse explanations face several conceptual pitfalls and can lead to problematic explanation hacking, which undermines their ethical status. As an alternative, we advocate that explanations of AI decisions should aim at understanding.

In a recent book and an article, Carl Craver construes the relations between different levels of a mechanism, which he also refers to as constitutive relations, in terms of mutual manipulability (MM). Interpreted metaphysically, MM implies that inter-level relations are symmetrical. MM thus violates one of the main desiderata of scientific explanation, namely explanatory asymmetry. Parts of Craver’s writings suggest a metaphysical interpretation of MM, and Craver explicitly commits to constitutive relationships being symmetrical. The paper furthermore explores the option (...) of interpreting MM epistemologically, as a means for individuating mechanisms. It is argued that MM then is redundant. MM should therefore better be abandoned. (shrink)

The purpose of this work is to address what notion of geometrical object and geometrical figure we have in different kinds of geometry: practical, pure, and applied. Also, we address the relation between geometrical objects and figures when this is possible, which is the case of pure and applied geometry. In practical geometry it turns out that there is no conception of geometrical object.

Much of our information comes to us indirectly, in the form of conclusions others have drawn from evidence they gathered. When we hear these conclusions, how can we modify our own opinions so as to gain the benefit of their evidence? In this paper we study the method known as geometric pooling. We consider two arguments in its favour, raising several objections to one, and proposing an amendment to the other.

Supra-Bayesianism is the Bayesian response to learning the opinions of others. Probability pooling constitutes an alternative response. One natural question is whether there are cases where probability pooling gives the supra-Bayesian result. This has been called the problem of Bayes-compatibility for pooling functions. It is known that in a common prior setting, under standard assumptions, linear pooling cannot be nontrivially Bayes-compatible. We show by contrast that geometric pooling can be nontrivially Bayes-compatible. Indeed, we show that, under certain assumptions, geometric and (...) Bayes-compatible pooling are equivalent. Granting supra-Bayesianism its usual normative status, one upshot of our study is thus that, in a certain class of epistemic contexts, geometric pooling enjoys a normative advantage over linear pooling as a social learning mechanism. We discuss the philosophical ramifications of this advantage, which we show to be robust to variations in our statement of the Bayes-compatibility problem. (shrink)

Due to the wide array of phenomena that are of interest to them, psychologists offer highly diverse and heterogeneous types of explanations. Initially, this suggests that the question "What is psychological explanation?" has no single answer. To provide appreciation of this diversity, we begin by noting some of the more common types of explanations that psychologists provide, with particular focus on classical examples of explanations advanced in three different areas of psychology: psychophysics, physiological psychology, and information-processing psychology. To analyze (...) what is involved in these types of explanations, we consider the ways in which law-like representations of regularities and representations of mechanisms factor in psychological explanations. This consideration directs us to certain fundamental questions, e.g., "To what extent are laws necessary for psychological explanations?" and "What do psychologists have in mind when they appeal to mechanisms in explanation?" In answering such questions, it appears that laws do play important roles in psychological explanations, although most explanations in psychology appeal to accounts of mechanisms. Consequently, we provide a unifying account of what psychological explanation is. (shrink)

The bridges of Königsberg case has been widely cited in recent philosophical discussions on scientific explanation as a potential example of a structural explanation of a physical phenomenon. However, when discussing this case, different authors have focused on two different versions, depending on what they take the explanandum to be. In one version, the explanandum is the _failure_ of a given individual in performing an Eulerian walk over the bridge system. In the other version, the explanandum is the (...) _impossibility_ of performing an Eulerian walk over the bridges. The goal of this paper is to show that only the latter version amounts to a real case of a structural explanation. I will also suggest how to fix the first version, and show how my remarks apply to other purported cases of structural explanations. (shrink)

This article defends the principle of sufficient reason (PSR) from a simple and direct valid argument according to which PSR implies that there is a truth that explains every truth, namely an omni-explainer. Many proponents of PSR may be willing to bite the bullet and maintain that, if PSR is true, then there is an omni-explainer. I object to this strategy by defending the principle that explanation is irreflexive. Then I argue that proponents of PSR can resist the conclusion (...) that there is an omni-explainer by denying that explanation distributes over conjunction. In the final section, I consider a plausible revision of this assumption. I argue there that, given the revised assumption, PSR seems to yield a striking picture of the explanatory structure of the universe. I explain why the resulting model does not appear utterly implausible to me. (shrink)

Scientific explanations must bear the proper relationship to the world: they must depict what, out in the world, is responsible for the explanandum. But explanations must also bear the proper relationship to their audience: they must be able to create human understanding. With few exceptions, philosophical accounts of explanation either ignore entirely the relationship between explanations and their audience or else demote this consideration to an ancillary role. In contrast, I argue that considering an explanation’s communicative role is (...) crucial to any satisfactory account of explanation. (shrink)

A compelling idea holds that reality has a layered structure. We often disagree about what inhabits the bottom layer, but we agree that higher up we find chemical, biological, geological, psychological, sociological, economic, /etc./, entities: molecules, human beings, diamonds, mental states, cities, interest rates, and so on. How is this intuitive talk of a layered structure of entities to be understood? Traditionally, philosophers have proposed to understand layered structure in terms of either reduction or supervenience. But these traditional views face (...) well-known problems. A plausible alternative is that layered structure is to be explicated by appeal to explanations of a certain sort, termed / grounding explanations/. Grounding explanations tell us what obtains in virtue of what. Unfortunately, the use of grounding explanations to articulate the layered conception faces a problem, which I call /the collapse/. The collapse turns on the question of how to ground the facts stated by the explanations themselves. In this paper I make a suggestion about how to ground explanations that avoids the collapse. Briefly, the suggestion is that the fact stated by a grounding explanation is grounded in its /explanans/. (shrink)

In this chapter, I first outline the public health workflow from assessment via goal definition and intervention to evaluation. Further, I discuss the types and subtypes of explanation used in public health research and practice: scientific, justificatory, methodological, and prospective. In doing this, I take the discussion far beyond the usual focus in philosophy of science as answers to “why?”-questions. The chapter ends with a few comments on my proposal.

Recent work on interpretability in machine learning and AI has focused on the building of simplified models that approximate the true criteria used to make decisions. These models are a useful pedagogical device for teaching trained professionals how to predict what decisions will be made by the complex system, and most importantly how the system might break. However, when considering any such model it’s important to remember Box’s maxim that "All models are wrong but some are useful." We focus on (...) the distinction between these models and explanations in philosophy and sociology. These models can be understood as a "do it yourself kit" for explanations, allowing a practitioner to directly answer "what if questions" or generate contrastive explanations without external assistance. Although a valuable ability, giving these models as explanations appears more difficult than necessary, and other forms of explanation may not have the same trade-offs. We contrast the different schools of thought on what makes an explanation, and suggest that machine learning might benefit from viewing the problem more broadly. (shrink)

Some properties are causally relevant for a certain effect, others are not. In this paper we describe a problem for our understanding of this notion and then offer a solution in terms of the notion of a program explanation.

I introduce an argument for Platonism based on intra-mathematical explanation: the explanation of one mathematical fact by another. The argument is important for two reasons. First, if the argument succeeds then it provides a basis for Platonism that does not proceed via standard indispensability considerations. Second, if the argument fails it can only do so for one of three reasons: either because there are no intra-mathematical explanations, or because not all explanations are backed by dependence relations, or because (...) some form of noneism-the view according to which non-existent entities possess properties and stand in relations-is true. The argument thus forces a choice between nominalism without noneism, intra-mathematical explanation and a backing conception of explanation. You can have any two, but not all three. (shrink)

How is knowledge of geometry developed and acquired? This central question in the philosophy of mathematics has received very different answers. Spelke and colleagues argue for a “core cognitivist”, nativist, view according to which geometric cognition is in an important way shaped by genetically determined abilities for shape recognition and orientation. Against the nativist position, Ferreirós and García-Pérez have argued for a “culturalist” account that takes geometric cognition to be fundamentally a culturally developed phenomenon. In this paper, I argue that (...) when understood as moderate versions supported by the state-of-the-art research, the nativist and culturalist views are in fact possible to reconcile. While Ferreirós and García-Pérez present the work of Spelke and colleagues as implying that geometric cognition is genetically determined, I argue that they fail to appreciate the role that Spelke and colleagues see for cultural factors. On this basis, I provide theoretical and terminological clarifications and show that moderate versions of the nativist and culturalist view are in fact consistent with each other. I then propose a unifying theoretical framework for future study that can integrate the two accounts in ontogeny by moving beyond the crude nature (nativism) vs. nurture (culturalism) dichotomy. (shrink)

This study presents a new type of foundational model unifying quantum theory, relativity theory and gravitational physics, with a novel cosmology. It proposes a six-dimensional geometric manifold as the foundational ontology for our universe. The theoretical unification is simple and powerful, and there are a number of novel empirical predictions and theoretical reductions that are strikingly accurate. It subsequently addresses a variety of current anomalies in physics. It shows how incomplete modern physics is by giving an example of a theory (...) that is genuinely unified. In doing this, it radically alters the metaphysical interpretation of the nature of time, space and matter currently interpreted from modern physics. It also profoundly challenges materialist expectations about a naturalistic account our own existence. I contend here that there is sufficient evidence to support this theory as a leading paradigm for a unified foundational theory. (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)

Humeanism about laws of nature — the view that the laws reduce to the Humean mosaic — is a popular view, but currently existing versions face powerful objections. The non-supervenience objection, the non-fundamentality objection and the explanatory circularity objection have all been thought to cause problems for the Humean. However, these objections share a guiding thought — they are all based on the idea that there is a certain kind of divergence between the practice of science and the metaphysical picture (...) suggested by Humeanism. -/- I suggest that the Humean can respond to these objections not by rejecting this divergence, but by arguing that is appropriate. In particular the Humean can, in the spirit of Loewer (2012), distinguish between scientific and metaphysical explanation — this is motivated by differing aims of explanation in science and metaphysics. And they can further leverage this into distinctions between scientific and metaphysical fundamentality and scientific and metaphysical possibility. We can use these distinctions to respond to the objections that the Humean faces. (shrink)

I discuss the methodological passage in the begin- ning of Ethica Eudemia I.6 (1216b26-35), which has received attention in connection with Aristotle’s notion of dialectic and his methodology in Ethics. My central focus is not to discuss whether Aristotle is prescribing and using what has been called the method of endoxa. I will focus on how this passage coheres with the remaining parts of the same chapter, which also are advancing methodological remarks. My claim is that the meth- od of (...) Ethica Eudemia I.6 is in agreement with many features of Aristotle’s theory of explanation as presented in the Posterior Analytics: Aristotle’s main concern is a warning against misuses of explanatory arguments. (shrink)

This book introduces readers to the topic of explanation. The insights of Plato, Aristotle, J.S. Mill and Carl Hempel are examined, and are used to argue against the view that explanation is merely a problem for the philosophy of science. Having established its importance for understanding knowledge in general, the book concludes with a bold and original explanation of explanation.

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic (...) theory faces. The final view appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

This paper offers a new account of metaphysical explanation. The account is modelled on Kitcher’s unificationist approach to scientific explanation. We begin, in Sect. 2, by briefly introducing the notion of metaphysical explanation and outlining the target of analysis. After that, we introduce a unificationist account of metaphysical explanation before arguing that such an account is capable of capturing four core features of metaphysical explanations: irreflexivity, non-monotonicity, asymmetry and relevance. Since the unificationist theory of metaphysical (...) class='Hi'>explanation inherits irreflexivity and non-monotonicity directly from the unificationist theory of scientific explanation that underwrites it, we focus on demonstrating how the account can secure asymmetry and relevance. (shrink)

Ground is all the rage in contemporary metaphysics. But what is its nature? Some metaphysicians defend what we could call, following Skiles and Trogdon (2021), the inheritance view: it is because constitutive forms of metaphysical explanation are such-and-such that we should believe that ground is so-and-so. However, many putative instances of inheritance are not primarily motivated by scientific considerations. This limitation is harmless if one thinks that ground and science are best kept apart. Contrary to this view, we believe (...) that ground is a highly serviceable tool for investigating metaphysical areas of science. In this paper, we defend a naturalistic version of the inheritance view which takes constitutive scientific explanation as a better guide to ground. After illustrating the approach and its merits, we discuss some implications of the emerging scientific conception for the theory of ground at large. (shrink)

The ontic conception of explanation is predicated on the proposition that “explanation is a relation between real objects in the world” and hence, according to this approach, scientific explanation cannot take place absent such a premise. Despite the fact that critics have emphasized several drawbacks of the ontic conception, as for example its inability to address the so-called “abstract explanations”, the debate is not settled and the ontic view can claim to capture cases of explanation that (...) are non-abstract, such as causal relations between events. However, by eliminating the distinction between abstract and non-abstract explanations, it follows that ontic and epistemic proposals can no longer contend to capture different cases of explanation and either all are captured by the ontic view or all are captured by the epistemic view. On closer inspection, it turns out that the ontic view deals with events that fall outside the scientists’ scope of observation and that it does not accommodate common instances of explanation such as explanations from false propositions and hence it cannot establish itself as the dominant philosophical stance with respect to explanation. On the contrary, the epistemic conception does account for almost all episodes of explanation and can be described as a relation between representations, whereby the explanans transmit information to the explanandum and that this information can come, dependent on context, in the form of any of the available theories of explanation (law-like, unificatory, causal and non-causal). The range of application of the ontic view thus is severely restricted to trivial cases of explanation that come through direct observation of the events involved in an explanation and explanation is to be mostly conceived epistemically. (shrink)

Normative explanations of why things are wrong, good, or unfair are ubiquitous in ordinary practice and normative theory. This paper argues that normative explanation is subject to a justification condition: a correct complete explanation of why a normative fact holds must identify features that would go at least some way towards justifying certain actions or attitudes. I first explain and motivate the condition I propose. I then support it by arguing that it fits well with various theories of (...) normative reasons, makes good sense of certain legitimate moves in ordinary normative explanatory discourse, and helps to make sense of our judgments about explanatory priority in certain cases of normative explanation. This last argument also helps to highlight respects in which normative explanation won’t be worryingly discontinuous with explanations in other domains even though these other explanations aren’t subject to the justification condition. Thus the paper aims not only to do some constructive theorizing about the relatively neglected topic of normative explanation but also to cast light on the broader question of how normative explanation may be similar to and different from explanations in other domains. (shrink)

It is often thought that metaphysical grounding underwrites a distinctive sort of metaphysical explanation. However, it would be a mistake to think that all metaphysical explanations are underwritten by metaphysical grounding. In service of this claim, I offer a novel kind of metaphysical explanation called metaphysical explanation by constraint, examples of which have been neglected in the literature. I argue that metaphysical explanations by constraint are not well understood as grounding explanations.