Kant was an idealist. His idealism was in some ways, it is true, less extreme than that of Berkeley. He distinguished his own by calling it ‘transcendental’. It is less extreme than Berkeley's in two ways. First, Kant does not assert that everything which exists is essentially mental, as Berkeley does. Second, those things which he does hold to be essentially mental, he holds to be so in a weaker fashion. Nevertheless he was an idealist; he held that neither intuition (...) nor thought could concern any object that was not essentially related to our minds. Since intuition and thought together provide knowledge, and since we can have no knowledge except through them, it follows that every object of which we have knowledge is essentially within our minds. Moreover, there is at least one respect in which his idealism is more extreme than that of Berkeley. He held that the objects of intuition were essentially within our minds, whereas Berkeley held only that they were essentially within some mind. (shrink)

Kant was an idealist. His idealism was in some ways, it is true, less extreme than that of Berkeley. He distinguished his own by calling it ‘transcendental’. It is less extreme than Berkeley's in two ways. First, Kant does not assert that everything which exists is essentially mental, as Berkeley does. Second, those things which he does hold to be essentially mental, he holds to be so in a weaker fashion. Nevertheless he was an idealist; he held that neither intuition (...) nor thought could concern any object that was not essentially related to our minds. Since intuition and thought together provide knowledge, and since we can have no knowledge except through them, it follows that every object of which we have knowledge is essentially within our minds. Moreover, there is at least one respect in which his idealism is more extreme than that of Berkeley. He held that the objects of intuition were essentially within our minds, whereas Berkeley held only that they were essentially within some mind. (shrink)

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 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.

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)

I begin by reviewing some recent work on the status of the geodesic principle in general relativity and the geometrized formulation of Newtonian gravitation. I then turn to the question of whether either of these theories might be said to ``explain'' inertial motion. I argue that there is a sense in which both theories may be understood to explain inertial motion, but that the sense of ``explain'' is rather different from what one might have expected. This sense of explanation (...) is connected with a view of theories---I call it the ``puzzleball view''---on which the foundations of a physical theory are best understood as a network of mutually interdependent principles and assumptions. (shrink)

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.

In this paper, we focus on the development of geometric cognition. We argue that to understand how geometric cognition has been constituted, one must appreciate not only individual cognitive factors, such as phylogenetically ancient and ontogenetically early core cognitive systems, but also the social history of the spread and use of cognitive artifacts. In particular, we show that the development of Greek mathematics, enshrined in Euclid’s Elements, was driven by the use of two tightly intertwined cognitive artifacts: the use of (...) lettered diagrams; and the creation of linguistic formulae. Together, these artifacts formed the professional language of geometry. In this respect, the case of Greek geometry clearly shows that explanations of geometric reasoning have to go beyond the confines of methodological individualism to account for how the distributed practice of artifact use has stabilized over time. This practice, as we suggest, has also contributed heavily to the understanding of what mathematical proof is; classically, it has been assumed that proofs are not merely deductively correct but also remain invariant over various individuals sharing the same cognitive practice. Cognitive artifacts in Greek geometry constrained the repertoire of admissible inferential operations, which made these proofs inter-subjectively testable and compelling. By focusing on the cognitive operations on artifacts, we also stress that mental mechanisms that contribute to these operations are still poorly understood, in contrast to those mechanisms which drive symbolic logical inference. (shrink)

We examine critically the interdependence between science and philosophy which Sklar asserts in Space, Time, and Spacetime. We find that such a view makes it difficult to criticize the ideas of science, like that of absolute space, on their own merits, without importing extraneous philosophical associations. It also impedes appreciation of the importance, and subtlety, of explanation in scientific theory. As a result, particular explanations, such as the one Newton offered of his bucket experiment, are dismissed facilely-- indeed, all (...) geometric explanation appears illegitimate--, and such general attitudes as conventionalism appear to lack rationale. (shrink)

This paper seeks to demonstrate that propositions 23–27 of the Euclidian Optics originated in the context of geometrical astronomy. These entries, which deal with the geometry of spheres and rays, present material that overlaps considerably with propositions 1–3 of Aristarchus of Samos’ On the Sizes and Distances of the Sun and the Moon. While all these theorems deal with material that could conceivably be native to celestial illumination, the proofs do not work for binocular vision. It therefore seems probable (...) that the proofs were borrowed from Aristarchus or, more likely, some common astronomical predecessor. As an extension of this observation, this paper argues that the Optics displays a far less unified purpose than has typically been assumed. Instead, it reflects a conglomerate of concerns, goals and dependencies, insofar as it presents multiple proofs that emerge from a set of geometrical concerns not directly related to vision as a physical phenomenon. These entries complicate the idea that the Optics is about explaining light, sight or false appearances in any straightforward sense. Instead, the text is oriented toward a somewhat more diffuse goal: simply to articulate the geometry of vision and rays. Optics as a discipline should should therefore be understood not as a purely mathematical explanation of vision, but as a target-field of explanation constituted by features and boundaries derived from extra-visual geometric concerns. (shrink)

I discuss the debate between dynamical versus geometrical approaches to spacetime theories, in the context of both special and general relativity, arguing that the debate takes a substantially different form in the two cases; different versions of the geometrical approach—only some of which are viable—should be distinguished; in general relativity, there is no difference between the most viable version of the geometrical approach and the dynamical approach. In addition, I demonstrate that what have previously been dubbed two (...) ‘miracles’ of general relativity admit of no resolution from within general relativity, on either the dynamical or ‘qualified’ geometrical approaches, modulo some possible hints that the second ‘miracle’ may be resolved by appeal to recent results regarding the ‘geodesic principle’ in GR. (shrink)

Pure mathematics can play an indispensable role explaining empirical phenomena if recent accounts of insect evolution are correct. In particular, the prime life cycles of cicadas and the geometric structure of honeycombs are taken to undergird an inference to the best explanation about mathematical entities. Neither example supports this inference or the mathematical realism it is intended to establish. Both incorrectly assume that facts about mathematical optimality drove selection for the respective traits and explain why they exist. We show (...) how this problem can be avoided, identify limitations of explanatory indispensability arguments, and attempt to clarify the nature of mathematical explanation. (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)

In his writings Philosophical Remarks, the Austrian-British Philosopher Ludwig Wittgenstein draws an octahedron with the words of pure colours such as “white”, “red” and “blue” at the corners and argues: “The colour octahedron is grammar, since it says that you can speak of a reddish blue but not of a reddish green, etc”. He uses the word “grammar” in such a specific way that the grammar or grammatical rules describe the meanings of words/expressions, in other words, how we use them (...) in our language. Accordingly, the colour octahedron can also be taken to represent grammatical rules about how we apply words of colour, e.g., that we can call a certain colour “reddish-blue”, but not “reddish-green”. In a different context, the Japanese philosopher Shūzō Kuki explores in his work The Structure of Iki what the Japanese word “iki” means. This word is often translated as “chic” or “stylistic” in English, but Kuki holds that it is an aesthetic Japanese concept that cannot be translated one-to-one, instead encompassing three aspects: “coquetry”, “pride and honour” and “resignation”. To explain the meanings of the word “iki” and other related words all of which Kuki calls “tastes”, he introduces a rectangular prism as a geometrical representation similar to Wittgenstein’s colour octahedron. In this paper, I argue that the rectangular prism does not solely explain how the modes of Japanese tastes are related to each other, but also has a grammatical character. On this score, I suggest that one can regard this rectangular prism as a description of the grammatical rules of the Japanese language. By appeal to the arguments of both philosophers and in comparison with them, I will not only clarify what they claim by geometrical representations but also examine what role this kind of representation plays as an explanation of grammar in general. Keywords: grammar, colour octahedron, rectangular prism, Shūzō Kuki, Wittgenstein. (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)

Philosophers have paid little attention to mathematical explanations . I present a variety of examples of mathematical explanation and examine two cases in detail. I argue that mathematical explanations have important implications for the philosophy of mathematics and of science. ;The first case study compares many proofs of Pick's theorem, a simple geometrical result. Though a simple proof surfaces to establish the result, some of the proofs explain the result better than others. The second case study comes from (...) George Polya's Mathematics and Plausible Reasoning. He gives a proof that, while entirely satisfactory in establishing its conclusion, is insufficiently explanatory. To provide a better explanation, he supplements the proof with additional exposition. ;These case studies illustrate at least two distinct explanatory virtues, and suggest there may be more. First, an explanatory improvement occurs when a sense of "arbitrariness" is reduced in the proofs. Proofs more explanatory in this way place greater restrictions on the steps that can be used to reach the conclusion. Second, explanatoriness is judged by directness of representation. More explanatory proofs allow one to ascribe geometric meaning to the terms of Pick's formula as they arise. ;I trace the lack of attention to mathematical explanations to an implicit assumption, justificationism, that only justificational aspects of mathematical reasoning are epistemically important. I propose an anti-justificationist epistemic position, the epistemic virtues view, which holds that justificational virtues, while important, are not the only ones of philosophical interest in mathematics. Indeed, explanatory benefits are rarely justificational. I show how the epistemic virtues view and the recognition of mathematical explanation can shed new light on philosophical debates. ;Mathematical explanations have consequences for philosophy of science as well. I show that mathematical explanations provide serious challenges to any theory, such as Bas van Fraassen's, that considers explanations to be fundamentally answers to why-questions. I urge a closer interaction between philosophy of mathematics and philosophy of science; both will be needed for a fuller understanding of mathematical explanation. (shrink)

In scientific explanations, the explanans theory is sometimes incommensurable with the explanandum empirical data. How is this possible, especially when the explanation is deductive in nature? This paper attempts to solve the puzzle without relying on any particular theory of reference. For us, it is rather obvious that the geometric idea of projection plays a key role in Keplers explanation of Tycho Brahes empirical data. We discover that a similar mechanism operates in theoretic explanations in general. In short, (...) all theoretic explanations are projective explanations. If so, there should be no logical reason why explanans theories cannot be incommensurable with explanandum data. For illustration, we analyse Einsteins explanation of the results of the Michelson–Morley experiment in some detail. (shrink)

Sen's classic social choice result supposedly demonstrates a conflict between Pareto and even minimal forms of liberalism. By providing the first direct mathematical proof of this seminal result, we underscore a significantly different interpretation: rather than conflicts among rights, Sen's result occurs because the liberalism assumption negates the assumption that voters have transitive preferences. This explanation enriches interpretations of Sen's conclusion by including radically new kinds of societal conflicts, it suggests ways to sidestep these difficulties, and it explains earlier (...) approaches to avoid the difficulties. (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)

Late ancient Platonists discuss two theories in which geometric entities xplain natural phenomena : the regular polyhedra of geometric atomism and the ccentrics and epicycles of astronomy. Simplicius explicitly compares the status of the first to the hypotheses of the astronomers. The point of omparison is the fallibility of both theories, not the reality of the entities postulated. Simplicius has strong realist commitments as far as astronomy is concerned. Syrianus and Proclus, too, do not consider the polyhedra as devoid of (...) physical reality. Proclus rejects epicycles and eccentrics, but accepts the reality of material homocentric spheres, moved by their own souls. The spheres move the astral objects contained in them, which, however, add motions caused by their own souls. The epicyclical and eccentric hypotheses are useful, as they help us to understand the complex motions resulting from the interplay of spherical motions and volitional motions of the planets. Yet astral souls do not think in accordance with human theoretical constructs, but rather grasp the complex patterns of their motions directly. Our understanding of astronomy depends upon our own cognition of intelligible patterns and their mathematical images. (shrink)

ABSTRACTThis paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on ‘algebraical geometry’ and ‘geometrical algebra’ in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but (...) also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the ‘abstract algebra’ and ‘abstract geometry’ of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to ‘algebraical geometry’ and ‘geometrical algebra’ of the second generation of ‘scientific’ symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s–1840s. (shrink)

A mechanistic explanation is a desired outcome in many studies of perception. Such explanations require discovering the processes that contribute to the realization of a perceptual phenomenon at different levels of information processing. The present analysis aims at investigating the obstacles to develop such mechanistic explanations and their potential solutions in the context of psychopathology. Geometric-Optical Illusions (GOIs) are among perceptual phenomena that have been studied to understand psychopathology in various clinical populations. In the present analysis, the GOIs will (...) be discussed as a methodological tool in the investigation of the mechanistic approach to understanding perception in such populations. I will argue that the fact that most such studies are mainly focused on the investigation of susceptibility to GOIs can only lead to classifying such phenomena or to reporting some correlations between different aspects of such phenomena with certain brain activities, without formulating a mechanistic explanation. The investigation of the variables that mediate perceptual phenomena at earlier stages of information processing is required for the formulation of such explanations. Thus, the objective of the present analysis is twofold: 1) showing how ignoring such variables have prevented the formulation of mechanistic explanations of responses to GOIs in clinical populations, such as schizophrenia and autism, 2) analyzing the methodological aspects of an effective approach in which a top-down guidance based on phenomenological insights (the first-person data) along with the insights provided by the existing theories help overcome obstacles in developing mechanistic explanations of GOIs. (shrink)

The much larger appearance of the moon near horizon than the perceived size of the moon at zenith has motivated many scientists to develop theories that aim at explaining this puzzling phenomenon. Considering that the size of retinal images of the moon in these positions are very similar, the explanation of difference in their apparent sizes has relied on perceptual cues of distance embedded in the retinal image of their respective contexts. Although this account of the moon illusion is (...) quite popular, it does not explain all aspects of this phenomenon. The theoretical formulation of the moon illusion based on other factors such as size contrast later may have had some advantages but has also created some new problems. Although the moon is perceived in a three-dimensional (3D) environment, the present analysis proposes that an explanation of the moon illusion based on two-dimensional (2D) cues can remove some of the unnecessary problems. The empty space and size contrast that have already been considered in explaining classic geometric-optical illusions play a parallel role in explaining the moon illusion. In other words, the role of open space in interaction with the image of the moon and different objects near horizon, all reflected on the retina, are considered as the main explaining factors. The advantages of this approach will be discussed and some of the facts pertaining to the moon illusion will be explained within this theoretical framework. (shrink)

One of several important issues that inform contemporary philosophy of chemistry is the issue of structural explanation, precisely because modern chemistry is primarily concerned with microstructure. This paper argues that concern over microstructure, albeit understood differently than it is today, also informs the chemical philosophy of Robert Boyle. According to Boyle, the specific microstructure of ‘chymical atoms’, understood in geometric terms, accounts for the unique essential properties of different chemical substances. Because he considers the microstructure of ‘chymical atoms’ as (...) semi-permanent, Boyle considers these stable entities as operationally irreducible, even if they are not ontologically fundamental. While it is generally believed that our contemporary concern over structural explanation is a function of modern chemistry’s emphasis on microstructure, this discussion of structural explanation in Boyle will serve as a case study to illustrate the manner in which many of our contemporary concerns have deeply historical origins and the manner in which the history of chemistry can substantively inform issues in contemporary philosophy of chemistry. (shrink)

One of several important issues that inform contemporary philosophy of chemistry is the issue of structural explanation, precisely because modern chemistry is primarily concerned with microstructure. This paper argues that concern over microstructure, albeit understood differently than it is today, also informs the chemical philosophy of Robert Boyle (1627–1691). According to Boyle, the specific microstructure of ‘chymical atoms’, understood in geometric terms, accounts for the unique essential properties of different chemical substances. Because he considers the microstructure of ‘chymical atoms’ (...) as semi-permanent, Boyle considers these stable entities as operationally irreducible, even if they are not ontologically fundamental. While it is generally believed that our contemporary concern over structural explanation is a function of modern chemistry’s emphasis on microstructure, this discussion of structural explanation in Boyle will serve as a case study to illustrate the manner in which many of our contemporary concerns have deeply historical origins and the manner in which the history of chemistry can substantively inform issues in contemporary philosophy of chemistry. (shrink)

The paper will be divided into two parts. In the first part concerning teleological order in nature, different types of order will be distinguished: beginning with order as structure and then proceeding to higher and stronger types of order, which include special arithmetical and geometrical relations and eventually also negentropy. It will be shown that certain processes of becoming can possess higher order in such a way that they can have teleological order. In the second part a definition of (...) teleological explanation will be given and the following two questions will be answered: Are there teleological explanations which are logically valid? Are there teleological explanations which are used by scientists? (shrink)

This is a revised and updated edition of Graham Nerlich's classic book The Shape of Space. It develops a metaphysical account of space which treats it as a real and concrete entity. In particular, it shows that the shape of space plays a key explanatory role in space and spacetime theories. Arguing that geometrical explanation is very like causal explanation, Professor Nerlich prepares the ground for philosophical argument, and, using a number of novel examples, investigates how different (...) spaces would affect perception differently. This leads naturally to conventionalism as a non-realist metaphysics of space, an account which Professor Nerlich criticises, rejecting its Kantian and positivistic roots along with Reichenbach's famous claim that even the topology of space is conventional. He concludes that there is, in fact, no problem of underdetermination for this aspect of spacetime theories, and offers an extensive discussion of the relativity of motion. (shrink)

A part of relativistic dynamics is axiomatized by simple and purely geometrical axioms formulated within first-order logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein's famous E = mc² . The connection of our geometrical axioms and the usual axioms on the conservation of mass, momentum and four-momentum is also investigated.

A part of relativistic dynamics is axiomatized by simple and purely geometrical axioms formulated within first-order logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein’s famous E = mc 2. The connection of our geometrical axioms and the usual axioms on the conservation of mass, momentum and four-momentum is also investigated.

The aim of the paper is to analyse the geometrical aspects of a series of modern paintings and to show the parallel between them and the development of modern geometry. It starts with El Greco, offering a geometrical explanation of his painting the figures in a prolonged manner. Further the analogy between the impressionist way of creating space and the geometrical idea of Cayley to use projective space as a basis for non-Euclidean geometry is reconstructed. Next (...) the paper describes the parallel between the creation of space in the paintings of Cézanne and Picasso and the concept of space in algebraic topology. In conclusion the paper deals with the analogy between Kandinski's abstract paintings and the set-theoretical foundations of geometry. (shrink)

It is difficult to evaluate the role of activity - of force or of that which has causal efficacy - in Descartes’ natural philosophy. On the one hand, Descartes claims to include in his natural philosophy only that which can be described geometrically, which amounts to matter (extended substance) in motion (where this motion is described kinematically).’ Yet on the other hand, rigorous adherence to a purely geometrical description of matter in motion would make it difficult to account for (...) the interactions among the particles that constitute Descartes’ universe, since the notions of extension and kinematical motion do not in themselves imply any causal agency. There is, after all, no reason to expect that a particle whose single essence is extension, even if we suppose it to be moving, should impart motion to another particle, while conversely, there is no reason to expect a resting particle to hinder the motion of an impinging one. Descartes' hankering for an austere ontology of matter in motion is in danger of excluding causal agency (force, conceived dynmically) from matter. To the modern reader it may seem obvious that Descartes did not get himself into such a fix, since matter in motion so readily reminds us of kinetic energy or of some more primitive notion of force. Yet by no means is it obvious that Descartes attributed causal efficacy to matter in motion per se. Serious scholars have held opposite positions on this issue. Westfall, Gabbey, and others’ have argued that although on the metaphysical plane Descartes attempted to eliminate force from his mechanical universe, nonetheless ‘force is a real feature’ of his mechanical world. The thrust of this view is that Descartes conceived of force as ‘the capacity of a body in motion to act’, by means of impact, upon other bodies. An opposing interpretation, defended here, is that Descartes did in fact deny causal agency to moving matter per se, restricting agency to immaterial substances such as the human mind, angels, and God. The intention of this interpretation is not to de-emphasize the role of matter in motion in Descartes’ explanation of nature, but rather to stress the fact that Descartes did not conceive of this moving matter dynamically. -/- Reprinted in Descartes, ed. by J. Cottingham, Oxford Readings in Philosophy (Oxford: Oxford University Press, 1998), 281–310. (shrink)

Reprint of: Gary Hatfield, Force (God) in Descartes' physics, Studies in History and Philosophy of Science Part A 10 (2):113-140 (1979) -/- Abstract. It is difficult to evaluate the role of activity - of force or of that which has causal efficacy - in Descartes’ natural philosophy. On the one hand, Descartes claims to include in his natural philosophy only that which can be described geometrically, which amounts to matter (extended substance) in motion (where this motion is described kinematically).’ Yet (...) on the other hand, rigorous adherence to a purely geometrical description of matter in motion would make it difficult to account for the interactions among the particles that constitute Descartes’ universe, since the notions of extension and kinematical motion do not in themselves imply any causal agency. There is, after all, no reason to expect that a particle whose single essence is extension, even if we suppose it to be moving, should impart motion to another particle, while conversely, there is no reason to expect a resting particle to hinder the motion of an impinging one. Descartes' hankering for an austere ontology of matter in motion is in danger of excluding causal agency (force, conceived dynmically) from matter. To the modern reader it may seem obvious that Descartes did not get himself into such a fix, since matter in motion so readily reminds us of kinetic energy or of some more primitive notion of force. Yet by no means is it obvious that Descartes attributed causal efficacy to matter in motion per se. Serious scholars have held opposite positions on this issue. Westfall, Gabbey, and others’ have argued that although on the metaphysical plane Descartes attempted to eliminate force from his mechanical universe, nonetheless ‘force is a real feature’ of his mechanical world. The thrust of this view is that Descartes conceived of force as ‘the capacity of a body in motion to act’, by means of impact, upon other bodies. An opposing interpretation, defended here, is that Descartes did in fact deny causal agency to moving matter per se, restricting agency to immaterial substances such as the human mind, angels, and God. The intention of this interpretation is not to de-emphasize the role of matter in motion in Descartes’ explanation of nature, but rather to stress the fact that Descartes did not conceive of this moving matter dynamically. (shrink)

Discussions of the metaphysical status of spacetime assume that a spacetime theory offers a causal explanation of phenomena of relative motion, and that the fundamental philosophical question is whether the inference to that explanation is warranted. I argue that those assumptions are mistaken, because they ignore the essential character of spacetime theory as a kind of physical geometry. As such, a spacetime theory does notcausally explain phenomena of motion, but uses them to construct physicaldefinitions of basic geometrical (...) structures by coordinating them with dynamical laws. I suggest that this view of spacetime theories leads to a clearer view of the philosophical foundations of general relativity and its place in the historical evolution of spacetime theory. I also argue that this view provides a much clearer and more defensible account of what is entailed by realism concerning spacetime. (shrink)

The current literature that attempts to bridge between geometric morphometrics (GMM) and finite element analyses (FEA) of CT-derived data from bones of living animals and fossils appears to lack a sound biotheoretical foundation. To supply the missing rigor, the present article demonstrates a new rhetoric of quantitative inference across the GMM–FEA bridge—a rhetoric bridging form to function when both have been quantified so stringently. The suggested approach is founded on diverse standard textbook examples of the relation between forms and the (...) way strains in them are produced by stresses imposed upon them. One potentially cogent approach to the explanatory purposes driving studies of this class arises from a close scrutiny of the way in which computations in both domains, shape and strain, can be couched as minimizations of a scalar quantity. For GMM, this is ordinary Procrustes shape distance; in FEA, it is the potential energy that is stored in the deformed configuration of the solid form. A hybrid statistical method is introduced requiring that all forms be subjected to the same detailed loading designs (the same “probes”) in a manner careful to accommodate the variations of those same forms before they were stressed. The proper role of GMM is argued to be the construction of regressions for strain energy density on the largest-scale relative warps in order that biological explanations may proceed in terms of the residuals from those regressions: the local residual features of strain energy density. The method, evidently a hierarchical one, might be intuitively apprehended as a geometrical approach to a formal allometric analysis of strain. The essay closes with an exhortation. (shrink)

In his Elementorum Myologiae Specimen, Steno geometrizes "the new fabric of muscles" and their movement of contraction, so as to refute the main contemporary hypothesis about the functioning of the muscles. This physiological refutation relies on an abstract representation of the muscular fibre as a parallelepiped of flesh transversally linked to the tendons. Those two features have been comprehensively studied. But the method used by Steno, as well as the way he has chosen to present his physiological results, have so (...) far been neglected. Yet, Steno's work follows a true synthetic order, which he conceives as a tool to separate uncertain anatomical "elements" from the certain ones. We will show that the true understanding of this "more geometrico" order is the only way to avoid frequent misconceptions of the scientific aim pursued by Steno, which is neither to give a mathematical explanation of the functioning of the muscles, nor to reduce the muscles to some mathematical shapes. (shrink)

This article examines the use of mathematical concepts in philosophy, focusing on topology, which may be viewed as a modern supplement to geometry. We show that Plato and Parmenides were already employing geometric ideas in their research, and discuss three examples of the application of topology to philosophical problems: the first concerns the analysis of the Cartesian distinction between res extensa and res cogitans, the second the ontology of possible worlds of Wittgenstein's Tractatus, and the third Leibniz's monadology. We also (...) consider the role of topology in mathematical explanations of the sort found in science, arguing that it can perform a role in philosophy that is of comparable importance. (shrink)

Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the Gödel–McKinsey–Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than in the (...) explanation of the rules. (shrink)

This paper examines geometrical arguments from Galileo's Mechanics and Two New Sciences to discern the influence of the Aristotelian Mechanical Problems on Galileo's dynamics. A common scientific procedure is found in the Aristotelian author's treatment of the balance and lever and in Galileo's rules concerning motion along inclined planes. This scientific procedure is understood as a development of Eudoxan proportional reasoning, as it was used in Eudoxan astronomy rather than simply as it appears in Euclid's Elements. Topics treated include (...) the significance of the circle in Galileo's demonstrations, the substitution of rectilinear elements for heterogeneous factors like weight and curvilinear distance, and the way in which elements of a motion are used to measure other elements of the same motion. The indirectness of Galileo's proofs, his conception of speed as relative and comparative, and the meaning of his concept of moment all come into clearer focus. Conclusions are drawn about Galilean idealization, and also about the contrast of literal versus figural modes of explanation in Galileo's science. (shrink)

As an explanation for order and long range correlations in living systems, Fröhlich (1968; 1970; 1975) proposed certain biomolecules pumped by metabolic processes could exhibit coherent phonon dynamics, perhaps even macroscopic quantum coherence akin to Bose Einstein condensation or lasers. The biomolecular requirements, according to Fröhlich, were: 1) a geometric array or lattice of dipoles constrained in a common voltage gradient, and 2) ample, non coherent biochemical energy. Eligible proposed candidates included membrane proteins, nucleic acids and cytoskeletal microtubules.

Quick reference guide includes illustrated explanations of the most common terms used in general math classes. Discusses how students can use manipulatives and basic math tools to improve their understanding. With measurement conversion tables, guides to geometric shapes, and more.

This paper examines geometrical arguments from Galileo's Mechanics and Two New Sciences to discern the influence of the Aristotelian Mechanical Problems on Galileo's dynamics. A common scientific procedure is found in the Aristotelian author's treatment of the balance and lever and in Galileo's rules concerning motion along inclined planes. This scientific procedure is understood as a development of Eudoxan proportional reasoning, as it was used in Eudoxan astronomy rather than simply as it appears in Euclid's Elements. Topics treated include (...) the significance of the circle in Galileo's demonstrations, the substitution of rectilinear elements for heterogeneous factors like weight and curvilinear distance, and the way in which elements of a motion are used to measure other elements of the same motion. The indirectness of Galileo's proofs, his conception of speed as relative and comparative, and the meaning of his concept of moment all come into clearer focus. Conclusions are drawn about Galilean idealization, and also about the contrast of literal versus figural modes of explanation in Galileo's science. (shrink)