In a recent paper, Takacs and Bourrat (Biol Philos 37:12, 2022) examine the use of geometric mean reproductive output as a measure of biological fitness. We welcome Takacs and Bourrat’s scrutiny of a fitness definition that some philosophers have adopted uncritically. We also welcome Takacs and Bourrat’s attempt to marry the philosophical literature on fitness with the biological literature on mathematical measures of fitness. However, some of the main claims made by Takacs and Bourrat are not correct, while others (...) are correct but not for the reasons they give. (shrink)

It is stated in many text books that the any metric appearing in general relativity should be locally Lorentzian i.e. of the type η μ ν =diag (1,−1,−1,−1) this is usually presented as an independent axiom of the theory, which can not be deduced from other assumptions. The meaning of this assertion is that a specific coordinate (the temporal coordinate) is given a unique significance with respect to the other spatial coordinates. In this work it is shown that the above (...) assertion is a consequence of requirement that the metric of empty space should be linearly stable and need not be assumed. (shrink)

The geometric mean is also called the mean proportional. This is how the mathematicians of the √ −1. 19th Century, such as Gauss, understood..

Showing that the arithmetic mean number of offspring for a trait type often fails to be a predictive measure of fitness was a welcome correction to the philosophical literature on fitness. While the higher mathematical moments of a probability-weighted offspring distribution can influence fitness measurement in distinct ways, the geometric mean number of offspring is commonly singled out as the most appropriate measure. For it is well-suited to a compounding process and is sensitive to variance in offspring number. The (...) geometric mean thus proves to be a predictively efficacious measure of fitness in examples featuring discrete generations and within- or between-generation variance in offspring output. Unfortunately, this advance has subsequently led some to conclude that the arithmetic mean is never a good measure of fitness and that the geometric mean should accordingly be the default measure of fitness. We show not only that the arithmetic mean is a perfectly reasonable measure of fitness so long as one is clear about what it refers to, but also that it functions as a more general measure when properly interpreted. It must suffice as a measure of fitness in any case where the geometric mean has been effectively deployed as a measure. We conclude with a discussion about why the mathematical equivalence we highlight cannot be dismissed as merely of mathematical interest. (shrink)

We discuss in this paper the question of the scope of the principle of tolerance about languages promoted in Carnap's The Logical Syntax of Language and the nature of the analogy between it and the rudimentary conventionalism purportedly exhibited in the work of Poincaré and Hilbert. We take it more or less for granted that Poincaré and Hilbert do argue for conventionalism. We begin by sketching Coffa's historical account, which suggests that tolerance be interpreted as a conventionalism that allows us (...) complete freedom to select whatever language we wish—an interpretation that generalizes the conventionalism promoted by Poincaré and Hilbert which allows us complete freedom to select whatever axiom system we wish for geometry. We argue that such an interpretation saddles Carnap with a theory of meaning that has unhappy consequences, a theory we believe he did not hold. We suggest that the principle of linguistic tolerance in fact has a more limited scope; but within that scope the analogy between tolerance and geometric conventionalism is quite tight. (shrink)

The object in this article is to discuss the philosophical bearing of recent inquiries concerning geometrical axioms and the possibility of working out analytically other systems of geometry with other axioms than Euclid's. Digital edition compiled by Gabriele Dörflinger, Heidelberg University Library.

The object in this article is to discuss the philosophical bearing of recent inquiries concerning geometrical axioms and the possibility of working out analytically other systems of geometry with other axioms than Euclid's. Digital edition compiled by Gabriele Dörflinger, Heidelberg University Library.

Luminance and color are strong and self-sufficient cues to pictorial depth in visual scenes and images. The present study investigates the conditions Under which luminance or color either strengthens or overrides geometric depth cues. We investigated how luminance contrasts associated with color contrast interact with relative height in the visual field, partial occlusion, and interposition in determining the probability that a given figure is perceived as ‘‘nearer’’ than another. Latencies of ‘‘near’’ responses were analyzed to test for effects of (...) attentional selection. Figures in a pair were supported by luminance contrast or isoluminant color contrast and combined with one of the three geometric cues. The results of Experiment 1 show that luminance contrasts associated with hue, when it does not interact with other hues, produces the same effects as achromatic luminance contrasts: The probability of‘‘near’’ increases with luminance contrast while the latencies for ‘‘near’’ responses decrease. Partial occlusion is found to be a strong enough pictorial cue to support a weaker red luminance contrast. Interposition cues lose out against cues of spatial position and partial occlusion. The results of Experiment 2, with isoluminant displays of varying color contrast, reveal that red color contrast on a light background supported by any of the three geometric cues wins over green or white supported by any of the three geometric cues. On a dark background, red color contrast supported by the interposition cue loses out against green or white color contrast supported by partial occlusion. These findings reveal that color is not an independent depth cue, but is strongly influenced by luminance contrast and stimulus geometry. Systematically shorter response latencies for stronger ‘‘near’’ percepts demonstrate that selective visual attention reliably detects the most likely depth cue combination in a given configuration. (shrink)

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)

The question of the existence of gravitational stress-energy in general relativity has exercised investigators in the field since the inception of the theory. Folklore has it that no adequate definition of a localized gravitational stress-energetic quantity can be given. Most arguments to that effect invoke one version or another of the Principle of Equivalence. I argue that not only are such arguments of necessity vague and hand-waving but, worse, are beside the point and do not address the heart of the (...) issue. Based on a novel analysis of what it may mean for one tensor to depend in the proper way on another, which, en passant, provides a precise characterization of the idea of a "geometric object", I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of the bundle of Lorentz metrics over spacetime. I conclude by showing that my results also imply that, under a few natural conditions, the Einstein field equation is the unique equation relating gravitational phenomena to spatiotemporal structure, and discuss how this relates to the non-localizability of gravitational stress-energy. (shrink)

We analyze the geometric foundations of classical Yang-Mills theory by studying the relationships between internal relativity, locality, global/local invariance, and background independence. We argue that internal relativity and background independence are the two independent defining principles of Yang-Mills theory. We show that local gauge invariance -heuristically implemented by means of the gauge argument- is a direct consequence of internal relativity. Finally, we analyze the conceptual meaning of BRST symmetry in terms of the invariance of the gauge fixed theory (...) under general local gauge transformations. (shrink)

We reformulate minimalist grammars as partial functions on term algebras for strings and trees. Using filler/role bindings and tensor product representations, we construct homomorphisms for these data structures into geometric vector spaces. We prove that the structure-building functions as well as simple processors for minimalist languages can be realized by piecewise linear operators in representation space. We also propose harmony, i.e. the distance of an intermediate processing step from the final well-formed state in representation space, as a measure of (...) processing complexity. Finally, we illustrate our findings by means of two particular arithmetic and fractal representations. (shrink)

It is argued that quantum mechanics is fundamentally a geometric theory. This is illustrated by means of the connection and symplectic structures associated with the projective Hilbert space, using which the geometric phase can be understood. A prescription is given for obtaining the geometric phase from the motion of a time dependent invariant along a closed curve in a parameter space, which may be finite dimensional even for nonadiabatic cyclic evolutions in an infinite dimensional Hilbert space. (...) Using the natural metric on the projective space, we reformulate Schrödinger's equation for an isolated system. This metric is generalized to the space of all density matrices, and a physical meaning is proposed. (shrink)

The formulas for the Lie covariant differentiation of spinors are deduced from an algebraic viewpoint. The Lie covariant derivative of the spinor connection is calculated, and is given a geometric meaning. A theorem about the Lie covariant derivative of an operator in spin space that was stated in Part I of this work is discussed.

The author was invited by the organizers of the Benin symposium on the encounter between rationalities to contribute from the particular perspective of his research experience in ethno-mathematics – the study of mathematical ideas and practices as embedded in their cultural contexts. In this article he tries to contribute to the understanding of mathematical reasoning, as embedded in cultural practices, by means of illuminating some complementary aspects of geometrical exploration in diverse cultural contexts. He ends by offering a few (...) comments on possible educational implications. (shrink)

Examples are given of applications by Pauling, Mulliken, Marcus and G.E.Kimball of the three Pythagorian means to formulate the scales of electronegativity of the elements, to the calculations of rate constants of electron transfer cross-reactions, to the calculation of the observed rate constant as function of activation and diffusion rate constants in the case of mixed reaction-diffusion rates and to the calculation of the effective diffusion coefficient in solution of a salt AB as a whole from the diffusion coefficients (...) of the ions in which it dissociates. (shrink)

In his commentary on Euclid, Proclus says both that the first principle of geometry are self-evident and that they are hypotheses received from the single, highest, unhypothetical science, which is probably dialectic. The implication of this seems to be that a geometer both does and does not know geometrical truths. This dilemma only exists if we assume that Proclus follows Aristotle in his understanding of these terms. This paper shows that this is not the case, and explains what Proclus himself (...) means by definition, hypothesis, axiom, postulate, and the self-evident, and how geometry is a science that receives its principles from dialectic. (shrink)

The fragmentary nature ofGSmakes it difficult to read as it stands, and for this reason, I have rearranged the material slightly so that it falls into four primary, reasonably coherent, parts. Their titles are: ‘The nature of mathematical objects’, ‘Thirteen propositions of Euclid 1’, ‘The philosophy of parallel lines’ and ‘On the algebra of geometrical figures’.GSactually starts with ‘Thirteen propositions of Euclid 1’. The justification for the reversal of order in the translation is to have Hegel's philosophical basis for geometry (...) available first, to provide a context for his detailed discussion of geometrical propositions. The reader can easily reconstruct the original order ofGSas follows. Interchange the first and second parts and also the third and fourth parts. Next, move the passages marked a …. a', b …. b',c….c'to the places in the text markeda-a',b-b'andc-c'. For the reader wishing to locate the original German passages inDok., the page and paragraph for each of these passages are given at the start of that passage.In addition, subtitles for sections in these parts have been added which do not appear in the original. They are included in the translation to help orient the reader. I have inserted in some places words not in the original in order to clarify the meaning of the text. These insertions as well as my comments on the text are also enclosed within square brackets. Further clarifications can be found in the appended endnotes. (shrink)

We take a fresh look at voting theory, in particular at the notion of manipulation, by employing the geometry of the Saari triangle. This yields a geometric proof of the Gibbard/Satterthwaite theorem, and new insight into what it means to manipulate the vote. Next, we propose two possible strengthenings of the notion of manipulability (or weakenings of the notion of non-manipulability), and analyze how these affect the impossibility proof for non-manipulable voting rules.

The theorem that the arithmetic mean is greater than or equal to the geometric mean is investigated for cardinal and ordinal numbers. It is shown that whereas the theorem of the means can be proved for n pairwise comparable cardinal numbers without the axiom of choice, the inequality a2 + b2 ≥ 2ab is equivalent to the axiom of choice. For ordinal numbers, the inequality α2 + β2 ≥ 2αβ is established and the conditions for equality are derived; (...) stronger inequalities are obtained for finite and infinite sequences of ordinals under suitable monotonicity hypotheses. MSC: 03E10, 04A10, 03E25, 04A25. (shrink)

This paper examines Aristotle's discussion of the priority of actuality to potentiality in geometry at Metaphysics Θ9, 1051a21–33. Many scholars have assumed what I call the "geometrical construction" interpretation, according to which his point here concerns the relation between an inquirer's thinking and a geometrical figure. In contrast, I defend what I call the "geometrical analysis" interpretation, according to which it concerns the asymmetrical relation between geometrical propositions in which one is proved by means of the other. His argument (...) as so construed is ultimately based on the asymmetrical relation between the corresponding geometrical facts. Then I explore this ontological priority in geometry by drawing attention to a parallel passage, Posterior Analytics II.11, 94a24–35, where Aristotle explains the relation between the same geometrical propositions in connection to material causation. (shrink)

Readers of Spinoza's philosophy have often been daunted, and sometimes been enchanted, by the geometrical method which he employs in his philosophical masterpiece the Ethics. In Meaning in Spinoza's Method Aaron Garrett examines this method and suggests that its purpose, in Spinoza's view, was not just to present claims and propositions but also in some sense to change the readers and allow them to look at themselves and the world in a different way. His discussion draws not only on Spinoza's (...) works but also on those of the philosophers who influenced Spinoza most strongly, including Hobbes, Descartes, Maimonides and Gersonides. This controversial book will be of interest to historians of philosophy and to anyone interested in the relation between form and content in philosophical works. (shrink)

Packing optimization problems have a wide spectrum of real-word applications. One of the applications of the problems is problem of placement of containers with spent nuclear fuel on the storage platform. The solution of the problem can be reduced to the solution of the problem of finding the optimal placement of a given set of congruent circles into a multiconnected domain taking into account technological restrictions. A mathematical model of the prob-lem is constructed and its peculiarities are considered. Our approach (...) is based on the mathematical modelling of rela-tions between geometric objects by means of phi-function technique. That allowed us to reduce the problem solving to nonlinear programming. Today, an important scientific problem is the problem of creating conditions for safe storage of spent nuclear fuel. In the process of creating any dry spent nuclear fuel storage, the following main stages can be identified: site selection, storage design, construction, operation and decommissioning. A full check for compliance of the repository and its elements with these standards usually begins at the design stage. At the stage of site selection, the inspection for compliance with safety standards is carried out only in terms of the impact of the repository as a whole on the environment. This approach cannot be considered fully appropriate, because, taking into account, for example, all the climatic features of the future storage site, it is possible to adjust the thermal storage regimes of spent nuclear fuel. Similarly, it can be considered necessary to analyze and select the shape of the storage site in order to accommo-date the maximum possible number of spent fuel containers. Such a choice, obviously, should be made taking into ac-count the norms of nuclear, radiation and thermal safety, as well as in compliance with technological limitations. The problem of finding the optimal placement of containers taking into account the given technological limitations can be formulated in the form of the problem of optimization of geometric design. Therefore, the purpose of the study is to build a mathematical model of the problem and study its characteristics to develop effective methods of solution. The proposed approach is based on mathematical modeling of relations between geometric objects using the method of phi-functions. This allowed to reduce the solution of the problem to the problem of nonlinear programming. (shrink)

This essay offers an in-depth reading of the geometrical illustration of Ethics IIP8S and shows how it can be used to explicate the whole architecture of Spinoza’s system by specifying the way in which all the key structural features of his basic ontology find their analogies in the example. The illustration can also throw light on Spinoza’s ontology of finite things and inform us about what is at stake when we form universal ideas. In general, my reading of IIP8S thus (...) elucidates what it means, for Spinoza, to think geometrically or to consider geometry as a model: fundamentally, geometricity is not a form of exposition but the way in which reality itself is structured. (shrink)

We discuss an exact solution to the simplest nontrivial example of a geometrical phase in quantum mechanics. By means of this example: (1) we elucidate the fundamental distinction between rays and vectors in describing quantum mechanical states; (2) we show that superposition of quantal states is invalid; only decomposition is allowed—which is adequate for the measurement process. Our example also shows that the origin of singularities in the analog vector potential is to be found in the unavoidable breaking of (...) projective symmetry caused by using the Schrödinger equation. (shrink)

We present a gravitational quantum dynamics theory that combines quantum field theory for particle dynamics in space-time with classical Einstein’s general relativity in a non-Riemannian Finsler space. This approach is based on the geometrization of quantum mechanics proposed in Tavernelli and combines quantum and gravitational effects into a global curvature of the Finsler space induced by the quantum potential associated to the matter quantum fields. In order to make this theory compatible with general relativity, the quantum effects are described in (...) the framework of quantum field theory, where a covariant definition of ‘simultaneity’ for many-body systems is introduced through the definition of a suited foliation of space-time. As in Einstein’s gravitation theory, the particle dynamics is finally described by means of a geodesic equation in a curved space-time manifold. (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)

In lieu of an abstract, here is a brief excerpt of the content:61. HUME'S GEOMETRIC "OBJECTS" Arithmetic and algebra allow of precision and certainty. The science of geometry is not likewise a perfect and infallible science. At any rate, this is Hume's teaching in the Treatise. When two numbers are so combin ' d, as that the one has always an unite answering to every unite of the other, we pronounce them equal; and 'tis for want of such a (...) standard of equality in extension, that geometry can scarce be esteem'd a perfect and infallible science. 1 The ideas of, say, equality and right line shade into· inequality and curve. It must not be supposed, however, that the vagueness of these ideas renders all judgments uncertain and fallible. 'Tie evident, that the eye, or rather the mind is often able at one view to determine the proportions of bodies....Such judgments are not only common, but in many cases certain and infallible. (T47) Nevertheless, geometry is a fallible science because its essential ideas are vague. Vagueness, due to lack of boundaries between ideas, is in the ideas themselves. For this reason, 'tis impossible to produce any definition of them, which will fix the precise boundaries betwixt them. (T49) (Yet, some forty pages earlier Hume argued at length that the mind cannot form any notion of quantity or quality without forming a precise notion of the degrees of each [T18]) As far as the Treatise goes, geometry is the science of measurement and as such it must allow for error. Even if precise geometric standards could be had, they would not make the propositions of geometry certain. Thus, we might precisely define a right line by a rule of order of indivisible points. But geometry, construed as the science of measurement, would not be perfected. Nor, if it were (the standard), is there any such firmness in our senses or imagination, as to determine when such an order is violated or preserv ' d. The original standard of a right line is in reality nothing but a certain general appearance. (T52) 62. In the Enquiries, Hume continues to hold that notions essential to geometry, for example, equality, right line, and so on, are sensibles. But they are no longer generic appearances. They are determinate. The great advantage of the mathematical sciences above the moral consists in this, that the ideas of the former, being sensible, are always clear and determinate... even when no definition is employed, the object itself may be presented to the senses, and by that means be steadily and clearly apprehended. 2 In addition, geometry is no longer merely the science of measurement. The truths of Euclid, Hume holds, would remain truths even if there were no figures in nature. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence. (EHU25) Professor Zabeeh, in his commentary on Hume, holds that Hume did not mean to imply that Euclid could dispense with the observation of figures. However, whenever possible Hume should be taken at his word: observation of figures is not a truth condition for propositions of geometry. A distinction must be drawn between discovery conditions and what Hume calls the "truth or evidence" of a geometric proposition. We may suppose that observation of figures is a discovery condition. But it is hardly a truth condition. The certainty, as opposed to the discovery, of geometric propositions does not depend on the existence in nature of any figure. I am mainly concerned, however, with Professor Zabeeh' s account of how, according to Hume, geometric propositions retain their certainty and evidence in the face of recalcitrant experience. Noting that Hume does not "fully explain," he gives the real reason for Hume's belief in the apodeictic certainty of mathematical truths. He appears to assume we never allow such truths to be controverted by empirical evidence. That is to say, if perchance we find, by measurement, that the sum of the angles of a Euclidean triangle 63. does not equal 180 degrees, either we say that we measured wrongly or we say that the triangle we have been measuring is not Euclidean.4 This formulation of Hume's reason avoids... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:61. HUME'S GEOMETRIC "OBJECTS" Arithmetic and algebra allow of precision and certainty. The science of geometry is not likewise a perfect and infallible science. At any rate, this is Hume's teaching in the Treatise. When two numbers are so combin ' d, as that the one has always an unite answering to every unite of the other, we pronounce them equal; and 'tis for want of such a (...) standard of equality in extension, that geometry can scarce be esteem'd a perfect and infallible science. 1 The ideas of, say, equality and right line shade into· inequality and curve. It must not be supposed, however, that the vagueness of these ideas renders all judgments uncertain and fallible. 'Tie evident, that the eye, or rather the mind is often able at one view to determine the proportions of bodies....Such judgments are not only common, but in many cases certain and infallible. (T47) Nevertheless, geometry is a fallible science because its essential ideas are vague. Vagueness, due to lack of boundaries between ideas, is in the ideas themselves. For this reason, 'tis impossible to produce any definition of them, which will fix the precise boundaries betwixt them. (T49) (Yet, some forty pages earlier Hume argued at length that the mind cannot form any notion of quantity or quality without forming a precise notion of the degrees of each [T18]) As far as the Treatise goes, geometry is the science of measurement and as such it must allow for error. Even if precise geometric standards could be had, they would not make the propositions of geometry certain. Thus, we might precisely define a right line by a rule of order of indivisible points. But geometry, construed as the science of measurement, would not be perfected. Nor, if it were (the standard), is there any such firmness in our senses or imagination, as to determine when such an order is violated or preserv ' d. The original standard of a right line is in reality nothing but a certain general appearance. (T52) 62. In the Enquiries, Hume continues to hold that notions essential to geometry, for example, equality, right line, and so on, are sensibles. But they are no longer generic appearances. They are determinate. The great advantage of the mathematical sciences above the moral consists in this, that the ideas of the former, being sensible, are always clear and determinate... even when no definition is employed, the object itself may be presented to the senses, and by that means be steadily and clearly apprehended. 2 In addition, geometry is no longer merely the science of measurement. The truths of Euclid, Hume holds, would remain truths even if there were no figures in nature. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence. (EHU25) Professor Zabeeh, in his commentary on Hume, holds that Hume did not mean to imply that Euclid could dispense with the observation of figures. However, whenever possible Hume should be taken at his word: observation of figures is not a truth condition for propositions of geometry. A distinction must be drawn between discovery conditions and what Hume calls the "truth or evidence" of a geometric proposition. We may suppose that observation of figures is a discovery condition. But it is hardly a truth condition. The certainty, as opposed to the discovery, of geometric propositions does not depend on the existence in nature of any figure. I am mainly concerned, however, with Professor Zabeeh' s account of how, according to Hume, geometric propositions retain their certainty and evidence in the face of recalcitrant experience. Noting that Hume does not "fully explain," he gives the real reason for Hume's belief in the apodeictic certainty of mathematical truths. He appears to assume we never allow such truths to be controverted by empirical evidence. That is to say, if perchance we find, by measurement, that the sum of the angles of a Euclidean triangle 63. does not equal 180 degrees, either we say that we measured wrongly or we say that the triangle we have been measuring is not Euclidean.4 This formulation of Hume's reason avoids... (shrink)

In this paper, we examine Sextus Empiricus' treatise Against the geometers . We first set this treatise in the overall context of the sceptic's polemics against the liberal arts. After a discussion of Sextus' attitude to the quadrivium , we discuss the structure, the sources and the target of the Against the geometers . It appears that Euclid is not Sextus' source, and neither he, nor the professional geometers, seem to be Sextus' main targets. Of course, Sextus never really makes (...) clear his precise target, but his attacks are rather directed against geometry as a means of modelling the physical world, thus ruining the support geometry was intended to bring to the physical part of dogmatic philosophy. (shrink)

We construct the Extended Relativity Theory in Born-Clifford-Phase spaces with an upper R and lower length λ scales (infrared/ultraviolet cutoff). The invariance symmetry leads naturally to the real Clifford algebra Cl (2, 6, R) and complexified Clifford Cl C (4) algebra related to Twistors. A unified theory of all Noncommutative branes in Clifford-spaces is developed based on the Moyal-Yang star product deformation quantization whose deformation parameter involves the lower/upper scale $$(\hbar \lambda / R)$$. Previous work led us to show from (...) first principles why the observed value of the vacuum energy density (cosmological constant) is given by a geometric mean relationship $$\rho \sim L_{\rm Planck}^{-2}R^{-2} = L_{P}^{-4} (L_{\rm Planck} / R)^{2} \sim 10^{-122}M_{\rm Planck}^4$$, and can be obtained when the infrared scale R is set to be of the order of the present value of the Hubble radius. We proceed with an extensive review of Smith’s 8D model based on the Clifford algebra Cl (1, 7) that reproduces at low energies the physics of the Standard Model and Gravity, including the derivation of all the coupling constants, particle masses, mixing angles,....with high precision. Geometric actions are presented like the Clifford-Space extension of Maxwell’s Electrodynamics, and Brandt’s action related to the 8D spacetime tangent-bundle involving coordinates and velocities (Finsler geometries). Finally we outline the reasons why a Clifford-Space Geometric Unification of all forces is a very reasonable avenue to consider and propose an Einstein-Hilbert type action in Clifford-Phase spaces (associated with the 8D Phase space) as a Unified Field theory action candidate that should reproduce the physics of the Standard Model plus Gravity in the low energy limit. (shrink)

EDMUND HUSSERL in his early writings on space distinguishes three kinds of problems surrounding the presentation of space: psychological, logical, and metaphysical. By the term "psychology" Husserl means a descriptive and genetic psychology which seeks to characterize the contents and structure of particular experiences and to investigate the genetic relations between different experiences. Included among the genetic questions concerning space is the problem of the origin of the science of space.

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

How to measure fitness in the theory of natural selection? A fitness measure that has been proposed in both the biological and the philosophical literature is the expected relative reproductive success. The aim of this article is to examine the relationship between expected relative reproductive success and future actual evolutionary success. Doing so will not only clarify the use of expected relative reproductive success as a fitness measure but also shed light on the role of fitness in the theory of (...) natural selection. _1_ Introduction _2_ The Role of Fitness _3_ Geometric Mean versus Expected Relative Reproductive Success _4_ Reconciliation _5_ Reconciliation Continued _6_ Alternative Justifications _7_ Conclusion. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Meaning in Spinoza’s MethodAlan Nelson and Noa SheinAaron V. Garrett. Meaning in Spinoza’s Method. New York: Cambridge University Press, 2003. Pp. xii + 240. Cloth, $60.00.This is a book about some fundamental aspects of Spinoza's mature metaphysics. The principal focus is on Part I of the Ethics concerning infinite substance, and on Part V concerning the intuitive knowledge that is the goal of philosophy. Within this focus, Garrett (...) concentrates on Spinoza's method. Garrett explains how he has been influenced by interpretations that stress the structural parallel between the synthetic, geometrical presentation of the Ethics and the relations among the ideas of the well-ordered mind. The Ethics proceeds by deriving propositions concerning substance's modes from definitions of substance and related notions while the mind, insofar as it is free from emotions, intuitively knows things as following from God. Garrett's distinctive thesis, however, is that Spinozistic enlightenment depends on the process of emending the intellect as described in one of Spinoza's earliest works, the Treatise on the Emendation of the Intellect. While this work strongly foreshadows many of Spinoza's mature doctrines, its exposition much more closely resembles the analytic, non-geometric approach of Descartes's Meditations. Spinoza's therapy for the emendation of our "inadequate" ideas closely parallels Descartes's analytic therapy for making what he calls confused ideas "clear and distinct."Garrett is on firm ground in arguing that the process of enlightenment cannot be an unbroken chain of intuitive deductions. For one thing, the definitions and axioms printed in a copy of the Ethics are first sensed by the reader and are not likely to produce immediately the requisite adequate ideas (172). An example of one of Garrett's interesting conclusions is that the definition of God as consisting of an infinity of attributes, the sixth of Part I of the Ethics (1D6), is preliminary and replaced in the process of therapeutic emendation by a definition of God as self caused. It is only after our ideas have become adequate at the end of the process that we are able to fully appreciate 1D6 (171).One might think that Garrett does not go far enough here because his point can be naturally extended. Definitions might assist us in the process of bringing our adequate ideas to light, but our cognitive goal is making these ideas themselves adequate. What then are the ideas we can make most adequate and exactly what distinguishes them from highly refined ideas that are nevertheless more inadequate than these? On one interpretation it is divine essence, that is, the attributes of thought and extension that are individually most adequately conceived after successful therapy. It is tempting to suppose that a single idea of God as both thinking and extended would be more adequate still, but Spinoza holds that our conceptions of these attributes are "really distinct" in a sense analogous to Descartes's(1P10). Spinoza's 1D6 definition of God as a substance consisting of both these attributes (and perhaps other inconceivable ones) thus seems to indicate a subsumption of the attributes we can conceive under the image of some printed or spoken words. Such an idea would be more inadequate than our adequate conceptions of the individual attributes. One is here strongly reminded of Descartes's characterization in Article 63 of the first Part of his Principles of Philosophy, "... we more readily understand extended substance or thinking substance than substance alone."In his seventh and concluding chapter, Garrett tackles the grand problems of intuitive knowledge or knowledge of "the third kind" and the sense in which Spinoza thought a part of the human mind is eternal. The discussion is necessarily complex, but one great advantage of Garrett's treatment is that he realizes intuitive knowledge cannot consist in the explicit deduction of particular things from God. The detailed determination of particular "everyday objects" requires sensory knowledge which is, of course, never perfectly determinate and therefore inherently inadequate. But we can acquire the "second kind" of adequate knowledge by understanding what our minds have in common with other determinate, finite things. This finally enables us to ascend to an appreciation of these... (shrink)

While “meaning negotiation” has become an ubiquitous term, its use is often confusing. A negotiation problem implies not only a convenience to agree, but also diverging interest on what to agree upon. It implies agreement but also the possibility of disagreement. In this chapter, we look at meaning negotiation as the process through which agents starting from different preferred conceptual representations of an object, an event or a more complex entity, converge to an agreement through some communication medium. We shortly (...) sketch the outline of a geometric view of meaning negotiation, based on conceptual spaces. We show that such view can inherit important structural elements from game theoretic models of bargaining – in particular, in the case when the protagonists have overlapping negotiation regions, we emphasize a parallel to the Nash solution in cooperative game theory. When acceptable solution regions of the protagonists are disjoint, we present several types of processes: changes in the salience of dimensions, dimensional projections and metaphorical space transformations. None of the latter processes are motivated by normative or rationality considerations, but presented as argumentation tools that we believe are used in actual situations of conceptual disagreement. (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)

This paper analyzes correspondence between Reichenbach and Einstein from the spring of 1926, concerning what it means to ‘geometrize’ a physical field. The content of a typewritten note that Reichenbach sent to Einstein on that occasion is reconstructed, showing that it was an early version of §49 of the untranslated Appendix to his Philosophie der Raum-Zeit-Lehre, on which Reichenbach was working at the time. This paper claims that the toy-geometrization of the electromagnetic field that Reichenbach presented in his note (...) should not be regarded as merely a virtuoso mathematical exercise, but as an additional argument supporting the core philosophical message of his 1928 monograph. This paper concludes by suggesting that Reichenbach’s infamous ‘relativization of geometry’ was only a stepping stone on the way to his main concern—the question of the ‘geometrization of gravitation’. (shrink)

Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific issues, (...) in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a “non-standard” approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which “ontological” interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax–semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century’s philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational “crisis”, as if algebraic geometry and synthetic differential geometry – not to mention algebraic topology – were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as ‘the’ philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; the time is ripe to realise that the same holds for classical topics of philosophy. (shrink)

In 1870, Hermann von Helmholtz criticized the Kantian conception of geometrical axioms as a priori synthetic judgments grounded in spatial intuition. However, during his dispute with Albrecht Krause (Kant und Helmholtz über den Ursprung und die Bedeutung der Raumanschauung und der geometrischen Axiome. Lahr, Schauenburg, 1878), Helmholtz maintained that space can be transcendental without the axioms being so. In this paper, I will analyze Helmholtz’s claim in connection with his theory of measurement. Helmholtz uses a Kantian argument that can be (...) summarized as follows: mathematical structures that can be defined independently of the objects we experience are necessary for judgments about magnitudes to be generally valid. I suggest that space is conceived by Helmholtz as one such structure. I will analyze his argument in its most detailed version, which is found in Helmholtz (Zählen und Messen, erkenntnistheoretisch betrachtet 1887. In: Schriften zur Erkenntnistheorie. Springer, Berlin, 1921, 70–97). In support of my view, I will consider alternative formulations of the same argument by Ernst Cassirer and Otto Hölder. (shrink)

Why is a red face not really red? How do we decide that this book is a textbook or not? Conceptual spaces provide the medium on which these computations are performed, but an additional operation is needed: Contrast. By contrasting a reddish face with a prototypical face, one gets a prototypical ‘red’. By contrasting this book with a prototypical textbook, the lack of exercises may pop out. Dynamic contrasting is an essential operation for converting perceptions into predicates. The existence of (...) dynamic contrasting may contribute to explaining why lexical meanings correspond to convex regions of conceptual spaces. But it also explains why predication is most of the time opportunistic, depending on context. While off-line conceptual similarity is a holistic operation, the contrast operation provides a context-dependent distance that creates ephemeral predicative judgments that are essential for interfacing conceptual spaces with natural language and with reasoning. (shrink)

The theory that ``consistency implies existence'' was put forward by Hilbert on various occasions around the start of the last century, and it was strongly and explicitly emphasized in his correspondence with Frege. Since (Gödel's) completeness theorem, abstractly speaking, forms the basis of this theory, it has become common practice to assume that Hilbert took for granted the semantic completeness of second order logic. In this paper I maintain that this widely held view is untrue to the facts, and that (...) the clue to explain what Hilbert meant by linking together consistency and existence is to be found in the role played by the completeness axiom within both geometrical and arithmetical axiom systems. (shrink)