Plusieurs textes de Giordano Bruno définissent le statut des mathématiques en reprenant la classification aristotélicienne des sciences théorétiques. Toutefois cette classification change radicalement de sens à la lumière du monisme brunien. Les mathématiques restent certes une science abstraite, mais, pour le métaphysicien, elles deviennent l’instrument qui permet de penser le rapport de la substance aux modes, le déploiement dans l’unité de la substance de la pluralité infinie des formes; en d’autres termes, elles deviennent une véritable logique de l’être.

In lieu of an abstract, here is a brief excerpt of the content:Notes and Discussions ARISTOTLE'S "DE MOTU ANIMALIUM" AND THE SEPARABILITY OF THE SCIENCES In contrast to Plato's vision of a unified science of reality and with a profound effect on subsequent natural science and philosophy, Aristotle urges in the Posterior Analytics and elsewhere that scientific knowledge is to be pursued in limited, separable domains, each with its own true and necessary first principles for the explanation of a discrete (...) range of phenomena, that is, accepted observations and beliefs, from which its investigations are launched (An. Pr. 46al7ff.; An. Post. 74b25, 76a26-3 o, 84b14-18, 88a31; Cael. 3o6a6ff.; Gen. An. 748a7ff.). In an early introduction to a course of lectures on the natural sciences, Aristotle also indicates the order in which those sciences ought to be presented (Mete. 338a2o-~9, 339a5-9; cf. Cael. 268al6, Part. An. 644b~2ff.). The plan is to start with a general discussion of change and motion, then progress to studies of specific natural phenomena, ending with the many species of plants and animals. In the theory of demonstration in Posterior Analytics 1, which is concerned with the organization, justification, and teaching of a finished science, Aristotle maintains that terms cannot ordinarily cross genera; for example, geometry cannot provide demonstrations of truths of arithmetic or aesthetics. Demonstration of a theorem of some one science by means of another can be accomplished when and only when the sciences are related as "subordinate " to "superior" (75b14-17), for example, as harmonics to arithmetic or optics to geometry. A superior science may supply the "reason" for a "fact" known to obtain in the subordinate science (78b34-79a6). Evidently at least part of what he has in mind is the relation between pure and applied sciences.' Aristotle's practice as well as a number of methodological remarks ' See ThomasAquinas,In Post. An. 50.1,15.131, 50.1.25,as wellasJ. Barnes'snoteson An. Post. 78b34,in Aristotle'sPosteriorAnalytics (Oxford:OxfordUniversityPress,1975).For relevant background see Ian Mueller,"Ascendingto Problems:Astronomyand Harmonicsin Republic VII," inJohn Anton,ed., Scienceand the Sciencesin Plato (Albany:StateUniversityof NewYork Press, 1979). [65] 66 HISTORY OF PHILOSOPHY suggest also that the strictures of the Posterior Analytics are not intended to prohibit the heuristic use of material from other disciplines in research (see e.g., Top. 1.14; Part An. 639b17ff., 641a6-14, 642alo-14, 645b15-2o; Ph. 192b8-34, 199a8-21, 199a33-b5). The requirements for science laid down here are very strict, and it is sometimes maintained that Aristotle violates them himself to some degree in other works, but it is not usually believed that he ever denies the separability of the sciences in general. Martha C. Nussbaum has recently presented a lively challenge to this concensus? She argues that Aristotle's late, little known work De Motu Animalium represents a radical but "deliberate and fruitful" rejection of his earlier philosophy of science as enunciated in the Organon and not seriously questioned in other, subsequent writings. Her claim is that in his mature thought about the sciences Aristotle arrives at a significantly "less departmental and more flexible picture of scientific study" (p. 113) and comes to hold that "no inquiry is genuinely separable from a whole group of interlocking studies, and no being can be extensively studied without an account of its placement in the whole of nature" (p. 164). Nussbaum's conclusion is probably not intended to be as Platonic as it may appear at first blush, for she seems actually to be speaking only of connections between sciences of different substances. There is no suggestion that the study of beauty or health or geometry, say, is on a par with the sciences of substances. Nor does she suggest that Aristotle wavers in his conviction that there are fundamental cleavages between the practical and theoretical sciences, which will make a full-blown science of ethics-politics look very different from the demonstrative science of meteorology, for example. Nevertheless, even if limited to the natural sciences of substances, her claim remains an important and interesting one. She holds, more specifically, that Aristotle departs from his earlier position as follows: (1) He recognizes that the biological study of various modes of local motion... (shrink)

Aristotle’s philosophy of geometry is widely interpreted as a reaction against a Platonic realist conception of mathematics. Here I argue to the contrary that Aristotle is concerned primarily with the methodological question of how universal inferences are warranted by particular geometrical constructions. His answer hinges on the concept of abstraction, an operation of “taking away” certain features of material particulars that makes perspicuous universal relations among magnitudes. On my reading, abstraction is a diagrammatic procedure for Aristotle, and it is through (...) imagined variation of diagrams that one can universalize spatial inferences. In order to shift the discussion of Aristotle’s work on geometry from ontological to methodological questions, I argue in the first section that geometrical properties for Aristotle are necessary accidents of the particulars, canonically diagrams, in which they inhere. I then consider a line of research that shows how diagrams in ancient geometry are not illustrations of the textual proof but are in part constitutive of geometrical inference, an insight I claim can shed light on Aristotle’s philosophical project. Next I substantiate this assertion by giving an alternative interpretation of abstraction according to which it is a diagrammatic procedure that allows a part of a particular diagram to stand in for a universal. In the last section, I consider Aristotle’s account of mathematical cognition, arguing that there is evidence for the view that imagination is necessary both for the construction of figures, and for their presentation as abstractions subject to universal inference. (shrink)

Edited by Richard McKeon, with an introduction by C.D.C. Reeve Preserved by Arabic mathematicians and canonized by Christian scholars, Aristotle’s works have shaped Western thought, science, and religion for nearly two thousand years. Richard McKeon’s The Basic Works of Aristotle—constituted out of the definitive Oxford translation and in print as a Random House hardcover for sixty years—has long been considered the best available one-volume Aristotle. Appearing in paperback at long last, this edition includes selections from the Organon, On the (...) Heavens, The Short Physical Treatises, Rhetoric, among others, and On the Soul, On Generation and Corruption, Physics, Metaphysics, Nicomachean Ethics, Politics, and Poetics in their entirety. (shrink)

Aristotle,Le confutazioni sofistiche. Organon VI. translated with notes by P. Fait. Bari: Laterza, 2007. lxi + 253 pp. [euro] 25.00. ISBN 978-88-420-8316-0. Reviewed by Annamaria Schiaparelli, The...

This collection of new essays by an international group of scholars closely examine the works of Aristotle's Organon. The Organon is the general title given to the collection of Aristotle's logical works: Categories, De Interpretatione, Prior Analytics, Posterior Analytics, Topics and Sophistical Refutations. This extremely influential collection gave Aristotle the reputation of being the founder of logic, and has helped shaped the development of logic for over two millennia. The chapters in this volume cover topics pertaining to each (...) of the six works traditionally included in the Organon as well as its manuscript tradition. In addition, a comprehensive introduction by the editors discusses Aristotle and logic, the composition and order of the Organon, and the authenticity, title, and chronology of the treatises that make up these works New Essays on Aristotle's Organon offers a valuable insight into this collection for students and scholars working on Aristotle, the works of the Organon, or the philosophy of logic more broadly. (shrink)

‘The Place of Geometry’ discusses the excursus on mathematics from Heidegger's 1924–25 lecture course on Platonic dialogues, which has been published as Volume 19 of the Gesamtausgabe as Plato's Sophist, as a starting point for an examination of geometry in Euclid, Aristotle and Descartes. One of the crucial points Heidegger makes is that in Aristotle there is a fundamental difference between arithmetic and geometry, because the mode of their connection is different. The units of geometry are positioned, the units of (...) arithmetic unpositioned. Following Heidegger's claim that the Greeks had no word for space, and David Lachterman's assertion that there is no term corresponding to or translatable as ‘space’ in Euclid's Elements, I examine when the term ‘space’ was introduced into Western thought. Descartes is central to understanding this shift, because his understanding of extension based in terms of mathematical co‐ordinates is a radical break with Greek thought. Not only does this introduce this word ‘space’ but, by conceiving of geometrical lines and shapes in terms of numerical co‐ordinates, which can be divided, it turns something that is positioned into unpositioned. Geometric problems can be reduced to equations, the length of lines: a problem of number. The continuum of geometry is transformed into a form of arithmetic. Geometry loses position just as the Greek notion of ‘place’ is transformed into the modern notion of space. (shrink)

‘The Place of Geometry’ discusses the excursus on mathematics from Heidegger's 1924–25 lecture course on Platonic dialogues, which has been published as Volume 19 of the Gesamtausgabe as Plato's Sophist, as a starting point for an examination of geometry in Euclid, Aristotle and Descartes. One of the crucial points Heidegger makes is that in Aristotle there is a fundamental difference between arithmetic and geometry, because the mode of their connection is different. The units of geometry are positioned, the units of (...) arithmetic unpositioned. Following Heidegger's claim that the Greeks had no word for space, and David Lachterman's assertion that there is no term corresponding to or translatable as ‘space’ in Euclid's Elements, I examine when the term ‘space’ was introduced into Western thought. Descartes is central to understanding this shift, because his understanding of extension based in terms of mathematical co‐ordinates is a radical break with Greek thought. Not only does this introduce this word ‘space’ but, by conceiving of geometrical lines and shapes in terms of numerical co‐ordinates, which can be divided, it turns something that is positioned into unpositioned. Geometric problems can be reduced to equations, the length of lines: a problem of number. The continuum of geometry is transformed into a form of arithmetic. Geometry loses position just as the Greek notion of ‘place’ is transformed into the modern notion of space. (shrink)

Categorie -- De interpretatione -- Analitici primi -- Analitici secondi -- Topici -- Confutazioni sofistiche.

Dans les Refutations sophistiques (sixieme et dernier des traites logiques rassembles sous le titre d'Organon), Aristote analyse et classe les differents types de paralogismes que commettent les sophistes qui s'emploient a refuter leurs interlocuteurs dans le cadre d'un echange dialectique. Assez curieusement, l'erudition contemporaine, qui a pourtant multiplie les etudes sur la dialectique d'Aristote, s'est peu interessee aux Refutations sophistiques, si bien que ce traite peut a bon droit etre tenu pour le parent pauvre de la recherche aristotelicienne. Les (...) plus recents commentaires des Refutations sophistiques remontent en effet a la fin du XIXe siecle. La presente etude, qui offre une longue introduction, une traduction inedite et un commentaire approfondi des Refutations sophistiques, vise a combler une importante lacune et a reveler l'interet d'une oeuvre qui a ete injustement negligee. (shrink)

A presentation showing how the statements which relate to microphysical objects as they are different from the statements of classical mechanics is made. The determinism of classical and of quantum-mechanical theories is qualified. A (crucial) distinction between causality and determinism is given. Detailed analyses of diffraction as a result of single and double-slit demonstrations point to paradoxes arising from the use of particle or wave models, respectively, for photons and electrons. The compromising wave-packet model is underscored. The meanings for the (...) Psi-function and for |ψ|2 are presented, with an explanation of the limitations that arise from giving an interpretation of |ψ|2 as a probability. The meaning of ψ -waves is analyzed. A case is made for the ψ -function's describing a well-ordered list of betting odds, based on previous statistical experiences, for the different results of specific experiments of a microscopic object of study with macroscopic apparatus. The Schrödinger wave equation is evaluated for meaning and judged as a deterministic expression. (shrink)

The main goal of part 1 is to challenge the widely held view that Poincaré orders the sciences in a hierarchy of dependence, such that all others presuppose arithmetic. Commentators have suggested that the intuition that grounds the use of induction in arithmetic also underlies the conception of a continuum, that the consistency of geometrical axioms must be proved through arithmetical induction, and that arithmetical induction licenses the supposition that certain operations form a group. I criticize each of these readings. (...) More fully, I argue that the justification Poincaré offers for the use of the group notion in geometry appears to extend to set-theoretic notions that would suffice to put arithmetic on a logical foundation, thus undermining his own case for the necessity of intuition in arithmetic. In part 2, I offer an interpretation of intuition’s role on which it justifies the use of group-theoretic, but not set-theoretic, notions. (shrink)

Part 1 of this article exposed a tension between Poincaré’s views of arithmetic and geometry and argued that it could not be resolved by taking geometry to depend on arithmetic. Part 2 aims to resolve the tension by supposing not merely that intuition’s role is to justify induction on the natural numbers but rather that it also functions to acquaint us with the unity of orders and structures and show practices to fit or harmonize with experience. I argue that in (...) this manner, intuition serves the epistemological function of warranting generalizations and justifying practices. In particular, it justifies the application of group-theoretic notions in geometry but not the use of set-theoretic notions in arithmetic. (shrink)

Hume’s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume’s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume’s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within (...) this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume’s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume’s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume’s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume’s most original contributions. First, Hume’s epistemological model invokes the “confounding” of indivisible minima to explain the appearance of spatial continuity. Second, Hume’s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the “loose judgements” of the vulgar. (shrink)

This thesis is directed towards a philosophy of music by attention to conceptions and perceptions of space. I focus on melody and harmony, and do not emphasise rhythm, which, as far as I can tell, concerns time rather than space. I seek a metaphysical account of Western Classical music in the diatonic tradition. More specifically, my interest is in wordless, untitled music, often called 'absolute' music. My aim is to elucidate a spatial approach to the world combined with a curiosity (...) about the nature of the Pythagorean intervals. Thus one question I seek to answer is: What do the Pythagorean intervals import to music? In their original formulation by Pythagoras, and further, by Plato, their import seemed mystical and analogous to a 'harmony of the spheres.' In this spatial approach, the Pythagorean intervals are indicative of infinite depth. The faculty of hearing alone does not ordinarily spatially locate sources. Hearing is ordinarily combined with sight in order to spatially locate a sound. Yet we talk about music as if it were spatially located, that is, as if it were a visible object with a surface and themes or 'objects'. I explore the faculty of vision and consider in what ways the processes of vision might be similar to and distinct from audition. My source for this approach is David Marr's book 'Vision' and his theory of the 2 1/2-dimensional sketch. In a medium in which there is no spatial location, it is important to situate the listener. Newton solved the problem of location by demonstrating the effects of gravity. P. F. Strawson claims that an analogy of distance is required in hearing to situate the listener, that is, a sense of nearer to and further from a source or object that permits a perception of distance. I claim that in music, the key signature and the scalar relations of musical themes provide this analogy of distance. The works of Plato, Aristotle, Descartes, Leibniz, and Newton are studied for their accounts of motion, bodies, and space. From these metaphysicians can be gleaned elements of music that include form, motion, timbre, dynamics, and attraction. By incorporating motion, I aim to show that although we may not know the true nature of space, we can learn from hearing music that space is perceptible in other than three dimensions. (shrink)

Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can be (...) also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

Much of Aristotle's thought developed in reaction to Plato's views, and this is certainly true of his philosophy of mathematics. To judge from his dialogue, the Meno, the first thing that struck Plato as an interesting and important feature of mathematics was its epistemology: in this subject we can apparently just “draw knowledge out of ourselves.” Aristotle certainly thinks that Plato was wrong to “separate” the objects of mathematics from the familiar objects that we experience in this world. His main (...) arguments on this point are in Chapter 2 of Book XIII of the Metaphysics. There are three distinct lines of argument: The first concerns the objects of geometry ; the second deals with the Platonist principles which are applied to arithmetic and geometry; the third is about substance as living things, especially animals, and perhaps man in particular. In addition to the above, this article also examines Aristotle's treatment of infinity. (shrink)

The fundamental question in reverse mathematics is to determine which set existence axioms are required to prove particular theorems of mathematics. In addition to being interesting in their own right, answers to this question have consequences in both effective mathematics and the foundations of mathematics. Before discussing these consequences, we need to be more specific about the motivating question.Reverse mathematics is useful for studying theorems of either countable or essentially countable mathematics. Essentially countable mathematics is a vague term that is (...) best explained by an example. Complete separable metric spaces are essentially countable because, although the spaces may be uncountable, they can be understood in terms of a countable basis. Simpson gives the following list of areas which can be analyzed by reverse mathematics: number theory, geometry, calculus, differential equations, real and complex analysis, combinatorics, countable algebra, separable Banach spaces, computability theory, and the topology of complete separable metric spaces. Reverse mathematics is less suited to theorems of abstract functional analysis, abstract set theory, universal algebra, or general topology.Section 2 introduces the major subsystems of second order arithmetic used in reverse mathematics: RCA0, WKL0, ACA0, ATR0 and – CA0. Sections 3 through 7 consider various theorems of ordered group theory from the perspective of reverse mathematics. Among the results considered are theorems on ordered quotient groups, groups and semigroup conditions which imply orderability, the orderability of free groups, Hölder's Theorem, Mal'tsev's classification of the order types of countable ordered groups. (shrink)

Kant's theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant's theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant's theory of arithmetic can (...) be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or "formulas"—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant's theory of addition as a form of synthetic operation. (shrink)

We describe a first order axiom set which yields the classical first order Euclidean geometry of Tarski when used with classical logic, and yields an intuitionistic Euclidean geometry when used with intuitionistic logic. The first order language has a single six place atomic predicate and no function symbols. The intuitionistic system has a computational interpretation in recursive function theory, that is, a realizability interpretation analogous to those given by Kleene for intuitionistic arithmetic and analysis. This interpretation shows the unprovability in (...) the intuitionistic theory of certain “nonconstructive” theorems of the classical geometry. (shrink)

Corcoran, John. 1974. Aristotelian Syllogisms: Valid arguments or true generalized conditionals?, Mind 83, 278–81. MR0532928 (58 #27178) This tightly-written and self-contained four-page paper must be studied and not just skimmed. It meticulously analyses quotations from Aristotle and Lukasiewicz to establish that Aristotle was using indirect deductions—as required by the natural-deduction interpretation—and not indirect proofs—as required by the axiomatic interpretation. Lukasiewicz was explicit and clear about the subtle fact that Aristotle’s practice could not be construed as correctly performed indirect proof. Lukasiewicz (...) evidence is presented fully; it is irrefutable. But, instead of considering the possibility that Aristotle’s discourses were not intended to express indirect proofs of universalized conditions presupposing axiomatic premises, Lukasiewicz came to the amazing conclusion that Aristotle did not understand indirect proof. This paper builds on the admirable Lukasiewicz scholarship to establish a conclusion diametrically opposed to the one Lukasiewicz asserted. This paper points out that if Aristotle had not been establishing an underlying logic but he was instead axiomatizing a theory of terms, Aristotelians could not claim for Aristotle the title Founder of Logic. People who take Euclid, Peano, and Zermelo to have first axiomatized geometry, arithmetic, and set theory, respectively, do not think that this qualifies them for the titles Founder of Geometry, Founder of Arithmetic, and Founder of Set Theory, respectively. Such people do think that this would qualify the three for the titles Founder of Axiomatic Geometry, Founder of Axiomatic Arithmetic, and Founder of Axiomatic Set Theory, respectively: titles that carry no honors in logic. Being the Founder of Axiomatic Term Theory would likewise carry no honors in the field of logic. CORRECTION: The paper inadvertently implied that Euclid was the first to axiomatize geometry. In order to understand Aristotle’s Analytics as a treatise on demonstration it helps to realize that axiomatized geometry was studied in Plato’s Academy when Aristotle was a student. Euclid was the author of the only ancient axiomatized geometry now available. (shrink)

The work of David R. Lachterman, The Ethics of Geometry, subtitled A Genealogy of Modernity, concerns essentially the status of geometry in Euclid’s Elements and in Descartes’s Geometry. It is a remarkable work, at once by the declared breadth of its ambitions and by the very great precision of its analyses, which are always supported by a prodigious philosophical culture. David Lachterman’s concern is to grasp, by way of an in-depth commentary of certain, particularly crucial passages of these two foundational (...) works, the change in geometry’s status from the one to the other, and this, as a sort of symptom of what the Moderns, since Descartes, will always experience as a necessary, inaugural rupture with regard to tradition. This change is underlaid, according to the author, by a profound change in the philosophical ethos, and in particular, in the geometer’s ethos. One must understand ethos, Lachterman explains, in the manner of Aristotle: those characteristic means that human beings have by which to act in the world or to behave in relation to one another, or to themselves. There is thus an “ethics” of geometry, namely, in the manner and the style of doing geometry, of behaving as a mathematician both toward apprentices and toward the veritable nature of the “entities” which are to be taught or learnt, and which give their discipline its name. That there is a difference of ethos between Euclid and Descartes implies, according to Lachterman, that there is also a difference in the source of intelligibility of the “mathematical,” understood in the most general sense. This in turn implies a deeper difference in the mode of being in general: it suffices to note that, in the ancient case, the source of intelligibility of the geometric figure lies in the nature of the figure itself, and that in the modern case, the same source is found in the “strategies” and “tactics” suited to bringing the figure to its visibility or to its embodiment, 315 in order to notice that this change in the mode of access to intelligibility must have had profound repercussions upon philosophical thought. It is in this sense that the author’s design is also, at the same time, not genetic, but genealogical. The break between ancient and modern mathematics pleas in favor of a consideration of the rupture of modern thought relative to ancient thought; thus, implicitly—and it is one of the great merits of this book that it shows this—against all homogenizing levelling, in Heidegger’s fashion, of the history of thought down to the history of “the” discipline of metaphysics. If there is something Heideggerian, in the best sense of the term, in Lachterman’s way of considering ethos as coextensive with modes of being, there is, on the other hand, and we ought to be very glad of it, something anti-Heideggerian in his original manner of bringing out the effective novelty of modern thought. For, according to him, modern thought is no longer to be defined, explicitly or exclusively, by the so-called “metaphysics of subjectivity.” Instead, it is defined by a self-regulated, operative, and constructive productivity of thought—and this as much with regard to itself as to its objects. To put it briefly, whereas ancient thought is rather polarized by the logico-eidetic, modern thought is polarized by a sort of methodical constructivism, the status of which is, in reality, very complex. If Descartes speaks of “the construction of a problem,” while Leibniz speaks of the “construction of an equation,” and Kant of the “construction of a concept,” everything depends—as we might guess—upon the multiple meanings that the term ‘construction’ may take on. Is it to be taken in the sense of an absolute creation, a quasi ex nihilo creation, which would render mathematics divine? Is it the creation of artefacts, or again, is it the methodical exploration of problems posed to thought by the existence of objects or corresponding ideas, themselves supposed to be somewhere in the divine understanding? We already discern the breadth and the depth of the debate, and we sense that the essential will be played out in Lachterman’s commentaries on the famous Cartesian solution of the problem of Pappus—the birth certificate, as we know, of analytic geometry. (shrink)

The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical responses to (...) these changes, focusing on the idea of modifying Kant’s conception of intuition in order to accommodate the increasing abstractness of mathematics. It is argued that far from clinging to an outdated paradigm, programs based on new conceptions of intuition should be seen as motivated by important philosophical desiderata, such as the truth, apriority, distinctiveness and autonomy of mathematics. (shrink)

I provide a survey of the contents of the works belonging to Aristotle's Organon in order to define their nature, in the light of his declared intentions and of other indications (mainly internal ones) about his purposes. No unifying conception of logic can be found in them, such as the traditional one, suggested by the very title Organon, of logic as a methodology of demonstration. Logic for him can also be formal logic (represented in the main by the (...) De Interpretatione), axiomatized syllogistic (represented in the main by the Prior Analytics) and a methodology of dialectical and rhetorical discussion. The consequent lack of unity presented by those works does not exclude that both the set of works called Analytics and the set of works concerning dialectic (Topics and Sophistici Elenchi) form a unity, and that a certain priority is attributed to the analytics with respect to dialectic. (shrink)

Kant’s theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant’s theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant’s theory of arithmetic can (...) be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or “formulas”—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant’s theory of addition as a form of synthetic operation. (shrink)

I examine the passages where Aristotle maintains that intellectual activity employs φαντάσματα (images) and argue that he requires awareness of the relevant images. This, together with Aristotle’s claims about the universality of understanding, gives us reason to reject the interpretation of Michael Wedin and Victor Caston, on which φαντάσματα serve as the material basis for thinking. I develop a new interpretation by unpacking the comparison Aristotle makes to the role of diagrams in doing geometry. In theoretical understanding of mathematical and (...) natural beings, we usually need to employ appropriate φαντάσματα in order to grasp explanatory connections. Aristotle does not, however, commit himself to thinking that images are required for exercising all theoretical understanding. Understanding immaterial things, in particular, may not involve employing phantasmata. Thus the connection that Aristotle makes between images and understanding does not rule out the possibility that human intellectual activity could occur apart from the body. (shrink)

This article addresses Rancière’s critique of Aristotle’s political theory as parapolitics in order to show that Aristotle is a resource for developing an inclusionary notion of political community. Rancière argues that Aristotle attempts to cut off politics and merely police (maintain) the community by eliminating the political claim of the poor by including it. I respond to three critiques that Rancière makes of Aristotle: that he ends the political dispute by including the demos in the government; that he includes the (...) free masses only incidentally because, by chance, the rule is political, not based on mastery; and that he attempts to close off politics by determining in advance what speech counts as speech. I argue that Aristotle attempts to institute this contest by accepting the incommensurable claims of arithmetic and geometric equality. Aristotle puts this conflict at the root of political life in such a way that serves Rancière’s larger project of perpetuating politics. (shrink)

This paper explores Aristotle's remarks in Posterior Analytics on certain special disciplines that are subordinate to pure mathematical sciences. Optics, harmonics and mechanics prove their own contents by means of premises belonging to arithmetic or geometry. Even though subaltern sciences are exceptions to the prohibition on kind crossing, the premises to their demonstrations are legitimately appropriate to the relative conclusions. In order to delineate the demonstrative structure of subordinate sciences, Aristotle introduces the distinction between knowledge of a fact and knowledge (...) of the reason for it. In his view, these two different levels of knowledge, characterizing respectively empirical and theoretical approaches, are closely related. Nautical astronomy, for instance, deals with the observation and recording of the astral motions, which are made intelligible by mathematical astronomy. The Aristotelian ideal of an explanatory connection between appearances and mathematical principles seems to be the main aspect in the treatment of subordinate sciences in Posterior Analytics. (shrink)

We carry out a systematic study of decidability for theories of real vector spaces, inner product spaces, and Hilbert spaces and of normed spaces, Banach spaces and metric spaces, all formalized using a 2-sorted first-order language. The theories for list turn out to be decidable while the theories for list are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic.We find that the purely universal and (...) purely existential fragments of the theory of normed spaces are decidable, as is the ∀∃ fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbertʼs 10th problem show that the ∃∀ fragments for metric and normed spaces and the ∀∃ fragment for normed spaces are all undecidable. (shrink)

In 1675, Leibniz elaborated his longest mathematical treatise he everwrote, the treatise ``On the arithmetical quadrature of the circle, theellipse, and the hyperbola. A corollary is a trigonometry withouttables''. It was unpublished until 1993, and represents a comprehensive discussion of infinitesimalgeometry. In this treatise, Leibniz laid the rigorous foundation of thetheory of infinitely small and infinite quantities or, in other words,of the theory of quantified indivisibles. In modern terms Leibnizintroduced `Riemannian sums' in order to demonstrate the integrabilityof continuous functions. The (...) article deals with this demonstration,with Leibniz's handling of infinitely small and infinite quantities,and with a general theorem regarding hyperboloids. (shrink)

A survey of Kant's views on space, time, geometry and the synthetic nature of mathematics. I concentrate mostly on geometry, but comment briefly on the syntheticity of logic and arithmetic as well. I believe the view of many that Kant's system denied the possibility of non-Euclidean geometries is clearly mistaken, as Kant himself used a non-Euclidean geometry (spherical geometry, used in his day for navigational purposes) in order to explain his idea, which amounts to an anticipation of the later discovery (...) of the general concept of non- Euclidean geometries. Kant's view of geometry and arithmetic as synthetic was, I believe, essentially correct, in that geometry and arithmetic are both synthetic a priori if considered as branches of mathematics independent of the rest of mathematics. However, the view that somehow logic is analytic, while mathematics is synthetic for Kantian reasons, is mistaken. All three disciplines—logic, arithmetic and geometry—are synthetic as disciplines independent from one another. However, they have a common basis, recursion theory, which I prefer to identify with mathematics as a whole. As a result, I do not say, as is often considered to be the Kantian view, that mathematics is synthetic while logic is analytic. Rather, I prefer to say that mathematics is analytic, while logic is synthetic. This is perfectly consistent with Kant's system, since it was arithmetic and geometry individually that he argued were synthetic. What Kant called the analytic is recursion theory, which could be considered as a basic formulation of mathematics or logic—or better, both mathematics and logic could be recognized as essentially the same discipline. However, if "logic" is taken to mean "predicate logic", as is often the case in modern times, then it is mathematics that is closer to Kant's analytic, not logic. Such ambiguities, of course, can be avoided by simply associating Kant's analytic with recursion theory, and avoiding the controversies as to what counts as mathematics or logic.. (shrink)

ABSTRACT This article addresses Rancière's critique of Aristotle's political theory as parapolitics in order to show that Aristotle is a resource for developing an inclusionary notion of political community. Rancière argues that Aristotle attempts to cut off politics and merely police the community by eliminating the political claim of the poor by including it. I respond to three critiques that Rancière makes of Aristotle: that he ends the political dispute by including the demos in the government; that he includes the (...) free masses only incidentally because, by chance, the rule is political, not based on mastery; and that he attempts to close off politics by determining in advance what speech counts as speech. I argue that Aristotle attempts to institute this contest by accepting the incommensurable claims of arithmetic and geometric equality. Aristotle puts this conflict at the root of political life in such a way that serves Rancière's larger project of perpetuating politics. (shrink)

We give a consistency proof within a weak fragment of arithmetic of elementary algebra and geometry. For this purpose, we use EFA (exponential function arithmetic), and various first order theories of algebraically closed fields and real closed fields.

This is the first critical edition of the first and second Epistles of the Brethren Purity--the Rasa 'il--in Arabic with a fully annotated English translation. It presents technical and epistemic analyses of mathematical concepts and their metaphysical bases, and an overview of the mathematical sciences within Islamic intellectual milieu.

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $$\Pi _2$$ -property formalized in an appropriate language for second order number theory is forcible (...) from some $$T\supseteq \mathsf {ZFC}+$$ large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of $$H_{\omega _1}$$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for $$\Delta _0$$ -properties and for all universally Baire sets of reals. We will extend these results also to the theory of $$H_{\aleph _2}$$ in a follow up of this paper. (shrink)

This paper examines the ontological commitments of the second-order language of arithmetic and argues that they do not extend beyond the first-order language. Then, building on an argument by George Boolos, we develop a Tarski-style definition of a truth predicate for the second-order language of arithmetic that does not involve the assignment of sets to second-order variables but rather uses the same class of assignments standardly used in a definition for the first-order language.

Since the discovery of incommensurability in ancient Greece, arithmeticism and geometricism constantly switched roles. After ninetieth century arithmeticism Frege eventually returned to the view that mathematics is really entirely geometry. Yet Poincaré, Brouwer, Weyl and Bernays are mathematicians opposed to the explication of the continuum purely in terms of the discrete. At the beginning of the twenty-first century ‘continuum theorists’ in France (Longo, Thom and others) believe that the continuum precedes the discrete. In addition the last 50 years witnessed the (...) revival of infinitesimals (Laugwitz and Robinson—non-standard analysis) and—based upon category theory—the rise of smooth infinitesimal analysis and differential geometry. The spatial whole-parts relation is irreducible (Russell) and correlated with the spatial order of simultaneity. The human imaginative capacities are connected to the characterization of points and lines (Euclid) and to the views of Aristotle (the irreducibility of the continuity of a line to its points), which remained in force until the ninetieth century. Although Bolzano once more launched an attempt to arithmetize continuity, it appears as if Weierstrass, Cantor and Dedekind finally succeeded in bringing this ideal to its completion. Their views are assessed by analyzing the contradiction present in Grünbaum’s attempt to explain the continuum as an aggregate of unextended elements (degenerate intervals). Alternatively a line-stretch is characterized as a one-dimensional spatial subject, given at once in its totality (as a whole) and delimited by two points—but it is neither a breadthless length nor the (shortest) distance between two points. The overall aim of this analysis is to account for the uniqueness of discreteness and continuity by highlighting their mutual interconnections exemplified in the nature of a line as a one-dimensional spatial subject, while acknowledging that points are merely spatial objects which are always dependent upon an extended spatial subject. Instead of attempting to reduce continuity to discreteness or discreteness to continuity, a third alternative is explored: accept the irreducibility of number and space and then proceed by analyzing their unbreakable coherence. The argument may be seen as exploring some implications of the view of John Bell, namely that the “continuous is an autonomous notion, not explicable in terms of the discrete.” Bell points out that initially Brouwer, in his dissertation of 1907, “regards continuity and discreteness as complementary notions, neither of which is reducible to each other.”. (shrink)

In the philosophy of mathematics, as in its a meta-domain, we find that the words as: consequentialism, implicativity, operationalism, creativism, fertility, … grasp at most of mathematical essence and that the questions of truthfulness, of common sense, or of possible models for (otherwise abstract) mathematical creations,i.e. of ontological status of mathematical entities etc. - of second order. Truthfulness of (necessary) succession of consequences from causes in the science of nature is violated yet with Hume, so that some traditional footings of (...) logico-mathematical conclusions should equally be falled under suspicion in the last century. We have in mind, say, strict-material implication which led the emergence of relevance logics, or the law of excluded middle that denied intuitionists i.e. paraconsistent logical systems where the contradiction is allowed, as well as the quantum logic which doesn't know, say, the definition of implication etc. Kant's beliefs miscarried hereafter that number (arithmetic) and form (geometry) would bring a (finite) truth on space and time, when they revealed relative and curvated, just as it is contradictory essentially understanding of basic phenomena in the nature: of light as an unity of wave – particle, or that both "exist" and "don't exist" numbers as powers of sets between 0א and c (the independence of continuum hypothesis) etc. Mathematical truths are ''truths of possible worlds'', in which we have only to believe that they will meet once recognizable models in reality. At last, we argue in favour of thesis that a possible representing "in relief" of mathematical entities and relations in the "noetic matter" (Aristotle) would be of a striking heuristic character for this science. (shrink)

IN BOTH THE EUDEMIAN ETHICS AND THE NICOMACHEAN ETHICS, Aristotle says that the aim of ethical inquiry is a practical one; we want to know what virtue is so that we may become good ourselves and thereby do well and be happy. By classifying ethical inquiry as a practical endeavor, Aristotle is rejecting a view that he attributes to Socrates according to which ethics is a kind of theoretical science. In theoretical sciences, such as geometry or astronomy, the knowledge of (...) a particular subject matter is sought as an end in itself, and the possession of such knowledge is sufficient to make one a geometer or an astronomer. In rejecting this model Aristotle argues that the knowledge of virtue is sought not solely for itself but in order to inform praxis and in order that we become virtuous and good, not by knowing what the virtues are but by cultivating them in practice. (shrink)

Constructive ZF + full separation is shown to be equiconsistent with Second Order Arithmetic.

We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third author in [V. (...) G. Kanovei, On descriptive forms of the countable axiom of choice, Investigations on nonclassical logics and set theory, Work Collect., Moscow, 3-136 ]. (shrink)