A simple interpretation of quantity calculus is given. Quantities are described as two-place functions from objects, states or processes (or some combination of them) into numbers that satisfy the mutual measurability property. Quantity calculus is based on a notational simplification of the concept of quantity. A key element of the simplification is that we consider units to be intentionally unspecified numbers that are measures of exactly specified objects, states or processes. This interpretation of quantity (...) calculus combines all the advantages of calculating with numerical values (since the values of quantities are numbers, we can do with them everything we do with numbers) and all the advantages of calculating with standardly conceived quantities (calculus is invariant to the choice of units and has built-in dimensional analysis). This also shows that the standard metaphysics and mathematics of quantities and their magnitudes is not needed for quantity calculus. At the end of the article, arguments are given that the concept of quantity as defined here is a pivotal concept in understanding the quantitative approach to nature. As an application of this interpretation of quantity calculus, an easy proof of dimensional homogeneity of physical laws is given. (shrink)

A mathematical model in science can be formulated as a counterfactual conditional, with the model’s assumptions in the antecedent and its predictions in the consequent. Interestingly, some of these models appear to have assumptions that are metaphysically impossible. Consider models in ecology that use differential equations to track the dynamics of some population of organisms. For the math to work, the model must assume that population size is a continuous quantity, despite that many organisms are necessarily discrete. This means (...) our counterfactual representation of the model can have an impossible antecedent, giving us a counterpossible. Analogous counterpossibles arise in other sciences, as we’ll see. According to a prominent view in counterfactual semantics, the vacuity thesis, all counterpossibles are vacuously true, that is, true merely because their antecedents are necessarily false. But some counterpossible formulations of differential equation models in science are not all vacuously true—some are non-vacuously true, and some are false. I go on to show how an alternative semantics, one that employs impossible worlds, can deliver this judgment. (shrink)

Thomists do not have an account of how modern mathematics relates to classical mathematics or more generally fits into the Aristotelian hierarchy of sciences. Rather than treat primarily of Aquinas’s theses on mathematical abstraction, I turn to considering what modern mathematics is in itself, seen from a broadly classical perspective. I argue that many modern quantities can be considered to be, not quantities as such or in themselves, but derived quantities, i.e., quantities that can be defined wholly in terms of (...) the principles of number or magnitude. I also interpret the parts of modern mathematics that study quantitative change as being properly-speaking parts of natural philosophy, for example, probability theory, statistics, calculus, etc. In conclusion, I consider the place that quantity as such has in the order of the world and why we should expect the world to be highly mathematical, as we have found it to be. (shrink)

In this paper, we propose a new method to deal with continuously varying quantity in the situation calculus based on the concept of the nonstandard analysis. The essential point of the method is to devise a new model called nonstandard situation calculus, which is an interpretation of the situation calculus in the set of hyperreals. This nonstandard model allows discrete but uncountable (hyperfinite) state transition, so that we can describe and reason about the continuous dynamics which (...) are usually treated with differential equations. In this enlarged perspective of the nonstandard situation calculus, we discuss about an infinitesimal action, infinite plan and an abnormality theory for the temporal prediction based on the differentiability. (shrink)

In his Foundations of a General Theory of Manifolds, Georg Cantor praised Bernard Bolzano as a clear defender of actual infinity who had the courage to work with infinite numbers. At the same time, he sharply criticized the way Bolzano dealt with them. Cantor’s concept was based on the existence of a one-to-one correspondence, while Bolzano insisted on Euclid’s Axiom of the whole being greater than a part. Cantor’s set theory has eventually prevailed, and became a formal basis of contemporary (...) mathematics, while Bolzano’s approach is generally considered a step in the wrong direction. In the present paper, we demonstrate that a fragment of Bolzano’s theory of infinite quantities retaining the part-whole principle can be extended to a consistent mathematical structure. It can be interpreted in several possible ways. We obtain either a linearly ordered ring of finite and infinitely great quantities, or a partially ordered ring containing infinitely small, finite and infinitely great quantities. These structures can be used as a basis of the infinitesimal calculus similarly as in non-standard analysis, whether in its full version employing ultrafilters due to Abraham Robinson, or in the recent “cheap version” avoiding ultrafilters due to Terence Tao. (shrink)

Newton's use of mathematics in mechanics was justified by him from his neo-platonician conception of the physical world that was going along with his «absolute, true and mathematical concepts» such as space, time, motion, force, etc. But physics, afterwards, although it was based on newtonian dynamics, meant differently the legitimacy of being mathematized, and this difference can be seen already in the works of eighteenth century «Geometers» such as Euler, Clairaut and d'Alembert (and later on Lagrange, Laplace and others). Despite (...) their inheritance of Newton's achievements, they understood differently the meaning and use of mathematical quantities for physics, in a way that was more neutral to metaphysics. The continental reception and assimilation of Newton's Principia had indeed occured as its budding onto Leibniz' calculus and a cartesian conception of rationality (spread in particular by the malebranchist disciples of Leibniz). This new thought of the legitimacy of mathematization is clearly at variance with Descartes' identification of physics with geometry, but it nevertheless can be traced back to Descartes' conception of magnitudes, as it was developed and analyzed from the notion of dimension in his Regulæ ad directionem ingenii (in particular, rule 14). This idea can be followed afterwards with further philosophical or mathematical specifications through authors such as Kant, Riemann and others. This inquiry into the original thought of magnitudes, and of physical magnitudes conceived through mathematization, leads us to suggest an extension of meaning for the concept of physical magnitude that puts emphasis on its relational and structural aspects rather than restraining it to a simple «numerically valued» acception. Such a broadening would have immediate implications on our comprehension of «non classical» aspects of contemporary physics in the quantum area and in dynamical systems. (shrink)

In this paper, we endeavour to give a historically accurate presentation of how Leibniz understood his infinitesimals, and how he justified their use. Some authors claim that when Leibniz called them “fictions” in response to the criticisms of the calculus by Rolle and others at the turn of the century, he had in mind a different meaning of “fiction” than in his earlier work, involving a commitment to their existence as non-Archimedean elements of the continuum. Against this, we show (...) that by 1676 Leibniz had already developed an interpretation from which he never wavered, according to which infinitesimals, like infinite wholes, cannot be regarded as existing because their concepts entail contradictions, even though they may be used as if they exist under certain specified conditions—a conception he later characterized as “syncategorematic”. Thus, one cannot infer the existence of infinitesimals from their successful use. By a detailed analysis of Leibniz’s arguments in his De quadratura of 1675–1676, we show that Leibniz had already presented there two strategies for presenting infinitesimalist methods, one in which one uses finite quantities that can be made as small as necessary in order for the error to be smaller than can be assigned, and thus zero; and another “direct” method in which the infinite and infinitely small are introduced by a fiction analogous to imaginary roots in algebra, and to points at infinity in projective geometry. We then show how in his mature papers the latter strategy, now articulated as based on the Law of Continuity, is presented to critics of the calculus as being equally constitutive for the foundations of algebra and geometry and also as being provably rigorous according to the accepted standards in keeping with the Archimedean axiom. (shrink)

The translation of Newton’s geometrical Propositions in the Principia into the language of the differential calculus in the form developed by Leibniz and his followers has been the subject of many scholarly articles and books. One of the most vexing problems in this translation concerns the transition from the discrete polygonal orbits and force impulses in Prop. 1 to the continuous orbits and forces in Prop. 6. Newton justified this transition by lemma 1 on prime and ultimate ratios which (...) was a concrete formulation of a limit, but it took another century before this concept was established on a rigorous mathematical basis. This difficulty was mirrored in the newly developed calculus which dealt with differentials that vanish in this limit, and therefore were considered to be fictional quantities by some mathematicians. Despite these problems, early practitioners of the differential calculus like Jacob Hermann, Pierre Varignon, and Johann Bernoulli succeeded without apparent difficulties in applying the differential calculus to the solution of the fundamental problem of orbital motion under the action of inverse square central forces. By following their calculations and describing some essential details that have been ignored in the past, I clarify the reason why the lack of rigor in establishing the continuum limit was not a practical problem. (shrink)

It is well known that over the eighteenth century the calculus moved away from its geometric origins; Euler, and later Lagrange, aspired to transform it into a “purely analytical” discipline. In the 1780 s, the Portuguese mathematician José Anastácio da Cunha developed an original version of the calculus whose interpretation in view of that process presents challenges. Cunha was a strong admirer of Newton (who famously favoured geometry over algebra) and criticized Euler’s faith in analysis. However, the fundamental (...) propositions of his calculus follow the analytical trend. This appears to have been possible due to a nominalistic conception of variable that allowed him to deal with expressions as names, rather than abstract quantities. Still, Cunha tried to keep the definition of fluxion directly applicable to geometrical magnitudes. According to a friend of Cunha’s, his calculus had an algebraic (analytical) branch and a geometrical branch, and it was because of this that his definition of fluxion appeared too complex to some contemporaries. (shrink)

If the import of a book can be assessed by the problem it takes on, how that problem unfolds, and the extent of the problem’s fruitfulness for further exploration and experimentation, then Duffy has produced a text worthy of much close attention. Duffy constructs an encounter between Deleuze’s creation of a concept of difference in Difference and Repetition (DR) and Deleuze’s reading of Spinoza in Expressionism in Philosophy: Spinoza (EP). It is surprising that such an encounter has not already been (...) explored, at least not to this extent and in this much detail. Since the two works were written simultaneously, as Deleuze’s primary and secondary dissertations, it is to be expected that there is much to learn from their interaction. Duffy proceeds by explicating, in terms of the differential calculus, a logic of what Deleuze in DR calls different/ciation, and then maps this onto Deleuze’s account of modal expression in EP. (shrink)

This book is part of a major project undertaken by the Centre for Studies in Civilizations , being one of a total of ninety-six planned volumes. The author is a statistician and computer scientist by training, who has concentrated on historical matters for the last ten years or so. The book has very ambitious aims, proposing an alternative philosophy of mathematics and a deviant history of the calculus. Throughout, there is an emphasis on the need to combine history and (...) philosophy of mathematics, especially in order to evaluate properly the history of mathematics in India, in particular the history of the calculus.The pivotal goals of the book are to oppose the Eurocentric account of the history of science in general and mathematics in particular; to avoid the usual philosophical idea of the centrality of proof for mathematical knowledge, in favour of the traditional Indian notion of pramāṇa [validation] encompassing empirical elements and emphasizing calculation; to analyze the thousand-year-long development of infinite series in India, starting in the fifth century; and to show ‘how and why the calculus was imported into Europe’ from about 1500. The result is a picture in which inputs from the Indian subcontinent and epistemology are the driving forces of the history of mathematics, as people in the European subcontinent struggle to adopt new calculation techniques from the East in spite of Western philosophico-religious biases . Thus, a ‘first math war’ involved the algorismus de numero indorum, adopted for practical reasons, which forced Europeans to modify their epistemology of number and quantities. A ‘second math war’ revolved around infinite series and the calculus, since the background of Western epistemology created ‘spurious difficulties’ about infinities and infinitesimals, partially resolved with theories of real numbers. And a ‘third math war’ is …. (shrink)

The base units of the SI include six units of continuous quantities and the mole, which is defined as proportional to the number of specified elementary entities in a sample. The existence of the mole as a unit has prompted comment in Metrologia that units of all enumerable entities should be defined though not listed as base units. In a similar vein, the BIPM defines numbers of entities as quantities of dimension one, although without admitting these entities as base units. (...) However, there is a basic ontological distinction between continuous quantities and enumerable aggregates. The distinction is the basis of the difference between real and natural numbers. This paper clarifies the nature of the distinction: (i) in terms of a set of measurement axioms stated by Hölder; and (ii) using the formalism known in metrology as quantity calculus. We argue that a clear and unambiguous scientific distinction should be made between measurement and enumeration. We examine confusion in metrological definitions and nomenclature concerning this distinction, and discuss the implications of this distinction for ontology and epistemology in all scientific disciplines. (shrink)

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, (...) contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality. (shrink)

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. (...) We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropie quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer's calculus of 'conditional density operators'. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category. (shrink)

Hegel’s conception of becoming can be said to arise from his intense confrontation with the debates of his time on continuous and mathematical infinite, in which also the thesis about time and movement of Aristotle’s Physics converge. According to Hegel, mathematical infinite already includes the true infinite, which is essentially relation. Basing his insights on the Newtonian rather than on the Leibnizian theory of differential calculus, Hegel draws the idea that the quantum exists only as a ratio; as such, (...) however, the quantum sublates itself into the quality. The continuous is for Hegel the concept of this unceasing sublating of the quantity into the quality : as some of Plato’s claims show in the Parmenides, the instant is the limit of this transition into the other. The Hegelian concept of Aufhebung is this operation of “passage to limit”. In it, the process of the quantitative increasing gives rise to a new figure, to something heterogeneous compared with the previous figure, as in the case of the polygon inscribed in a circle, whose sides are multiplied to infinity until it becomes the circle itself. So understood, the becoming – as well as its metaphysical presupposition, i.e.the true infinite – is an unceasing “passage into the other”: the continuous is not a succession of homogeneous entities, but the process of the unceasing differentiation of the real. In this paper, I argue that this conception of the becoming is the presupposition of the process of liberation that, according to Hegel, is immanent to the becoming itself: if the differentiation happens at every instant, the liberation is, correspondently, possible at every instant. (shrink)

Hegel on Being provides an authoritative treatment of Hegel's entire logic of being. Stephen Houlgate presents the Science of Logic as an important and neglected text within Hegel's oeuvre that should hold a more significant place in the history of philosophy. In the Science of Logic, Hegel set forth a distinctive conception of the most fundamental forms of being through ideas on quality, quantity and measure. Exploring the full trajectory of Hegel's logic of being from quality to measure, this (...) two-volume work by preeminent Hegel scholar, Houlgate situates Hegel's text in relation to the work of Plato, Aristotle, Descartes, Spinoza, Kant, and Frege. Volume I: Quality and the Birth of Quantity in Hegel's 'Science of Logic' covers all material on the purpose and method of Hegel's dialectical logic and charts the crucial transition from the concept of quality to that of quantity, as well as providing an original account of Hegel's critique of Kant's antinomies across two chapters. Volume II: Quantity and Measure in Hegel's 'Science of Logic' continues the discussion of Hegel's logic of being and considers all aspects of quantity and measure in his logic, including his basic categories of being, writings on calculus, philosophy of mathematics, as well as a comparative study of Hegel and Frege's approach to logic. Lucidly written, with characteristic philosophical depth and analysis, Houlgate's Hegel on Being explicates one of Hegel's most complex works, providing a vital reference for a generation of Hegel scholars and a major contribution to the literature on 19th century German philosophy. (shrink)

Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his (...) Leibnizian sympathies, Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity. (shrink)

Starting from the geometric calculus based on Clifford algebra, the idea that physical quantities are Clifford aggregates (“polyvectors”) is explored. A generalized point particle action (“polyvector action”) is proposed. It is shown that the polyvector action, because of the presence of a scalar (more precisely a pseudoscalar) variable, can be reduced to the well known, unconstrained, Stueckelberg action which involves an invariant evolution parameter. It is pointed out that, starting from a different direction, DeWitt and Rovelli postulated the existence (...) of a clock variable attached to particles which serve as a reference system for identification of spacetime points. The action they postulated is equivalent to the polyvector action. Relativistic dynamics (with an invariant evolution parameter) is thus shown to be based on even stronger theoretical and conceptual foundations than usually believed. (shrink)

This chapter focuses on one of the common fallacies in Western philosophy, “two wrongs make a right”. If the notion that “two wrongs make a right” seems familiar and peculiarly stated, it may be because we moreover hear it in other more commonly rendered forms. To say “two wrongs do not make a right”, necessarily implies a wholesale condemnation of retributive justice. Retributive justice, despite its largely sanitized form in contemporary society, retains the core idea that justice can be achieved (...) by the extraction of some quantity of something, such as time. With restorative justice the aim is that both parties are meant to achieve some kind of healing and consolation. But the essentially rhetorical nature of the claim, “two wrongs make a right”, assures us that no such logic is in place, and worse, that the so‐called justifications for committing wrongful acts are not derived from a rigorous calculus but rather through blunt insistence. (shrink)

Preface. I. BASICS OF LOGIC. Introduction. The Structure of Simple Statements. The Structure of Complex Statements. Simple and Complex Properties. Validity. 2. PROBABILITY AND INDUCTIVE LOGIC. Introduction. Arguments. Logic. Inductive versus Deductive Logic. Epistemic Probability. Probability and the Problems of Inductive Logic. 3. THE TRADITIONAL PROBLEM OF INDUCTION. Introduction. Hume’s Argument. The Inductive Justification of Induction. The Pragmatic Justification of Induction. Summary. IV. THE GOODMAN PARADOX AND THE NEW RIDDLE OF INDUCTION. Introduction. Regularities and Projection. The Goodman Paradox. The Goodman (...) Paradox, Regularity, and the Principle of the Uniformity of Nature. Summary. 5. MILL’S METHODS OF EXPERIMENTAL INQUIRY AND THE NATURE OF CAUSALITY. Introduction. Causality and Necessary and Sufficient Conditions. Mill’s Methods. The Direct Method of Agreement. The Inverse Method of Agreement. The Method of Difference. The Combined Methods. The Application of Mill’s Methods. Sufficient Conditions and Functional Relationships. Lawlike and Accidental Conditions. 6. THE PROBABILITY CALCULUS. Introduction. Probability, Arguments, Statements, and Properties. Disjunction and Negation Rules. Conjunction Rules and Conditional Probability. Expected Value of a Gamble. Bayes’ Theorem. Probability and Causality. 7. KINDS OF PROBABILITY. Introduction. Rational Degree of Belief. Utility. Ramsey. Relative Frequency. Chance. 8. PROBABILITY AND SCIENTIFIC INDUCTIVE LOGIC. Introduction. Hypothesis and Deduction. Quantity and Variety of Evidence. Total Evidence. Convergence to the Truth. ANSWERS TO SELECTED EXERCISES. INDEX. (shrink)

The paper aims to put certain basic mathematical elements and operations into an empirical perspective, evaluate the empirical status of various analytic operations widely used within psychology and suggest alternatives to procedures criticized as inadequate. Experimentation shows the "manyness" of items to be a perceptual quality for both young children and animals and that natural operations are performed by naive children analogous to those performed by persons tutored in arithmetic. Number, counting, arithmetic operations therefore can make distinctions that are not (...) inevitably arbitrary, and conceptual operations can obviously have a status as natural events with psychology. If the elements and conceptual operations involved in mathematical systems were not inherent in physiological process, various primitive discriminations could not be possible. Also, since some calculi have a natural status in a given empirical circumstance, the axioms of others can not be satisfied. Therefore the psychologist when acting as an empirical scientist seeks a calculus having a structure whose elements are isomorphic with natural units of stimulus and response and whose operations are isomorphic with whatever natural processes are involved. Measurement poses a special problem for the empirical scientist. It concerns but a single class of natural qualities and this only in a limited way. The concept of quantity has a natural counterpart but quantity and measurement are not wholly analogous. Measurement is defined, as H. S. Leonard suggests, as a theoretical activity. Measurement theory has a formal structure but empirical end. Measurement hypothesizes about the position of a particular quality within a definite range of qualities. It therefore is beholden to definite empirical restrictions. Some hypotheses-making systems use terms and relations per se as the context and starting point for dealing with discriminable events. Such procedures are 'transcendent." In empirical science, questions are part of problem-solving activity and their reference is naturally restricted. In providing description and explanation, psychological researchers frequently use calculi in a transcendent way. This results in theories that are only quasi-empirical and "half" true. The roles measurement plays in psychology are discussed. Of particular concern are those cases in which the results of measuring or a theory of measurement are used to define the "real" units, or the "real" relations involved in problematic psychological events, and thence to describe and explain behaviors of interest. Metaphysical or ontological usages of measurement sometimes occur. The implication of these arguments with regard to a view of empirical science is discussed. (shrink)

Integrated information theory (IIT) starts from the essential properties of experience and translates them into requirements that any physical system must satisfy to be conscious. It argues that the physical substrate of consciousness (PSC) must constitute a maximum of irreducible, internal cause‐effect power of a specific form, and provides a calculus to determine, in principle, both the quality and the quantity of an experience. Applied to the brain, IIT predicts that the spatio‐temporal grain of the neural units constituting (...) the PSC, and the relevant neural states, are those that maximize cause‐effect power. Moreover, the PSC can shrink, move, split and disintegrate depending on various anatomical and physiological parameters. These predictions are testable with brain stimulation and recording experiments. The theory can explain parsimoniously many known facts about the relationship between consciousness and the brain, including its association with certain cortical structures, its breakdown in deep sleep, anesthesia and seizures, and its return in dreams. (shrink)

In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal (...) Analysis, as propounded by John Bell. I find some salient differences, especially with regard to higher-order infinitesimals. I illustrate these differences by a consideration of how each approach might be applied to propositions of Newton’s Principia concerning the derivation of force laws for bodies orbiting in a circle and an ellipse. “If the Leibnizian calculus needs a rehabilitation because of too severe treatment by historians in the past half century, as Robinson suggests (1966, 250), I feel that the legitimate grounds for such a rehabilitation are to be found in the Leibnizian theory itself.”—(Bos 1974–1975, 82–83). (shrink)

In this paper, we will consider the theoretical aspects of Leibniz’s thought on infinitely small and infinite quantities in the context of the natural philosophy developed by him in the Parisian period. We will hold that in the texts of this period an attempt of problematizing concepts of infinitary mathematics is found, which is not in the strictly mathematical texts. In this perspective, we also propose that there is in Leibniz a “double methodological record” concerning the question of the infinity (...) and of the infinitesimal. As a mathematician, Leibniz is concerned about technical questions related to the construction of the infinitesimal calculus, in its different aspects, without be concerned about the philosophical challenges its adoption implies. However, as a philosopher and a metaphysician, Leibniz is obliged to deal with epistemological and ontological problems that emerge from the adoption of technical resources in matters of infinitesimal mathematics. (shrink)

François Viète is often regarded as the first modern mathematician on the grounds that he was the first to develop the literal notation, that is, the use of two sorts of letters, one for the unknown and the other for the known parameters of a problem. The fact that he achieved neither a modern conception of quantity nor a modern understanding of curves, both of which are explicit in Descartes’ Geometry, is to be explained on this view “by an (...) incomplete symbolization rather than by any obstacle intrinsic in the system.” Descartes’ Geometry provides only a “clearer expression” of themes already sounded in Viète’s work, one that perfects Viète’s literal calculus and gives it “its modern form”; it merely continues the “‘new’ and ‘pure’ algebra which Viète first established as the ‘general analytic art’.” It can seem, furthermore, that this must be right, that had there been some obstacle intrinsic to Viète’s system that barred the way to a modern conception of quantity and a modern understanding of curves, then Descartes’ Geometry would have had to have taken a very different form than it did. As it was, Descartes had only to improve Viète’s symbolism, free himself of the last vestiges of the ancient view of geometrical and arithmetical objects, and apply the new symbolism to the study of curves in order to achieve what Viète did not but could have. But what, really, is the status of this “could have”; what would it actually have taken for Viète to achieve Descartes’ results in the Geometry? The interest of the question lies in its potential to better our understanding of the nature of modern mathematics. (shrink)

While many Christians accept the claim that giving to support the poor and needy is a core moral and religious obligation, most Christian giving is usually not very efficient in EA terms. In this paper, I explore possibilities for productive collaboration between effective altruists and Christian givers. I argue that Christians are obligated from their own perspective to give radically in terms of quantity and scope to alleviate the suffering of the poor and needy. I raise two important potential (...) stumbling blocks for EAChristian collaboration. First, Christians cannot assess outcomes using a straightforward utilitarian calculus of the sort preferred by many EAs, lest they run into a reductio. Second, Christians will want to give to support aims such as worship and evangelism that are not shared by secular EAs and that are not easily commensurable, making the allocation of giving resources more difficult. I conclude with some tentative suggestions about how Christians who are sympathetic to EA might become more effective in their giving. (shrink)

Gottfried Wilhelm Leibniz (1646–1716) was a universal genius, making original contributions to law, mathematics, philosophy, politics, languages, and many areas of science, including what we would now call physics, biology, chemistry, and geology. By profession he was a court counselor, librarian, and historian, and thus much of his intellectual activity had to be fit around his professional duties. Leibniz’s fame and reputation among his contemporaries rested largely on his innovations in the field of mathematics, in particular his discovery of the (...) calculus in 1675. Another of his enduring mathematical contributions was his invention of binary arithmetic, though the significance of this was not recognized until the 20th century. These days, a good proportion of scholarly interest in Leibniz is focused on his philosophy. Among his signature philosophical doctrines are the pre-established harmony, the theory of monads, and the claim that ours is the best of all possible worlds, which forms the central plank of his theodicy. For Leibniz, philosophy was not the discovery of deep truths of interest only to other philosophers, but a practical discipline with the means to increase happiness and well-being. Philosophical truths, he believed, revealed the beauty and rational order of the universe, and the justice and wisdom of its creator, and accordingly could inspire contentment and peace of mind. Leibniz’s other intellectual projects were likewise geared toward the improvement of the human condition. He lobbied tirelessly for the establishment of scientific societies, devised measures to improve public health, and was actively engaged in projects to unite the churches and so end the religious strife that marred the Europe of his day. He was also engaged in politics for much of his career, and often took on a diplomatic role, sometimes officially and other times not. In the political sphere, Leibniz did not wield true power but was a man with influence, obtained in no small part by his cultivation of relationships with leaders and sovereigns both inside and outside Germany. The sheer range of Leibniz’s interests, projects, and activities can make him a difficult figure to study, and the vast quantity of his writings only compounds the problem (around fifty thousand of his writings survive). Nevertheless, even a sampling of Leibniz’s work is enough to get a sense of his vision, originality, and intellectual depth, and good secondary literature will only enhance this. The items in this bibliography were chosen with this in mind. (shrink)

Sufficient conditions for first-order-based sequent calculi to admit cut elimination by a Schütte–Tait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related to the calculus. The conditions are general enough to be satisfied by a wide class of sequent calculi encompassing, among others, some sequent calculi presentations for the first order and the propositional versions of classical and intuitionistic logic, classical and (...) intuitionistic modal logic S4, and classical and intuitionistic linear logic and some of its fragments. Moreover the conditions are such that there is an algorithm for checking if they are satisfied by a sequent calculus. (shrink)

Asymptotic extremal combinatorics deals with questions that in the language of model theory can be re-stated as follows. For finite models M, N of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let p(M, N) be the probability that a randomly chosen sub-model of N with |M| elements is isomorphic to M. Which asymptotic relations exist between the quantities p(M₁, N),...,p(Mh, N), where M₁,...,Mh are fixed "template" models and |N| grows to infinity? In this paper (...) we develop a formal calculus that captures many standard arguments in the area, both previously known and apparently new. We give the first application of this formalism by presenting a new simple proof of a result by Fisher about the minimal possible density of triangles in a graph with given edge density. (shrink)

The infinitesimal methods commonly used in the 17th and 18th centuries to solve analytical problems had a great deal of elegance and intuitive appeal. But the notion of infinitesimal itself was flawed by contradictions. These arose as a result of attempting to representchange in terms ofstatic conceptions. Now, one may regard infinitesimals as the residual traces of change after the process of change has been terminated. The difficulty was that these residual traces could not logically coexist with the static quantities (...) traditionally employed by mathematics. The solution to this difficulty, as it turns out, is to regard these quantities asalso being subject to (a form of) change, for then they will have the same nature as the infinitesimals representing the residual traces of change, and will become,ipso facto, compatible with these latter.In fact, the category-theoretic models which realize the Principle of Infinitesimal Linearity may themselves be regarded as representations of a general concept of variation (cf. Bell (1986)). While the static set-theoretical models represent change or motion by making a detour through the actual (but static) infinite, the varying category-theoretic models enable such change to be representeddirectly, thus permitting the introduction of geometric infinitesimals and, as we have attempted to demonstrate in this paper, the virtually complete incorporation of the methods of the early calculus.It is surely a remarkable — even an ironic — fact that the contradiction between the flux of the objective world and the stasis of mathematical entities has found its resolution in category theory, a branch of mathematics commonly, and, as one now sees, mistakenly, regarded as the summit of gratuitous abstraction. (shrink)

In Descartes, the concept of a ‘universal science’ differs from that of a ‘mathesis universalis’, in that the latter is simply a general theory of quantities and proportions. Mathesis universalis is closely linked with mathematical analysis; the theorem to be proved is taken as given, and the analyst seeks to discover that from which the theorem follows. Though the analytic method is followed in the Meditations, Descartes is not concerned with a mathematisation of method; mathematics merely provides him with examples. (...) Leibniz, on the other hand, stressed the importance of a calculus as a way of representing and adding to what is known, and tried to construct a ‘universal calculus’ as part of his proposed universal symbolism, his ‘characteristica universalis’. The characteristica universalis was never completed—it proved impossible, for example, to list its basic terms, the ‘alphabet of human thoughts’—but parts of it did come to fruition, in the shape of Leibniz's infinitesimal calculus and his various logical calculi. By his construction of these calculi, Leibniz proved that it is possible to operate with concepts in a purely formal way. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Bishop Berkeley Exorcises the Infinite: Fuzzy Consequences of Strict Finitism1 David M. Levy Introduction It all began simply enough when Molyneux asked the wonderful question whether a person born blind, now able to see, would recognize by sight what he knew by touch (Davis 1960). After George Berkeley elaborated an answer, that we learn to perceive by heuristics, the foundations ofcontemporarymathematics wereinruin. Contemporary mathematicians waved their hands and changed (...) the subject.2 Berkeley's answer received a much more positive response from economists. Adam Smith,in particular, seizedupon Berkeley's doctrine that welearn to perceive distance tobuild anelaborate system in which one learns to perceive one's self-interest.3 Perhaps because older histories of mathematics are a positive hindrance in helping us understand the importance of Berkeley's argument against infinitesimals,4 its consequences for economics have passed unnoticed. If infinitesimal numbers are ruled out, and we wish to maintain an algebra, then we rule out infinite numbers. This implication of Berkeley's argumenthas a dramatic implication for our understanding ofthe history ofideas about rule-directed action. Let me motivate this claim with a well-known historical problem. The social doctrine set forth by John Locke outside the Two Treatises depends critically upon individual belief about possible states of infinite pain or pleasure. The critical issue for the intolerance of atheism in Locke's system is his claim that without beUefin the infinite bliss of heaven forgone by a criminal, there is no reason to think a calculating individual will avoid crime.5 Locke's discussion is so different from the discussion of religion by David Hume and Smith. Here, questions of the substance of belief—the infinite worth of Heaven—have largely vanished.6 What happened between Locke and Hume-Smith? One answer is Pierre Bayle and Bernard Mandeville. Berkeley's doctrine that only the finite is perceived is, I think, a better answer. Out of sight, out ofmind. In this paper I propose to give a close look at the content and consequences ofBishop Berkeley's once famous Towards a New Theory ofVision. While there are other aspects ofBerkeley's work that attract Volume XVIII Number 2 511 DAVID M. LEVY attention from historians ofeconomics,7 1 shall claim that Berkeley's insight into human perception in the Theory of Vision is central to Adam Smith's attempt to found economic behaviour on the self-awareness of systematic illusion. There is no doubt that Smith finds that much ofhuman behaviour is characterized by illusion.8 The difficulty arises, or so it seems to me, from modern unwilUngness to beheve that illusion can be systematic. There are four pieces to my argument. First, we consider Berkeley's statementofwhat we shall call his doctrine ofstrict finitism in perception.9 Berkeley puts forward two postulates: 1) there exists some minimum perceptible quantity, and 2) distance is not perceived directly but by experience. Second, we define strict finitism and demonstrate that it translates into the claim that perception is fuzzy in a technical sense. If Berkeleian perception is fuzzy, then some modern criticism of Berkeley's strict finitism is based upon a simple misunderstandingofthe propertiesoffuzzyrelations. Thisis quite easy to set right. (Getting right the doctrine of those who implicitly take perception to be fuzzy—Adam Smith is my prime candidate—will be muchharder.)Third, wereconsiderBerkeley's dispute withMandeville on the role ofinfinite gain in choice. Here, we find Mandeville putting forward the Berkeleian position that if infinite amounts mattered, people would not behave as they do. Berkeley's position against Mandeville, however, seems to depend upon the supposition that infinite distance is sensible.10 Earlier, however, the position that Mandeville argued for was defended by Berkeley. Fourth, we ask why all this isn't well known. The debate between Berkeley and Mandeville has been well studied, as has been the link between Berkeley and the Scots. In fact, our work focuses on one aspect of Berkeley's work that led to his challenge of the contemporary foundations of the calculus. If sense is a matter of perception, and infinitesimals cannot be perceived, then they are quite literally nonsense. Armed with this insight, Berkeley refuted contemporary mathematics. Nevertheless, the calculus was too... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Bishop Berkeley Exorcises the Infinite: Fuzzy Consequences of Strict Finitism1 David M. Levy Introduction It all began simply enough when Molyneux asked the wonderful question whether a person born blind, now able to see, would recognize by sight what he knew by touch (Davis 1960). After George Berkeley elaborated an answer, that we learn to perceive by heuristics, the foundations ofcontemporarymathematics wereinruin. Contemporary mathematicians waved their hands and changed (...) the subject.2 Berkeley's answer received a much more positive response from economists. Adam Smith,in particular, seizedupon Berkeley's doctrine that welearn to perceive distance tobuild anelaborate system in which one learns to perceive one's self-interest.3 Perhaps because older histories of mathematics are a positive hindrance in helping us understand the importance of Berkeley's argument against infinitesimals,4 its consequences for economics have passed unnoticed. If infinitesimal numbers are ruled out, and we wish to maintain an algebra, then we rule out infinite numbers. This implication of Berkeley's argumenthas a dramatic implication for our understanding ofthe history ofideas about rule-directed action. Let me motivate this claim with a well-known historical problem. The social doctrine set forth by John Locke outside the Two Treatises depends critically upon individual belief about possible states of infinite pain or pleasure. The critical issue for the intolerance of atheism in Locke's system is his claim that without beUefin the infinite bliss of heaven forgone by a criminal, there is no reason to think a calculating individual will avoid crime.5 Locke's discussion is so different from the discussion of religion by David Hume and Smith. Here, questions of the substance of belief—the infinite worth of Heaven—have largely vanished.6 What happened between Locke and Hume-Smith? One answer is Pierre Bayle and Bernard Mandeville. Berkeley's doctrine that only the finite is perceived is, I think, a better answer. Out of sight, out ofmind. In this paper I propose to give a close look at the content and consequences ofBishop Berkeley's once famous Towards a New Theory ofVision. While there are other aspects ofBerkeley's work that attract Volume XVIII Number 2 511 DAVID M. LEVY attention from historians ofeconomics,7 1 shall claim that Berkeley's insight into human perception in the Theory of Vision is central to Adam Smith's attempt to found economic behaviour on the self-awareness of systematic illusion. There is no doubt that Smith finds that much ofhuman behaviour is characterized by illusion.8 The difficulty arises, or so it seems to me, from modern unwilUngness to beheve that illusion can be systematic. There are four pieces to my argument. First, we consider Berkeley's statementofwhat we shall call his doctrine ofstrict finitism in perception.9 Berkeley puts forward two postulates: 1) there exists some minimum perceptible quantity, and 2) distance is not perceived directly but by experience. Second, we define strict finitism and demonstrate that it translates into the claim that perception is fuzzy in a technical sense. If Berkeleian perception is fuzzy, then some modern criticism of Berkeley's strict finitism is based upon a simple misunderstandingofthe propertiesoffuzzyrelations. Thisis quite easy to set right. (Getting right the doctrine of those who implicitly take perception to be fuzzy—Adam Smith is my prime candidate—will be muchharder.)Third, wereconsiderBerkeley's dispute withMandeville on the role ofinfinite gain in choice. Here, we find Mandeville putting forward the Berkeleian position that if infinite amounts mattered, people would not behave as they do. Berkeley's position against Mandeville, however, seems to depend upon the supposition that infinite distance is sensible.10 Earlier, however, the position that Mandeville argued for was defended by Berkeley. Fourth, we ask why all this isn't well known. The debate between Berkeley and Mandeville has been well studied, as has been the link between Berkeley and the Scots. In fact, our work focuses on one aspect of Berkeley's work that led to his challenge of the contemporary foundations of the calculus. If sense is a matter of perception, and infinitesimals cannot be perceived, then they are quite literally nonsense. Armed with this insight, Berkeley refuted contemporary mathematics. Nevertheless, the calculus was too... (shrink)

There are two common assumptions about well-being that I am especially concerned to dispute in this dissertation. The first assumption is that differences in kinds of prudential values can be reduced to differences in amount of prudential value. That is, that differences in the qualities of values can reliably be reduced to mere differences in quantity. The second assumption is that well-being is the appropriate object of moral concern. Consequentialist moral theories typically argue that morality requires the maximization of (...) well-being across persons. Thus, such consequentialists suggest, the way to take another into account morally is to promote that person's well-being. ;Against the first assumption I take issue with hedonism, decision theory, and full information accounts of well-being. I argue that none of these popular methods of commensurating well-being is adequate to the task. Against the second assumption, I argue that, properly construed, one's well-being does not adequately capture the full range of one's concerns. I argue that this is a crucial fault with well-being promoting ethical theories. I suggest instead that we allow people to throw their weight in the moral calculus where they choose, provided that they understand the aggregation procedure. (shrink)

A theorem from Archimedes on the area of a circle is proved in a setting where some inconsistency is permissible, by using paraconsistent reasoning. The new proof emphasizes that the famous method of exhaustion gives approximations of areas closer than any consistent quantity. This is equivalent to the classical theorem in a classical context, but not in a context where it is possible that there are inconsistent innitesimals. The area of the circle is taken 'up to inconsistency'. The fact (...) that the core of Archimedes's proof still works in a weaker logic is evidence that the integral calculus and analysis more generally are still practicable even in the event of inconsistency. (shrink)

With Hegel on Being, Stephen Houlgate presents an impressive philosophical analysis of one of the most obscure, but also most important texts of the whole Western philosophical tradition. Houlgate's book is an in-depth systematic investigation of the entire doctrine of being of Hegel's mature logical system. This work takes up and develops a series of reflections presented in The Opening of Hegel's Logic (2006), which were dedicated to the first two chapters of the section on quality in the Science of (...) Logic, and extends them to the further two sections of the first part of Hegel's mature system. Houlgate's text is thus particularly interesting, especially if one considers that not many Hegel scholars have dealt with the difficulties of the sections dedicated to quantity and measure. This is probably due to the complexity of the dialectic of the categories involved in these sections of the Logic and to the necessity of approaching Hegel's dialogue with the exact and natural sciences of his time. On the interaction of Hegel's philosophy with these scientific contributions, Houlgate offers us particularly significant pages. Suffice to mention the excursus on differential calculus at the end of the section dedicated to quantity, or the analysis of the relevance of Galileo Galilei's and Johannes Kepler's discoveries in the context of the analysis of realized measure. (shrink)

Several philosophers have argued that interpersonal comparisons of utility are problematic or even impossible, and that this poses a problem for the thesis that pleasure is a legitimate, measurable quantity. This, in turn, is thought to pose a problem of some kind for a variety of normative ethical and axiological theories. Perhaps it is supposed to show that utilitarianism or hedonism is false, or is supposed to show that there is no genuine hedonic calculus, or that any view (...) that presupposes that pleasure and pain are morally relevant is false or untenable. Its proponents begin by noticing that interpersonal comparisons of pleasure and pain are impossible in some relevant sense. This impossibility has been exploited by different philosophers for different purposes. It is clear, however, that a great many ethical theories rely in some way on the thesis that pleasure is measurable, and that its incoherence therefore threatens a great many theories in ethics. The purpose of this paper is to survey several of the most interesting versions of this argument; then to explain several attempts to respond to the problem, and to explain why each attempt fails; then to present, explain, and defend the correct solution to the problem. (shrink)

“Ordinary algebra in its modern developments,” Whitehead observed in 1897, “is studied as being a large body of propositions, inter-related by deductive reasoning, and based upon conventional definitions which are generalizations of fundamental conceptions.” The use of ‘based upon’ here is perhaps too weak, for some “propositions” must of course be picked out as determinative of the kind of algebra in question by way of axioms. The definitions are then ancillary devices of notational abbreviation and may or may not be (...) of terms or phrases previously introduced, and may or may not be of a “generalized” form. “Thus a science is being created [ordinary algebra in its modern developments],” Whitehead continues, “which by reason of its fundamental character has relation to almost every event, phenomenal or intellectual, which can occur…. Such algebras are mathematical sciences which are not essentially concerned with number or quantity; and this bold extension beyond the traditional domain of pure quantity forms their peculiar interest. The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thought, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus.”. (shrink)

Druga knjiga Petrićeve Pancosmije potpuno nam otkriva što je Petrić mislio de continuo ili de divisibilitate quantitatis te nam ujedno nudi mnoge detalje Petrićeve neuspješne strategije pri osporavanju Aristotelovih pojmova neprekidnine i potencijalne beskonačnine. Prigovarajući Aristotelu, Petrić i ne htijući upozorava na glavne domete Aristotelova nauka o neprekidnini, ali nudi svoja, drugačija rješenja, poput zamisli o najmanjoj nedjeljivoj crti. Iako svojim rješenjima ne uspijeva postići ono što je Aristotel blistavo postigao pojmom neprekidnine u tumačenju prirode i matematike, Petrić unutar polemike (...) s Aristotelom postavlja nova pitanja kad se bavi računom s beskonačninama i genealogijom znanosti.The second book of Pancosmia provides full insight into Petrić’s view of the continuum or de divisibilitate quantitatis, offering us many details on Petrić’s futile strategy in refuting Aristotle’s notions of the continuum and the potential infinite. In his objections to Aristotle, Petrić implicitly points the major contributions of Aristotle’s doctrine of continuum, but also submits his own solutions such as the idea of the minimal indivisible line. Although his solutions failed to bring him the results as brilliant as those of Aristotle with the notion of the continuum in the interpretation of nature and mathematics, within his polemic with Aristotle, Petrić comes forward with new questions dealing with the calculus of infinite quantities and genealogy of science. (shrink)

Modern mathematical physics uses real-number variables, and therefore presupposes set theory. (A real number is defined as a certain kind of set or sequence of natural or rational numbers.) Set theory is also used to define the operations of differential calculus, needed to state physical laws as differential equations constraining the numerical variables representing physical quantities. The derivative f' = df(t)/dt is defined as the limit of an infinite sequence of terms [f(t+e)-f(t)]/e as e → 0, and this definition (...) can only be expressed in a language allowing reference to infinite sequences. Moreover, closure under these limit operations again requires the underlying field of numbers to be Dedekind complete, hence to include all of the reals.The possibility that mathematical physics depends essentially upon set theory is disturbing, for two distinct reasons. From a logical viewpoint, it is disturbing that an important branch of natural science should depend upon a part of logic or mathematics whose status remains uncertain and controversial. (shrink)

In this contribution, we focus on probabilistic problems with a denumerably or non-denumerably infinite number of possible outcomes. Kolmogorov (1933) provided an axiomatic basis for probability theory, presented as a part of measure theory, which is a branch of standard analysis or calculus. Since standard analysis does not allow for non-Archimedean quantities (i.e. infinitesimals), we may call Kolmogorov's approach "Archimedean probability theory". We show that allowing non-Archimedean probability values may have considerable epistemological advantages in the infinite case. The current (...) paper focuses on the motivation for our new axiomatization. (shrink)

In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal (...) Analysis (SIA), as propounded by John Bell. Despite many parallels between SIA and Leibniz’s approach —the non-punctiform nature of infinitesimals, their acting as parts of the continuum, the dependence on variables (as opposed to the static quantities of both Standard and Non-standard Analysis), the resolution of curves into infinitesided polygons, and the finessing of a commitment to the existence of infinitesimals— I find some salient differences, especially with regard to higher-order infinitesimals. These differences are illustrated by a consideration of how each approach might be applied to Newton’s Proposition 6 of the Principia, and the derivation from it of the v2/r law for the centripetal force on a body orbiting around a centre of force. It is found that while Leibniz’s syncategorematic approach is adequate to ground a Leibnizian version of the v2/r law for the “solicitation” ddr experienced by the orbiting body, there is no corresponding possibility for a derivation of the law by nilsquare infinitesimals; and while SIA can allow for second order differentials if nilcube infinitesimals are assumed, difficulties remain concerning the compatibility of nilcube infinitesimals with the principles of SIA, and in any case render the type of infinitesimal analysis adopted dependent on its applicability to the problem at hand. (shrink)

From ancient times to the beginning of the nineteenth century, mathematics was commonly viewed as the general science of quantity, with two main branches: geometry, which deals with continuous quantities, and arithmetic, which deals with quantities that are discrete. Mathematical logic does not fit neatly into this taxonomy. In 1847, George Boole [1] offered an alternative characterization of the subject in order to make room for this new discipline: mathematics should be understood to include the use of any symbolic (...) calculus “whose laws of combination are known and general, and whose results admit of a consistent interpretation.” Depending on the laws chosen, symbols can just as well be used to represent propositions instead of quantities; in that way, one can consider calculations with propositions on par with more familiar arithmetic calculations. Despite Boole’s efforts, logic has always held an uncertain place in the mathematical community. This can partially be attributed to its youth; while number theory and geometry have their origins in antiquity, mathematical logic did not emerge as a recognizable discipline until the latter half of the nineteenth century, and so lacks the provenance and prestige of its elder companions. Furthermore, the nature of the subject matter serves to set it apart; as objects of mathematical study, formulas and proofs do not have the same character as groups, topological spaces, and measures. Even distinctly mathematical branches of contemporary logic, like model theory and set theory, tend to employ linguistic classifications and measures of complexity that are alien to other mathematical disciplines. The Birth of Model Theory, by Calixto Badesa, describes a seminal step in the emergence of logic as a mature mathematical discipline. As its title suggests, the focus is on one particular result, now referred to as the L¨owenheim-Skolem theorem, as presented in a paper by L¨owenheim in 1915. But the book is, more broadly, about the story of the logic community’s gradual coming to the modern distinction between syntax and semantics, that is, between systems of symbolic expressions and the meanings that can be assigned to them.. (shrink)

What is common to all languages is notation, so Universal Grammar can be understood as a system of notational types. Given that infants acquire language, it can be assumed to arise from some a priori mental structure. Viewing language as having the two layers of calculus and protocol, we can set aside the communicative habits of speakers. Accordingly, an analysis of notation results in the three types of Identifier, Modifier and Connective. Modifiers are further interpreted as Quantifiers and Qualifiers. (...) The resulting four notational types constitute the categories of Universal Grammar. Its ontology is argued to consist in the underlying cognitive schema of Essence, Quantity, Quality and Relation. The four categories of Universal Grammar are structured as polysemous fields and are each constituted as a radial network centred on some root concept which, however, need not be lexicalized. The branches spread out along troponymic vectors and together map out all possible lexemes. The notational typology of Universal Grammar is applied in a linguistic analysis of the ‘parts of speech’ using the English language. The analysis constitutes a ‘proof of concept’ in (1) showing how the schema of Universal Grammar is capable of classifying the so-called ‘parts of speech’, (2) presenting a coherent analysis of the verb, and (3) showing how the underlying cognitive schema allows for a sub-classification of the auxiliaries. (shrink)

This article tackles an old, classical problem, which is acquiring a new epochal relevance with the techno-aesthetic processing of form and substance, expression and content. The field of digital architecture is embarked in the ancient controversy between the line and the curve, binary communication and fuzzy logic. Since the 1990s, the speculative qualities of digital architecture have exposed spatial design to the qualities of growing or breeding, rather than planning. However, such qualities still deploy the tension between discrete spaces and (...) continual curving. In this context, the article suggests the computational coexistence of discrete coding with continual morphing, defying any easy resolution for an aesthetic of continuity or discontinuity, the superiority of the analog or the meta-logic of the digital. The metaphysical dimension of such coexistence needs to include the abstract capacities of experiencing the transition from one state to another as the registering of algorithmic processing. Computation is intrinsic to microperceptions, incomputable quantities deploying the infectious property of the digital code. The article draws on the digital architecture of Greg Lynn to explore whether the computational nature of the digital calculus has the potential to challenge the bifurcation between the biological and the mathematical, the physical and the mental. (shrink)

What is common to all languages is notation, so Universal Grammar can be understood as a system of notational types. Given that infants acquire language, it can be assumed to arise from some a priori mental structure. Viewing language as having the two layers of calculus and protocol, we can set aside the communicative habits of speakers. Accordingly, an analysis of notation results in the three types of Identifier, Modifier and Connective. Modifiers are further interpreted as Quantifiers and Qualifiers. (...) The resulting four notational types constitute the categories of Universal Grammar. Its ontology is argued to consist in the underlying cognitive schema of Essence, Quantity, Quality and Relation. The four categories of Universal Grammar are structured as polysemous fields and are each constituted as a radial network centred on some root concept which, however, need not be lexicalized. The branches spread out along troponymic vectors and together map out all possible lexemes. The notational typology of Universal Grammar is applied in a linguistic analysis of the ‘parts of speech’ using the English language. The analysis constitutes a ‘proof of concept’ in (1) showing how the schema of Universal Grammar is capable of classifying the so-called ‘parts of speech’, (2) presenting a coherent analysis of the verb, and (3) showing how the underlying cognitive schema allows for a sub-classification of the auxiliaries. (shrink)

An integrated view concerning the probabilistic organization of quantum mechanics is obtained by systematic confrontation of the Kolmogorov formulation of the abstract theory of probabilities, with the quantum mechanical representationand its factual counterparts. Because these factual counterparts possess a peculiar spacetime structure stemming from the operations by which the observer produces the studied states (operations of state preparation) and the qualifications of these (operations of measurement), the approach brings forth “probability trees,” complex constructs with treelike spacetime support.Though it is strictly (...) entailed by confrontation with the abstract theory of probabilities as it now stands, the construct of a quantum mechanical probability treetransgresses this theory. It indicates the possibility of an extended abstract theory of probabilities including explicit representations of the cognitive operations involved in the probabilistic descriptions. So quantum mechanics appears to be neither a “normal” probabilistic theory nor an “abnormal” one, but a pioneering particular realization of afuture extended abstract theory of probabilities.The consequences of the integrated perception of the probabilistic organization of quantum mechanics are developed constructively. The current identifications of spectral decompositions, with superpositions of states, are removed. Then: (a) Inside the frontiers of the purely operational-observational orthodox formalism, operators of state preparation and the calculus with these are defined consistently with the definition and the calculus of quantum mechanical operators representing measurable dynamical quantities. This permits to grasp the physical meaning of superselection rules. Furthermore, a complement to the quantum theory of measurements is obtained. These prolongations of the orthodox formalism bring forth a “probabilistic incompleteness” of the quantum theory. (b) Beyond quantum mechanics as it now stands, a model is outlined that removes this probabilistic incompleteness, “the [particle + medium] individual model,”microscopic by certain aspects andcosmic by others.Globally, the approach draws attention upon the possibility and the interest of a general representation of the descriptions of any kind founded upon the explicit specification of the epistemic operations—with their spacetime features—by which the observer, who always is involved, produces the objects to be qualified and the qualifications of these. (shrink)