The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to (...) accept intuitively. Next, the geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction. (shrink)

In Hegel ou Spinoza,1 Pierre Macherey challenges the influence of Hegel’s reading of Spinoza by stressing the degree to which Spinoza eludes the grasp of the Hegelian dialectical progression of the history of philosophy. He argues that Hegel provides a defensive misreading of Spinoza, and that he had to “misread him” in order to maintain his subjective idealism. The suggestion being that Spinoza’s philosophy represents, not a moment that can simply be sublated and subsumed within the dialectical progression of the (...) history of philosophy, but rather an alternative point of view for the development of a philosophy that overcomes Hegelian idealism. Gilles Deleuze also considers Spinoza’s philosophy to resist the totalising effects of the dialectic. Indeed, Deleuze demonstrates, by means of Spinoza, that a more complex philosophy antedates Hegel’s, which cannot be supplanted by it. Spinoza therefore becomes a significant figure in Deleuze’s project of tracing an alternative lineage in the history of philosophy, which, by distancing itself from Hegelian idealism, culminates in the construction of a philosophy of difference. It is Spinoza’s role in this project that will be demonstrated in this paper by differentiating Deleuze’s interpretation of the geometrical example of Spinoza’s Letter XII (on the problem of the infinite) in Expressionism in Philosophy, Spinoza,2 from that which Hegel presents in the Science of Logic.3. (shrink)

One shortcoming of the chain rule is that it does not iterate: it gives the derivative of f(g(x)), but not (directly) the second or higher-order derivatives. We present iterated differentials and a version of the multivariable chain rule which iterates to any desired level of derivative. We first present this material informally, and later discuss how to make it rigorous (a discussion which touches on formal foundations of calculus). We also suggest a finite calculus chain rule (contrary to (...) Graham, Knuth and Patashnik's claim that "there's no corresponding chain rule of finite calculus"). (shrink)

In this paper I present an interpretation of Deleuze's concept of the virtual. I argue that this concept is best understood in relation to the problematic of individuation or differentiation, which Deleuze inherits from Duns Scotus. After analysing Scotus' critique of Aristotelian or hylomorphic approaches to the problem of individuation, I turn to Deleuze's account of differentiation and his interpretation of the calculus in chapter 4 of Difference and Repetition. The paper seeks thereby to explicate Deleuze's dialectics or theory (...) of Ideas, his approach to the principle of sufficient reason, and the concept of the virtual itself. (shrink)

Maimon’s theory of the differential has proved to be a rather enigmatic aspect of his philosophy. By drawing upon mathematical developments that had occurred earlier in the century and that, by virtue of the arguments presented in the Essay and comments elsewhere in his writing, I suggest Maimon would have been aware of, what I propose to offer in this paper is a study of the differential and the role that it plays in the Essay on Transcendental Philosophy (...) (1790). In order to do so, this paper focuses upon Maimon’s criticism of the role played by mathematics in Kant’s philosophy, to which Maimon offers a Leibnizian solution based on the infinitesimal calculus. The main difficulties that Maimon has with Kant’s system, the second of which will be the focus of this paper, include the presumption of the existence of synthetic a priori judgments, i.e. the question quid facti, and the question of whether the fact of our use of a priori concepts in experience is justified, i.e. the question quid juris. Maimon deploys mathematics, specifically arithmetic, against Kant to show how it is possible to understand objects as having been constituted by the very relations between them, and he proposes an alternative solution to the question quid juris, which relies on the concept of the differential. However, despite these arguments, Maimon remains sceptical with respect to the question quid facti. (shrink)

The aim of this paper is to explore the uses made of the calculus by Gilles Deleuze and G. W. F. Hegel. I show how both Deleuze and Hegel see the calculus as providing a way of thinking outside of finite representation. For Hegel, this involves attempting to show that the foundations of the calculus cannot be thought by the finite understanding, and necessitate a move to the standpoint of infinite reason. I analyse Hegel’s justification for this (...) introduction of dialectical reason by looking at his responses to Berkeley’s criticisms of the calculus. For Deleuze, instead, I show that the differential must be understood as escaping from both finite and infinite representation. By highlighting the sub-representational character of the differential in his system, I show how the differential is a key moment in Deleuze’s formulation of a transcendental empiricism. I conclude by dealing with some of the common misunderstandings that occur when Deleuze is read as endorsing a modern mathematical interpretation of the calculus. (shrink)

In this perspective paper, I tried to explain that what will be the possible prospect of multiple functions in one and another through the chain rule of differentiation? The chain rule is a formula to compute the derivative of the functional composition of two or more functions. The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. The chain rule (...) in that what changes I noted, how it can be modified to reduce the differentiation process of multiple functions in one and another and implementing that process in the inverse of multiple functions in one and another, further proceedings them through the chain rule of differentiation with all prospects. Here I claim that the differentiation’s chain rule is not changed but the sequences of finding the derivative of multiple functions are changed. (shrink)

In the paper “Math Anxiety,” Aden Evens explores the manner by means of which concepts are implicated in the problematic Idea according to the philosophy of Gilles Deleuze. The example that Evens draws from Difference and Repetition in order to demonstrate this relation is a mathematics problem, the elements of which are the differentials of the differential calculus. What I would like to offer in the present paper is an historical account of the mathematical problematic that Deleuze deploys (...) in his philosophy, and an introduction to the role that this problematic plays in the develop- ment of his philosophy of difference. One of the points of departure that I will take from the Evens paper is the theme of “power series.”2 This will involve a detailed elaboration of the mechanism by means of which power series operate in the differential calculus deployed by Deleuze in Difference and Repetition. Deleuze actually constructs an alternative history of mathematics that establishes an historical conti- nuity between the differential point of view of the infinitesimal calculus and modern theories of the differential calculus. It is in relation to the differential point of view of the infinitesimal calculus that Deleuze determines a differential logic which he deploys, in the form of a logic of different/ciation, in the development of his proj- ect of constructing a philosophy of difference. (shrink)

The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses recently introduced infinite and infinitesimal numbers being in accordance with the principle ‘The part is less than the whole’ observed in the physical world around us. These numbers have a strong practical advantage with respect to traditional approaches: they are representable at a new (...) kind of a computer – the Infinity Computer – able to work numerically with all of them. An introduction to the theory of physical and mathematical continuity and differentiation (including subdifferentials) for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains is developed in the paper. This theory allows one to work with derivatives that can assume not only finite but infinite and infinitesimal values, as well. It is emphasized that the newly introduced notion of the physical continuity allows one to see the same mathematical object as a continuous or a discrete one, in dependence on the wish of the researcher, i.e., as it happens in the physical world where the same object can be viewed as a continuous or a discrete in dependence on the instrument of the observation used by the researcher. Connections between pure mathematical concepts and their computational realizations are continuously emphasized through the text. Numerous examples are given. (shrink)

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt what may (...) be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (shrink)

What is cognition? Equivalently, what is cognition good for? Or, what is it that would not be but for human cognition? But for human cognition, there would not be science. Based on this kinship between individual cognition and collective science, here we put forward Isbell conjugacy---the adjointness between objective geometry and subjective algebra---as a scientific method for developing cognitive science. We begin with the correspondence between categorical perception and category theory. Next, we show how the Gestalt maxim is subsumed by (...) the mathematical construct of colimit, a generalization of summation. The universal mapping property definitions of mathematical constructs, by virtue of being the best with respect to the universe of discourse, can be learned using reinforcement learning algorithms, which raises the possibility of abstracting the architecture of mathematics by artificial intelligence. Subsequently, we present naturality (to be contrasted with miracles), understood as 'Becoming consistent with Being', which governs the transformations of both things and their theories, as the zeroth law of change. Furthermore, the contrast---physical [mechanism] vs. biological [organism]---is smoothed via natural transformation, wherein transformations are respectful of the cohesion of the objects transformed. In closing, upon recognizing the scientific value of learning difficult-to-master differential calculus by physicists, of learning a strange four-letter language by biologists, and of learning the grammar of our respective mother tongues, we make a case for learning the theory of naturality / category theory for developing cognitive science. (shrink)

Digital practices in design, together with computer-assisted manufacturing (CAM), have inspired the reflection of philosophers, theorists, and historians over the last decades. Gilles Deleuze’s The Fold: Leibniz and the Baroque (1988) presents one of the first and most successful concepts created to think about these new design and manufacturing practices.1 Deleuze proposed a new concept of the technological object, which was inspired by Bernard Cache’s digital design practices and computer-assisted manufacturing. Deleuze compared Cache’s practices to Leibniz’s differential calculus-based (...) notion of the parametric curve. From this perspective, the object is no longer an essential form: rather, it is functional, defined by a family of parametric curves. In Deleuze’s terms, it is an objectile. This definition grasps a particular aspect of modernity in the technological object, rendering possible the industrial production of “the unique object” (la pièce unique), while making obsolete the homogeneity that comes with industrial standardization. What follows introduces developmental biology into this design matrix, interrogating the relationship between parametric design and computer-assisted manufactory on the one hand and biological morphogenetic processes on the other. Is one necessary for the other? Does an architect need computation in order to render morphogenetic shapes? In answering these questions, this chapter displaces the centrality of digital technology in the design of shape-shifting forms within architecture and calls for a rethinking of design by way of analog prototypes. I argue that “thinking through analogs” offers another means of generating parametrically based morphogenetic forms. (shrink)

This essay describes and evaluates the conception of mereological structure that underpins Deleuze’s account of ontogenesis in Difference and Repetition. A theory of mereology is a theory of composition: it asks what it is to be a part making a whole, what it is to be a whole collecting its parts; in short, in what the relation of making or composing consists. The locus classicus for modern mereology is the third of Husserl’s Logical Investigations (‘On the Theory of Wholes and (...) Parts’), which deduces the definitions, axioms and principles governing the relation of parthood and associated notions, such as wholeness, separation, simplicity and complexity. My question is this: what happens to these notions when they are no longer the terms of a ‘calculus of individuals’ and, instead, become the terms of a differential calculus of individuation? (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)

In this paper, I want to look at the way in which Deleuze's reading of Kant's transcendental dialectic influences some of the key thèmes of Différence and Répétition. As we shall see, in the transcendental dialectic, Kant takes the step of claiming that reason, in its natural functioning, is prone to misadventures. Whereas for Descartes, for instance, error takes place between two faculties, such as when reason (wrongly) infers that a stick in water is bent on the basis of sensé (...) impressions, Kant postulâtes that reason générâtes illusions internally purely in the course of its natural function. It is thèse illusions which lead reason into antinomy, as on the basis of thèse illusions, it is led to posit an illegitimate concept of the world as a totality. Further, for Kant, the antinomies represent an indirect proof of transcendental idealism, as it is only with the additional assumption of the noumenon, as that which falls outside of appearance, that we are able to résolve the antinomies. Deleuze's work on the image of thought clearly owes a great deal to Kant's theory of transcendental illusion, but the connections between Kant's transcendental dialectic and the structure of Différence and Répétition go deeper than this. Whereas Kant's problem is that reason générâtes contradictions when it assumes that the unconditioned can be given to reason, Deleuze's problem is the impossibility of developing a concept of différence within représentation. Between thèse two problems, there are significant structural parallels - in particular, the attempt to think outside the dichotomy of the finite and the infinité, and the attempt to prevent the application of spatio-temporal predicates to the noumenon. The antinomy of représentation for Deleuze is the inability of représentation to think différence apart from as purely representational or as undifferenciated abyss. As we shall see, Deleuze gives an explanation of this antinomy in terms of the differential calculus, and the notion of the differential in particular. While thèse parallels exist between Kant and Deleuze's thought, there are also some important différences. Although the differential is not determined in relation to représentation, this does not mean that it lacks ail détermination. This opens up a possibility not available in Kant's philosophy, that is, a thinking beyond the limit of représentation. As we shall see, Kant closes off this possibility by giving reason a heuristic function which in effect reinstates représentation. Kant makes this move precisely because the lack of spatiotemporal déterminations of the noumenal means for Kant that the noumenal lacks déterminations altogether. (shrink)

In the chapter of Difference and Repetition entitled ‘Ideas and the synthesis of difference,’ Deleuze mobilizes mathematics to develop a ‘calculus of problems’ that is based on the mathematical philosophy of Albert Lautman. Deleuze explicates this process by referring to the operation of certain conceptual couples in the field of contemporary mathematics: most notably the continuous and the discontinuous, the infinite and the finite, and the global and the local. The two mathematical theories that Deleuze draws upon for this (...) purpose are the differential calculus and the theory of dynamical systems, and Galois’ theory of polynomial equations. For the purposes of this paper I will only treat the first of these, which is based on the idea that the singularities of vector fields determine the local trajectories of solution curves, or their ‘topological behaviour’. These singularities can be described in terms of the given mathematical problematic, that is for example, how to solve two divergent series in the same field, and in terms of the solutions, as the trajectories of the solution curves to the problem. What actually counts as a solution to a problem is determined by the specific characteristics of the problem itself, typically by the singularities of this problem and the way in which they are distributed in a system. Deleuze understands the differential calculus essentially as a ‘calculus of problems’, and the theory of dynamical systems as the qualitative and topological theory of problems, which, when connected together, are determinative of the complex logic of different/ciation. (DR 209). Deleuze develops the concept of a problematic idea from the differential calculus, and following Lautman considers the concept of genesis in mathematics to ‘play the role of model ... with respect to all other domains of incarnation’. While Lautman explicated the philosophical logic of the actualization of ideas within the framework of mathematics, Deleuze (along with Guattari) follows Lautman’s suggestion and explicates the operation of this logic within the framework of a multiplicity of domains, including for example philosophy, science and art in What is Philosophy?, and the variety of domains which characterise the plateaus in A Thousand Plateaus. While for Lautman, a mathematical problem is resolved by the development of a new mathematical theory, for Deleuze, it is the construction of a concept that offers a solution to a philosophical problem; even if this newly constructed concept is characteristic of, or modelled on the new mathematical theory. (shrink)

The role of mathematics in the development of Gilles Deleuze's (1925-95) philosophy of difference as an alternative to the dialectical philosophy determined by the Hegelian dialectic logic is demonstrated in this paper by differentiating Deleuze's interpretation of the problem of the infinitesimal in Difference and Repetition from that which G. W. F Hegel (1770-1831) presents in the Science of Logic . Each deploys the operation of integration as conceived at different stages in the development of the infinitesimal calculus in (...) his treatment of the problem of the infinitesimal. Against the role that Hegel assigns to integration as the inverse transformation of differentiation in the development of his dialectical logic, Deleuze strategically redeploys Leibniz's account of integration as a method of summation in the form of a series in the development of his philosophy of difference. By demonstrating the relation between the differential point of view of the Leibnizian infinitesimal calculus and the differential calculus of contemporary mathematics, I argue that Deleuze effectively bypasses the methods of the differential calculus which Hegel uses to support the development of the dialectical logic, and by doing so, sets up the critical perspective from which to construct an alternative logic of relations characteristic of a philosophy of difference. The mode of operation of this logic is then demonstrated by drawing upon the mathematical philosophy of Albert Lautman (1908-44), which plays a significant role in Deleuze's project of constructing a philosophy of difference. Indeed, the logic of relations that Deleuze constructs is dialectical in the Lautmanian sense. (shrink)

This paper offers an overview of various alternative formulations for Analysis, the theory of Integral and Differential Calculus, and its diverging conceptions of the topological structure of the continuum. We pay particularly attention to Smooth Analysis, a proposal created by William Lawvere and Anders Kock based on Grothendieck’s work on a categorical algebraic geometry. The role of Heyting’s logic, common to all these alternatives is emphasized.

Based on the various documents, 1989-2002, through the original texts, in addition to the author's contributions, this paper presents the refutation of the mathematicians and physicists A. Logunov and M. Mestvirishvil of A. Einstein's "general relativity", from the relativistic theory of gravitation of these authors, who applying the fundamental principle of the science of physics of the conservation of the energy-momentum and using absolute differential calculus they rigorously perform their mathematical tests. It is conclusively shown that, from the (...) Einstein-Grossman-Hilbert equations, gravity is absurdly a metric field devoid of physical reality unlike all other fields in nature that are material fields, interrupting the chain of transformations between the different existing fields. Also, in Einstein's theory the proved "inertial mass" equal to gravitational mass has no physical meaning. Therefore, "general relativity" does not obey the correspondence principle with Newton's gravity. (shrink)

In this paper, I trace the three-fold essence of “return”—a generating trope of identity and difference, through which formal aspects of the theory of relativity, the movement of language and emergence in evolution might converge. The trope of return is contrasted with the more common two-fold structure of relatedness underwriting differential calculus, propositional semantics and reductionism, which privileges space over time, identity over difference, self over creation.

According to current history of science, Levi-Civita introduced parallel transport solely to give a geometrical interpretation to the covariant derivative of absolute differential calculus. Levi-Civita, however, searched a simple computation of the curvature of a Riemannian manifold, basing on notions of the Italian school of mathematical physics of his time: holonomic constraints, virtual displacements and work, which so have a remarkable, if not dominant, role in the origin of parallel transport.

Description In this paper, we propose an advanced mathematical framework centered around the Energy Number Field (E), which fundamentally avoids the conventional concept of zero by introducing a neutral ele- ment, νE. Through this approach, we redefine core mathematical constructs, including limits, continuity, differentiation, integration, and series summation, ensuring they operate seamlessly within a zero-less paradigm. We address and redefine matrix operations, topology, metric spaces, and complex analysis, aligning them with the principles of E. Additionally, we explore non-mappable properties of (...) energy numbers in the context of symmetry groups, non-standard fields, fractional calculus, and non-Euclidean geometries. By employing νE as the central element, our framework resolves conventional zero-related sin- gularities and computational anomalies, offering a novel perspective that bridges advanced mathematical theories with practical applications in physics, computational algorithms, and topological transforma- tions. This work paves the way for future research to experimentally validate these formulations and explore potential applications in advanced physics, quantum mechanics, and beyond. (shrink)

Higher - dimensional calculus and integral transformation play crucial roles in advancing our understanding of complex systems in mathematics and theoretical physics . Integral transformations are instrumental in simplifying complex differential equations, enabling the resolution of multi - dimensional problems that arise in various scientific fields . This paper aims to delve into a specific higher - dimensional integral transformation defined by the axioms \(F[q, s, l, \alpha]\) and \(G[q, s, l, \beta, c]\) . We start by outlining (...) the axioms which define the functions \(F\) and \(G\) . Specifically, Axiom 1 defines \(F\) as a function of four variables : \(q\), \(s\), \(l\), and \( \alpha\), whereas Axiom 2 defines \(G\) as a function that additionally includes variables \( \beta\) and \(c\) . Axiom 3 relates \(h\) and \(l\) via a sine function . The core of our investigation is the integral transformation expressed as a five - dimensional integral involving \(G\) and proving its equivalence to \(F\), provided a specific condition on \(c\) holds . We approach this problem by first deriving the expression for \(c\) through detailed differentiation of \(F\) and equating it to \(G\) . The derivation involves advanced calculus techniques and symbolic mathematics to solve the resulting equations . We then verify the derived expression for \(c\) by substituting it back into the relationship between \(F\) and \(G\), ensuring that the equality holds under integral transformation . Finally, to corroborate our findings, we employ visualizations through multidimensional contour plots to illustrate the relationship between the derived expressions . This provides an intuitive confirmation of the mathematical consistency and validity of the transformation . This paper contributes to the field by providing a nuanced and detailed examination of higher - dimensional integral transformations and their underlying mathematical structures . The results have potential implications for theoretical physics, particularly in areas involving complex systems and multi - dimensional analyses . (shrink)

An important lesson that philosophy can learn from the Turing Test and computer science more generally concerns the careful use of the method of Levels of Abstraction (LoA). In this paper, the method is first briefly summarised. The constituents of the method are “observables”, collected together and moderated by predicates restraining their “behaviour”. The resulting collection of sets of observables is called a “gradient of abstractions” and it formalises the minimum consistency conditions that the chosen abstractions must satisfy. Two useful (...) kinds of gradient of abstraction – disjoint and nested – are identified. It is then argued that in any discrete (as distinct from analogue) domain of discourse, a complex phenomenon may be explicated in terms of simple approximations organised together in a gradient of abstractions. Thus, the method replaces, for discrete disciplines, the differential and integral calculus, which form the basis for understanding the complex analogue phenomena of science and engineering. The result formalises an approach that is rather common in computer science but has hitherto found little application in philosophy. So the philosophical value of the method is demonstrated by showing how making the LoA of discourse explicit can be fruitful for phenomenological and conceptual analysis. To this end, the method is applied to the Turing Test, the concept of agenthood, the definition of emergence, the notion of artificial life, quantum observation and decidable observation. It is hoped that this treatment will promote the use of the method in certain areas of the humanities and especially in philosophy. (shrink)

The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a central issue in the applicable mathematics of the last hundred years. (...) This article explains the distinction and why it has proved to be one of the great organizing themes of mathematics. (shrink)

In the article we consider the relationship of traditional provisions of basic logical concepts and confront them with new and modern approaches to the same concepts. Logic is characterized in different ways when it is associated with syllogistics (referential – semantical model of logic) or with symbolic logic (inferential – syntactical model of logic). This is not only a difference in the logical calculation of (1) concepts, (2) statements, and (3) predicates, but this difference also appears in the treatment of (...) the calculative abilities of logical forms, the ontological-referential status of conceptual content and the inferential-categorical status of logical forms. The basic markers or basic ideas that separate ontologically oriented logic from categorically oriented logic are the (1) concept of truth, the (2) concept of meaning, the (3) concept of identity, and the (4) concept of predication. Here, this differences are explicitly demonstrated by the introduction of differential terminology. From this differential methodology follows a new set of characterizations of logic. (shrink)

We present a novel approach to defining the mathematical constant π through the infinite den- sification of a sweeping net, which approximates a circle as the net becomes infinitely dense. By developing and enhancing notation related to sweeping nets and saddle maps, we establish a rigor- ous framework for expressing π in terms of the densification process using reverse integration. This method, inspired by the concept that numbers ”come from infinity,” leverages a reverse integral approach to model the transition from (...) infinite densification to the finite circle. Our work not only offers a new perspective on the geometric interpretation of π but also provides insights into reverse integration techniques and their applications in mathematical analysis. (shrink)

General Introduction: In [1] a Calculus of Qualia (CQ) was proposed. The key idea is that, for example, blackness is radically different than █. The former term, “blackness” refers to or is about a quale, whereas the latter term, “█” instantiates a quale in the reader’s mind and is non-referential; it does not even refer to itself. The meaning and behavior of these terms is radically different. All of philosophy, from Plato through Descartes through Chalmers, including hieroglyphics and emojis, (...) used referential terms up until CQ. This paper in this series of papers includes a relatively short but comprehensive discussion with the AI Claude 3.5 Sonnet on how the CQ impacts various theories of consciousness, including Materialism, Dualism, Idealism, and 14 others. (shrink)

General Introduction: In [1] a Calculus of Qualia (CQ) was proposed. The key idea is that, for example, blackness is radically different than █. The former term, “blackness” refers to or is about a quale, whereas the latter term, “█” instantiates a quale in the reader’s mind and is non-referential; it does not even refer to itself. The meaning and behavior of these terms is radically different. All of philosophy, from Plato through Descartes through Chalmers, including hieroglyphics and emojis, (...) used referential terms up until CQ. This paper in this series of papers addresses the question of why is there something rather than nothing, at length, and gives a radical new answer. It has to do with the fact in CQ that, while blackness is contingently possible, █ is necessarily actual, as it was in this sentence. (shrink)

In recent years, the e ffort to formalize erotetic inferences (i.e., inferences to and from questions) has become a central concern for those working in erotetic logic. However, few have sought to formulate a proof theory for these inferences. To fill this lacuna, we construct a calculus for (classes of) sequents that are sound and complete for two species of erotetic inferences studied by Inferential Erotetic Logic (IEL): erotetic evocation and regular erotetic implication. While an attempt has been made (...) to axiomatize the former in a sequent system, there is currently no proof theory for the latter. Moreover, the extant axiomatization of erotetic evocation fails to capture its defeasible character and provides no rules for introducing or eliminating question-forming operators. In contrast, our calculus encodes defeasibility conditions on sequents and provides rules governing the introduction and elimination of erotetic formulas. We demonstrate that an elimination theorem holds for a version of the cut rule that applies to both declarative and erotetic formulas and that the rules for the axiomatic account of question evocation in IEL are admissible in our system. (shrink)

This paper aims to address the difficulties of high school students in bridging their computational understanding with their visualization skills in understanding the notion of the limits in their calculus class. This research used a pre-experimental one-group pretest-posttest design research on 62 grade 10 students enrolled in the Science, Technology, and Engineering Program (STEP) in one of the public high schools in Zamboanga del Sur, Philippines. A series of remedial sessions were given to help them understand the function values, (...) one-sided limits, limits of a function, and its discontinuities with the aid of the GeoGebra. Results revealed that there is a significant improvement in their mathematics performance before and after the remediation, t(61) = -19.99, p<.05. The real-time feedback provided by the drawing interface of GeoGebra has provided the students the insights into how objects behave in geometrical space, bridging their understanding in locating the values shown by the computational algebraic processes. Additionally, the significant difference in the achievement reveals a large effect size (Cohen’s =2.54) which could be accounted for by the remediation using interactive activities in the GeoGebra. (shrink)

The purpose of this paper is to introduce a bi-intuitionistic sequent calculus and to give proofs of admissibility for its structural rules. The calculus I will present, called SC2Int, is a sequent calculus for the bi-intuitionistic logic 2Int, which Wansing presents in [2016a]. There he also gives a natural deduction system for this logic, N2Int, to which SC2Int is equivalent in terms of what is derivable. What is important is that these calculi represent a kind of bilateralist (...) reasoning, since they do not only internalize processes of verifcation or provability but also the dual processes in terms of falsifcation or what is called dual provability. In [Wansing, 2017] a normal form theorem for N2Int is stated, here, I want to prove a cut-elimination theorem for SC2Int, i.e., if successful, this would extend the results existing so far. (shrink)

The basic idea is to put qualia into equations (broadly understood) to get what might as well be called qualations. Qualations arguably have different truth behaviors than the analogous equations. Thus ‘black’ has a different behavior than ‘ █ ’. This is a step in the direction of a ‘calculus of qualia’. It might help clarify some issues.

The University of Tehran English Proficiency Test (UTEPT) is a high-stakes entrance examination taken by more than 10,000 master’s degree holders annually. The examinees’ scores have a significant influence on the final decisions concerning admission to the University of Tehran Ph.D. programs. As a test validation investigation, the present study, which is a bias detection research in nature, utilized multistep logistic regression (LR) procedure to examine the presence of gender differential item functioning (DIF) in the UTEPT with a sample (...) of 6,555 examinees who took the test in November 2006. Specifically, the LR DIF two-degree of freedom Chi-squared test of significance was employed to test the significance of DIF. Following what has been recently suggested in the literature, the test of significance for DIF was accompanied by a measure of magnitude, namely the R-squared effect size for LR DIF, for the interpretation of which the two widely accepted classification schemes were used comparatively. The results reveal that 39 of the 100 items in the test display significant gender differences. However, these group differences are viewed as “negligible” based on both of the schemes. Accordingly, it could be argued that the UTEPT is a gender DIF-free test, though the Reading Comprehension section of the test remains in need of further analysis as it seems that the general trend of DIF indices at the item level may hint at an inclination towards males. (shrink)

One way in which the practice of inclusion can be actualised in classrooms is through the use of consistent, appropriate differentiated instruction. What remains elusive, however, is insight into what teachers in different contexts think and believe about differentiation, how consistently they differentiate instruction and what challenges they experience in doing so. In the study reported on here high school classrooms in a private and a government school in Lesotho were compared in order to determine teachers’ thoughts and beliefs about (...) differentiation, the frequency of differentiated instruction, and the challenges faced by teachers who implement this inclusive practice. Sampled teachers offered their views on what they understood differentiated instruction to be, the frequency of differentiated instruction, and identified challenges via an administered questionnaire. Data analysis was based on frequency counts and bar charts for comparative purposes. Findings indicate that private school teachers have a higher frequency of differentiated teaching practice, with time constraints indicated as the main challenge. Government school teachers had a lower frequency of differentiation, and identified a lack of resources, and the learner-teacher ratio as challenges, among others. In the study we highlighted the critical role that private schools can play in the national call for the implementation of inclusive teaching in Lesotho, in terms of active collaboration with surrounding government schools. Private schools, with their resources and access to professional development opportunities, can become catalysts in the implementation of inclusive teaching practices. (shrink)

The idea of this paper is to put actual qualia into equations (broadly understood) to get what might be called qualations. Qualations arguably have different meanings and truth behaviors than the analogous equations. For example, the term ‘ black ’ arguably has a different meaning and behavior than the term ‘ █ ’. This is a step in the direction of a ‘calculus of qualia’ and of expanding science to include 1st-person phenomena.

In this paper we introduce a Gentzen calculus for (a functionally complete variant of) Belnap's logic in which establishing the provability of a sequent in general requires \emph{two} proof trees, one establishing that whenever all premises are true some conclusion is true and one that guarantees the falsity of at least one premise if all conclusions are false. The calculus can also be put to use in proving that one statement \emph{necessarily approximates} another, where necessary approximation is a (...) natural dual of entailment. The calculus, and its tableau variant, not only capture the classical connectives, but also the `information' connectives of four-valued Belnap logics. This answers a question by Avron. (shrink)

John Venn has the “uneasy suspicion” that the stagnation in mathematical logic between J. H. Lambert and George Boole was due to Kant’s “disastrous effect on logical method,” namely the “strictest preservation [of logic] from mathematical encroachment.” Kant’s actual position is more nuanced, however. In this chapter, I tease out the nuances by examining his use of Leonhard Euler’s circles and comparing it with Euler’s own use. I do so in light of the developments in logical calculus from G. (...) W. Leibniz to Lambert and Gottfried Ploucquet. While Kant is evidently open to using mathematical tools in logic, his main concern is to clarify what mathematical tools can be used to achieve. For without such clarification, all efforts at introducing mathematical tools into logic would be blind if not complete waste of time. In the end, Kant would stress, the means provided by formal logic at best help us to express and order what we already know in some sense. No matter how much mathematical notations may enhance the precision of this function of formal logic, it does not change the fact that no truths can, strictly speaking, be revealed or established by means of those notations. (shrink)

The study examined the differential item functioning (DIF) of 2018 Basic Education Certificate examination (BECE) in Mathematics tests of National Examination Council (NECO) and BECE of Akwa Ibom State government in Nigeria. The invariance in the tests with regards to sex was considered using Item Response Theory (IRT) approach. The study area was Akwa Ibom state of Nigeria having a student population of 58,281 for the examination. The sample was made of up 3810 students drawn through a multi-stage sampling (...) approach. The multidimensional IRT (MIRT) package implemented in R-programming language software was applied in analyzing the data. The findings reveal that BECE of NECO displayed 23(38.3%) DIF items while BECE of Akwa Ibom State had 37 (61.7%) DIF items in terms of sex. The findings also revealed that, in the two examinations, more items favoured the male candidates more than the female candidates in terms of performance. It was recommended that IRT model should be adopted by test developers to determine item parameters for selection of good items to ensure quality of items before administration. Test equating of students who write equivalent form examinations conducted by different examining bodies was also recommended for admission and placement of candidates to determine actual group differences in performance. The study posits that research on Differential Item Functioning is inconclusive, so should be encouraged. (shrink)

It is fair to say that Georg Wilhelm Friedrich Hegel's philosophy of mathematics and his interpretation of the calculus in particular have not been popular topics of conversation since the early part of the twentieth century. Changes in mathematics in the late nineteenth century, the new set-theoretical approach to understanding its foundations, and the rise of a sympathetic philosophical logic have all conspired to give prior philosophies of mathematics (including Hegel's) the untimely appearance of naïveté. The common view was (...) expressed by Bertrand Russell: -/- The great [mathematicians] of the seventeenth and eighteenth centuries were so much impressed by the results of their new methods that they did not trouble to examine their foundations. Although their arguments were fallacious, a special Providence saw to it that their conclusions were more or less true. Hegel fastened upon the obscurities in the foundations of mathematics, turned them into dialectical contradictions, and resolved them by nonsensical syntheses. . . .The resulting puzzles [of mathematics] were all cleared up during the nineteenth century, not by heroic philosophical doctrines such as that of Kant or that of Hegel, but by patient attention to detail (1956, 368–69). (shrink)

A new proof style adequate for modal logics is defined from the polynomial ring calculus. The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra???Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S 5, and can be easily extended to other (...) modal logics. (shrink)

'A Differential Play of Forces' is a study of transcendental empiricism in musical contexts. It presents a reading of Gilles Deleuze and Félix Guattari’s philosophical apparatus and explores how music can be thought of as functioning in the operation Deleuze terms transcendental empiricism. Central to transcendental empiricism is the idea of an encounter with intensive difference and the consequent experience of intensive and virtual forces. The thesis sets out to explore this idea in three interwoven steps. First, it develops (...) transcendental empiricism as a philosophical framework, expounding central notions related to time and aesthetic experience. In the context of this philosophical apparatus, music is then discussed as an encounter with intensity and what this may entail for our understanding of musical experience and its potential. The third part of the thesis explores the idea of experiencing intensive forces through music in the thinking and poietic of three composers of the twentieth century: Arnold Schönberg, Olivier Messiaen and Giacinto Scelsi. The thesis develops three ‘images of thought’ corresponding, respectively, to the composers’ expressed theoretical ideas, identifying central concepts from Deleuze and Guattari’s philosophy. A mutual illumination of ideas about music and transcendental empiricism is thus presented, pointing towards a possible convergence between theory and practice, thinking and aesthetic experience as a differential play of intensive forces. (shrink)

Building on the work of Peter Hinst and Geo Siegwart, we develop a pragmatised natural deduction calculus, i.e. a natural deduction calculus that incorporates illocutionary operators at the formal level, and prove its adequacy. In contrast to other linear calculi of natural deduction, derivations in this calculus are sequences of object-language sentences which do not require graphical or other means of commentary in order to keep track of assumptions or to indicate subproofs. (Translation of our German paper (...) "Ein Redehandlungskalkül. Ein pragmatisierter Kalkül des natürlichen Schließens nebst Metatheorie"; online available at http://philpapers.org/rec/CORERE.). (shrink)

Saturated free fatty acids-induced hepatocyte lipoapoptosis plays a pivotal role in non-alcoholic steatohepatitis. Theactivation of endoplasmic reticulum (ER) stress isinvolved in hepatocyte lipoapoptosis induced by thesaturated free fatty acidpalmitate (PA). However, the underlying mechanismsof the role of ER stress in hepatocyte lipoapoptosis remain largely unclear.In this study, we showed that PA and tunicamycin (Tun), a classic ER stress inducer, resulted in differential activation of ERstress pathways. Our data revealed that PA inducedchronic and persistent ER stress response, but Tuninduced acute (...) and transientER stress response. (shrink)

This paper examines systematically which features of a life story (or history) make it good for the subject herself - not aesthetically or morally good, but prudentially good. The tentative narrative calculus presented claims that the prudential narrative value of an event is a function of the extent to which it contributes to her concurrent and non-concurrent goals, the value of those goals, and the degree to which success in reaching the goals is deserved in virtue of exercising agency. (...) The narrative value of a life is a simple sum of the values of individual events that comprise it. I claim that this view best explains and support common intuitions about the significance of the shape of a life. (shrink)

Hilbert's ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand (...) disjunctions of existential theorems obtained by this elimination procedure. (shrink)

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian (...) infinitesimal calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

We provide a logical matrix semantics and a Gentzen-style sequent calculus for the first-degree entailments valid in W. T. Parry’s logic of Analytic Implication. We achieve the former by introducing a logical matrix closely related to that inducing paracomplete weak Kleene logic, and the latter by presenting a calculus where the initial sequents and the left and right rules for negation are subject to linguistic constraints.

I am presenting a sequent calculus that extends a nonmonotonic consequence relation over an atomic language to a logically complex language. The system is in line with two guiding philosophical ideas: (i) logical inferentialism and (ii) logical expressivism. The extension defined by the sequent rules is conservative. The conditional tracks the consequence relation and negation tracks incoherence. Besides the ordinary propositional connectives, the sequent calculus introduces a new kind of modal operator that marks implications that hold monotonically. Transitivity (...) fails, but for good reasons. Intuitionism and classical logic can easily be recovered from the system. (shrink)

Schizophrenia is often associated with other physical and mental problems. Generalized anxiety disorder is notably one of the comorbid disorders which is often linked to schizophrenia. The association of polythematic delusions and of ideas resulting from generalized anxiety disorder complicates the exercise of the corresponding cognitive therapy, for the resulting ideas are most often inextricably intertwined. In what follows, we endeavour to propose a methodology for the differential treatment of polythematic delusions inherent to schizophrenia when combined with ideas originating (...) from generalized anxiety disorder. We propose, with regard to the corresponding content of delusions, an analysis which allows under certain conditions, to separate the content associated with polythematic delusions and the one that relates to generalized anxiety disorder, in order to facilitate the exercise of the corresponding cognitive therapy. (shrink)