Daniel Sutherland - Kant on Arithmetic, Algebra, and the Theory of Proportions - Journal of the History of Philosophy 44:4 Journal of the History of Philosophy 44.4 533-558 Muse Search Journals This Journal Contents Kant on Arithmetic, Algebra, and the Theory of Proportions Daniel Sutherland Kant's philosophy of mathematics has both enthralled and exercised philosophers since the appearance of the Critique of Pure Reason. Neither the Critique nor any other work provides a sustained and focused account of his (...) mature views on mathematical cognition, forcing readers to glean what they can from disparate contexts. Despite these hurdles, Kant's views have been of great interest to philosophers of mathematics. They have also been of interest to philosophers wishing to understand Kant's philosophy more generally, since Kant maintains that mathematical judgments are synthetic a priori and that the type of synthesis underlying mathematics is the same as that underlying the perception of objects . Our understanding of Kant's views has been greatly improved during the last four decades by work on Kant's philosophy of mathematics. Despite these advances, however, I think the recent work has largely neglected the importance of Kant's theory of magnitudes and the extent to which that theory is influenced by the Eudoxian theory of proportions presented in Euclid's Elements. As a consequence, an important feature of Kant's.. (shrink)

Humans communicate with different modalities. We offer an account of multi-modal meaning coordination, taking speech-gesture meaning coordination as a prototypical case. We argue that temporal synchrony (plus prosody) does not determine how to coordinate speech meaning and gesture meaning. Challenging cases are asynchrony and broadcasting cases, which are illustrated with empirical data. We propose that a process algebra account satisfies the desiderata. It models gesture and speech as independent but concurrent processes that can communicate flexibly with each other and (...) exchange the same information more than once. The account utilizes the ψ-calculus, allowing for agents, input-output-channels, concurrent processes, and data transport of typed λ-terms. A multi-modal meaning is produced integrating speech meaning and gesture meaning into one semantic package. Two cases of meaning coordination are handled in some detail: the asynchrony between gesture and speech, and the broadcasting of gesture meaning across several dialogue contributions. This account can be generalized to other cases of multi-modal meaning. (shrink)

It has been said that pragmatism's "merely instrumental" truths fail to motivate radical change whereas absolute ideals make excellent guiding and driving forces for justice. However, in Radical Equations: Math Literacy and Civil Rights, Robert Moses speaks of the radical success of pragmatic principles, used in the Civil Rights Movement, that are continued today in the Algebra Project. This paper applies Dewey's claims about education and community to Moses's own arguments as a means of depicting the role that pragmatic (...) ideals play in achieving radical social change. (shrink)

Heinrich Behmann (1891-1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved - independently of Löwenheim and Skolem's earlier work - the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks (...) on the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included. (shrink)

This article explores the complementary strengths and weaknesses of grounded and abstract representations in the domain of early algebra. Abstract representations, such as algebraic symbols, are concise and easy to manipulate but are distanced from any physical referents. Grounded representations, such as verbal descriptions of situations, are more concrete and familiar, and they are more similar to physical objects and everyday experience. The complementary computational characteristics of grounded and abstract representations lead to trade‐offs in problem‐solving performance. In prior research (...) with high school students solving relatively simple problems, Koedinger and Nathan (2004) demonstrated performance benefits of grounded representations over abstract representations—students were better at solving simple story problems than the analogous equations. This article extends this prior work to examine both simple and more complex problems in two samples of college students. On complex problems with two references to the unknown, a “symbolic advantage” emerged, such that students were better at solving equations than analogous story problems. Furthermore, the previously observed “verbal advantage” on simple problems was replicated. We thus provide empirical support for a trade‐off between grounded, verbal representations, which show advantages on simpler problems, and abstract, symbolic representations, which show advantages on more complex problems. (shrink)

In this paper we prove that, for n > 1, the n-generated free algebra in any locally finite subvariety of HoRA can be written in a unique nontrivial way as Ł2 × A′, where A′ is a directly indecomposable algebra in . More precisely, we prove that the unique nontrivial pair of factor congruences of is given by the filters and , where the element is recursively defined from the term introduced by W. H. Cornish. As an additional (...) result we obtain a characterization of minimal irreducible filters of in terms of its coatoms. (shrink)

A study of the making of George Peacock's highly influential, yet disturbingly split, 1830 account of algebra as an entanglement of two separate undertakings: arithmetical and symbolical or formal.

Through our work to examine mathematical and computational learning in authentic and convivial contexts that requires creativity, imagination, reasoning, and discourse, we have theorized an experiential learning cycle that attends to the development of voice, agency, and identity needed in young people for an earned insurgency—the right to demand change. Our work underscores how the current situation that many students face in classrooms amounts to a type of cognitive segregation that denies these students access to authentic and empowering intellectual agency. (...) By facilitating a process whereby students, using their own creative and imaginative means, intentionally develop a type of ownership over the exploration and application of the mathematical concepts they are being taught, we help students move from simple surface level, syntactic understandings, to deeper semantic learning that is more personally significant and meaningful. (shrink)

A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a (...) Łukasiewicz–Moisil Topos with an n-valued Łukasiewicz–Moisil Algebraic Logic subobject classifier description that represents non-random and non-linear network activities as well as their transformations in developmental processes and carcinogenesis. The unification of the theories of organismic sets, molecular sets and Robert Rosen’s (M,R)-systems is also considered here in terms of natural transformations of organismal structures which generate higher dimensional algebras based on consistent axioms, thus avoiding well known logical paradoxes occurring with sets. Quantum bionetworks, such as quantum neural nets and quantum genetic networks, are also discussed and their underlying, non-commutative quantum logics are considered in the context of an emerging Quantum Relational Biology. (shrink)

In this paper, I define and study an abstract algebraic structure, the dimensive algebra, which embodies the most general features of the algebra of dimensional physical quantities. I prove some elementary results about dimensive algebras and suggest some directions for future work.

Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition. Bibliography.

In this essay I will defend three points, the first being that Descartes- unlike the aristotelian traditon- maintained that abstraction is not a operation in which the intellect builds the mathematical object resorting to sensible ob- jects. Secondly I will demonstrate that, according to cartesian philosophy, the faculty of understanding has the ability to instatiate- within the process of abstraction- mathematical symbols that represent the relation between quantities, whether magnitude or multitude.And finally I will advocate that the lack of onthological (...) commitment with sensible experience found in cartesian philosophy of mathematics allows for the creation of a mathematical language that regards the objects of geometry and arithmetics through a system of rules and notations, in other words, algebra. (shrink)

We study the theory of a Hilbert space H as a module for a unital C*-algebra ${\mathcal{A}}$ from the point of view of continuous logic. We give an explicit axiomatization for this theory and describe the structure of all the representations which are elementary equivalent to it. Also, we show that this theory has quantifier elimination and we characterize the model companion of the incomplete theory of all non-degenerate representations of ${\mathcal{A}}$ . Finally, we show that there is an (...) homeomorphism between the space of types of norm less than 1 in this model companion, and the space of quasistates of the C*-algebra ${\mathcal{A}}$. (shrink)

The purpose of this article is to analyse the mathematical practices leading to Rafael Bombelli’s L’algebra (1572). The context for the analysis is the Italian algebra practiced by abbacus masters and Renaissance mathematicians of the fourteenth to sixteenth centuries. We will focus here on the semiotic aspects of algebraic practices and on the organisation of knowledge. Our purpose is to show how symbols that stand for underdetermined meanings combine with shifting principles of organisation to change the character of (...) algebra. (shrink)

Various representation results have been established for logics of belief revision, in terms of remainder sets, epistemic entrenchment, systems of spheres and so on. In this paper I present another representation for logics of belief revision, as an algebra of theories. I show that an algebra of theories, enriched with a set of rejection operations, provides a suitable algebraic framework to characterize the theory change operations of systems of belief revision. The theory change operations arise as power operations (...) of the conjunction and disjunction connectives of the underlying logic. (shrink)

This is the fth and last installment in series dealing with the Wittgen- steinian notion of a situation . All proofs and most lemmas have been omitted. They are contained in a comprehensive paper on the ontol- ogy of situations to be submitted to Studia Logica.

In this article, the concepts of Nth Power Set of a Set, Super-Hyper-Oper-Operation, Super-Hyper-Axiom, SuperHyper-Algebra, and their corresponding Neutrosophic Super-Hyper-Oper-Operation, Neutrosophic Super-Hyper-Axiom and Neutrosophic Super-Hyper-Algebra are reviewed. In general, in any field of knowledge, really what are found are Super-HyperStructures (or more specifically Super-Hyper-Structures (m, n)).

We give an example of a variety of Heyting algebras and of a splitting algebra in this variety that is not finitely presentable. Moreover, we show that the corresponding splitting pair cannot be defined by any finitely presentable algebra. Also, using the Gödel-McKinsey-Tarski translation and the Blok-Esakia theorem, we construct a variety of Grzegorczyk algebras with similar properties.

This paper argues that Noether's axiomatic method in algebra cannot be assimilated to Weyl's late view on axiomatics, for his acquiescence to a phenomenological epistemology of correctness led Weyl to resist Noether's principle of detachment.

This chapter discusses the complex conditions for the emergence of 19th-century symbolic logic. The main scope will be on the mathematical motives leading to the interest in logic; the philosophical context will be dealt with only in passing. The main object of study will be the algebra of logic in its British and German versions. Special emphasis will be laid on the systems of George Boole and above all of his German follower Ernst Schröder.

Within the scope of interest of deontic logic, systems in which names of actions are arguments of deontic operators (deontic action logic) have attracted less interest than purely propositional systems. However, in our opinion, they are even more interesting from both theoretical and practical point of view. The fundament for contemporary research was established by K. Segerberg, who introduced his systems of basic deontic logic of urn model actions in early 1980s. Nowadays such logics are considered mainly within propositional dynamic (...) logic (PDL). Two approaches can be distinguished: in one of them deontic operators are introduced using dynamic operators and the notion of violation, in the other at least some of them are taken as primitive. The second approach may be further divided into the systems based on Boolean algebra of actions and the systems built on the top of standard PDL. In the present paper we are interested in the systems of deontic action logic based on Boolean algebra. We present axiomatizations of six systems and set theoretical models for them. We also show the relations among them and the position of some existing theories on the resulting picture. Such a presentation allows the reader to see the spectrum of possibilities of formalization of the subject. (shrink)

In a recent paper, we have shown that the class of Boolean contact algebras (BCAs) has the hereditary property, the joint embedding property and the amalgamation property. By Fraïssé’s theorem, this shows that there is a unique countable homogeneous BCA. This paper investigates this algebra and the relation algebra generated by its contact relation. We first show that the algebra can be partitioned into four sets {0}, {1}, K, and L, which are the only orbits of the (...) group of base automorphisms of the algebra, and then show that the contact relation algebra of this algebra is finite, which is the first non-trivial extensional BCA we know which has this property. (shrink)