The limits of ‘criterial rationality’ (that is, rationality as rule‐following) have been extensively explored in the philosophy of science by Kuhn and others. In this paper I attempt to extend this line of enquiry into mathematics by means of a pair of case studies in early algebra. The first case is the Ars Magna (Nuremburg 1545) by Jerome Cardan (1501–1576), in which a then recently‐discovered formula for finding the roots of some cubic equations is extended to cover all cubics (...) and proved. The second is the formulation by Albert Girard (1595–1632) of an early version of the fundamental theorem of algebra in his L'invention nouvelle en l'algèbre (Amsterdam, 1629). I conclude that in these cases at least, the questions raised in the philosophy of science debate can also be asked of the history of mathematics, and that a modest methodological anarchism is the appropriate stance. (shrink)

‘‘What Is Algebra?-Why This Book?’’ This is the amazing prelude to Taming the Unknown by Victor J. Katz, emeritus professor of mathematics at the University of the District of Columbia and Karen Hunger Parshall, professor of history of mathematics at the University of Virginia. This is an excellent book; its accurate historical and pedagogical purpose offers an accessible read for historians and mathematicians. [continue...].

Relation algebras were invented by Tarski and his collaborators in the middle of the 20th century. The concept of integrality arose naturally early in the history of the subject, as did various constructions of finite integral relation algebras. Later the concept of finite-dimensionality was introduced for classifying nonrepresentable relation algebras. This concept is closely connected to the number of variables used in proofs in first-order logic. Some results on these topics are presented in chronological order.

By the middle of the seventeenth century we that find that algebra is able to offer proofs in its own right. That is, by that time algebraic argument had achieved the status of proof. How did this transformation come about?

Davide Bondoni, La teoria delle relazioni nell'algebra della logica schroderiana. Milan: LED Edizioni Universitarie Lettere Economia Diritto, 2007. 95 pp. [euro]13.00. ISBN 978-88-7916-349-1. Revie...

Among the Arabs, wijdân means a particularly rich and moving relationship between essence and existence (and vice versa). In recent periods of their history it has been interpreted as mass movements, emotional violence, and the impulse to change life. Is this but one aspect, one of several effects, or is it the substance of revolutions? Does the concept “to change life” have a value by itself, or is it only a symptom or corollary of more hidden movements which justify (...) such an abstraction?Etymologically thawra in arabic means “effervescence.” Moreover, it does not just partially include, but surpasses our understanding of revolutions. We have become accustomed to defining them as changes in class relationships. The exercise of a secret logic can be seen which would determine, in the end, the fortunes of production relationships. This is, at any rate, what Marxist doctrine maintained, until recent studies revealed more shades of meaning and completed it. This is not the place to take up in our turn the debate about genesis/structure, structure/superstructure, the predominant factor and its consequences or effects, or “superdetermination,” which is the contingency to use ruses. (shrink)

According to the algebraic approach to spacetime, a thoroughgoing dynamicism, physical fields exist without an underlying manifold. This view is usually implemented by postulating an algebraic structure (e.g., commutative ring) of scalar-valued functions, which can be interpreted as representing a scalar field, and deriving other structures from it. In this work, we point out that this leads to the unjustified primacy of an undetermined scalar field. Instead, we propose to consider algebraic structures in which all (and only) physical fields are (...) primitive. We explain how the theory of natural operations in differential geometry—the modern formalism behind classifying diffeomorphism-invariant constructions—can be used to obtain concrete implementations of this idea for any given collection of fields. For concrete examples, we illustrate how our approach applies to a number of particular physical fields, including electrodynamics coupled to a Weyl spinor. (shrink)

Eine Reihe von mittelalterlichen Schriften zur Landmessung (vom 9. islamischen Jahrhundert bis zu Fibonacci und Pacioli) enthält eine besondere Art von „algebraischen” Aufgaben. Darin werden z.B. die Summe der Fläche und einer oder alle vier Seiten eines Quadrates beschrieben und nach der Seite gefragt. Es zeigt sich erstens, daß dieser Aufgabentyp mindestens seit dem frühesten 2. vorchristlichen Jahrtausend von geometrischen Praktikern tradiert wurde, und zweitens, daß er die Entwicklung einer „Algebra” in der altbabylonischen Schreiberschule inspirierte. Der Aufsatz untersucht, in (...) welcher Weise die Überführung der „sub-wissenschaftlichen” Praktikertradition in einen systematischen Schulunterricht den mathematischen Inhalt und den Denkstil künftiger Praktiker präge. Im letzten Kapitel wird diskutiert, inwieweit es sinnvoll ist, innerhalb dieser Schulalgebra von „Schulen” zu sprechen. (shrink)

In this paper, we examine the relationship between general relativity and the theory of Einstein algebras. We show that according to a formal criterion for theoretical equivalence recently proposed by Halvorson and Weatherall, the two are equivalent theories.

This book celebrates the work of Don Pigozzi on the occasion of his 80th birthday. In addition to articles written by leading specialists and his disciples, it presents Pigozzi’s scientific output and discusses his impact on the development of science. The book both catalogues his works and offers an extensive profile of Pigozzi as a person, sketching the most important events, not only related to his scientific activity, but also from his personal life. It reflects Pigozzi's contribution to the rise (...) and development of areas such as abstract algebraic logic, universal algebra and computer science, and introduces new scientific results. Some of the papers also present chronologically ordered facts relating to the development of the disciplines he contributed to, especially abstract algebraic logic. The book offers valuable source material for historians of science, especially those interested in history of mathematics and logic. (shrink)

The CPT theorem states that any causal, Lorentz-invariant, thermodynamically well-behaved quantum field theory must also be invariant under a reflection symmetry that reverses the direction of time, flips spatial parity, and conjugates charge. Although its physical basis remains obscure, CPT symmetry appears to be necessary in order to unify quantum mechanics with relativity. This paper attempts to decipher the physical reasoning behind proofs of the CPT theorem in algebraic quantum field theory. Ultimately, CPT symmetry is linked to a systematic reversal (...) of the C*-algebraic Lie product that encodes the generating relationship between observables and symmetries. In any physically reasonable relativistic quantum field theory it is always possible to systematically reverse this generating relationship while preserving the dynamics, spectra, and localization properties of physical systems. Rather than the product of three separate reflections, CPT symmetry is revealed to be a single global reflection of the theory’s state space. (shrink)

This article reviews Hestenes' work on the Dirac theory, where his main achievement is a real formulation of the theory within thereal Clifford algebra Cl 1,3 ≃ M2 (H). Hestenes invented first in 1966 hisideal spinors $\phi \in Cl_{1,3 _2}^1 (1 - \gamma _{03} )$ and later 1967/75 he recognized the importance of hisoperator spinors ψ ∈ Cl 1,3 + ≃ M2 (C).This article starts from the conventional Dirac equation as presented with matrices by Bjorken-Drell. Explicit mappings are given (...) for a passage between Hestenes' operator spinors and Dirac's column spinors. Hestenes' operator spinors are seen to be multiples of even parts of real parts of Dirac spinors (real part in the decompositionC ⊗ Cl 1,3 andnot inC ⊗ M4 (R)=M4 (C)). It will become apparent that the standard matrix formulation contains superfluous parts, which ought to be cut out by Occam's razor.Fierz identities of bilinear covariants are known to be sufficient to study the non-null case but are seen to be insufficient for the null case ψ†γ0ψ=0, ψ†γ0γ0123ψ=0. The null case is thoroughly scrutinized for the first time with a new concept calledboomerang. This permits a new intrinsically geometric classification of spinors. This in turn reveals a new class of spinors which has not been discussed before. This class supplements the spinors of Dirac, Weyl, and Majorana; it describes neither the electron nor the neutron; it is awaiting a physical interpretation and a possible observation.Projection operators P±, Σ± are resettled among their new relatives in End(Cl 1,3 ). Finally, a new mapping, calledtilt, is introduced to enable a transition from Cl 1,3 to the (graded) opposite algebra Cl 3,1 without resorting to complex numbers, that is, not using a replacement γμ →iγμ. (shrink)

In this paper, we use the differential forms of three-dimensional Euclidean space to realize a Clifford algebra which is isomorphic to the algebra of the Pauli matrices or the complex quaternions. This is an associative vector-antisymmetric tensor algebra with division: We provide the algebraic inverse of an eight-component spinor field which is the sum of a scalar + vector + pseudovector + pseudoscalar. A surface of singularities is defined naturally by the inverse of an eight-component spinor and (...) corresponds to a generalized Minkowski “double” light cone in the parameter space. A general description of finite spatial rotations, which utilizes the Baker-Campbell-Hausdorff formula, generalizes the usual infinitesimal treatments of the rotation group. We derive an explicit expression for the angle corresponding to two successive finite rotations in any direction. We also discuss Lorentz transformations and duality rotations of the electromagnetic field and exhibit a relationship between the algebraic inverse and a duality rotated field. Using a combined transformation, one can always transform an arbitrary electromagnetic field (E≠0) into a pure electric field, but never into a pure magnetic field. (shrink)

In Plato's Parmenides we find on the one hand that the One is denied every property , and on the other hand that the One is attributed every property . In the course of the history of Platonism , these assertions - probably meant by Plato as ontological statements of an entirely formal nature - were repeatedly made the starting points of metaphysical speculations. In the Mystical Theology of the Pseudo-Dionysius they became principles of Christian mysticism and negative theology. (...) I shall show that the two assertions can each be interpreted within the ontological framework of ancient and medieval logic in such a manner that it becomes true, and I shall make plausible that they were understood, with regard to their logical core, by pagan and Christian Platonic metaphysicians just as is indicated by that interpretation. The mentioned ontological framework is basically the Boolean algebra of first-order properties. The main points of the interpretation are on the one hand the identification of the One with the maximal element of the algebra of properties, and on the other hand two alternative intuitively prominent mereological definitions of ontic predication. (shrink)

The invention of symbolic algebra in the sixteenth and seventeenth centuries fundamentally changed the way we do mathematics. If we want to understand this change and appreciate its importance, we must analyze it on two levels. One concerns the compositional function of algebraic symbols as tools for representing complexity; the other concerns the referential function of algebraic symbols, which enables their use as tools for describing objects (such as polynomials), properties (such as irreducibility), relations (such as divisibility), and operations (...) (such as factorization). The reconstruction of both the compositional function and the referential function of algebraic symbols requires the use of different analytic tools and the taking of different temporal perspectives. In this chapter, we offer both: a reconstruction of the compositional function of algebraic symbols, and a reconstruction of their referential function. (shrink)

Bell inequalities, understood as constraints between classical conditional probabilities, can be derived from a set of assumptions representing a common causal explanation of classical correlations. A similar derivation, however, is not known for Bell inequalities in algebraic quantum field theories establishing constraints for the expectation of specific linear combinations of projections in a quantum state. In the paper we address the question as to whether a ‘common causal justification’ of these non-classical Bell inequalities is possible. We will show that although (...) the classical notion of common causal explanation can readily be generalized for the non-classical case, the Bell inequalities used in quantum theories cannot be derived from these non-classical common causes. Just the opposite is true: for a set of correlations there can be given a non-classical common causal explanation even if they violate the Bell inequalities. This shows that the range of common causal explanations in the non-classical case is wider than that restricted by the Bell inequalities. (shrink)

Gravitomagnetic equations result from applying quaternionic differential operators to the energy–momentum tensor. These equations are similar to the Maxwell’s EM equations. Both sets of the equations are isomorphic after changing orientation of either the gravitomagnetic orbital force or the magnetic induction. The gravitomagnetic equations turn out to be parent equations generating the following set of equations: the vorticity equation giving solutions of vortices with nonzero vortex cores and with infinite lifetime; the Hamilton–Jacobi equation loaded by the quantum potential. This equation (...) in pair with the continuity equation leads to getting the Schrödinger equation describing a state of the superfluid quantum medium ; gravitomagnetic wave equations loaded by forces acting on the outer space. These waves obey to the Planck’s law of radiation. (shrink)

It is shown in this article that the two sides of an equation in the medieval Arabic algebra are aggregations of the algebraic “numbers” (powers) with no operations present. Unlike an expression such as our 3x + 4, the Arabic polynomial “three things and four dirhams” is merely a collection of seven objects of two different types. Ideally, the two sides of an equation were polynomials so the Arabic algebraists preferred to work out all operations of the enunciation to (...) a problem before stating an equation. Some difficult problems which involve square roots and divisions cannot be handled nicely by this basic method, so we do find square roots of polynomials and expressions of the form “A divided by B” in some equations. But rather than initiate a reconsideration of the notion of equation, these developments were used only for particularly complex problems. Also, the algebraic notation practiced in the Maghreb in the later middle ages was developed with the “aggregations” interpretation in mind, so it had no noticeable impact on the concept of polynomial. Arabic algebraists continued to solve problems by working operations before setting up an equation to the end of the medieval period. (shrink)

Ghost Dirac’s fermions are a manifestation of virtual particles. One fermion is the particle whose companion is the antiparticle. An ensemble of these fermions coupled in pairs represents the Bose-Einstein condensate. This condensate forms the superfluid ether. Due to the Meissner effect inherent in a superfluid medium, the paired fermions are inaccessible for instrument observation. For that reason, the ghost particles can pose the dark matter that, together with the dark energy, can be the fundamental basis of physical reality. In (...) the article, we consider this item through the prism of the Dirac equation stemming from the superfluid quantum ether. The latter is the inherent ubiquitous parent of fermion particle-antiparticle pairs. (shrink)

Françios Viète was a geometer in search of better techniques for astronomical calculation. Through his theorem on angular sections he found a use for higher-dimensional geometric magnitudes which allowed him to create an algebra for geometry. We show that unlike traditional numerical algebra, the knowns and unknowns in Viète’s logistice speciosa are the relative sizes of non-arithmetized magnitudes in which the “calculations” must respect dimension. Along with this foundational shift Viète adopted a radically new notation based in Greek (...) geometric equalities. His letters stand for values rather than types, and his given values are undetermined. Where previously algebra was founded in polynomials as aggregations, Viète became the first modern algebraist in working with polynomials built from operations, and the notations reflect these conceptions. Viète’s innovations are situated in the context of sixteenth-century practice, and we examine the interpretation of Jacob Klein, the only historian to have conducted a serious inquiry into the ontology of Viète’s “species”. (shrink)

De Morgan's Formal Logic, which was published on virtually the same day in 1847 as Boole's The Mathematical Analysis of Logic, contains a logic of complex terms (LCT) which has been sadly neglected. It is surprising to find that LCT contains almost a full theory of Boolean algebra. This paper will: (1) provide some background to LCT; (2) outline its main features; (3) point out some gaps in it; (4) compare it with Boole's algebra; (5) show that it (...) is a lattice-theoretical formulation of Boolean algebra; (6) discuss some issues of historical priority; and (7) conclude with the puzzle of LCT's lack of influence. (shrink)

A short seventeenth-century text, sometimes cited as one of the first essays in mathematical logic, is introduced, translated and evaluated. Although by no means sharing the depth and magnitude of the investigations by Leibniz being undertaken at the same time, and although in particular not yet applying algebraic symbolism to logical structures, the treatise is of historical interest as an early published attempt to trace out analogies between logical and mathematical form, and may be viewed as a preliminary step toward (...) the formalization of logic. (shrink)

It can be said that Formal Logic begun by studying an idealization of the statements ’if p, then q’, something coming from long ago in both Greek and Scholastic Philosophy. Nevertheless, only in the XX Century it arrived at a stage of formalization once in 1910 Russell introduced and identified the ’material conditional’ with the expresion ”not p or q”. In 1918, and from paradoxical conditionals like ”If the Moon is a cheese, it is a Lyon’s face”, Lewis critiziced the (...) material conditional and introduced the so-called ’strict conditional’ as the modal necessity of the material one. In 1934, Huntington proved that both material and strict implications are coincidental in the setting of Boolean algebras and, hence, that they only can be actually different in algebraic structures weaker than Boolean algebras. Boolean algebras have too much laws for supporting the difference of both conditionals. This paper is nothing else than an algebraic trial to find simple binary Boolean operations able to express Lewis’ strict implication, and contains a new and simpler proof than that of Huntington, made by identifying ’possible’ with ’non self-contradictory’. By the way, it is shown that this proof is only valid in Boolean algebras, but neither in proper De Morgan algebras, nor in proper Ortholattices, with which it still remains open where the two conditional relations are actually different. (shrink)

Karl Eugen Müller's contribution to the development of the algebra of logic is perhaps the most important part of his scientific work. Müller, who became Gymnasialprofessor after his university studies, was a student of Ernst Schröder's friend, the mathematician Jakob Lüroth. As a result of publishing two papers on problems related to Schröder's monumental Vorlesungen iiber die Algebra der Logik, Müller was commissioned by the Deutsche Mathematiker- Vereinigung with the editing of the unpublished parts of the Vorlesungen from (...) Schröder's Nachla?. Müller worked on Schröder's papers until 1910, but did not bring this work to a conclusion. Müller's own Nachla?, including those parts of Schröder's papers still in his possession, was destroyed in Frankfurt a.M. in 1943, so there remains no hope of finding through Müller any part of the missing Nachla? of Schröder. (shrink)

The Russian philosopher and cultural theorist Mikhail Bakhtin has traditionally been seen as the leading figure in the group of intellectuals known as the Bakhtin Circle. The writings of other members of the Circle are considered much less important than his work, while Bakhtin's achievement has been exaggerated in proportion to the downgrading of the thinkers with whom he associated in the 1920s. This volume, which includes new translations and studies of the work of the most important members of the (...) Circle, sets out to correct the distortions in the established representations of its activity. The original contributions to literary and linguistic theory made by Valentin Voloshinov and Pavel Medvedev (but frequently credited to Bakhtin) are assessed, and the distinctiveness of their approaches is highlighted. (shrink)

We define a finite relation algebra and show that the network satisfaction problem is undecidable for this algebra.

Katie Steele and H. Orri Stefánsson argue that, to reflect an agent’s limited awareness, the algebra of propositions on which that agent’s credences are defined should be relativised to their awareness state. I argue that this produces insurmountable difficulties. But the project of relativising the agent’s algebra to reflect their partial perspective need not be abandoned: the algebra can be relativised, not to the agent’s awareness state, but to what we might call their subjective modality.