The paper aims to establish if Grassmann’s notion of an extensive form involved an epistemological change in the understanding of geometry and of mathematical knowledge. Firstly, it will examine if an ontological shift in geometry is determined by the vectorial representation of extended magnitudes. Giving up homogeneity, and considering geometry as an application of extension theory, Grassmann developed a different notion of a geometrical object, based on abstract constraints concerning the construction of forms rather than on the homogeneity conditions required (...) by the modern version of the theory of proportions. Secondly, Grassmann’s conception of mathematical knowledge will be investigated. Parting from the traditional definition of mathematics as a science of magnitudes, Grassmann considered mathematical forms as particulars rather than universals: the classification of the branches of mathematics was thus based on different operational rules, rather than on empirical criteria of abstraction or on the distinction of different species belonging to a common genus. It will be argued that a different notion of generalization is thus involved, and that the knowledge of mathematical forms relies on the understanding of the rules of generation of the forms themselves. Finally, the paper will analyse if Grassmann’s approach in the first edition of the Ausdehnungslehre should be explained in terms of the notion of purity of method, and if it clashes with Grassmann’s later conventionalism. Although in the second edition the features of the operations are chosen by convention, as it is the case for the anti-commutative property of the multiplication, the choice is oriented by our understanding of the resulting forms: a simplification in the algebraic calculus need not correspond to a simplification in the ‘dimensional’ interpretation of the result of the multiplicative operation. (shrink)

To analyze the task of mental arithmetic with external representations in different number systems we model algorithms for addition and multiplication with Arabic and Roman numerals. This demonstrates that Roman numerals are not only informationally equivalent to Arabic ones but also computationally similar—a claim that is widely disputed. An analysis of our models' elementary processing steps reveals intricate tradeoffs between problem representation, algorithm, and interactive resources. Our simulations allow for a more nuanced view of the received wisdom on Roman numerals. (...) While symbolic computation with Roman numerals requires fewer internal resources than with Arabic ones, the large number of needed symbols inflates the number of external processing steps. (shrink)

Pourquoi une université « cheap »? Pourquoi nous en faut-il davantage? Les membres de The Cheapest University nous expliquent les enjeux et les tactiques qui structurent leur plateforme d’initiatives, où se mélangent création artistique, activité pédagogique et pratiques de convivialité.

The paper deals with the study of the existence of weak positive solutions for a new class of the system of elliptic differential equations with respect to the symmetry conditions and the right hand side which has been defined as multiplication of two separate functions by using the sub-supersolutions method.

Number facts are commonly assumed to be verbally stored in an associative multiplication fact retrieval network. Prominent evidence for this assumption comes from so-called operand-related errors (e.g. 4 × 6 = 28). However, little is known about the development of this network in children and its relation to verbal and non-verbal memories. In a longitudinal design, we explored elementary school children from grades 3 and 4 in a multiplication verification task with the operand-related and -unrelated distractors. We examined the contribution (...) of multiplicative fact retrieval by verbal and visuo-spatial short-term and working memory. Children in grade 4 showed smaller reaction times in all conditions. However, there was no significant difference in errors between grades. Contribution of verbal and visuo-spatial working memory also changed with grade. Multiplication correlated with verbal working memory and performance in grade 3 but with visuo-spatial working memory and performance in grade 4. We suggest that the relation to verbal working memory in grade 3 indicates primary linguistic learning of and access to multiplication in grade 3 which is probably based on verbal repetition of the multiplication table heavily practiced in grades 2 and 3. However, the relation to visuo-spatial semantic working memory in grade 4 suggests that there is a shift from verbal to visual and semantic learning in grade 4. This shifting may be induced because later in elementary school, multiplication problems are rather carried out via more written, i.e., visual tasks, which also involve executive functions. More generally, the current data indicates that mathematical development is not generally characterized by a steady progress in performance; rather verbal and non-verbal memory contributions of performance shift over time, probably due to different learning contents. (shrink)

The evolution of research on the biochemistry of virus multiplication cannot be understood without knowing something of the structure of biochemistry and of virology before, during and immediately after World War II. My own research on virus multiplication began after studies on plant viruses and wartime research on the rickettsial components of the typhus vaccine, all of which involved work on the nucleic acids. Interest in the chemotherapy of virus disease led to a search for a model system. A simple (...) methodology for handling phage systems enabled biochemists to examine infected cells. The fortuitous use of an extremely virulent phage group (the T‐even phages) which evoke marked changes in polymeric products without affecting the production of energy and small building blocks facilitated these studies. An exaggerated synthesis of viral DNA by these phages concentrated our attention on the origins of this substance, which proved to contain a new pyrimidine deoxyribonucleotide, synthesized by a novel virus‐determined enzyme. The generality of the existence of virus‐induced enzymes and other proteins may be the key to a chemotherapy of virus infection. (shrink)

Frank Ramsey in his paper ‘Truth and Probability’ was the first to develop a theory of utility based on a representation theorem, and a theory of partial belief based on utility-valued odds. But his proof of the multiplication theorem, on which in his system the law of addition depends, contains a step for which there seems to be no justification, and Ramsey provided no clue as to how to supply one. I conjecture that the missing justification appeals naturally to a (...) three-valued logic that de Finetti later described in his own discussion of the multiplication theorem. If this is the case then Ramsey’s paper is even more strikingly original than has been thought. (shrink)

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o‐minimal structure. This fact together with the results in a previous paper implies a tame dimension theory and a decomposition theorem into good‐shaped definable subsets called quasi‐special submanifolds. Using this fact, we investigate definably complete locally o‐minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o‐minimal expansions of ordered (...) fields, Łojasiewicz's inequality, Tietze's extension theorem and affiness of pseudo‐definable spaces hold true for such structures under the extra assumption that the domains of definition and the pseudo‐definable spaces are definably compact. Here, a pseudo‐definable space is a topological space having finite definable atlases. We also demonstrate Michael's selection theorem for definable set‐valued functions with definably compact domains of definition. (shrink)

The article examines the transformations of the philosophical concept of world as it appears in the cosmopolitan tradition of political thought and its relation to the problem of the border. It focuses particularly on how world is understood as either lost or multiplied in the contexts of modernity, globalisation, and migration. The article discusses postcolonial conceptions of cosmopolitics and the political philosophy of Hannah Arendt to show how the universal ideal of the world is replaced by singular constructions of worlds (...) in terms of the experience of migrants and refugees or the phenomenological horizon of political action. I conclude by suggesting that Jacques Rancière’s understanding of politics as a conflict of worlds can take us beyond the traps of both cosmopolitan universalism and the phenomenological singularity of being-in-the-world. (shrink)

In this paper, we introduce for the first time the neutrosophic quadruple numbers (of the form a + bT + cI + dF) and the refined neutrosophic quadruple numbers. Then we define an absorbance law, based on a prevalence order, both of them in order to multiply the neutrosophic components T, I, F, or their sub-components tJ, Ik, Fl and thus to construct the multiplication of neutrosophic quadruple numbers.

The topic of this paper is Hegel's claim in the Philosophy of Right that, within the modern social world, human needs tend to be endlessly expanded. Unlike the role that the system of needs plays in the formation of its participants' psychological makeup and the problem of poverty and the rabble, the topic of the expansion of needs remains underdiscussed in the recent Hegel literature on the virtues and vices of civil society. My discussion of the topic aims to answer (...) the following two sets of questions: How does it come to pass that individuals' needs are endlessly expanded in this way? And is that expansion a phenomenon to be applauded or condemned? In particular, does the endless expansion of needs aid or obstruct the realization of social members' freedom? In answer to the first question, I argue that for Hegel the endless expansion of needs results from the level of specialization and division of labor distinctive of the modern market economy, the human capacity for a certain kind of abstraction, and the desire to be recognized by other participants in the market system. In answer to the second set of questions, and despite Hegel's own apparent ambivalence, I argue on his behalf that the endless expansion of needs represents an obstacle to the realization of freedom, and is on that ground a phenomenon to be condemned, for the following two reasons: First, the endless expansion of needs increases the influence or “pressure” of desire on the members of civil society that are subject to that expansion. Second, that expansion leads to widespread frustration, understood as the inability on the part of the members of civil society to ever fully realize their ends or satisfy their desires. I end by briefly considering two Hegelian solutions to the pernicious effects of the endless expansion of needs. (shrink)

The topic of this paper is Hegel's claim in the Philosophy of Right that, within the modern social world, human needs tend to be endlessly expanded. Unlike the role that the system of needs plays in the formation of its participants' psychological makeup and the problem of poverty and the rabble, the topic of the expansion of needs remains underdiscussed in the recent Hegel literature on the virtues and vices of civil society. My discussion of the topic aims to answer (...) the following two sets of questions: How does it come to pass that individuals' needs are endlessly expanded in this way? And is that expansion a phenomenon to be applauded or condemned? In particular, does the endless expansion of needs aid or obstruct the realization of social members' freedom? In answer to the first question, I argue that for Hegel the endless expansion of needs results from the level of specialization and division of labor distinctive of the modern market economy, the human capacity for a certain kind of abstraction, and the desire to be recognized by other participants in the market system. In answer to the second set of questions, and despite Hegel's own apparent ambivalence, I argue on his behalf that the endless expansion of needs represents an obstacle to the realization of freedom, and is on that ground a phenomenon to be condemned, for the following two reasons: First, the endless expansion of needs increases the influence or “pressure” of desire on the members of civil society that are subject to that expansion. Second, that expansion leads to widespread frustration, understood as the inability on the part of the members of civil society to ever fully realize their ends or satisfy their desires. I end by briefly considering two Hegelian solutions to the pernicious effects of the endless expansion of needs. (shrink)

In this paper, we make a short history about: the neutrosophic set, neutrosophic numerical components and neutrosophic literal components, neutrosophic numbers, neutrosophic intervals, neutrosophic hypercomplex numbers of dimension n, and elementary neutrosophic algebraic structures. Afterwards, their generalizations to refined neutrosophic set, respectively refined neutrosophic numerical and literal components, then refined neutrosophic numbers and refined neutrosophic algebraic structures.

SummaryIn the 1940s a controversy developed between John H. Northrop (Nobel Laureate, 1946) and Max Delbrück (Nobel Laureate, 1969) on the formation of bacteriophage. From the historiography of molecular biology there emerges a picture of an obstinate Northrop who repudiated the ‘correct’ insights revealed by the experiments of Delbrück. The established reputation of Northrop confronts one with the question of why Delbrück's epoch-making experiments were not convincing for Northrop. It will be argued that this was a consequence of local incommensurability (...) between Northrop's physiological/chemical research style and Delbrück's bacteriological/genetic research style. The analysis of this controversy reveals the elements that made up these two research styles. (shrink)

Multiple realization historically mandated the autonomy of psychology, and its principled irreducibility to neuroscience. Recently, multiple realization and its implications for the reducibility of psychology to neuroscience have been challenged. One challenge concerns the proper understanding of reduction. Another concerns whether multiple realization is as pervasive as is alleged. I focus on the latter question. I illustrate multiple realization with actual, rather than hypothetical, cases of multiple realization from within the biological sciences. Though they do support a degree of autonomy (...) for higher levels of explanation and organization, they do not have the dire consequences critics of multiple realization fear. †To contact the author, please write to: Department of Philosophy, University of Cincinnati, P.O. Box 210374, Cincinnati, OH 45221‐0374; e‐mail: [email protected]. (shrink)

Since Hilary Putnam offered multiple realization as an empirical hypothesis in the 1960s, philosophical consensus has turned against the idea that mental processes are identifiable with brain processes, and multiple realization has become the keystone of the 'antireductive consensus' across philosophy of science. Thomas W. Polger and Lawrence A. Shapiro offer the first book-length investigation of multiple realization, which serves as a starting point to a series of philosophically sophisticated and empirically informed arguments that cast doubt on the generality of (...) multiple realization in the cognitive sciences. They argue that mind-brain identities have played an important role in the growth and achievements of the cognitive sciences, and suggest that there is little prospect for multiple realization in an empirically-based theory of mind. This leads Polger and Shapiro to offer an alternative framework for understanding explanations in the cognitive sciences, as well as in chemistry, biology, and other non-basic sciences. (shrink)

One way that scientifically recognized properties are multiply realized is by “compensatory differences” among realizing properties. If a property G is jointly realized by two properties F1 and F2, then G can be multiply realized by having changes in the property F1 offset changes in the property F2. In some cases, there are scientific laws that articulate how distinct combinations of physical quantities can determine one and the same value of some other physical quantity. One moral to draw is that (...) in such cases we have the multiple realization of a single determinate, “fine grained” property instance that is exactly similar to another instance. As simple as this moral is, it has ramifications for a number of recent discussions of multiple realization in science. Taken collectively, these ramifications indicate that multiple realization by compensatory adjustments merits greater attention in the philosophy of science literature than it has hitherto received. (shrink)

The Multiple Reality theory represents a key piece to understand the socio-phenomenology. It is sustained by a triangulation of the phenomenology of Edmund Husserl, the vitalism of Henry Bergson and the pragmatism of William James (and it has as a background the praxeology of Ludwig Von Mises and ..