This edited volume, aimed at both students and researchers in philosophy, mathematics and history of science, highlights leading developments in the overlapping areas of philosophy and the history of modern mathematics. It is a coherent, wide ranging account of how a number of topics in the philosophy of mathematics must be reconsidered in the light of the latest historical research and how a number of historical accounts can be deepened by embracing philosophical questions.

Architecture of Mathematics describes the logical structure of Mathematics from its foundations to its real-world applications. It describes the many interweaving relationships between different areas of mathematics and its practical applications, and as such provides unique reading for professional mathematicians and nonmathematicians alike. This book can be a very important resource both for the teaching of mathematics and as a means to outline the research links between different subjects within and beyond the subject. Features All notions and properties are (...) introduced logically and sequentially, to help the reader gradually build understanding. Focusses on illustrative examples that explain the meaning of mathematical objects and their properties. Suitable as a supplementary resource for teaching undergraduate mathematics, and as an aid to interdisciplinary research. Forming the reader's understanding of Mathematics as a unified science, the book helps to increase his general mathematical culture. (shrink)

During his stay in Padua ca. 1592–1610, Galileo Galilei (1564–1642) was a lecturer of mathematics at the University of Padua and a tutor to private students of military architecture and fortifications. He carried out these activities at the Academia degli Artisti. At the same time, and in relation to his teaching activities, he began to study the equilibrium of bodies and strength of materials, later better structured and completed in his Dialogues Concerning Two New Sciences of 1638. This paper (...) examines important details of four works dating to the Paduan period: Breve instruzione dell’architettura militare; Trattato di Fortificazione; Le Mecaniche; Le operazioni del compasso geometrico et militare. The two works on military architecture and fortifications were compiled from notes taken by students, and are not by Galileo’s hand, but are still illustrative of his work and thinking at the time. (shrink)

New technologies, such as functional magnetic resonance imaging and transcranial magnetic stimulation, are currently touted as, not only giving us a better picture of the structure of the brain, but also a better understanding of our thinking. As Alan Snyder demonstrates when he claims his aim is to understand the ‘architecture of thought’ by investigating the brain. Against this backdrop, I will argue that new technologies present a worrying extension of mathematical natural science into the domain of human (...) affairs. Extrapolating upon Heidegger, I will put forward that neuroscientific experiments force thinking to conform to the mathematical conception of nature, rather than reveal something about the ‘true’ nature of our thinking. In a time when the expansion of mathematical natural science threatens to reduce every domain to that which is quantifiable, I will conclude by suggesting that the responsibility of the philosopher is to question the presuppositions of modern science and psychology.Nuevas tecnologías como la imagen de resonancia magnética funcional y la estimulación magnética transcranial se consideran presumiblemente capaces de darnos no solo una mejor imagen de nuestro cerebro, sino también una mejor comprensión de nuestro pensamiento, tal como demuestra Alan Snyder cuando afirma que su objetivo es entender la “arquitectura del pensamiento” mediante la investigación del cerebro. Contra este marco general, argumentaré que las nuevas tecnologías nos presentan una preocupante extensión de la ciencia natural matemática al dominio de los asuntos humanos. Extrapolando a partir de Heidegger, propondré que los experimentos neurocientíficos fuerzan al pensamiento a conformarse a la concepción matemática de la naturaleza en vez de revelar algo sobre la verdadera naturaleza de nuestro pensamiento. Nos encontramos en un momento en que o la expansión de la ciencia natural matemática amenaza con reducir todos los dominios a lo que es cuantificable. Concluiré sugiriendo que la responsabilidad del filósofo es cuestionar los presupuestos de la ciencia moderna y de la psicología. (shrink)

Open peer commentary on the article “Radical Constructivist Structural Design Education for Large Cohorts of Chinese Learners” by Christiane M. Herr. Upshot: In the target article, Christiane Herr offers an insightful characterization of how von Glasersfeld’s radical constructivism can be implemented in structural design education in architecture. In this commentary, I articulate possible connections between research on problem solving and problem posing in mathematics education and design processes in structural design education as described in the target article.

In our contribution we will observe phenomenal architecture of a mind and operational architectonics of the brain and will show their intimate connectedness within a single integrated metastable continuum. The notion of operation of different complexity is the fundamental and central one in bridging the gap between brain and mind: it is precisely by means of this notion that it is possible to identify what at the same time belongs to the phenomenal conscious level and to the neurophysiological level (...) of brain activity organization, and what mediates between them. Implications for linguistic semantics, self-organized distributed computing algorithms, artificial machine consciousness, and diagnosis of dynamic brain diseases will be discussed briefly. (shrink)

This collection of essays explores what makes modern mathematics ‘modern’, where ‘modern mathematics’ is understood as the mathematics done in the West from roughly 1800 to 1970. This is not the trivial matter of exploring what makes recent mathematics recent. The term ‘modern’ (or ‘modernism’) is used widely in the humanities to describe the era since about 1900, exemplified by Picasso or Kandinsky in the visual arts, Rilke or Pound in poetry, or Le Corbusier or Loos in architecture (a (...) building by the latter graces the cover of this book’s dust jacket). (shrink)

The city of Kruja dates back to its existence in the 5th and 6th centuries. In the inner city are preserved great historical, cultural, and architectural values that are inherited from generation to generation. In the city interact and coexist three different typologies of dwellings: historic buildings that belong to the XIII, XIV, XV, XIII, XIX centuries (built using the foundations of previous buildings); socialist buildings dating back to the Second World War until 1990; and modern buildings which were built (...) from 1990 onwards. According to the questionnaires, the creation of mathematical models applied to each category will result in contradictory attitudes but also fairy ones based on different percentages. There are underlined 5 quality of life indicators(questions) from a total of 30 questions of the questionnaire, which are involved in mathematical regression and are statistically significant with a significance level p<5% (with a reliability of 95%), which interact for all three categories of buildings. The quality of life indicators: dwelling area, heating mode during winter, level of dwelling improvement, time spent in the dwelling, and monthly electricity payments, are the main actors who will be compared in order to draw conclusions. According to the statistical calculations trend, it is noted that the socialist buildings category does not directly participate in the debate between historic and modern buildings by means of the quality of life indicators (questions). (shrink)

The ARXMLIV corpus is a remarkable collection of text containing scientific mathematical discourse. With more than half a million documents, it is an ambitious target for large scale linguistic and semantic analysis, requiring a generalized and distributed approach. In this paper we implement an architecture which solves and automates the issues of knowledge representation and knowledge management, providing an abstraction layer for distributed development of semantic analysis tools. Furthermore, we enable document interaction and visualization and present current implementations (...) of semantic tools and follow-up applications using this architecture. We identify five different stages, or purposes, which such architecture needs to address, encapsulating each in an independent module. These stages are determined by the different properties of the document formats used, as well as the state of processing and linguistic enrichment introduced so far. We discuss the need of migration between XML representations and the challenges it would pose on our system, revealing the benefits and trade-off of each format we employ. In the heart of the architecture lies the Semantic Blackboard module. The Semantic Blackboard comprises a system based on a centralized RDF database which can facilitate distributed corpus analysis of arbitrary applications, or analysis modules. This is achieved by providing a document abstraction layer and a mechanism for storing, reusing and communicating results via RDF stand-off annotations deposited in the central database. Achieving a properly encapsulated and automated pipeline from the input corpus document to a semantically enriched output in a state-of-the-art representation is the task of the Preprocessing, Semantic Result and Output Generation modules. Each of them addresses the task of format migration and enhances the document for further semantic enrichment or aggregation. The fifth module, targeting Visualization and Feedback, enables user interaction and display of different stages of processing.. (shrink)

_ Source: _Volume 46, Issue 1, pp 35 - 53 In a letter written at the end of July 1930, Jean Cavaillès singled out two of his successful students at the _Ecole Normale_, Merleau-Ponty and Lautman, “full of interest in the philosophy of mathematics”. While both would play an important role in French philosophy in the coming decades, one almost never thinks of their names together. Indeed, only rarely do we think of Merleau-Ponty and Cavaillès together. This paper will argue (...) against this rarity. Cavaillès would write a treatise on logic that in retrospect is often viewed to advance French philosophy—but precisely at the expense phenomenology. I will argue that this evaluation is overblown. Merleau-Ponty’s philosophy, on the other hand, is often viewed in opposition to this as defending a Husserlian account of perception, a view equally too narrow. Between them, instead, I argue, we can see both transforming the Husserlian account of a transcendental logic and a transcendental aesthetic. Both provide resources for understanding phenomenology as a dialectical and historically emergent theory. Notwithstanding Cavaillès initial characterization of Husserl’s “exhorbitant use of the Cogito”, or his “aggressive” description of acts, to use Merleau-Ponty’s term, the result, in both cases, focuses on phenomenology as an emergent experience of philosophy as an “architecture” of signs. (shrink)

Iannis Xenakis (1922–2001), composer, architect, engineer and media artist, designed together with Le Corbusier the Philips-pavilion for the 1958 Brussels World Fair. This pavilion is an early example of (“hybrid”) combined media and architectural space as it contained a Poème Électronique, an electronic synthesis of visual projections (conceived by Le Corbusier ) and acoustic events (composed by Varèse). The pavilion's architecture with its hyperbolic-paraboloid shells had a dynamic expression. Xenakis continued this research into complex material architectural forms. He also (...) worked on the complex ephemeral architectures of light and sound events. What is specific to Xenakis is the way he used forms in different fields and transferred them from one field to another, from engineering to music, from music to architecture and visual events. This experience of working simultaneously and applying the same (mental) structures in different fields opened him the way (partly supported by the universal computer instrument) for the practice of transferring mathematical-scientific structures into artistic production. In this context Xenakis pleads for the development of a “General Morphology”, a research concerned with the understanding of form and its generation. Xenakis's ‘material’ architectural work is to be seen in continuity with his architectures of music and light. (shrink)

In Résolution des quatre principaux problèmes d’architecture (1673) then in Cours d’architecture (1683), the architect–mathematician Nicolas-François Blondel addresses one of the most famous architectural problems of all times, that of the reduction in columns (entasis). The interest of the text lies in the variety of subjects that are linked to this issue. (1) The text is a response to the challenge launched by Curabelle in 1664 under the name Étrenne à tous les architectes; (2) Blondel mathematicizes the problem (...) in the “style of the Ancients”; (3) The problem is reformulated and solved through the continuous drawing of the curve; (4) Blondel refutes the uniqueness of the curve by enumerating a variety of solutions (conchoid, spiral, parabola, ellipse, circle, hyperbola). This exuberance responds to an intention that does not coincide with the state of the art of mathematics at the end of the seventeenth century, nor with the taste for geometry of the Ancients, nor with any pedagogical project. This feature is explained by Blondel’s plan to found architecture on scientific bases. The reasons for his failure are analysed. (shrink)

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

This work, in continuity with the article published by Ferdinando Casolaro and Giovanna Della Vecchia in Vol 5, 2017 of this series, in which we noted that in the centuries since the eight century B.C. at the 13th century A.D. the evolution of Astronomy and historical events have influenced the development of Mathematics, intends to demonstrate how the Architecture and Literature of the following centuries have further conditioned the development of the sciences in Italy and, in particular, of Mathematics, (...) identifying the affinities and, in some cases, the coincidences between the different forms of thought that invite us to make a serious reflection on the uniqueness of culture. (shrink)

In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its “architecture” or “unity”. This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the (...) agents who produced them. The starting point is to recognize that to each mathematical proof corresponds a proof activity which consists of a sequence of deductive inferences—i.e., a sequence of epistemic actions—and that any written mathematical proof is only a report of its corresponding proof activity. The main idea to be developed is that the plan of a mathematical proof is to be conceived and analyzed as the plan of the agent(s) who carried out the corresponding proof activity. The core of the paper is thus devoted to the development of an account of plans and planning in the context of proof activities. The account is based on the theory of planning agency developed by Michael Bratman in the philosophy of action. It is fleshed out by providing an analysis of the notions of intention—the elementary components of plans—and practical reasoning—the process by which plans are constructed—in the context of proof activities. These two notions are then used to offer a precise characterization of the desired notion of plan for proof activities. A fruitful connection can then be established between the resulting framework and the recent theme of modularity in mathematics introduced by Jeremy Avigad. This connection is exploited to yield the concept of modular presentations of mathematical proofs which has direct implications for how to write and present mathematical proofs so as to deliver various epistemic benefits. The account is finally compared to the technique of proof planning developed by Alan Bundy and colleagues in the field of automated theorem proving. The paper concludes with some remarks on how the framework can be used to provide an analysis of understanding and explanation in the context of mathematical proofs. (shrink)

There are many modal epistemologies available to us. Which should we endorse? According to Bob Hale, we can start to answer this question by examining the architecture of modal knowledge. That is, we can try to decide between the following claims: knowing that p is possible is essentially a matter of having a well-founded belief that there are no conflicting necessities—a necessity-based approach—and knowing that p is necessary is essentially a matter of having a well-founded belief that there are (...) no conflicting possibilities—a possibility-based approach. Hale argues for the first of these claims, but I contend that his arguments fail. However, it doesn't follow that we should endorse a possibility-based approach. I repurpose Hale's arguments to show those who would endorse possibility-based approaches need to say more about our modal knowledge concerning logic and mathematics; if they can't, then they ought to endorse a hybrid modal epistemology that doesn't give priority to one modality across the board. Additionally, those who endorse possibility-based views may be committed to Peter van Inwagen's modal skepticism. (shrink)

In the 1990s, Building Information Modeling (BIM) software significantly altered architectural approaches to planning and building. Based on parametric methods, BIM technologies sought to simulate the construction process prior to a building’s realisation. These computer simulations challenged the existing practice of representing a building through plan, section and elevation, proposing that one computational model could create a more efficient way of building. The history of BIM explorations and applications, while hardly linear, can be traced back to developments in computing since (...) the post-war period. This article maps some of these histories by examining how the computational model became an organisational infrastructure, collecting data about design and building parameters, and facilitating knowledge transfer across industries. Special attention will be given to the foundational role of Charles Eastman’s work on a Building Description System (BDS) in the 1970s, as well as Robert Aish’s contribution to RUCAPS, one of the earliest applications of Building Modeling for the design of parametric structures. I will further address research on interface technologies and computational curve modelling as well as the popularisation of Building Information systems through the office of Gehry Partners. By highlighting the interrelated nature of technology and cultural shifts in the making of BIM, this contribution sheds light on the epistemic status of computer simulations in architecture, and the dynamics of the design and building processes in which they are used. (shrink)

Mathematical explanations are poorly understood. Although mathematicians seem to regularly suggest that some proofs are explanatory whereas others are not, none of the philosophical accounts of what such claims mean has become widely accepted. In this paper we explore Wilkenfeld’s suggestion that explanations are those sorts of things that generate understanding. By considering a basic model of human cognitive architecture, we suggest that existing accounts of mathematical explanation are all derivable consequences of Wilkenfeld’s ‘functional explanation’ proposal. We (...) therefore argue that the explanatory criteria offered by earlier accounts can all be thought of as features that make it more likely that a mathematical proof will generate understanding. On the functional account, features such as characterising properties, unification, and salience correlate with explanatoriness, but they do not define explanatoriness. (shrink)

In this paper, I explore general features of the “architecture” (relations of white space, diagram, and text on the page) of medieval manuscripts and early printed editions of Euclidean geometry. My focus is primarily on diagrams in the Arabic transmission, although I use some examples from both Byzantine Greek and medieval Latin manuscripts as a foil to throw light on distinctive features of the Arabic transmission. My investigations suggest that the “architecture” often takes shape against the backdrop of (...) an educational landscape. The constraints of the economic marketplace and cultural aesthetic ideals also appear to play a role in determining the “architecture” of both manuscripts and early printed editions. (shrink)

The object lesson concerns the passage from the foundational aims for which various branches of modern logic were originally developed to the discovery of areas and problems for which logical methods are effective tools. The main point stressed here is that this passage did not consist of successive refinements, a gradual evolution by adaptation as it were, but required radical changes of direction, to be compared to evolution by migration. These conflicts are illustrated by reference to set theory, model theory, (...) recursion theory, and proof theory. At the end there is a brief autobiographical note, including the touchy point to what extent the original aims of logical foundations are adequate for the broad question of the heroic tradition in the philosophy of mathematics concerned with the nature of the latter or, in modern jargon, with the architecture of mathematics and our intuitive resonances to it. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 44.2 (2006) 321-322 [Access article in PDF] Thomas Holden, The Architecture of Matter: Galileo to Kant. Oxford: Clarendon Press, 2004. Pp. x + 305. Cloth $74.00. Most scholars believe that the problem of infinite divisibility that plagued early modern natural philosophy was an entirely mathematical issue and, therefore, resulted from the short-comings of early modern mathematics. Accordingly, advances in geometry, topology (...) and the logic of infinity are often thought to lay this debate to rest, thereby reducing it to some quaint footnote in the history of natural philosophy. Thomas Holden, however, through thorough textual and philosophical analysis, convincingly argues that this debate was more metaphysical in nature than previously supposed. He shows how the various paradoxes concerning the structure of matter stem from two widely-accepted theses central to the new science. The first resulted from the early modern method of applying the principles of geometry to physics. This geometrical method dictated that bodies are metaphysically divisible (m-divisible) in infinitum, because they occupy regions of space that can, at least by the power of God, be subdivided into those parts occupying corresponding sub-regions of space, and those sub-parts into those sub-sub-parts occupying their respective corresponding sub-sub-regions of space in infinitum. The other thesis is that "the parts of material bodies are each concrete existents in their own right" (3). Such actual parts exist prior to any act of division, whereas parts that are only potential would not exist prior to division but only after a divisive act has occurred. Holden argues that the actual parts doctrine was a metaphysical thesis just as fundamental to Enlightenment natural philosophers as the geometrization of matter. Indeed, it is these two theses that came into conflict and gave rise to a plethora of antimonies at the heart of the new science.Holden should be applauded for the skill he employs in challenging the received scholarship. In chapter 1, after some historical and terminological preliminaries, Holden reconstructs several of the more popular paradoxes of material structure found throughout the period's vast literature and successfully shows how these paradoxes would not arise if the problems of material structure were merely mathematical. He shows how the metaphysical doctrine of actual parts gave rise to these difficulties because of its conflict with the assumed geometrical doctrine that extension is infinitely m-divisible. For example, some thought it followed from the actual parts doctrine that there must be an actual infinity of parts, because actual parts exist prior to any process of division and, therefore, in order for both actual parts and infinite m-divisibility to be true, there must be an actual infinity of parts. This raised issues about the intelligibility and piety of an actual infinite as well as issues about equal and unequal infinities, e.g., half an infinite number of parts would still be infinite but half as much as the whole collection of parts. Hence, this and other paradoxes would not arise in the absence of the actual parts doctrine, and so the debate is more metaphysical in nature than previously supposed. [End Page 321]Chapter 2 examines in more detail the actual parts metaphysic and how it leads to tension with infinite m-divisibility. Chapter 3 examines various "short-circuit" arguments designed to undermine the actual parts doctrine. And chapter 4 examines the actual parts doctrine via the argument from composition found in Kant's pre-critical work, Physical Monadology. In chapter 5, Holden briefly examines the potential parts doctrine and how some actual-parts partisans tried to undermine it. It is eventually concluded that actual-parts and potential-parts advocates are at an impasse due to the question-begging nature of their respective arguments. Finally, chapter 6 is "something of an addendum" (74), since it examines a fascinating synthesis of the two conflicting positions, viz. the Kant-Boscovich force-shell atom theory.Given that the book is about material structure, it is a propos that my critical comment is about the structure of the book. It seems... (shrink)

In my dissertation I analyze the structure of fundamental physical theories. I start with an analysis of what an adequate primitive ontology is, discussing the measurement problem in quantum mechanics and theirs solutions. It is commonly said that these theories have little in common. I argue instead that the moral of the measurement problem is that the wave function cannot represent physical objects and a common structure between these solutions can be recognized: each of them is about a clear three-dimensional (...) primitive ontology that evolves according to a law determined by the wave function. The primitive ontology is what matter is made of while the wave function tells the matter how to move. One might think that what is important in the notion of primitive ontology is their three-dimensionality. If so, in a theory like classical electrodynamics electromagnetic fields would be part of the primitive ontology. I argue that, reflecting on what the purpose of a fundamental physical theory is, namely to explain the behavior of objects in three--dimensional space, one can recognize that a fundamental physical theory has a particular architecture. If so, electromagnetic fields play a different role in the theory than the particles and therefore should be considered, like the wave function, as part of the law. Therefore, we can characterize the general structure of a fundamental physical theory as a mathematical structure grounded on a primitive ontology. I explore this idea to better understand theories like classical mechanics and relativity, emphasizing that primitive ontology is crucial in the process of building new theories, being fundamental in identifying the symmetries. Finally, I analyze what it means to explain the word around us in terms of the notion of primitive ontology in the case of regularities of statistical character. Here is where the notion of typicality comes into play: we have explained a phenomenon if the typical histories of the primitive ontology give rise to the statistical regularities we observe. (shrink)

Current understanding of the development of quantity representations is based primarily on performance in the number‐line task. We posit that the data from number‐line tasks reflect the observer's underlying representation of quantity, together with the cognitive strategies and skills required to equate line length and quantity. Here, we specify a unified theory linking the underlying psychological representation of quantity and the associated strategies in four variations of the number‐line task: the production and estimation variations of the bounded and unbounded number‐line (...) tasks. Comparison of performance in the bounded and unbounded number‐line tasks provides a unique and direct way to assess the role of strategy in number‐line completion. Each task produces a distinct pattern of data, yet each pattern is hypothesized to arise, at least in part, from the same underlying psychological representation of quantity. Our model predicts that the estimated biases from each task should be equivalent if the different completion strategies are modeled appropriately and no other influences are at play. We test this equivalence hypothesis in two experiments. The data reveal all variations of the number‐line task produce equivalent biases except for one: the estimation variation of the bounded number‐line task. We discuss the important implications of these findings. (shrink)

Mathematics stand in a privileged relationship with aesthetics: a relationship that follows two main directions. The first concerns the introduction of mathematical considerations into aesthetic discourse. For instance, it is common to mention the mathematical architecture of certain artistic productions. The second leads from aesthetics to mathematics. In this case, the question is that of the role and meaning that aesthetic considerations may assume in mathematics. It is indeed a widely held view among mathematicians, of whatever socio-historical (...) context, not only to see their discipline as presenting a strong aesthetic dimension, but also to consider that this dimension plays a fundamental role in the process of developing and understanding mathematics. The main ambition of this paper is to show how Nelson Goodman’s aesthetics can be used to justify this point of view and to propose a thesis concerning the aesthetic functioning of mathematics. This first result allows to resituate Goodman’s aesthetics within a very classical tradition that will be described. Finally, the underlying ambition is to show the keys provided by Goodman’s theory for the philosophy of mathematics. (shrink)

This article develops a framework for analysing how digital software and models become mediums for creative imagination in architectural design. To understand the hermeneutics of these relationships, we develop key concepts from Material Engagement Theory (MET) and Postphenomenology (PP). To push these frameworks into the realm of digital design, we develop the concept of Digital Materiality. Digital Materiality describes the way successive layers of mathematics, code, and software come to mediate enactive perception, and the possibilities of creative material engagement actualised (...) in work with software. Just as molecular materials come to transform action with material objects, so digital materiality comes to enable and transform creative practices with computers. Digital architectural design form a new space for ongoing enactive discovery and creativity through manipulation of digital models and their underlying software environments. By shifting relationships within their digital models, architects can direct their attention, intention, and imagination towards widely different aspects of the model. Here, creative imagination becomes a fundamentally situated activity where mind emerges through dynamic interaction between a variety of embodied, material, and cultural domains. (shrink)

In The Aesthetics of Architecture, Roger Scruton makes at least four claims about rightness of architectural proportion. The present paper lists those claims, briefly discusses the way they are related, and, finally, selects one as the topic of discussion: the claim that there cannot be an exact, mathematical definition of rightness of proportion. Scruton’s arguments for this claim are reviewed. The first is found to be substantially correct, whereas the second is found to rely on a mistaken assumption, (...) namely the assumption that rightness of proportion is relative to a point of view. The paper ends by arguing that either the real or the apparent proportions of a building have to be definitely right, and that neither can be allowed to be definitely wrong. (shrink)

Previous research has suggested that in cultural production fields the concatenation of eminence explains success, defined as influence and innovation. We propose that individuals in fields as diverse as philosophy, literature, mathematics, painting, or architecture gain visibility by cumulating the eminence of others connected to them across and within generations. We draw on interaction ritual chain and social movement theories, and use evidence from the field of modernist architecture, to formulate a model of how networks of very strong (...) ties generate motivations and emotional enthusiasm, change reputations, and form collective movements that over time transform the structure of cultural fields. Because major aesthetic innovations break sharply with older styles, they need very strong group solidarity over a long period of time to propagate a new standard of practice. We propose mutual halo effects, i.e., the reciprocal reinforcement of upstream and downstream prestige on a given individual node, as the key factor accounting for success in a cultural production field. We discuss the relevance of these results for building a model of influence and innovation in cultural production fields in which networks—reshaped by shifting technological, political, and economic conditions—trigger new styles. (shrink)

In this contribution we propose an agent architecture for theorem proving which we intend to investigate in depth in the future. The work reported in this paper is in an early state, and by no means finished. We present and discuss our proposal in order to get feedback from the Calculemus community.

Evolution produced many species whose members are pre-programmed with almost all the competences and knowledge they will ever need. Others appear to start with very little and learn what they need, but appearances can deceive. I conjecture that evolution produced powerful innate meta-knowledge about a class of environments containing 3- D structures and processes involving materials of many kinds. In humans and several other species these innate learning mechanisms seem initially to use exploration techniques to capture a variety of useful (...) generalisations after which there is a "phase transition" in which learnt generalisations are displaced by a new generative architecture that allows novel situations and problems to be dealt with by reasoning -- a pre-cursor to explicit mathematical theorem proving in topology, geometry, arithmetic, and kinematics. This process seems to occur in some non-human animals and in preverbal human toddlers, but is clearest in the switch from pattern-based to syntax-based language use. The discovery of non-linguistic toddler theorems has largely gone unnoticed, though Piaget investigated some of the phenomena, and creative problem solving in some other animals also provides clues. A later evolutionary development seems to have enabled humans to cope with domains that involve both regularities and exceptions, explaining "U-shaped" language learning. Only humans appear to be able to develop meta-meta-competences needed for teaching learnt "theorems" and their proofs. I'll sketch a speculative theory, present examples, and propose a research programme, reducing the 'G' in AGI, while promising increased power in return. (shrink)

ArgumentTextual and material evidence suggests that early Byzantine architects, known asmechanikoi, were comprehensively educated in the mathematical sciences according to contemporary standards. This paper explores the significance of the astronomical and optical sciences for the working methods of the twomechanikoiof Hagia Sophia in Constantinople, Anthemios of Tralles and Isidoros of Miletus. It argues that one major concern in the sixth-century architectural design of the Great Church was the visual effect of its sacred interior, particularly the luminosity within. Anthemios and (...) Isidoros seem to have been thoroughly conversant with the ancient corpus of astronomical and optical writings and, as will be shown, implemented their theoretical knowledge in the design of Hagia Sophia. Specifically, the paper demonstrates that the orientation of the building's longitudinal axis coincides with the sunrise on the winter solstice according to ancient computations, implying that the orientation was intentionally calculated in order to secure an advantageous natural illumination of the interior. Light and visual effects served to reinforce the symbolic significance of the sacred space that furthermore provides evidence for optical considerations with respect to late antique concepts of light and vision. (shrink)

The paper considers the intellectual computer mathematics system InparSolver, which is designed to automatically explore and solve basic classes of computational mathematics problems on multi-core computers with graphics accelerators. The problems of results reliability of solving problems with approximate input data are outlined. The features of the use of existing computer mathematics systems are analyzed, their weaknesses are found. The functionality of InparSolver, some innovative approaches to the implementation of effective solutions to problems in a hybrid architecture are described. (...) Examples of applied usage of InparSolver for processes mathematical modeling in various subject areas are given. Nowadays, new more complex objects and phenomena in many subject areas are constantly emerging, which are subject to mathematical research on a computer. This encourages the development of new numerical methods and technologies of mathematical modeling, as well as the creation of more powerful computers for their implementation. With the advent and constant development of supercomputers of various architectures, the problems of their effective use, expansion of tasks range should be solved, ensuring the reliability of computer results and increasing the level of intellectual information support for users ‒ specialists in various fields. Today, the issues of solving these problems are given special attention by many specialists in the fields of information technology and parallel programming. The world's leadingscientists in the field of computer technology see the solution to the problems of efficient usage of modern supercomputers in algorithmic software creation that easily adapts to different computer architectures with different types of memory and coprocessors, supports efficient parallelism on millions of cores etc. In addition, improving the efficiency of high-performance computing on modern supercomputers is provided by their intellectualization, transferring to the computer to perform a significant part of the functions. The industry of development and usage of intelligent computer technologies is one of the main directions of science and technology development in modern society. (shrink)

We present a search engine for mathematical formulae. The MathWebSearch system harvests the web for content representations (currently MathML and OpenMath) of formulae and indexes them with substitution tree indexing, a technique originally developed for accessing intermediate results in automated theorem provers. For querying, we present a generic language extension approach that allows constructing queries by minimally annotating existing representations. First experiments show that this architecture results in a scalable application.

MKM has been defined as the quest for technologies to manage mathematical knowledge. MKM “in the small” is well-studied, so the real problem is to scale up to large, highly interconnected corpora: “MKM in the large”. We contend that advances in two areas are needed to reach this goal. We need representation languages that support incremental processing of all primitive MKM operations, and we need software architectures and implementations that implement these operations scalably on large knowledge bases. We present (...) instances of both in this paper: the MMT framework for modular theory-graphs that integrates meta-logical foundations, which forms the base of the next OMDOC version; and TNTBase, a versioned storage system for XML- based document formats. TNTBase becomes an MMT database by instantiating it with special MKM operations for MMT. (shrink)

Mark Wilson argues that the standard categorizations of "Theory T thinking"— logic-centered conceptions of scientific organization (canonized via logical empiricists in the mid-twentieth century)—dampens the understanding and appreciation of those strategic subtleties working within science. By "Theory T thinking," we mean to describe the simplistic methodology in which mathematical science allegedly supplies ‘processes’ that parallel nature's own in a tidily isomorphic fashion, wherein "Theory T’s" feigned rigor and methodological dogmas advance inadequate discrimination that fails to distinguish between explanatory structures (...) that are architecturally distinct. One of Wilson's main goals is to reverse such premature exclusions and, thus, early on Wilson returns to John Locke's original physical concerns regarding material science and the congeries of descriptive concern insofar as capturing varied phenomena (i.e., cohesion, elasticity, fracture, and the transmission of coherent work) encountered amongst ordinary solids like wood and steel are concerned. Of course, Wilson methodologically updates such a purview by appealing to multiscalar techniques of modern computing, drawing from Robert Batterman's work on the greediness of scales and Jim Woodward's insights on causation. (shrink)

This paper is concerned with the status of mathematical fictions in Leibniz’s work and especially with infinitary quantities as fictions. Thus, it is maintained that mathematical fictions constitute a kind of symbolic notion that implies various degrees of impossibility. With this framework, different kinds of notions of possibility and impossibility are proposed, reviewing the usual interpretation of both modal concepts, which appeals to the consistency property. Thus, three concepts of the possibility/impossibility pair are distinguished; they give rise, in (...) turn, to three concepts of mathematical fictions. Moreover, such a distinction is the base for the claim that infinitesimal quantities, as mathematical fictions, do not imply an absolute impossibility, resulting from self-contradiction, but a relative impossibility, founded on irrepresentability and on the fact that it does not conform to architectural principles. In conclusion, this “soft” impossibility of infinitesimals yields them, in Leibniz view, a presumptive or “conjectural” status. (shrink)

A long history of research has revealed many neurophysiological changes and concomitant behavioral impacts of sleep deprivation, sleep restriction, and circadian rhythms. Little research, however, has been conducted in the area of computational cognitive modeling to understand the information processing mechanisms through which neurobehavioral factors operate to produce degradations in human performance. Our approach to understanding this relationship is to link predictions of overall cognitive functioning, or alertness, from existing biomathematical models to information processing parameters in a cognitive architecture, (...) leveraging the strengths from each to develop a more comprehensive explanation. The integration of these methodologies is used to account for changes in human performance on a sustained attention task across 88 h of total sleep deprivation. The integrated model captures changes due to time awake and circadian rhythms, and it also provides an account for underlying changes in the cognitive processes that give rise to those effects. The results show the potential for developing mechanistic accounts of how fatigue impacts cognition, and they illustrate the increased explanatory power that is possible by combining theoretical insights from multiple methodologies. (shrink)

The article analyses innovative ethno-geographical systems of the first century BC. During Hellenistic times, the science of geography made use of increasingly advanced mathematical and astronomical skills to ensure a scientific basis for the cartographical project; however, this geographical research apparently disregarded the natural and human environments. There is a paradigm change in the referred century. The Stoic Posidonius focuses on the concept of zones found in the early philosophers and finds a compromise between the ‘scientific’ and the ‘descriptive’ (...) geographies. Likewise, Vitruvius conveys a geographical system which associates climatic, somatic, and psychic features. (shrink)

In 1900 Wallace Sabine, a physicist at Harvard University, published a mathematical formula for calculating the reverberation time in a room, a measure of how quickly or slowly sound energy dies away in an enclosed space. In 1930 Carl Eyring, a physicist working in the Sound Motion Picture Studio at Bell Telephone Laboratories, revised Sabine's equation. This essay examines material changes in the practice of architectural acoustics in order to explain how and why Eyring was motivated to reformulate the (...) Sabine equation. Sabine's equation was the product of experimentation in highly reverberant rooms. Eyring worked in a world increasingly constructed of sound-absorbing building materials--a world of acoustically "dead" rooms in which Sabine's original assumptions were no longer valid. Further, Eyring's world was filled with electroacoustic devices that had not existed in 1900. "Live" wires powered new tools for producing, measuring, and controlling sound, and the new electroacoustic technologies additionally provided a new conceptual framework for thinking about the behavior of sound. Eyring's equation is shown to be a direct product of these new material conditions of the science and practice of architectural acoustics. -/- . (shrink)

L’objet de cet essai est l’accueil des entités mathématiques dans l’architecture des modes d’existence proposée par Bruno Latour dans le cadre de son ontologie pluraliste du monde moderne [Latour 2012]. Les travaux de Reviel Netz sur l’émergence des mathématiques grecques [Netz 1999] et de Charles Sanders Peirce sur la dimension diagrammatique de l’activité mathématique [Peirce 1933-1958], [Peirce 1976] sont employés pour proposer une réponse dans le cadre d’une conception empirique des mathématiques basée sur la notion d’expérience chère à William (...) James [James 2007] et inspirée par certains aspects de la philosophie de Per Martin-Löf [Martin-Löf 1987]. Cette approche permet de penser la solide certitude dont la démonstration dote les résultats mathématiques tout en invalidant son interprétation comme la marque d’un accès direct à une vérité absolue et transcendante. (shrink)

Once the Luwian hieroglyphics for God “” and Gate “” were discovered at Göbekli Tepe, this author was able to directly link the site’s carved pillars and pillar enclosures to the Abrahamic/Mosaic “Word of God”,. Archaeologists and anthropologists have long viewed the Bible as mankind’s best guide to prehistoric religion, however, archaeologist Klaus Schmidt had no reason to believe that the site he spent years excavating at Göbekli Tepe might be the legendary “Pillars of Enoch”, carved by the first Biblical (...) holy man and scribe. To make this connection, the author first had to decipher Judaism’s Razah D’Oraytah (Secret of Knowledge) from the Sefer Yetzirah (Book of Creation). The Book of Creation is the only text to explain the Secret of Knowledge, and the only text attributed to the prophet Abraham within the rabbinical community. The Zohar, Judaism’s most accessible Kabbalistic text, admits that it lacks knowledge of “deep and hidden things which issue from God’s thought and are taken up by the Voice [of God] which are not disclosed till the Word [] reveals them”. Abrahamic/Mosaic “Word of God” encrypts the mathematics of sound that emanates from God’s “Voice” in terms of ancient string theory. Just as modern string theory defines a Theory of Everything (ToE) within theoretical physics, so too does this ancient string theory define the Theory of Everything that Abraham named the “Living God”. The author’s previous paper ties Jewish sources to Göbekli Tepe, while the current paper applies this sonic “God Table” to the textual exegesis and hermeneutics structuring all religious creation allegories, including the writings of Sumer, Assyria, Babylonia, Vedic India, pre-Dynastic Egypt, Ugarit, and Israel. This paper will apply Göbekli Tepe’s structural template to archaeology’s oldest legible cuneiform tablets, including the Kesh Temple Hymn, Sumerian King List, and the Akkadian Ark Tablet. It will also be demonstrated that Göbekli Tepe holds the key to the Rig Veda, Hinduism’s oldest text, and defines the essence of Plato’s metaphysics. (shrink)