Claude P. Bruter (editor), Mathematics in Art: Mathematical Visualization in Art and Education, Springer-Verlag, New York, 2002, pp. X + 337, ISBN 3-540-43422-4.

Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show (...) that diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability. I then consider whether diagrams can be essential to the proofs in which they appear. (shrink)

Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working definition that carves out the phenomena that are of most importance (...) for a taxonomy of diagrams in the context of a practice-based philosophy of mathematics, privileging examples from contemporary mathematics. In doing so, I move away from vague, ordinary notions. I define mathematical diagrams as forming notational systems and as being geometric/topological representations or two-dimensional representations. I also examine the relationship between mathematical diagrams and spatiotemporal intuition. By proposing an explication of diagrams, I explain certain controversies in the existing literature. Moreover, I shed light on why mathematical diagrams are so effective in certain instances, and, at other times, dangerously misleading. (shrink)

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...) are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

The mental risk poses a high threat to the individuals, especially overseas demographic, including expatriates in comparison to the general Arab demographic. Since Arab countries are renowned for their multicultural environment with half of the population of students and faculties being international, this paper focuses on a comprehensive analysis of mental health problems such as depression, stress, anxiety, isolation, and other unfortunate conditions. The dataset is developed from a web-based survey. The detailed exploratory data analysis is conducted on the dataset (...) collected from Arab countries to study an individual’s mental health and indicative help-seeking pointers based on their responses to specific pre-defined questions in a multicultural society. The proposed model validates the claims mathematically and uses different machine learning classifiers to identify individuals who are either currently or previously diagnosed with depression or demonstrate unintentional “save our souls” (SOS) behaviors for an early prediction to prevent risks of danger in life going forward. The accuracy is measured by comparing with the classifiers using several visualization tools. This analysis provides the claims and authentic sources for further research in the multicultural public medical sector and decision-making rules by the government. (shrink)

This commentary is a reflection on a collaboration with the artist Rossella Biscotti and comments on how artistic research and logico-mathematical methods can be used to contribute to the development of critical perspectives on contemporary data practices.

The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, (...) the representations used in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in low-dimensional topology experts exploit a particular type of manipulative imagination which is connected to intuition of two- and three-dimensional space and motor agency. This imagination allows recognizing the transformations which connect different pictures in an argument. Third, the epistemic—and inferential—actions performed are permissible only within a specific practice: this form of reasoning is subject-matter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in low-dimensional topology. (shrink)

Just before the Scientific Revolution, there was a "Mathematical Revolution", heavily based on geometrical and machine diagrams. The "faculty of imagination" (now called scientific visualization) was developed to allow 3D understanding of planetary motion, human anatomy and the workings of machines. 1543 saw the publication of the heavily geometrical work of Copernicus and Vesalius, as well as the first Italian translation of Euclid.

The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will (...) be argued that one of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation. (shrink)

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

This paper explores the mathematical details behind the geometric transformation of folding a circle into a cone. It includes detailed theo- rems, proofs, and Python implementations for visualization, providing a thorough analysis of the transformation process.

We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which (...) the brain can transform and organize its perceptual intake. It is not necessary for a geometrical form to be perfectly instantiated in order for perception of such a form to be the basis of a geometrical concept. (shrink)

This book analyses the impact computerization has had on contemporary science and explains the origins, technical nature and epistemological consequences of the current decisive interplay between technology and science: an intertwining of formalism, computation, data acquisition, data and visualization and how these factors have led to the spread of simulation models since the 1950s. -/- Using historical, comparative and interpretative case studies from a range of disciplines, with a particular emphasis on the case of plant studies, the author shows (...) how and why computers, data treatment devices and programming languages have occasioned a gradual but irresistible and massive shift from mathematical models to computer simulations. -/- . (shrink)

Lucid dreaming (LD) is a fun and interesting activity, but most participants have difficulties in attaining lucidity, retaining it during the dream, concentrating on the needed task and remembering the results. This motivates to search for a new way to enhance lucid dreaming via different induction techniques, including chemicals and electric brain stimulation. However, results are still unstable. An alternative approach is to reach the lucid dreaming-like states via altered state of consciousness not related to dreaming. Several methods such as (...) guided visualization, internal dialog, creative writing, hypnosis, hypnagogia, daydreaming, DMT trips, voice dialog, shamanic journey, rebirthing, and “forcing” tulpas can help in attaining such states. One of the most promising of them is Jungian “active imagination” (AIM) technique, which allows unconscious content to build up inside some mental frames. This article explores the hypothesis of replacing lucid dreaming research with active imagination, and the conditions and ways to accomplish it. Method: An open label pilot experiment was performed in 2004-2005 in Moscow, Russia with 100 participants. Results: The results show that there are two groups of people: ones with “visual imagination screen” and others have “mental imagination screen”. AIM works perfectly as a replacement for lucid dreams only for the first group of people. For the second group, it created interesting content, but not visual or emotional intensity equal to enter lucid dreaming like state. No known instruments helped to move the person from one group to another. The first group consisted of young females, while the second mostly contained males with rational and mathematical type of personality. Conclusion: AIM partly works as a replacement for LD, as it works great only for half of people, and it requires a sitter. However, AIM outperforms LD in reliability and availability in any circumstance: it could be performed even by text chat or in a crowd. It is also better than LD in retaining concentration on topic and the easiness of memorizing the results (which could be recorded). Self-performed AIM is less effective. AIM can be improved by intelligent chat bots as sitters and weak brain stimulation that can increase the probability of attaining something like hypnogogic state. (shrink)

We present a formal mechanical analysis using sweeping net methods to approximate surfacing singularities of saddle maps. By constructing densified sweeping subnets for individual vertices and integrating them, we create a comprehensive approximation of singularities. This approach utilizes geometric concepts, analytical methods, and theorems that demonstrate the robustness and stability of the nets under perturbations. Through detailed proofs and visualizations, we provide a new perspective on singularities and their approximations in analytic geometry.

The type-level abstraction is a formal way to represent molecular structures in biological practice. Graphical representations of molecular structures of biological objects are also used to identify functional processes of things. This paper will reveal that category theory is a formal mathematical language not only to visualize molecular structures of biological objects as type-level abstraction formally but also to understand how to infer biological functions from the molecular structures of biological objects. Category theory is a toolkit to understand biological (...) knowledge at the type-level formally, not individual token-level, as well as typical heuristic strategies in molecular biology. (shrink)

Philosophers have relied on visual metaphors to analyse ideas and explain their theories at least since Plato. Descartes is famous for his system of axes, and Wittgenstein for his first design of truth table diagrams. Today, visualisation is a form of ‘computer-aided seeing’ information in data. Hence, information is the fundamental ‘currency’ exchanged through a visualisation pipeline. In this article, we examine the types of information that may occur at different stages of a general visualization pipeline. We do so (...) from a quantitative and a qualitative perspective. The quantitative analysis is developed on the basis of Shannon’s information theory. The qualitative analysis is developed on the basis of Floridi’s taxonomy in the philosophy of information. We then discuss in detail how the condition of the ‘data processing inequality’ can be broken in a visualisation pipeline. This theoretic finding underlines the usefulness and importance of visualisation in dealing with the increasing problem of data deluge. We show that the subject of visualisation should be studied using both qualitative and quantitative approaches, preferably in an interdisciplinary synergy between information theory and the philosophy of information. (shrink)

Nền kinh tế nước ta đang chuyển biến mạnh mẽ từ nền kinh tế bao cấp sang kinh tế thị trường, nhất là từ khi nước ta gia nhập WTO. Đảng và chính phủ đã đề ra rất nhiều các chính sách để cải tiến các thể chế quản lý nền kinh tế và tài chính. Thị trường chứng khoán Việt Nam đã ra đời và đang đóng một vai trò quan trọng trong việc huy động vốn phục vụ cho (...) việc phát triển nền kinh tế quốc dân. Việc phân tích tác động của các chính sách đến nền kinh tế vĩ mô và nền tài chính, những đối sách của chính phủ ta và chính phủ các nước đối với cuộc khủng khoảng tài chính toàn cầu là các vấn đề rất thời sự đang thu hút sự chú ý của nhiều nhà kinh tế và toán học. (shrink)

The article proposes systematisation and development of the discourse of the East European methodological traditions regarding application of the systematic approach as a way of subject localisation in psychological research. In particular, the author’s version of systematic localisation of psychological research subjects by means of structural and ontological visualisations has been developed. The procedure proposed for systematic localisation of the researched subject includes four subsequent stages: 1) fixation of the borders and structure of the ontological field which is being studied; (...) 2) segment analysis of the obtained structure; 3) study of specifics and contents of structural and functional ties; 4) visual mapping of genesis of the core processes. The article describes specifics of application of systematic localisation of the subject of research at the example of such researched object as personal socialisation. (shrink)

Traditionally, vision science and information/data visualization have interacted by using knowledge of human vision to help design effective displays. It is argued here, however, that this interaction can also go in the opposite direction: the investigation of successful visualizations can lead to the discovery of interesting new issues and phenomena in visual perception. Various studies are reviewed showing how this has been done for two areas of visualization, namely, graphical representations and interaction, which lend themselves to work on (...) visual processing and the control of visual operations, respectively. The results of these studies have provided new insights into aspects of vision such as grouping, attentional selection and the sequencing of visual operations. More generally yet, such results support the view that the perception of visualizations can be a useful domain for exploring the nature of visual cognition, inspiring new kinds of questions as well as casting new light on the limits to which information can be conveyed visually. (shrink)

Visual imagination (or visualization) is peculiar in being both free, in that what we imagine is up to us, and useful to a wide variety of practical reasoning tasks. How can we rely upon our visualizations in practical reasoning if what we imagine is subject to our whims? The key to answering this puzzle, I argue, is to provide an account of what constrains the sequence in which the representations featured in visualization unfold—an account that is consistent with (...) its freedom. Three different proposals are outlined, building on theories that link visualization to sensorimotor predictive mechanisms (e.g., efference copies, forward models ). Each sees visualization as a kind of reasoning, where its freedom consists in our ability to choose the topic of the reasoning. Of the three options, I argue that the approach many will find most attractive—that visualization is a kind of off-line perception, and is therefore in some sense misrepresentational—should be rejected. The two remaining proposals both conceive of visualization as a form of sensorimotor reasoning that is constitutive of one’s commitments concerning the way certain kinds of visuomotor scenarios unfold. According to the first, these commitments impinge on one’s web of belief from without, in the manner of normal perceptual experience; according to the second, these commitments just are one’s (occurrent) beliefs about such generalizations. I conclude that, despite being initially counterintuitive, the view of visualization as a kind of occurrent belief is the most promising. (shrink)

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of (...) class='Hi'>mathematical theorems can cover at most one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

This article explores how readers recognize their personal identities represented through data visualizations. The recognition is investigated starting from three definitions captured by the philosopher Paul Ricoeur: the identification with the visualization, the recognition of someone in the visualization, and the mutual recognition that happens between readers. Whereas these notions were initially applied to study the role of the book reader, two further concepts complete the shift to data visualization: the digital identity stays for the present-day passport (...) of human actions and the promise is the intimate reflection that projects readers towards their own future. This article reflects on the delicate meaning of digital identity and the way of representing it according to this structure: From Personal Identity to Media is a historical introduction to self-recognition, Data Visualization for Representing Identities moves the focus to visual representation, and The Course of Recognition breaks the self-recognition in through the five concepts above just before the Conclusion. (shrink)

Socialist buildings in the city of Kruja (Albania) date back after the Second World War between the years 1945-1990. These buildings were built during the time of the socialist Albanian dictatorship and the totalitarian communist regime. A questionnaire with 30 questions was conducted and 14 people were interviewed. The interviewed residents belong to a certain area of the city of Kruja. Based on the results obtained, diagrams have been conceived and mathematical regression models have been developed which will serve (...) as a base for drawing conclusions. The purpose of this study is to find a relationship that specifies the role of the inhabitants, the role of the dwellings, the physical-mechanical characteristics, and their energy efficiency, in order to improve the living conditions of the inhabitants. The independent variables interact with each other assuming positive or negative values depending on the probabilistic mathematical regression, giving different results based on the calculated data. The variables that co-exist among them are living space, quality of life, time spent at home, social interaction of residents, dwelling humidity level, dwelling orientation, satisfaction level of the inhabitants, dwelling improvements demand, and monthly electricity bills. (shrink)

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical (...) ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies. (shrink)

Bioinformatics scientists often describe their own scientific activities as the practice of working with large amounts of data using computing devices. An essential part of their self-identification is also the development of ways to visually represent the results of this work. Some of these methods are aimed at building convenient representations of data and demonstrating patterns present in them (graphics, diagrams, graphs). Others are ways of visualizing objects that are not directly accessible to human perception (microphotography, X-ray). Both the construction (...) of visualizations and (especially) the creation of new computer visualization methods are considered in bioinformatics as significant scientific achievements. Representations of the three-dimensional structure of protein molecules play a special role in the inquiries of bioinformatics scientists. 3D-visualization of a macromolecule, on the one hand, is, like a graph, a representation of the results of computer processing of data arrays obtained by material methods – spatiotemporal coordinates of structural elements of the molecule. On the other hand, like microphotography, these 3D structures should serve as accurate representations of specific scientific objects. This leads to the parallel existence of two contradictory epistemic regimes: creative arbitrariness in making convenient, communicatively successful models, is combined with commitment to the object “as it really is”. The paradox is reinforced by the fact that the scientific study of objects in question (determining the properties of the structure, its functions, comparison with other structures) by means of computers does not require visualization at all. Its obviously high value for bioinformatics does not look justified if we take into account the prominent artificiality and artistry of the resulting images. However, the status of these images becomes clearer if we relate them to earlier notions of the role of the visual in scientific discovery. The highest estimation of visualization as the final result of scientific research was characteristic of Renaissance science. The artistic representation of ideal essential properties, instead of a strict correspondence to a particular biological object, is an epistemic virtue typical of the naturalists of the 17th and 18th centuries. Both suggested a close collaboration between the scientist and the artist; and standards for visualizing macromolecules in bioinformatics grow out of a similar collaboration (Geis’ drawings). The desire for maximum accuracy and detail inherits the regulation of “mechanical objectivity” (as Daston and Galison put it into words), for which it is also important to eliminate humans from the image production process (in bioinformatics, to transfer these functions to computer programs). Thus, 3D-visualization of protein structures bears traces of historically different value orientations, but the scientific practice of the 20th and 21st centuries, supplemented by computer technologies, allows them to be intertwined in particular disciplinary units. (shrink)

Randomness, a core concept of gambling, is seen in problem gambling as responsible for the formation of the math-related cognitive distortions, especially the Gambler’s Fallacy. In problem-gambling research, the concept of randomness was traditionally referred to as having a mathematical nature and categorized and approached as such. Randomness is not a mathematical concept, and I argue that its weak mathematical dimension is not decisive at all for the randomness-related issues in gambling and problem gambling, including the correction (...) of the misconceptions and fallacies about probability and statistical concepts applied in gambling. I distinguish between mathematical and nonmathematical dimensions of randomness (the epistemic, the theoretical-methodological, the functional, and the ethical) falling within the general concept, and I argue that both the studies having as object the math-related cognitive distortions among gamblers and the educational programs aiming at correcting them should employ this distinction in their design and content. (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)

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the (...) basic theory faces. The final view appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

Computational systems biologists create and manipulate computational models of biological systems, but they do not always have straightforward epistemic access to the content and behavioural profile of such models because of their length, coding idiosyncrasies, and formal complexity. This creates difficulties both for modellers in their research groups and for their bioscience collaborators who rely on these models. In this paper we introduce a new kind of visualization that was developed to address just this sort of epistemic opacity. The (...) visualization is unusual in that it depicts the dynamics and structure of a computer model instead of that model’s target system, and because it is generated algorithmically. Using considerations from epistemology and aesthetics, we explore how this new kind of visualization increases scientific understanding of the content and function of computer models in systems biology to reduce epistemic opacity. (shrink)

Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.

We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content to key (...) intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as explanatory generality is concerned. (shrink)

I argue that certain species of belief, such as mathematical, logical, and normative beliefs, are insulated from a form of Harman-style debunking argument whereas moral beliefs, the primary target of such arguments, are not. Harman-style arguments have been misunderstood as attempts to directly undermine our moral beliefs. They are rather best given as burden-shifting arguments, concluding that we need additional reasons to maintain our moral beliefs. If we understand them this way, then we can see why moral beliefs are (...) vulnerable to such arguments while mathematical, logical, and normative beliefs are not—the very construction of Harman-style skeptical arguments requires the truth of significant fragments of our mathematical, logical, and normative beliefs, but requires no such thing of our moral beliefs. Given this property, Harman-style skeptical arguments against logical, mathematical, and normative beliefs are self-effacing; doubting these beliefs on the basis of such arguments results in the loss of our reasons for doubt. But we can cleanly doubt the truth of morality. (shrink)

In this position paper, we describe a number of methodological and philosophical challenges that arose within our interdisciplinary Digital Humanities project CatVis, which is a collaboration between applied geometric algorithms and visualization researchers, data scientists working at OCLC, and philosophers who have a strong interest in the methodological foundations of visualization research. The challenges we describe concern aspects of one single epistemic need: that of methodologically securing (an increase in) trust in visualizations. We discuss the lack of ground (...) truths in the (digital) humanities and argue that trust in visualizations requires that we evaluate visualizations on the basis of ground truths that humanities scholars themselves create. We further argue that trust in visualizations requires that a visualization provides provable guarantees on the faithfulness of the visual representation and that we must clearly communicate to the users which part of the visualization can be trusted and how much. Finally, we discuss transparency and accessibility in visualization research and provide measures for securing transparency and accessibility. (shrink)

Vehicle externalism maintains that the vehicles of our mental representations can be located outside of the head, that is, they need not be instantiated by neurons located inside the brain of the cogniser. But some disagree, insisting that ‘non-derived’, or ‘original’, content is the mark of the cognitive and that only biologically instantiated representational vehicles can have non-derived content, while the contents of all extra-neural representational vehicles are derived and thus lie outside the scope of the cognitive. In this paper (...) we develop one aspect of Menary’s vehicle externalist theory of cognitive integration—the process of enculturation—to respond to this longstanding objection. We offer examples of how expert mathematicians introduce new symbols to represent new mathematical possibilities that are not yet understood, and we argue that these new symbols have genuine non-derived content, that is, content that is not dependent on an act of interpretation by a cognitive agent and that does not derive from conventional associations, as many linguistic representations do. (shrink)

With the rapidly growing amounts of information, visualization is becoming increasingly important, as it allows users to easily explore and understand large amounts of information. However the field of information visualiza- tion currently lacks sufficient theoretical foundations. This article addresses foundational questions connecting information visualization with computing and philosophy studies. The idea of multiscale information granula- tion is described based on two fundamental concepts: information (structure) and computation (process). A new information processing paradigm of Granular Computing enables stepwise (...) increase of granulation/aggregation of information on different levels of resolution, which makes possible dynamical viewing of data. Information produced by Google Earth is an illustration of visualization based on clustering (granulation) of information on a succession of layers. Depending on level, specific emergent properties become visible as a result of different ways of aggregation of data/information. As information visualization ultimately aims at amplifying cognition, we discuss the process of simulation and emulation in relation to cognition, and in particular visual cognition. (shrink)

Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols (...) are not only used to express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. (shrink)

The triumph of scientific materialism in the Seventeenth Century not only bifurcated nature into matter and mind and primary and secondary qualities, as Alfred North Whitehead pointed out in Science and the Modern World. It divided science and the humanities. The core of science is the effort to comprehend the cosmos through mathematics. The core of the humanities is the effort to comprehend history and human nature through narratives. The life sciences can be seen as the zone in which the (...) conflict between these two very different ways of comprehending the world collide. Evolutionary theory as defended by Schelling developed out of natural history, but efforts have been made to formulate neo-Darwinism through mathematical models. However, it is impossible to eliminate stories from biology. As Stuart Kauffman argued, mathematical models attempt to pre-state all possibilities, but in evolution there can be adjacent possibles that can be embraced by organisms but cannot be pre-stated. To account for such actions it is necessary to tell stories. Mathematics provides analytic precision allowing long chains of deductions, but tends to deny temporal becoming and cannot do justice to the openness of the future, while narratives focus on processes and events, but lack exactitude that would provide precise deductions and predictions. In advancing mathematics adequate to life, Robert Rosen argued that living beings as anticipatory systems must have models of themselves, and strove to develop a form of mathematics able to model life itself. It has been convincingly argued that narratives are central to human self-creation and they are lived out before being explicitly told. Their models of themselves are first and foremost, stories or narratives. If this is the case, might not living beings as biological entities be characterized by proto-stories or narratives in their models of themselves? Biosemiotics, largely inspired by C.S. Peirce, provides a bridge between mathematical and narrative comprehension, conceiving them as different forms of semiosis. The study of life through biosemiotics could reveal how mathematics and narratives can be understood in relation to each other. This could have implications for how we understand science and the humanities and their relationship to each other. In this paper I will examine work in theoretical biology that might advance these efforts. (shrink)

The evaluation of universities from different perspectives is important for their scientific development. Analyzing the scientific papers of a university under the bibliometric approach is one main evaluative approach. The aim of this study was to conduct a bibliometric analysis and visualization of papers published by Hamadan University of Medical Science (HUMS), Iran, during 1992-2018. This study used bibliometric and visualization techniques. Scopus database was used for data collection. 3753 papers were retrieved by applying Affiliation Search in Scopus (...) advanced search section. Excel and VOSviewer software packages were used for data analysis and bibliometric indicator extraction. An increasing trend was seen in the numbers of HUMS's published papers and received citations. The highest rate of collaboration in national level was with Tehran University of Medical Sciences. Internationally, HUMS's researchers had the highest collaboration with the authors from the United States, the United Kingdom and Switzerland, respectively. All highly-cited papers were published in high level Q1 journals. Term clustering demonstrated four main clusters: epidemiological studies, laboratory studies, pharmacological studies, and microbiological studies. The results of this study can be beneficial to the policy-makers of this university. In addition, researchers and bibliometricians can use this study as a pattern for studying and visualizing the bibliometric indicators of other universities and research institutions. (shrink)

Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of (...) class='Hi'>mathematical instrumentalism are defeated by Gödel’s theorem, not all are. By considering inductive reasons in mathematics, we show that some mathematical instrumentalisms survive the theorem. (shrink)

Progress in philosophy is difficult to achieve because our methods are evidentially and rhetorically weak. In the last two decades, experimental philosophers have begun to employ the methods of the social sciences to address philosophical questions. However, the adequacy of these methods has been called into question by repeated failures of replication. Experimental philosophers need to incorporate more robust methods to achieve a multi-modal perspective. In this chapter, we describe and showcase cutting-edge methods for data-mining and visualization. Big data (...) is a useful investigatory tool for moral psychology, and it fits well with the Ramsification method the first author advances in a series of recent papers. The guiding insight of these papers is that we can infer the meaning and structure of concepts from patterns of assertions and inferential associations in natural language. (shrink)

Andr ́e Weil viewed mathematics as deeply intertwined with metaphysics. In his essay ”From Metaphysics to Mathematics,” he illustrates how mathematical ideas often arise from vague, metaphysical analogies and reflections that guide researchers toward new theories. For instance, Weil discusses how analogies between different areas, such as number theory and algebraic functions, have led to significant breakthroughs. These metaphysical underpinnings provide a fertile ground for mathematical creativity, eventually transforming into rigorous mathematical structures. -/- Alexander Grothendieck’s work, particularly (...) in ”R ́ecoltes et Semailles,” resonates with Weil’s ideas by emphasizing the organic and generative aspects of mathematical creation. Grothendieck sees mathematical work as akin to cultivating a garden, where ideas grow and develop in a nurturing environment. He describes the process of mathematical discovery as a deeply personal and creative journey, involving both solitary reflection and collaborative effort. -/- Grothendieck’s concept of creating mathematical ”houses” aligns with his broader philosophical view that mathematics provides structures within which new ideas can flourish. His work in category theory and algebraic geometry exemplifies this approach, where he developed vast, interconnected frameworks that have become foundational in modern mathematics. Grothendieck’s analogy of constructing houses for others to live in highlights the mathematician’s role in creating abstract frameworks that others can inhabit and explore. This metaphor emphasizes the constructive nature of mathematics, where researchers build theoretical structures that form the basis for further exploration and application by the mathematical community. (shrink)

Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) (...) class='Hi'>mathematical truths are not truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. (shrink)

The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This article analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humor is associated with risky forms of inference, which (...) are essential to creative mathematics. The components of the joke are explicated by argumentation schemes devised for application to topic-neutral reasoning. These in turn are classified under seven headings: retroduction, citation, intuition, meta-argument, closure, generalization, and definition. Finally, the wider significance of this account for the cognitive science of mathematics is discussed. (shrink)

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, (...) I argue that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

We give a mathematical definition of the present or 'what is real' and its duration on McTaggart's A-series future/present/past. This is applicable to at least one conception of the block-world, the growing-block, and presentism.

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these (...) options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

This contribution aims to explore the historical predecessors of the Five Percenter model of self-realization, as popularized by Hip Hop artists such as Supreme Team, Rakim Allah, Brand Nubian, Wu-Tang Clan, or Sunz of Man. As compared to frequent considerations of the phenomenon as a creative mythological background for a socio-political struggle, Five Percenter teachings shall be discussed as contemporary interpretations of historical models of self-realization in various philosophical, religious, and esoteric systems. By putting the coded system of the tenfold (...) Supreme Mathematics as one of its core teachings in context with the Pythagorean Tetractys, an arrangement of ten points in four lines, the commonalities between the sequence and concepts attributed to the respective numbers will be demonstrated. (shrink)

There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. We focus on Marc Lange’s account of distinctively mathematical explanations to argue that purported mathematical explanations are essentially causal explanations in disguise and are no different from ordinary applications of mathematics. This is because these explanations work not by (...) appealing to what the world must be like as a matter of mathematical necessity but by appealing to various contingent causal facts. (shrink)

Callard (2007) argues that it is metaphysically possible that a mathematical object, although abstract, causally affects the brain. I raise the following objections. First, a successful defence of mathematical realism requires not merely the metaphysical possibility but rather the actuality that a mathematical object affects the brain. Second, mathematical realists need to confront a set of three pertinent issues: why a mathematical object does not affect other concrete objects and other mathematical objects, what counts (...) as a mathematical object, and how we can have knowledge about an unchanging object. (shrink)