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)

There are two important ways in which, when dealing with documents, we go beyond the boundaries of linear text. First, by incorporating diagrams into documents, and second, by creating complexes of intermeshed documents which may be extended in space and evolve and grow through time. The thesis of this paper is that such aggregations of documents are today indispensable to practically all complex human achievements from law and finance to orchestral performance and organized warfare. Documents provide for what (...) we can think of as a division of intellectual, instructional, and deontic labour, allowing plans, orders, and obligations to be enmeshed together in a way that often involves the use of diagrammatic elements, as for example in a musical score. (shrink)

My first aim in this paper is to use time diagrams in the style of Brentano to analyze constructions in Brouwer's separable mathematics more precisely. I argue that constructions must involve not only pairing and projecting as basic operations guaranteed by the intuition of twoity, as sometimes assumed in the literature, but also a recalling operation. My second aim is to argue that Brouwer's views on the intuition of twoity and arithmetic lead to an ontological explosion. Redeveloping the (...) constructions of natural numbers and systems sketched in an appendix to Brouwer's Cambridge lectures, I observe that the only plausible way he can make some elementary arithmetic in his separable mathematics is by allowing for the same canonical number to be determined by multiple separable entities, resulting in an overabundant mathematical ontology. (shrink)

In a time which it is not amiss to term “the Dark Ages of logic”, Karl Christian Friedrich Krause stayed not only true to logic but actually did something for its advancement. Besides making systematic use of Venn-diagrams long before Venn, Krause — once more taking his inspiration from Leibniz — propounded what appears to be the first completely symbolic systematic representation of logical forms, strongly suggestive of the powerful symbolic languages that have become the mainstay of logic (...) since the beginning of the 20th century. However, Krause’s limits in logic are also clearly visible: Krause’s method in logic is, in the main, not axiomatic; it is combinatorial. More importantly, Krause remained entirely within the confines of traditional syllogistics, neglecting propositional logic and, of course, first-order relational terms. (shrink)

Whereas decades of research have cataloged striking errors in physical reasoning, a resurgence of interest in intuitive physics has revealed humans’ remarkable ability to successfully predict the unfolding of physical scenes. A leading interpretation intended to resolve these opposing results is that physical reasoning recruits a general-purpose mechanism that reliably models physical scenarios (explaining recent successes), but overly contrived tasks or impoverished and ecologically invalid stimuli can produce poor performance (accounting for earlier failures). But might there be tasks that persistently (...) strain physical understanding, even in naturalistic contexts? Here, we explore this question by introducing a new intuitive physics task: evaluating the strength of knots and tangles. Knots are ubiquitous across cultures and time-periods, and evaluating them correctly often spells the difference between safety and peril. Despite this, 5 experiments show that observers fail to discern even very large differences in strength between knots. In a series of two-alternative forced-choice tasks, observers viewed a variety of simple “bends” (knots joining two pieces of thread) and decided which would require more force to undo. Though the strength of these knots is well-documented, observers’ judgments completely failed to reflect these distinctions, across naturalistic photographs (E1), idealized renderings (E2), dynamic videos (E3), and even when accompanied by schematic diagrams of the knots’ structures (E4). Moreover, these failures persisted despite accurate identification of the topological differences between the knots (E5); in other words, even when observers correctly perceived the underlying structure of the knot, they failed to correctly judge its strength. These results expose a blindspot in physical reasoning, placing new constraints on general-purpose theories of scene understanding. (shrink)

Among the possible options for the origin of the universe the most sensible one is nothingness, because it is without a need for any other beginning. It must be possible for nothingness to have a structure so that we can speak about it. The structure of nothingness can be constructed by using inward-outward vanishing points, with a guiding principle of conservation of nothingness. When taken all at once, the inward-outward vanishing points remain as they are—nothing; but when they are taken (...) step-by-step, they become something. The idea of a step-by-step move introduces the idea of time. So time is the first one to emerge as a real-worldly concept from the reading of the structure of nothingness. The Emergence of time gives rise to other real-worldly concepts. What we consider as inward-outward vanishing in the realm of nothingness can now be taken as a turn-by-turn state of expansion and contraction in terms of real world perspectives. And what makes such dynamics possible can be considered as energy. And what has been labelled as conservation of nothingness can now be taken as conservation of energy. And the span of events that emerge due to the introduction of time gives us space. And in space matter is produced. The emergence of matter has only been alluded in the paper, but the possible ingredients and their possible combinations are partially manifested in the field diagrams that are integral parts of the paper, and are also thought to be suitable to the task of mathematization of the metaphysical ideas of the work. (shrink)

Lays out the problem of sentience in a physical world and the solution based on the event ontology of Russell and Whitehead. The second edition includes the construction of physics from arrow diagrams, stemming from the discovery that the arrows of time form frequency ratios, which serve to define energy ratios and the quantum.

Karl Pfeifer attempts to present a coherent view of panentheism that eschews Pickwickian senses of “in” and aligns itself with, and builds upon, familiar diagrammed portrayals of panentheism. The account is accordingly spatial-locative and moreover accepts the proposal of R.T. Mullins that absolute space and time be regarded as attributes of God. In addition, however, it argues that a substantive parthood relation between the world and God is required. Pfeifer’s preferred version of panpsychism, viz. panintentionalism, is thrown into the (...) mix as an optional add-on. On this account, God is conceived of as a “spiritual field” whose nature can be made more intelligible by regarding “God” as having a mass-noun sense in some contexts. Pfeifer closes with the suggestion that we look to topology and mereology for further development of the position outlined in his paper. (shrink)

Human beings have the ability to ‘augment’ reality by superimposing mental imagery on the visually perceived scene. For example, when deciding how to arrange furniture in a new home, one might project the image of an armchair into an empty corner or the image of a painting onto a wall. The experience of noticing a constellation in the sky at night is also perceptual-imaginative amalgam: it involves both seeing the stars in the constellation and imagining the lines that connect them (...) at the same time. I here refer to such hybrid experiences – involving both a bottom-up, externally generated component and a top-down, internally generated component – as make-perceive (Briscoe 2008, 2011). My discussion in this paper has two parts. In the first part, I show that make-perceive enables human beings to solve certain problems and pursue certain projects more effectively than bottom-up perceiving or top-down visualization alone. To this end, the skillful use of projected mental imagery is surveyed in a variety of contexts, including action planning, the interpretation of static mechanical diagrams, and non-instrumental navigation. In the second part, I address the question of whether make-perceive may help to account for the “phenomenal presence” of occluded or otherwise hidden features of perceived objects. I argue that phenomenal presence is not well explained by the hypothesis that hidden features are represented using projected mental images. In defending this position, I point to important phenomenological and functional differences between the way hidden object features are represented respectively in mental imagery and amodal completion. (shrink)

Some have suggested that images can be arguments. Images can certainly bolster the acceptability of individual premises. We worry, though, that the static nature of images prevents them from ever playing a genuinely argumentative role. To show this, we call attention to a dilemma. The conclusion of a visual argument will either be explicit or implicit. If a visual argument includes its conclusion, then that conclusion must be demarcated from the premise or otherwise the argument will beg the question. If (...) a visual argument does not include its conclusion, then the premises on display must license that specific conclusion and not its opposite, in accordance with some demonstrable rationale. We show how major examples from the literature fail to escape this dilemma. Drawing inspiration from the graphical logic of C. S. Peirce, we suggest instead that images can be manipulated in a way that overcomes the dilemma. Diagrammatic reasoning can take one stepwise from an initial visual layout to a conclusion—thereby providing a principled rationale that bars opposite conclusions—and the visual inscription of this correct conclusion can come afterward in time—thereby distinguishing the conclusion from the premises. Even though this practical application of Peirce’s logical ideas to informal contexts requires that one make adjustments, we believe it points to a dynamic conception of visual argumentation that will prove more fertile in the long run. (shrink)

When Gilles Deleuze, in his book on Michel Foucault, asks, ‘who would think of looking for life among the archives?’, he uncovers something particular to Foucault's philosophy, but also to his own: a commitment to the question of what it means to think, and think politically. Although Foucault and Deleuze, who first met in 1952, immediately felt fondness for each other, a growing animosity had settled into the friendship by the end of the 1970s – a rift deepened by theoretical (...) differences. Notwithstanding these difficulties, Foucault and Deleuze shared a love for Nietzsche, as well as a curious fascination with Kant. Kant's influence, which was met with more opposition, is precisely the tension I tease out in this article, showing how Foucault critiques Kant's a priori through his own concept of the historical a priori, along with regularity and resistance, while Deleuze and Guattari, if more obliquely, critique Kant's a priori by further developing Foucault's notion of the historical a priori through their own method of pragmatics, particularly in three chapters of A Thousand Plateaus, namely ‘The Geology of Morals’, ‘Postulates of Linguistics’ and ‘On Several Regimes of Signs’. What I am interested in, then, is Deleuze and Guattari's treatment of redundancy, a concept mentioned seventy times in A Thousand Plateaus. Although not one of their main or most developed concepts, redundancy is a thread that can be traced back to Deleuze's first book, Empiricism and Subjectivity. Moreover, their more developed concepts, for example the diagram, the abstract machine, becoming, micro-politics and the ritornello, as well as their political philosophy, are grounded in their understanding of redundancy, which moves their philosophy beyond mapping out the conditions of possibility and demonstrating that the historical a priori emerges as a contingent function of its own expression to the genesis of thought. (shrink)

The discussions which follow rest on a distinction, first expounded by Husserl, between formal logic and formal ontology. The former concerns itself with (formal) meaning-structures; the latter with formal structures amongst objects and their parts. The paper attempts to show how, when formal ontological considerations are brought into play, contemporary extensionalist theories of part and whole, and above all the mereology of Leniewski, can be generalised to embrace not only relations between concrete objects and object-pieces, but also relations between what (...) we shall call dependent parts or moments. A two-dimensional formal language is canvassed for the resultant ontological theory, a language which owes more to the tradition of Euler, Boole and Venn than to the quantifier-centred languages which have predominated amongst analytic philosophers since the time of Frege and Russell. Analytic philosophical arguments against moments, and against the entire project of a formal ontology, are considered and rejected. The paper concludes with a brief account of some applications of the theory presented. (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)

Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy in Kyoto's (...) school. For the proving the consistency of such systems, it is crucial to prove the well-foundedness of ordinals called "ordinal diagrams" developed for it. Takeuti presented such arguments several times in order to show that they are admitted in his stand point. As a starting point of investigating his finitist stand point, we formulate the system of ordinal notations up to ε0 and reconstruct the well-foundedness arguments of them. (shrink)

Popular culture continues fuelling public imagination with things, human and non-human, that we might beco-me or confront. Besides robots, other significant tropes in popular fiction that generated images include non-human humans and cyborgs, wired into his-torically varying sociocultural realities. Robots and artificial intelligence are re-defining the natural order and its hierar-chical structure. This is not surprising, as natural order is always in flux, shaped by new scientific discoveries, especially the reading of the genetic code, that reveal and redefine relationships between (...) life forms. However, for the first time, a new, artificial, species is being introduced into the existing diagrams, and, for the first time, it appears to hybridise the anthro-pocentric natural world order. (shrink)

Introduction to Philosophy: Epistemology engages first-time philosophy readers on a guided tour through the core concepts, questions, methods, arguments, and theories of epistemology—the branch of philosophy devoted to the study of knowledge. After a brief overview of the field, the book progresses systematically while placing central ideas and thinkers in historical and contemporary context. The chapters cover the analysis of knowledge, the nature of epistemic justification, rationalism vs. empiricism, skepticism, the value of knowledge, the ethics of belief, Bayesian epistemology, (...) social epistemology, and feminist epistemologies. Along the way, instructors and students will encounter a wealth of additional resources and tools: chapter learning outcomes, key terms, images of philosophers and related art, useful diagrams and tables, boxes containing excerpts and other supplementary material, questions for reflection, suggestions for further reading, and a glossary. For an undergraduate survey epistemology course, Introduction to Philosophy: Epistemology is ideal when used as a main text paired with primary sources and scholarly articles. For an introductory philosophy course, select book chapters are best used in combination with chapters from other books in the Introduction to Philosophy open textbook series (edited by Christina Hendricks). (shrink)

Berkeley in his Introduction to the Principles of Human knowledge uses geometrical examples to illustrate a way of generating “universal ideas,” which allegedly account for the existence of general terms. In doing proofs we might, for example, selectively attend to the triangular shape of a diagram. Presumably what we prove using just that property applies to all triangles.I contend, rather, that given Berkeley’s view of extension, no Euclidean triangles exist to attend to. Rather proof, as Berkeley would normally assume, requires (...) idealizing diagrams; treating them as if they obeyed Euclidean constraints. This convention solves the problem of representative generalization. View HTML Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.Berkeley and Proof in GeometryVolume 51, Issue 3RICHARD J. BROOK DOI: https://doi.org/10.1017/S0012217312000686Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. Berkeley and Proof in GeometryVolume 51, Issue 3RICHARD J. BROOK DOI: https://doi.org/10.1017/S0012217312000686Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. Berkeley and Proof in GeometryVolume 51, Issue 3RICHARD J. BROOK DOI: https://doi.org/10.1017/S0012217312000686Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)

Essential Microeconomics is designed to help students deepen their understanding of the core theory of microeconomics. Unlike other texts, this book focuses on the most important ideas and does not attempt to be encyclopedic. Two-thirds of the textbook focuses on price theory. As well as taking a new look at standard equilibrium theory, there is extensive examination of equilibrium under uncertainty, the capital asset pricing model, and arbitrage pricing theory. Choice over time is given extensive coverage and includes a (...) basic introduction to control theory. The final third of the book, on game theory, provides a comprehensive introduction to models with asymmetric information. Topics such as auctions, signaling and mechanism design are made accessible to students who have a basic rather than a deep understanding of mathematics. Examples and diagrams are used to illustrate issues as well as formal derivations. (shrink)

The essay analyses the contribution of contemporary analytical metaphysics to socialand legal ontology. In particular, the focus is on two authors: Brian Epstein and JonathanSchaffer. I discuss Epstein’s use of analytical metaphysics notions to explain the structureof social kinds and facts, providing a unique model based on three relations: grounding,anchoring, and framing (GAF).This model offers a new reading of the origin and nature ofsocial entities and brings innovative arguments to the debate in legal ontology. Schaffer’sviews represent a competing thesis, which (...) can be included among the so-called conjunctivisttheories. Epstein and Schaffer’s analyses converge in some respects and both theories appro-ach social and legal ontology with new, fine-grained tools. However, I defend the epistemicand practical value of Epstein’s model. At the same time, I claim that it would bring moreclarity to Epstein’s model if the framing relation was rejected. For this reason, I suggestconsidering a social reality model relying on the grounding-anchoring diagram alone. (shrink)

This is one of the best popular cosmology books ever written and Guth is now (2016) a top physics Professor at MIT. He tells the extremely complex story of inflation and related areas of particle physics in such an absorbing style that it reads like a detective novel-in fact, it is a detective novel-how he and others found out how the universe started! The interweaving of his personal story and that of many colleagues along with their photos and many wonderfully (...) clear diagrams allows just the right amount of relaxation from the intensity of the physics. In places the style reminds one of Watson´s famous book ``The Double Helix``. He tells how his work on magnetic monopoles and spontaneous symmetry breaking led to the discovery of the inflationary theory of the very early universe (ca. 10 to minus 35 seconds!). -/- Along the way you will learn many gems that should stay with you a long time such as: the observed universe(e.g., everything the Hubble telescope etc. can see out to ca. 15 billion light years when the universe began) is likely just a vanishingly tiny part of the entire inhomogeneous universe which is about 10 to the 23rd times larger; the big bang probably took place simultaneously and homogeneously in our observed universe; there probably have been and will continue to be an infinite number of big bangs in an infinite number of universes for an infinite time; when a bang happens, everything(space, time, all the elements) from the previous universe are destroyed; the stretching of space can happen at speeds much greater than the speed of light; our entire observed universe lies in a single bubble out of an endless number so there may be trillions of trillions just in our own entire(pocket) universe(and there may be an endless number of such); none of these infinite number of universes interact-i.e., we can never find out anything about the others; each universe started with its own big bang and will eventually collapse to create a new big bang; all this implies that the whole universe is fractal in nature and thus infinitely regresses to ever more universes(which can lead one to thinking of it as a giant hologram); disagreements between the endless(hundreds at least) variations of inflation are sometimes due to lack of awareness that different definitions of time are being used; some theories suggest that there was a first big bang but we can never find out what happened before it; nevertheless it appears increasingly plausible that there was no beginning but rather an eternal cycle of the destruction and creation, each being the beginning of spacetime for that universe; to start a universe you need about 25g of matter in a 10 to minus 26cm diameter sphere with a false vacuum and a singularity(white hole). -/- He deliberately spends little time on the endless variants of inflation such as chaotic, expanded and supernatural inflation or on dark matter´, supersymmetry and string theory, though they were well known at the time as you can find by reading other books such as Michio Kaku´s `Hyperspace` (see my review) and countless others. Of course much has happened since this book appeared but it still serves as an excellent background volume so cheap now it’s free for the cost of mailing. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019). (shrink)

(in lieu of abstract, first paragraphs here) For philosophers, reading Richard Dawkins is often a frustrating experience. Many of Dawkins’ writings treat important philosophical topics, such as the existence of God, the meaning of life, the relationship of randomness to order. Dawkins has original ideas, but he lacks the philosophical training and vocabulary to articulate these ideas properly and to develop them coherently. In Believing in Dawkins, Eric Steinhart sets himself an ambitious task: to use the writings of Dawkins to (...) present a coherent, naturalistic alternative to religious metaphysics, specifically to Christian theism. Using an architectural metaphor, he summarizes the project of Believing in Dawkins as follows -/- It is helpful to think of the Dawkinsian texts in architectural terms. His fragmentary sketches for spiritual naturalism are like architectural drawings. Sometimes they depict little windows, while other times they portray enormous spires reaching towards the stars. The edifice is vast. But these architectural diagrams are often unclear, incomplete, and inconsistent. I want to clarify them, fill in their missing parts, and resolve their conflicts. So I’m using his writings to construct a novel building. … To fit his fragmentary plans into a coherent and self-supporting structure, I will add some large-scale frameworks (10). -/- Steinhart’s wide-ranging book covers many topics: complexity, ontology, possibility, humanity, spirituality, among many others. He does not sneak theism into Dawkins’ writings. In fact, Steinhart argues that many practices and ideas that he uses to construct the edifice of Dawkins’ spiritual naturalism have been hijacked by theism, such as the notion of an afterlife, sacredness, holiness, and even the notion of gratitude. (shrink)

This paper analyses a hitherto unknown technique of using logic diagrams to create argument maps in eristic dialectics. The method was invented in the 1810s and -20s by Arthur Schopenhauer, who is considered the originator of modern eristic. This technique of Schopenhauer could be interesting for several branches of research in the field of argumentation: Firstly, for the field of argument mapping, since here a hitherto unknown diagrammatic technique is shown in order to visualise possible situations of arguments in (...) a dialogical controversy. Secondly, the art of controversy or eristic, since the diagrams do not analyse the truth of judgements and the validity of inferences, but the persuasiveness of arguments in a dialogue. (shrink)

In a multi-study naturalistic quasi-experiment involving 269 students in a semester-long introductory philosophy course, we investigated the effect of teaching argument diagramming on students’ scores on argument analysis tasks. An argument diagram is a visual representation of the content and structure of an argument. In each study, all of the students completed pre- and posttests containing argument analysis tasks. During the semester, the treatment group was taught AD, while the control group was not. The results were that among the different (...) pretest achievement levels, the scores of low-achieving students who were taught AD increased significantly more than the scores of low-achieving students who were not taught AD, while the scores of the intermediate- and high-achieving students did not differ significantly between the treatment and control groups. The implication of these studies is that learning AD significantly improves low-achieving students’ ability to analyze arguments. (shrink)

In this paper, we present a survey of the development of the technique of argument diagramming covering not only the fields in which it originated - informal logic, argumentation theory, evidence law and legal reasoning – but also more recent work in applying and developing it in computer science and artificial intelligence. Beginning with a simple example of an everyday argument, we present an analysis of it visualised as an argument diagram constructed using a software tool. In the context of (...) a brief history of the development of diagramming, it is then shown how argument diagrams have been used to analyze and work with argumentation in law, philosophy and artificial intelligence. (shrink)

There is substantial evidence from many domains that visual representations aid various forms of cognition. We aimed to determine whether visual representations of argument structure enhanced the acquisition and development of critical thinking skills within the context of an introductory philosophy course. We found a significant effect of the use of argument diagrams, and this effect was stable even when multiple plausible correlates were controlled for. These results suggest that natural⎯and relatively minor⎯modifications to standard critical thinking courses could provide (...) substantial increases in student learning and performance. (shrink)

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)

Historians occasionally use timelines, but many seem to regard such signs merely as ways of visually summarizing results that are presumably better expressed in prose. Challenging this language-centered view, I suggest that timelines might assist the generation of novel historical insights. To show this, I begin by looking at studies confirming the cognitive benefits of diagrams like timelines. I then try to survey the remarkable diversity of timelines by analyzing actual examples. Finally, having conveyed this (mostly untapped) potential, I (...) argue that neglecting timelines might mean neglecting significant aspects of reality that are revealed only by those signs. My overall message is that once we accept that relations are as important for the mind as what they relate, we have to pay closer attention to any semiotic device that enables or facilitates the discernment of new relations. (shrink)

Using as case studies two early diagrams that represent mechanisms of the cell division cycle, we aim to extend prior philosophical analyses of the roles of diagrams in scientific reasoning, and specifically their role in biological reasoning. The diagrams we discuss are, in practice, integral and indispensible elements of reasoning from experimental data about the cell division cycle to mathematical models of the cycle’s molecular mechanisms. In accordance with prior analyses, the diagrams provide functional explanations of (...) the cell cycle and facilitate the construction of mathematical models of the cell cycle. But, extending beyond those analyses, we show how diagrams facilitate the construction of mathematical models, and we argue that the diagrams permit nomological explanations of the cell cycle. We further argue that what makes diagrams integral and indispensible for explanation and model construction is their nature as locality aids: they group together information that is to be used together in a way that sentential representations do not. (shrink)

Logic diagrams have been increasingly studied and applied for a few decades, not only in logic, but also in many other fields of science. The history of logic diagrams is an important subject, as many current systems and applications of logic diagrams are based on historical predecessors. While traditional histories of logic diagrams cite pioneers such as Leibniz, Euler, Venn, and Peirce, it is not widely known that Kant and the early Kantians in Germany and England (...) played a crucial role in popularising Euler(-type) diagrams. In this paper, the role of the Kantians in the late eighteenth and early nineteenth centuries will be analysed in more detail. It shows that diagrams (or intuition in general) were a highly contentious topic that depend on the philosophical attitude and went beyond logic to touch on issues of physics, metaphysics, linguistics and, above all, mathematics. (shrink)

In his 1903 Syllabus, Charles S. Peirce makes a distinction between icons and iconic signs, or hypoicons, and briefly introduces a division of the latter into images, diagrams, and metaphors. Peirce scholars have tried to make better sense of those concepts by understanding iconic signs in the context of the ten classes of signs described in the same Syllabus. We will argue, however, that the three kinds of hypoicons can better be understood in the context of Peirce's sixty-six classes (...) of signs. We analyze examples of hypoicons taken from the field of information design, describing them in the framework of the sixty-six classes, and discuss the consequences of those descriptions to the debate about the order of determination of the 10 trichotomies that form those classes. (shrink)

Reism or concretism are the labels for a position in ontology and semantics that is represented by various philosophers. As Kazimierz Ajdukiewicz and Jan Woleński have shown, there are two dimensions with which the abstract expression of reism can be made concrete: The ontological dimension of reism says that only things exist; the semantic dimension of reism says that all concepts must be reduced to concrete terms in order to be meaningful. In this paper we argue for the following two (...) theses: (1) Arthur Schopenhauer has advocated a reistic philosophy of language which says that all concepts must ultimately be based on concrete intuition in order to be meaningful. (2) In his semantics, Schopenhauer developed a theory of logic diagrams that can be interpreted by modern means in order to concretize the abstract position of reism. Thus we are not only enhancing Jan Woleński’s list of well-known reists, but we are also adding a diagrammatic dimension to concretism, represented by Schopenhauer. (shrink)

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)

In this paper, we analyze and discuss Schopenhauer’s n-term diagrams for eristic dialectics from a graph-theoretical perspective. Unlike logic, eristic dialectics does not examine the validity of an isolated argument, but the progression and persuasiveness of an argument in the context of a dialogue or even controversy. To represent these dialogue situations, Schopenhauer created large maps with concepts and Euler-type diagrams, which from today’s perspective are a specific form of graphs. We first present the original method with Euler-type (...) diagrams, then give the most important graph-theoretical definitions, then discuss Schopenhauer’s diagrams graph-theoretically and finally give an example of how the graphs or diagrams can be used to analyze dialogues. (shrink)

Robert Brandom’s expressivism argues that not all semantic content may be made fully explicit. This view connects in interesting ways with recent movements in philosophy of mathematics and logic (e.g. Brown, Shin, Giaquinto) to take diagrams seriously - as more than a mere “heuristic aid” to proof, but either proofs themselves, or irreducible components of such. However what exactly is a diagram in logic? Does this constitute a semiotic natural kind? The paper will argue that such a natural kind (...) does exist in Charles Peirce’s conception of iconic signs, but that fully understood, logical diagrams involve a structured array of normative reasoning practices, as well as just a “picture on a page”. (shrink)

Whatever its ultimate philosophical merits, it is often thought that the growing block theory presents an intuitive picture of reality that accords well with our pre-reflective or folk view of time, and of the past, present, and future. This is partly motivated by the idea that we find it intuitive that, in some sense, the future is open and the past closed, and that the growing block theory is particularly well suited to accommodate this being so. In this paper, (...) we empirically investigate three claims. First, that people's intuitive or pre-reflective sense that the future is open is at least partly captured by there being truth-gaps for future contingents: what we call alethic openness. Second, that people, perhaps tacitly, believe that the fact that the future is alethically open is a reason to endorse the growing block theory; and third, that part of what explains why people tend to naively represent our world as a growing block is that they represent the future as alethically open. We found evidence in support of the first two claims. We consider the implications of these findings for several debates in the philosophy of time. (shrink)

This document diagrams the forms OOA, OOE, OOI, and OOO, including all four figures. Each form and figure has the following information: (1) Premises as stated: Venn diagram showing what the premises say; (2) Purported conclusion: diagram showing what the premises claim to say; (3) Relation of premises to conclusion: intended to describe how the premises and conclusion relate to each other, such as validity or contradiction. Used in only a few examples; (4) Distribution: intended to create a system (...) in which each syllogism has a unique code. In each premise and conclusion, the terms are each assigned a one or a zero, based on whether the term is distributed; (5) Rules: lists the rules of the syllogism and shows whether that particular syllogism follows, violates, or is unaffected by, each rule. (shrink)

Logic diagrams have seen a resurgence in their application in a range of fields, including logic, biology, media science, computer science and philosophy. Consequently, understanding the history and philosophy of these diagrams has become crucial. As many current diagrammatic systems in logic are based on ideas that originated in the 18th and 19th centuries, it is important to consider what motivated the use of logic diagrams in the past and whether these reasons are still valid today. This (...) paper proposes that transcendental philosophy was a key inspiration for the development of logic diagrams and that such diagrams can be employed in transcendental arguments, even after the linguistic turn. (shrink)

The argument diagramming method developed by Monroe C. Beardsley in his (1950) book Practical Logic, which has since become the gold standard for diagramming arguments in informal logic, makes it possible to map the relation between premises and conclusions of a chain of reasoning in relatively complex ways. The method has since been adapted and developed in a number of directions by many contemporary informal logicians and argumentation theorists. It has proved useful in practical applications and especially pedagogically in teaching (...) basic logic and critical reasoning skills at all levels of scientific education. I propose in this essay to build on Beardsley diagramming techniques to refine and supplement their structural tools for visualizing logical relationships in a number of categories not originally accommodated by Beardsley diagramming, including circular reasoning, reductio ad absurdum arguments, and efforts to dispute and contradict arguments, with applications and analysis. (shrink)

In his article “Insight: Genesis and Ongoing Context,” Fred Crowe calls out Lonergan’s line “the diagram is more important than…is ordinarily believed” as the “philosophical understatement of the century.” Sixteen pages later he identifies elaborating an invariant cognitional theory to underlie generalized emergent probability and thus “the immanent order of the universe of proportionate being,” as “our challenge,” “but given the difficulty” he does not “see any prospect for an immediate answer.” Could this have something to do with the lack (...) of a comprehensive diagram of cognitional theory? Appendix A of The Boston College Lectures on Mathematical Logic and Existentialism offers diagrams of the dynamics of knowing and doing perhaps copied from Lonergan’s own blackboard work, but they do not distinguish explanatory and descriptive insights, let alone statistical insights, and do not illustrate the pull upwards or the fusing of routinized insights. Before we can effectively relate cognitional theory to generalized emergent probability, we must have an adequately rigorous and precise cognitional theory. I firmly believe in the truth of Lonergan’s fundamental insights, but in order to rigorously undergird generalized emergent probability there are many pertinent questions about his cognitional theory which must be asked and answered. In this paper I (1) review some of Lonergan’s attempts to diagram cognitional theory and discuss what insights they do and do not express, (2) elaborate and defend principles for making our formulations of insight into insight rigorous and clear, and (3) attempt to build up a diagram which makes full use of those principles in a maximally expressive way. (shrink)

This essay provides an analysis of the role of affectivity in Martin Heidegger's writings from the mid to late 1920s. We begin by situating his account of mood within the context of his project of fundamental ontology in Being and Time. We then discuss the role of Befindlichkeit and Stimmung in his account of human existence, explicate the relationship between the former and the latter, and consider the ways in which the former discloses the world. To give a more (...) vivid and comprehensive picture of Heidegger's account of mood, we focus on the experience of anxiety by articulating both its function within fundamental ontology and, relatedly, its revelatory nature. We conclude by considering the place of emotions in Heidegger's thinking from this period. In a companion essay, ‘Affectivity in Heidegger II: Temporality, Boredom, and Beyond’, we complement our present analysis by revisiting the issue of affectivity in terms of Heidegger's discussion of temporality in Division II of Being and Time. We also expand our present discussion by considering the fundamental mood of boredom and other specific moods that Heidegger considers within his later thinking. (shrink)

The recent wave of data on exoplanets lends support to METI ventures (Messaging to Extra-Terrestrial Intelligence), insofar as the more exoplanets we find, the more likely it is that “exominds” await our messages. Yet, despite these astronomical advances, there are presently no well-confirmed tests against which to check the design of interstellar messages. In the meantime, the best we can do is distance ourselves from terracentric assumptions. There is no reason, for example, to assume that all inferential abilities are language-like. (...) With that in mind, I argue that logical reasoning does not have to be couched in symbolic notation. In diagrammatic reasoning, inferences are underwritten, not by rules, but by transformations of self-same qualitative signs. I use the Existential Graphs of C. S. Peirce to show this. Since diagrams are less dependent on convention and might even be generalized to cover non-visual senses, I argue that METI researchers should add some form of diagrammatic representations to their repertoire. Doing so can shed light, not just on alien minds, but on the deepest structures of reasoning itself. (shrink)

The following technique has proven effective in helping beginning logic students locate the sections of a three-circled Venn Diagram in which they are to represent a categorical sentence. Very often theses students are unable to identify the parts of the diagram they are to shade or bar.

This paper discusses similarities of both form and meaning between two symbolic structures: the Diagram of the Supreme Pole of Song Neo-Confucianism and the Kabbalistic Tree of medieval Jewish mysticism. These similarities are remarkable in the light of the many differences that exist between Chinese and Judaic thought, which also manifest in the two symbols. Intercultural influence might account for the similarities, but there is no historical evidence for such influence. An alternative explanation would attribute the similarities to the ubiquity (...) of religious-philosophical ideas about hierarchy, polarity, and macrocosm-microcosm parallelism, but this does not adequately account for the similar overall structure of the symbols. The question of how to understand these similarities remains open. (shrink)

In this paper I consider two strategies for providing tenseless truth-conditions for tensed sentences: the token-reflexive theory and the date theory. Both theories have faced a number of objections by prominent A-theorists such as Quentin Smith and William Lane Craig. Traditionally, these two theories have been viewed as rival methods for providing truth-conditions for tensed sentences. I argue that the debate over whether the token-reflexive theory or the date theory is true has arisen from a failure to distinguish between conditions (...) for the truth of tensed tokens and conditions for the truth of propositions expressed by tensed tokens. I demonstrate that there is a true formulation of the token-reflexive theory that provides necessary and sufficient conditions for the truth of tensed tokens, and there is a true formulation of the date theory that provides necessary and sufficient conditions for the truth of propositions expressed by tensed tokens. I argue that once the views are properly formulated, the A-theorist’s objections fail to make their mark. However, I conclude by claiming that even though there is a true formulation of the token-reflexive theory and a true formulation of the date theory, the New B-theory nonetheless fails to provide a complete account of the truth and falsity of tensed sentences. (shrink)

The master argument posits a metaphysical thesis: Diodorus does away with Aristotle’s dunamis understood as a power simultaneously oriented toward being and non-being and proclaims that possibilities that fail to actualize are simply nothing. My contention is that this claim is not a mere application of Diodorus’ contribution to modal logic. Rather, Diodorus creates an ontologico-temporal concept of possibility and impossibility. Diodorus envisions the future as the past that the future will become. Since what will have been can never be (...) the accomplishment of a possibility that did not obtain, and since the future is destined to become past, any futural possibility that doesn’t actualize is neither futural nor even possible. (shrink)

Abstract: There are two traditionally rival views about the nature of time: substantivalism that takes time to be a substance that exists independently of events located in it, and relationism that takes time to be constructed out of events. In this paper, first, I want to make some progress with respect to the debate between these two views, and I do this mainly by examining the strategies they use to face the possibilities of ‘empty time’ and (...) ‘time without change’. As we shall see, the two allegedly very different rival views are much less different than has been thought: their structure is extremely similar, their strategies are extremely similar, and they can both face the possibilities of ‘empty time’ and ‘time without change’ in the same way. Thus, I argue in favour of a certain kind of equivalence between the two views; I discuss a Strong and a Weak version of this claim; and I provide reasons for endorsing the former. I also discuss the parallel between this pair of views about the nature of time and another analogous pair of views: the bundle theory and the substratum theory about the nature of material objects, with respect to the problem with Identity of Indiscernibles. (shrink)

This paper presents the results of two corpus studies investigating the discourse surrounding conspiracy theories and genuine theories. The results of these studies show that conspiracy theories lack the epistemic and scientific standing characteristic of theories more generally. Instead, our findings indicate that conspiracy theories are spread in a manner that resembles the dissemination of rumors and falsehoods. Based on these empirical results, we argue that it is time for both re-engineering conspiracy theory and for relabeling "conspiracy theory". We (...) propose relabelling conspiracy theories as either "conspiracy stories" or "conspiracy narratives" to more accurately convey their true nature. (shrink)