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 “mathematicaldiagram” 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)
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)
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)
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)
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)
If the allusive stratagems can claim to define a new type of systematicity, it is because they give access to a space where the singularity, the diagram and the metaphor may interlace, to penetrate further into the physico-mathematic intuition and the discipline of the gestures which precede and accompany ‘formalisation’. This interlacing is an operation where each component backs up the others: without the diagram, the metaphor would only be a short-lived fulguration because it would be unable to (...) operate: without the metaphor, the diagram would only be a frozen icon, unable to jump over its bold features which represent the images of an already acquired knowledge; without the subversion of the functional by the singular, nothing would come to oppose the force of habit. (shrink)
In this paper I begin by extending two results of Solovay; the first characterizes the possible Turing degrees of models of True Arithmetic (TA), the complete first-order theory of the standard model of PA, while the second characterizes the possible Turing degrees of arbitrary completions of P. I extend these two results to characterize the possible Turing degrees of m-diagrams of models of TA and of arbitrary complete extensions of PA. I next give a construction showing that the conditions Solovay (...) identified for his characterization of degrees of models of arbitrary completions of PA cannot be dropped (I showed that these conditions cannot be simplified in the paper. (shrink)
Rigorous proof is supposed to guarantee that the premises invoked imply the conclusion reached, and the problem of rigor may be described as that of bringing together the perspectives of formal logic and mathematical practice on how this is to be achieved. This problem has recently raised a lot of discussion among philosophers of mathematics. We survey some possible solutions and argue that failure to understand its terms properly has led to misunderstandings in the literature.
It seems possible to know that a mathematical claim is necessarily true by inspecting a diagrammatic proof. Yet how does this work, given that human perception seems to just (as Hume assumed) ‘show us particular objects in front of us’? I draw on Peirce’s account of perception to answer this question. Peirce considered mathematics as experimental a science as physics. Drawing on an example, I highlight the existence of a primitive constraint or blocking function in our thinking which we (...) might call ‘the hardness of the mathematical must’. (shrink)
This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their first-order axiomatization. We focus on classes given by universal and universal-existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTL-algebra, from which analogues of the Łoś–Tarski and the Chang–Łoś–Suszko preservation theorems follow.
The objective of this article is to take into account the functioning of representational cognitive tools, and in particular of notations and visualizations in mathematics. In order to explain their functioning, formulas in algebra and logic and diagrams in topology will be presented as case studies and the notion of manipulative imagination as proposed in previous work will be discussed. To better characterize the analysis, the notions of material anchor and representational affordance will be introduced.
I examine the passages where Aristotle maintains that intellectual activity employs φαντάσματα (images) and argue that he requires awareness of the relevant images. This, together with Aristotle’s claims about the universality of understanding, gives us reason to reject the interpretation of Michael Wedin and Victor Caston, on which φαντάσματα serve as the material basis for thinking. I develop a new interpretation by unpacking the comparison Aristotle makes to the role of diagrams in doing geometry. In theoretical understanding of mathematical (...) and natural beings, we usually need to employ appropriate φαντάσματα in order to grasp explanatory connections. Aristotle does not, however, commit himself to thinking that images are required for exercising all theoretical understanding. Understanding immaterial things, in particular, may not involve employing phantasmata. Thus the connection that Aristotle makes between images and understanding does not rule out the possibility that human intellectual activity could occur apart from the body. (shrink)
For an Aristotelian observer, the halo is a puzzling phenomenon since it is apparently sublunary, and yet perfectly circular. This paper studies Aristotle's explanation of the halo in Meteorology III 2-3 as an optical illusion, as opposed to a substantial thing (like a cloud), as was thought by his predecessors and even many successors. Aristotle's explanation follows the method of explanation of the Posterior Analytics for "subordinate" or "mixed" mathematical-physical sciences. The accompanying diagram described by Aristotle is one (...) of the earliest lettered geometrical diagrams, in particular of a terrestrial phenomenon, and versions of it can still be found in modern textbooks on meteorological optics. (shrink)
Resumen: En la actualidad uno de los libros más usados para dar lógica elemental es el de Irving Copi y Carl Cohen (Introducción a la lógica, 2001), allí se presentan unas reglas para decidir la validez de los silogismos categóricos de forma estándar. Pero en tal texto ni en ninguno que nosotros conozcamos se ofrece una fundamentación de las mismas. Es decir, una demostración de que ellas son realmente una condición necesaria y suficiente de la validez de un silogismo categórico (...) de forma estándar, lo único que hemos leído es su motivación como condición necesaria. El objetivo de este trabajo es ofrecer una fundamentación de las mismas usando la noción de conjunto (o clase) y la técnica de diagramas de Venn, un concepto y una técnica que están expuestos en los principales textos de lógica elemental, incluyendo el referido anteriormente. -/- Abstract: Currently, one of the most widely used textbooks in elementary logic is the one by Irving Copi and Carl Cohen (Introduction to Logic, 2001). They offer rules for deciding the validity of standard categorical syllogisms. But neither that textbook, nor any other we know of, provides the foundations of these rules; that is to say, the demonstration that they are a necessary and sufficient condition for the validity of a standard categorical syllogism. We have only found motivations for the necessary condition. The main goal of this work is to provide a foundation of these rules using the notion of set as well as Venn diagrams, a concept and a technique exposed in the main textbooks of logic, including the aforementioned. (shrink)
In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume's maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or (...) localized a faculty to register them. We defend the perception of necessity against such Humeanism, drawing on examples from mathematics. (shrink)
Schemes of diagrammatic representation have been so familiarly introduced into logical treatises during the last century or so, that many readers, even of those who have made no professional study of logic, may be supposed to be acquainted with the general nature and object of such devices. Of these schemes one only, viz. that commonly called "Eulerian circles," has met with any general acceptance. A variety of others indeed have been proposed by ingenious and celebrated logicians, several of which would (...) claim notice in a historical treatment of the subject; but they mostly do not seem to me to differ in any essential respect from that of Euler. They rest upon the same leading principle, and are subject all alike to the same restrictions and defects. We must therefore cast about for some new scheme of diagrammatic representation which shall be competent to indicate imperfect knowledge on our part; for this will at once enable us to appeal to it step by step in the process of working out our conclusions. I have never seen any hint at such a scheme, though the want seems so evident that one would suppose that something of the kind must have been proposed before. (shrink)
In the course of daily life we solve problems often enough that there is a special term to characterize the activity and the right to expect a scientific theory to explain its dynamics. The classical view in psychology is that to solve a problem a subject must frame it by creating an internal representation of the problem’s structure, usually called a problem space. This space is an internally generable representation that is mathematically identical to a graph structure with nodes and (...) links. The nodes can be annotated with useful information, and the whole representation can be distributed over internal and external structures such as symbolic notations on paper or diagrams. If the representation is distributed across internal and external structures the subject must be able to keep track of activity in the distributed structure. Problem solving proceeds as the subject works from an initial state in mentally supported space, actively constructing possible solution paths, evaluating them and heuristically choosing the best. Control of this exploratory process is not well understood, as it is not always systematic, but various heuristic search algorithms have been proposed and some experimental support has been provided for them. (shrink)
In demonstration, speakers use real-world activity both for its practical effects and to help make their points. The demonstrations of origami mathematics, for example, reconfigure pieces of paper by folding, while simultaneously allowing their author to signal geometric inferences. Demonstration challenges us to explain how practical actions can get such precise significance and how this meaning compares with that of other representations. In this paper, we propose an explanation inspired by David Lewis’s characterizations of coordination and scorekeeping in conversation. In (...) particular, we argue that words, gestures, diagrams and demonstrations can function together as integrated ensembles that contribute to conversation, because interlocutors use them in parallel ways to coordinate updates to the conversational record. (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.
Few metaphors in biology are more enduring than the idea of Adaptive Landscapes, originally proposed by Sewall Wright (1932) as a way to visually present to an audience of typically non- mathematically savvy biologists his ideas about the relative role of natural selection and genetic drift in the course of evolution. The metaphor, how- ever, was born troubled, not the least reason for which is the fact that Wright presented different diagrams in his original paper that simply can- not refer (...) to the same concept and are therefore hard to reconcile with each other (Pigliucci 2008). For instance, in some usages, the landscape’s non- fitness axes represent combinations of individual genotypes (which cannot sensibly be aligned on a linear axis, and accordingly were drawn by Wright as polyhedrons of increasing dimensionality). In other usages, however, the points on the diagram represent allele or genotypic frequencies, and so are actually populations, not individuals (and these can indeed be coherently represented along continuous axes). (shrink)
This is a defence of the authenticity of Plato’s Epistula vii against the recent onslaught by Frede and Burnyeat (2015). It focusses on what Ep. vii has to say about writing and the embedded philosophical Digression and evaluates this in the context of other mainly late dialogues. In the Cratylus, Socrates ends with resignation regarding the potential of language study as a source of truth. This is also the case in Ep. vii, where the four means of knowledge (names, definitions, (...) images (diagrams) and knowledge/insight/true opinion) do not offer the essence of reality due to the weakness of language, in this case owing to a diagram with contrary properties. Consequently, name and definition are impermanent and conventional. Contra Burnyeat, definition is not impossible but useful in the acquisition of knowledge. Only after dialectical examination comes a flash of insight into reality. So, reality (truth) must, as in the Cratylus, be studied directly. Significantly for Plato (as opposed to Socrates), the epistemology is illustrated by a mathematical case. Hence it is relevant to look at mathematical procedure in middle and late dialogues. Moreover, the possible role of division is considered. Finally, the critique of writing justified by the Digression is placed in the context of the general aversion to the written word. The epistemology of the Digression is shown to be a fair and competent synopsis of the later Plato’s epistemology. Hence, there seems to be no reason thus far to doubt that Ep. vii is a genuine work of Plato. (shrink)
From my ongoing "Metalogical Plato" project. The aim of the diagram is to make reasonably intuitive how the Socratic elenchos (the logic of refutation applied to candidate formulations of virtues or ruling knowledges) looks and works as a whole structure. This is my starting point in the project, in part because of its great familiarity and arguable claim to being the inauguration of western philosophy; getting this point less wrong would have broad and deep consequences, including for philosophy’s self-understanding. (...) -/- (i.) is the first pass at elenchos in which the Socratic interlocutor does not reflect on knowledge being the crux of the problem. (ii.) is the second, rarer, reflective pass in which they are, making the investigation explicitly about knowledge. Its centrality in the Charmides makes that neglected dialogue of superlative importance. This structure is also the gateway through which the discussion/dialectic crosses into the Agathology (discussion of the form of the good) at Republic 505. -/- The problem of elenchos, then, grasped as a whole structure, is that it seems that knowledge can neither satisfactorily be included in nor excluded from its own scope. The development of the ti esti (“what is ---?”) question leads to the introduction of knowledge into its own scope (i. implicitly, as goodness contrasted with blind rule-following ii. explicitly qua knowledge) while the development of the dual peri tinos (“----about what?) question leads to K’s elimination from its own scope. The introductions are motivated to avoid contradiction, but produce regress; the eliminations are motivated to avoid regress, but produce contradiction. In scholarship, and in the history of philosophy, the ti esti question is universally recognized, to the point of being identified with philosophy’s origin and essence; the peri tinos question is neglected textually and never recognized as the equal dual to the ti esti. This, I claim, has blocked the development of a nontrivial logical appreciation of what Plato's Socrates is up to. (One rather disastrous effect of this neglect is taking Aristotle as the beginning of the development of logic, rather than, correctly, for the beginning of logic’s fatal separation from mathematics and dialectic.) -/- Further, because of this monopticism of the ti esti, the function of consistency in the elenchos has not been understood, even with respect to the ti esti. The ti esti is actually in search of completeness, given a norm of consistency; the peri tinos is in search of consistency, given a norm of completeness. Only appreciating the two questions as dual allows space in the structure to clarify these different orientations relative to consistency. (And, dually, to completeness, whose function in the elenchos is generally entirely missed by scholars and relegated to discussions of eros; it's not out of place there, of course, but its significance is secured here.) Recognizing the duality of the ti esti and peri tinos questions is thus the royal road, in the Socratic-Platonic context, to catching sight of what we post-Cantorians can recognize as the metalogical duality of consistency and completeness. (The salutary disruptive effects of this Plato-Cantor proximity have, of course, been traced in complementary ways by Badiou.) -/- Note that the diagram is supposed to provide a relatively accessible orientation, not to stand on its own, and certainly not to be the last word on any subject. An important qualification (telegraphed in the previous paragraph) is that what elenchos shows is not finally circular or paradoxical, though the problem first presents as such (stubbornly, obdurately, as "difficult" Plato's Socrates always says with characteristic understatement). What is depicted here is meant, at a first pass, to be the shape of that first presentation, the form of the problem of elenchos, rather than of its solution. It's not an accident that this problem strongly resembles "Russell's paradox" (not Russell's not a paradox.) Problem is to solution as RP is to the diagonal theorems. (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)
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)
This is a defence of the authenticity of Plato’s Epistula vii against the recent onslaught by Frede and Burnyeat (2015). It focusses on what Ep. vii has to say about writing and the embedded philosophical Digression and evaluates this in the context of other mainly late dialogues. In the Cratylus, Socrates ends with resignation regarding the potential of language study as a source of truth. This is also the case in Ep. vii, where the four means of knowledge (names, definitions, (...) images (diagrams) and knowledge/insight/true opinion) do not offer the essence of reality due to the weakness of language, in this case owing to a diagram with contrary properties. Consequently, name and definition are impermanent and conventional. Contra Burnyeat, definition is not impossible but useful in the acquisition of knowledge. Only after dialectical examination comes a flash of insight into reality. So, reality (truth) must, as in the Cratylus, be studied directly. Significantly for Plato (as opposed to Socrates), the epistemology is illustrated by a mathematical case. Hence it is relevant to look at mathematical procedure in middle and late dialogues. Moreover, the possible role of division is considered. Finally, the critique of writing justified by the Digression is placed in the context of the general aversion to the written word. The epistemology of the Digression is shown to be a fair and competent synopsis of the later Plato’s epistemology. Hence, there seems to be no reason thus far to doubt that Ep. vii is a genuine work of Plato. (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)
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 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)
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)
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)
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)
An Aid to Venn Diagrams.Robert Allen - 1997 - American Philosophical Association Newsletter on Teaching Philosophy 96 (Spring 1997):104-105.details
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.
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)
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)
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 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)
The aim of this article is to explain why knot diagrams are an effective notation in topology. Their cognitive features and epistemic roles will be assessed. First, it will be argued that different interpretations of a figure give rise to different diagrams and as a consequence various levels of representation for knots will be identified. Second, it will be shown that knot diagrams are dynamic by pointing at the moves which are commonly applied to them. For this reason, experts must (...) develop a specific form of enhanced manipulative imagination, in order to draw inferences from knot diagrams by performing epistemic actions. Moreover, it will be argued that knot diagrams not only can promote discovery, but also provide evidence. This case study is an experimentation ground to evaluate the role of space and action in making inferences by reasoning diagrammatically. (shrink)
This article explores aspects of Rosenzweig’s Star of Redemption from the perspective of systems theory. Mosès, Pollock, and others have noted the systematic character of the Star. While “systematic” does not mean “systems theoretic,” the philosophical theology of the Star encompasses ideas that are salient in systems theory. The Magen David star to which the title refers, and which deeply structures Rosenzweig’s thought, fits the classic definition of “system” – a set of elements and relations between the elements. The Yes (...) and No of the elements and their reversals illustrate the bridging of element and relation with the third category of “attribute,” a notion also central to the definition of “system.” In the diachronics of “the All,” the relations actualize what is only potential in the elements in their primordial state and thus remedy the incompleteness of these elements, fusing them into an integrated whole. Incompleteness is a major theme of systems theory, which also explicitly examines the relations between wholes and parts and offers a formal framework for expressing such fusions. In this article, the systems character of Parts I & II of the Star is explored through extensive use of diagrams; a systems exploration of Part III is left for future work. Remarkably, given its highly architectonic character, diagrams are absent in Rosenzweig’s book, except for the triangle of elements, the triangle of relations, and the hexadic star, which are presented on the opening page of each part of the book. While structures can be explicated entirely in words, diagrams are a visual medium of communication that supplements words and supports a nonverbal understanding that structures both thought and experience. (shrink)
While mechanistic explanation and, to a lesser extent, nomological explanation are well-explored topics in the philosophy of biology, topological explanation is not. Nor is the role of diagrams in topological explanations. These explanations do not appeal to the operation of mechanisms or laws, and extant accounts of the role of diagrams in biological science explain neither why scientists might prefer diagrammatic representations of topological information to sentential equivalents nor how such representations might facilitate important processes of explanatory reasoning unavailable to (...) scientists who restrict themselves to sentential representations. Accordingly, relying upon a case study about immune system vulnerability to attacks on CD4+ T-cells, I argue that diagrams group together information in a way that avoids repetition in representing topological structure, facilitate identification of specific topological properties of those structures, and make available to controlled processing explanatorily salient counterfactual information about topological structures, all in ways that sentential counterparts of diagrams do not. (shrink)
In these days, there is an increasing technological development in intelligent tutoring systems. This field has become interesting to many researchers. In this paper, we present an intelligent tutoring system for teaching mathematics that help students understand the basics of math and that helps a lot of students of all ages to understand the topic because it's important for students of adding and subtracting. Through which the student will be able to study the course and solve related problems. An evaluation (...) of the intelligent tutoring systems was carried out and the results were encouraging. (shrink)
In this paper we argue that the different positions taken by Dyson and Feynman on Feynman diagrams’ representational role depend on different styles of scientific thinking. We begin by criticizing the idea that Feynman Diagrams can be considered to be pictures or depictions of actual physical processes. We then show that the best interpretation of the role they play in quantum field theory and quantum electrodynamics is captured by Hughes' Denotation, Deduction and Interpretation theory of models (DDI), where “models” are (...) to be interpreted as inferential, non-representational devices constructed in given social contexts by the community of physicists. (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)
The aim of this paper is to elucidate the relationship between Aristotelian conceptual oppositions, commutative diagrams of relational structures, and Galois connections.This is done by investigating in detail some examples of Aristotelian conceptual oppositions arising from topological spaces and similarity structures. The main technical device for this endeavor is the notion of Galois connections of order structures.
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)
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)
Create an account to enable off-campus access through your institution's proxy server.
Monitor this page
Be alerted of all new items appearing on this page. Choose how you want to monitor it:
Email
RSS feed
About us
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.