In the period 1893–1897 Charles Dodgson, writing as Lewis Carroll, published two books and two articles on logic topics. Manuscript material first published in 1977 together with letters and diary entries provide evidence that he was working toward a visual proof system for complex syllogistic propositional logic based on a mechanical tree method that he devised.

Reviewed Works:Reuben Hersh, Proving is Convincing and Explaining.Philip J. Davis, Visual Theorems.Gila Hanna, H. Niels Jahnke, Proof and Application.Daniel Chazan, High School Geometry Students' Justification for Their Views of Empirical Evidence and Mathematical Proof.

Transition to Real Analysis with Proof provides undergraduate students with an introduction to analysis including an introduction to proof. The text combines the topics covered in a transition course to lead into a first course on analysis. This combined approach allows instructors to teach a single course where two were offered. The text opens with an introduction to basic logic and set theory, setting students up to succeed in the study of analysis. Each section is followed by graduated exercises that (...) both guide and challenge students. The author includes examples and illustrations that appeal to the visual side of analysis. The accessible structure of the book makes it an ideal refence for later years of study or professional work. (shrink)

Marcus Giaquinto presents an investigation into the different kinds of visual thinking involved in mathematical thought, drawing on work in cognitive psychology, philosophy, and mathematics. He argues that mental images and physical diagrams are rarely just superfluous aids: they are often a means of discovery, understanding, and even proof.

The past twenty years have witnessed extensive collaborative research between computer scientists, logicians, linguists, philosophers, and psychologists. These interdisciplinary studies stem from the realization that researchers drawn from all fields are studying the same problem. Specifically, a common concern amongst researchers today is how logic sheds light on the nature of information. Ancient questions concerning how humans communicate, reason and decide, and modern questions about how computers should communicate, reason and decide are of prime interest to researchers in various disciplines. (...) _Words, Proofs and Diagrams_ is a collection of papers covering active research areas at the interface of logic, computer science, and linguistics. Readers of the volume will find traditional research on process logics, issues in formal semantics, and language processing. In addition, the volume also highlights a particularly new area where all three disciplines meet—the study of images and graphics as information carriers and the diagrammatic reasoning supported by them. (shrink)

During the first phase of Greek mathematics a proof consisted in showing or making visible the truth of a statement. This was the epagogic method. This first phase was followed by an apagogic or deductive phase. During this phase visual evidence was rejected and Greek mathematics became a deductive system. Now epagoge and apagoge, apart from being distinguished, roughly according to the modern distinction between inductive and deductive procedures, were also identified on account of the conception of generality as (...) continuity. Epistemology of mathematics today only remembers the distinction, forgetting where they agreed, in this manner not only destroying the unity of the perceptual and conceptual but also forgetting what could be gained from Aristotelian demonstrative science. (shrink)

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late 19th century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis received much attention in the 19th century. They helped to instigate what Hans Hahn called a ‘crisis of intuition’, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this ‘crisis’ as follows : " Mathematicians (...) had for a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof … "Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called ‘logicization’, ‘when the discipline requires nothing but purely logical fundamental concepts and propositions for its development’. On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals …. (shrink)

This book's activities highlight the important cycle of exploration, conjecture, and justification in all five mathematical strands. Students recognize patterns and make conjectures, learn the value of a counterexample, explore the strengths and weaknesses of visual proofs, discover the power of algebraic representations, and learn that theoretical approaches can substantiate empirical results. The supplemental CD-ROM features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers. --publisher description.

Mathematicians sometimes judge a mathematical proof to be beautiful and in doing so seem to be making a judgement of the same kind as aesthetic judgements of works of visual art, music or literature. Mathematical proofs are also appraised for explanatoriness: some proofs merely establish their conclusions as true, while others also show why their conclusions are true. This paper will focus on the prima facie plausible assumption that, for mathematical proofs, beauty and explanatoriness tend to (...) go together. To make headway we need to have some grip on what it is for a proof to be beautiful, and for that we need some account of judgements of beauty in general. That is the concern of the first section. The second section faces the problem that it is far from obvious how abstract entities, such as mathematical proofs, can be beautiful, strictly and literally speaking. Reasons are given for the view that they can be. The third section introduces the distinction between proofs which explain their conclusions and proofs which do not. Finally, the question whether, for mathematical proofs, the beautiful and the explanatory tend to coincide is addressed. It is argued that we have reason to doubt that explanatory proofs tend to be beautiful, and insufficient reason to believe or disbelieve that beautiful proofs tend to be explanatory. (shrink)

This work examines how people interpret the sentential connective “or”, which can be viewed either inclusively (A or B or both) or exclusively (A or B but not both). Following up on prior work concerning quantifiers (Noveck, 2001; Noveck & Posada, 2003; Bott & Noveck, 2004) which shows that the common pragmatic interpretation of “some,” some but not all, is conveyed as part of an effortful step, we investigate how extra effort applied to disjunctive statements leads to a pragmatic interpretation (...) of “or”, or but not both. Experiment 1 compelled participants to wait for three seconds before answering, hence giving them the opportunity to process the utterance more deeply. Experiments 2 and 3 emphasized “or”, either by visual means (“OR”) or by prosodic means (contrastive stress) as another way to encourage participants to apply more effort. Following a relevance-theoretic line of argument, we hypothesized that conditions encouraging more processing effort would give rise to more pragmatic inferences and hence to more exclusive interpretations of the disjunction. This prediction was confirmed in the three experiments. (shrink)

Can visual thinking be a means of discovery in elementary analysis, as well as a means of illustration and a stimulus to discovery? The answer to the corresponding question for geometry and arithmetic seems to be ‘yes’ (Giaquinto [1992], [1993]), and so a positive answer might be expected for elementary analysis too. But I argue here that only in a severely restricted range of cases can visual thinking be a means of discovery in analysis. Examination of persuasive (...) class='Hi'>visual routes to two simple theorems (Rolle, Bolzano) shows that they are not ways of discovering the theorems; the type of visual thinking involved can never be used to discover analytic theorems of a certain generality. The hypothesis that visual thinking is never a means of discovering the existence or nature of the limit of some infinite process is considered, and a likely counter-example is set out. It is still possible that restricted theorems can be discovered visually: an example from Littlewood is examined in detail and not found wanting. Even when visual thinking is not a means of discovery it can provide the idea for a proof in a direct way; an example is presented (Intermediate Value Theorem). In conclusion: it may be possible to discover theorems in elementary real analysis by visual means, but only theorems of a restricted kind; however, visual thinking in analysis can be very useful in other ways. (shrink)

Diagrams are hybrid entities, which incorporate both linguistic and pictorial elements, and are crucial to any account of scientific and mathematical reasoning. Hence, they offer a rich source of examples to examine the relation between model-theoretic considerations and linguistic features. Diagrams also play different roles in different fields. In scientific practice, their role tends not to be evidential in nature, and includes: highlighting relevant relations in a micrograph ; sketching the plan for an experiment; and expressing expected visually salient information (...) about the outcome of an experiment. None of these traits are evidential; rather they are all pragmatic. In contrast, in mathematical practice, diagrams are used as heuristic tools in proof construction ; notational devices; and full-blown proof procedures. Some of these traits are evidential. After assessing these different roles, I explain why diagrams are used in the way they are in these two fields. The result leads to an account of different styles of scientific reasoning within a broadly model-based conception. (shrink)

Students of geometry do not prove Euclid's first theorem by examining an accompanying diagram, or by visualizing the construction of a figure. The original proof of Euclid's first theorem is incomplete, and this gap in logic is undetected by visual imagination. While cognition involves truth values, vision does not: the notions of inference and proof are foreign to vision.

The paper raises the question of the function of visual representations in medieval cosmographical texts. It proposes to view diverse functions of figures in relation to changing discursive environments, including differing philosophical positions and changing social and intellectual contexts. It further suggests a distinction between figures that were elaborated within the highly specialized disciplines of mathematics and philosophy of nature in Greek Antiquity and figures that were instrumental in transmitting accepted world models, thus avoiding the opposition between scientific and (...) unscientific types of verbal and pictorial documents. Simplifying changes, when figures are abstracted form their geometrical context and accompany doxographical, descriptive accounts, are characterized in terms of schematization. Concomitantly, mathematical and philosophical demonstrations tend to give way to proofs of a predominantly rhetorical nature: images are verbally construed and, in order to enhance these, actual visual figures— mostly linear, diagrammatic constructs—are added. With regard to the Middle Ages, the paper distinguishes two principal periods: the period from the seventh to the eleventh century and the period of the so-called twelfth-century Renaissance. First, the verbal and pictorial cosmological corpus of Roman origin gave rise to explanations and variations but not to consequential theoretical developments and cosmological diagrams tended to fuse with summarizing tables at this time. Then, during the twelfth century, mathematical and philosophical documents of a specialized kind that were translated from the Arabic and also from the Greek became available in the Latin West. In mathematics, specialized types of study remained, however, sparse. Continuous elaborations of the assimilated material set in later only, within the thirteenth-century university context. Nevertheless, twelfth-century authors of cosmographical accounts became increasingly aware that their expositions and visual figures were ultimately derived from geometrical models of the universe. More diversified types of demonstration and corresponding visual figures were being used, as exemplified by William of Conches’ Dragmaticon philosophiae. (shrink)

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late 19th century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis received much attention in the 19th century. They helped to instigate what Hans Hahn called a ‘crisis of intuition’, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this ‘crisis’ as follows : " Mathematicians (...) had for a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof … "Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called ‘logicization’, ‘when the discipline requires nothing but purely logical fundamental concepts and propositions for its development’. On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals …. (shrink)

John Venn and Charles L. Dodgson (Lewis Carroll) created systems of logic diagrams capable of representing classes (sets) and their relations in the form of propositions. Each is a proof method for syllogisms, and Carroll's is a sound and complete system. For a large number of sets, Carroll diagrams are easier to draw because of their self-similarity and algorithmic construction. This regularity makes it easier to locate and thereby to erase cells corresponding with classes destroyed by the premises of an (...) argument, a particularly difficult task in Venn diagrams for more than four sets. Carroll diagrams can represent existential propositions easily, so they are capable of clearly representing more complex problems than Venn's system can. Finally, both Carroll and Venn diagrams are maximal, in the sense that no additional logic information like inclusive disjunctions is able to be represented by them. Carroll's logic diagrams and logic trees constitute his visual logic system. (shrink)

Music can be interpreted by attributing syntactic relationships to sequential musical events, and, computationally, such musical interpretation represents an analogous combinatorial task to syntactic processing in language. While this perspective has been primarily addressed in the domain of harmony, we focus here on rhythm in the Western tonal idiom, and we propose for the first time a framework for modeling the moment‐by‐moment execution of processing operations involved in the interpretation of music. Our approach is based on (1) a music‐theoretically motivated (...) grammar formalizing the competence of rhythmic interpretation in terms of three basic types of dependency (preparation, syncopation, and split; Rohrmeier, 2020), and (2) psychologically plausible predictions about the complexity of structural integration and memory storage operations, necessary for parsing hierarchical dependencies, derived from the dependency locality theory (Gibson, 2000). With a behavioral experiment, we exemplify an empirical implementation of the proposed theoretical framework. One hundred listeners were asked to reproduce the location of a visual flash presented while listening to three rhythmic excerpts, each exemplifying a different interpretation under the formal grammar. The hypothesized execution of syntactic‐processing operations was found to be a significant predictor of the observed displacement between the reported and the objective location of the flashes. Overall, this study presents a theoretical approach and a first empirical proof‐of‐concept for modeling the cognitive process resulting in such interpretation as a form of syntactic parsing with algorithmic similarities to its linguistic counterpart. Results from the present small‐scale experiment should not be read as a final test of the theory, but they are consistent with the theoretical predictions after controlling for several possible confounding factors and may form the basis for further large‐scale and ecological testing. (shrink)

The issue of shifting scales between the microscopic and the macroscopic dimensions is a recurrent one in the history of science, and in particular the history of microscopy. But it took on new dimensions in the context of early twentieth-century microscophysics, with the progressive realisation that the physical laws governing the macroscopic world were not always adequate for describing the sub-microscopic one. The paper focuses on the researches of Jean Perrin in the 1900s, in particular his use of Brownian motion (...) to produce evidence of the existence of atoms and in favour of the kinetic theory. His results were described by many contemporaries, and subsequently by historians, as the first direct proof of atomic and molecular reality. The paper examines the different strategies developed by Perrin for bridging the macro and sub-microphysical realms and making the latter accessible to the senses; even though neither atoms nor molecules were ever actually seen, and in fact very few visual representations were shown and published in connection with these experiments. This case provides a good example of how visualizing, representing and convincing could be interwoven in the production of evidence about the sub-microphysical realm circa 1900.Keywords: Jean Perrin; Brownian motion; Ultramicroscope; Evidence; Atom; Atomism. (shrink)

Interpreting a seemingly simple function word like “or,” “behind,” or “more” can require logical, numerical, and relational reasoning. How are such words learned by children? Prior acquisition theories have often relied on positing a foundation of innate knowledge. Yet recent neural‐network‐based visual question answering models apparently can learn to use function words as part of answering questions about complex visual scenes. In this paper, we study what these models learn about function words, in the hope of better understanding (...) how the meanings of these words can be learned by both models and children. We show that recurrent models trained on visually grounded language learn gradient semantics for function words requiring spatial and numerical reasoning. Furthermore, we find that these models can learn the meanings of logical connectives and and or without any prior knowledge of logical reasoning as well as early evidence that they are sensitive to alternative expressions when interpreting language. Finally, we show that word learning difficulty is dependent on the frequency of models' input. Our findings offer proof‐of‐concept evidence that it is possible to learn the nuanced interpretations of function words in a visually grounded context by using non‐symbolic general statistical learning algorithms, without any prior knowledge of linguistic meaning. (shrink)

Recent decades have witnessed the rapid evolution of robotic applications and their expansion into a variety of spheres with remarkable achievements. This article researches a crucial technique of robot manipulators referred to as visual servoing, which relies on the visual feedback to respond to the external information. In this regard, the visual servoing issue is tactfully transformed into a quadratic programming problem with equality and inequality constraints. Differing from the traditional methods, a gradient-based recurrent neural network for (...) solving the visual servoing issue is newly proposed in this article in the light of the gradient descent method. Then, the stability proof is presented in theory with the pixel error convergent exponentially to zero. Specifically speaking, the proposed method is able to impel the manipulator to approach the desired static point while maintaining physical constraints considered. After that, the feasibility and superiority of the proposed GRNN are verified by simulative experiments. Significantly, the proposed visual servo method can be leveraged to medical robots and rehabilitation robots to further assist doctors in treating patients remotely. (shrink)

Brimming with visual examples of concepts, derivation rules, and proof strategies, this introductory text is ideal for students with no previous experience in logic. Students will learn translation both from formal language into English and from English into formal language; how to use truth trees and truth tables to test propositions for logical properties; and how to construct and strategically use derivation rules in proofs.

This thesis challenges a general prejudice against visualization in the history of logic and mathematics, by providing a semantic analysis of two graphical representation systems--a traditional Venn diagram representation system and an extension of it. While Venn diagrams have been used to solve problems in set theory and to test the validity of syllogisms in logic, they have not been considered valid proofs but heuristic tools for finding valid formal proofs. ;I present Venn diagrams which have been used (...) in logic as a formal system of representation equipped with its own syntax and semantics. I also show that Venn-I, with the rules of transformation that I specify, is sound and complete. The soundness of Venn-I refutes a worry that mathematicians and logicians have about graphical representation: Fallacies often have arisen from the misuse of diagrams in mathematical proofs, especially in geometry. However, the validity of the transformation rules assures us that the correct application of these rules will not lead to fallacies. ;I extend Venn-I so that this new system has a general way to convey disjunctive information. A new formation rule is introduced and new transformation rules are added. I call this extended system Venn-II. The semantics of Venn-II is formalized as an extension of the semantics of Venn-I. The soundness and completeness of Venn-II are also proven. I then show that Venn-II and a monadic first-order language are equivalent in their expressiveness. ;The analysis of Venn-I and Venn-II leads to interesting issues which have their analogues in other deductive systems. I conclude the thesis with a discussion of some fundamental differences between graphical systems and linguistic systems. (shrink)

Most clinical neurofeedback studies based on functional magnetic resonance imaging use the patient's own neural activity as feedback. The objective of this study was to create a subject-independent brain state classifier as part of a real-time fMRI neurofeedback system that can guide patients with depression in achieving a healthy brain state, and then to examine subsequent clinical changes. In a first step, a brain classifier based on a support vector machine was trained from the neural information of happy autobiographical imagery (...) and motor imagery blocks received from a healthy female participant during an MRI session. In the second step, 7 right-handed female patients with mild or moderate depressive symptoms were trained to match their own neural activity with the neural activity corresponding to the “happiness emotional brain state” of the healthy participant. The training was carried out using the rt-fMRI NF system guided by the brain-state classifier we had created. Thus, the informative voxels previously obtained in the first step, using SVM classification and Effect Mapping, were used to classify the Blood-Oxygen-Level Dependent activity of the patients and converted into real-time visual feedback during the neurofeedback training runs. Improvements in the classifier accuracy toward the end of the training were observed in all the patients [Session 4–1 Median = 6.563%; Range = 4.10–27.34; Wilcoxon Test, 2-tailed p = 0.031]. Clinical improvement also was observed in a blind standardized clinical evaluation [HDRS CE2-1 Median = 7; Range 2 to 15; Wilcoxon Test, 2-tailed p = 0.016], and in self-report assessments [BDI-II CE2-1 Median = 8; Range 1–15; Wilcoxon Test, 2-tailed p = 0.031]. In addition, the clinical improvement was still present 10 days after the intervention [BDI-II CE3-2_Median = 0; Range −1 to 2; Wilcoxon Test, 2-tailed p = 0.50/ HDRS CE3-2 Median = 0; Range −1 to 2; Wilcoxon Test, 2-tailed p = 0.625]. Although the number of participants needs to be increased and a control group included to confirm these findings, the results suggest a novel option for neural modulation and clinical alleviation in depression using noninvasive stimulation technologies. (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)

Advertising is one of the commonly visible elements of the urban landscape (real and virtual). It also does not require proof that advertisements of cities as such are also part of their “cityscape.” Since at least the nineteenth century, cities have advertised themselves as attractive places to live, visit, or do business. Therefore, the following research question can be asked: How do Polish cities present themselves in advertisements one can find in the landscape? The study assumes that each advertisement should (...) clearly identify the sender and indicate its specific characteristics, distinguishing itself from its competitors. Polish cities began to use logos and slogans as a mecha­nism of description and distinction after 1990, when socioeconomic changes started. There are quite a few studies on this activity, but most of them are single case studies. Therefore, the authors decided to examine a relatively large sample of Polish cities, which allows for statistical analysis. Analysis of the logo and slogan content allows the authors to examine the desired image or the projected identity of Polish cities. The methods were chosen because they admit qualitative and quantitative analysis, especially when there is a sufficiently large sample. Therefore, the survey covered all towns and cities with more than 20,000 inhabitants in Poland. There are 218 such towns in total. The analysis was carried out in the first half of 2021. In conclusion, the authors find that the advertising of Polish cities is embedded in the past, promotes resources (substance), sometimes geographical location, but rarely refers to famous characters and human potential. And such a picture of these cities one may find on outdoor advertisements, which sometimes produce dissonance when accompanying modern buildings or new transport solutions in the city. (shrink)

In his long-awaited new edition of Philosophy of Mathematics, James Robert Brown tackles important new as well as enduring questions in the mathematical sciences. Can pictures go beyond being merely suggestive and actually prove anything? Are mathematical results certain? Are experiments of any real value?" "This clear and engaging book takes a unique approach, encompassing nonstandard topics such as the role of visual reasoning, the importance of notation, and the place of computers in mathematics, as well as traditional topics (...) such as formalism, Platonism, and constructivism. The combination of topics and clarity of presentation make it suitable for beginners and experts alike. The revised and updated second edition of Philosophy of Mathematics contains more examples, suggestions for further reading, and expanded material on several topics including a novel approach to the continuum hypothesis."--BOOK JACKET. (shrink)

This monograph treats the important topic of the epistemology of diagrams in Euclidean geometry. Norman argues that diagrams play a genuine justificatory role in traditional Euclidean arguments, and he aims to account for these roles from a modified Kantian perspective. Norman considers himself a semi-Kantian in the following broad sense: he believes that Kant was right that ostensive constructions are necessary in order to follow traditional Euclidean proofs, but he wants to avoid appealing to Kantian a priori intuition as (...) the epistemological background for these constructions.Norman's main argument is limited to the thesis that certain Euclidean arguments—in particular, that of Proposition 1.32, the internal-angle-sum theorem—require inferences from diagrams. Interestingly, Norman is not committed to the view that these arguments are proofs. This becomes clear only quite late in the book, when he distinguishes argument from proofs, remarking that the argument he has been focusing on is not rigorous, and so is not a proof. Norman does not, however, explicitly classify all Euclidean arguments as non-proofs. His view is that diagrammatic reasoning can in principle feature in rigorous proofs, but he is not committed to the thesis that any particular argument, Euclidean or otherwise, provides an example of this. Rather than proofs in particular, Norman is more interested in the general issue of justification in mathematics.The argument has three components. First, Norman argues against competing accounts of Euclidean arguments such as empiricism, and ‘Leibnizianism’—the view that diagrams play only a heuristic role. Second, he provides a more direct, or positive, argument that the way we actually follow the standard argument for 1.32 does appeal to the diagram. This is an appeal to the phenomenology of following the argument. Third, he articulates and defends his semi-Kantian position against some objections.The book has a very careful and …. (shrink)

Suppose one has a visual experience as of having hands, and then reasons as follows: I have hands, If I have hands an external world exists; An external world exists. Suppose one’s visual experience gives one defeasible perceptual warrant, or justification, to believe – that is, one’s experience makes it epistemically appropriate to believe . And suppose one comes to believe on the basis of this visual experience. The conditional premise is knowable a priori. And can be (...) established by modus ponens inference. If one reasons thus, say one’s engaged in -reasoning. What, if anything, is wrong with -reasoning? I consider two prominent responses to this question – the dogmatists’ and Crispin Wright’s. Each finds fault in -reasoning, but on different grounds. I argue Wright’s response faces a problem which is standardly only taken to be faced by dogmatists. (shrink)

The first English-language publication of a classic French book on the relationship between the development of photography and of the medical category of hysteria. In this classic of French cultural studies, Georges Didi-Huberman traces the intimate and reciprocal relationship between the disciplines of psychiatry and photography in the late nineteenth century. Focusing on the immense photographic output of the Salpetriere hospital, the notorious Parisian asylum for insane and incurable women, Didi-Huberman shows the crucial role played by photography in the invention (...) of the category of hysteria. Under the direction of the medical teacher and clinician Jean-Martin Charcot, the inmates of Salpetriere identified as hysterics were methodically photographed, providing skeptical colleagues with visual proof of hysteria's specific form. These images, many of which appear in this book, provided the materials for the multivolume album Iconographie photographique de la Salpetriere. As Didi-Huberman shows, these photographs were far from simply objective documentation. The subjects were required to portray their hysterical "type"—they performed their own hysteria. Bribed by the special status they enjoyed in the purgatory of experimentation and threatened with transfer back to the inferno of the incurables, the women patiently posed for the photographs and submitted to presentations of hysterical attacks before the crowds that gathered for Charcot's "Tuesday Lectures." Charcot did not stop at voyeuristic observation. Through techniques such as hypnosis, electroshock therapy, and genital manipulation, he instigated the hysterical symptoms in his patients, eventually giving rise to hatred and resistance on their part. Didi-Huberman follows this path from complicity to antipathy in one of Charcot's favorite "cases," that of Augustine, whose image crops up again and again in the Iconographie. Augustine's virtuosic performance of hysteria ultimately became one of self-sacrifice, seen in pictures of ecstasy, crucifixion, and silent cries. (shrink)

This monograph treats the important topic of the epistemology of diagrams in Euclidean geometry. Norman argues that diagrams play a genuine justificatory role in traditional Euclidean arguments, and he aims to account for these roles from a modified Kantian perspective. Norman considers himself a semi-Kantian in the following broad sense: he believes that Kant was right that ostensive constructions are necessary in order to follow traditional Euclidean proofs, but he wants to avoid appealing to Kantian a priori intuition as (...) the epistemological background for these constructions.Norman's main argument is limited to the thesis that certain Euclidean arguments—in particular, that of Proposition 1.32, the internal-angle-sum theorem—require inferences from diagrams. Interestingly, Norman is not committed to the view that these arguments are proofs. This becomes clear only quite late in the book, when he distinguishes argument from proofs, remarking that the argument he has been focusing on is not rigorous, and so is not a proof . Norman does not, however, explicitly classify all Euclidean arguments as non-proofs. His view is that diagrammatic reasoning can in principle feature in rigorous proofs, but he is not committed to the thesis that any particular argument, Euclidean or otherwise, provides an example of this. Rather than proofs in particular, Norman is more interested in the general issue of justification in mathematics.The argument has three components. First, Norman argues against competing accounts of Euclidean arguments such as empiricism, and ‘Leibnizianism’—the view that diagrams play only a heuristic role. Second, he provides a more direct, or positive, argument that the way we actually follow the standard argument for 1.32 does appeal to the diagram. This is an appeal to the phenomenology of following the argument. Third, he articulates and defends his semi-Kantian position against some objections.The book has a very careful and …. (shrink)

For a decade or so there has been a new ‘hype’ in educational research: it is called educational neuroscience or even neuroeducation —there are numerous publications, special journals, and an abundance of research projects together with the advertisement of many positions at renowned research centres worldwide. After a brief introduction of what is going on in the ‘emerging sub-discipline’, a number of characterisations are offered of what is envisaged by authors working in this field. In the discussion that follows various (...) problems are listed: the assumption that ‘visual proof’ of brain activity is supposedly given; the correlational nature of this kind of research; the nature of the concepts that are used; the lack of addressing and possibly influencing the neurological mechanism; and finally the need for other insights in educational contexts. Following Bakhurst and others, a number of crucially relevant philosophical issues are highlighted. It is argued that though there are cases where neuroscience insights may be helpful, these are scarce. In general, it is concluded, not a lot may be expected from this discipline for education and educational research. A reminder is offered that the promise of neurophilia may be just another neuromyth, which needs to be addressed by philosophy and education. (shrink)

When people make sense of situations, illustrations, instructions and problems they do more than just think with their heads. They gesture, talk, point, annotate, make notes and so on. What extra do they get from interacting with their environment in this way? To study this fundamental problem, I looked at how people project structure onto geometric drawings, visual proofs, and games like tic tac toe. Two experiments were run to learn more about projection. Projection is a special capacity, (...) similar to perception, but less tied to what is in the environment. Projection, unlike pure imagery, requires external structure to anchor it, but it adds ‘mental’ structure to the external scene much like an augmented reality system adds structure to an outside scene. A person projects when they look at a chessboard and can see where a knight may be moved. Because of the cognitive costs of sustaining and extending projection, humans make some of their projections real. They create structure externally. They move the piece, they talk, point, notate, represent. Much of our interactivity during sense making and problem solving involves a cycle of projecting then creating structure. (shrink)

Visual thinking in mathematics is widespread; it also has diverse kinds and uses. Which of these uses is legitimate? What epistemic roles, if any, can visualization play in mathematics? These are the central philosophical questions in this area. In this introduction I aim to show that visual thinking does have epistemically significant uses. The discussion focuses mainly on visual thinking in proof and discovery and touches lightly on its role in understanding.

Introduction: image ethics -- Harnessing the visual: from illustration to ekphrasis -- From visible to invisible: Spenser's Aprill and messianic ethics -- Looking for ethics in Spenser's Faerie queene -- "To look, but with another's eyes": translating vision in A midsummer night's dream -- The ethics of temporality in Measure for measure -- "Ocular proof" and the dangers of the perceptual faith -- "Disliken the truth of your own seeming": visual and ethical truth in The winter's tale.

Euler diagrams often require several figures to adequately represent propositions and syllogisms. Euler’s followers, notably Friedrich Ueberweg, endeavored to overcome this difficulty with the use of dotted lines to express uncertainty about the relation between the terms of a proposition. Subsequently, Venn regarded such attempts as ineffectual and went to construct his own celebrated scheme. In this paper, we argue that Ueberweg’s method could be expanded to meet Venn’s expectations, and hence, produce alternative Venn-like diagrams.

We conducted a computer-based psychological experiment in which a random mix of 40 tautologies and 40 non-tautologies were presented to the participants, who were asked to determine which ones of the formulas were tautologies. The participants were eight university students in computer science who had received tuition in propositional logic. The formulas appeared one by one, a time-limit of 45 s applied to each formula and no aids were allowed. For each formula we recorded the proportion of the participants who (...) classified the formula correctly before timeout and the mean response time among those participants. We propose a new proof formalism for modeling propositional reasoning with bounded cognitive resources. It models declarative memory, visual memory, working memory, and procedural memory according to the memory model of Atkinson and Shiffrin and reasoning processes according to the model of Newell and Simon. We also define two particular proof systems, T and NT, for showing propositional formulas to be tautologies and non-tautologies, respectively. The accuracy was found to be higher for non-tautologies than for tautologies ( p T was.89 and for non-tautologies the correlation between latency and minimum proof length in NT was.87. (shrink)

Diagrams have played an important role throughout the entire history of differential equations. Geometrical intuition, visual thinking, experimentation on diagrams, conceptions of algorithms and instruments to construct these diagrams, heuristic proofs based on diagrams, have interacted with the development of analytical abstract theories. We aim to analyze these interactions during the two centuries the classical theory of differential equations was developed. They are intimately connected to the difficulties faced in defining what the solution of a differential equation is (...) and in describing the global behavior of such a solution. (shrink)

"Intuition" has perhaps been the least understood and the most abused term in philosophy. It is often the term used when one has no plausible explanation for the source of a given belief or opinion. According to some sceptics, it is understood only in terms of what it is not, and it is not any of the better understood means for acquiring knowledge. In mathematics the term has also unfortunately been used in this way. Thus, intuition is sometimes portrayed as (...) if it were the Third Eye, something only mathematical "mystics", like Ramanujan, possess. In mathematics the notion has also been used in a host of other senses: by "intuitive" one might mean informal, or non-rigourous, or visual, or holistic, or incomplete, or perhaps even convincing in spite of lack of proof. My aim in this book is to sweep all of this aside, to argue that there is a perfectly coherent, philosophically respectable notion of mathematical intuition according to which intuition is a condition necessary for mathemati cal knowledge. I shall argue that mathematical intuition is not any special or mysterious kind of faculty, and that it is possible to make progress in the philosophical analysis of this notion. This kind of undertaking has a precedent in the philosophy of Kant. While I shall be mostly developing ideas about intuition due to Edmund Husser! there will be a kind of Kantian argument underlying the entire book. (shrink)

One sceptical rejoinder to those who claim that sensory perception is cognitively penetrable is to appeal to the involvement of attention. So, while a phenomenon might initially look like one where, say, a perceiver’s beliefs are influencing her visual experience, another interpretation is that because the perceiver believes and desires as she does, she consequently shifts her spatial attention so as to change what she senses visually. But, the sceptic will urge, this is an entirely familiar phenomenon, and it (...) hardly involves some special or theoretically important cognitive effect on sensory perception. Even supposing that the sceptic is correct about cases that are accurately described in this way, the rejoinder oversimplifies the possible roles that attention may play in mediating cognition and perception. This paper aims to identify these different roles, and by emphasis on empirical research on feature-based and object-based attention. What emerges is a plausible and well- evidenced mental schema that describes attention -mediated cognitive penetration. At the very least, the burden of proof is shifted to the sceptic, as he then must show that there are no mental phenomena involving attention in the more nuanced ways described here. One additional benefit of this analysis is that it illuminates various features of attention and its relation to both cognition and phenomenal consciousness. Therefore the analysis should be of interest to a broad range of theorists of the mind, and not just those invested in the cognitive penetration debate. (shrink)

The current paper highlights a new, interactive Shiny App that can be used to aid in understanding and teaching the important task of conducting a prior sensitivity analysis when implementing Bayesian estimation methods. In this paper, we discuss the importance of examining prior distributions through a sensitivity analysis. We argue that conducting a prior sensitivity analysis is equally important when so-called diffuse priors are implemented as it is with subjective priors. As a proof of concept, we conducted a small simulation (...) study, which illustrates the impact of priors on final model estimates. The findings from the simulation study highlight the importance of conducting a sensitivity analysis of priors. This concept is further extended through an interactive Shiny App that we developed. The Shiny App allows users to explore the impact of various forms of priors using empirical data. We introduce this Shiny App and thoroughly detail an example using a simple multiple regression model that users at all levels can understand. In this paper, we highlight how to determine the different settings for a prior sensitivity analysis, how to visually and statistically compare results obtained in the sensitivity analysis, and how to display findings and write up disparate results obtained across the sensitivity analysis. The goal is that novice users can follow the process outlined here and work within the interactive Shiny App to gain a deeper understanding of the role of prior distributions and the importance of a sensitivity analysis when implementing Bayesian methods. The intended audience is broad and can include those with limited exposure to Bayesian methods or the specific model presented here. (shrink)

Gestures are usually regarded as a casual element of communication processes between logicians. By contrast, we aim to show that gestures have played a significant role in logic. We argue that the development of communication techniques and their standardization have led to the rise of formal notation systems commonly used in logic today. In order to substantiate this claim, the historical development of the use of gestures in (early) modern logic is investigated. This investigation uncovers exemplary communication and proof techniques (...) that illustrate the efficacy of gestures as representational media in formal logic. Revisiting the tradition of gestures in logic offers promising paths and insights for today’s logic as a communication practice in the digital age. (shrink)

This book explores the results of applying empirical methods to the philosophy of logic and mathematics. Much of the work that has earned experimental philosophy a prominent place in twenty-first century philosophy is concerned with ethics or epistemology. But, as this book shows, empirical methods are just as much at home in logic and the philosophy of mathematics. -/- Chapters demonstrate and discuss the applicability of a wide range of empirical methods including experiments, surveys, interviews, and data-mining. Distinct themes emerge (...) that reflect recent developments in the field, such as issues concerning the logic of conditionals and the role played by visual elements in some mathematical proofs. -/- Featuring leading figures from experimental philosophy and the fields of philosophy of logic and mathematics, this collection reveals that empirical work in these disciplines has been quietly thriving for some time and stresses the importance of collaboration between philosophers and researchers in mathematics education and mathematical cognition. (shrink)

The paper explores the potential of Byzantine diagrams in syllogistic logic. Byzantine diagrams are originated by Byzantine scholars in the early modern period to use as tools for teaching and studying Aristotelian logic. This paper presents pioneering work on employing Byzantine diagrams for checking syllogistic validity through reduction.

This paper seeks to demonstrate that propositions 23–27 of the Euclidian Optics originated in the context of geometrical astronomy. These entries, which deal with the geometry of spheres and rays, present material that overlaps considerably with propositions 1–3 of Aristarchus of Samos’ On the Sizes and Distances of the Sun and the Moon. While all these theorems deal with material that could conceivably be native to celestial illumination, the proofs do not work for binocular vision. It therefore seems probable (...) that the proofs were borrowed from Aristarchus or, more likely, some common astronomical predecessor. As an extension of this observation, this paper argues that the Optics displays a far less unified purpose than has typically been assumed. Instead, it reflects a conglomerate of concerns, goals and dependencies, insofar as it presents multiple proofs that emerge from a set of geometrical concerns not directly related to vision as a physical phenomenon. These entries complicate the idea that the Optics is about explaining light, sight or false appearances in any straightforward sense. Instead, the text is oriented toward a somewhat more diffuse goal: simply to articulate the geometry of vision and rays. Optics as a discipline should should therefore be understood not as a purely mathematical explanation of vision, but as a target-field of explanation constituted by features and boundaries derived from extra-visual geometric concerns. (shrink)

Le motif du langage visuel, qui traverse l'ensemble de l'oeuvre de Berkeley, n'est pas seulement le noyau de sa philosophie de la perception. Il est aussi le préréquisit d'une preuve originale de l'existence de Dieu, une évaluation spécifique de la nature de l'expérience commune et de la portée de l'explication scientifique, et il a des conséquences singulières quant à la doctrine de la création du monde. La première conclusion de l'article est qu'en dépit du rejet berkeleyen du mécanisme, on peut (...) avec raison comparer sa pensée avec la conception cybernétique des machines à information; le langage visuel est, dans une certaine mesure, la source d'un réseau global de communication. Mais rien n'est plus important pour Berkeley que la question de la mort, et c'est pourquoi il doit rejeter toute forme d'objectivisme. L'immatérialiste est celui qui affirme que le monde ne peut pas survivre à son propre anéantissement sans l'aide d'un autre esprit, dont la vision est la substance même de toute chose. The design of visual language, which pervades the entire work of Berkeley, is not only the core of his philosophy of perception. It is also the prerequisite of an original proof of God's existence, a specific understanding of both the nature of common experience and the scope of scientific explanation, and it involves unusual consequences upon the doctrine of the creation of the world. The first conclusion of the paper is that, despite Berkeley's rejection of mechanicism, we have a right to confront his views with the cybernetic perspective on information processing machines : visual language is, to a certain extent, the source of a global communication network. But there is nothing more important to Berkeley than the topical question of death, and this is why he is compelled to rejecting whatever kind of objectivism. The immaterialist is he who acknowledges that the world cannot survive his own annihilation without the help of another spirit, whose vision is the very substance of all things. (shrink)

The project of this Précis de philosophie de la logique et des mathématiques (vol. 1 under the direction of F. Poggiolesi and P. Wagner, vol. 2 under the direction of A. Arana and M. Panza) aims to offer a rich, systematic and clear introduction to the main contemporary debates in the philosophy of mathematics and logic. The two volumes bring together the contributions of thirty researchers (twelve for the philosophy of logic and eighteen for the philosophy of mathematics), specialists in (...) the history or philosophy of logic or mathematics. Each volume consists of ten chapters; each chapter is about forty pages, is independent from the others and deals with a particular philosophical question about logic or mathematics. The objective is to offer to the French-speaking reader a reference book. On many of the issues addressed, this Précis offers the first clear and thorough synthesis that is available in French. -/- This book is intended for third year / master's students, but also for teachers of philosophy in secondary or higher education, for logicians and mathematicians willing to take a philosophical look at the fundamental concepts of their discipline, and for any reader interested in the basic issues discussed in the philosophy of logic and mathematics. Each chapter is self-contained, although some basic training in logic (or in mathematics, as the case may be) is required. The book should be looked at as a comprehensive framework for contemporary discussions in the philosophy of logic and mathematics, while also giving some guidance on the disciplinary content required by these discussions. -/- The first three chapters of the present volume (on the philosophy of mathematics) are devoted to the history of the philosophy of mathematics: from antiquity to the modern era, and from this period to the foundational crisis of the nineteenth century, to the twentieth century. Next, four chapters deal with the crucial questions for the philosophy of mathematics of the twentieth century: the opposition and/or comparison of set theory and category theory as foundational frameworks for mathematics; mathematical constructivism; the analysis of calculability; and Benaceraff's problem. The two following chapters are focused on the philosophy of mathematical practice, by treating the notion of ideals of proof, in particular explanation and purity, and the notion of informal proof and the use of visual artifacts in mathematical argumentation. Finally the last chapter deals with the applicability of mathematics, including the role of probability. -/- In Chapter 1, Sébastien Maronne and David Rabouin present the history of the philosophy of mathematics from antiquity to the modern era. The goal is to show that the philosophy of mathematics has a history as ancient as mathematics and philosophy itself, a history largely continuous with the latter, often by staying out of sync with respect to the former. In Chapter 2, Sébastien Gandon discusses the question of the foundations of mathematics from Kant to the end of the nineteenth century. By tying this question to Kant, he is able to show that the discussion of the foundations of mathematics that occupied a large part of these two centuries has much older roots. Accent is placed on the differences between the framework proposed by Kant for making sense of classical mathematics and the use made by Frege and then Russell of logical tools for giving a different reading of arithmetic and real analysis. The link between between these different conceptions and the evolution of mathematics itself is also underlined. -/- Chapter 3, written by Hourya Benis-Sinaceur and Mirna Džamonja, deals with the evolution of mathematics in the twentieth century, notably the structuralist turn of Dedekind, E. Noether, and Bourbaki, and certain more recent developments. They devote themselves, among other things, to explaining how these developments can be understood as responses to and extensions of questions posed during the foundational debate of the immediately preceding era. -/- In Chapter 4, Jean-Pierre Marquis and Jean-Jacques Szczeciniarz discuss the different foundational options coming from set theory and category theory. They discuss, among other things, Lawvere's program to replace ZF with an axiomatization of the category of sets, the latter judged better adapted to the needs of contemporary mathematics. Different reactions, as much philosophical as technical, raised by this program are also taken into account. -/- Constructive mathematics are the object of Chapter 5, in which Gerhard Heinzmann and Mark van Atten examine in a critical and comparative manner a variety of constructivisms, including intuitionism, constructive type theory, predicativism, and finitism. -/- In Chapter 6, Guido Gherardi and Maël Pegny present computability theory, stressing its philosophical consequences. They discuss the basic concepts of this theory, for the computability of both the natural numbers and the real numbers. They then tackle the theory of computational complexity. -/- In Chapter 7, Andrea Sereni and Fabrice Pataut present Benacerraf's problem, opposing every possible account of mathematical knowledge to the availability of a theory of truth founded on an abstract ontology of mathematical objects. They give an overview of the original problem and its reformulation by Hartry Field, and reconstruct the debate that this problem has raised, by meeting at once the major opposing positions today of the analytic affiliation of the philosophy of mathematics, as much those on the platonist side as on that of nominalism. -/- In Chapter 8, Valeria Giardino and Yacin Hamami treat certain crucial aspects of the philosophy of mathematical practice. They deal in particular with the notion of an informal proof and on the role of artifacts in mathematical reasoning. They consider how such proofs, making use of these means, can contribute to the understanding of theorems and mathematical theories, thanks also to the role of visualisation and the aid of computer tools. -/- Why do mathematics often give several proofs of the same theorem? This is the question that Andrew Arana raises in Chapter 9, introducing the notion of an epistemic ideal and discuss two such ideals, the explanatoriness and purity of proof. -/- Chapter 10 is devoted to the applicability of mathematics in the study of empirical phenomena. After summarizing the history of this question, going back to Plato and Aristotle and passing by Mill and Kant, Daniele Molinini and Marco Panza reconstruct the contemporary debate, in relation, among other things, to questions raised in the philosophy of science. -/- Two appendices complete the volume. In the first Frédéric Patras treats the French tradition in philosophy of mathematics of the 20th century. In the second, Maria Carla Galavotti discusses the notion of probability and its use in science. (shrink)

Hutton made a geological collection to illustrate his theory of the Earth, and frequently cited phenomena displayed by specimens in it to support his arguments. His followers also considered that the evidence provided by the collection would help to establish his views. After Hutton's death it was given to the Royal Society of Edinburgh which, however, under the terms of its charter, was obliged to lodge it in the Natural History Museum of the University. The Museum's curator, the Wernerian, Robert (...) Jameson, thought the collection of little account, and neither displayed it nor kept it together. By the mid-1830s most of it had been dispersed or thrown away, and nothing is known of it since. The list of specimens compiled by its first trustees has also disappeared. By piecing together information in Hutton's writings and letters, and in those of his friends, we learn something about the contents of the collection and how it was formed, and also add to our knowledge of his acquaintance. The final section of the article brings together all Hutton's descriptions of his rocks and minerals and, in doing so, demonstrates the weight of proof that he rested on careful visual analysis of hand specimens. (shrink)

A set of images can be considered as meaningfully different for an observer if they can be distinguished phenomenally from one another. Each phenomenal difference must be supported by some neurophysiological differences. Differentiation analysis aims to quantify neurophysiological differentiation evoked by a given set of stimuli to assess its meaningfulness to the individual observer. As a proof of concept using high-density EEG, we show increased neurophysiological differentiation for a set of natural, meaningfully different images in contrast to another set of (...) artificially generated, meaninglessly different images in nine participants. Stimulus-evoked neurophysiological differentiation (over 257 channels, 800 ms) was systematically greater for meaningful vs. meaningless stimulus categories both at the group level and for individual subjects. Spatial breakdown showed a central-posterior peak of differentiation, consistent with the visual nature of the stimulus sets. Temporal breakdown revealed an early peak of differentiation around 110 ms, prominent in the central-posterior region; and a later, longer-lasting peak at 300–500 ms that was spatially more distributed. The early peak of differentiation was not accompanied by changes in mean ERP amplitude, whereas the later peak was associated with a higher amplitude ERP for meaningful images. An ERP component similar to visual-awareness-negativity occurred during the nadir of differentiation across all image types. Control stimulus sets and further analysis indicate that changes in neurophysiological differentiation between meaningful and meaningless stimulus sets could not be accounted for by spatial properties of the stimuli or by stimulus novelty and predictability. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Two Thought Experiments in the Dissoi LogoiDeborah Levine GeraRecent scholarship has stressed that it is not useful to speak of Greek scientific experimentation in sweeping fashion. The Greeks did perform scientific experiments, but the quantity, quality, and areas explored varied over different periods. Thus, while at certain times such testing procedures flourished, at other times very few actual experiments were performed. So, too, certain fields were more fruitful or (...) feasible for experimentation than others.1 Those who tend to play down the quantity and quality of Greek scientific experimentation usually point to the Greeks' preference for speculation and theory over observation and performance of manual tasks.2 Thought experiments allow the mind to range freely without leaving one's armchair or getting one's hands dirty, and that may be why such mental tests were favored by Greek thinkers, from Xenophanes onward. My purpose here is to study two thought experiments found in the Dissoi Logoi in conjunction with several related trials found in Herodotus, Xenophon, Plato, and Xenophanes. All these thought experiments make use of hypothetical human beings and deal with educational and ethical matters. Generally speaking, these hypothetical tests contrast two types of people, either individuals or larger groups, in an attempt to determine the roles played by heredity and environment in the acquisition of knowledge and moral values.It is notoriously difficult to arrive at a precise definition of thought experiments,3 and the following is meant to be no more than a working definition. Thought experiments are, first of all, experiments.4 In ancient [End Page 21] Greece scientific experiments were not used as a neutral means to decide between two competing theories but rather were intended to prove or refute a hypothesis, and this is true of Greek thought experiments as well.5 Experiments must involve a procedure of some kind, a test consisting of at least one change deliberately introduced into the initial situation.6 Normally, then, an experiment is composed of (1) a thesis to be proven (or refuted), (2) a carefully contrived starting point, (3) at least one further action to be undertaken, an action affecting the original situation, and (4) a test of the results of the action(s). In a thought experiment these steps are carried out in thought, in the laboratory of the mind.7 The thought experimenter visualizes a situation, mentally carries out an operation, and then sees, in the mind's eye, the results. A Gedankenexperiment is not actually performed, either because there is no need to do so—common sense, intuition, or experience is sufficient to supply the results of the procedure—or because the test is one that cannot, in fact, be executed in the real world. Such experiments are not intended to prove fanciful, unlikely theories, nor are they meant to yield results which are intrinsically implausible. Generally, thought experiments are carefully imagined scenarios, which are convincing both because they have some basis in empirical observation and because they use specific details, often colorful ones, to appeal to the audience's intuition.8 These well-conceived tests teach us new things, even though their execution does not supply us with new empirical data of any kind.Let us begin by looking at one such hypothetical trial in the Dissoi Logoi, an anonymous sophistic composition, written mainly in Doric, which is generally dated to the beginning of the fourth century.9 This [End Page 22] brief work is comprised of nine chapters, often presented in the form of double (or opposing) arguments, on subjects such as good and evil, justice and injustice, truth and falsehood, etc. Of these, chapter 6 discusses whether wisdom and virtue can be taught, and here the anonymous author touches upon the acquisition of language, while contrasting systematic teaching with informal instruction. He argues that we learn language and are not born knowing it, and offers an imaginary skeptical opponent the following proof of his claim (, 6.12): if one were to send a newborn Greek infant off to Persia and raise him there, without his hearing Greek, he would speak Persian; and if one were to bring a Persian baby to Greece, he would speak Greek (, 6.12... (shrink)