An examination of the emergence of the phenomenon of deductive argument in classical Greek mathematics.

Wittgenstein finished part 1 of the Philosophical Investigations in the spring of 1945. From 1946 to 1949 he worked on the philosophy of psychology almost without interruption. The present two-volume work comprises many of his writings over this period. Some of the remarks contained here were culled for part 2 of the Investigations ; others were set aside and appear in the collection known as Zettel . The great majority, however, although of excellent quality, have hitherto remained unpublished. This bilingual (...) edition of the Remarks on the Philosophy of Psychology presents the first English translation of an essential body of Wittegenstein's work. It elaborates Wittgenstein's views on psychological concepts such as expectation, sensation, knowing how to follow a rule, and knowledge of the sensations of other persons. It also shows strong emphasis on the "anthropological" aspect of Wittgenstein's thought. Philosophers, as well as anthropologists, psychologists, and sociologists will welcome this important publication. (shrink)

Parallelism has been drawn between modes of representation and problem-sloving processes: Diagrams are more useful for brainstorming while symbolic representation is more welcomed in a formal proof. The paper gets to the root of this clear-cut dualistic picture and argues that the strength of diagrammatic reasoning in the brainstorming process does not have to be abandoned at the stage of proof, but instead should be appreciated and could be preserved in mathematical proofs.

Philosophers from Hume, Kant, and Wittgenstein to the recent realists and antirealists have sought to answer the question, What are concepts? This book provides a detailed, systematic, and accessible introduction to an original philosophical theory of concepts that Christopher Peacocke has developed in recent years to explain facts about the nature of thought, including its systematic character, its relations to truth and reference, and its normative dimension. Particular concepts are also treated within the general framework: perceptual concepts, logical concepts, and (...) the concept of belief are discussed in detail. The general theory is further applied in answering the question of how the ontology of concepts can be of use in classifying mental states, and in discussing the proper relation between philosophical and psychological theories of concepts. Finally, the theory of concepts is used to motivate a nonverificationist theory of the limits of intelligible thought. Peacocke treats content as broad rather than narrow, and his account is nonreductive and non-Quinean. Yet Peacocke also argues for an interactive relationship between philosophical and psychological theories of concepts, and he plots many connections with work in cognitive psychology. (shrink)

Proposition I.1 is, by far, the most popular example used to justify the thesis that many of Euclid’s geometric arguments are diagram-based. Many scholars have recently articulated this thesis in different ways and argued for it. My purpose is to reformulate it in a quite general way, by describing what I take to be the twofold role that diagrams play in Euclid’s plane geometry (EPG). Euclid’s arguments are object-dependent. They are about geometric objects. Hence, they cannot be diagram-based unless diagrams (...) are supposed to have an appropriate relation with these objects. I take this relation to be a quite peculiar sort of representation. Its peculiarity depends on the two following claims that I shall argue for: ( i ) The identity conditions of EPG objects are provided by the identity conditions of the diagrams that represent them; ( ii ) EPG objects inherit some properties and relations from these diagrams. (shrink)

The standard reading of Kuhn's philosophy attributes to him the view that the incommensurability of rival theories and theory-ladenness of observation make rational debate about competing paradigms nearly impossible. If this reflects his real view, then he has claimed something prima facie absurd, and easily refuted with historical counter-examples. It is not the incommensurability thesis per se that is easily refutable, but Kuhn's gestelt interpretation of it. The gestalt interpretation, moreover misrepresents his more fundamental ideas on paradigms, and is in (...) itself confused. The incommensurability thesis can be explained and defended without invoking gestalts, and this reconstructed view can be used to show that familiar criticisms, such as those of Davidson and Laudan, and unwelcome endorsements, such as that of Barnes, which are based on the assumption that Kuhn must be an extreme relativist, are all directed at views that he need not, and should not, hold. (shrink)

In Part III of his 1879 logic Frege proves a theorem in the theory of sequences on the basis of four definitions. He claims in Grundlagen that this proof, despite being strictly deductive, constitutes a real extension of our knowledge, that it is ampliative rather than merely explicative. Frege furthermore connects this idea of ampliative deductive proof to what he thinks of as a fruitful definition, one that draws new lines. My aim is to show that we can make good (...) sense of these claims if we read Frege’s notation diagrammatically, in particular, if we take that notation to have been designed to enable one to exhibit the (inferentially articulated) contents of concepts in a way that allows one to reason deductively on the basis of those contents. (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)

In this paper I address the question of whether the fact that our colour experiences have a finer‐grained content than our ordinary colour concepts allow us to represent should be taken as a threat to theories of the conceptual content of experience. In particular, I consider and criticise McDowell's response to that argument and propose a possible development of it. As a consequence, I claim that the role of the argument from the finer‐grained content of experience has to be redefined. (...) In particular, I acknowledge that this problem is helpful in order to bring to the fore the issue of the proper characterisation of the constraints upon the possession conditions of perceptual demonstrative concepts. Yet, I contend that, in light of the foregoing discussion, it is neutral with respect to the dispute between conceptual and non‐conceptual theorists. For that dispute hinges on whether it is possible to have experiences with a certain content independently of having the concepts, which are needed for its canonical specification and not on whether those experiences are conceptualisable in all their finesse of grain. (shrink)

According to a very natural picture of rational belief, we aim to believe only what is true. However, as Bernard Williams used to say, the world does not just inscribe itself onto our minds. Rather, we have to try to figure out what is true from the evidence available to us. To do this, we rely on a set of epistemic rules that tell us in some general way what it would be most rational to believe under various epistemic circumstances. (...) We reason about what to believe; and we do so by relying on a set of rules.... (shrink)

We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.

Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking in (...) mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. He shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. Visual Thinking in Mathematics reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences. Clear and concise throughout, it will appeal to scholars and students of philosophy, mathematics, and psychology, as well as anyone with an interest in mathematical thinking. (shrink)

Wittgenstein finished part 1 of the Philosophical Investigations in the spring of 1945. From 1946 to 1949 he worked on the philosophy of psychology almost without interruption. The present two-volume work comprises many of his writings over this period. Some of the remarks contained here were culled for part 2 of the Investigations; others were set aside and appear in the collection known as Zettel. The great majority, however, although of excellent quality, have hitherto remained unpublished. This bilingual edition of (...) the Remarks on the Philosophy of Psychology presents the first English translation of an essential body of Wittgenstein's work. It elaborates Wittgenstein's views on psychological concepts such as expectation, sensation, knowing how to follow a rule, and knowledge of the sensations of other persons. It also shows strong emphasis on the "anthropological" aspect of Wittgenstein's thought. Philosophers, as well as anthropologists, psychologists, and sociologists will welcome this important publication. (shrink)

This chapter provides a survey of issues about diagrams in traditional geometrical reasoning. After briefly refuting several common philosophical objections, and giving a sketch of diagram-based reasoning practice in Euclidean plane geometry, discussion focuses first on problems of diagram sensitivity, and then on the relationship between uniform treatment and geometrical generality. Here, one finds a balance between representationally enforced unresponsiveness (to differences among diagrams) and the intellectual agent's contribution to such unresponsiveness that is somewhat different from what one has come (...) to expect in modern logic. Finally, challenges and opportunities for further work are indicated. (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.

This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is necessary (...) for this to work. Case-branching occurs when a construction renders a diagram un-representative. The roles of diagrams in reductio arguments, and of objection in probing a demonstration, are discussed. (shrink)