I argue that the technique Socrates describes in the Meno at 86e-87c allows him to make progress without definitions, even while accepting that definitions are necessary for knowledge. Some contend that the technique involves provisionally accepting a claim. I argue, instead, that it provides a secure biconditional that one can use to reduce the question one cares care about to a new question that one thinks will be easier to answer.

In my dissertation, I present Hermann Cohen's foundation for the history and philosophy of science. My investigation begins with Cohen's formulation of a neo-Kantian epistemology. I analyze Cohen's early work, especially his contributions to 19th century debates about the theory of knowledge. I conclude by examining Cohen's mature theory of science in two works, The Principle of the Infinitesimal Method and its History of 1883, and Cohen's extensive 1914 Introduction to Friedrich Lange's History of Materialism. In the former, Cohen gives (...) an historical and philosophical analysis of the foundations of the infinitesimal method in mathematics. In the latter, Cohen presents a detailed account of Heinrich Hertz's Principles of Mechanics of 1894. Hertz considers a series of possible foundations for mechanics, in the interest of finding a secure conceptual basis for mechanical theories. Cohen argues that Hertz's analysis can be completed, and his goal achieved, by means of a philosophical examination of the role of mathematical principles and fundamental concepts in scientific theories. (shrink)

Plato articulates a deep perplexity about inquiry in ?Meno's Paradox??the claim that one can inquire neither into what one knows, nor into what one does not know. Although some commentators have wrestled with the paradox itself, many suppose that the paradox of inquiry is special to Plato, arising from peculiarities of the Socratic elenchus or of Platonic epistemology. But there is nothing peculiarly Platonic in this puzzle. For it arises, too, in classical Indian philosophical discussions, where it is formulated with (...) great clarity, and analysed in a way that casts it in a new light. We present three treatments of the puzzle in Indian philosophy, as a way of refining and sharpening our understanding of the paradox, before turning to the most radical of the Indian philosophers to tackle it. The Indian philosophers who are optimistic that the paradox can be resolved appeal to the existence of prior beliefs, and to the resources embedded in language to explain how we can investigate, and so move from ignorance to knowledge. Highlighting this structural feature of inquiry, however, allows the pessimist philosopher to demonstrate that the paradox stands. The incoherence of inquiry is rooted in the very idea of aiming our desires at the unknown. Asking questions and giving answers rests on referential intentions targeting objects in a region of epistemic darkness, and so our ?inquiry sceptic? also finds structurally similar forms of incoherence in the pragmatics of interrogative discourse. (shrink)

Analysis has always been at the heart of philosophical method, but it has been understood and practised in many different ways. Perhaps, in its broadest sense, it might be defined as a process of isolating or working back to what is more fundamental by means of which something, initially taken as given, can be explained or reconstructed. The explanation or reconstruction is often then exhibited in a corresponding process of synthesis. This allows great variation in specific method, however. The aim (...) may be to get back to basics, but there may be all sorts of ways of doing this, each of which might be called ‘analysis’. The dominance of ‘analytic’ philosophy in the Englishspeaking world, and increasingly now in the rest of the world, might suggest that a consensus has formed concerning the role and importance of analysis. This assumes, though, that there is agreement on what ‘analysis’ means, and this is far from clear. On the other hand, Wittgenstein's later critique of analysis in the early (logical atomist) period of analytic philosophy, and Quine's attack on the analytic-synthetic distinction, for example, have led some to claim that we are now in a ‘post-analytic’ age. Such criticisms, however, are only directed at particular conceptions of analysis. If we look at the history of philosophy, and even if we just look at the history of analytic philosophy, we find a rich and extensive repertoire of conceptions of analysis which philosophers have continually drawn upon and reconfigured in different ways. Analytic philosophy is alive and well precisely because of the range of conceptions of analysis that it involves. It may have fragmented into various interlocking subtraditions, but those subtraditions are held together by both their shared history and their methodological interconnections. It is the aim of this article to indicate something of.. (shrink)

The author discusses Aristotle’s notion of deliberation and shows that it differs considerably from the model of deliberation as is common in contemporary discussions of free will and moral responsibility. As opposed to the contemporary model, Aristotle’s account does not require that the deliberator has any belief (or lack thereof) concerning the availability of possible courses of action. However, the action chosen by deliberation, before it is performed, is still contingent––i.e. such that it can both be and not be done––and (...) up to us. Moreover, the action’s being up to us can be seen as grounded in our having rational capacities that are necessarily two-sided. This might suggest that the agent can do otherwise than she has decided by deliberation. The author argues that this is not the case: after deliberation, or after forming the relevant desire, the agent can actualize only one arm of her two-sided capacity, and hence, she cannot act differently than as decided by deliberation. If it makes sense to say that she can act differently, it is only because there may occur, in the interval between deliberation and action, some other desire which takes over a role of the decisive factor. (shrink)

The purpose of this paper is to reassess some mathematical examples in Plato’s dialogues which at a first glance may appear to be nothing more than trivial puzzles. In order to provide the necessary background for this analysis, I shall begin by sketching a brief overview of Plato’s mathematical passages and discuss the criteria for aptly selecting them. Second, I shall explain what I mean by ‘mathematical examples,’ and reflect on their function in light of the discussion on παραδείγματα outlined (...) in theSophistand theStatesman. Against this framework, I shall move to a close examination of specific examples drawn from theTheaetetus(154c–55d), theRepublic(523c–24d), and thePhaedo(96d–97b, 100e–101d). By placing these examples in the broader context of pre-Euclidean mathematics, I shall show that their mathematical content is often less elementary than might appear at first sight. Moreover, by placing emphasis on the specific philosophical concerns that motivated their introduction, I shall argue that the examples are not merely nonsensical jokes. Even if their illustrative purpose is not always immediately clear, and even if they can sound playful and bizarre, they in fact fulfil an important function: by virtue of their power to generate wonder or confusion, they serve as exercises and act as a trigger with respect to the difficult philosophical issues they are intended to clarify. (shrink)

In the first book of the Prior Analytics, Aristotle sets out, for the first time in Greek philosophy, a logical system. It consists of a deductive system (I.4-22), meta-logical results (I.23-26), and a method for finding and giving deductions (I.27-29) that can apply in “any art or science whatsoever” (I.30). After this, Aristotle compares this method with Plato’s method of division, a procedure designed to find essences of natural kinds through systematic classification. This critical comparison in APr I.31 raises an (...) interpretive puzzle: how can Aristotle reasonably juxtapose two methods that differ so much in their aims and approach? What can be gained by doing so? Previous interpreters have failed to show how this comparison is legitimate or what important point Aristotle is making. The goal of this paper is to resolve the puzzle. In resolving this puzzle we not only learn more about the relationship be- tween division and the syllogistic in Aristotle. We will also learn something about the motivation for the syllogistic itself, by seeing the role that it plays in his philosophy of science. (shrink)

At least since Socrates, philosophy has been understood as the desire for acquiring a special kind of knowledge, namely wisdom, a kind of knowledge that human beings ordinarily do not possess. According to ancient thinkers this desire may result from a variety of causes: wonder or astonishment, the bothersome or even painful realization that one lacks wisdom, or encountering certain hard perplexities or aporiai. As a result of this basic understanding of philosophy, Greek thinkers tended to regard philosophy as an (...) activity of inquiry (zētēsis) rather than as a specific discipline. Discussions concerning the right manner of engaging in philosophical inquiry – what methodoi or routes of inquiry were best suited to lead one to wisdom – became an integral part of ancient philosophy, as did the question how such manners or modes of inquiry are related to, and differ from, other types of inquiry, for instance medical or mathematical. In this special issue of History of Philosophy & Logical Analysis, we wish to concentrate in particular on ancient modes of inquiry. (shrink)

Observing certain affinities with Plato’s Alcibiades I , this paper argues that a distinction between care (epimeleia ) of the soul and philosophy as its art (technê ) is reflected in Aristotle’s Protrepticus . On the basis of this distinction, it claims that two notions of philosophy can be distinguished in the Protrepticus : philosophy as epistêmê and philosophy as technê . The former has the function of contemplating the truth of nature, and Aristotle praises it as the natural telos (...) of human beings; whereas philosophy as technê helps nature to accomplish the end it designed for human beings. It emerges that according to Aristotle in the Protrepticus philosophy is the art of making oneself coincide with one’s nature as a human being. (shrink)

Most historians and philosophers of philosophy and history of mathematics hold one interpretation or the other of the nature of method of analysis and synthesis in itself and in its historical development. In this paper, I am trying to prove – through three points – that, in fact, there were two understandings of that method in Greek mathematics and philosophy, and which were reflected in Arabic mathematical science and philosophy; this reflection is considered as proof also of this double nature (...) of that method. Thus, we have to rethink the nature of Arabic philosophy systems. (shrink)

The paper examines Posterior Analytics II 11, 94a20-36 and makes three points. (1) The confusing formula ‘given what things, is it necessary for this to be’ [τίνων ὄντων ἀνάγκη τοῦτ᾿ εἶναι] at a21-22 introduces material cause, not syllogistic necessity. (2) When biological material necessitation is the only causal factor, Aristotle is reluctant to formalize it in syllogistic terms, and this helps to explain why, in II 11, he turns to geometry in order to illustrate a kind of material cause that (...) can be expressed as the middle term of an explanatory syllogism. (3) If geometrical proof is viewed as a complex construction built on simpler constructions, it can in effect be described as a case of purely material constitution. (shrink)

This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors tried to naturalize (...) mathematics by relying on evolutionism. But several difficulties arise when we try to do this. This chapter suggests that, in order to naturalize mathematics, it is better to take the method of mathematics to be the analytic method, rather than the axiomatic method, and thus conceive of mathematical knowledge as plausible knowledge. (shrink)

Aristotle’s philosophy of geometry is widely interpreted as a reaction against a Platonic realist conception of mathematics. Here I argue to the contrary that Aristotle is concerned primarily with the methodological question of how universal inferences are warranted by particular geometrical constructions. His answer hinges on the concept of abstraction, an operation of “taking away” certain features of material particulars that makes perspicuous universal relations among magnitudes. On my reading, abstraction is a diagrammatic procedure for Aristotle, and it is through (...) imagined variation of diagrams that one can universalize spatial inferences. In order to shift the discussion of Aristotle’s work on geometry from ontological to methodological questions, I argue in the first section that geometrical properties for Aristotle are necessary accidents of the particulars, canonically diagrams, in which they inhere. I then consider a line of research that shows how diagrams in ancient geometry are not illustrations of the textual proof but are in part constitutive of geometrical inference, an insight I claim can shed light on Aristotle’s philosophical project. Next I substantiate this assertion by giving an alternative interpretation of abstraction according to which it is a diagrammatic procedure that allows a part of a particular diagram to stand in for a universal. In the last section, I consider Aristotle’s account of mathematical cognition, arguing that there is evidence for the view that imagination is necessary both for the construction of figures, and for their presentation as abstractions subject to universal inference. (shrink)

This paper argues that the hypothesis proposed in the Meno is the proposition ‘virtue is good’ alone, and that its epistemic nature is essentially insecure. It has been an object of huge scholarly debate which other hypothesis Socrates posited with regard to the relationship between virtue and knowledge. This debate is, however, misleading in the sense of making us believe that the hypothesis that virtue is good is regarded as a truism in the light of the process of positing a (...) higher hypothesis described in the Phaedo and the Republic. I argue that the hypothesis in question is presented as a result of the earlier discussion of Meno's third definition of virtue, which implies the consequence that morally correct actions are beneficial whether or not accompanied by conventional goods such as wealth and honour. The underlying role of presenting it as a hypothesis is therefore to postpone demonstrating the truth of a controversial thesis which needs a substantial justification. In conclusion, I s... (shrink)

In the nineteenth century, the separation of naturalist or psychological accounts of validity from normative validity came into question. In his 1877 Logical Studies (Logische Studien), Friedrich Albert Lange argues that the basis for necessary inference is demonstration, which takes place by spatially delimiting the extension of concepts using imagined or physical diagrams. These diagrams are signs or indications of concepts' extension, but do not represent their content. Only the inference as a whole captures the objective content of the proof. (...) Thus, Lange argues, the necessity of an inference is independent of psychological accounts of how we grasp the content of a proposition. (shrink)

Scholars ubiquitously refer to the method εξ υποθεσεως, introduced at Meno 86e1-87d8, as a method of hypothesis. In contrast, this paper argues that the method εξ υποθεσεως in Meno is not a hypothetical method. On the contrary, in the Meno passage, υποθεσις means “postulate”, that is, cognitively secure proposition. Furthermore, the method εξ υποθεσεως is derived from the method of geometrical analysis. More precisely, it is derived from the use of geometrical analysis to achieve reduction, that is, reduction of a (...) less tractable problem to a more tractable problem. As such, the method εξ υποθεσεως does not by itself serve to solve problems. (shrink)

Aristotle contrasts episteme and doxa through the key notions of universal and necessary. These notions have played a central role in Aristotle’s characterization of scientific knowledge in the previous chapters of APo. They are not spelled out in APo I.33, but work as a sort of reminder that packs an adequate characterization of scientific knowledge and thereby gives a highly specified context for Aristotle’s contrast between episteme and doxa. I will try to show that this context introduces a contrast in (...) terms of explanatory claims: on the one hand, episteme covers those claims which capture explanatory connections that are universal and necessary and thereby deliver scientific understanding; on the other hand, doxa covers the explanatory attempts that fail at doing so. (shrink)

This argues for a reading of PA I.1, 639b11-640a9 as a continuous argument, which I divide into 3 main sections. Aristotle’s point in the first section is that teleological explanations should precede non-teleological explanations in the order of exposition. His reasoning is that the ends cited in teleological explanations are definitions, and definitions—which are not subject to further explanation—are appropriate starting points, insofar as they prevent explanations from going on ad infinitum. Moreover, I argue that Aristotle proceeds in the following (...) two sections to criticize certain non-teleological accounts offered by his predecessors on the grounds that they are explanatorily defective: I take Aristotle’s complaints to be that those accounts—unlike teleological explanations—neither begin from appropriate starting points nor entail the phenomena that they purport to explain. Along the way, I am defending an alternative way of understanding what "hypothetical necessity" refers to, for Aristotle. Comments are welcome! (shrink)

In Aristotle’s writings, there is a current relationship between investigation and deliberation. This paper will make a reassessment of such relationship and it will try to reject a mere analogical relationship between investigation and deliberation, which, as will be explained, is founded upon a strong distinction between theoretical and practical reason. This paper will try to prove a stronger relationship between investigation and deliberation, showing that there is neither their object nor their rational and cognitive abilities what differentiate one from (...) another, but simply their aim: investigation is the genus of deliberation, and subsequently, a distinction should be made between theoretical and practical investigation or deliberation, the latter being different from the former because its aim is to find the way to reach a given state of affairs, which depends not on the object itself, like in theoretical investigation, but on the agent. (shrink)

The Socratic elenchus is a method of philosophical analysis which Plato largely dropped in his middle and later writings, with two exceptions, Republic 1 and the Theaetetus. But it is a mistake to describe these as elenctic dialogues, which typically seek an analysis of a virtue in terms of necessary and sufficient conditions, by questioning some alleged expert about its essence. Republic 1 does not follow this pattern: Thrasymachus fundamentally objects to such a procedure and the presuppositions underlying it, while (...) Glaucon and Socrates turn to developing their own theories of justice. The Theaetetus is likewise concerned with exploring and testing theories, in this case of knowledge. The Socratic elenchus cannot produce any philosophically interesting theories, let alone establish their truth, but at most refute them. As Plato increasingly sought out such theories, the kind of analysis at issue in the Socratic elenchus came to interest him less. (shrink)

The article examines the relevance of Aristotle’s analysis that concerns the syllogistic figures. On the assumption that Aristotle’s analytics was inspired by the method of geometric analysis, we show how Aristotle used the three terms, when he formulated the three syllogistic figures. So far it has not been appropriately recognized that the three terms — the major, the middle and the minor one — were viewed by Aristotle syntactically and predicatively in the form of diagrams. Many scholars have misunderstood Aristotle (...) in that in the second and third figure the middle term is outside and that in the second figure the major term is next to the middle one, whereas in the third figure it is further from it. By means of diagrams, we have elucidated how this perfectly accords with Aristotle's planar and graphic arrangement. In the light of these diagrams, one can appropriately capture the definition of syllogism as a predicative set of terms. Irrespective of the tricky question concerning the abbreviations that Aristotle himself used with reference to these types of predication, the reconstructed figures allow us better to comprehend the reductions of syllogism to the first figure. We assume that the figures of syllogism are analogous to the figures of categorical predication, i.e., they are specific syntactic and semantic models. Aristotle demanded certain logical and methodological competence within analytics, which reflects his great commitment and contribution to the field. (shrink)

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

This paper re-constructs Plato's ‘philosophy of geometry’ by arguing that he uses a geometrical method of hypothesis in his account of the cosmos’ generation in the Timaeus. Commentators on Plato's philosophy of mathematics often start from Aristotle's report in the Metaphysics that Plato admitted the existence of mathematical objects in-between ( metaxu) Forms and sensible particulars ( Meta. 1.6, 987b14–18). I argue, however, that Plato's interest in mathematics was centred on its methodological usefulness for philosophical inquiry, rather than on questions (...) of mathematical ontology. My key passage of interest is Timaeus’ account of the generation of the primary bodies in the cosmos, i.e. fire, air, water and earth ( Tim. 48b–c, 53b–56c). Timaeus explains the primary bodies’ origin by hypothesising two right-angled triangles as their starting-point ( arkhê) and describing their individual geometrical constitution. This hypothetical operation recalls the hypothetical method which Socrates introduces in the Meno (86e–87b), as well as the use of hypotheses by mathematicians which is described in the Republic (510b–c). Throughout the passage, Timaeus is focussed on explicating the bodies in terms of their formal structure, without however considering the ontological status of the triangles in relation to the physical world. (shrink)

I argue that, in the Prior Analytics, higher and above the well-known ‘reduction through impossibility’ of figures, Aristotle is resorting to a general procedure of demonstrating through impossibility in various contexts. This is shown from the analysis of the role of adunaton in conversions of premises and other demonstrations where modal or truth-value consistency is indirectly shown to be valid through impossibility. Following the meaning of impossible as ‘non-existent’, the system is also completed by rejecting any invalid combinations of terms (...) in deductions or conversions. The notion of impossibility reaches the core of Aristotle's system in the Prior Analytics. On the one hand, the use of adunaton shows that he is following one of the two requisites for demonstrative science formulated in the Posterior Analytics, i.e. to demonstrate that it is impossible for things to be otherwise than stated. On the other hand, that demonstrations through impossibility are rooted in the notion of contradiction supp.. (shrink)

Readers who are introduced to philosophical analysis by reading the early Platonic dialogues may be puzzled to find that Plato, in his middle and late periods, largely abandons the style of analysis characteristic of early Plato, namely, the 'Socratic elenchus'. This paper undertakes to solve the puzzle. In contrast to what is popularly called 'the Socratic method', the elenchus requires that Socrates, the lead investigator, not have a satisfactory answer to his 'What is F-ness?' question. Here is the bind. Part (...) of what motivates the elenctic inquiry is the natural assumption that one cannot identify F-things unless one has a satisfactory analysis of what it is to be F. But to test the adequacy of suggested analyses of F-ness one needs to be able to identify counterexamples. Together these two points present us with a 'catch-22', which is something the 'paradox of inquiry' in the Meno brings out. In the Theaetetus Plato makes clear that, although the elenchus, including Socratic ignorance, can refute philosophical theses arrived at by other means, it cannot, by itself, give birth to viable philosophical theses. Its legitimate role is therefore only propaedeutic. (shrink)