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)

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)

Relying upon a very close reading of all of the definitions given in Euclid’s Elements, I argue that this mathematical treatise contains a philosophical treatment of mathematical objects. Specifically, I show that Euclid draws elaborate metaphysical distinctions between substances and non-substantial attributes of substances, different kinds of substance, and different kinds of non-substance. While the general metaphysical theory adopted in the Elements resembles that of Aristotle in many respects, Euclid does not employ Aristotle’s terminology, or indeed, any philosophical terminology at (...) all. Instead, Euclid systematically uses different types of definition to distinguish between metaphysically different kinds of mathematical object. (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)

On the basis of a wide range of historical examples various features of axioms are discussed in relation to their use in mathematical practice. A very general framework for this discussion is provided, and it is argued that axioms can play many roles in mathematics and that viewing them as self-evident truths does not do justice to the ways in which mathematicians employ axioms. Possible origins of axioms and criteria for choosing axioms are also examined. The distinctions introduced aim at (...) clarifying discussions in philosophy of mathematics and contributing towards a more refined view of mathematical practice. (shrink)

In order to understand Hegel's approach to philosophy, we need to ask why, and how, he reacts to the well-known criticism of German Romantics, like Novalis and Friedrich Schlegel, against philosophical system building in general, and against Kant's system in particular. Hegel's encyclopedic system is a topical ordering of categorically different ontological realms, corresponding to different conceptual forms of representation and knowledge. All in all it turns into a systematic defense of Fichte's doctrine concerning the primacy of us as actors (...) with respect to any knowledge claim or scientific theory. Hegel's limitations of the principle of causality and of the possibility of using mathematical methods in science show, in fact, how a merely compatibilist solution of Kant's third antinomy can be overcome. (shrink)

In this paper I show that proofs by contradiction were a serious problem in seventeenth century mathematics and philosophy. Their status was put into question and positive mathematical developments emerged from such reflections. I analyse how mathematics, logic, and epistemology are intertwined in the issue at hand. The mathematical part describes Cavalieri's and Guldin's mathematical programmes of providing a development of parts of geometry free of proofs by contradiction. The logical part shows how the traditional Aristotelean doctrine that perfect demonstrations (...) are causal demonstrations influenced the reflection on proofs by contradiction. The main protagonist of this part is Wallis. Finally, I analyse some epistemological developments arising from the Cartesian tradition. In particular, I look at Arnauld's programme of providing an epistemologically motivated reformulation of Geometry free of proofs by contradiction. The conclusion explains in which sense these epistemological reflections can be compared with those informing contemporary intuitionism. (shrink)

In her book Abductive Reasoning Atocha Aliseda stresses the attention to the logical models of abduction, centering on the semantic tableaux as a method for extending and improving both the whole cognitive/philosophical view on it and on other more restricted logical approaches. I will describe the importance of increasing logical knowledge on abduction also taking advantage of some ideas coming from the so-called distributed cognition where logical models are seen as forms of cognitive externalizations of preexistent in-formal human reasoning performances.

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)

In this paper, I aim to address a specific issue underpinning Cartesian metaphysics since its first public appearance in the Discourse right up until the Meditations, but which definitely came to the surface in the Second and Fifth Replies. It involves the possibility that to be thinking and to be extended do not actually contrast as two entirely different properties; hence, these two essences cannot serve as the basis for a disjunctive, real distinction between two corresponding substances, the mind and (...) the body. I dub this problem the ‘thinking matter issue.’ I suggest that Descartes’s concerns about the ‘thinking matter issue’ characterizes and structures the entirety of Meditation Two and its connection with Meditation Six, especially in the attempt to covertly implement what I refer to as Descartes’s ‘prejudice strategy.’ The core of the ‘prejudice strategy’ lies in the idea that the ‘thinking matter issue’ is just a false problem, one raised by the inadequate notions of the mind and the body that we apply to this problem. In section 1, I set out the way in which the ‘thinking matter issue’ emerged in Descartes’s philosophy after the first exposition of his metaphysics in the Discourse. Section 2 deals with the new argumentative path Descartes draws for the Meditations and with the new role that he assigns to the inference about the mind’s non-physical nature after the cogito. In section 3, I contend that the ‘prejudice strategy’ structures Meditation Two and, partially, Six. Section 4 shows that Descartes himself reveals this in his Replies to the Second and the Fifth Set of Objections. In section 5, I delve into Descartes’s foundational theory of the ‘prejudice strategy,’ i.e. his theory of infancy as set out especially in the Replies to the Sixth Set of Objections. -/- . (shrink)

It is shown how the historiographic purport of Lakatosian methodology of mathematics is structured on the theme of analysis and synthesis. This theme is explored and extended to the revolutionary phase around 1800. On the basis of this historical investigation it is argued that major innovations, crucial to the appraisal of mathematical progress, defy reconstruction as irreducibly rational processes and should instead essentially be understood as processes of social-cognitive interaction. A model of conceptual change is developed whose essential ingredients are (...) the variability of rational responses to new intellectual and practical challenges arising in the cultural environment of mathematics, and the shifting selective pressure of society. The resulting view of mathematical development is compared with Kuhn's theory of scientific paradigms in the light of some personal communications. (shrink)

(1986). The method of analysis‐synthesis and the structure of causal explanation in Newton. International Studies in the Philosophy of Science: Vol. 1, No. 1, pp. 60-84.

In this article I attempt to reconstruct David Hume's use of the label ?experimental? to characterise his method in the Treatise. Although its meaning may strike the present-day reader as unusual, such a reconstruction is possible from the background of eighteenth-century practices and concepts of natural inquiry. As I argue, Hume's inquiries into human nature are experimental not primarily because of the way the empirical data he uses are produced, but because of the way those data are theoretically processed. He (...) seems to follow a method of analysis and synthesis quite similar to the one advertised in Newton's Opticks, which profoundly influenced eighteenth-century natural and moral philosophy. This method brings him much closer to the methods of qualitative, chemical investigations than to mechanical approaches to both nature and human nature. (shrink)

This article argues that early modern philosophy should be seen as an integrated enterprise of moral and natural philosophy. Consequently, early modern moral and natural philosophy should be taught as intellectual enterprises that developed hand in hand. Further, the article argues that the unity of these two fields can be best introduced through methodological ideas. It illustrates these theses through a case study on Scottish Newtonianism, starting with visions concerning the unity of philosophy and then turning to a discussion of (...) how methodological ideas figure in those visions. Finally, the article argues that methodological considerations can serve as good starting points to introduce and discuss central topics and canonical figures of the early modern period. (shrink)

In this paper, we provide a detailed critical review of current approaches to ecthesis in Aristotle’s Prior Analytics, with a view to motivate a new approach, which builds upon previous work by Marion & Rückert (2016) on the dictum de omni. This approach sets Aristotle’s work within the context of dialectic and uses Lorenzen’s dialogical logic, hereby reframed with use of Martin-Löf's constructive type theory as ‘immanent reasoning’. We then provide rules of syllogistic for the latter, and provide proofs of (...) e-conversion, Darapti and Bocardo and e-subalternation, while showing how close to Aristotle’s text these proofs remain. (shrink)

Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, on the (...) epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (shrink)

The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method (...) of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Gödel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable. (shrink)

This paper addresses the black-box problem in artificial intelligence (AI), and the related problem of explainability of AI in the legal context. We argue, first, that the black box problem is, in fact, a superficial one as it results from an overlap of four different – albeit interconnected – issues: the opacity problem, the strangeness problem, the unpredictability problem, and the justification problem. Thus, we propose a framework for discussing both the black box problem and the explainability of AI. We (...) argue further that contrary to often defended claims the opacity issue is not a genuine problem. We also dismiss the justification problem. Further, we describe the tensions involved in the strangeness and unpredictability problems and suggest some ways to alleviate them. (shrink)

In this paper I describe the birth of Russell’s notion of a propositional function on 3 May 1902 and its immediate context and implications. In particular, I consider its significance in relation to the development of his views on analysis.

The logic of discovering and that of justifying have been a permanent source of debate in mathematics, because of their different and apparently contradictory features within the processes of production of mathematical sentences. In fact, a fundamental unity appears as soon as one investigates deeply the phenomenology of conjecturing and proving using concrete examples. In this paper it is shown that abduction, in the sense of Peirce, is an essential unifying activity, ruling such phenomena. Abduction is the major ingredient in (...) a theoretical model suitable for describing the tran-sition from the conjecturing to the proving phase. In the paper such a model is introduced and worked out to test Lakatos' machinery of proofs and refutations from a new point of view. Abduction and its categorical counter-part, adjunction, allow to explain within a unifying framework most of the phenomenology of conjectures and proofs, encompassing also the method of Greek analysis-synthesis. (shrink)

This article focusses on the generality of the entities involved in a geometric proof of the kind found in ancient Greek treatises: it shows that the standard modern translation of Greek mathematical propositions falsifies crucial syntactical elements, and employs an incorrect conception of the denotative letters in a Greek geometric proof; epigraphic evidence is adduced to show that these denotative letters are ‘letter-labels’. On this basis, the article explores the consequences of seeing that a Greek mathematical proposition is fully general, (...) and the ontological commitments underlying the stylistic practice. (shrink)

In her book Abductive Reasoning Atocha Aliseda (2006) stresses the attention to the logical models of abduction, centering on the semantic tableaux as a method for extending and improving both the whole cognitive/philosophical view on it and on other more restricted logical approaches. I will provide further insight on two aspects. The first is re-lated to the importance of increasing logical knowledge on abduction: Aliseda clearly shows how the logical study on abduction in turn helps us to extend and modernize (...) the classical and received idea of logic. The second refers to some ideas coming from the so-called distributed cognition and concerns the role of logical models as forms of cognitive exter-nalizations of preexistent in-formal human reasoning performances. The logical externalization in objective systems, communicable and sharable, is able to grant stable perspectives endowed with symbolic, abstract, and rigorous cogni-tive features. I will also emphasize that Aliseda especially stresses that this character of stability and objectivity of logical achievements are not usually present in models of abduction that are merely cognitive and epistemological, and of ex-treme importance from the computational point of view. (shrink)

Leibniz frequently argued that reasons are to be weighed against each other as in a pair of scales, as Professor Marcelo Dascal has shown in his article "The Balance of Reason." In this kind of weighing it is not necessary to reach demonstrative certainty – one need only judge whether the reasons weigh more on behalf of one or the other option However, a different kind of account about rational decision-making can be found in some of Leibniz's writings. In his (...) article "Was Leibniz's Deity an Akrates?" Professor Jaakko Hintikka has argued that Leibniz developed a new vectorial model for rational decisions which is better suited to complicated decisions, where values are complementary to each other. This model, related closely to his work in metaphysics and the philosophy of mind, is a heuristic device which helps in finding rational combinations - and in an ideal case an optimum - between plural inclinations to the good. I shall argue that Leibniz applies more or less implicitly both of these models in his practical rationality. In simple situations he applied the pair of scales model and in more complicated situations he applied the vectorial model. (shrink)

In the disciplines related to the design of products and services, such as New Product Development and Design Science, there is a lack of a commonly accepted theoretical and methodical basis. This papers starts with the proposition that the ancient method of analysis and synthesis, developed originally by Greek geometers, is the basis of models that have been used to classify and describe the ill structured design problem. In this paper, we examine the possibility of improving our understanding of the (...) design process and therefore lean design management by bringing to light a discussion about the concepts of analysis and synthesis and how these have been interpreted through time. Also, how this concept has been used within engineering design methods. To do so, we investigate how analysis and synthesis have been understood in the literature, indicating similarities and differences between ancient and current understandings. (shrink)