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This chapter investigates deductive practices in what is arguably their main current instantiation, namely practices of mathematical proofs. The dialogical hypothesis delivers a compelling account of a number of features of these practices; indeed, the fictive characters Prover and Skeptic can be viewed as embodied by real-life mathematicians. The chapter includes a discussion of the ontological status of proofs, the functions of proofs, practices of mathematicians such as peer review and collaboration, and a brief discussion of probabilistic and computational proofs. It also discusses three case studies: the reception of Gödel’s incompleteness results, a failed proof of the inconsistency of Peano Arithmetic, and a purported proof of the ABC conjecture.
Throughout this book, deduction has been examined and discussed from many angles and perspectives. However, one question has remained conspicuously unaddressed until now: Is deduction a correct, reliable method for reasoning? In other words, is deduction justified (Dummett, 1978)?
This investigation has focused extensively on the social conditions and factors influencing the emergence of deduction, both historically and ontogenetically. It is thus reasonable to ask whether it offers a social constructivist account of deduction, which in turn has implications for the justification problem. Indeed, on at least some versions of social constructivism, the very question of the correctness of deductive reasoning as a scientific method, understood in absolute terms, is seen as misguided.
This chapter examines the historical roots of deduction in Ancient Greek philosophy and mathematics. It relies extensively on the work of G.E.R. Lloyd and Reviel Netz to argue that dialogical debating practices in a democratic city-state like Athens were causally instrumental for the emergence of the axiomatic-deductive method in mathematics. The same sociocultural political background was decisive for the emergence of practices of dialectic, the kinds of dialogical interactions famously portrayed in Plato’s dialogues. In turn, dialectic provided the background for the emergence of the first fully-fledged theory of deduction in history, namely Aristotle’s syllogistic.
This chapter focuses on the ‘phylogeny’ of deduction, i.e. how deductive reasoning may have emerged given the genetically endowed cognitive apparatus of humans. It discusses reasoning in non-human animals, Mercier and Sperber’s account of the evolution of reasoning, Heyes’ concept of cognitive gadgets, and neurological studies of deductive reasoning. It is argued that the emergence of deduction should not be viewed as genetically encoded in humans but rather as a product of cultural processes, roughly as described by the cognitive gadgets model.
This chapter argues that Aristotle’s syllogistic emerged from a dialectical matrix as well as from considerations pertaining to scientific demonstration and demonstration in mathematics. This means that, even early on, non-dialogical components motivated and were integrated into theories and practices of deduction. The chapter also briefly discusses two other formidable ancient intellectual traditions and their reflections on logic and reasoning, namely the Indian tradition and the Chinese tradition. It is argued that, while these were indeed highly sophisticated, fully-fledged theories of deduction are not to be found in classical Indian or classical Chinese thought.
This chapter defines and introduces the explanandum of the book, i.e. the phenomenon (or phenomena) that it is about: deductive reasoning and argumentation. It presents deduction as having three main characteristics: necessary truth-preservation – which is perhaps the most central one, distinguishing deduction from other forms of inference and argument such as induction and abduction – perspicuity, and belief-bracketing. It also discusses a number of puzzling features of deduction, i.e. philosophical issues pertaining to deduction that remain open questions, as they have not yet been adequately ‘solved.’ These are: the range and scope of deductive reasoning and argumentation, the nature of deductive necessity, and the function(s) of deduction.
This chapter critically discusses the prominent dialogical accounts of logic and deduction proposed by Lorenzen, Hintikka, and Lakatos. It is argued that, while they contain valuable insights, Lorenzen’s dialogical logic and Hintikka’s game-theoretical semantics ultimately both fail to provide a satisfactory philosophical account of logic and deduction in dialogical terms. This critical evaluation then leads to a precise formulation of the dialogical model defended in the book, the Prover–Skeptic model, which is by and large inspired by Lakatos’ ‘proofs and refutations’ model, but with some important modifications.
This chapter retraces the genealogical development of deduction in the Latin and Arabic medieval traditions and in the early modern period, and finally the emergence of mathematical logic in the nineteenth century. It is shown that dialogical conceptions of logic remained pervasive in the Latin medieval tradition, but that they coexisted with other, non-dialogical conceptualizations, in part because of the influence of Arabic logic. In the modern period, however, mentalistic conceptions of logic and deduction became increasingly prominent. The chapter thus explains why we (i.e. twenty-first-century philosophers) have by and large forgotten the dialogical roots of deduction.
This chapter returns to the three main features of deduction defined in Chapter 1 from a cognitive, empirically informed perspective: necessary truth-preservation, perspicuity, and belief-bracketing. It discusses experimental findings that lend support to the dialogical conceptualization of these three features presented in Chapter 4. It also discusses the notion of internalization as formulated by Lev Vygotsky, which allows for an explanation of how deductive practices can also take place in purely mono-agent situations: as an intrapersonal enactment of interpersonal dialogues. The upshot is that framing deductive practices dialogically provides cognitive scaffolding that facilitates the ontogenetic development of deductive reasoning in an individual.
In this chapter, it is argued that what is needed to make progress on the issues described in Chapter 1 is a ‘roots’ approach, i.e. going back to the roots of deduction. The distinction between phylogenetic, ontogenetic, and historical roots is introduced, and it is argued that all three perspectives must be taken into account. The chapter further briefly presents the four main senses in which deduction has dialogical roots treated in this book: philosophical roots, historical roots, cognitive roots, and with respect to mathematical practices.
This chapter presents an overview of experimental work on deductive reasoning, which has shown that human reasoners do not seem to reason spontaneously according to the deduction canons. However, there are also experimental results suggesting that, when tackling deductive tasks in groups, performance comes much closer to the canons. These findings offer a partial vindication of the dialogical conception of deduction insofar as they show that, when given the opportunity to engage in dialogues with others, humans become better deductive reasoners.
This chapter presents a dialogical rationale based on the Prover–Skeptic model for the three main features of deduction identified in Chapter 1: necessary truth-preservation, perspicuity, and belief-bracketing. Moreover, it addresses four important ongoing debates in the philosophy of logic: the normativity of logic, logical pluralism, logical paradoxes, and logical consequence. It is shown that the Prover–Skeptic model provides a promising vantage point to address the questions raised in these debates.