Complete inferential rigour is achieved by breaking down arguments into steps that are as small as possible: inferential ‘atoms’. For example, a mathematical or philosophical argument may be made completely inferentially rigorous by decomposing its inferential steps into the type of step found in a natural deduction system. It is commonly thought that atomization, paradigmatically in mathematics but also more generally, is pro tanto epistemically valuable. The paper considers some plausible candidates for the epistemic value arising from atomization and finds (...) that none of them fits the bill. In particular, atomized arguments do not ground the correctness of their unatomized counterparts; they are typically not credence-preserving; and they need not reveal the source of inferential disagreement. The moral this suggests is that complete rigour is not even a defeasible epistemic ideal. (shrink)

Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical instrumentalism (...) are defeated by Gödel’s theorem, not all are. By considering inductive reasons in mathematics, we show that some mathematical instrumentalisms survive the theorem. (shrink)

Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathematical proposition. We also show that (...) some of what mathematicians take to be deductive knowledge is in fact non-deductive. 1 Introduction2 Why It Might Matter3 Two Further Examples and Preliminaries4 An Exclusive Epistemic Virtue of Proof?5 Analyses of Knowledge6 The Inductive Basis of Deduction7 Physical to Mathematical Linkages8 Conclusion. (shrink)

This paper proposes an account of why proof is the only way to acquire knowledge of some mathematical proposition’s truth. Admittedly, non-deductive arguments for mathematical propositions can be strong and play important roles in mathematics. But this paper proposes a necessary condition for knowledge that can be satisfied by putative proofs (and proof sketches), as well as by non-deductive arguments in science, but not by non-deductive arguments from mathematical evidence. The necessary condition concerns whether we can justly expect that if (...) the mathematical proposition is actually false, despite the evidence for its truth, then this fact has an explanation. (shrink)

Easwaran has given a definition of transferability and argued that, under this definition, randomized arguments are not transferable. I show that certain aspects of his definition are not suitable for addressing the underlying question of whether or not there is an epistemic distinction between randomized and deductive arguments. Furthermore, I demonstrate that for any suitable definition, randomized arguments are in fact transferable.

In “How to Collaborate,” Paul Thagard tries to explain why there is so much collaboration in science, and so little collaboration in philosophy, by giving an epistemic cost-benefit analysis. In this paper, I argue that an adequate explanation requires a more fully developed epistemic value theory than Thagard utilizes. In addition, I offer an alternative to Thagard’s explanation of the lack of collaboration in philosophy. He appeals to its lack of a tradition of collaboration and to the a priori nature (...) of much philosophical research. I claim that philosophers rarely collaborate simply because they can usually get the benefits without paying the costs of actually collaborating. (shrink)

In a series of papers, Don Fallis points out that although mathematicians are generally unwilling to accept merely probabilistic proofs, they do accept proofs that are incomplete, long and complicated, or partly carried out by computers. He argues that there are no epistemic grounds on which probabilistic proofs can be rejected while these other proofs are accepted. I defend the practice by presenting a property I call ‘transferability’, which probabilistic proofs lack and acceptable proofs have. I also consider what this (...) says about the similarities between mathematics and, on the one hand natural sciences, and on the other hand philosophy. (shrink)

This paper investigates an important aspect of mathematical practice: that proof is required for a finished piece of mathematics. If follows that non-deductive arguments — however convincing — are never sufficient. I explore four aspects of mathematical research that have facilitated the impressive success of the discipline. These I call the Practical Virtues: Permanence, Reliability, Autonomy, and Consensus. I then argue that permitting results to become established on the basis of non-deductive evidence alone would lead to their deterioration. This furnishes (...) us with a partial rational justification for mathematicians strict insistence on proof. (shrink)

In his article Open Problems in the Philosophy of Information [1] Luciano Floridi presented a Philosophy of Information research program in the form of eighteen open problems, covering the following fundamental areas: Information definition, information semantics, intelligence/cognition, informational universe/nature and values/ethics. We revisit Floridi’s program, highlighting some of the major advances, commenting on unsolved problems and rendering the new landscape of the Philosophy of Information (PI) emerging at present. As we analyze the progress of PI we try to situate Floridi’s (...) program in the context of scientific and technological development that have been made last ten years. We emphasize that Philosophy of Information is a huge and vibrant research field, with its origins dating before Open Problems, and its domains extending even outside their scope. In this paper, we have been able only to sketch some of the developments during the past ten years. Our hope is that, even if fragmentary, this review may serve as a contribution to the effort of understanding the present state of the art and the paths of development of Philosophy of Information as seen through the lens of Open Problems. (shrink)

In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character, although they do not fall squarely under a traditional philosophical purview. The goal of this article is to try to articulate some of these questions more clearly, and assess the philosophical methods that may be brought to bear. In Section 3, I note that most of the issues can be classified under two headings: some (...) deal with the ability of computers to deliver appropriate “evidence” for mathematical assertions, a notion that is explored in Section 4, while others deal with the ability of computers to deliver appropriate mathematical “understanding,” a notion that is considered in Section 5. Final thoughts are provided in Section 6. (shrink)

In his article Open Problems in the Philosophy of Information 1 Luciano Floridi presented a Philosophy of Information research program in the form of eighteen open problems, covering the following fundamental areas: Information definition, information semantics, intelligence/cognition, informational universe/nature and values/ethics. We revisit Floridis program, highlighting some of the major advances, commenting on unsolved problems and rendering the new landscape of the Philosophy of Information emerging at present. As we analyze the progress of PI we try to situate Floridis program (...) in the context of scientific and technological development that have been made last ten years. We emphasize that Philosophy of Information is a huge and vibrant research field, with its origins dating before Open Problems, and its domains extending even outside their scope. In this paper, we have been able only to sketch some of the developments during the past ten years. Our hope is that, even if fragmentary, this review may serve as a contribution to the effort of understanding the present state of the art and the paths of development of Philosophy of Information as seen through the lens of Open Problems. (shrink)