Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer.

Wagenknecht a récemment introduit une distinction conceptuelle entre dépendance épistémique translucide et dépendance épistémique opaque, dans le but de mieux rendre compte de la diversité des relations de dépendance épistémique au sein des pratiques collaboratives de recherche. Dans la continuité de son travail, mon but est d’expliciter les différents types d’expertise requis lorsque sont employés instruments et ordinateurs dans la production de connaissance, et d’identifier des sources potentielles d’opacité. Mon analyse s’appuie sur un cas contemporain de création de connaissance scientifique, (...) à savoir le traitement de données astrophysiques. (shrink)

This essay explores the following position: Ultimate moral principles are a priori truths; hence, it is irrational to assign a non-zero credence to any proposition that is incompatible with these ultimate moral principles ; and this sort of irrationality, if it could have been avoided, is in a sense inexcusable. So—at least if moral relativism is false—in any disagreement about ultimate moral principles, at least one party to the disagreement is inexcusably irrational. This position may seem extreme, but it is (...) argued that it is more plausible than it first appears. (shrink)

This paper argues that a priori justification is, in principle, compatible with naturalism—if the a priori is understood in a way that is free of the inessential properties that, historically, have been associated with the concept. I argue that empirical indefeasibility is essential to the primary notion of the a priori ; however, the indefeasibility requirement should be interpreted in such a way that we can be fallibilist about apriori-justified claims. This fallibilist notion of the a priori accords with the (...) naturalist’s commitment to scientific methodology in that it allows for apriori-justified claims to be sensitive to further conceptual developments and the expansion of evidence. The fallibilist apriorist allows that an a priori claim is revisable in only a purely epistemic sense. This modal claim is weaker than what is required for a revisability thesis to establish empiricism, so fallibilist apriorism represents a distinct position. (shrink)

What does it mean to trust the results of a computer simulation? This paper argues that trust in simulations should be grounded in empirical evidence, good engineering practice, and established theoretical principles. Without these constraints, computer simulation risks becoming little more than speculation. We argue against two prominent positions in the epistemology of computer simulation and defend a conservative view that emphasizes the difference between the norms governing scientific investigation and those governing ordinary epistemic practices.

In 1977, the first computer-assisted proof of a mathematical theorem was presented by K. Appel and W. Haken. The proof was met with a lot of criticism from both mathematicians and philosophers. In this paper, I present some examples of computer-assisted proofs, including Appel and Haken’s work. Then, I analyze the most famous arguments against the equal acceptance of computer-based and human-based proofs in mathematics and examine the philosophical assumptions behind the presented criticism. In the conclusion, I talk about whether (...) the philosophical assumptions are justified as they are, or one needs to take a specific philosophical position to accept them. (shrink)

Evidential holism begins with something like the claim that “it is only jointly as a theory that scientific statements imply their observable consequences.” This is the holistic claim that Elliott Sober tells us is an “unexceptional observation”. But variations on this “unexceptional” claim feature as a premise in a series of controversial arguments for radical conclusions, such as that there is no analytic or synthetic distinction that the meaning of a sentence cannot be understood without understanding the whole language of (...) which it is a part and that all knowledge is empirical knowledge. This paper is a survey of what evidential holism is, how plausible it is, and what consequences it has. Section 1 will distinguish a range of different holistic claims, Sections 2 and 3 explore how well motivated they are and how they relate to one another, and Section 4 returns to the arguments listed above and uses the distinctions from the previous sections to identify holism's role in each case. (shrink)

The paper argues that the use of epistemic terms, prominently “… knows” and even “… knows a priori/a posteriori” is context-sensitive along several dimensions. Besides the best known dimension of quality of evidence (lower quality for less demanding context, and higher one for more demanding), there is a dimension of depth (shallow justification for superficial evaluation, and deeper justification for deeper probing evaluation contexts). This claim is illustrated by context-dependent ascription of apriority and aposteriority. The argument proposed here focuses upon (...) the status of propositions that are analytic in empirical concepts (like “Whales are animals”). It is a commonplace in epistemology that any analytic proposition (including e-analytic ones) is a priori. The paper claims that propositions analyzing empirical concepts are an interesting counterexample. It develops the following argument: Many such propositions have empirical counterparts that are expressed by the same form-of-words. (E.g., the form of words “Whales are mammals” can express both an e-analytic proposition and an empirical statement.) They normally derive from their empirical counterparts. Beliefs in such propositions, can be explicitly justified either a priori, by pointing out their conceptual, analytic status, or by reverting to their empirical counterparts. In contexts of very superficial evaluation, one may justify such an analytic belief in the first, conceptual way. In most contexts a belief in a proposition analyzing an empirical concept is being justified by appeal to its empirical counterparts. The empirical justification is normally taken as being ultimate. Empirical counterparts are derivationally deeper than the corresponding analytic propositions, and empirical justification is deeper than a priori one as well. Therefore, propositions analyzing empirical concepts are deeply a posteriori and superficially a priori. (shrink)

Participants on both sides of the contemporary debate between reductionism and anti-reductionism about testimony commonly describe testimonial knowledge as knowledge acquired at second hand. I argue that fully appreciating the distinctive sense in which testimonial knowledge is secondhand supports anti-reductionism over reductionism but also that it supports a particular kind of anti-reductionism very different from that typically offered in the literature. Testimonial knowledge is secondhand in the demanding sense of being justified by the authority of a speaker where this requires (...) that epistemic responsibility for meeting challenges to the audience's testimonial knowledge is shared between speaker and audience. The epistemic credentials of testimonial knowledge are in this sense importantly interpersonal. (shrink)

The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.

There are many tangled normative and technical questions involved in evaluating the quality of software used in epidemiological simulations. In this paper we answer some of these questions and offer practical guidance to practitioners, funders, scientific journals, and consumers of epidemiological research. The heart of our paper is a case study of the Imperial College London covid-19 simulator, set in the context of recent work in epistemology of simulation and philosophy of epidemiology.

Epistemology has recently witnessed a number of efforts to rehabilitate rationalism, to defend the existence and importance of a priori knowledge or warrant construed as the product of rational insight or apprehension (Bealer 1987; Bigelow 1992; BonJour 1992, 1998; Burge 1998; Butchvarov 1970; Katz 1998; Plantinga 1993). This effort has sometimes been coupled with an attack on naturalistic epistemology, especially in BonJour 1994 and Katz 1998. Such coupling is not surprising, because naturalistic epistemology is often associated with thoroughgoing empiricism and (...) the rejection of the a priori. In this paper, however, I shall present a conception of naturalistic epistemology that is entirely compatible with a priori justification or knowledge. The resulting conception, I claim, gives us a better appreciation of the respective merits of the rational and the empirical, as well as a better understanding of how moderate epistemological naturalism comports, at least in principle, with moderate rationalism. This paper defends moderate rationalism; but it does not defend everything rationalists have often wanted, only what it is reasonable to grant them. (shrink)

Most explorations of the epistemic implications of Semantic Anti- Individualism (SAI) focus on issues of self-knowledge (first-person au- thority) and/or external-world skepticism. Less explored has been SAIs implications forthe epistemology of reasoning. In this paperI argue that SAI has some nontrivial implications on this score. I bring these out by reflecting on a problem first raised by Boghossian (1992). Whereas Boghos- sians main interest was in establishing the incompatibility of SAI and the a priority of logical abilities (Boghossian 1992: 22), (...) I argue that Boghossians argument is better interpreted as pointing to SAIs implications for the na- ture of discursive justification. (shrink)

We must allow that knowledge can be transmitted. But to allow this is to allow that an individual can know a proposition despite lacking any evidence for it and reaching belief by an unreliable means. So some explanation is required as to how knowledge rather than belief is transmitted. This paper considers two non-individualistic explanations: one in terms of knowledge existing autonomously, the other in terms of it existing as a property of communities. And it attempts to decide what is (...) at issue between these explanations. (shrink)

As a belief source, testimony has long been held by theorists of the mind to play a deeply important role in human cognition. It is unclear, however, just why testimony has been afforded such cognitive importance. We distinguish three suggestions on the matter: the number claim, which takes testimony’s cognitive importance to be a function of the number of beliefs it typically yields, relative to other belief sources; the reliability claim, which ties the importance of testimony to its relative truth-conduciveness; (...) and the scope claim, according to which testimony’s importance is a function of its relative representational power, non-numerically conceived. After laying out these three suggestions, we go on to argue that there is little hope of grounding testimony’s cognitive importance in either the number claim or the reliability claim. We conclude with a tentative exploration of the basis and plausibility of the scope claim. (shrink)

The debate around the notions of a priori knowledge and a posteriori knowledge has proven crucial for the development of many fields in philosophy, such as metaphysics, epistemology, metametaphysics etc. We advocate that the recent debate on the two notions is also fruitful for man-made distributed computing systems and for the epistemic analysis thereof. Following a recently proposed modal and fallibilistic account of a priori knowledge, we elaborate the corresponding concept of a priori belief: We propose a rich taxonomy of (...) types of a priori beliefs and their role for the different agents that participate in the system engineering process, which match the existing view exceedingly well and are particularly promising for explaining and dealing with unexpected behaviors in fault-tolerant distributed systems. Developing such a philosophical foundation will provide a sound basis for eventually implementing our ideas in a suitable epistemic reasoning and analysis framework and, hence, constitutes a mandatory first step for developing methods and tools to cope with the various challenges that emerge in such systems. (shrink)

Jennifer Lackey’s case “Creationist Teacher,” in which students acquire knowledge of evolutionary theory from a teacher who does not herself believe the theory, has been discussed widely as a counterexample to so-called transmission theories of testimonial knowledge and justification. The case purports to show that a speaker need not herself have knowledge or justification in order to enable listeners to acquire knowledge or justification from her assertion. The original case has been criticized on the ground that it does not really (...) refute the transmission theory, because there is still somebody in a chain of testifiers—the person from whom the creationist teacher acquired what she testifies—who knows the truth of the testified statements. In this paper, we provide a kind of pattern for generating counterexample cases, one that avoids objections discussed by Peter Graham and others in relation to such cases. (shrink)

The original proof of the four-color theorem by Appel and Haken sparked a controversy when Tymoczko used it to argue that the justification provided by unsurveyable proofs carried out by computers cannot be a priori. It also created a lingering impression to the effect that such proofs depend heavily for their soundness on large amounts of computation-intensive custom-built software. Contra Tymoczko, we argue that the justification provided by certain computerized mathematical proofs is not fundamentally different from that provided by surveyable (...) proofs, and can be sensibly regarded as a priori. We also show that the aforementioned impression is mistaken because it fails to distinguish between proof search (the context of discovery) and proof checking (the context of justification). By using mechanized proof assistants capable of producing certificates that can be independently checked, it is possible to carry out complex proofs without the need to trust arbitrary custom-written code. We only need to trust one fixed, small, and simple piece of software: the proof checker. This is not only possible in principle, but is in fact becoming a viable methodology for performing complicated mathematical reasoning. This is evinced by a new proof of the four-color theorem that appeared in 2005, and which was developed and checked in its entirety by a mechanical proof system. (shrink)

Tyler Burge famously argues in his 1993 paper "Content Preservation" that it is not only a priori true that we enjoy a prima facie warrant to take what others assert as true, but also that there our warrant to believe what we are told in certain special cases is a priori. So just as our warrant for believing certain mathematical truths might be a priori, so too there are cases of belief through testimony that are a priori. Then in a (...) 2013 Postscript to "Content Preservation" he took it all back. All of Burge's ideas and arguments for and against are interesting and important, and deepen our understanding of testimony, comprehension, warrant, and the a priori. But his ideas and arguments are hard to understand. This paper explains all of ideas and arguments, both for and against, in a clear and comprehensive way. Along the way, Anna-Sara Malmgren's interpretation and criticism of Burge's position is also expounded and criticized. (shrink)