Pascal’s Wager does not exist in a Platonic world of possible gods, abstract probabilities and arbitrary payoffs. Real decision-makers, such as Pascal’s “man of the world” of 1660, face a range of religious options they take to be serious, with fixed probabilities grounded in their evidence, and with utilities that are fixed quantities in actual minds. The many ingenious objections to the Wager dreamed up by philosophers do not apply in such a real decision matrix. In the situation Pascal addresses, (...) the Wager is a good bet. In the situation of a modern Western intellectual, the reasoning of the Wager is still powerful, though the range of options and the actions indicated are not the same as in Pascal’s day. (shrink)

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...) are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

The focus of this study is cognitive choice: the selection of one cognitive option (a hypothesis, a theory, or an axiom, for instance) rather than another. The study proposes that cognitive choice should be based on the plausibilities of states posited by rival cognitive options and the utilities of these options' information outcomes. The proposal introduces a form of decision theory that is novel because comparative; it permits many choices among cognitive options to be based on merely comparative plausibilities and (...) utilities. This form of decision theory intersects with recommendations by advocates of decision theory for cognitive choice, on the one hand, and defenders of comparative evaluation of scientific hypotheses and theories, on the other. But it differs from prior decision-theoretic proposals because it requires no more than minimal precision in specifying plausibilities and utilities. And it differs from comparative proposals because none has shown how comparative evaluations can be carried out within a decision-theoretic framework. (shrink)

The purpose of this paper is to discuss the role of financial practice in the development of mathematics as applied in human judgement. The basis of the paper is in historical research from the 1990s that argues that the monetisation of western commerce, which abstracted value into quantified price, was synthesised with scholastic analysis resulting in a “mathematical mechanistic world picture” that led to the widespread use of mathematics in science from the seventeenth century. An aspect of this process was (...) the quantification of chance that led to the development of mathematical probability, the branch of mathematics most relevant to judgement and the focus of this paper. Ideas from this historical research are related to the fundamental theorem of asset pricing, the foundational theory of contemporary financial mathematics. The paper observes that vestiges of medieval scholastic attitudes to financial ethics can be discerned in the FTAP, offering a novel interpretation of the mathematical theorem. The paper then considers the Dutch book argument, the most popular justification for subjective probability. The paper’s main contribution is in describing the significance of financial practice in validating the DBA and the paper explains how the FTAP addresses some criticisms of how the DBA represents beliefs. The conclusions emphasise the distinction between pure and practical reasoning and that this should be mirrored in a distinction between the mathematics of physica and practica. This point is important as mathematics is becoming more widespread in modelling, and directing, social systems. (shrink)

This paper argues that the subsistence of the fundamental theorem of contemporary financial mathematics is the ethical concept ‘reciprocity’. The argument is based on identifying an equivalence between the contemporary, and ostensibly ‘value neutral’, Fundamental Theory of Asset Pricing with theories of mathematical probability that emerged in the seventeenth century in the context of the ethical assessment of commercial contracts in a framework of Aristotelian ethics. This observation, the main claim of the paper, is justified on the basis of results (...) from the Ultimatum Game and is analysed within a framework of Pragmatic philosophy. The analysis leads to the explanatory hypothesis that markets are centres of communicative action with reciprocity as a rule of discourse. The purpose of the paper is to reorientate financial economics to emphasise the objectives of cooperation and social cohesion and to this end, we offer specific policy advice. (shrink)

In this article we explore an argumentative pattern that provides a normative justification for expected utility functions grounded on empirical evidence, showing how it worked in three different episodes of their development. The argument claims that we should prudentially maximize our expected utility since this is the criterion effectively applied by those who are considered wisest in making risky choices (be it gamblers or businessmen). Yet, to justify the adoption of this rule, it should be proven that this is empirically (...) true: i.e. that a given function allows us to predict the choices of that particular class of agents. We show how expected utility functions were introduced and contested in accordance with this pattern in the 18th century and how it recurred in the 1950s when Allais made his case against the neo-Bernoullians. (shrink)

In actuarial parlance, the price of an insurance policy is considered fair if customers bearing the same risk are charged the same price. The estimate of this fair amount hinges on the expected value obtained by weighting the different claims by their probability. We argue that, historically, this concept of actuarial fairness originates in an Aristotelian principle of justice in exchange (equality in risk). We will examine how this principle was formalized in the 16th century and shaped in life insurance (...) during the following two hundred years, in two different interpretations. The Domatian account of actuarial fairness relied on subjective uncertainty: An agreement on risk was fair if both parties were equally ignorant about the chances of an uncertain event. The objectivist version grounded any agreement on an objective risk estimate drawn from a mortality table. We will show how the objectivist approach collapsed in the market for life annuities during the 18th century, leaving open the question of why we still speak of actuarial fairness as if it were an objective expected value. (shrink)

This paper explores two persistent questions in the literature on presumptions: the place and the nature of presumptions in law and legal argumentation. These questions were originally raised by James Bradley Thayer, one of the masters of the Law of Evidence and the author of the classic chapter devoted this subject in A preliminary treatise on Evidence. Like Thayer, I believe that these questions deserve attention. First the paper shows that the connection between presumptions and argumentation is a constant feature (...) in the literature on presumptions, since its foundation in the Middle Ages to modern times. James Bradley Thayer was probably the last jurist who clearly saw that presumptions belong to argumentation. Second, the paper examines two sources of controversy in the language of presumptions. First, “presumption” is an ambiguous word in the legal discourse. As a result, it is almost impossible to provide a clear and succinct answer to the question “What is a presumption?”. Second, there are at least four reconstructions of the concept of presumption whose merits and shortcomings are relevant to explore. The analysis presented here may be of interest for legal and non-legal scholars and, hopefully, it may help to shed light on the possibilities and limits of an interdisciplinary dialogue about presumptions. (shrink)

Review of James Franklin, The Science of Conjecture: Evidence and Probability Before Pascal (Johns Hopkins University Press, 2001).

The late scholastics, from the fourteenth to the seventeenth centuries, contributed to many fields of knowledge other than philosophy. They developed a method of conceptual analysis that was very productive in those disciplines in which theory is relatively more important than empirical results. That includes mathematics, where the scholastics developed the analysis of continuous motion, which fed into the calculus, and the theory of risk and probability. The method came to the fore especially in the social sciences. In legal theory (...) they developed, for example, the ethical analyses of the conditions of validity of contracts, and natural rights theory. In political theory, they introduced constitutionalism and the thought experiment of a “state of nature”. Their contributions to economics included concepts still regarded as basic, such as demand, capital, labour, and scarcity. Faculty psychology and semiotics are other areas of significance. In such disciplines, later developments rely crucially on scholastic concepts and vocabulary. (shrink)

According to epistemic fallibilism, we cannot be certain of anything. According to the Christian tradition, faith comes with certainty. I develop this dilemma from recent accounts of fallibilism and various representatives of the Christian tradition. I then argue that on John Henry Newman's account of faith the dilemma is merely apparent. Finally, I develop Newman's account of the certainty that accompanies faith and is compatible with fallibilism.

The central thesis of this book is that the argument that probability is insufficient to handle uncertainty in artificial intelligence (AI) is metaphysical in nature. Piscopo calls this argument against probability the non-adequacy claim and provides this summary of it [which first appeared in (Piscopo and Birattari 2008)]:Probability theory is not suitable to handle uncertainty in AI because it has been developed to deal with intrinsically stochastic phenomena, while in AI, uncertainty has an epistemic nature. (Piscopo (3))Piscopo uses the term (...) “metaphysical” in the Popperian sense: the non-adequacy claim is metaphysical (rather than scientific) since it cannot be disproved (much less established) by observation or experiment.In Chapter 2, Piscopo recounts some of the relevant history and issues in philosophy of science. There is little that is controversial here. One problem is that when addressing the question of how a particular probabilistic model is tested, Piscopo limits herself .. (shrink)

I address the philosophical debate over whether the mathematical theory of probability arose on the basis of empirical observations or of purely theoretical speculations. The debate tends to pose a strict dichotomy between empirical problem-solving and pure theorizing. I alternatively suggest that, in the case of mathematical probability, an empirical problem-context acted as an enabling condition for the possibility of mathematical innovation, but that the activity of the early mathematical probabilists gradually became the study of a theoretical system of ideas. (...) This case has some implications for a more general philosophical view on the context of mathematical discovery. ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ----------------------------------------. (shrink)

This volume presents mathematical game theory as an interface between logic and philosophy.

Inductive generalization asserts that what obtains in known instances can be generalized to all. Its original form is enumerative induction, the earliest form of inductive inference, and it has been elaborated in various ways, largely with the goal of extending its reach. Its principal problem is that it supplies no intrinsic notion of strength of support so that one cannot tell if the generalization has weak or strong support.

methods that have shown promise for improving extreme risk analysis, particularly for assessing the risks of invasive pests and pathogens associated with international trade. We describe the legally inspired regulatory regime for banks, where these methods have been brought to bear on extreme ‘operational risks’. We argue that an ‘advocacy model’ similar to that used in the Basel II compliance regime for bank operational risks and to a lesser extent in biosecurity import risk analyses is ideal for permitting the diversity (...) of relevant evidence about invasive species to be presented and soundly evaluated. We recommend that the process be enhanced in ways that enable invasion ecology to make more explicit use of the methods found successful.. (shrink)

"Does torture work?" is a factual rather than ethical or legal question. But legal and ethical discussions of torture should be informed by knowledge of the answer to the factual question of the reliability of torture as an interrogation technique. The question as to whether torture works should be asked before that of its legal admissibility—if it is not useful to interrogators, there is no point considering its legality in court.

Aristotle divided arguments that persuade into the rhetorical (which happen to persuade), the dialectical (which are strong so ought to persuade to some degree) and the demonstrative (which must persuade if rightly understood). Dialectical arguments were long neglected, partly because Aristotle did not write a book about them. But in the sixteenth and seventeenth century late scholastic authors such as Medina, Cano and Soto developed a sound theory of probable arguments, those that have logical and not merely psychological force but (...) fall short of demonstration. Informed by late medieval treatments of the law of evidence and problems in moral theology and aleatory contracts, they considered the reasons that could render legal, moral, theological, commercial and historical arguments strong though not demonstrative. At the same time, demonstrative arguments became better understood as Galileo and other figures of the Scientific Revolution used mathematical proof in arguments in physics. Galileo moved both dialectical and demonstrative arguments into mathematical territory. (shrink)

Fifty years of effort in artificial intelligence (AI) and the formalization of legal reasoning have produced both successes and failures. Considerable success in organizing and displaying evidence and its interrelationships has been accompanied by failure to achieve the original ambition of AI as applied to law: fully automated legal decision-making. The obstacles to formalizing legal reasoning have proved to be the same ones that make the formalization of commonsense reasoning so difficult, and are most evident where legal reasoning has to (...) meld with the vast web of ordinary human knowledge of the world. Underlying many of the problems is the mismatch between the discreteness of symbol manipulation and the continuous nature of imprecise natural language, of degrees of similarity and analogy, and of probabilities. (shrink)