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Quantity is the first category that Aristotle lists after substance. It has extraordinary epistemological clarity: "2+2=4" is the model of a selfevident and universally known truth. Continuous quantities such as the ratio of circumference to diameter of a circle are as clearly known as discrete ones. The theory that mathematics was "the science of quantity" was once the leading philosophy of mathematics. The article looks at puzzles in the classification and epistemology of quantity. 

Written by experts in the field, this volume presents a comprehensive investigation into the relationship between argumentation theory and the philosophy of mathematical practice. Argumentation theory studies reasoning and argument, and especially those aspects not addressed, or not addressed well, by formal deduction. The philosophy of mathematical practice diverges from mainstream philosophy of mathematics in the emphasis it places on what the majority of working mathematicians actually do, rather than on mathematical foundations. / The book begins by first challenging the (...) 

Epistemic theories of objective chance hold that chances are idealised epistemic probabilities of some sort. After giving a brief history of this approach to objective chance, I argue for a particular version of this view, that the chance of an event E is its epistemic probability, given maximal knowledge of the possible causes of E. The main argument for this view is the demonstration that it entails all of the commonlyaccepted properties of chance. For example, this analysis entails that chances (...) 

Contemporary debates about scientific institutions and practice feature many proposed reforms. Most of these require increased efforts from scientists. But how do scientists’ incentives for effort interact? How can scientific institutions encourage scientists to invest effort in research? We explore these questions using a gametheoretic model of publication markets. We employ a base game between authors and reviewers, before assessing some of its tendencies by means of analysis and simulations. We compare how the effort expenditures of these groups interact in (...) 

Mathematicians often speak of conjectures, yet unproved, as probable or wellconfirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) 



I argue that the constitutive aim of belief and the constitutive aim of science are both knowledge. The ‘aim of belief’, understood as the correctness conditions of belief, is to be identified with the product of properly functioning cognitive systems. Science is an institution that is the social functional analogue of a cognitive system, and its aim is the same as that of belief. In both cases it is knowledge rather than true belief that is the product of proper functioning. 



Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as `subjective', `heuristic' or (...) 

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 decisionmaking. 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 (...) 