This paper is a response to Paul Bartha’s ‘Making Do Without Expectations’. We provide an assessment of the strengths and limitations of two notable extensions of standard decision theory: relative expectation theory and Paul Bartha’s relative utility theory. These extensions are designed to provide intuitive answers to some well-known problems in decision theory involving gaps in expectations. We argue that both RET and RUT go some way towards providing solutions to the problems in question but neither extension solves all the (...) relevant problems. (shrink)

We sometimes joke about the world conspiring against us. For instance, I might suggest that it will rain today because I’m without my umbrella or that my football team will play poorly this weekend because I’m attending the game. But of course the weather is not influenced by my umbrella, and football teams do not perform any differently when I’m present. Any perceived correlations here are most likely illusory or, if real, mere accidents. In other words, the probability of a (...) fine day and the probability of my team playing well are independent of my actions. (shrink)

This chapter contains section titled: Introduction Laws in Ecology Mathematical Models What is the Reason for Population Cycles? The Balance of Nature Debate Socio‐Political Aspects of Population Ecology Ecology and Evolution Acknowledgments References Further Reading.

In this paper we explore the connections between ethics and decision theory. In particular, we consider the question of whether decision theory carries with it a bias towards consequentialist ethical theories. We argue that there are plausible versions of the other ethical theories that can be accommodated by “standard” decision theory, but there are also variations of these ethical theories that are less easily accommodated. So while “standard” decision theory is not exclusively consequentialist, it is not necessarily ethically neutral. Moreover, (...) even if our decision-theoretic models get the right answers vis-`a-vis morally correct action, the question remains as to whether the motivation for the non-consequentialist theories and the psychological processes of the agents who subscribe to those ethical theories are lost or poorly represented in the resulting models. (shrink)

The main focus of the book is the presentation of the 'inertial' view of population growth. This view provides a rather simple model for complex population dynamics, and is achieved at the level of the single species without invoking species interactions. An important part of this account is the maternal effect. Investment of mothers in the quality of their daughters makes the rate of reproduction of the current generation depend not only on the current environment, but also on the environment (...) experienced by the previous generation. (shrink)

Recently Colin Cheyne and Charles Pigden have challenged supporters of mathematical indispensability arguments to give an account of how causally inert mathematical entities could be indispensable to science. Failing to meet this challenge, claim Cheyne and Pigden, would place Platonism in a no win situation: either there is no good reason to believe in mathematical entities or mathematical entities are not causally inert. The present paper argues that Platonism is well equipped to meet this challenge.

This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.

ON DECEMBER 10, 1991 Charles Shonubi, a Nigerian citizen but a resident of the USA, was arrested at John F. Kennedy International Airport for the importation of heroin into the United States.1 Shonubi's modus operandi was ``balloon swallowing.'' That is, heroin was mixed with another substance to form a paste and this paste was sealed in balloons which were then swallowed. The idea was that once the illegal substance was safely inside the USA, the smuggler would pass the balloons and (...) recover the heroin. On the date of his arrest, Shonubi was found to have swallowed 103 balloons containing a total of 427.4 grams of heroin. There was little doubt about Shonubi's guilt. In fact, there was considerable evidence that he had made at least seven prior heroin-smuggling trips to the USA (although he was not tried for these). In October 1992 Shonubi was convicted in a United States District Court for possessing and importing heroin. Although the conviction was only for crimes associated with Shonubi's arrest date of December 10, 1991, the sentencing judge, Jack B. Weinstein, also made a ®nding that Shonubi had indeed made seven prior drug-smuggling trips to the USA. The interesting part of this case was in the sentencing. According to the federal sentencing guidelines, the sentence in cases such as this should depend on the total quantity of heroin involved. This instruction was interpreted rather broadly.. (shrink)

Mathematics has a great variety ofapplications in the physical sciences.This simple, undeniable fact, however,gives rise to an interestingphilosophical problem:why should physical scientistsfind that they are unable to evenstate their theories without theresources of abstract mathematicaltheories? Moreover, theformulation of physical theories inthe language of mathematicsoften leads to new physical predictionswhich were quite unexpected onpurely physical grounds. It is thought by somethat the puzzles the applications of mathematicspresent are artefacts of out-dated philosophical theories about thenature of mathematics. In this paper I argue (...) that this is not so.I outline two contemporary philosophical accounts of mathematics thatpay a great deal of attention to the applicability of mathematics and showthat even these leave a large part of the puzzles in question unexplained. (shrink)

This chapter contains sections titled: A Lover's Question The Game of Love Where Did Our Love Go? Love is Strange.

What do submarine attacks, ant trails, and dating have in common? Not much, except that they are all instances of pursuit and evasion problems and all submit to elegant mathematical treatments. The mathematics involved in such problems is varied and interesting in its own right, but the applications breathe life into the mathematics and invite wider engagement—as the intense interest of the military in such problems, especially during wartime, demonstrates. Consider the problem of a submarine commander about to fire on (...) an enemy warship. The enemy warship, of course, refuses to co operate, and the warship keeps up its course across the open sea. The submarine commander needs a strategy to ensure that the torpedoes arrive where the warship will be by the time the torpedoes arrive. How does the commander do this, without knowing the warship’s course? After all, there is no guarantee that the warship will travel in a straight line or at a constant speed. One way for the submarine commander to proceed is to keep the torpedo always aimed at the ship, instantaneously adjusting its direction to changes in the warship’s position. If this is done, the torpedo will (typically) trace out an arc and (typically) strike the ship from behind. This is called a pure pursuit strategy and is important in many military applications (such as Second World War air battles). (shrink)

ABSTRACTIn the target article ‘If-Thenism’, Stephen Yablo develops a novel form of if-thenism, that appeals to the notion of logical subtraction. In this commentary, I explore the limits of Yablo's proposed subtraction procedure, by leaning on an analogy with photographic subtraction. In particular, I will argue that there will be cases when there's nothing interesting left after the subtraction and, as a consequence, there are serious limits to the applicability of the subtraction procedure.

Indispensability arguments for realism about mathematical entities have come under serious attack in recent years. To my mind the most profound attack has come from Penelope Maddy, who argues that scientific/mathematical practice doesn't support the key premise of the indispensability argument, that is, that we ought to have ontological commitment to those entities that are indispensable to our best scientific theories. In this paper I defend the Quine/Putnam indispensability argument against Maddy's objections.

In this paper I examine Quine''s indispensability argument, with particular emphasis on what is meant by ''indispensable''. I show that confirmation theory plays a crucial role in answering this question and that once indispensability is understood in this light, Quine''s argument is seen to be a serious stumbling block for any scientific realist wishing to maintain an anti-realist position with regard to mathematical entities.

Hartry Field has shown us a way to be nominalists: we must purge our scientific theories of quantification over abstracta and we must prove the appropriate conservativeness results. This is not a path for the faint hearted. Indeed, the substantial technical difficulties facing Field's project have led some to explore other, easier options. Recently, Jody Azzouni, Joseph Melia, and Stephen Yablo have argued that it is a mistake to read our ontological commitments simply from what the quantifiers of our best (...) scientific theories range over. In this paper, I argue that all three arguments fail and they fail for much the same reason; would-be nominalists are thus left facing Field's hard road. (shrink)

The Pasadena paradox presents a serious challenge for decision theory. The paradox arises from a game that has well-defined probabilities and utilities for each outcome, yet, apparently, does not have a well-defined expectation. In this paper, I argue that this paradox highlights a limitation of standard decision theory. This limitation can be (largely) overcome by embracing dominance reasoning and, in particular, by recognising that dominance reasoning can deliver the correct results in situations where standard decision theory fails. This, in turn, (...) pushes us towards pluralism about decision rules. (shrink)

Various real and imagined criminal law cases rest on “naked statistical evidence”. That is, they rest more or less entirely on a probability for guilt/liability derived from a single statistical model. The intuition is that there is something missing in these cases, high as the probability for guilt/liability may be, such that the relevant standard for legal proof is not met. Here we contribute to the considerable debate about how this intuition is best explained and what it teaches us about (...) evidential reasoning in the legal setting. We part ways with the recent scholarship, however. Unlike most others, our diagnosis is not that there is an important qualitative property that some evidence or bodies of evidence have, and that naked statistical evidence, most strikingly, lacks. Rather, we see cases resting on naked statistical evidence as lying at the extreme of a continuum of cases that are vulnerable to challenge due to “meta-uncertainty” about the underlying model. (shrink)

This paper considers a generalisation of the sorites paradox, in which only topological notions are employed. We argue that by increasing the level of abstraction in this way, we see the sorites paradox in a new, more revealing light—a light that forces attention on cut-off points of vague predicates. The generalised sorites paradox presented here also gives rise to a new, more tractable definition of vagueness.

This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from both the contemporary literature and (...) older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics. (shrink)

Journal of Political Philosophy, EarlyView.

How is that when scientists need some piece of mathematics through which to frame their theory, it is there to hand? What has been called 'the unreasonable effectiveness of mathematics' sets a challenge for philosophers. Some have responded to that challenge by arguing that mathematics is essentially anthropocentric in character, whereas others have pointed to the range of structures that mathematics offers. Otavio Bueno and Steven French offer a middle way, which focuses on the moves that have to be made (...) in both the mathematics and the relevant physics in order to bring the two into appropriate relation. This relation can be captured via the inferential conception of the applicability of mathematics, which is formulated in terms of immersion, inference, and interpretation. In particular, the roles of idealisations and of surplus structure in science and mathematics respectively are brought to the fore and captured via an approach to models and theories that emphasize the partiality of the available information: the partial structures approach. The discussion as a whole is grounded in a number of case studies drawn from the history of quantum physics, and extended to contest recent claims that the explanatory role of certain mathematical structures in scientific practice supports a realist attitude towards them. The overall conclusion is that the effectiveness of mathematics does not seem unreasonable at all once close attention is paid to how it is actually applied in practice. (shrink)

This paper examines the paradox of revisability. This paradox was proposed by Jerrold Katz as a problem for Quinean naturalised epistemology. Katz employs diagonalisation to demonstrate what he takes to be an inconsistency in the constitutive principles of Quine's epistemology. Specifically, the problem seems to rest with the principle of universal revisability which states that no statement is immune to revision. In this paper it is argued that although there is something odd about employing universal revisability to revise itself, there (...) is nothing paradoxical about this. At least, there is no paradox along the lines suggested by Katz. (shrink)

A number of people have recently argued for a structural approach to accounting for the applications of mathematics. Such an approach has been called "the mapping account". According to this view, the applicability of mathematics is fully accounted for by appreciating the relevant structural similarities between the empirical system under study and the mathematics used in the investigation ofthat system. This account of applications requires the truth of applied mathematical assertions, but it does not require the existence of mathematical objects. (...) In this paper, we discuss the shortcomings of this account, and show how these shortcomings can be overcome by a broader view of the application of mathematics: the inferential conception. (shrink)

The Quine-Putnam indispensability argument urges us to place mathematical entities on the same ontological footing as (other) theoretical entities of empirical science. Recently this argument has attracted much criticism, and in this paper I address one criticism due to Elliott Sober. Sober argues that mathematical theories cannot share the empirical support accrued by our best scientific theories, since mathematical propositions are not being tested in the same way as the clearly empirical propositions of science. In this paper I defend the (...) Quine-Putnam argument against Sober's objections. (shrink)

The standard mathematical models in population ecology assume that a population's growth rate is a function of its environment. In this paper we investigate an alternative proposal according to which the rate of change of the growth rate is a function of the environment and of environmental change. We focus on the philosophical issues involved in such a fundamental shift in theoretical assumptions, as well as on the explanations the two theories offer for some of the key data such as (...) cyclic populations. We also discuss the relationship between this move in population ecology and a similar move from first-order to second-order differential equations championed by Galileo and Newton in celestial mechanics. (shrink)

In this paper we argue that there is a kind of moral disagreement that survives the Rawlsian veil of ignorance. While a veil of ignorance eliminates sources of disagreement stemming from self-interest, it does not do anything to eliminate deeper sources of disagreement. These disagreements not only persist, but transform their structure once behind the veil of ignorance. We consider formal frameworks for exploring these differences in structure between interested and disinterested disagreement, and argue that consensus models offer us a (...) solution concept for disagreements behind the veil of ignorance. (shrink)

In this paper I reply to Jody Azzouni, Otávio Bueno, Mary Leng, David Liggins, and Stephen Yablo, who offer defences of so-called ‘ easy road ’ nominalist strategies in the philosophy of mathematics.

The Eleatic Principle or causal criterion is a causal test that entities must pass in order to gain admission to some philosophers’ ontology.1 This principle justifies belief in only those entities to which causal power can be attributed, that is, to those entities which can bring about changes in the world. The idea of such a test is rather important in modern ontology, since it is neither without intuitive appeal nor without influential supporters. Its supporters have included David Armstrong (1978, (...) Vol 2, 5), Brian Ellis (1990, 22) and Hartry Field2 (1989, 68) to name but a few. (shrink)

One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. (...) Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics. (shrink)

David Malament argued that Hartry Field's nominalisation program is unlikely to be able to deal with non-space-time theories such as phase-space theories. We give a specific example of such a phase-space theory and argue that this presentation of the theory delivers explanations that are not available in the classical presentation of the theory. This suggests that even if phase-space theories can be nominalised, the resulting theory will not have the explanatory power of the original. Phase-space theories thus raise problems for (...) nominalists that go beyond Malament's initial concerns. Thanks to Mark Steiner, Jens Christian Bjerring, Ben Fraser, John Mathewson, and two anonymous referees for helpful comments on an earlier draft of this paper. CiteULike Connotea Del.icio.us What's this? (shrink)

Games such as the St. Petersburg game present serious problems for decision theory.1 The St. Petersburg game invokes an unbounded utility function to produce an infinite expectation for playing the game. The problem is usually presented as a clash between decision theory and intuition: most people are not prepared to pay a large finite sum to buy into this game, yet this is precisely what decision theory suggests we ought to do. But there is another problem associated with the St. (...) Petersburg game. The problem is that standard decision theory counsels us to be indifferent between any two actions that have infinite expected utility. So, for example, consider the decision problem of whether to play the St. Petersburg game or a game where every payoff is $1 higher. Let’s call this second game the Petrograd game (it’s the same as St. Petersburg but with a bit of twentieth century inflation). Standard decision theory is indifferent between these two options. Indeed, it might be argued that any intuition that the Petrograd game is better than the St. Petersburg game is a result of misguided and na¨ıve intuitions about infinity.2 But this argument against the intuition in question is misguided. The Petrograd game is clearly better than the St. Petersburg game. And what is more, there is no confusion about infinity involved in thinking this. When the series of coin tosses comes to an end (and it comes to an end with probability 1), no matter how many tails precede the first head, the payoff for the Petrograd game is one dollar higher than the St. Petersburg game. Whatever the outcome, you are better off playing the Petrograd game. Infinity has nothing to do with it. Indeed, a straightforward application of dominance reasoning backs up this line of reasoning.3 Standard decision theory. (shrink)

In philosophy of logic and elsewhere, it is generally thought that similar problems should be solved by similar means. This advice is sometimes elevated to the status of a principle: the principle of uniform solution. In this paper I will explore the question of what counts as a similar problem and consider reasons for subscribing to the principle of uniform solution.

The argument from fine tuning is supposed to establish the existence of God from the fact that the evolution of carbon-based life requires the laws of physics and the boundary conditions of the universe to be more or less as they are. We demonstrate that this argument fails. In particular, we focus on problems associated with the role probabilities play in the argument. We show that, even granting the fine tuning of the universe, it does not follow that the universe (...) is improbable, thus no explanation of the fine tuning, theistic or otherwise, is required. (shrink)

Standard approaches to counterfactuals in the philosophy of explanation are geared toward causal explanation. We show how to extend the counterfactual theory of explanation to non-causal cases, involving extra-mathematical explanation: the explanation of physical facts by mathematical facts. Using a structural equation framework, we model impossible perturbations to mathematics and the resulting differences made to physical explananda in two important cases of extra-mathematical explanation. We address some objections to our approach.

A common response to those who question the Law of Non-Contradiction is that it is impossible to debate such a fundamental law of logic. The reasons for this response vary, but what seems to underlie them is the thought that there is a minimal set of logical resources without which rational debate is impossible. This chapter argues that this response is misguided. First, it defends non-apriorism in logic: the view that logic is in the same epistemic boat as other scientific (...) theories. It then offers an account of logical theory change in terms of this epistemology. The LNC is discussed in terms of this account of logical theory change, and it is shown that rational debate over this law can, and does, proceed. Finally, arguments for and against the LNC are discussed, and how and where non-a priori considerations arise in these arguments are illustrated. (shrink)

The Quine-Putnam Indispensability argument is the argument for treating mathematical entities on a par with other theoretical entities of our best scientific theories. This argument is usually taken to be an argument for mathematical realism. In this chapter I will argue that the proper way to understand this argument is as putting pressure on the viability of the marriage of scientific realism and mathematical nominalism. Although such a marriage is a popular option amongst philosophers of science and mathematics, in light (...) of the indispensability argument, the marriage is seen to be very unstable. Unless one is careful about how the Quine-Putnam argument is disarmed, one can be forced to either mathematical realism or, alternatively, scientific instrumentalism. I will explore the various options: (i) finding a way to reconcile the two partners in the marriage by disarming the indispensability argument (Jody Azzouni [2], Hartry Field [13, 14], Alan Musgrave [18, 19], David Papineau [21]); (ii) embracing mathematical realism (W.V.O. Quine [23], Michael Resnik [25], J.J.C. Smart [27]); and (iii) embracing some form of scientific instrumentalism (Ot´ avio Bueno [7, 8], Bas van Fraassen [30]). Elsewhere [11], I have argued for option (ii) and I won’t repeat those arguments here. Instead, I will consider the difficulties for each of the three options just mentioned, with special attention to option (i). In relation to the latter, I will discuss an argument due to Alan Musgrave [19] for why option (i) is a plausible and promising approach. From the discussion of Musgrave’s argument, it will emerge that the issue of holist versus separatist theories of confirmation plays a curious role in the realism–antirealism debate in the philosophy of mathematics. I will argue that if you take confirmation to be an holistic matter—it’s whole theories (or significant parts thereof) that are confirmed in any experiment—then there’s an inclination to opt for (ii) in order to resolve the marital tension outlined above.. (shrink)