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)
“Offsetting” habitat destruction has widespread appeal as an instrument for balancing economic growth with biodiversity conservation. Requiring proponents to pay the nontrivial costs of habitat loss encourages sensitive planning approaches. Offsetting, biobanking, and biodiverse carbon sequestration schemes will play an important role in conserving biodiversity under increasing human pressures. However, untenable assumptions in existing schemes are undermining their beneﬁts. Policies that allow habitat destruction to be offset by the protection of existing habitat are guaranteed to result in further loss of (...) biodiversity. Similarly, schemes that allow trading the immediate loss of existing habitat for restoration projects that promise future habitat will, at best, result in time lags in the availability of habitat that increases extinction risks, or at worst, fail to achieve the offset at all. We detail concerns about existing approaches and describe how offsetting and trading policies can be improved to provide genuine beneﬁts for biodiversity. Due to uncertainties about the way in which restored vegetation matures, we propose that the biodiversity bank should be a savings bank. Accrued biodiversity values should be demonstrated before they can be used to offset biodiversity losses. We provide recommendations about how this could be achieved in practice. (shrink)
Yablo’s paradox is generated by the following (infinite) list of sentences (called the Yablo list): (s1) For all k > 1, sk is not true. (s2) For all k > 2, sk is not true. (s3) For all k > 3, sk is not true. . . . . . . . .
At various times, mathematicians have been forced to work with inconsistent mathematical theories. Sometimes the inconsistency of the theory in question was apparent (e.g. the early calculus), while at other times it was not (e.g. pre-paradox na¨ıve set theory). The way mathematicians confronted such difficulties is the subject of a great deal of interesting work in the history of mathematics but, apart from the crisis in set theory, there has been very little philosophical work on the topic of inconsistent mathematics. (...) In this paper I will address a couple of philosophical issues arising from the applications of inconsistent mathematics. The first is the issue of whether finding applications for inconsistent mathematics commits us to the existence of inconsistent objects. I then consider what we can learn about a general philosophical account of the applicability of mathematics from successful applications of inconsistent mathematics. (shrink)
Time is perceived very differently from different vantage points. A year in the life of a primary-school student, for instance, is a very long time—somewhere between 1/5 and 1/ 12 of a primary-school child’s life. When you tlirow in the massive amount a child learns in any one year, compared with the diminishing returns that conspire against us later in life, a child’s year is more like a decade in adult years. But for a primary-school teacher, a school year is (...) just another ten or so months spent trying to remember names, delivering lessons, writing report cards, and endeavoring to shepherd students through the educational system. And, of course, the classroom is just one part of a teacher’s life. Teachers are juggling other serious and timeconsuming matters such as relationships, mortgages, further study, family and so forth. The best teachers, however, don’t let on about these asymmetries between the child’s world and their own; they conceal the differences in temporal perception and they give no clue that each student is just a small part of their life. Such sleight of hand can’t be easy, yet all my primary school teachers pulled it off to perfection. One, however, deserves special mention: Mr. Williams. In 1969 I turned 11 and was in 6th class at the Armidale Demonstration.. (shrink)
The business of Selective Realism, is to distinguish the denoting terms from the nondenoting terms in our best scientific theories. This is no easy matter, and despite agreement amongst many philosophers of science that at least some of our scientific vocabulary denotes and some does not, there is very little agreement about how the demarcation in question is to be affected.1 One strategy that enjoys fairly widespread support, however, is the appeal to a causal test.2 According to this view, the (...) only terms that are taken to denote are those concerning causally active entities. Such an approach rules against the existence of abstracta but maintains realism about theoretical entities such as electrons and the like. In this paper I will consider one important defence of such a test due to David Armstrong. I argue that this defence fails because of its reliance on the assumption that only causally active entities can have explanatory power.. (shrink)
I was interested to read Greg Pritchard’s articles ‘Civilised Lands’ in past issues of your magazine. In general, I think he gave a good overview of places of interest and tips for an overseas visitor on a climbing holiday to Australia. He failed, however, to warn visitors of the Australian pastime of sandbagging (which, I might add, Mr. Pritchard is a deft exponent of himself). I don’t know what state sandbagging has reached in your country but in Australia it has (...) become a refined art form. It is for this reason that I feel compelled to write this article, so as to warn otherwise unsuspecting English climbers of the possible pitfalls (or perhaps ground-falls would be more appropriate here) they may face on an Australian holiday. Firstly, an outline of what sandbagging is. The basic aim of the game is for the bagger to get an unsuspecting victim (also known as a baggee or sometimes a bunny) to do a climb (which, in the context of the game, is known as “the bag”) which they (the baggee) will not enjoy for one or a combination of three basic reasons. The first is that the climb may be positively dangerous, either loose and/or unprotected. Secondly, the climb may be much harder than either the grade or appearances indicate. Thirdly, the climb may be simply repulsive; for example, it might look like a nice diagonal layback flake and yet turn out to be a smooth flared overhanging off-width. The baggee, naturally enough, has a really miserable time and of course this gives the bagger great delight—in perfect accord with Toadie’s Law of the Conservation of Happiness (“happiness can be neither created nor destroyed”). Now, different baggers operate with different emphasis on each of the three basic bagging criteria, and have as their goal different levels of baggee misery. For example, there are those extremists, operating largely on criterion one, who are not completely happy unless they have witnessed a ground-fall from a reasonable height, in which case, if you are the baggee, your Australian holiday may consist of a few weeks in traction in Natimuk Bush Nursing Hospital.. (shrink)
It seems one can’t open a climbing magazine these days without encountering a barrage of duty statements such as “It is wrong to retro-bolt” or “It is wrong to bolt a new route too close to a naturally protected route”. Such statements are often referred to as examples of ethical debate, however, as we shall see, they are more properly referred to as moral debate. The distinction is not just a pedantic piece of linguistics either, it is, I believe, essential (...) to understanding the true nature of these disputes, and it is the nature of these disputes which I am concerned with in this article. The distinction between ethics and morality was first brought to my attention in an article by Dr Green called ‘The Ethics of Climbing’ in Screamer 9 (1981). In this article Dr Green explains how ‘ethics’ derives from the Greek ‘ethikos’ which pertains to the spirit of the thing in question, so the ethics of climbing are concerned with the spirit of climbing. ‘Morality’, on the other hand, is is derived from the Latin ‘moralis’ pertaining to right conduct, so a morality is a set of commands, usually used to encapsulate a particular ethic. Dr Green goes on to suggest that just as many Christians’ obsession with the ten commandments is symptomatic of a failure by those individuals to grasp the spirit or ethos of Christianity, so too modern rockclimbers’ obsession with the morals of climbing signifies a shift in the ethos of climbing. I wish to examine this claim a little more closely. I believe that the legitimate role of morals is in the teaching of ethics. For example, it would be difficult to teach a child the abstract ethos of “caring for your fellow human being’s welfare” without first giving some concrete examples in the form of commands such as “Don’t hit other children at school”. It is by learning these moral commands.. (shrink)
Environmental ethics concerns itself with ethical issues arising from the relationship between humans and the natural environment. Of particular interest are ethical considerations in relation to human efforts to conserve the natural environment. Some of the key environmental ethics issues are whether environmental value is intrinsic or instrumental, whether biodiversity is valuable in itself or whether it is an indicator of some other value(s), and what the appropriate time scale is for conservation planning. But there is much more to environmental (...) philosophy than environmental ethics. For a start, environmental philosophy covers a whole raft of issues in philosophy of science such as the role of mathematical models in population ecology,1 the relationship between the stability of ecosystems and the complexity of those ecosystems,2 the representation and treatment of uncertainty in ecological and conservation biology applications,3 and whether ecology has laws4. None of these issues has anything to do with ethics. But there is another sense in which environmental philosophy is much broader than environmental ethics: even in relation to topics where there are value or ethical issues, there are other philosophical issues that we would do well to disentangle from the ethics. I will argue that it is a mistake to think of the philosophical issues in question as merely environmental ethics. (shrink)
Evidence-based policy is gaining support in many areas of government and in public affairs more generally. In this paper we outline what evidence—based policy is then discuss its strengths and weaknesses. In particular, we argue that it faces a serious challenge to provide a plausible account of evidence. This account needs to be at least in the spirit of the hierarchy of evidence subscribed to by evidence-based medicine (from which evidence—based policy derives its name and inspiration). Yet evidence-based policy’s hierarchy (...) needs to be tailored to the kinds of evidence relevant and available to the policy arena. The evidence required for policy decisions does not easily lend itself to randomised controlled trials (the "gold standard" in evidence-based medicine), nor, for that matter, being listed in a single all—purpose hierarchy. (shrink)
In this article, I discuss an argument that purports to prove that probability theory is the only sensible means of dealing with uncertainty. I show that this argument can succeed only if some rather controversial assumptions about the nature of uncertainty are accepted. I discuss these assumptions and provide reasons for rejecting them. I also present examples of what I take to..
There has been a long history of discussion on the usefulness of formal methods in legal settings.1 Some of the recent debate has focussed on foundational issues in statistics, in particular, how the reference-class problem affects legal decisions based on certain types of statistical evidence.2 Here we examine aspects of this debate, stressing why the reference-class problem presents serious difficulties for the kinds of statistical inferences under consideration and the relevance of this for the use of statistics in the courtroom. (...) We also consider the relevance of foundational statistical issues in the broader context of.. (shrink)
It is often assumed that empiricism in the philosophy of mathematics was laid to rest by Frege’s stinging attack on Mill. I will argue that empiricism is alive and well and able to deal with almost everything that’s thrown at it. In particular, I will show how the brand of empiricism I subscribe to is able to give a satisfying account of mathematical knowledge. This brand of mathematical empiricism has a rather curious feature though: some parts of mathematics (e.g., analysis, (...) modern algebra, ZFC set theory) are taken to be theories about which we have genuine mathematical knowledge, while others (e.g., set theory with large cardinal axioms) are (following Quine) treated as “mathematical recreation”. I will defend this demarcation against some recent criticisms from Mary Leng. (shrink)
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In this book Noel Curran suggests that considerations in the philosophy of mathematics—in particular, the proper interpretation of quaternions—leads to a “new” philosophy of space and time. According to Curran: space is Euclidean; time is absolute, flows and has a beginning; and God created the universe at the beginning of time.
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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 , Hartry Field [13, 14], Alan Musgrave [18, 19], David Papineau ); (ii) embracing mathematical realism (W.V.O. Quine , Michael Resnik , J.J.C. Smart ); and (iii) embracing some form of scientific instrumentalism (Ot´ avio Bueno [7, 8], Bas van Fraassen ). Elsewhere , 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  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)
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)
The recent publication of a couple of guidebooks to some of the many crags around Armidale (in the New England area of northern New South Wales) has resulted in a bit of interest from outof-towners. (So far guides have been published on Dome Wall and Moonbi, arguably the best two crags in the district.) This article aims to give a bit of inside information on some of the climbs and, hopefully, entice some new blood (and splintered bone) to the area. (...) Fortunately, however, from your point of view as baggees and potential baggees, Armidale is a very good place to be— there are no scary routes and everything is fairly graded (particularly if you are in the back bar of the Wicklow Hotel). This, naturally enough, makes my job as bagger very difficult; however, I’ll do my best. (shrink)
With Fermat’s Last Theorem finally disposed of by Andrew Wiles in 1994, it’s only natural that popular attention should turn to arguably the most outstanding unsolved problem in mathematics: the Riemann Hypothesis. Unlike Fermat’s Last Theorem, however, the Riemann Hypothesis requires quite a bit of mathematical background to even understand what it says. And of course both require a great deal of background in order to understand their significance. The Riemann Hypothesis was first articulated by Bernhard Riemann in an address (...) to the Berlin Academy in 1859. The address was called “On the Number of Prime Numbers Less Than a Given Quantity” and among the many interesting results and methods contained in that paper was Riemann’s famous hypothesis: all non-trivial zeros of the zeta function, ζ(s) = ∞ n=1 n−s, have real part 1/2. Although the zeta function as stated and considered as a real-valued function is defined only for s > 1, it can be suitably extended. It can, as a matter of fact, be extended to have as its domain all the complex numbers (numbers of the form x + yi, where x and y √ −1) with the exception of 1 + 0i (at which point are real numbers and i =. (shrink)
Cox’s theorem states that, under certain assumptions, any measure of belief is isomorphic to a probability measure. This theorem, although intended as a justification of the subjectivist interpretation of probability theory, is sometimes presented as an argument for more controversial theses. Of particular interest is the thesis that the only coherent means of representing uncertainty is via the probability calculus. In this paper I examine the logical assumptions of Cox’s theorem and I show how these impinge on the philosophical conclusions (...) thought to be supported by the theorem. I show that the more controversial thesis is not supported by Cox’s theorem. (shrink)
In many of the special sciences, mathematical models are used to provide information about specified target systems. For instance, population models are used in ecology to make predictions about the abundance of real populations of particular organisms. The status of mathematical models, though, is unclear and their use is hotly contested by some practitioners. A common objection levelled against the use of these models is that they ignore all the known, causally-relevant details of the often complex target systems. Indeed, the (...) objection continues, mathematical models, by their very nature, abstract away from what matters and thus cannot be relied upon to provide any useful information about the systems they are supposed to represent. In this paper, I will examine the role of some typical mathematical models in population ecology and elsewhere. I argue that while, in a sense, these models do ignore the causal details, this move can not only be justified, it is necessary. I will argue that idealising away from complicating causal details often gives a clearer view of what really matters. And often what really matters is not the push and shove of base-level causal processes, but higher-level predictions and (non-causal) explanations. (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.
We consider several ways in which a good understanding of modern techniques and principles in physics can elucidate ecology. We focus on analogical reasoning between these two branches of science. This style of reasoning requires an understanding of both sciences and an appreciation of the similarities and points of contact between the two. In the current ecological literature on the relationship between ecology and physics, there has been some misunderstanding about the nature of modern physics and its methods. Physics is (...) seen as being much cleaner and tidier than ecology. When ecology is compared to this idealised, fictional version of physics, ecology looks very different, and the prospect of ecology and physics learning from one another is questionable. We argue that physics, once properly appreciated, is more like ecology than ecologists have thus far appreciated. Physicists and ecologists can and do learn from each other, and in this paper we outline how analogical reasoning can facilitate such exchanges. (shrink)
A good philosophical understanding of ecology is important for a number of reasons. First, ecology is an important and fascinating branch of biology, with distinctive philosophical issues. Second, ecology is only one small step away from urgent political, ethical, and management decisions about how best to live in an apparently fragile and increasingly-degraded environment. Third, philosophy of ecology, properly conceived, can contribute directly to both our understanding of ecology and help with its advancement. Philosophy of ecology can thus be seen (...) as part of the emerging discipline of “biohumanities”, where biology and humanities disciplines together advance our understanding and knowledge of biology (Stotz and Griffiths 2008). In this paper, we focus primarily on this third role of the philosophy of ecology and consider a number of places where philosophy can play an important role in ecology. In the process, we survey some of the current research being done in philosophy of ecology, as well as make suggestions about the agenda for future research in this area. We also hope to help clarify what philosophy of ecology is and what it should aspire to be. In what follows, we discuss several topics in the philosophy of ecology and conservation biology, starting with the role and understanding of mathematical models. This is followed by a discussion of a couple of practical problems involving the standard model of hypothesis testing and the use of decision-theoretic methods in environmental science. We then move on to discuss the issue of how we should understand biodiversity, and why this matters for conservation management. Finally, we look at environmental ethics and its relationship with ecology and conservation biology. These four topics were chosen because they are all of contemporary interest in philosophy of ecology circles and are ones where there is much fruitful work still to be done. The topics in question are also useful vehicles for highlighting the variety of issues in ecology and conservation biology where philosophy might prove useful.. (shrink)
Philosophical interest in ecology is relatively new. Standard texts in the philosophy of biology pay little or no attention to ecology (though Sterelny and Griffiths 1999 is an exception). This is in part because the science of ecology itself is relatively new, but whatever the reasons for the neglect in the past, the situation must change. A good philosophical understanding of ecology is important for a number of reasons. First, ecology is an important and fascinating branch of biology with distinctive (...) philosophical issues that arise from its study. Second, ecology is only one small step away from urgent political, ethical, and management decisions about how best to live in an apparently increasingly-fragile environment. Third, philosophy of ecology, properly conceived, can contribute directly to both our understanding of ecology and help with its advancement. Philosophy of ecology can thus be seen as part of the emerging discipline of “biohumanities”, where biology and humanities disciplines together advance our understanding and knowledge of biology (Stotz and Griffiths forthcoming). In this paper, we focus primarily on this third role of the philosophy of ecology and consider a number of places where philosophy can play an important role in ecology. In the process, we.. (shrink)
It has been argued in the conservation literature that giving conservation absolute priority over competing interests would best protect the environment. Attributing infinite value to the environment or claiming it is ‘priceless’ are two ways of ensuring this priority (e.g. Hargrove 1989; Bulte and van Kooten 2000; Ackerman and Heinzerling 2002; McCauley 2006; Halsing and Moore 2008). But such proposals would paralyse conservation efforts. We describe the serious problems with these proposals and what they mean for practical applications, and we (...) diagnose and resolve some conceptual confusions permeating the literature on this topic. (shrink)
On the face of it, ethics and decision theory give quite different advice about what the best course of action is in a given situation. In this paper we examine this alleged conflict in the realm of environmental decision-making. We focus on a couple of places where ethics and decision theory might be thought to be offering conflicting advice: environmental triage and carbon trading. We argue that the conflict can be seen as conflicts about other things (like appropriate temporal scales (...) for value assignments, idealisations of the decision situation, whether the conservation budget really is fixed and the like). The good news is that there is no conflict between decision theory and environmental ethics. The bad news is that a great deal of environmental decision modelling may be rather simple minded, in that it does not fully incorporate some of these broader issues about temporal scales and the dynamics of many of the decision situations. (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)
Mark Colyvan (University of Sydney)∗ Stefan Linquist (University of Queensland) William Grey (University of Queensland) Paul E. Griffiths (University of Sydney) Jay Odenbaugh (Lewis and Clark College).
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In this paper I discuss the kinds of idealisations invoked in normative theories—logic, epistemology, and decision theory. I argue that very often the so-called norms of rationality are in fact mere idealisations invoked to make life easier. As such, these idealisations are not too different from various idealisations employed in scientific modelling. Examples of the latter include: fluids are incompressible (in fluid mechanics), growth rates are constant (in population ecology), and the gravitational influence of distant bodies can be ignored (in (...) celestial mechanics). Thinking of logic, epistemology, and decision theory as normative models employing various idealisations of these kinds, changes the way we approach the justification of the models in question. (shrink)
This paper explores the scope and limits of rational consensus through mutual respect, with the primary focus on the best known formal model of consensus: the Lehrer–Wagner model. We consider various arguments against the rationality of the Lehrer–Wagner model as a model of consensus about factual matters. We conclude that models such as this face problems in achieving rational consensus on disagreements about unknown factual matters, but that they hold considerable promise as models of how to rationally resolve non-factual disagreements.
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)
We argue that standard definitions of ‘vagueness’ prejudice the question of how best to deal with the phenomenon of vagueness. In particular, the usual understanding of ‘vagueness’ in terms of borderline cases, where the latter are thought of as truth-value gaps, begs the question against the subvaluational approach. According to this latter approach, borderline cases are inconsistent (i.e., glutty not gappy). We suggest that a definition of ‘vagueness’ should be general enough to accommodate any genuine contender in the debate over (...) how to best deal with the sorites paradox. Moreover, a definition of ‘vagueness’ must be able to accommodate the variety of forms sorites arguments can take. These include numerical, total-ordered sorites arguments, discrete versions, continuous versions, as well as others without any obvious metric structure at all. After considering the shortcomings of various definitions of ‘vagueness’, we propose a very general non-question-begging definition. (shrink)
Machine generated contents note: 1. Mathematics and its philosophy; 2. The limits of mathematics; 3. Plato's heaven; 4. Fiction, metaphor, and partial truths; 5. Mathematical explanation; 6. The applicability of mathematics; 7. Who's afraid of inconsistent mathematics?; 8. A rose by any other name; 9. Epilogue: desert island theorems.
We discuss a recent attempt by Chris Daly and Simon Langford to do away with mathematical explanations of physical phenomena. Daly and Langford suggest that mathematics merely indexes parts of the physical world, and on this understanding of the role of mathematics in science, there is no need to countenance mathematical explanation of physical facts. We argue that their strategy is at best a sketch and only looks plausible in simple cases. We also draw attention to how frequently Daly and (...) Langford find themselves in conflict with mathematical and scientific practice. (shrink)
Fictionalism in the philosophy of mathematics is the view that mathematical statements, such as ‘8+5=13’ and ‘π is irrational’, are to be interpreted at face value and, thus interpreted, are false. Fictionalists are typically driven to reject the truth of such mathematical statements because these statements imply the existence of mathematical entities, and according to fictionalists there are no such entities. Fictionalism is a nominalist (or anti-realist) account of mathematics in that it denies the existence of a realm of abstract (...) mathematical entities. It should be contrasted with mathematical realism (or Platonism) where mathematical statements are taken to be true, and, moreover, are taken to be truths about mathematical entities. Fictionalism should also be contrasted with other nominalist philosophical accounts of mathematics that propose a reinterpretation of mathematical statements, according to which the statements in question are true but no longer about mathematical entities. Fictionalism is thus an error theory of mathematical discourse: at face value mathematical discourse commits us to mathematical entities and although we normally take many of the statements of this discourse to be true, in doing so we are in error (cf. error theories in ethics). (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.
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 (in different ways) 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)
In this paper I discuss the problem of providing an account of the normative force of theories of rationality. The theories considered are theories of rational inference, rational belief and rational decision— logic, probability theory and decision theory, respectively. I provide a naturalistic account of the normativity of these theories that is not viciously circular. The account offered does have its limitations though: it delivers a defeasible account of rationality. On this view, theories of rational inference, belief and decision are (...) not a priori . Rather, they are a posteriori and may change over time. Finally, I compare this approach with another that emerges from the Ramsey-Lewis approach to defining theoretical terms. (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)
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)