As more young people feel safe to outwardly identify as transgender or gender expansive (TGE), meaning that their gender identity does not align with the sex they were assigned at birth, an increasing number of youth who identify as TGE seek gender-affirming medical care (GAMC). GAMC raises a number of ethical questions, such as the capacity of a minor to assent or consent, the role of parents or legal guardians in decisions about treatment, and implications for equitable access to care (...) when differing parental or custodial viewpoints are present. These questions are further complicated by the difficulties in explaining the limits of long-term research in GAMC, with regard to the preservation of fertility, for example. We present two de-identified composite case studies to highlight dilemmas that may arise and offer recommendations to better support patient- and family-centered decision making for GAMC. These include employing a multidisciplinUniverary shared decision-making approach, disentangling informed consent and assent from chronological age, developing a consistent approach to the assessment of decisional capacity, and developing age-appropriate informational materials. (shrink)

We discuss from a philosophical perspective the way in which the normal concept of time might be said to `emerge' in a quantum theory of gravity. After an introduction, we briefly discuss the notion of emergence, without regard to time. We then introduce the search for a quantum theory of gravity ; and review some general interpretative issues about space, time and matter. We then discuss the emergence of time in simple quantum geometrodynamics, and in the Euclidean approach. Section 6 (...) concludes. (shrink)

Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. It has been common since the turn of the twentieth century to take correctness to be underwritten by the existence of formal derivations in a suitable axiomatic foundation, but then it is hard to see how this normative standard can be met, given the differences between informal proofs and formal derivations, and given the inherent fragility and complexity of (...) the latter. This essay describes some of the ways that mathematical practice makes it possible to reliably and robustly meet the formal standard, preserving the standard normative account while doing justice to epistemically important features of informal mathematical justification. (shrink)

The main aim of this paper is to make a remark about the relation between dualities between theories, as `duality' is understood in physics and equivalence of theories, as `equivalence' is understood in logic and philosophy. The remark is that in physics, two theories can be dual, and accordingly get called `the same theory', though we interpret them as disagreeing---so that they are certainly not equivalent, as `equivalent' is normally understood. So the remark is simple: but, I shall argue, worth (...) stressing---since often neglected. My argument for this is based on the account of duality developed by De Haro: which is illustrated here with several examples, from both elementary physics and string theory. Thus I argue that in some examples, including in string theory, two dual theories disagree in their claims about the world. I also spell out how this remark implies a limitation of proposals to understand theoretical equivalence as either logical equivalence or a weakening of it. (shrink)

I have two main aims. The first is general, and more philosophical. The second is specific, and more closely related to physics. The first aim is to state my general views about laws and causation at different ”levels’. The main task is to understand how the higher levels sustain notions of law and causation that ”ride free’ of reductions to the lower level or levels. I endeavour to relate my views to those of other symposiasts. The second aim is to (...) give a framework for describing dynamics at different levels, emphasizing how the various levels’ dynamics can mesh or fail to mesh. This framework is essentially that of elementary dynamical systems theory. The main idea will be, for simplicity, to work with just two levels, dubbed ”micro’ and ”macro’, which are related by coarse-graining. I use this framework to describe, in part, the first four of Ellis’ five types of top-down causation. (shrink)

We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.

In a wide range of fields, the word “modular” is used to describe complex systems that can be decomposed into smaller systems with limited interactions between them. This essay argues that mathematical knowledge can fruitfully be understood as having a modular structure and explores the ways in which modularity in mathematics is epistemically advantageous.

The epsilon calculus is a logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term εx A denotes some x satisfying A(x), if there is one. In Hilbert's Program, the epsilon terms play the role of ideal elements; the aim of Hilbert's finitistic consistency proofs is to give a procedure which removes such terms (...) from a formal proof. The procedures by which this is to be carried out are based on Hilbert's epsilon substitution method. The epsilon calculus, however, has applications in other contexts as well. The first general application of the epsilon calculus was in Hilbert's epsilon theorems, which in turn provide the basis for the first correct proof of Herbrand's theorem. More recently, variants of the epsilon operator have been applied in linguistics and linguistic philosophy to deal with anaphoric pronouns. (shrink)

This paper expounds the modern theory of symplectic reduction in finite-dimensional Hamiltonian mechanics. This theory generalizes the well-known connection between continuous symmetries and conserved quantities, i.e. Noether's theorem. It also illustrates one of mechanics' grand themes: exploiting a symmetry so as to reduce the number of variables needed to treat a problem. The exposition emphasises how the theory provides insights about the rotation group and the rigid body. The theory's device of quotienting a state space also casts light on philosophical (...) issues about whether two apparently distinct but utterly indiscernible possibilities should be ruled to be one and the same. These issues are illustrated using ``relationist'' mechanics. (shrink)

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that (...) standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect. (shrink)

"Arguing that debates over legitimacy, political violence, freedom, and justice would benefit greatly from cross-tradition theorizing, this book shows how putting analytic and continental political theory in conversation would help us to overcome these intractable problems"--.

“Now, in calm weather, to swim in the open ocean is as easy to the practised swimmer as to ride in a spring-carriage ashore. But the awful lonesomeness is intolerable. The intense concentration of self in the middle of such a heartless immensity, my God! who can tell it? Mark, how when sailors in a dead calm bathe in the open sea—mark how closely they hug their ship and only coast along her sides.” (Herman Melville, Moby Dick, Chapter 94).

We show that certain model-theoretic forcing arguments involving subsystems of second-order arithmetic can be formalized in the base theory, thereby converting them to effective proof-theoretic arguments. We use this method to sharpen the conservation theorems of Harrington and Brown-Simpson, giving an effective proof that WKL+0 is conservative over RCA0 with no significant increase in the lengths of proofs.

The concept of minimal risk has been used to regulate and limit participation by adolescents in clinical trials. It can be understood as setting an absolute standard of what risks are considered minimal or it can be interpreted as relative to the actual risks faced by members of the host community for the trial. While commentators have almost universally opposed a relative interpretation of the environmental risks faced by potential adolescent trial participants, we argue that the ethical concerns against the (...) relative standard may not be as convincing as these commentators believe. Our aim is to present the case for a relative standard of environmental risk in order to open a debate on this subject. We conclude by discussing how a relative standard of environmental risk could be defended in the specific case of an HIV vaccine trial among adolescents in South Africa. (shrink)

This paper provides a new analysis and critique of Rawlsian public reason’s handling of the abortion question. It is often claimed that public reason is indeterminate on abortion, because it cannot say enough about prenatal moral status, or give content to the (allegedly) political value which Rawls calls ‘respect for human life’. I argue that public reason requires much greater argumentative restraint from citizens debating abortion than critics have acknowledged. Beyond the preliminary observation that fetuses do not meet the criteria (...) of political personhood, I contend, public reasoners may say nothing about prenatal moral status. Indeed, respect for human life (as distinct from respect for persons) is not, I show, a genuine political value, whose interpretation lies within public reason’s remit. This fact does not, however, prevent deliberators from drawing determinate conclusions regarding the law of abortion. Instead, I show, public reason has radically permissive implications for the legal regulation of feticide, inclining heavily towards the view that abortion should be allowed with little or no qualification, right until birth. I end by giving grounds for thinking that, even if the argumentative restraint which public reason enjoins over prenatal moral status does not generate indeterminacy, it is nonetheless objectionable. (shrink)

Many psychological studies of categorization and reasoning use undergraduates to make claims about human conceptualization. Generalizability of findings to other populations is often assumed but rarely tested. Even when comparative studies are conducted, it may be challenging to interpret differences. As a partial remedy, in the present studies we adopt a 'triangulation strategy' to evaluate the ways expertise and culturally different belief systems can lead to different ways of conceptualizing the biological world. We use three groups (US bird experts, US (...) undergraduates, and ordinary Itza' Maya) and two sets of birds (North American and Central American). Categorization tasks show considerable similarity among the three groups' taxonomic sorts, but also systematic differences. Notably, US expert categorization is more similar to Itza' than to US novice categorization. The differences are magnified on inductive reasoning tasks where only undergraduates show patterns of judgment that are largely consistent with current models of category-based taxonomic inference. The Maya commonly employ causal and ecological reasoning rather than taxonomic reasoning. Experts use a mixture of strategies (including causal and ecological reasoning), only some of which current models explain. US and Itza' informants differed markedly when reasoning about passerines (songbirds), reflecting the somewhat different role that songbirds play in the two cultures. The results call into question the importance of similarity-based notions of typicality and central tendency in natural categorization and reasoning. These findings also show that relative expertise leads to a convergence of thought that transcends cultural boundaries and shared experiences. (shrink)

The purpose of this paper is to think about the various methods of attempting to make money in the capital markets (“investing”). I suggest that though running a betting system on a Roulette wheel is silly, running a betting system on the capital markets may be a good idea.

A notion called Herbrand saturation is shown to provide the model-theoretic analogue of a proof-theoretic method, Herbrand analysis, yielding uniform model-theoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a third approach, which is finitary but retains the semantic flavor of the model-theoretic version.

is a fragment of first-order aritlimetic so weak that it cannot prove the totality of an iterated exponential fimction. Surprisingly, however, the theory is remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.

This paper expounds the relations between continuous symmetries and conserved quantities, i.e. Noether's ``first theorem'', in both the Lagrangian and Hamiltonian frameworks for classical mechanics. This illustrates one of mechanics' grand themes: exploiting a symmetry so as to reduce the number of variables needed to treat a problem. I emphasise that, for both frameworks, the theorem is underpinned by the idea of cyclic coordinates; and that the Hamiltonian theorem is more powerful. The Lagrangian theorem's main ``ingredient'', apart from cyclic coordinates, (...) is the rectification of vector fields afforded by the local existence and uniqueness of solutions to ordinary differential equations. For the Hamiltonian theorem, the main extra ingredients are the asymmetry of the Poisson bracket, and the fact that a vector field generates canonical transformations iff it is Hamiltonian. (shrink)

Paul Cohen’s method of forcing, together with Saul Kripke’s related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects (...) of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments. (shrink)

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses ofDirichlet characters in presentations of Dirichlet’s proof in the nineteenth and early twentieth centuries, with an eye toward understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method.

This paper offers a new critical evaluation of the Rawlsian model of global public reason (‘GPR’), focusing on its ability to serve as a normative standard for guiding international diplomacy and deliberation in matters of war. My thesis is that, where war is concerned, the model manifests two fatal weaknesses. First, because it demands extensive neutrality over the moral status of persons – and in particular over whether they possess equal basic worth or value – out of respect for the (...) beliefs of inegalitarian yet ‘decent’ societies, or ‘peoples’, Rawlsian GPR renders calculations of proportionality in war impossible. Second, because its content is provided by a conception of global justice (the so-called ‘Law of Peoples’) whose injunctions are addressed exclusively to peoples, as corporate agents, Rawlsian GPR pushes the moral evaluation of the independent wartime choices of individuals off the agenda of the global public forum altogether. (shrink)

In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character, although they do not fall squarely under a traditional philosophical purview. The goal of this article is to try to articulate some of these questions more clearly, and assess the philosophical methods that may be brought to bear. In Section 3, I note that most of the issues can be classified under two headings: some (...) deal with the ability of computers to deliver appropriate “evidence” for mathematical assertions, a notion that is explored in Section 4, while others deal with the ability of computers to deliver appropriate mathematical “understanding,” a notion that is considered in Section 5. Final thoughts are provided in Section 6. (shrink)

The purpose of this paper is to point the way to the money tree. Currently, almost all investment professionals think that outperformance requires an “edge,” that is, the ability to predict the future to some degree. In this paper, I suggest that money can be made in a 0, or even slightly negative, expected value environment by carefully choosing investment/bet sizes. Philosophical considerations are found mainly in Section 4.

Some recent philosophical debate about persistence has focussed on an argument against perdurantism that discusses rotating perfectly homogeneous discs. The argument has been mostly discussed by metaphysicians, though it appeals to ideas from classical mechanics, especially about rotation. In contrast, I assess the RDA from the perspective of the philosophy of physics. After introducing the argument and emphasizing the relevance of physics, I review some metaphysicians' replies to the argument, especially those by Callender, Lewis, Robinson and Sider. Thereafter, I argue (...) for three main conclusions. They all arise from the fact, emphasized in Section 2, that classical mechanics is both more subtle, and more problematic, than philosophers generally realize. The first conclusion is that the RDA can be formulated more strongly than is usually recognized: it is not necessary to ``imagine away'' the dynamical effects of rotation. The second is that in general relativity, the RDA fails because of frame-dragging. The third is that even setting aside general relativity, the strong formulation of the RDA can after all be defeated. Namely, by the perdurantist taking objects in classical mechanics to have only temporally extended, i.e. non-instantaneous, temporal parts: which immediately blocks the RDA. Admittedly, this version of perdurantism defines persistence in a weaker sense of `definition' than pointilliste versions that aim to define persistence assuming only instantaneous temporal parts. But I argue that temporally extended temporal parts: can do the jobs within the endurantism- perdurantism debate that the perdurantist wants temporal parts to do; and are supported by both classical and quantum mechanics. (shrink)

Algebraic proofs of the cut-elimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the double-negation translation is also discussed: if ϕ is provable classically, then ¬(¬ϕ)nf is provable in minimal logic, where θnf denotes the negation-normal form of θ. The translation is used to show that cut-elimination theorems for classical logic can be viewed as special (...) cases of the cut-elimination theorems for intuitionistic logic. (shrink)

We analyze the pointwise convergence of a sequence of computable elements of L1 in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory RCA0, each is equivalent to the assertion that every Gδ subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak Königʼs lemma relativized to the Turing jump of any set. It is also equivalent to the (...) conjunction of the statement asserting the existence of a 2-random relative to any given set and the principle of Σ2 collection. (shrink)

This is a discussion of some themes in Max Tegmark’s recent book, Our Mathematical Universe. It was written as a review for Plus Magazine, the online magazine of the UK’s national mathematics education and outreach project, the Mathematics Millennium Project. Since some of the discussion---about symmetry breaking, and Pythagoreanism in the philosophy of mathematics---went beyond reviewing Tegmark’s book, the material was divided into three online articles. This version combines those three articles, and adds some other material, in particular a brief (...) defence of quidditism about properties. It also adds some references, to other Plus articles as well as academic articles. But it retains the informal style of Plus. (shrink)

A general method of interpreting weak higher-type theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomial-time computable arithmetic. A means of formalizing basic real analysis in such theories is sketched.

Can there be `peaceful coexistence' between quantum theory and special relativity? Thirty years ago, Shimony hoped that isolating the culprit in proofs of Bell inequalities as Outcome Independence would secure such peaceful coexistence: or, if not secure it, at least show a way---maybe the best or only way---to secure it. In this paper, I begin by being sceptical of Shimony's approach, urging that we need a relativistic solution to the quantum measurement problem. Then I analyse Outcome Independence in Kent's realist (...) one-world Lorentz-invariant interpretation of quantum theory. Then I consider Shimony's other condition, Parameter Independence, both in Kent's proposal and more generally, in the light of recent remarkable theorems by Colbeck, Renner and Leegwater. For both Outcome Independence and Parameter Independence, there is a striking analogy with the situation in pilot-wave theory. Finally, I will suggest that these recent theorems make some kind of peaceful coexistence mandatory for someone who, like Shimony, endorses Parameter Independence. (shrink)

We investigate the entrepreneurial opportunities and ethical dilemmas presented by technological turbulence. More specifically we investigate the line between Baumol’s [J. Polit. Econ. 98 (1990) 893] productive (e.g. innovation), unproductive (e.g. rent seeking) and destructive (e.g. criminal) entrepreneurship through three examples of Internet innovation – spam (destructive), music file sharing (unproductive), and Internet pharmacies (potentially productive). The emergence of accessible Internet technologies, under present norms, has created the potential for all three entrepreneurial activities. Because of the propensity for self-serving biases (...) and for bending the rules, the need for creativity in overcoming obstacles and overall liabilities of newness, entrepreneurs are likely to challenge established industrial morals and laws. Unlike new entrants, incumbents must abide by the currently accepted norms, and thus suffer from “liabilities of oldness”. The challenge for new entrants is to change sociopolitical legitimacy, whereas incumbents need to defend the established norms. We discuss competitive and other issues that result from technological turbulence and innovation. (shrink)

We describe a model-theoretic approach to ordinal analysis via the finite combinatorial notion of an α-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first- and second-order arithmetic.

We develop fundamental aspects of the theory of metric, Hilbert, and Banach spaces in the context of subsystems of second-order arithmetic. In particular, we explore issues having to do with distances, closed subsets and subspaces, closures, bases, norms, and projections. We pay close attention to variations that arise when formalizing definitions and theorems, and study the relationships between them.

The 1-consistency of arithmetic is shown to be equivalent to the existence of fixed points of a certain type of update procedure, which is implicit in the epsilon-substitution method.

This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes that we view mathematics as a system of conventions and norms that is designed to help us make sense of the world and reason efficiently. Like any designed system, it can perform well or poorly, and the philosophy of mathematics has a role to play in helping (...) us understand the general principles by which it serves its purposes well. (shrink)

This paper gives a technically elementary treatment of some aspects of Hamilton -Jacobi theory, especially in relation to the calculus of variations. The second half of the paper describes the application to geometric optics, the optico-mechanical analogy and the transition to quantum mechanics. Finally, I report recent work of Holland providing a Hamiltonian formulation of the pilot-wave theory.

The purpose of this note is to contrast a Cantorian outlook with a non-Cantorian one and to present a picture that provides support for the latter. In particular, I suggest that: i) infinite hyperreal numbers are the (actual, determined) infinite numbers, ii) ω is merely potentially infinite, and iii) infinitesimals should not be used in the di Finetti lottery. Though most Cantorians will likely maintain a Cantorian outlook, the picture is meant to motivate the obvious nature of the non-Cantorian outlook.

Solovay has shown that if $\cal{O}$ is an open subset of $P(\omega)$ with code $S$ and no infinite set avoids $\cal{O}$ , then there is an infinite set hyperarithmetic in $S$ that lands in $\cal{O}$ . We provide a direct proof of this theorem that is easily formalizable in $ATR_0$.

Extending Gödel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.

Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to the counsel (...) of philosophers, in much the same way as a nation considering an action of dubious international legality is called upon to turn to the United Nations for guidance. After due consideration of appropriate regulations and guidelines (and, possibly, debate between representatives of different philosophical factions), the philosophers render a decision, by which the dutiful mathematician abides. Quine was famously critical of such dreams of a ‘first philosophy.’ At the opposite extreme, our hypothetical mathematician answers only to the subject’s internal concerns, blithely or brashly indifferent to philosophical approval. What is at stake to our mathematician friend is whether the questionable practice provides a proper mathematical solution to the problem at hand, or an appropriate mathematical understanding; or, in pragmatic terms, whether it will make it past a journal referee. In short, mathematics is as mathematics does, and the philosopher’s task is simply to make sense of the subject as it evolves and certify practices that are already in place. In his textbook on the philosophy of mathematics (Shapiro, 2000), Stewart Shapiro characterizes this attitude as ‘philosophy last, if at all.’. (shrink)