Parmenides of Elea Parmenides of Elea was a Presocratic Greek philosopher. As the first philosopher to inquire into the nature of existence itself, he is incontrovertibly credited as the “Father of Metaphysics.” As the first to employ deductive, a priori arguments to justify his claims, he competes with Aristotle … Continue reading Parmenides →.

Warranting further examination is how the nascent philosophical tradition initially spread to this region from its Ionian provenance. Despite numerous ancient attestations that Parmenides of Elea was influenced, or even directly instructed, by the Ionian-born Xenophanes, many modern scholars remain skeptical of this historical association. The extent of this skepticism ranges from cautious uncertainty to outright denial of any historical plausibility. The skeptical grounds similarly vary, from distrusting the historical veracity of late and/or perhaps biased commentators, to understanding these thinkers (...) as involved in radically different projects. This essay aims to challenge the skeptical position, and establish a direct link disseminating Ionian philosophy to Magna Graecia via Xenophanes and Parmenides. The argument is straightforward. First, the ancient geographical and temporal evidence is noted, establishing that it was possible for Parmenides to have been influenced and/or taught by Xenophanes. Next, the metaphysical and epistemological parallels between these thinkers are considered. Despite notable differences, on balance, these close parallels suggest against the skeptical view, making it quite plausible to impute a direct intellectual link between these thinkers. Third, I consider ancient claims that both thinkers were engaging with religious topics, offering a sort of “rational theology.” This evidence for a close intellectual relationship between these thinkers has been entirely ignored by modern scholars, and orthodox interpretative models cannot readily provide a charitable explanation for them. However, by reconsidering the theistic content in Parmenides’s poem, a new interpretative approach is revealed which can. Once this evidence is considered in its totality, the case for imputing a close and direct intellectual heritage from Xenophanes to Parmenides proves quite substantial. (shrink)

This paper analyzes the ethical themes and theory portrayals by particular characters in Star Trek: Into Darkness. It is concluded that the film can be understood as explicating the pros and cons of both "male" and "female" ethical perspectives, and that a comprehensive understanding of morality requires some synthesis of both perspectives.

Evidence for a Parmenidean influence on Plato’s Republic typically focuses on content from Bks. V-VI, and the development of Plato’s Theory of Forms. This essay aims to suggest that Plato’s censorship of poetic content in Bks. II-III—particularly the rules for portraying divine nature (376e-383c)—also draw heavily upon the Eleatic tradition, particularly Parmenides’s. Identifying this further Eleatic influence will be enhanced by my own reading of Parmenides. This reading advocates understanding Parmenides in a more Xenophanean-vein—i.e. by taking What-Is to be an (...) explication of the essential qualities of divine nature, and the overall poem as rejecting traditional, mythopoetic accounts of divinity. Recognizing this Eleatic influence on the censorship of poetic content, a tension arises. For Plato infamously censors poetic styles next, concluding that mimetic dialogue may only be rarely employed, and only then in imitation of virtuous persons and actions (392c-398b). This would entail banning all poetic works relying exclusively on mimetic dialogue. Yet, not only do Plato’s own dialogues entirely consist of mimetic dialogue, so does Parmenides’s proto-dialogue. Furthermore, by so closely imitating Parmenides’s thought and language in Republic, has not Plato himself engaged in a type of intellectual and compositional mίμησις? Just as it would be strange to ban the very dialogue (Republic) which outlines and justifies Kallipolis in the first place, it would also be troubling to ban a philosophical work (i.e. Parmenides’s poem) which Republic is so heavily indebted to. Such a ban would also seem strongly at odds with Plato’s general reverence for Parmenides. In an attempt to address these tensions, I suggest that in Republic II-III, Plato’s lack of concern for banning philosophical works along with mimetic poetry should further suggest that he intends the ban to be far narrower than it first appears: as a rejection of performative, rather than compositional, mίμησις. (shrink)

This essay explicates the primary interpretative import of B1: 31-32 in Parmenides poem (On Nature)—lines which have radical implications for the overall argument, and which the traditional arrangement forces into an irreconcilable dilemma. I argue that the “negative” reading of lines 31-32 is preferable, even on the traditional arrangement. This negative reading denies that a third thing is to be taught to the reader by the goddess—a positive account of how the apparent world is to be “acceptably” understood. I then (...) suggest that a rearrangement of the fragments would make more sense overall, while further supporting the “negative” reading as more natural and coherent. In particular, the rearrangement dispels the objection that, “if mortal opinions were not true, why would Parmenides include such a lengthy false account of the apparent world--an account which explicitly denies the conclusions of the earlier section, Truth?”. (shrink)

Mimesis can refer to imitation, emulation, representation, or reenactment - and it is a concept that links together many aspects of ancient Greek Culture. The Western Greek bell-krater on the cover, for example, is painted with a scene from a phlyax play with performers imitating mythical characters drawn from poetry, which also represent collective cultural beliefs and practices. One figure is shown playing a flute, the music from which might imitate nature, or represent deeper truths of the cosmos based upon (...) Pythagorean views (which were widespread in Western Greece at the time). The idea that mimesis should be restricted to ideals was made famous by Plato (whose connections to Pythagoreanism and Siracusa are well-known), and famously challenged by his student Aristotle (not to mention by the mimetic character of Plato’s own poetry). This volume gathers essays not only on the philosophical debate about mimesis, but also on its use in architecture, drama, poetry, history, music, ritual, and visual art. The emphasis is on examples from Hellenic cities in Southern Italy and Sicily, but the insights apply far beyond – even to modern times. Contributors include: Thomas Noble Howe, Francisco J. Gonzalez, Gene Fendt, Guilherme Domingues da Motta, Jeremy DeLong, Carolina Araújo, Marie-Élise Zovko, Lidia Palumbo, Sean Driscoll, Konstantinos Gkaleas, Anna Motta, Jure Zovko, Alexander H. Zistakis, Christos C. Evangeliou, Dorota Tymura, Iris Sulimani, Elliott Domagola, Jonah Radding, Giulia Corrente, Laura Tisi, Ewa Osek, Argyri G. Karanasiou, Rocío Manuela Cuadra Rubio, Jorge Tomás García, Aura Piccioni, and José Miguel Puebla Morón. (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)

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)

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"--.

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)

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.

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)

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.

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)

Is the mechanism behind presupposition projection and filtering fundamentally asymmetric or symmetric? This is a foundational question for the theory of presupposition which has been at the centre of attention in recent literature :287–316, 2008b. https://doi.org/10.1515/THLI.2008.021, Semant Pragmat 2:1–78, 2009. https://doi.org/10.3765/sp.2.3; Rothschild in Semant Pragmat 4:1–43, 2011/2015. https://doi.org/10.3765/sp.4.3 a.o.). It also bears on broader issues concerning the source of asymmetries observed in natural language: are these simply rooted in superficial asymmetries of language use ; or are they, at least in (...) part, directly encoded in linguistic knowledge and representations? In this paper we aim to make progress on these questions by exploring presupposition projection across conjunction, which has traditionally been taken as a central piece of evidence that presupposition filtering is asymmetric in general. As a number of authors have recently pointed out, however, the evidence which has typically been used to support this conclusion is muddied by independent issues concerning redundancy; additional concerns have to do with the possibility of local accommodation. We report on a series of experiments, building on previous work by Chemla and Schlenker :177–226, 2012. https://doi.org/10.1007/s11050-012-9080-7) and Schwarz Experimental perspectives on presuppositions, Springer, Cham, 2015), using inference and acceptability tasks, which aim to control for both of these potential confounds. In our results, we find strong evidence for left-to-right filtering across conjunctions, but no evidence for right-to-left filtering—even when right-to-left filtering would, if available, rescue an otherwise unacceptable sentence. These results suggest that presupposition filtering across conjunction is asymmetric, contra suggestions in the recent literature :157–212, 2008a. https://doi.org/10.1515/THLI.2008.013, 2009 a.o.), and pave the way for the investigation of further questions about the nature of this asymmetry and presupposition projection more generally. Our results also have broader implications for the study of presupposition: we find important differences in the verdicts of acceptability versus inference tasks in testing for projected content, which has both methodological ramifications for the question of how to distinguish presupposed content, and theoretical repercussions for understanding the nature of projection and presuppositions more generally. (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.

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)

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)

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)

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.

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.

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 $ATR_0$.

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)

In this paper, I suggest that infinite numbers are large finite numbers, and that infinite numbers, properly understood, are 1) of the structure omega + (omega* + omega)Ө + omega*, and 2) the part is smaller than the whole. I present an explanation of these claims in terms of epistemic limitations. I then consider the importance, part of which is demonstrating the contradiction that lies at the heart of Cantorian set theory: the natural numbers are too large to be counted (...) by any finite number, but too small to be counted by any infinite number – there is no number of natural numbers. (shrink)

The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1/ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdos in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.

Fogelin’s Wittgensteinian view of deep disagreement as allowing no rational resolution has been criticized from both argumentation theoretic and epistemological perspectives. These criticisms typically do not recognize how his point applies to the very argumentative resources on which they rely. Additionally, more extremely than Fogelin himself argues, the conditions of deep disagreement make each position literally unintelligible to the other, again disallowing rational resolution. In turn, however, this failure of sense is so extreme that it partly cancels its own meaning (...) as a failure of sense. Consequently, it paradoxically opens new possibilities for sense and therefore rationally unexpected resolutions. (shrink)

We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue measure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages Anf do not converge to a computable element of L2([0, 1]). In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, (...) we show that, for any nonexpansive linear operator T on a separable Hilbert space and any element f , it is possible to compute a bound on the rate of convergence of Anf from T , f , and the norm f ∗ of the limit. In particular, if T is the Koopman operator arising from a computable ergodic measure preserving transformation of a probability space X and f is any computable element of L2 (X ), then there is a computable bound on the rate of convergence of the sequence Anf. (shrink)

From proofs in any classical first-order theory that proves the existence of at least two elements, one can eliminate definitions in polynomial time. From proofs in any classical first-order theory strong enough to code finite functions, including sequential theories, one can also eliminate Skolem functions in polynomial time.

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.

We discuss the legacy of Alan Turing and his impact on computability and analysis.

A multiverse is comprised of many universes, which quickly leads to the question: How many universes? There are either finitely many or infinitely many universes. The purpose of this paper is to discuss two conceptions of infinite number and their relationship to multiverses. The first conception is the standard Cantorian view. But recent work has suggested a second conception of infinite number, on which infinite numbers behave very much like finite numbers. I will argue that that this second conception of (...) infinite number is the correct one, and analyze what this means for multiverses. (shrink)

In place of Tomasello's explanation for the source of moral obligation, we suggest that it develops from the concern for others already implicit in the human developmental system. Mutual affection and caring make the development of communication and thinking possible. Humans develop as persons within such relationships and this develops into respect and moral obligation.

The purpose of this paper is to suggest that we are in the midst of a Cantorian bubble, just as, for example, there was a dot com bubble in the late 1990’s.

A seminal theorem due to Weyl [14] states that if $(a_n)$ is any sequence of distinct integers, then, for almost every $x \in \mathbb{R}$, the sequence $(a_n x)$ is uniformly distributed modulo one. In particular, for almost every $x$ in the unit interval, the sequence $(a_n x)$ is uniformly distributed modulo one for every computable sequence $(a_n)$ of distinct integers. Call such an $x$ UD random. Here it is shown that every Schnorr random real is UD random, but there are (...) Kurtz random reals that are not UD random. On the other hand, Weyl's theorem still holds relative to a particular effectively closed null set, so there are UD random reals that are not Kurtz random. (shrink)