Hellman [2003] raises interesting challenges to categorical structuralism. He starts citing Awodey [1996] which, as Hellman sees, is not intended as a foundation for mathematics. It offers a structuralist framework which could denned in any of many different foundations. But Hellman says Awodey's work is 'naturally viewed in the context of Mac Lane's repeated claim that category theory provides an autonomous foundation for mathematics as an alternative to set theory' (p. 129). Most of Hellman's paper 'scrutinizes the formulation of category (...) theory' specifically in 'its alleged role as providing a foundation' (p. 130). I will also focus on the foundational question. Page numbers in parentheses are page references to Hellman [2003]. (shrink)

Linnebo and Pettigrew's critique in this journal of categorical foundations well emphasizes that the particulars of various categorical foundations matter, and that mathematical practice must be a major consideration. But several categorists named by the authors as proposing categorical foundations do not propose foundations, notably Awodey, and the article's description of current textbook practice seems inaccurate. They say that categorical foundations have justificatory autonomy if and only if mathematics can be justified simply by its practice. Do they seriously believe philosophers (...) may be able to justify some currently unpractised mathematics better than current practice justifies itself? (shrink)

The unifying theme of this issue is Plato’s dialectical view of mathematical progress and hypotheses. Besides provisional propositions, he calls concepts and goals also hypotheses. He knew mathematicians create new concepts and goals as well as theorems, and abandon many along the way, and erase the creative process from their proofs. So the hypotheses of mathematics necessarily change through use — unless Benson is correct that Plato believed mathematics could reach the unhypothetical goals of dialectic. Landry discusses Plato on mathematical (...) and philosophical problem solving. Arsen and White relate Plato’s philosophy to mathematics in his time, and to Aristotle. (shrink)

We are used to seeing foundations linked to the mainstream mathematics of the late nineteenth century: the arithmetization of analysis, non-Euclidean geometry, and the rise of abstract structures in algebra. And a growing number of case studies bring a more philosophy-of-science viewpoint to the latest mathematics, as in [Carter, 2005; Corfield, 2006; Krieger, 2003; Leng, 2002]. Mac Lane's autobiography is a valuable bridge between these, recounting his experience of how the mid- and late-twentieth-century mainstream grew especially through Hilbert's school.An autobiography (...) at age 94 obviously has a lot of ground to cover. Mac Lane entered Yale in 1926 to study chemistry as a good practical career field but by the end of his first year he had won a $50 prize in mathematics and learned that you could make a living with it—as an actuary. He decided to do that. The next year he was excited to learn from his philosophy professor F.S.C. Northrop that you can also pursue new mathematical discoveries! Northrop had studied with A.N. Whitehead and sold Mac Lane on the excitement of Principia Mathematica. In a pattern that foreshadowed his career Mac Lane bought and annotated a copy of the first volume of Principia but his planned tutorial study of the book turned into a study of [Hausdorff, 1914] applying set theory to topology and analysis. As a graduate student at Chicago he was powerfully influenced by E.H. Moore, who taught the new axiomatic methods as tools for unifying and advancing the most classical analysis . Then he went to Göttingen. He heard Hilbert lecture on philosophy. He studied mathematics and its philosophy with Weyl. He followed Noether's lectures on abstract algebra for number theory. He had gone there to study logic and he wrote a …. (shrink)

For 300 years, Fermat’s Last Theorem seemed to be pure arithmetic little connected even to other problems in arithmetic. But the last decades of the twentieth century saw the discovery of very special cubic curves, and the rise of the huge theoretical Langlands Program. The Langlands perspective showed those curves are so special they cannot exist, and thus proved Fermat’s Last Theorem. With many great contributors, the proof ended in a deep and widely applicable geometric result relating nice curves in (...) rational coordinates to very nice surfaces in complex coordinates. This geometry is the only proof strategy for FLT yet known. Outstanding but not atypical of current mathematics, the proof challenges common philosophic views on abstraction and structure and raises still-open logical questions in arithmetic. (shrink)

This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.

The view that toposes originated as generalized set theory is a figment of set theoretically educated common sense. This false history obstructs understanding of category theory and especially of categorical foundations for mathematics. Problems in geometry, topology, and related algebra led to categories and toposes. Elementary toposes arose when Lawvere's interest in the foundations of physics and Tierney's in the foundations of topology led both to study Grothendieck's foundations for algebraic geometry. I end with remarks on a categorical view of (...) the history of set theory, including a false history plausible from that point of view that would make it helpful to introduce toposes as a generalization from set theory. (shrink)

Games may seem like a waste of time, where we struggle under artificial rules for arbitrary goals. The author suggests that the rules and goals of games are not arbitrary at all. They are a way of specifying particular modes of agency. This is what make games a distinctive art form. Game designers designate goals and abilities for the player; they shape the agential skeleton which the player will inhabit during the game. Game designers work in the medium of agency. (...) Game-playing, then, illuminates a distinctive human capacity. We can take on ends temporarily for the sake of the experience of pursuing them. Game play shows that our agency is significantly more modular and more fluid than we might have thought. It also demonstrates our capacity to take on an inverted motivational structure. Sometimes we can take on an end for the sake of the activity of pursuing that end. (shrink)

Now available in paperback, this acclaimed book introduces categories and elementary toposes in a manner requiring little mathematical background. It defines the key concepts and gives complete elementary proofs of theorems, including the fundamental theorem of toposes and the sheafification theorem. It ends with topos theoretic descriptions of sets, of basic differential geometry, and of recursive analysis.

Elementary axioms describe a category of categories. Theorems of category theory follow, including some on adjunctions and triples. A new result is that associativity of composition in categories follows from cartesian closedness of the category of categories. The axioms plus an axiom of infinity are consistent iff the axioms for a well-pointed topos with separation axiom and natural numbers are. The theory is not finitely axiomatizable. Each axiom is independent of the others. Further independence and definability results are proved. Relations (...) between categories and sets, the latter defined as discrete categories, are described, and applications to foundations are discussed. (shrink)

There seems to be a deep tension between two aspects of aesthetic appreciation. On the one hand, we care about getting things right. On the other hand, we demand autonomy. We want appreciators to arrive at their aesthetic judgments through their own cognitive efforts, rather than deferring to experts. These two demands seem to be in tension; after all, if we want to get the right judgments, we should defer to the judgments of experts. The best explanation, I suggest, is (...) that aesthetic appreciation is something like a game. When we play a game, we try to win. But often, winning isn’t the point; playing is. Aesthetic appreciation involves the same flipped motivational structure: we aim at the goal of correctness, but having correct judgments isn’t the point. The point is the engaged process of interpreting, investigating, and exploring the aesthetic object. Deferring to aesthetic testimony, then, makes the same mistake as looking up the answer to a puzzle, rather than solving it for oneself. The shortcut defeats the whole point. This suggests a new account of aesthetic value: the engagement account. The primary value of the activity of aesthetic appreciation lies in the process of trying to generate correct judgments, and not in having correct judgments. -/- *There is an audio version available: look for the Soundcloud link, below.*. (shrink)

I propose to study one problem for epistemic dependence on experts: how to locate experts on what I will call cognitive islands. Cognitive islands are those domains for knowledge in which expertise is required to evaluate other experts. They exist under two conditions: first, that there is no test for expertise available to the inexpert; and second, that the domain is not linked to another domain with such a test. Cognitive islands are the places where we have the fewest resources (...) for evaluating experts, which makes our expert dependences particularly risky. -/- Some have argued that cognitive islands lead to the complete unusability of expert testimony: that anybody who needs expert advice on a cognitive island will be entirely unable to find it. I argue against this radical form of pessimism, but propose a more moderate alternative. I demonstrate that we have some resources for finding experts on cognitive islands, but that cognitive islands leave us vulnerable to an epistemic trap which I will call runaway echo chambers. In a runaway echo chamber, our inexpertise may lead us to pick out bad experts, which will simply reinforce our mistaken beliefs and sensibilities. (shrink)

Glaucon in Plato's _Republic_ fails to grasp intermediates. He confuses pursuing a goal with achieving it, and so he adopts ‘mathematical platonism’. He says mathematical objects are eternal. Socrates urges a seriously debatable, and seriously defensible, alternative centered on the destruction of hypotheses. He offers his version of geometry and astronomy as refuting the charge that he impiously ‘ponders things up in the sky and investigates things under the earth and makes the weaker argument the stronger’. We relate his account (...) briefly to mathematical developments by Plato's associates Theaetetus and Eudoxus, and then to the past 200 years' developments in geometry. (shrink)

While Saunders Mac Lane studied for his D.Phil in Göttingen, he heard David Hilbert's weekly lectures on philosophy, talked philosophy with Hermann Weyl, and studied it with Moritz Geiger. Their philosophies and Emmy Noether's algebra all influenced his conception of category theory, which has become the working structure theory of mathematics. His practice has constantly affirmed that a proper large-scale organization for mathematics is the most efficient path to valuable specific results—while he sees that the question of which results are (...) valuable has an ineliminable philosophic aspect. His philosophy relies on the ideas of truth and existence he studied in Göttingen. His career is a case study relating naturalism in philosophy of mathematics to philosophy as it naturally arises in mathematics. Introduction Structures and Morphisms Varieties of Structuralism Göttingen Logic: Mac Lane's Dissertation Emmy Noether Natural Transformations Grothendieck: Toposes and Universes Lawvere and Foundations Truth and Existence Naturalism Austere Forms of Beauty. (shrink)

We can learn from questions as well as from their answers. This paper urges some things to learn from questions about categorical foundations for mathematics raised by Geoffrey Hellman and from ones he invokes from Solomon Feferman.

Of the seven $1,000,000 Clay Millennium Prize Problems in mathematics, just one would immediately appeal to Leonard Euler. That is “Existence and Smoothness of the Navier-Stokes Equation” (Fefferman 2000). Euler gave the basic equation in the 1750s. The work to this day shows Euler’s intuitive, vividly physical sense of mathematics.

An adequate theory of rights ought to forbid the harming of animals to promote trivial interests of humans, as is often done in the animal-user industries. But what should the rights view say about situations in which harming some animals is necessary to prevent intolerable injustices to other animals? I develop an account of respectful treatment on which, under certain conditions, it’s justified to intentionally harm some individuals to prevent serious harm to others. This can be compatible with recognizing the (...) inherent value of the ones who are harmed. My theory has important implications for contemporary moral issues in nonhuman animal ethics, such as the development of cultured meat and animal research. (shrink)

often insisted existence in mathematics means logical consistency, and formal logic is the sole guarantor of rigor. The paper joins this to his view of intuition and his own mathematics. It looks at predicativity and the infinite, Poincaré's early endorsement of the axiom of choice, and Cantor's set theory versus Zermelo's axioms. Poincaré discussed constructivism sympathetically only once, a few months before his death, and conspicuously avoided committing himself. We end with Poincaré on Couturat, Russell, and Hilbert.

Value capture occurs when an agent’s values are rich and subtle; they enter a social environment that presents simplified — typically quantified — versions of those values; and those simplified articulations come to dominate their practical reasoning. Examples include becoming motivated by FitBit’s step counts, Twitter Likes and Re-tweets, citation rates, ranked lists of best schools, and Grade Point Averages. We are vulnerable to value capture because of the competitive advantage that such crisp and clear expressions of value have in (...) our private reasoning and our public justification. There is, however, a price. In value capture, we take a central component of our autonomy — our ongoing deliberation over the exact articulation of our values — and we outsource it. And the metrics to which we outsource usually engineered for the interests of some external force, like a large-scale institution’s interest in cross-contextual comprehensibility and quick aggregability. That outsourcing cuts off one of the key benefits to personal deliberation. In value capture, we no longer adjust our values and their articulations in light of own rich experience of the world. Our values should often be carefully tailored to our particular selves or our small-scale communities, but in value capture, we buy our values off the rack. In some cases – like decreasing CO2 emissions – the costs of non-tailored values are outweighed by the benefit of precise collective coordination. In other cases, like in our aesthetic lives, they are not. This suggests that we should want different values suited to different scales. We should want value federalism. Some values are perhaps best pursued at the largest-scale level, others at smaller scales. The problem occurs when we exhibit an excess preference for the largest-scale values – when we consistently let the universal metrics swamp our quieter interests. (shrink)

We offer an account of the generic use of the term “porn”, as seen in recent usages such as “food porn” and “real estate porn”. We offer a definition adapted from earlier accounts of sexual pornography. On our account, a representation is used as generic porn when it is engaged with primarily for the sake of a gratifying reaction, freed from the usual costs and consequences of engaging with the represented content. We demonstrate the usefulness of the concept of generic (...) porn by using it to isolate a new type of such porn: moral outrage porn. Moral outrage porn is representations of moral outrage, engaged with primarily for the sake of the resulting gratification, freed from the usual costs and consequences of engaging with morally outrageous content. Moral outrage porn is dangerous because it encourages the instrumentalization of one’s empirical and moral beliefs, manipulating their content for the sake of gratification. Finally, we suggest that when using porn is wrong, it is often wrong because it instrumentalizes what ought not to be instrumentalized. (shrink)

This note argues against Barwise and Etchemendy's claim that their semantics for self-reference requires use of Aczel's anti-foundational set theory, AFA, semantics for self-reference requires use of Aczel's anti-foundational set theory, AFA, ones irrelevant to the task at hand" (The Liar, p. 35). Switching from ZF to AFA neither adds nor precludes any isomorphism types of sets. So it makes no difference to ordinary mathematics. I argue against the author's claim that a certain kind of 'naturalness' nevertheless makes AFA preferable (...) to ZF for their purposes. I cast their semantics in a natural, isomorphism invariant form with self-reference as a fixed point property for propositional operators. Independent of the particulars of any set theory, this form is somewhat simpler than theirs and easier to adapt to other theories of self-reference. (shrink)

Structuralism in philosophy of mathematics has largely focused on arithmetic, algebra, and basic analysis. Some have doubted whether distinctively structural working methods have any impact in other fields such as differential equations. We show narrowly construed structuralism as offered by Benacerraf has no practical role in differential equations. But Dedekind’s approach to the continuum already did not fit that narrow sense, and little of mathematics today does. We draw on one calculus textbook, one celebrated analysis textbook, and a monograph on (...) the Navier–Stokes equation to show structural methods like Dedekind’s have long been central to differential equations, and have philosophically respectable ontology and epistemology. (shrink)

The article looks briefly at Fefermans own foundations. Among many different senses of foundations, the one that mathematics needs in practice is a recognized body of truths adequate to organize definitions and proofs. Finding concise principles of this kind has been a huge achievement by mathematicians and logicians. We put ZFC and categorical foundations both into this context.

The large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in practice, yet published foundations for them go beyond ZFC in logical strength. We reduce the gap by founding all the theorems of Grothendieck’s SGA, plus derived categories, at the level of Finite-Order Arithmetic, far below ZFC. This is the weakest possible foundation for the large-structure tools because one elementary topos of sets with infinity is already this strong.

In The Case for Animal Rights, Tom Regan argues that although all subjects-of-a-life have equal inherent value, there are often differences in the value of lives. According to Regan, lives that have the highest value are lives which have more possible sources of satisfaction. Regan claims that the highest source of satisfaction, which is available to only rational beings, is the satisfaction associated with thinking impartially about moral choices. Since rational beings can bring impartial reasons to bear on decision making, (...) Regan maintains that they have an additional possible source of satisfaction that nonrational beings do not have and, consequently, the lives of rational beings turn out to have greater value.. (shrink)

What is a game? What are we doing when we play a game? What is the value of playing games? Several different philosophical subdisciplines have attempted to answer these questions using very distinctive frameworks. Some have approached games as something like a text, deploying theoretical frameworks from the study of narrative, fiction, and rhetoric to interrogate games for their representational content. Others have approached games as artworks and asked questions about the authorship of games, about the ontology of the work (...) and its performance. Yet others, from the philosophy of sport, have focused on normative issues of fairness, rule application, and competition. The primary purpose of this article is to provide an overview of several different philosophical approaches to games and, hopefully, demonstrate the relevance and value of the different approaches to each other. Early academic attempts to cope with games tried to treat games as a subtype of narrative and to interpret games exactly as one might interpret a static, linear narrative. A faction of game studies, self-described as “ludologists,” argued that games were a substantially novel form and could not be treated with traditional tools for narrative analysis. In traditional narrative, an audience is told and interprets the story, where in a game, the player enacts and creates the story. Since that early debate, theorists have attempted to offer more nuanced accounts of how games might achieve similar ends to more traditional texts. For example, games might be seen as a novel type of fiction, which uses interactive techniques to achieve immersion in a fictional world. Alternately, games might be seen as a new way to represent causal systems, and so a new way to criticize social and political entities. Work from contemporary analytic philosophy of art has, on the other hand, asked questions whether games could be artworks and, if so, what kind. Much of this debate has concerned the precise nature of the artwork, and the relationship between the artist and the audience. Some have claimed that the audience is a cocreator of the artwork, and so games are a uniquely unfinished and cooperative art form. Others have claimed that, instead, the audience does not help create the artwork; rather, interacting with the artwork is how an audience member appreciates the artist's finished production. Other streams of work have focused less on the game as a text or work, and more on game play as a kind of activity. One common view is that game play occurs in a “magic circle.” Inside the magic circle, players take on new roles, follow different rules, and actions have different meanings. Actions inside the magic circle do not have their usual consequences for the rest of life. Enemies of the magic circle view have claimed that the view ignores the deep integration of game life from ordinary life and point to gambling, gold farming, and the status effects of sports. Philosophers of sport, on the other hand, have approached games with an entirely different framework. This has lead into investigations about the normative nature of games—what guides the applications of rules and how those rules might be applied, interpreted, or even changed. Furthermore, they have investigated games as social practices and as forms of life. (shrink)

Анотація. У статті проаналізовані досягнення Соціального Вчительства Католицької Церкви, репрезентовані працями Лева ХІІІ, Пія ХІ, Пія ХІІ, Івана Павла ІІ, що розкривають змістовні характеристики поняття «принцип субсидіарності», його роль і значення в системі християнських цінностей. Принцип субсидіарності робить можливими такі взаємовідносини в соціальному житті, коли спільнота вищого порядку не втручається у внутрішнє життя спільноти нижчого порядку, перебираючи на себе належні тій функції; заради спільного добра, спільного блага вона надає їй у разі потреби підтримку й допомогу, узгоджуючи у такий спосіб її (...) взаємодію з іншими соціальними структурами. Принцип субсидіарності скеровує соціальну практику на утвердження спільного блага в людському співтоваристві. Поширення і застосування принципу субсидіарності протистоїть небезпеці «одержавлення» суспільства та найзагрозливіших проявів колективізму, обмежує абсолютизацію влади, бюрократизацію державних і соціокультурних структур, стає одним із гарантів дотримання прав і свобод громадян своєї країни. (shrink)

This book goes far beyond its title. Landry indeed surveys current definitions of “mathematical platonism” to show nothing like them applies to Socrates in Plat.

In her BBC Reith Lectures on Trust, Onora O’Neill offers a short, but biting, criticism of transparency. People think that trust and transparency go together but in reality, says O'Neill, they are deeply opposed. Transparency forces people to conceal their actual reasons for action and invent different ones for public consumption. Transparency forces deception. I work out the details of her argument and worsen her conclusion. I focus on public transparency – that is, transparency to the public over expert domains. (...) I offer two versions of the criticism. First, the epistemic intrusion argument: The drive to transparency forces experts to explain their reasoning to non-experts. But expert reasons are, by their nature, often inaccessible to non-experts. So the demand for transparency can pressure experts to act only in those ways for which they can offer public justification. Second, the intimate reasons argument: In many cases of practical deliberation, the relevant reasons are intimate to a community and not easily explicable to those who lack a particular shared background. The demand for transparency, then, pressures community members to abandon the special understanding and sensitivity that arises from their particular experiences. Transparency, it turns out, is a form of surveillance. By forcing reasoning into the explicit and public sphere, transparency roots out corruption — but it also inhibits the full application of expert skill, sensitivity, and subtle shared understandings. The difficulty here arises from the basic fact that human knowledge vastly outstrips any individual’s capacities. We all depend on experts, which makes us vulnerable to their biases and corruption. But if we try to wholly secure our trust — if we leash groups of experts to pursuing only the goals and taking only the actions that can be justified to the non-expert public — then we will undermine their expertise. We need both trust and transparency, but they are in essential tension. This is a deep practical dilemma; it admits of no neat resolution, but only painful compromise. (shrink)

The article surveys some past and present debates within mathematics over the meaning of category theory. It argues that such conceptual analyses, applied to a field still under active development, must be in large part either predictions of, or calls for, certain programs of further work.

Category theory has two unexpected links to constructivism: First, why is topos logic so close to intuitionistic logic? The paper argues that in part the resemblance is superficial, in part it is due to selective attention, and in part topos theory is objectively tied to the motives for later intuitionistic logic little related to Brouwer’s own stated motives. Second, why is so much of general category theory somehow constructive? The paper aims to synthesize three hypotheses on why it would be (...) so, with three that suggest it is not. (shrink)

Category theory has two unexpected links to constructivism: First, why is topos logic so close to intuitionistic logic? The paper argues that in part the resemblance is superficial, in part it is due to selective attention, and in part topos theory is objectively tied to the motives for later intuitionistic logic little related to Brouwer’s own stated motives. Second, why is so much of general category theory somehow constructive? The paper aims to synthesize three hypotheses on why it would be (...) so, with three that suggest it is not. (shrink)

Recent discussions in the just war literature suggest that soldiers have a duty to assume certain risks in order to protect the lives of all innocent civilians. I challenge this principle of risk by arguing that it is justified neither as a principle that guides the conduct of combat soldiers, nor as a principle that guides commanders in the US military. I demonstrate that the principle of risk fails on the first account because it requires soldiers both to violate their (...) strict duty of obedience and loyalty and to exceed their special obligations to protect their fellow comrades, the state, the state's constituents and other protected civilians. I then illustrate that the principle of risk fails on the second account since it conflicts with the commander's primary obligation to protect and promote the welfare and lives of his or her soldiers. I conclude by arguing that we cannot reasonably expect soldiers and commanders to adhere to the principle of risk until there is a radical, institutional-level transformation of militaristic goals, values, strategies, policies, warrior codes and expectations of service members in the US Armed Forces. (shrink)

Alexandre Grothendieck foi um dos maiores matemáticos do século 20 e um dos mais atípicos. Nascido na Alemanha a um pai anarquista de origem russa, sua infância foi marcada pela militância política dos seus pais, assim passando por revoluções, guerras e sobrevivência. Descoberto por sua precocidade matemática por Henri Cartan, Grothendieck fez seu doutorado sob orientação de Laurent Schwartz e Jean Dieudonné. As principais contribuições dele são na área da topologia e na geometria algébrica, assim como na teoria das categorias. (...) No final dos anos de 1960, ele se dedicou à militância política e ecológica, organizando a revista Survivre durante três anos. Em 1986, publicou um manuscrito autobiográfico de 1000 páginas, Récoltes et semailles, em que ele descreve sua experiência e sua prática da matemática, assim suas contribuições à comunidade matemática francesa. Pouco comentado na filosofia, as implicações dos seus descobrimentos fora mais recentemente discutidas por Alain Badiou na sua "fenômeno-lógica", em Logiques des mondes e Arkady Plonitsky, Mathgematics, Science and postclassical Theory, pesquisa trata da semelhança entre os aspectos formais da filosofia de Gilles Deleuze e da topologia de Grothendieck. (shrink)

Emmy Noether’s many articles around the time that Felix Klein and David Hilbert were arranging her invitation to Göttingen include a short but brilliant note on invariants of finite groups highlighting her creativity and perspicacity in algebra. Contrary to the idea that Noether abandoned Paul Gordan’s style of mathematics for Hilbert’s, this note shows her combining them in a way she continued throughout her mature abstract algebra.