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)

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.

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)

“Probably Peirce’s best-known works are the first two articles in a series of six that originally were collectively entitled Illustrations of the Logic of Science and published in Popular Science Monthly from November 1877 through August 1878. The first is entitled ‘The Fixation of Belief’ and the second is entitled ‘How to Make Our Ideas Clear.’ In the first of these papers Peirce defended, in a manner consistent with not accepting naive realism, the superiority of the scientific method over other (...) methods of overcoming doubt and ‘fixing belief.’” — Robert Burch, “Charles Sanders Peirce,” entry in the Stanford Encyclopedia of Philosophy (2021 revision) -/- “Pragmatist epistemologies often explore how we can carry out inquiries in a self-controlled and fruitful way. (Where much analytic epistemology centres around the concept of knowledge, considered as an idealised end-point of human thought, pragmatist epistemology centres around the concept of inquiry, considered as the process of knowledge-seeking and how we can improve it.) So pragmatists often provide rich accounts of the capacities or virtues that we must possess in order to inquire well, and the rules or guiding principles that we should adopt. A canonical account is Peirce’s classic early paper ‘The Fixation of Belief‘. Here Peirce states that inquiry is a struggle to replace doubt with “settled belief“, and that the only method of inquiry that can make sense of the fact that at least some of us are disturbed by inconsistent beliefs, and will subsequently reflect upon which methods of fixing belief are correct is the Method of Science, which draws on the Pragmatic Maxim described above. This contrasts with three other methods of fixing belief: i) refusing to consider evidence contrary to one’s favored beliefs (the Method of Tenacity), ii) accepting an institution’s dictates (the Method of Authority), iii) developing the most rationally coherent or elegant-seeming belief-set (the A Priori Method).“ — Catherine Legg, “Pragmatism,” entry in the Stanford Encyclopedia of Philosophy (2021 revision). (shrink)

According to most accounts of trust, you can only trust other people (or groups of people). To trust is to think that another has goodwill, or something to that effect. I sketch a different form of trust: the unquestioning attitude. What it is to trust, in this sense, is to settle one’s mind about something, to stop questioning it. To trust is to rely on a resource while suspending deliberation over its reliability. Trust lowers the barrier of monitoring, challenging, checking, (...) and questioning. Trust sets up open pipelines between yourself and parts of the external world. Trust permits external resources to have a similar relationship to one as one’s internal cognitive faculties. This creates efficiency, but at the price of exquisite vulnerability. We must trust in this way because we are cognitively limited beings in a cognitively overwhelming world. Crucially, we can hold the unquestioning attitude towards objects and technologies. When I trust my climbing rope, I stop worrying about its reliability. When I trust my online calendaring system, I simply go to the events indicated, without question. But, one might worry, how could one ever hold such a normatively loaded attitude as trust towards mere objects? How could it ever make sense to feel betrayed by an object? Trust is our engine for expanding and outsourcing our agency — for binding external processes into our practical selves. Thus, we can be betrayed by our smartphones in the same way that we can be betrayed by our memory. When we trust, we try to make something a part of our agency, and we are betrayed when our part lets us down. To unquestioningly trust something is to let it in—to attempt to bring it inside one’s practical functioning. This suggests a new form of gullibility: agential gullibility, which occurs when agents too hastily and carelessly integrate external resources into their own agency. (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)

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)

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)

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.

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.

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.

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)

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.

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.

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)

Recent conversation has blurred two very different social epistemic phenomena: echo chambers and epistemic bubbles. Members of epistemic bubbles merely lack exposure to relevant information and arguments. Members of echo chambers, on the other hand, have been brought to systematically distrust all outside sources. In epistemic bubbles, other voices are not heard; in echo chambers, other voices are actively undermined. It is crucial to keep these phenomena distinct. First, echo chambers can explain the post-truth phenomena in a way that epistemic (...) bubbles cannot. Second, each type of structure requires a distinct intervention. Mere exposure to evidence can shatter an epistemic bubble, but may actually reinforce an echo chamber. Finally, echo chambers are much harder to escape. Once in their grip, an agent may act with epistemic virtue, but social context will pervert those actions. Escape from an echo chamber may require a radical rebooting of one's belief system. (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)

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)

This note describes Saunders Mac Lane as a philosopher, and indeed as a paragon naturalist philosopher. He approaches philosophy as a mathematician. But, more than that, he learned philosophy from David Hilbert’s lectures on it, and by discussing it with Hermann Weyl, as much as he did by studying it with the mathematically informed Göttingen Philosophy professor Moritz Geiger.

The creation of algebraic topology required “all the energy and the temperament of Emmy Noether” according to topologists Paul Alexandroff and Heinz Hopf. Alexandroff stressed Noether's radical pro-Russian politics, which her colleagues found in “poor taste”; yet he found “a bright trait of character.” She joined the Independent Social Democrats in 1919. They were tiny in Göttingen until that year when their vote soared as they called for a dictatorship of the proletariat. The Minister of the Army and many Göttingen (...) students called them Bolshevist terrorists. Noether's colleague Richard Courant criticized USPD radicalism. Her colleague Hermann Weyl downplayed her radicalism and that view remains influential but the evidence favors Alexandroff. Weyl was ambivalent in parallel ways about her mathematics and her politics. He deeply admired her yet he found her abstractness and her politics excessive and even dangerous. (shrink)

Why do moral people so often fail to act morally? Standard scientific answers point to poor moral judgment (based on deficient character development, reason, or intuition) or to situational pressure. I consider a third possibility: a relative lack of truly moral motivation and emotion. What has been taken for moral motivation is often instead a subtle form of egoism. Recent research provides considerable evidence for moral hypocrisy—motivation to appear moral while, if possible, avoid the cost of actually being moral—but very (...) little evidence for moral integrity—motivation to actually be moral. The lack of truly moral motivation may, in turn, be linked to a lack of truly moral emotion, at least in response to violation of certain moral standards. (shrink)

forthcoming in R. Cameron, B. Hale and A. Hoffmann (ed.s), The Logic, Epistemology and Metaphysics of Modality, Oxford University Press. Presents a concept-grounding account of modal knowledge.