Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these conditions . Instead of diagrams, pairs of derivations in Intuitionistic Multiplicative Linear logic can be considered . Two derivations of the same sequent are equivalent if and only if (...) all their interpretations in K are equal. In fact, the assignment of values to atoms is defined constructively for each pair of derivations. Taking into account a mistake in R. Voreadou's proof of the “abstract coherence theorem” found by the author, it was necessary to modify her description of the class of non-commutative diagrams in SMC categories; our proof of S. Mac Lane conjecture also proves the correctness of the modified description. (shrink)

Saunders Mac Lane famously remarked that "Bourbaki just missed" formulating adjoints in a 1948 appendix (written no doubt by Pierre Samuel) to an early draft of Algebre--which then had to wait until Daniel Kan's 1958 paper on adjoint functors. But Mac Lane was using the orthodox treatment of adjoints that only contemplates the object-to-object morphisms within a category, i.e., homomorphisms. When Samuel's treatment is reconsidered in view of the treatment of adjoints using heteromorphisms or hets (object-to-object morphisms between (...) objects in different categories), then he, in effect, isolated the concept of a left representation solving a universal mapping problem. When dualized to obtain the concept of a right representation, the two halves only need to be united to obtain an adjunction. Thus Samuel was only a now-simple dualization away for formulating adjoints in 1948. Apparently, Bodo Pareigis' 1970 text was the first and perhaps only text to give the heterodox "new characterization" (i.e., heteromorphic treatment) of adjoints. Orthodox category theory uses various relatively artificial devices to avoid formally recognizing hets--even though hets are routinely used by the working mathematician. Finally we consider a "philosophical" question as to whether the most important concept in category theory is the notion of an adjunction or the notion of a representation giving a universal mapping property (where adjunctions arise as the special case of a bi-representation of dual universal mapping problems). (shrink)

Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one (...) that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory , and obtain an apparently new proof that is not finitely axiomatisable; we study Friedman's strengthening of , and the Forster–Kaye subsystem of , and use forcing over ill-founded models and forcing to establish independence results concerning and ; we show, again using ill-founded models, that proves the consistency of ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that proves a weak form of Stratified Collection, and that is a conservative extension of for stratified sentences, from which we deduce that proves a strong stratified version of ; we analyse the known equiconsistency of with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion—namely that there exists an infinite set of infinite sets, no two of which have the same cardinal—and use it to establish the failure of the full schema of Stratified Collection in ; and we determine the point of failure of various other schemata in . The paper closes with some philosophical remarks. (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)

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)

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.

Inducing the rubber hand illusion (RHI) requires that participants look at an imitation hand while it is stroked in synchrony with their occluded biological hand. Previous explanations of the RHI have emphasized multisensory integration, and excluded higher cognitive functions. We investigated the relationship between the RHI and higher cognitive functions by experimentally testing task switch (as measured by switch cost) and mind wandering (as measured by SART score); we also included a questionnaire for attentional control that comprises two subscales, attention-shift (...) and attention-focus. To assess experience of RHI, the Botvinick and Cohen (1998) questionnaire was used and illusion onset time was recorded. Our results indicate that rapidity of onset reliably indicates illusion strength. Regression analysis revealed that participants evincing less switch cost and higher attention-shift scores had faster RHI onset times, and that those with higher attention-shift scores experienced the RHI more vividly. These results suggest that the multi-sensory hypothesis is not sufficient to explain the illusion: higher cognitive functions should be taken into account when explaining variation in the experience of ownership for the rubber hand. (shrink)

This paper considers various objections to Carnap’s logical syntax definition of ’logical expression’, including those by Saunders Mac Lane and W. V. O. Quine. While the specific objections of these two authors can be answered, if necessary by a slight modification of Carnap’s definition, there are other objections that I do not see how to meet. I also consider the proposal by Denis Bonnay for avoiding the objections to Carnap’s definition. In light of the unresolved problems with Carnap’s definition, (...) I go on to consider what Carnap’s assumptions must have been in framing his definition and to assess how much damage is caused by this failure of Carnap’s definition. This damage is not as much as might be assumed. (shrink)

Among Plato's works, the Statesman is usually seen as transitional between the Republic and the Laws. This book argues that the dialogue deserves a special place of its own. Whereas Plato is usually thought of as defending unchanging knowledge, Dr Lane demonstrates how, by placing change at the heart of political affairs, Plato reconceives the link between knowledge and authority. The statesman is shown to master the timing of affairs of state, and to use this expertise in managing the (...) conflict of opposed civic factions. To this political argument corresponds a methodological approach which is seen to rely not only on the familiar method of 'division', but equally on the unfamiliar centrality of the use of 'example'. The demonstration that method and politics are interrelated transforms our understanding of the Statesman and its fellow dialogues. (shrink)

Socrates wrote nothing; Plato's accounts of Socrates helped to establish western politics, ethics, and metaphysics. Both have played crucial and dramatically changing roles in western culture. In the last two centuries, the triumph of democracy has led many to side with the Athenians against a Socrates whom they were right to kill. Meanwhile the Cold War gave us polar images of Plato as both a dangerous totalitarian and an escapist intellectual. And visions of Plato have proliferated at the heart of (...) postmodern critiques of the very idea of metaphysics and politics. Plato's Progeny begins with an account of modern responses to the trial of Socrates and the controversial question of Socrates' relation to Plato. At its centre are two chapters exploring the idea of Platonic origins in and for philosophy, and of Platonic foundations for philosophical politics. Exploring unfamiliar as well as familiar invocations of Plato, Melissa Lane argues that twentieth-century ideological battles have obscured the importance of Socratic individualism, the nature of Platonic ethics, and the value of Platonic politics. Succinct and clearly written, this is an ideal guide for everyone interested in the way philosophers are still writing footnotes to Plato. (shrink)

It is argued that the Crito hinges on the relation between words and deeds. Socrates sets out a standard of agreement reached through persuasive argument or words. In this case the argument is deliberative: a general shared principle (do not do wrong) is juxtaposed to a particular minor premise (this act of escape is wrong) to reach a conclusion (do not escape). Crito baulks at the perception of the minor premise. At this juncture the Laws of Athens are introduced, who (...) set out a standard of agreement established instead by deeds. Both standards apply to Socrates, constituting the drama of the dialogue. (shrink)

"This edition of Eco-Republic is published by arrangement with Peter Lang Ltd; first published in 2011 by Peter Lang Ltd"--T.p. verso.

This paper presents a coherence theorem for star-autonomous categories exactly analogous to Kelly and Mac Lane’s coherence theorem for symmetric monoidal closed categories. The proof of this theorem is based on a categorial cut-elimination result, which is presented in some detail.

This article looks at Bernard Williams’s relevance to particular debates in constitutional theory about the legitimacy of two competing models of institutional design: political constitutionalism which endorses giving the final say on the meaning of constitutional rights to legislatures; and legal constitutionalism which endorses giving the final say on the meaning of rights to courts. Recent defences of political constitutionalism have made claims about the realism of their accounts when compared with legal constitutionalism and have co-opted Bernard Williams’s realism to (...) support their case. This article examines these claims, concluding that these accounts of political constitutionalism rely on a distinctly non-Williamsian form of political moralism in that they assume a legitimacy for political constitutionalism which is prior to politics and political disagreement. It offers an alternative defence of political constitutionalism, a partial defence, which, it argues, is closer to the realism of Bernard Williams than these accounts. (shrink)

It is shown that all the assumptions for symmetric monoidal categories follow from a unifying principle involving natural isomorphisms of the type →, called medial commutativity. Medial commutativity in the presence of the unit object enables us to define associativity and commutativity natural isomorphisms. In particular, Mac Lane’s pentagonal and hexagonal coherence conditions for associativity and commutativity are derived from the preservation up to a natural isomorphism of medial commutativity by the biendofunctor . This preservation boils down to an (...) isomorphic representation of the Yang–Baxter equation of symmetric and braid groups. The assumptions of monoidal categories, and in particular Mac Lane’s pentagonal coherence condition, are explained in the absence of commutativity, and also of the unit object, by a similar preservation of associativity by the biendofunctor . In the final section one finds coherence conditions for medial commutativity in the absence of the unit object. These conditions are obtained by taking the direct product of the symmetric groups for 0≤i≤n. (shrink)

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)

"Plato's Statesman reconsiders many questions familiar to readers of the Republic: questions in political theory - such as the qualifications for the leadership of a state and the best from of constitution (politeia) - as well as questions of philosophical methodology and epistemology. Instead of the theory of Forms that is the centrepiece of the epistemology of the Republic, the emphasis here is on the dialectical practice of collection and division (diairesis), in whose service the interlocutors also deploy the ancillary (...) methods of myth and of models (paradeigmata). Plato here introduces the doctrine of due measure (to metrion) and a conception of statecraft (politikē) as an architectonic expertise that governs subordinate disciplines such as rhetoric and the military - doctrines later developed by Aristotle. Readers will find a sustained defence of the importance of expertise (technē or epistēmē) in the conduct of affairs of state, a robust (although not unqualified) defence of the rule of law, and an unsparing but nuanced critique of democratic government. The chapters in this volume provide a comprehensive and detailed philosophical engagement with the entirety of Plato's wide-ranging dialogue, with successive chapters devoted to the sections of the dialogue as it unfolds, and an introduction that places the dialogue in the context of Plato's philosophy as a whole. While not a commentary in the traditional sense, the volume engages with Plato's Statesman in its entirety" -- Publisher's description. (shrink)

This study reports the results of a survey designed to assess the impact of business education on the ethical beliefs of business students. The study examines the beliefs of graduate and undergraduate students about ethical behavior in educational settings. The investigation indicates that the behavior which students learn or perceive is required to succeed in business schools may run counter to the ethical sanctions of society and the business community.