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.

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)

The article considers structuralism as a philosophy of mathematics, as based on the commonly accepted explicit mathematical concept of a structure. Such a structure consists of a set with specified functions and relations satisfying specified axioms, which describe the type of the structure. Examples of such structures such as groups and spaces, are described. The viewpoint is now dominant in organizing much of mathematics, but does not cover all mathematics, in particular most applications. It does not explain why certain structures (...) are dominant, not why the same mathematical structure can have so many different and protean realizations. ‘structure’ is just one part of the full situation, which must somehow connect the ideal structures with their varied examples. (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)

The Protean Character of Mathematics SAUNDERS MAC LANE (Chicago) 1. Introduction The thesis of this paper is that mathematics is protean. ...

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)

A vibrant portrait of four college friends-Iris Murdoch, Philippa Foot, Elizabeth Anscombe, and Mary Midgley-who formed a new philosophical tradition while Oxford's men were away at war.

It is shown that the Hilbert-Bernays-Quine principle of identity of indiscernibles applies uniformly to all the contentious cases of symmetries in physics, including permutation symmetry in classical and quantum mechanics. It follows that there is no special problem with the notion of objecthood in physics. Leibniz's principle of sufficient reason is considered as well; this too applies uniformly. But given the new principle of identity, it no longer implies that space, or atoms, are unreal.

But this picture of a ‘block universe’, composed of a timeless web of ‘world-lines’ in a four-dimensional space, however strongly suggested by the theory of relativity, is a piece of gratuitous metaphysics. Since the concept of change, of something happening, is an inseparable component of the common-sense concept of time and a necessary component of the scientist's view of reality, it is quite out of the question that theoretical physics should require us to hold the Eleatic view that nothing happens (...) in ‘the objective world’. Here, as so often in the philosophy of science, a useful limitation in the form of representation is mistaken for a deficiency of the universe. (shrink)

The globalized nature of modern organizations presents new and intimidating challenges for effective relationship building. Organizations and their employees are increasingly being asked to manage unfamiliar relationships with unfamiliar parties. These relationships not only involve working across different national cultures, but also dealing with different organizational cultures, different professional cultures and even different internal constituencies. Managing such differences demands trust. This book brings together research findings on organizational trust-building across cultures. Established trust scholars from around the world consider the development (...) and maintenance of trust between, for example, management consultants and their clients, senior international managers from different nationalities, different internal organizational groupings during times of change, international joint ventures, and service suppliers and the local communities they serve. These studies, set in a wide variety of national settings, are an important resource for academics, students and practitioners who wish to know more about the nature of cross-cultural trust-building in organizations"--Provided by publisher. (shrink)

Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint (...) pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general­ ized monoid. Chapters VI and VII explore this notion and its generaliza­ tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure lead inter alia to the study of more convenient categories of topological spaces. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy's Role in Psychopathology Back to Jaspers and an Appeal to Grow PracticalThe author reports no conflicts of interest.In "Philosophy's role in theorizing psychopathology," Gibson presents a defense of the continued relevance of philosophy to psychopathology, and a non-exhaustive framework for the role of philosophy in this domain (Gibson, 2024). I find it hard to disagree that psychopathology is soaked in philosophy from its origins, and that to try (...) and dry it out would be both unwise and impossible.In this commentary, I go back to Karl Jaspers' account of the role of philosophy in his General Psychopathology. In doing so, I do not find that Gibson adds anything new, but instead gives us a well-illustrated reminder that philosophy "is operative within every living science" (Jaspers, 1997, p. 770).Situated Psychopathological TheorizingGibson addresses a broad 'we' who might be engaged with psychopathological theorizing. This seems to encompass three main settings, from which examples are drawn.1. Everyday and regulative social practices (ordinary and legal setting).2. Empirically driven disciplines which work with psychopathological constructs (academic setting).3. Psychiatry and psychotherapy (clinical setting).In my view, it is a mistake not to map the terrain where psychopathological theorizing is situated. In each setting psychopathological theorizing has its own set of aims and constraints. In the ordinary setting, the aim might be to know whether to blame a loved one for their action. In the legal setting it might be to decide whether treatment or punishment is appropriate. In the clinical setting, the aim is usually a therapeutic assessment, and moral stance taking is often best kept at bay. In the academic setting, we might want to ensure our concepts are reliable and valid, opening them up to investigation using objective scientific methods. [End Page 13]Beyond what can be said in the most general terms, the role of philosophy will look different in these different situations.Understanding Is Not (Just) a Philosophical ToolUnderstanding is a mode of knowing that is concerned with the particular, rather than the general, and accessed inter-relationally, relying on tacit knowledge and empathetic imagination. Gibson introduces the concept as an elementary form of social intuition, used in ordinary life to make sense of social relations and other people's behaviors and reactions. I agree with Gibson, drawing on Strawson, that Understanding is a mode of knowledge that involves the subject of enquiry as a participant (Strawson, 1982). There is a moral imperative to take this stance, which drives us to value the expansion of what we are able to understand of other people's mental states. This at least partly motivates the agenda when we theorize about psychopathology.At times throughout his argument, Gibson seems to be arguing for the role of Understanding in psychopathological theorizing, in place of his explicit aim of arguing for the role of philosophy. However it is not at all clear that this replacement can be made. Accepting the picture drawn, Understanding is not any more the expertise or domain of the philosopher than the author, police officer, or sociologist. We are surely not meant to take Understanding as a uniquely philosophical tool, but rather a mode of knowing available to all social agents, integral to social reasoning in ordinary life and in the human studies.What philosophy can distinctively do is point to and defend the existence of forms of intersubjective and tacit knowledge within human studies such as psychopathology. For Jaspers, Understanding can be a scientific tool, albeit one that reminds us that knowledge in psychopathology is not all of one kind. One role for philosophy therefore is to help us to "distinguish the different modes of knowing," so that we might "order knowledge into a comprehensive whole" (Jaspers, 1997, p. 769). This is akin to the 'synthetic' role that Gibson proposes.Philosophy and ScienceJaspers defines scientific knowledge broadly as knowledge with "general validity, convincing insights (which can be proved), clarity of method, and the possibility for a meaningful discussion" (Jaspers, 1997, p. 768).We do not get such a definition from Gibson, but in the sections of... (shrink)

Let's learn about the things all around us, with this fun, bright book of first concepts.

In this chapter, we compare Kant and James’ accounts of freedom. Despite both thinkers’ rejecting compatibilism for the sake of practical reason, there are two striking differences in their stances. The first concerns whether or not freedom requires the possibility of an open future. James holds that morality hinges on the real possibility that the future can be affected by our actions. Kant, on the other hand, seems to maintain that we can still be free in the crucial sense, even (...) if none of our actions can have any effect on the future. The second difference between them is related, and concerns the location of freedom. Kant views experience as determined by natural necessity, and locates freedom outside of it, in things-in-themselves. James, on the other hand, has a richer conception of experience than Kant, and holds that we can locate our freedom within experience alone. In the end, we contend that James has a better account of how freedom relates to our experience, but this comes at a cost. For while Kant's account struggles with the relationship between freedom and experience, it has the advantage of insulating our freedom from potential empirical challenges. (shrink)