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)

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 Politics is one of the most influential texts in the history of political thought, and it raises issues which still confront anyone who wants to think seriously about the ways in which human societies are organized and governed. The work of one of the world's greatest philosophers, it draws on Aristotle's own great knowledge of the political and constitutional affairs of the Greek cities. By examining the way societies are run - from households to city states - Aristotle establishes (...) how successful constitutions can best be initiated and upheld. For this edition Sir Ernest Barker's fine translation, which has been widely used for nearly half a century, has been extensively revised to meet the needs of the modern reader. The accessible introduction and clear notes by R F Stalley examine the historical and philosophical background of the work and discuss its significance for modern political thought. (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 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)

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)

For millennia self has been conjectured to be necessary for consciousness. But scant empirical evidence has been adduced to support this hypothesis. Inconsistent explications of “self” and failure to design apt experiments have impeded progress. Advocates of phenomenological psychiatry, however, have helped explicate “self,” and employed it to explain some psychopathological symptoms. In those studies, “self” is understood in a minimalist sense, sheer “for-me-ness.” Unfortunately, explication of the “minimal self” (MS) has relied on conceptual analysis, and applications to psychopathology have (...) been hermeneutic, allowing for many degrees of interpretive latitude. The result is that MS’s current scientific status is analogous to that of the “atom,” at the time when “atom” was just beginning to undergo transformation from a philosophical to a scientific concept. Fortunately, there is now an opportunity to promote a similar transformation for “MS.” Discovery of the brain’s Default Mode Network (DMN) opened the door to neuroimaging investigations of self. Taking the DMN and other forms of intrinsic activity as a starting point, an empirical foothold can be established, one that spurs experimental research and that enables extension of research into multiple phenomena. New experimental protocols that posit “MS” can help explain phenomena hitherto not thought to be related to self, thereby hastening development of a mature science of self. In particular, targeting phenomena wherein consciousness is lost and recovered, as in some cases of Unresponsive Wakefulness Syndrome (UWS), allow for design of neuroimaging probes that enable detection of MS during non-conscious states. These probes, as well as other experimental protocols applied to NREM Sleep, General Anesthesia (GA), and the waking state, provide some evidence to suggest that not only can self and consciousness dissociate, MS might be a necessary precondition for conscious experience. Finally, these findings have implications for the science of consciousness: it has been suggested that “levels of consciousness” (LoC) is not a legitimate concept for the science of consciousness. But because we have the conceptual and methodological tools with which to refine investigations of MS, we have the means to identify a possible foundation—a bifurcation point—for consciousness, as well as the means by which to measure degrees of distance from that foundation. These neuroimaging investigations of MS position us to better assess whether LoC has a role to play in a mature science of consciousness. (shrink)

Freedom after Kant situates Kant's concept of freedom in relation to leading philosophers of the period to trace a detailed history of philosophical thinking on freedom from the 18th to the 20th century. Beginning with German Idealism, the volume presents Kant's writings on freedom and their reception by contemporaries, successors, followers and critics. From exchanges of philosophical ideas on freedom between Kant and his contemporaries, Reinhold and Fichte, through to Kant's ideas on rational self-determination in Hegel and Schelling, we see (...) Kant's original arguments transformed through concepts of autonomy, freedom and absolutes. The political aspect of Kant's freedom finds further articulation in chapters on Marx and Mill who developed their own notions of political freedom after Kant. Revealing how Kant's concept of freedom shaped the history of philosophy in the broadest sense, contributors chart the development of an ethics of freedom in the 20th century which brings Kant into conversation with Heidegger, Beauvoir, Sartre, Levinas and Murdoch. This line of thinking on freedom signals a new departure for Kantian studies which brings his ideas into the present day and traverses major schools of thought including Idealism, Marxism, existentialism and moral philosophy. (shrink)

In this review of the Korean cloning scandal involving Woo-Suk Hwang, the nature of the disaster is documented and reasons why it occurred are suggested. The general problems it raises for scientific research are highlighted and six possible ways of improving practice are offered in the light of this case: better education of science students; independent monitoring and validation; guidelines for tissue donation for research; fostering of debate about ethically contentious research in science journals; development of an international code of (...) ethical research practice; fostering of public involvement in ethical review and debate through the web. (shrink)

The paper defends a view of structural realism similar to that of French and Ladyman, although it differs from theirs in an important respect: I do not take indistinguishabiity of particles in quantum mechanics to have the significance they think it has. It also differs from Cao's view of structural realism, criticized in my "Critical Notice: Cao's `The Conceptual Development of 20th Century Field Theories".

Drawing on compelling material from research interviews with former hostages and political prisoners, Guy Saunders reworks three classic thought experiment stories: Parfit's 'Teleporter', Nagel's 'What is it like to be a bat?' and Jackson's 'Mary the colour scientist' to form a fresh look at the study of consciousness. By examining consciousness from a social psychology perspective, Saunders develops a 'cubist psychology of consciousness' through which he challenges the accepted wisdom of mainstream approaches by arguing that people can act (...) freely. What makes 'cubist psychology' is both the many examples taken from different viewpoints and the multiple ways of looking at the key issues of person, mind and world. This is a unique and engaging book that will appeal to students and academics in the field of consciousness studies and other readers with an interest in consciousness. (shrink)