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)

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)

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)

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)

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)

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.

Previous research has suggested that bodily signals from internal organs are associated with diverse cortical and subcortical processes involved in sensory-motor functions, beyond homeostatic reflexes. For instance, a recent study demonstrated that the preparation and execution of voluntary actions, as well as its underlying neural activity, are coupled with the breathing cycle. In the current study, we investigated whether such breathing-action coupling is limited to voluntary motor action or whether it is also present for mental actions not involving any overt (...) bodily movement. To answer this question, we recorded electroencephalography (EEG), electromyography (EMG), and respiratory signals while participants were conducting a voluntary action paradigm including self-initiated motor execution (ME), motor imagery (MI), and visual imagery (VI) tasks. We observed that the voluntary initiation of ME, MI, and VI are similarly coupled with the respiration phase. In addition, EEG analysis revealed the existence of readiness potential (RP) waveforms in all three tasks (i.e., ME, MI, VI), as well as a coupling between the RP amplitude and the respiratory phase. Our findings show that the voluntary initiation of both imagined and overt action is coupled with respiration, and further suggest that the breathing system is involved in preparatory processes of voluntary action by contributing to the temporal decision of when to initiate the action plan, regardless of whether this culminates in overt movements. -/- . (shrink)

This article rejects the claim made by other scholars that Plato in the Statesman, by employing the so-called ‘architect’ (ὁ ἀρχιτέκτων) in one of the early divisions leading to the definition of political expertise, prefigured and anticipated the architectonic conception of political expertise advanced by Aristotle. It argues for an alternative reading in which Plato in the Statesman, and in the only other of his works (Gorgias) in which the word appears, closely tracks the existing social role of the architektōn, (...) who was designated as such only in virtue of appointment by a city to a role that was crucially defined as epitactic, involving overseeing the workers on site engaged in constructing some civic building works. It is this epitactic dimension of the role on which Plato relies in the Statesman, as opposed to the kind of claim to overarching integrative expertise that Aristotle would use the figure of architectonic political knowledge to make. (shrink)

It is often said nowadays that to understand pleasure we must understand it as affording us a reason for or an explanation of action. It is only from the standpoint of the agent that we can avoid being misled. Both Professor Nowell-Smith and Mr. Manser have argued along these lines; and Dr. Kenny has written that “pleasure is always a reason for action” and has elucidated what he means by a footnote: “I do not mean that a thing’s being pleasant (...) is always a sufficient reason for doing it; there may be strong reasons for abstaining. I mean merely that it is always silly to ask a man why he wants pleasure.” When I first saw this point made, I had the perhaps not uncommon philosophical experience of immediately finding it both lucid and convincing, but then, afterwards gradually becoming less and less clear about the source of my conviction. The reasons for my obfuscation are as follows. In one crucial sense anything can afford an agent with a reason for action. It depends upon what the agent wants and upon the projects in which he is engaged. Moreover, without having any strong reasons for abstaining from what will give me pleasure, I may not be at all moved by the prospect of pleasure. As I write this paper I can list a dozen activities or experiences which would afford me pleasure. I have no strong reason for abstaining; I do not particularly enjoy writing papers; I am not writing with a great sense of urgency; I have the time and the money to indulge myself. Yet I do not rush to open a Guinness or Mr. Alfred Grossman’s new novel. So that far from it being silly to ask me why I do not apparently want pleasure at the moment, it is a question that I find forced upon me. But if this question makes sense, there is at least a problem as to why “it is always silly to ask a man if he wants pleasure.” For one might expect the two questions to stand or fall together. This is the problem to which I address myself in this paper. But in order to do so I must first take up certain points from recent discussions which are in danger of preventing a solution. (shrink)

Finance is an area that, in practice, is plagued by accusations of unethical activity; the study of finance had adopted a largely nonbehavioral approach to business ethics research. We address this gap in by assessing whether individual ethical orientations predict the acceptability of questionable decisions about financial issues. Results show that individual ethical orientations are associated with different levels of acceptability of questionable decisions about financial issues, though the pattern of these differences varies across individual ethical orientations assessed. These results (...) represent evidence that ethical individual differences are associated with the acceptability of questionable finance decisions and are discussed in terms of methodological limitations and future directions in finance ethics research. (shrink)

Neurovascular coupling (NVC), the transient regional hyperemia following the evoked neuronal responses, is the basis of blood oxygenation level-dependent techniques and is generally adopted across physiological conditions, including the intrinsic resting state. However, the possibility of neurovascular dissociations across physiological alterations is indicated in the literature. To examine the NVC stability across sleep–wake states, we used electroencephalography (EEG) as the index of neural activity and functional magnetic resonance imaging (fMRI) as the measure of cerebrovascular response. Eight healthy adults were recruited (...) for simultaneous EEG-fMRI recordings in nocturnal sleep. We compared the cross-modality (EEG vs. fMRI) consistency of functional indices (spectral amplitude and functional connectivity) among five states of wakefulness and sleep (state effect). We also segregated the brain into three main partitions (anterior, middle and posterior) for spatial assessments (regional effect). Significant state effects were found on δ, α and fMRI indices and regional effects on the α and fMRI indices. However, the cross-state EEG changes in spectral amplitude and functional connectivity did not consistently match the changes in the fMRI indices across sleep–wake states. In spectral amplitude, the δ band peaked at the N3 stage for all brain regions, while the fMRI fluctuation amplitude peaked at the N2 stage in the central and posterior regions. In regional connectivity, the inter-hemispheric connectivity of the δ band peaked at the N3 stage for all regions, but the bilateral fMRI connectivity showed the state changes in the anterior and central regions. The cross-modality inconsistencies across sleep–wake states provided preliminary evidence that the neurovascular relationship may not change in a linear consistency during NREM sleep. Thus, caution shall be exercised when applying the NVC presumption to investigating sleep/wake transitions, even among healthy young adults. (shrink)

This selection is an argument for the pharmacology of cannabinoids. It also provides a phenotypical account of the botany of cannabis.

Objective Manic and depressive phases of bipolar disorder (BD) show opposite psychomotor symptoms. Neuronally, these may depend on altered relationships between sensorimotor network (SMN) and subcortical structures. The study aimed to investigate the functional relationships of SMN with substantia nigra (SN) and raphe nuclei (RN) via subcortical-cortical loops, and their alteration in bipolar mania and depression, as characterized by psychomotor excitation and inhibition. -/- Method In this resting-state functional magnetic resonance imaging (fMRI) study on healthy (n = 67) and BD (...) patients (n = 100), (1) functional connectivity (FC) between thalamus and SMN was calculated and correlated with FC from SN or RN to basal ganglia (BG)/thalamus in healthy; (2) using an a-priori-driven approach, thalamus-SMN FC, SN-BG/thalamus FC, and RN-BG/thalamus FC were compared between healthy and BD, focusing on manic (n = 34) and inhibited depressed (n = 21) patients. -/- Results (1) In healthy, the thalamus-SMN FC showed a quadratic correlation with SN-BG/thalamus FC and a linear negative correlation with RN-BG/thalamus FC. Accordingly, the SN-related FC appears to enable the thalamus-SMN coupling, while the RN-related FC affects it favoring anti-correlation. (2) In BD, mania showed an increase in thalamus-SMN FC toward positive values (ie, thalamus-SMN abnormal coupling) paralleled by reduction of RN-BG/thalamus FC. By contrast, inhibited depression showed a decrease in thalamus-SMN FC toward around-zero values (ie, thalamus-SMN disconnection) paralleled by reduction of SN-BG/thalamus FC (and RN-BG/thalamus FC). The results were replicated in independent HC and BD datasets. -/- Conclusions These findings suggest an abnormal relationship of SMN with neurotransmitters-related areas via subcortical-cortical loops in mania and inhibited depression, finally resulting in psychomotor alterations. (shrink)

Augustine is the person from the ancient world about whom we know most. He is the author of an intimate masterpiece, the Confessions, which continues to delight its many admirers. In it he writes about his infancy and his schooling in the classics in late Roman North Africa, his remarkable mother, his sexual sins ('Give me chastity, but not yet,' he famously prayed), his time in an outlawed heretical sect, his worldly career and friendships and his gradual return to God. (...) His account of his own eventual conversion is a classic study of anguish, hesitation and what he believes to be God's intervention. It has inspired philosophers, Christian thinkers and monastic followers, but it still leaves readers wondering why exactly Augustine chose to compose a work like none before it. Robin Lane Fox follows Augustine on a brilliantly described journey, combining the latest scholarship with recently found letters and sermons by Augustine himself to give a portrait of his subject which is subtly different from older biographies. Augustine's heretical years as a Manichaean, his relation to non-Christian philosophy, his mystical aspirations and the nature of his conversion are among the aspects of his life which stand out in a sharper light. For the first time Lane Fox compares him with two contemporaries, an older pagan and a younger Christian, each of whom also wrote about themselves and who illumine Augustine's life and writings by their different choices. More than a decade passed between Augustine's conversion and his beginning the Confessions. Lane Fox argues that the Confessions and their thinking were the results of a long gestation over these years, not a sudden change of perspective, but that they were then written as a single swift composition and that its final books are a coherent consummation of its scriptural meditation and personal biography. This exceptional study reminds us why we are so excited and so moved by Augustine's story. Praise for Pagans and Christians 'This open-hearted and learned book is one that any scholar of the ancient world and of early Christianity would be proud to have written... Lane Fox has opened his pages to let in an entirely new world.' Peter Brown, New York Review of Books 'A work of generous humanity and recondite erudition... of consuming interest to all of us who want to know and understand the consequences of the ancient world.' Henry Chadwick, Financial Times Praise for The Unauthorised Version 'I don't know where one would turn for an account of the Bible so clear, so lively and, in many respects, so just.' Frank Kermode, Guardian 'Magnificent... delivered with authority and verve. Learned but never pedantic, he is an unfailingly incisive, thought-provoking, humane, courteous, and often entertaining guide.' Economist. (shrink)