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)

I show that frequentism, as an explanation of probability in classical statistical mechanics, can be extended in a natural way to a decoherent quantum history space, the analogue of a classical phase space. The result is a form of finite frequentism, in which Gibbs’ concept of an infinite ensemble of gases is replaced by the quantum state expressed as a superposition of a finite number of decohering microstates. It is a form of finite and actual frequentism (as opposed to hypothetical (...) frequentism) insofar as all the microstates exist, in keeping with the decoherence-based Everett interpretation, and some versions of pilot-wave theory. (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)

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)

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".

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)

Book Information Intercultural Philosophy. By Ram Adhar Mall. Rowman & Littlefield. 2000. Pp. xiii + 152. Hardback, US$62.00. Paperback, US$16.95.

Ces études débordent le problème du temps chez les Stoïciens et se proposent d’établir que cette question, en apparence modeste, permet d’éclairer, et en même temps commande l’ensemble du système.Au premier abord en effet, la théorie du temps se présente comme une simple section d’un chapitre de la Physique, celui qui traite des Incorporels. Mais déjà la simple interprétation des textes transmis fait voir que cette théorie tient étroitement à d’autres théories, comme celle des incorporels en général, des catégories, de (...) la substance. Abordant ensuite les thèses majeures de la logique et de l’éthique, on s’aperçoit qu’à leur tour elles sont solidaires de la conception du temps qui, dans bien des cas, fait évanouir leur apparence de paradoxe. A cela même, il n’y a rien d’étrange, si l’on se souvient que le stoïcisme, plus que tout autre philosophie antique, constitue un système cohérent. Mais plus particulièrement, le problème du temps a dû former comme le noeud de la réflexion stoïcienne qui vise, contre les lourdes autorités de Platon et d’Aristote, à rétablir dans sa réalité et dans sa dignité, le concret, le sensible, disqualifié comme « sujet à la génération et à la corruption », c’est-à-dire, en un mot, comme « l’être dans le temps ». (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)

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)

Time is an essential feature in bipolar disorder (BP). Manic and depressed BP patients perceive the speed of time as either too fast or too slow. The present article combines theoretical and empirical approaches to integrate phenomenological, psychological, and neuroscientific accounts of abnormal time perception in BP. Phenomenology distinguishes between perception of inner time, ie, self-time, and outer time, ie, world-time, that desynchronize or dissociate from each other in BP: inner time speed is abnormally slow (as in depression) or fast (...) (as in mania) and, by taking on the role as default-mode function, impacts and modulates the perception of outer time speed in an opposite way, ie, as too fast in depression and too slow in mania. Complementing, psychological investigation show opposite results in time perception, ie, time estimation and reproduction, in manic and depressed BP. Neuronally, time speed can be indexed by neuronal variability, ie, SD. Our own empirical data show opposite changes in manic and depressed BP (and major depressive disorder [MDD]) with abnormal SD balance, ie, SD ratio, between somatomotor and sensory networks that can be associated with inner and outer time. Taken together, our combined theoretical-empirical approach demonstrates that desynchronization or dissociation between inner and outer time in BP can be traced to opposite neuronal variability patterns in somatomotor and sensory networks. This opens the door for individualized therapeutic “normalization” of neuronal variability pattern in somatomotor and sensory networks by stimulation with TMS and/or tDCS. (shrink)