We investigated binocular rivalry in the twocerebral hemispheres of callosotomized(split-brain) observers. We found that rivalryoccurs for complex stimuli in split-brainobservers, and that it is similar in the twohemispheres. This poses difficulties for twotheories of rivalry: (1) that rivalry occursbecause of switching of activity between thetwo hemispheres, and (2) that rivalry iscontrolled by a structure in the rightfrontoparietal cortex. Instead, similar rivalryfrom the two hemispheres is consistent with atheory that its mechanism is low in the visualsystem, at which each hemisphere conducts (...) asimilar analysis of its half of visual space. (shrink)

No satisfactory answers were received for the following questions.

Selon M. E. Alflatt, ibid., 20, 1974, p. 113-134, Augustin d'Hippone dit que l'acte involontaire peut être un péché au sens propre du terme. L'A. montre que cette conclusion repose sur une interprétation erronée d'une traduction anglaise de « De lib. arb. » III, 19, 54, traduction faite par J. H. S. Burleigh et parue dans la « Library of Christian Classics », London, SCM Press, 1953.

In lieu of an abstract, here is a brief excerpt of the content:The Adoro Te Devote of St. Thomas AquinasSr. Lucia Marie of the Visitation Langford O.P.The Adoro te devote is perhaps the most well-beloved Eucharistic hymn of our time, popularly attributed to St. Thomas Aquinas, the medieval Dominican friar known for his theological treatises as well as his Eucharistic hymnography. Unlike most of Aquinas's work, the poem reveals the intensely personal side of his faith. Rich in theological content and (...) inspiring deep Eucharistic devotion, this poem amply repays a closer investigation. We will look first at the questions of authorship, background, and structure before moving on to a discussion of the text itself.The "Disputed Question" of AuthorshipWhile the Adoro te has traditionally been attributed to St. Thomas Aquinas, the Thomistic authorship of the text was contested in the twentieth century, with two main objections being raised. The first claims that the content of the poem contradicts Thomas's own teachings and cannot have flowed from the same pen. The second objection is historical, claiming that a lack of evidence makes Thomistic authorship improbable.The first objection, claiming that the content of the poem is irreconcilable with Thomistic authorship, takes objection to a line in the poem which seems to claim that sense perception is unreliable and inaccurate, at least with regard to the Eucharist. The text of the Adoro te uses the word fallitur (deceived) with reference to the senses: "Visus, tactus, gustus in te fallitur"—"Sight, touch, taste in you are deceived." Some scholars have claimed that this line contradicts the teaching of Aquinas that the senses are [End Page 365] infallible in regard to their proper sensibles.1 Robert Wielockx addresses this charge in his study of the Adoro te, arguing that "the text does not contradict the continuous teaching of Thomas according to which the senses are not wrong when they judge their proper object, which as far as the Eucharist is concerned, is only the sacramental species."2 The substantial presence of Christ in the Eucharist is a mystery which is not knowable by the senses, and as Paul Murray, O.P., observes: "That's surely the point St. Thomas is making."3 However, the Thomistic insistence that all knowledge comes through our senses is not missing even here. While sight, touch, and taste fall short of comprehending the truth before them, the Adoro te itself proclaims that it is another sense—hearing—which reveals the mystery: "sed auditu solo tute creditur"—"but by hearing alone it is safely believed."4Thus there is no solid objection to Thomistic authorship based on the poem's content. But what about the historical evidence? Dom André Wilmart published a comprehensive study of the manuscript tradition in 1929. Noting that there were no extant copies of the text dating to Aquinas's lifetime and that the biography of Aquinas written by his own Dominican confrere and contemporary, William of Tocco, made no mention of the Adoro te, he considered it most probable that the poem was authored at a later date and ascribed to Aquinas.5However, Dom Wilmart did not know that he was not working with William of Tocco's final version of his work. The publication of a critical edition of William of Tocco's Ystoria Sancti Thome de Aquino in 1996 demonstrated that Tocco published his work in four successive redactions. The fourth and final version of his text (unknown to Dom Wilmart) was finished just before his death in 1323, and in this final version, Tocco includes the text of the Adoro te, attributing its authorship to St. Thomas.6 [End Page 366] Furthermore, Wielockx published a study of the Adoro te in 1998; his investigation of the manuscript tradition found that, of fifty-one sources, forty-four explicitly cite St. Thomas as their author. The remaining sources simply lacked any attribution. We may thus conclude with Wielockx: "External evidence, which must be given priority in sound criticism, wholly favors Thomas Aquinas's authorship of the poem."7BackgroundWith the attribution of the poem to St. Thomas Aquinas on secure footing, we now move to the background of the poem. Unlike Aquinas's Eucharistic hymns such as... (shrink)

Research on social influences often distinguishes between social and quality incentives to ascribe meaning to the value that popularity conveys. This study examines the neural correlates of those incentives through which popularity influences preferences. This research reports an functional magnetic resonance imaging experiment and a behavioral task in which respondents evaluated popular products with three focus perspectives; unspecified focus, focus on social aspects, and focus on quality. The results show that value derived with a social focus reflects inferences of approval (...) and reward value, and positively affects preferences. Value derived with a quality focus reflects inferences of quality and negatively affects preferences. This study provides evidence of two distinct inferential routes on both a neurological level, represented by different regions in the brain, and a behavioral level. These results provide the first evidence that a single popularity cue can in different ways influence the value derived from product popularity. (shrink)

The Initiation Biblique of Robert-Tricot was first published in 1939, with an enlarged edition in 1948, and was made available to a wider public in 1951 when it appeared in English translation as Guide to the Bible. The third French edition was a thorough-going revision—some chapters were entirely re-written. The most important change was, beyond all question, the chapter on Inspiration by Father P. Benoit. As an exegete and a theologian, keenly aware both of the doctrinal principles involved and (...) of the real difficulties in the Bible, his approach to the question of inspiration and inerrancy of Scripture is truly illuminating; it disposes beforehand of a host of false problems which inevitably bedevil a notion of inspiration that is too rigid and, in fact, ‘unscriptural’. A thorough understanding of Benoit’s theory of Inspiration will rid one of a certain vague uneasiness and will leave one free to savour to the full the beauty and the riches, human as well as divine, of the Bible. The second edition of Guide to the Bible, Vol. I gives us the first nine chapters of the new Initiation Biblique. True to the spirit of the original it is not a mere translation. The bibliography has been consistently brought up to date and the translators have added very many footnotes. Besides, there are appendices which cover the idea of inspiration in Protestantism and Islam, and which give a detailed analysis of the books of the Pentateuch and an extended treatment of the Dead Sea Scrolls. It would be superfluous to emphasise the excellent qualities of the work. The appearance of this new edition should be a reminder to Catholics of our incalculable debt to French biblical scholarship. (shrink)

We make three points. First, even if Finlay et al.'s proposed developmental mechanisms hold, there remains great scope for selection on specific brain structures. Second, the positive covariance among the size of brain structures provides far less support for the proposed developmental mechanisms than Finlay et al. acknowledge. Third, even if the proposed mechanisms are the primary size determinants for most brain structures, these structures should not be considered.

Despite the frequency of stillbirths, the subsequent implications are overlooked and underappreciated. We present findings from comprehensive, systematic literature reviews, and new analyses of published and unpublished data, to establish the effect of stillbirth on parents, families, health-care providers, and societies worldwide. Data for direct costs of this event are sparse but suggest that a stillbirth needs more resources than a livebirth, both in the perinatal period and in additional surveillance during subsequent pregnancies. Indirect and intangible costs of stillbirth are (...) extensive and are usually met by families alone. This issue is particularly onerous for those with few resources. Negative effects, particularly on parental mental health, might be moderated by empathic attitudes of care providers and tailored interventions. The value of the baby, as well as the associated costs for parents, families, care providers, communities, and society, should be considered to prevent stillbirths and reduce associated morbidity. (shrink)

This collection of recent articles by leading scholars is designed to illuminate one of the greatest and most influential philosophical books of all time. It includes incisive commentary on every major theme and argument in the Meditations, and will be valuable not only to philosophers but to historians, theologians, literary scholars, and interested general readers.

Genomic research results and incidental findings with health implications for a research participant are of potential interest not only to the participant, but also to the participant's family. Yet investigators lack guidance on return of results to relatives, including after the participant's death. In this paper, a national working group offers consensus analysis and recommendations, including an ethical framework to guide investigators in managing this challenging issue, before and after the participant's death.

This article is the lead piece in a special report that presents the results of a bioethical investigation into chimeric research, which involves the insertion of human cells into nonhuman animals and nonhuman animal embryos, including into their brains. Rapid scientific developments in this field may advance knowledge and could lead to new therapies for humans. They also reveal the conceptual, ethical, and procedural limitations of existing ethics guidance for human‐nonhuman chimeric research. Led by bioethics researchers working closely with an (...) interdisciplinary work group, the investigation focused on generating conceptual clarity and identifying improvements to governance approaches, with the goal of helping scholars, funders, scientists, institutional leaders, and oversight bodies (embryonic stem cell research oversight [ESCRO] committees and institutional animal care and use committees [IACUCs]) deliver principled and trustworthy oversight of this area of science. The article, which focuses on human‐nonhuman animal chimeric research that is stem cell based, identifies key ethical issues in and offers ten recommendations regarding the ethics and oversight of this research. Turning from bioethics’ previous focus on human‐centered questions about the ethics of “humanization” and this research's potential impact on concepts like human dignity, this article emphasizes the importance of nonhuman animal welfare concerns in chimeric research and argues for less‐siloed governance and oversight and more‐comprehensive public communication. (shrink)

Returning genomic research results to family members raises complex questions. Genomic research on life-limiting conditions such as cancer, and research involving storage and reanalysis of data and specimens long into the future, makes these questions pressing. This author group, funded by an NIH grant, published consensus recommendations presenting a framework. This follow-up paper offers concrete guidance and tools for implementation. The group collected and analyzed relevant documents and guidance, including tools from the Clinical Sequencing Exploratory Research Consortium. The authors then (...) negotiated a consensus toolkit of processes and documents. That toolkit offers sample consent and notification documents plus decision flow-charts to address return of results to family of living and deceased participants, in adult and pediatric research. Core concerns are eliciting participant preferences on sharing results with family and on choice of a representative to make decisions about sharing after participant death. (shrink)

Pancreatitis is a complex, progressively destructive inflammatory disorder. Alcohol was long thought to be the primary causative agent, but genetic contributions have been of interest since the discovery that rare PRSS1, CFTR and SPINK1 variants were associated with pancreatitis risk. We now report two associations at genome-wide significance identified and replicated at PRSS1-PRSS2 and X-linked CLDN2 through a two-stage genome-wide study. The PRSS1 variant likely affects disease susceptibility by altering expression of the primary trypsinogen gene. The CLDN2 risk allele is (...) associated with atypical localization of claudin-2 in pancreatic acinar cells. The homozygous CLDN2 genotype confers the greatest risk, and its alleles interact with alcohol consumption to amplify risk. These results could partially explain the high frequency of alcohol-related pancreatitis in men. © 2012 Nature America, Inc. All rights reserved. (shrink)

Johnson and Stricker published an opinion piece in the Journal of Medical Ethics presenting their perspective on the 2008 agreement between the Infectious Diseases Society of America (IDSA) and the Connecticut Attorney General with regard to the 2006 IDSA treatment guideline for Lyme disease. Their writings indicate that these authors hold unconventional views of a relatively common tick-transmitted bacterial infection caused by the spirochete Borrelia burgdorferi. Therefore, it should come as no surprise that their opinions would clash with the IDSA's (...) evidence-based guidelines for the diagnosis and treatment of Lyme disease. Their allegations of conflict of interest against the IDSA resemble those made against the National Institutes of Health, the Food and Drug Administration and the Centers for Disease Control and Prevention in 2000, which were found to be baseless. It is the responsibility of all physicians and medical scientists to stand up to antiscientific, baseless and unethical attacks on those who support an evidence-based approach to caring for patients. (shrink)

Robert Denis Collison Black was internationally recognized as the authority on Jevons, and in particular on the centrally important elements of Benthamite Utilitarianism in Jevons' thought. Jevons' Theory Political Economy was, Black argued, a Benthamite exercise, not a systematic treatise on value and distribution. This in turn explained why Jevons' theory of production was essentially classical, and why he had no theory of aggregate distribution. Black's work on Jevons also threw light on the professionalization of economics. Black was the (...) well-merited recipient of many honours. In 1974 he was elected both a Fellow of the British Academy and a Member of the Royal Irish Academy. He became an Honorary Fellow of Trinity College, Dublin in 1982; President of the Statistical and Social Inquiry Society of Ireland over the years 1983 to 1986; acted as President of Section F of the British Association in 1984–5; was elected a Distinguished Fellow of the History of Economics Society in 1987; and in 1988, Queen's University bestowed upon him an Hon. D.Sc. Econ. (shrink)

Founded or unfounded, these objections have not as yet received an adequate answer, i.e., an explanation of the possibility of a philosophy of subjectivity as constituting a reasonable addition to the philosophia perennis, a certain broadening of its perspective, without amounting instead to a simple jettisoning of the thought and gains of centuries. The writings of a Marcel, for example, do not provide such an explanation. Composed wholly within the perspective that is in question, and a little too cavalier in (...) their dismissal of scholastic thought, they cannot really come to grips with that thought, or thereby allay its reasonable suspicions. Even the work of P. de Finance is not wholly satisfactory on this point. As a reviewer of his latest book points out, P. de Finance makes constant and brilliant use of insights gained from the perspective of subjectivity, but he skirts around, without ever really meeting head on, the problems it raises. (shrink)

Terence Wilmot Hutchison, a Fellow of the British Academy, was a historian of economics, methodologist, and acerbic critic of hubris and pretension amongst economists. He was born at Bournemouth and grew up in London. Hutchison's father was the flamboyant and much married Robert Langton Douglas, while his mother was Grace Hutchison. It was as a classicist that he went to the University of Cambridge in 1931. But Hutchison quickly lost interest in a subject that seemed to him to have (...) little relevance to the economic turmoil of the world, and switched to economics, graduating in 1934 with a First. He left Cambridge in 1934 and registered as an occasional student at the London School of Economics. This chapter presents a biography of Hutchison and also narrates his trips to Germany, Iraq, and India, as well as his stints at the University of Hull, LSE, and the University of Birmingham. (shrink)

Is it the case that God, human beings, and air all share the same capacity for cognition, differing only in the degree to which they engage in cognitive acts? Robert Pasnau has recently argued that according to St. Thomas Aquinas they do, a conclusion that for Pasnau follows straightforwardly from Aquinas’s discussion of God’s cognition in the first part of the Summa theologiae. Further, Pasnau holds that Aquinas’s relation to contemporary cognitive theory should be understood in light of the (...) discussion of God. This essay argues that Pasnau’s analysis is mistaken. It begins by explaining Pasnau’s position. It then considers the problems this reading introduces into Aquinas’s discussion of God’s cognition, as well as those it faces when addressed to air and other cognitive media. Finally, it shows the role that Aquinas’s doctrine of analogy plays in understanding how “cognition” is said of human beings, how it is said of God, and how it is not said in the case of air and other cognitive media. It concludes by suggesting that the logic of analogy is Aquinas’s most crucial contribution to contemporary discussions of mind and cognition. (shrink)

Is it the case that God, human beings, and air all share the same capacity for cognition, differing only in the degree to which they engage in cognitive acts? Robert Pasnau has recently argued that according to St. Thomas Aquinas they do, a conclusion that for Pasnau follows straightforwardly from Aquinas’s discussion of God’s cognition in the first part of the Summa theologiae. Further, Pasnau holds that Aquinas’s relation to contemporary cognitive theory should be understood in light of the (...) discussion of God. This essay argues that Pasnau’s analysis is mistaken. It begins by explaining Pasnau’s position. It then considers the problems this reading introduces into Aquinas’s discussion of God’s cognition, as well as those it faces when addressed to air and other cognitive media. Finally, it shows the role that Aquinas’s doctrine of analogy plays in understanding how “cognition” is said of human beings, how it is said of God, and how it is not said in the case of air and other cognitive media. It concludes by suggesting that the logic of analogy is Aquinas’s most crucial contribution to contemporary discussions of mind and cognition. (shrink)

BackgroundThe increased risk of developing attention-deficit hyperactivity disorder (ADHD) in extremely preterm infants is well-documented. Better understanding of perinatal risk factors, particularly those that are modifiable, can inform prevention efforts.MethodsWe examined data from the Extremely Low Gestational Age Newborns (ELGAN) Study. Participants were screened for ADHD at age 10 with the Child Symptom Inventory-4 (N = 734) and assessed at age 15 with a structured diagnostic interview (MINI-KID) to evaluate for the diagnosis of ADHD (N = 575). We studied associations (...) of pre-pregnancy maternal body mass index (BMI), pregestational and/or gestational diabetes, maternal smoking during pregnancy (MSDP), and hypertensive disorders of pregnancy (HDP) with 10-year and 15-year ADHD outcomes. Relative risks were calculated using Poisson regression models with robust error variance, adjusted for maternal age, maternal educational status, use of food stamps, public insurance status, marital status at birth, and family history of ADHD. We defined ADHD as a positive screen on the CSI-4 at age 10 and/or meeting DSM-5 criteria at age 15 on the MINI-KID. We evaluated the robustness of the associations to broadening or restricting the definition of ADHD. We limited the analysis to individuals with IQ ≥ 70 to decrease confounding by cognitive functioning. We evaluated interactions between maternal BMI and diabetes status. We assessed for mediation of risk increase by alterations in inflammatory or neurotrophic protein levels in the first week of life.ResultsElevated maternal BMI and maternal diabetes were each associated with a 55–65% increase in risk of ADHD, with evidence of both additive and multiplicative interactions between the two exposures. MSDP and HDP were not associated with the risk of ADHD outcomes. There was some evidence for association of ADHD outcomes with high levels of inflammatory proteins or moderate levels of neurotrophic proteins, but there was no evidence that these mediated the risk associated with maternal BMI or diabetes.ConclusionContrary to previous population-based studies, MSDP and HDP did not predict ADHD outcomes in this extremely preterm cohort, but elevated maternal pre-pregnancy BMI, maternal diabetes, and perinatal inflammatory markers were associated with increased risk of ADHD at age 10 and/or 15, with positive interaction between pre-pregnancy BMI and maternal diabetes. (shrink)

Change blindness illustrates a remarkable limitation in visual processing by demonstrating that substantial changes in a visual scene can go undetected. Because these changes can ultimately be detected using top–down driven search processes, many theories assign a central role to spatial attention in overcoming change blindness. Surprisingly, it has been reported that change blindness can occur during blink-contingent changes even when observers fixate the changing location [O’Regan, J. K., Deubel, H., Clark, J. J., & Rensink, R. A. . Picture changes (...) during blinks: Looking without seeing and seeing without looking. Visual Cognition, 7, 191–212]. However, eye blinks produce a transient disruption of vision that is independent of any associated changes in the retinal image. We determined whether these ‘attentive blank stares’ could occur in the absence of blink-mediated visual suppression. Using a flicker change-blindness paradigm we confirm that despite direct attentive fixations, obvious scene changes often remain undetected. We conclude that change detection involves object or feature based attentional mechanisms, which can be ‘misdirected’ despite the allocation of spatial attention to the position of the change. (shrink)

Introducing Applied Ethics Edited by Brenda Almond, Blackwell, 1995. Pp. 375. ISBN 0-631-19389-8. 45.00 (hbk), 14.99 (pbk). Environmental Ethics Edited by Robert Elliot, Oxford University Press, 1995. Pp. 255. ISBN 9-19-875144-3. 9.95 (pbk) Medicine and Moral Reasoning Edited by K.W.M. Fulford, Grant Gillett and Janet Martin Soskice Cambridge University Press, 1994. Pp. 207. ISBN 0-521-45325-9 37.50 (hbk), 12.95 (pbk). Enlightenment and Religion. Rational Dissent in Eighteenth-century Britain Edited by Knud Haakonssen, Cambridge University Press, 1996. Pp. xii + 348. ISBN (...) 0-521-56060-8. 40.00. Dialettica, Arte e Societ : Saggio su Theodor W. Adorno By Giacomo Rinaldi, Quattroventi, Urbino, 1994. Pp. 205. L. 30,000. Relevance: Communication and Cognition, new revised edition, By Dan Sperber and Deirdre Wilson, Blackwell, 1995. Pp. 326. ISBN 0-631-19878-4. 15.99. Autobiographical Reflections By Eric Voegelin (Edited, with Introduction, by Ellis Sandoz), Louisiana State University Press, 1996. Pp. 131. ISBN 0807120766 $10.95. (shrink)

Multiple realization historically mandated the autonomy of psychology, and its principled irreducibility to neuroscience. Recently, multiple realization and its implications for the reducibility of psychology to neuroscience have been challenged. One challenge concerns the proper understanding of reduction. Another concerns whether multiple realization is as pervasive as is alleged. I focus on the latter question. I illustrate multiple realization with actual, rather than hypothetical, cases of multiple realization from within the biological sciences. Though they do support a degree of autonomy (...) for higher levels of explanation and organization, they do not have the dire consequences critics of multiple realization fear. †To contact the author, please write to: Department of Philosophy, University of Cincinnati, P.O. Box 210374, Cincinnati, OH 45221‐0374; e‐mail: robert[email protected]. (shrink)

D O N A L D D AV I D S O N’S “ Meaning and Truth,” re vo l u t i o n i zed our conception of how truth and meaning are related (Davidson    ). In that famous art i c l e , Davidson put forw a rd the bold conjecture that meanings are satisfaction conditions, and that a Tarskian theory of truth for a language is a theory of meaning for that language. (...) In “Meaning and Truth,” Davidson proposed only that a Tarskian truth theory is a theory of meaning. But in “Theories of Me a n i n g and Learnable Languages,” he argued that the finite base of a Tarskian theory, together with the now familiar combinatorics, would explain how a language with unbounded expre s s i ve capacity could be learned with finite means ( Davidson    ). This certainly seems to imply that learning a language is, in p a rt at least, learning a Tarskian truth theory for it, or, at least, learning what is specified by such a theory. Davisdon was cagey about committing to the view that meanings actually a re satisfaction conditions, but subsequent followers had no such scru p l e s . We can sum this up in a trio of claims: Davidson’s Conjecture () A theory of meaning for L is a truth-conditional semantics for L. () To know the meaning of an expression in L is to know a satisfaction condition for that expression. () Meanings are satisfaction conditions. For the most part, it will not matter in what follows which of these claims is at stake. I will simply take the three to be different ways of formulating what I will call Davidson’s Conjecture (or sometimes just The Conjecture). Davidson’s Conjecture was a very bold conjecture. I think we are now in a.. (shrink)

Formalising deontic concepts, such as obligation, prohibition and permission, is normally carried out in a modal logic with a possible world semantics, in which some worlds are better than others. The main focus in these logics is on inferring logical consequences, for example inferring that the obligation O q is a logical consequence of the obligations O p and O. In this paper we propose a non-modal approach in which obligations are preferred ways of satisfying goals expressed in first-order logic. (...) To say that p is obligatory, but may be violated, resulting in a less than ideal situation s, means that the task is to satisfy the goal p ∨ s, and that models in which p is true are preferred to models in which s is true. Whereas, in modal logic, the preference relation between possible worlds is part of the semantics of the logic, in this non-modal approach, the preference relation between first-order models is external to the logic. Although our main focus is on satisfying goals, we also formulate a notion of logical consequence, which is comparable to the notion of logical consequence in modal deontic logic. In this formalisation, an obligation O p is a logical consequence of goals G, when p is true in all best models of G. We show how this non-modal approach to the treatment of deontic concepts deals with problems of contrary-to-duty obligations and normative conflicts, and argue that the approach is useful for many other applications, including abductive explanations, defeasible reasoning, combinatorial optimisation, and reactive systems of the production system variety. (shrink)

Let M be a smooth, compact manifold of dimension n ≥ 5 and sectional curvature | K | ≤ 1. Let Met (M) = Riem(M)/Diff(M) be the space of Riemannian metrics on M modulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met (M) such as the diameter. They showed that for every Turing machine T e , e ∈ ω, there is a sequence (uniformly effective in e) of homology (...) n-spheres {P k e } k ∈ ω which are also hypersurfaces, such that P k e is diffeomorphic to the standard n-sphere S n (denoted P k e ≈ diff S n ) iff T e halts on input k, and in this case the connected sum N k e =M ♯ P k e ≈ diff M , so N k e ∈ Met(M), and N k e is associated with a local minimum of the diameter function on Met(M) whose depth is roughly equal to the settling time σ e (k) of T e on inputs y i } ∈ ω of c.e. sets so that for all i the settling time of the associated Turing machine for A i dominates that for A i + 1 , even when the latter is composed with an arbitrary computable function. From this, Nabutovsky and Weinberger showed that the basins exhibit a "fractal" like behavior with extremely big basins, and very much smaller basins coming off them, and so on. This reveals what Nabutovsky and Weinberger describe in their paper on fractals as "the astonishing richness of the space of Riemannian metrics on a smooth manifold, up to reparametrization." From the point of view of logic and computability, the Nabutovsky-Weinberger results are especially interesting because: (1) they use c.e. sets to prove structural complexity of the geometry and topology, not merely undecidability results as in the word problem for groups, Hilbert's Tenth Problem, or most other applications; (2) they use nontrivial information about c.e. sets, the Soare sequence {A i } i ∈ ω above, not merely G öodel's c.e. noncomputable set K of the 1930's; and (3) without using computability theory there is no known proof that local minima exist even for simple manifolds like the torus T 5 (see §). (shrink)