Leibniz and the rationality of the infinite

Part 1 Historically the term 'infinite' has had two apparently contrary meanings. On the one hand, it was taken by metaphysicians like Plotinus to mean that which " ... has never known measure and stands outside number, and so is under no limit either in regard to anything external or internal ... " (Branford 1949, V.5.11). 'Infinite' in this sense means 'irrevocably complete'. On the other hand, Aristotle defined it in this way: "A quantity is infinite if it is such that we can always take a part outside what has already been taken." (Hardie and Gaye 1941, 207a, 5-10) 'Infinite', in this second sense means, 'irrevocably incomplete'. Leibniz is someone who uses both these meanings. In particular, he identifies the irrevocably complete with God and the irrevocably incomplete with the world (as we know it). Given, firstly, that what is irrevocably complete includes everything and, secondly, that it excludes anything incomplete, the following conclusion can be drawn: Leibniz's philosophy of the infinite makes of the-world-as-we-know-it something that is logically dependent on God, but also something that exists in contradiction to 'him'. Leibniz cannot escape a kind of contradiction in what he says about God and the world but this is not a straightforward case of self-refutation. The reason turns on the consideration that to divorce the concept of the irrevocably complete from its object is to deprive this concept of its sense, specifically of its sense of completeness. For if the two are distinct, then there is something beyond the irrevocably complete, namely, how this is independently of its concept. It follows that to deny the irrevocably complete is, in the same breath, to affirm that very thing. Yet if we cannot quite deny the irrevocable complete, neither can we as human beings quite affirm it either-for the human mind is, we do not doubt, a limited one. Thus the irrevocably complete can neither be affirmed nor denied without contradiction. There is a strong resemblance between this paradox and the paradox of the liar: in both cases there is a thesis that says of itself that it is untrue and, in both cases, thesis and antithesis tum out to be equivalent. Part 2 Kant offers some powerful reasons to think that the paradox discussed in Part 1 involves no real contradiction. The critical philosophy suggests that the apparent contradiction is real, only if, per impossibile, we have some way to positively employ the concept of the world as it is independently of our conceptions of it. Kant's view of the infinite shares with Leibniz's the vice (if it is one) that it is paradoxical: both philosophers make use of a concept that cannot, strictly speaking, be possessed by the human mind. However each view has the significant virtue that it shows the difference between the irrevocably complete and the irrevocably incomplete to be not simply a logical difference. My overall conclusion is based on a synthesis of the Leibnizian and the Kantian philosophies of the infinite. According to Leibniz, neither the irrevocably complete, nor the irrevocably incomplete, can be eliminated from philosophy. According to Kant, infinity is-from a human perspective at least-something prior to conception; putting Leibniz and Kant together, I conclude that these modes of infinity combine to produce finitude, that they are the joint conditions under which difference, and therefore finitude, is possible. In particular, I argue that the irrevocably complete is the infinity of fullness, and that the irrevocably incomplete is the infinity of emptiness, and that logic is blind to any difference there might be between these, since both are, by definition, undifferentiated. Given that ethics, as well as logic, is dependent on finitude, I conclude, finally, that the perennial ambition to eliminate either the irrevocably complete or the irrevocably incomplete from philosophy is, not merely unrealisable, but potentially dangerous

Clinical and echocardiographic characteristics and cardiovascular outcomes according to diabetes status in patients with heart failure and preserved ejection fraction. A report from the Irbesartan in Heart Failure with Preserved Ejection Fraction Trial (I-Preserve)

Background—In patients with HF and preserved ejection fraction (HFpEF), little is known about the characteristics of and outcomes in those with and without diabetes. Methods—We examined clinical and echocardiographic characteristics and outcomes in the Irbesartan in Heart Failure with Preserved Ejection Fraction trial (I-Preserve), according to history of diabetes. Cox regression models were used to estimate hazard ratios (HR) for cardiovascular outcomes adjusted for known predictors, including age, sex, natriuretic peptides, and comorbidity. Echocardiographic data were available in 745 patients and were additionally adjusted for in supplementary analyses. Results—Overall, 1134 of 4128 patients (27%) had diabetes. Compared to those without diabetes, they were more likely to have a history of myocardial infarction (28% vs. 22%), higher BMI (31kg/m2 vs. 29kg/m2), worse Minnesota living with HF score (48 vs. 40), higher median NT-proBNP concentration (403 vs 320 pg/ml; all p<0.01), more signs of congestion but no significant difference in LVEF. Patients with diabetes had a greater left ventricular (LV) mass and left atrial area than patients without diabetes. Doppler E wave velocity (86 vs 76 cm/sec, p<0.0001) and the ratio of E/e' (11.7 vs 10.4, p=0.010) were higher in patients with diabetes. Over a median follow-up of 4.1 years, cardiovascular death or HF hospitalization occurred in 34% of patients with diabetes vs. 22% of those without diabetes; adjusted HR 1.75 (95% CI 1.49-2.05) and 28% vs. 19% of patients with and without diabetes died; adjusted HR 1.59 (1.33-1.91). Conclusions—In HFpEF, patients with diabetes have more signs of congestion, worse quality of life, higher NT-proBNP levels, and a poorer prognosis. They also display greater structural and functional echocardiographic abnormalities. Further investigation is needed to determine the mediators of the adverse impact of diabetes on outcomes in HFPEF, and whether they are modifiable

Geodynamic implications of synchronous Norite and TTG formation in the 3 Ga Maniitsoq Norite Belt, West Greenland

This study was supported by Villum Fonden through grant VKR18978 to K.S. Funding for article fees was supplied by the Ministry of Mineral Resources, Government of Greenland.We present new data for the ∼3.0 Ga Maniitsoq Norite Belt of the Akia Terrane, West Greenland, with the aim of understanding its petrogenesis. The Maniitsoq Norite Belt is hosted in regional tonalite-trondhjemite-granodiorite (TTG) and dioritic orthogneisses, intruded by later sheets of TTG and granite pegmatites, and comprises two main rock types: plagioclase-rich “norites” and pyroxene-rich “melanorites”. Both norites and melanorites have high SiO2 contents (52–60 wt% SiO2), high bulk rock Mg# (0.57–0.83), and low TiO2 contents (0.1–0.7 wt%). Their trace element patterns are defined by depleted heavy Rare-Earth elements, highly enriched light Rare-Earth elements, negative anomalies in Nb, Ta, and Ti, and variable anomalies in Zr, Hf, and Eu. New zircon U-Pb geochronology data and previously published ages establish an emplacement age of 3,013 ± 1 Ma for the majority of the Maniitsoq Norite Belt, with magmatism continuing until 3,001 ± 3 Ma. This ∼12 Myr period of norite magmatism is coeval with an ongoing period of TTG production in the Akia Terrane. Norite Belt emplacement was closely followed by high temperature, low pressure granulite-facies metamorphism at ∼800°C and 900°C/GPa) and that the norite magmas were emplaced into thin crust and lithosphere. Compositions of the norites and melanorites can be explained by derivation from a single mafic parental melt (∼13 wt% MgO), with the norites predominantly accumulating plagioclase and the melanorites predominantly accumulating pyroxene. Evidence from field relationships, the presence of xenocrystic zircon, major element compositions and combined trace element and Hf-isotope modelling suggests the norites were contaminated by assimilation of ∼20–30% continental TTG crust. Geochemical and Hf-Nd isotopic constraints indicate that the norite mantle source was depleted, and that this depletion occurred significantly before the emplacement of the norite magmas. Contemporaneous production of both TTGs and norite, their emplacement in thin crust, and the rapid transition to high temperature, low pressure granulite-facies metamorphism is best explained by their formation in an ultra-hot orogeny. Formation of norites in this setting may be restricted to >2.7 Ga, when geothermal gradients were higher on Earth.Publisher PDFPeer reviewe

The screening and management of pituitary dysfunction following traumatic brain injury in adults: British Neurotrauma Group guidance.

Pituitary dysfunction is a recognised, but potentially underdiagnosed complication of traumatic brain injury (TBI). Post-traumatic hypopituitarism (PTHP) can have major consequences for patients physically, psychologically, emotionally and socially, leading to reduced quality of life, depression and poor rehabilitation outcome. However, studies on the incidence of PTHP have yielded highly variable findings. The risk factors and pathophysiology of this condition are also not yet fully understood. There is currently no national consensus for the screening and detection of PTHP in patients with TBI, with practice likely varying significantly between centres. In view of this, a guidance development group consisting of expert clinicians involved in the care of patients with TBI, including neurosurgeons, neurologists, neurointensivists and endocrinologists, was convened to formulate national guidance with the aim of facilitating consistency and uniformity in the care of patients with TBI, and ensuring timely detection or exclusion of PTHP where appropriate. This article summarises the current literature on PTHP, and sets out guidance for the screening and management of pituitary dysfunction in adult patients with TBI. It is hoped that future research will lead to more definitive recommendations in the form of guidelines

Seismic imaging in Long Valley, California, by surface and borehole techniques: An investigation of active tectonics

The search for silicic magma in the upper crust is converging on the Long Valley Caldera of eastern California, where several lines of geophysical evidence show that an active magma chamber exists at mid‐to lower‐crustal depths. There are also other strong indications that magma may be present at depths no greater than about 5 km below the surface. In this paper, we review the history of the search for magma at Long Valley. We also present the preliminary results from a coordinated suite of seismic experiments, conducted by a consortium of institutions in the summer and fall of 1984, that were designed to refine our knowledge of the upper extent of the magma chamber. Major funding for the experiments was provided by the Geothermal Research Program of the U.S. Geological Survey (USGS) and by the Magma Energy Technology Program of the U.S. Department of Energy (DOE), a program to develop the technology necessary to extract energy directly from crustal magma. Additional funding came from DOE's Office of Basic Energy Sciences and the National Science Foundation (NSF). Also, because extensive use was made of a 0.9‐km‐deep well lent to us by Santa Fe Geothermal, Inc., the project was conducted partly under the auspices of the Continental Scientific Drilling Program (CSDP). As an integrated seismic study of the crust within the caldera that involved the close cooperation of a large number of institutions, the project was moreover viewed as a prototype for future scientific experiments to be conducted under the Program for Array Seismic Studies of the Continental Lithosphere (PASSCAL). The experiment thus represented a unique blend of CSDP and PASSCAL methods, and achieved goals consistent with both programs

Respiratory and Bronchitic Symptoms Predict Intention to Quit Smoking among Current Smokers with, and at Risk for, Chronic Obstructive Pulmonary Disease

Rationale: Smoking cessation is the most important intervention for patients with chronic obstructive pulmonary disease (COPD). What leads smokers with COPD to quit smoking remains unknown

Cause of Death in Patients With Acute Heart Failure: Insights From RELAX-AHF-2

OBJECTIVES: This study sought to better understand the discrepant results of 2 trials of serelaxin on acute heart failure (AHF) and short-term mortality after AHF by analyzing causes of death of patients in the RELAX-AHF-2 (Efficacy, Safety and Tolerability of Serelaxin When Added to Standard Therapy in AHF-2) trial. BACKGROUND: Patients with AHF continue to suffer significant short-term mortality, but limited systematic analyses of causes of death in this patient population are available. METHODS: Adjudicated cause of death of patients in RELAX-AHF-2, a randomized, double-blind, placebo-controlled trial of serelaxin in patients with AHF across the spectrum of ejection fraction (EF), was analyzed. RESULTS: By 180 days of follow-up, 11.5% of patients in RELAX-AHF-2 died, primarily due to heart failure (HF) (38% of all deaths). Unlike RELAX-AHF, there was no apparent effect of treatment with serelaxin on any category of cause of death. Older patients (≥75 years) had higher rates of mortality (14.2% vs. 8.8%) and noncardiovascular (CV) death (27% vs. 19%) compared to younger patients. Patients with preserved EF (≥50%) had lower rates of HF-related mortality (30% vs. 40%) but higher non-CV mortality (36% vs. 20%) compared to patients with reduced EF. CONCLUSIONS: Despite previous data suggesting benefit of serelaxin in AHF, treatment with serelaxin was not found to improve overall mortality or have an effect on any category of cause of death in RELAX-AHF-2. Careful adjudication of events in the serelaxin trials showed that older patients and those with preserved EF had fewer deaths from HF or sudden death and more deaths from other CV causes and from noncardiac causes. (Efficacy, Safety and Tolerability of Serelaxin When Added to Standard Therapy in AHF [RELAX-AHF-2]; NCT01870778)