More Trouble for Regular Probabilities

In standard probability theory, probability zero is not the same as impossibility. However, many have suggested that it should be—that only impossible events should have probability zero. In cases where infinitely many outcomes have equal probability, regularity requires that some probabilities are infinitesimal, but merely introducing infinitesimals does not solve all of the problems with regularity. We will see that regular probabilities are not invariant over rigid transformations, even for simple, bounded, countable, constructive, and disjoint sets of possible outcomes. Hence, regular chances cannot be determined by space-time invariant physical laws, and regular credences cannot satisfy seemingly reasonable symmetry principles. Moreover, the examples here are immune to the objections against Williamson’s infinite coin flips

Philosophical Method and Galileo's Paradox of Infinity

We consider an approach to some philosophical problems that I call the Method of Conceptual Articulation: to recognize that a question may lack any determinate answer, and to re-engineer concepts so that the question acquires a definite answer in such a way as to serve the epistemic motivations behind the question. As a case study we examine “Galileo’s Paradox”, that the perfect square numbers seem to be at once as numerous as the whole numbers, by one-to-one correspondence, and yet less numerous, being a proper subset. I argue that Cantor resolved this paradox by a method at least close to that proposed—not by discovering the true nature of cardinal number, but by articulating several useful and appealing extensions of number to the infinite. Galileo was right to suggest that the concept of relative size did not apply to the infinite, for the concept he possessed did not. Nor was Bolzano simply wrong to reject Hume’s Principle (that one-to-one correspondence implies equal number) in the infinitary case, in favor of Euclid’s Common Notion 5 (that the whole is greater than the part), for the concept of cardinal number (in the sense of “number of elements”) was not clearly defined for infinite collections. Order extension theorems now suggest that a theory of cardinality upholding Euclid’s principle instead of Hume’s is possible. Cantor’s refinements of number are not the only ones possible, and they appear to have been shaped by motivations and fruitfulness, for they evolved in discernible stages correlated with emerging applications and results. Galileo, Bolzano, and Cantor shared interests in the particulate analysis of the continuum and in physical applications. Cantor’s concepts proved fruitful for those pursuits. Finally, Gödel was mistaken to claim that Cantor’s concept of cardinality is forced on us; though Gödel gives an intuitively compelling argument, he ignores the fact that Euclid’s Common Notion is also intuitively compelling, and we are therefore forced to make a choice. The success of Cantor’s concept of cardinality lies not in its truth (for concepts are not true or false), nor its uniqueness (for it is not the only extension of number possible), but in its intuitive appeal, and most of all, its usefulness to the understanding

Molecular Mechanisms of Neuropilin-Ligand Binding

Neuropilin (Nrp) is an essential cell surface receptor with dual functionality in the cardiovascular and nervous systems. The first identified Nrp-ligand family was the Semaphorin-3 (Sema3) family of axon repulsion molecules. Subsequently, Nrp was found to serve as a receptor for the vascular endothelial growth factor (VEGF) family of pro-angiogenic cytokines. In addition to its physiological role, VEGF signaling via Nrp directly contributes to cancer stemness, growth, and metastasis. Thus, the Nrp/VEGF signaling axis is a promising anti-cancer therapeutic target. Interestingly, it has recently been shown that Sema3 and VEGF are functionally opposed to one another, with Sema3 possessing potent endogenous anti-angiogenic activity and VEGF serving as an attractive cue for neuronal axons. We hypothesized that direct competition for an overlapping binding site within the Nrp extracellular domain may explain the observed functional competition between VEGF and Sema3. To test this hypothesis we have separately investigated the mechanisms of VEGF and Sema3 binding to Nrp. Utilizing structural biology coupled with biophysics and biochemistry we have identified both distinct and common mechanisms that facilitate the interaction between Nrp and these two ligand families. Specifically, we have identified an Nrp binding pocket to which these ligands competitively bind. The Sema3 family uniquely requires proteolytic activation in order to engage this overlapping binding site. These findings provide critical mechanistic insight into VEGF and Sema3 mediated physiology. Additionally, these data have informed the development of small molecules, peptides, and soluble receptor fragments that function as potent and selective inhibitors of VEGF/Nrp binding with exciting therapeutic potential for treating cancer

Philosophical Method and Galileo's Paradox of Infinity

We consider an approach to some philosophical problems that I call the Method of Conceptual Articulation: to recognize that a question may lack any determinate answer, and to re-engineer concepts so that the question acquires a definite answer in such a way as to serve the epistemic motivations behind the question. As a case study we examine “Galileo’s Paradox”, that the perfect square numbers seem to be at once as numerous as the whole numbers, by one-to-one correspondence, and yet less numerous, being a proper subset. I argue that Cantor resolved this paradox by a method at least close to that proposed—not by discovering the true nature of cardinal number, but by articulating several useful and appealing extensions of number to the infinite. Galileo was right to suggest that the concept of relative size did not apply to the infinite, for the concept he possessed did not. Nor was Bolzano simply wrong to reject Hume’s Principle (that one-to-one correspondence implies equal number) in the infinitary case, in favor of Euclid’s Common Notion 5 (that the whole is greater than the part), for the concept of cardinal number (in the sense of “number of elements”) was not clearly defined for infinite collections. Order extension theorems now suggest that a theory of cardinality upholding Euclid’s principle instead of Hume’s is possible. Cantor’s refinements of number are not the only ones possible, and they appear to have been shaped by motivations and fruitfulness, for they evolved in discernible stages correlated with emerging applications and results. Galileo, Bolzano, and Cantor shared interests in the particulate analysis of the continuum and in physical applications. Cantor’s concepts proved fruitful for those pursuits. Finally, Gödel was mistaken to claim that Cantor’s concept of cardinality is forced on us; though Gödel gives an intuitively compelling argument, he ignores the fact that Euclid’s Common Notion is also intuitively compelling, and we are therefore forced to make a choice. The success of Cantor’s concept of cardinality lies not in its truth (for concepts are not true or false), nor its uniqueness (for it is not the only extension of number possible), but in its intuitive appeal, and most of all, its usefulness to the understanding

The Established and Evolving Role of Nailfold Capillaroscopy in Connective-Tissue Disease

Nailfold capillaroscopy (NFC) is a low-cost, non-invasive, rapid, highly specific and reproducible investigation well established in the diagnosis of systemic sclerosis and related conditions. This chapter will detail the relevant underlying scientific principles that underpin the investigation, the methods for performing NFC, the range of abnormalities that can be present and the currently available classification criteria before moving on to discuss the various established and emerging applications as relevant to the connective tissue diseases. In addition to its role in the diagnosis of SSc, highlighted by its inclusion in the most recent ACR/EULAR consensus classification criteria, NFC has been shown to predict disease activity, many organ-specific complications such as digital ulcers, pulmonary hypertension and interstitial lung disease, and even mortality. It is emerging as a useful investigation in other CTDs characterised by microvasculopathy, such as in the idiopathic inflammatory myopathies and mixed connective tissue disease, as well as being studied as a serial investigation in patients to act as a potential biomarker and measure of treatment efficacy. NFC can contribute to the earlier identification of patients with CTDs with clinically important complications and if applied accurately, therefore, can help improve outcomes in these often challenging diseases

Control of Cellular Motility by Neuropilin-Mediated Physical Interactions

The neuropilin (Nrp) family consists of multifunctional cell surface receptors with critical roles in a number of different cell and tissue types. A core aspect of Nrp function is in ligand-dependent cellular migration, where it controls the multistep process of cellular motility through integration of ligand binding and receptor signaling. At a molecular level, the role of Nrp in migration is intimately connected to the control of adhesive interactions and cytoskeletal reorganization. Here, we review the physiological role of Nrp in cellular adhesion and motility in the cardiovascular and nervous systems. We also discuss the emerging pathological role of Nrp in tumor cell migration and metastasis, providing motivation for continued efforts toward developing Nrp inhibitors

Undecidable Long-term Behavior in Classical Physics: Foundations, Results, and Interpretation

The behavior of some systems is non-computable in a precise new sense. One infamous problem is that of the stability of the solar system: Given the initial positions and velocities of several mutually gravitating bodies, will any eventually collide or be thrown off to infinity? Many have made vague suggestions that this and similar problems are undecidable: no finite procedure can reliably determine whether a given configuration will eventually prove unstable. But taken in the most natural way, this is trivial. The state of a system corresponds to a point in a continuous space, and virtually no set of points in space is strictly decidable. A new, more pragmatic concept is therefore introduced: a set is decidable up to measure zero (d.m.z.) if there is a procedure to decide whether a point is in that set and it only fails on some points that form a set of zero volume. This volume and probability: we can ignore a zero-volume set of states because the state of an arbitrary system almost certainly will not fall in that set. D.m.z. is also closer to the intuition of decidability than other notions in the literature, which are either less strict or apply only to special sets, like closed sets. Certain complicated sets are not d.m.z., most remarkably including the set of known stable orbits for planetary systems (the KAM tori). This suggests that the stability problem is indeed undecidable in the precise sense of d.m.z. Carefully extending decidability concepts from idealized models to actual systems, we see that even deterministic aspects of physical behavior can be undecidable in a clear and significant sense

Undecidable Long-term Behavior in Classical Physics: Foundations, Results, and Interpretation

The behavior of some systems is non-computable in a precise new sense. One infamous problem is that of the stability of the solar system: Given the initial positions and velocities of several mutually gravitating bodies, will any eventually collide or be thrown off to infinity? Many have made vague suggestions that this and similar problems are undecidable: no finite procedure can reliably determine whether a given configuration will eventually prove unstable. But taken in the most natural way, this is trivial. The state of a system corresponds to a point in a continuous space, and virtually no set of points in space is strictly decidable. A new, more pragmatic concept is therefore introduced: a set is decidable up to measure zero (d.m.z.) if there is a procedure to decide whether a point is in that set and it only fails on some points that form a set of zero volume. This volume and probability: we can ignore a zero-volume set of states because the state of an arbitrary system almost certainly will not fall in that set. D.m.z. is also closer to the intuition of decidability than other notions in the literature, which are either less strict or apply only to special sets, like closed sets. Certain complicated sets are not d.m.z., most remarkably including the set of known stable orbits for planetary systems (the KAM tori). This suggests that the stability problem is indeed undecidable in the precise sense of d.m.z. Carefully extending decidability concepts from idealized models to actual systems, we see that even deterministic aspects of physical behavior can be undecidable in a clear and significant sense

Tropical North Atlantic Subsurface Warming Events as a Fingerprint for AMOC Variability During Marine Isotope Stage 3

The role of Atlantic Meridional Overturning Circulation (AMOC) as the driver of Dansgaard-Oeschger (DO) variability that characterized Marine Isotope Stage 3 (MIS 3) has long been hypothesized. Although there is ample proxy evidence suggesting that DO events were robust features of glacial climate, there is little data supporting a link with AMOC. Recently, modeling studies and subsurface temperature reconstructions have suggested that subsurface warming across the tropical North Atlantic can be used to fingerprint a weakened AMOC during the deglacial because a reduction in the strength of the western boundary current allows warm salinity maximum water of the subtropical gyre to enter the deep tropics. To determine if AMOC variability played a role during the DO cycles of MIS 3, we present new, high-resolution Mg/Ca and δ18O records spanning 24-52 kyr from the near-surface dwelling planktonic foraminifera Globigerinoides ruber and the lower thermocline dwelling planktonic foraminifera Globorotalia truncatulinoides in Southern Caribbean core VM12-107 (11.33°N, 66.63°W, 1079m depth). Our subsurfaceMg/Ca record reveals abrupt increases in Mg/Ca ratios (the largest equal to a 4°C warming) during the interstadial-stadial transition of most DO events during this period. This change is consistent with reconstructions of subsurface warming events associated with cold events across the deglacial using the same core. Additionally, our data support the conclusion reached by a recently published study from the Florida Straits that AMOC did not undergo significant reductions during Heinrich events 2 and 3. This record presents some of the first high-resolution marine sediment derived evidence for variable AMOC during MIS 3

Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system

Some have suggested that certain classical physical systems have undecidable long-term behavior, without specifying an appropriate notion of decidability over the reals. We introduce such a notion, decidability in μ (or d-μ) for any measure μ, which is particularly appropriate for physics and in some ways more intuitive than Ko’s (1991) recursive approximability (r.a.). For Lebesgue measure λ, d-λ implies r.a. Sets with positive λ-measure that are sufficiently “riddled” with holes are never d-λ but are often r.a. This explicates Sommerer and Ott’s (1996) claim of uncomputable behavior in a system with riddled basins of attraction. Furthermore, it clarifies speculations that the stability of the solar system (and similar systems) may be undecidable, for the invariant tori established by KAM theory form sets that are not d-λ