Hume, the Occult, and the Substance of the School

I have not been able to locate any critique of Hume on substance by a Schoolman, at least in English, dating from Hume's period or shortly thereafter. I have, therefore, constructed my own critique as an exercise in ??post facto history??. This is what a late eighteenth-century/early nineteenth-century Scholastic could, would, and should have said in response to Hume's attack on substance should they have been minded to do so. That no one did is somewhat mysterious. My critique is precisely in the language of the period, using solely the conceptual resources available to a Schoolman at that time. The arguments, however, are as sound now as they were then, and in this sense the paper performs a dual role??contributing to the defence of substance contra Hume, and filling, albeit two hundred?years or so too late, a gap in the historical record.     

Groups, measures, and the NIP

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups in saturated -minimal structures to compact Lie groups. We also prove some other structural results about such , for example the existence of a left invariant finitely additive probability measure on definable subsets of . We finally introduce the new notion of ``compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the -minimal case.

     

On the Partially Ordered Semigroup Generated by the Class Operators I,R,H,S,P

Let I, H, S, P be the usual class operators on universal algebras. For a class K of universal algebras of the same type, let R({K}) be the class of all algebras isomorphic to a retract of a member of K and let R denote the corresponding class operator. In this paper the semigroup generated by class operators I, R, H, S, P and the corresponding partially ordered set are described. Also the standard semigroups of the above operators are determined for some varieties.     

Wheels, wheeling, and the Kontsevich integral of the Unknot

We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne [De]) for the relation between the two natural products on the space of uni-trivalent diagrams. The two formulas use the related notions of “Wheels” and “Wheeing”. We prove these formulas ‘on the level of Lie algebras’ using standard techniques from the theory of Vassiliev invariants and the theory of Lie algebras. In a brief epilogue we report on recent proofs of our full conjectures, by Kontsevich [Ko2] and by DBN, DPT, and T. Q. T. Le, [BLT]. This paper is available electronically     

Kant, Spinoza, and the Metaphysics of the Ontological Proof

This paper provides an interpretation and evaluation of Spinoza’s highly original version of the ontological proof in terms of the concept of substance instead of the concept of perfection in the first book of his Ethics. Taking the lead from Kant’s critique of ontological arguments in the Critique of Pure Reason, the paper explores the underlying ontological and epistemological presuppositions of Spinoza’s proof. The main topics of consideration are the nature of Spinoza’s definitions, the way he conceives of the relation between a substance and its essence, and his conception of existence. Once clarity is shed upon these fundamental issues, it becomes possible to address the proof in its own terms. It is then easy to see that Kant’s objections miss their target and that the same is true of those advanced by another of the ontological argument’s most famous critics, Bertrand Russell. Finally, several interpretations of Spinoza’s proof are proposed and critically evaluated; on all of them, the argument turns out to be either invalid or question-begging.     

Tilings,packings, coverings,and the approximation of functions

A packing (resp. covering) ? of a normed space X consisting of unit balls is called completely saturated (resp. completely reduced) if no finite set of its members can be replaced by a more numerous (resp. less numerous) set of unit balls of X without losing the packing property (resp. covering property) of ?. We show that a normed space X admits completely saturated packings with disjoint closed unit balls as well as completely reduced coverings with open unit balls, provided that there exists a tiling of X with unit balls. Completely reduced coverings by open balls are of interest in the context of an approximation theory for continuous real‐valued functions that rests on so‐called controllable coverings of compact metric spaces. The close relation between controllable coverings and completely reduced coverings allows an extension of the approximation theory to non‐compact spaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantum groups, the loop Grassmannian, and the Springer resolution

We establish equivalences of the following three triangulated categories:

Here, is the derived category of the principal block of finite-dimensional representations of the quantized enveloping algebra (at an odd root of unity) of a complex semisimple Lie algebra ; the category is defined in terms of coherent sheaves on the cotangent bundle on the (finite-dimensional) flag manifold for ( semisimple group with Lie algebra ), and the category is the derived category of perverse sheaves on the Grassmannian associated with the loop group , where is the Langlands dual group, smooth along the Schubert stratification.

The equivalence between and is an ``enhancement' of the known expression (due to Ginzburg and Kumar) for quantum group cohomology in terms of nilpotent variety. The equivalence between and can be viewed as a ``categorification' of the isomorphism between two completely different geometric realizations of the (fundamental polynomial representation of the) affine Hecke algebra that has played a key role in the proof of the Deligne-Langlands-Lusztig conjecture. One realization is in terms of locally constant functions on the flag manifold of a -adic reductive group, while the other is in terms of equivariant -theory of a complex (Steinberg) variety for the dual group.

The composite of the two equivalences above yields an equivalence between abelian categories of quantum group representations and perverse sheaves. A similar equivalence at an even root of unity can be deduced, following the Lusztig program, from earlier deep results of Kazhdan-Lusztig and Kashiwara-Tanisaki. Our approach is independent of these results and is totally different (it does not rely on the representation theory of Kac-Moody algebras). It also gives way to proving Humphreys' conjectures on tilting -modules, as will be explained in a separate paper.     

Concerning the Collins, Reed, Roscoe, Rudin Metrisation Theorem

New criteria, in terms of the Collins-Roscoe structuring mechanism,are given for a space to be semi-stratifiable. Known characterisationsof metrisable, developable and stratifiable spaces are deduced.     

Collusion,quarrel, and the Banzhaf value

We provide new, concise characterizations of the Banzhaf value on a fixed player set employing just the standard dummy player property and one of the collusion properties suggested by Haller (Int J Game Theory 23:261–281, 1994) and Malawski (Int J Game Theory 31:47–67, 2002). Within these characterizations, any of the collusion properties can be replaced by additivity and the quarrel property due to the latter author.     

Maximax, leximax, and the demanding criterion

Following the characterizations provided by Barberà and Jackson [Barberà, S., Jackson, M.O., 1988. Journal of Economic Theory 46, 34–44] for the maximin, the leximin, and the protective criterion, we examine the consequences of replacing convexity in their list of axioms by concavity, which means considering risk-loving instead of risk-averse agents. This yields characterizations of the maximax, the leximax and a new criterion which we term demanding criterion. We concentrate on the latter and demonstrate the independence of the axioms used for its characterization.     

Wronskians, Cumulants, and the Riemann Hypothesis

This paper proposes a new, analytic approach to the resolution of the Riemann hypothesis. The method has its origins in the 1986 results and techniques of Csordas, Norfolk, and Varga (CNV) in their proof of the Turán inequalities. Here, the mathematical structure of their work has been significantly extended and generalized. We make frequent use of the ideas on the sign-regularity of kernels found in the book of Karlin. The notion of cumulants plays an important role. The final step is to prove that a doubly infinite set of determinants are all positive. We present a conjecture, supported by computations, about the sign-regularity of a set of cumulants of the function called Φ(t) by CNV. To illustrate the ideas, the conjecture is proved for an early member of the set. We describe a new method, superior to the Karlin method used by CNV, for proving positivity of the determinants, but some cases remain to be treated.     

Reduced games,consistency, and the core

This paper establishes an axiomatization of the core by means of an internal consistency property with respect to a new reduced game introduced by Moulin (1985). Given a payoff vector chosen by a solution for some game, and given a subgroup of agents, we define thereduced game as that in which each coalition in the subgroup could attain payoffs to its members only if they are compatible with the initial payoffs toall the members outside of the subgroup. The solution isconsistent if it selects the same payoff distribution for the reduced game as initially. We show that consistency together with individual rationality characterizes the core of both transferable and non-transferable utility games.     

Entitlement, Justification, and the Bootstrapping Problem

According to the bootstrapping problem, any view that allows for basic knowledge (knowledge obtained from a reliable source prior to one??s knowing that that source is reliable) is forced to accept that one can utilize a track-record argument to acquire justification for believing that one??s belief source is reliable; yet, we tend to think that acquiring justification in this way is too easy. In this paper I argue, first, that those who respond to the bootstrapping problem by denying basic knowledge succumb to over-intellectualizing epistemology, and secondly, reliabilist views avoid over-intellectualization only at the expense of sanctioning bootstrapping as a benign procedure. Both of these outcomes are difficult to bear. To ward off each of these unsavory outcomes, I propose an alternative solution that draws on a distinction between two separate epistemic concepts: entitlement and justification.     

Johri's general dual,the Lagrangian dual,and the surrogate dual

Duality formulations can be derived from a nonlinear primal optimization problem in several ways. One abstract theoretical concept presented by Johri is the framework of general dual problems. They provide the tightest of specific bounds on the primal optimum generated by dual subproblems which relax the primal problem with respect to the objective function or to the feasible set or even to both. The well-known Lagrangian dual and surrogate dual are shown to be special cases. Dominating functions and including sets which are the two relaxation devices of Johri's general dual turn out to be the most general formulations of augmented Lagrangian functions and augmented surrogate regions.     

Indestructibility,HOD, and the Ground Axiom

Let φ₁ stand for the statement V = HOD and φ₂ stand for the Ground Axiom. Suppose T_i for i = 1, …, 4 are the theories “ZFC + φ₁ + φ₂,” “ZFC + ¬φ₁ + φ₂,” “ZFC + φ₁ + ¬φ₂,” and “ZFC + ¬φ₁ + ¬φ₂” respectively. We show that if κ is indestructibly supercompact and λ > κ is inaccessible, then for i = 1, …, 4, A_i = _df{δ < κ∣δ is an inaccessible cardinal which is not a limit of inaccessible cardinals and V_δ?T_i} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing four models in which A_i = ?? for i = 1, …, 4. In each of these models, there is an indestructibly supercompact cardinal κ, and no cardinal δ > κ is inaccessible. We show it is also the case that if κ is indestructibly supercompact, then V_κ?T₁, so by reflection, B₁ = _df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and V_δ?T₁} is unbounded in κ. Consequently, it is not possible to construct a model in which κ is indestructibly supercompact and B₁ = ??. On the other hand, assuming κ is supercompact and no cardinal δ > κ is inaccessible, we demonstrate that it is possible to construct a model in which κ is indestructibly supercompact and for every inaccessible cardinal δ < κ, V_δ?T₁. It is thus not possible to prove in ZFC that B_i = _df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and V_δ?T_i} for i = 2, …, 4 is unbounded in κ if κ is indestructibly supercompact. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Digital Mathematics Libraries: The Good,the Bad,the Ugly

The idea of a World digital mathematics library (DML) has been around since the turn of the 21th century. We feel that it is time to make it a reality, starting in a modest way from successful bricks that have already been built, but with an ambitious goal in mind. After a brief historical overview of publishing mathematics, an estimate of the size and a characterisation of the bulk of documents to be included in the DML, we turn to proposing a model for a Reference Digital Mathematics Library—a network of institutions where the digital documents would be physically archived. This pattern based rather on the bottom-up strategy seems to be more practicable and consistent with the digital nature of the DML. After describing the model we summarise what can and should be done in order to accomplish the vision. The current state of some of the local libraries that could contribute to the global views are described with more details.