The Pepsi Putnam Challenge

Opening a copy of The Mathematical Intelligencer you may ask yourself uneasily, “What is this anyway—a mathematical journal, or what?” Or you may ask, “Where am I?” Or even “Who am I?” This sense of disorientation is at its most acute when you open to Cohn Adam’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be hannless.     

Expressing and Describing Experiences. A Case of Showing Versus Saying

Experiences are interpreted as conscious mental occurrences that are of phenomenal character. There is already a kind of (weak) intentionality involved with this phenomenal interpretation. A stricter conception of experiences distinguishes between purely phenomenal experiences and intentional experiences in a narrow sense. Wittgenstein’s account of psychological (experiential) verbs is taken over: Usually, expressing mental states verbally is not describing them. According to this, “I believe” can be seen as an expression of one’s own belief, but not as an expression of a belief about one’s belief. Hence, the utterance “I believe it is raining” shows that I believe that it is raining, although it is not said by these words that I believe that it is raining. Thinking thoughts such as “I believe it is raining, but it is not raining” (a variant of Moore’s paradox) is an absurdity between what is already said by silently uttering “It is not raining” and what is shown by silently uttering “I believe it is raining.” The paper agrees with a main result of Wittgenstein’s considerations of Moore’s paradox, namely the view that logical structure, deducibility, and consistency cannot be reduced solely to propositions—besides a logic of propositions, there is, for example, a logic of assertions and of imperatives, respectively.     

A Probabilistic Approach to Carne’s Bound

Carne’s bound is a sharp inequality controlling the transition probabilities for a discrete reversible Markov chain (Section 1). Its ordinary proof uses spectral techniques which look as efficient as miraculous. Here we present a new proof, comparing a “drift” for ways “out” and “back”, to get the gaussian part of the bound (Section 2), and using a conditioning technique to get the flight factor (Section 4). Moreover we show how our proof is more “supple” than Carne’s one and may generalize (Section 3.2).      

Local Topological Toughness and Local Factors

We localize and strengthen Katona’s idea of an edge-toughness to a local topological toughness. We disprove a conjecture of Katona concerning the conection between edge-toughness and factors. For the topological toughness we prove a theorem similar to Katona’s 2k-factor-conjecture, which turned out to be false for his edge-toughness. We prove, that besides this the topological toughness has nearly all known nice properties of Katona’s edge-toughness and therefore is worth to be considered. Research supported by the “Mathematics in Information Society” project carried out by Alfréd Rényi Institute of Mathematics - Hungarian Academy of Sciences, in the framework of the European Community’s “Confirming the International Role of Community Research” programme. Research supported by the Ministry of Education OTKA grant OTKA T 043520.     

Monadic theory of order and topology, 1

We deal with the monadic theory of linearly ordered sets and topological spaces, disprove two of Shelah’s conjectures and prove some more results. In particular, if the Continuum Hypothesis holds, then there exist monadic formulae expressing the predicates “X is countable” and “X is meager” in the real line and in Cantor’s Discontinuum.     

On the Number of Facets of Three-Dimensional Dirichlet Stereohedra III: Full Cubic Groups

We are interested in the maximum possible number of facets that Dirichlet stereohedra for three-dimensional crystallographic groups can have. In two previous papers, D. Bochiş and the second author studied the problem for noncubic groups. This paper deals with “full” cubic groups, while “quarter” cubic groups are left for a subsequent paper. Here, “full” and “quarter” refers to the recent classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston. This paper’s main result is that Dirichlet stereohedra for any of the 27 full groups cannot have more than 25 facets. We also find stereohedra with 17 facets for one of these groups. Research partially supported by the Spanish Ministry of Education and Science, grant number MTM2005-08618-C02-02.     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

What Mary’s Aboutness Is About

The aim of this paper is to reinforce anti-physicalism by extending the “hard problem” to a specific kind of intentional states. For reaching this target, I investigate the mental content of the new intentional states of Jackson’s Mary. I proceed in the following way: I start analyzing the knowledge argument, which highlights the “hard problem” tied to phenomenal consciousness. In a second step, I investigate a powerful physicalist reply to this argument: the phenomenal concept strategy. In a third step, I propose a constitutional account of phenomenal concepts that captures the Mary scenario adequately, but implies anti-physicalist referents. In a last step, I point at the ramifications constitutional phenomenal concepts have on the constitution of Mary’s new intentional states. Therefore, by focusing the attention on phenomenal concepts, the so-called “hard problem” of consciousness will be carried over to the alleged “easy problem” of intentional states as well.     

On the algebraic integrals of the dynamical equations

We bring to the reader’s attention a translation of Levi-Civita’s work “Sugli integrali algebrici delle equazioni dinamiche” (1896).     

A note on Furstenberg’s filtering problem

This note gives a positive answer to an old question in elementary probability theory that arose in Furstenberg’s seminal article “Disjointness in Ergodic Theory.” As a consequence, Furstenberg’s filtering theorem holds without any integrability assumption.     

A consistency result on thin-very tall Boolean algebras

It was proved by Baumgartner and Shelah that Con (ZFC)→Con (ZFC + “there is a superatomic Boolean algebra of width ω and height ω₂”). In this paper we improve Baumgartner-Shelah’s theorem, showing that Con (ZFC)→Con (ZFC+“for every α<ω₃ there is a superatomic Boolean algebra of width ω and height α”). The preparation of this paper was supported by DGICYT Grant PB98-1231.     

Cantor set and interpolation

In 1998, Y. Benyamini published interesting results concerning interpolation of sequences using continuous functions ℝ → ℝ. In particular, he proved that there exists a continuous function ℝ → ℝ which in some sense “interpolates” all sequences (x _n ) _n∈ℤ ∈ [0, 1]^ℤ “simultaneously.” In 2005, M.R. Naulin and C. Uzcátegui unified and generalized Benyamini’s results. In this paper, the case of topological spaces X and Y with an Abelian group acting on X is considered. A similar problem of “simultaneous interpolation” of all “generalized sequences” using continuous mappings X → Y is posed. Further generalizations of Naulin-Uncátegui theorems, in particular, multidimensional analogues of Benyamini’s results are obtained.     

Local Search Proximal Algorithms as Decision Dynamics with Costs to Move

Acceptable moves for the “worthwhile-to-move” incremental principle are such that “advantages-to-move” are higher than some fraction of “costs-to-move”. When combined with optimization, this principle gives raise to adaptive local search proximal algorithms. Convergence results are given in two distinctive cases, namely low local costs-to-move and high local costs-to-move. In this last case, one obtains a dynamic cognitive approach to Ekeland’s ϵ-variational principle. Introduction of costs-to-move in the algorithms yields robustness and stability properties.     

Graph factors

This exposition is concerned with the main theorems of graph-factor theory, Hall’s and Ore’s Theorems in the bipartite case, and in the general case Petersen’s Theorem, the 1-Factor Theorem and thef-Factor Theorem. Some published extensions of these theorems are discussed and are shown to be consequences rather than generalizations of thef-Factor Theorem. The bipartite case is dealt with in Section 2. For the proper presentation of the general case a preliminary theory of “G-triples” and “f-barriers” is needed, and this is set out in the next three Sections. Thef-Factor Theorem is then proved by an argument of T. Gallai in a generalized form. Gallai’s original proof derives the 1-Factor Theorem from Hall’s Theorem. The generalization proceeds analogously from Ore’s Theorem to thef-Factor Theorem.     

Righting the early history of computing,or how sausage was made

The short answer to the question just posed seems to be, “Not much.” Since I have given “the long answer” elsewhere,¹² I can summarize it here. Berg could see no point in writing Bromley. What could he write to someone he believed guilty of plagiarism? What could such a letter accomplish? He did, however, write to New York University Press; to all the universities involved, and to the Works’ English publisher (Pickering and Chatto), who said they passed the letter on to Campbell-Kelly (30 June 1990); to a great many professional societies in Australia, England, and the United States; to a great many governmental agencies and some politicians in those countries; to some publications, both academic and popular; to the Pope and several cardinals; and to a miscellany of other individuals. Generally, those in the best position to do something—for example, the three universities involved —did not even answer Berg’s letter. Others often did answer, but their answer was generally that they were in no position to do anything. That was how matters stood when I published my first article on “the Berg Affair”.₁₂ Its publication finally roused those best positioned to answer. Late in 1993, Galler, Bromley, and Campbell-Kelly wrote letters to the editor of Accountability in Research criticizing me for not getting their side of the story before I published Berg’s. Campbell-Kelly threatened the journal’s publisher with a lawsuit if I (or it) did not retract. The three also provided some insight into what their explanation of events might be. Bromley, though listed prominently in ads for the Works, claimed to have had only a small part, merely advising Campbell-Kelly on selection and arrangement of the papers printed in Volumes 2 and 3. Campbell-Kelly confirmed that Bromley took no part in the detailed editing or in the provision of documents. That work was performed by one C.J.D. (“Jim”) Roberts, a “London-based independent scholar” who was “editorial consultant to the Works” (and, apparently, worked directly under Campbell-Kelly). Roberts seems to deserve more public credit than he has so far received. According to Campbell-Kelly, it was Roberts who, making a systematic search for unknown holdings of Babbage, turned up the original of the letter to Quetelet by writing the Royal Library (one “tiny triumph” among many). Campbell-Kelly also claimed that neither he nor Roberts knew of Berg’s prior discovery.     

Irreflexive Banach spaces are imperfect

It is shown that irreflexive Banach spaces are imperfect: thus the “perfect” spaces are exactly the reflexive Banach spaces and “mixed” spaces do not exist. This paper is part of the author’s M.Sc. dissertation, performed under guidance of Dr. D. Amir at the Technion — Israel Institute of Technology. The author wishes to thank Dr. Amir for his help.     

On the homology of mapping spaces

Following a Bendersky-Gitler idea, we construct an isomorphism between Anderson’s and Arone’s complexes modelling the chain complex of a mapping space. This allows us to apply Shipley’s convergence theorem to Arone’s model. As a corollary, we reduce the problem of homotopy equivalence for certain “toy” spaces to a problem in homological algebra.     

Stein domains and branched shadows of 4-manifolds

We provide sufficient conditions assuring that a suitably decorated 2-polyhedron can be thickened to a compact four-dimensional Stein domain. We also study a class of flat polyhedra in 4-manifolds and find conditions assuring that they admit Stein, compact neighborhoods. We base our calculations on Turaev’s shadows suitably “smoothed”; the conditions we find are purely algebraic and combinatorial. Applying our results, we provide examples of hyperbolic 3-manifolds admitting “many” positive and negative Stein fillable contact structures, and prove a four-dimensional analog of Oertel’s result on incompressibility of surfaces carried by branched polyhedra.      

About premise reduction of fuzzy inference algorithm

The comprehensive model with “weighted-objective nearness degree” is introduced in the process of multi-objective decision-making, by which a reduction problem of inference antecedents is studied in traditional fuzzy inference method. Moreover, based on the comprehensive model with “weighted-objective nearness degree”, SMTT fuzzy inference algorithm is proposed. This algorithm not only shows the relative importance of every antecedent component in fuzzy inference, but also considers the influence of nearness degree between every antecedent component’s evaluation and inference objective on inference conclusions. The enactment of inference objective reflects the preference degree of decision maker to every antecedent component’s evaluation. Therefore, it is much fitter for the demands of practical inference.     

The sweet mystery of compatibilism

Any satisfactory account of freedom must capture, or at least permit, the mysteriousness of freedom—a “sweet” mystery involving a certain kind of ignorance rather than a “sour” mystery of unintelligibility, incoherence, or unjustifiedness. I argue that compatibilism can capture the sweet mystery of freedom. I argue first that an action is free if and only if a certain “rationality constraint” is satisfied, and that nothing in standard libertarian accounts of freedom entails its satisfaction. Satisfaction of this constraint is consistent with the universal causal predetermination of action (UCP). If UCP is true and the rationality constraint satisfied, there’s a sense in which our actions are explanatorily (though not necessarily causally) overdetermined. While it seems plausible (given UCP) that our actions are so overdetermined, it seems utterly mysterious why they should be so overdetermined. Compatibilism’s capacity to accommodate this mystery is a mark in its favor.