Ball,sphere, and all,all, all

The relationship of multidimensional geometry with statistical thermodynamics and with laws of large numbers is described.     

Contextualism,art, and rigidity: Levinson,Currie and Davies

The topic of this paper is the role played by context in art. In this regard I examine three theories linked to the names of J. Levinson, G. Currie and D. Davies. Levinson’s arguments undermine the structural theory. He finds it objectionable because it makes the individuation of artworks independent of their histories. Secondly, such a consequence is unacceptable because it fails to recognise that works are created rather than discovered. But, if certain general features of provenance are always work-constitutive, as it seems that Levinson is willing to claim, these features must always be essential properties of works. On the other hand, consideration of our modal practice suggests that whether a given general feature of provenance is essential or non-essential depends upon the particular work in question or is “work relative”. D. Davies builds his performance theory on the basis of the critical evaluation of Currie’s action-type hypotheses (ATH). Performances, says Davies, are not to be identified with “basic actions” to which their times belong essentially, but with “doings” that permit of the sorts of variation in modal properties required by the work-relativity of modality. He is also a fierce critic of the contextualist account. Contextualism is in his view unable to reflect the fact that aspects of provenance bear upon our modal judgements with variable force.In the second part of the paper I consider Davies’s “modality principle”. Davies is inclined to defend the claim that labels used for designation of works are rigid designators. Such a view offers a ground for discussion about the historicity of art. What has been meant when people claim that art is an historical concept? I argue that any historical theory implies a two-dimensional notion of “art”. At the end of the paper I suggest that Davies should embrace the theory of contingent identity and not the colocationist view about the relationship that exists between a particular artwork and its physical bearer.     

Woodhouse, babbage, peacock, and modern Algebra

In a recent article, J. M. Dubbey [Historia Mathematica 4 (1977), 295–302] showed that George Peacock's A Treatise on Algebra (1830) was similar to an unpublished work written by Charles Babbage in 1821. Evidently perplexed about the absence of a dispute over priority, Dubbey concluded that Peacock had unconsciously assimilated Babbage's ideas, and that Babbage was too busy with other activities to be concerned. The thesis of this article is that the innovative aspects of the work of both Babbage and Peacock are extensions of ideas put forth in 1803 by Robert Woodhouse, and that probably neither Babbage nor Peacock was overly concerned with acknowledgments because their approach to algebra was not unique at Cambridge.     

Diagrams,orientations, and varieties

One of the central problems in the theory of ordered sets is to describe the orientations of the covering graph of an ordered set. We show that the particular operation called inversion, together with the classical constructions of retraction and product, provide a context for the classification of all such orientations. AMS subject classifications (1980). 06A10, 05C75, 05C20.     

Game,set, and graph

In the twentieth century the theory of games was transformed. It began as an amusing pastime, and ended as a major branch of mathematical research and a key paradigm of economic theory. Here it will be argued that the transformation was the result of the work of mathematicians, such as Ernst Zermelo, John von Neumann and Dénes K?nig, who also contributed to two other areas of mathematics that were emerging at the same time: the theory of sets and the theory of graphs.     

Breaks, cuts, and patterns

We generalize the concept of a break by considering pairs of arbitrary rounds. We show that a set of home-away patterns minimizing the number of generalized breaks cannot be found in polynomial time, unless P=NP. When all teams have the same break set, the decision version becomes easy; optimizing remains NP-hard.     

Constructivism,Factoring, and Beliefs

The errors made by remedial intermediate algebra students in factoring polynomials were analyzed in the light of student definitions of factoring. Certain belief sets about factoring are found to logically entail many of the errors made. The vocabulary of epistemic semantics is used to describe how beliefs are changed. It is suggested that belief-based teaching can be successful in teaching factoring.     

Scarcity,competition, and value

We suggest a value for finite coalitional games with transferable utility that are enriched by non-negative weights for the players. In contrast to other weighted values, players stand for types of agents and weights are intended to represent the population sizes of these types. Therefore, weights do not only affect individual payoffs but also the joint payoff. Two principles guide the behavior of this value. Scarcity: the generation of worth is restricted by the scarcest type. Competition: only scarce types are rewarded. We find that the types’ payoffs for this value coincide with the payoffs assigned by the Mertens value to their type populations in an associated infinite game.     

Worms,gaps, and hydras

We define a direct translation from finite rooted trees to finite natural functions which shows that the Worm Principle introduced by Lev Beklemishev is equivalent to a very slight variant of the well‐known Kirby‐Paris' Hydra Game. We further show that the elements in a reduction sequence of the Worm Principle determine a bad sequence in the well‐quasi‐ordering of finite sequences of natural numbers with respect to Friedman's gapembeddability. A characterization of gap‐embeddability in terms of provability logic due to Lev Beklemishev is also presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Puzzles,tableaux, and mosaics

We define mosaics, which are naturally in bijection with Knutson-Tao puzzles. We define an operation on mosaics, which shows they are also in bijection with Littlewood-Richardson skew-tableaux. Another consequence of this construction is that we obtain bijective proofs of commutativity and associativity for the ring structures defined either of these objects. In particular, we obtain a new, easy proof of the Littlewood-Richardson rule. Finally we discuss how our operation is related to other known constructions, particularly jeu de taquin.     

Permutations,periodicity, and chaos

Let f: I → I be a continuous function on a closed interval I. If there exists x?I which has period 3 with respect to f, then Li and Yorke [1] proved that f is chaotic in the sense that there are not only points x?I of arbitrarily large period, but also uncountably many points x?I which are not even asymptotically periodic with respect to f. By using only elementary combinatorial facts about permutations, it is shown that if there is a point x?I of period p with respect to f, where p is divisible by 3, 5, or 7, then f is chaotic. The proof is followed by a study of some related combinatorial problems in symmetric groups.     

Ownership, Agency, and Defeat

In this article, I respond to Jennifer Duke-Yonge’s (2012) discussion of my article ‘Reflective Luck and Belief Ownership’ (Breyer, Acta Analytica, 25:133–154, 2010) and defend my Taking Responsibility account of belief ownership against her insightful criticisms.     

Contractualism, Politics, and Morality

Rawls developed a contractualist theory of social justice and Scanlon attempted to extend the Rawlsian framework to develop a theory of rightness, or morality more generally. I argue that there are some good reasons to adopt a contractualist theory of social justice, but that it is a mistake to adopt a contractualist theory of rightness. I begin by illustrating the major shared features of Scanlon and Rawls’ theories. I then show that the justification for these features in Rawls’ theory, the centrality of cooperative fairness to social justice, cannot be used to defend their use in Scanlon’s. Finally, I argue that Scanlon has not provided an adequate alternative defense of these features, and show that they create problems when contractualists try to explain major features of our common-sense morality.