The co-evolution of number concepts and counting words

Humans possess a number concept that differs from its predecessors in animal cognition in two crucial respects: (1) it is based on a numerical sequence whose elements are not confined to quantitative contexts, but can indicate cardinal/quantitative as well as ordinal and even nominal properties of empirical objects (e.g. ‘five buses’: cardinal; ‘the fifth bus’: ordinal; ‘the #5 bus’: nominal), and (2) it can involve recursion and, via recursion, discrete infinity. In contrast to that, the predecessors of numerical cognition that we find in animals and human infants rely on finite and iconic representations that are limited to cardinality and do not support a unified concept of number. In this paper, I argue that the way such a unified number concept could evolve in humans is via verbal sequences that are employed as numerical tools, that is, sequences of words whose elements are associated with empirical objects in number assignments. In particular, I show that a certain kind of number words, namely the counting sequences of natural languages, can be characterised as a central instance of verbal numerical tools. I describe a possible scenario for the emergence of such verbal numerical tools in human history that starts from iconic roots and that suggests that in a process of co-evolution, the gradual emergence of counting sequences and the development of an increasingly comprehensive number concept supported each other. On this account, it is language that opened the way for numerical cognition, suggesting that it is no accident that the same species that possesses the language faculty as a unique trait, should also be the one that developed a systematic concept of number

Using homological duality in consecutive pattern avoidance

Using the approach suggested in [arXiv:1002.2761] we present below a sufficient condition guaranteeing that two collections of patterns of permutations have the same exponential generating functions for the number of permutations avoiding elements of these collections as consecutive patterns. In short, the coincidence of the latter generating functions is guaranteed by a length-preserving bijection of patterns in these collections which is identical on the overlappings of pairs of patterns where the overlappings are considered as unordered sets. Our proof is based on a direct algorithm for the computation of the inverse generating functions. As an application we present a large class of patterns where this algorithm is fast and, in particular, allows to obtain a linear ordinary differential equation with polynomial coefficients satisfied by the inverse generating function.Comment: 12 pages, 1 figur

On Algorithms and Complexity for Sets with Cardinality Constraints

Typestate systems ensure many desirable properties of imperative programs, including initialization of object fields and correct use of stateful library interfaces. Abstract sets with cardinality constraints naturally generalize typestate properties: relationships between the typestates of objects can be expressed as subset and disjointness relations on sets, and elements of sets can be represented as sets of cardinality one. Motivated by these applications, this paper presents new algorithms and new complexity results for constraints on sets and their cardinalities. We study several classes of constraints and demonstrate a trade-off between their expressive power and their complexity. Our first result concerns a quantifier-free fragment of Boolean Algebra with Presburger Arithmetic. We give a nondeterministic polynomial-time algorithm for reducing the satisfiability of sets with symbolic cardinalities to constraints on constant cardinalities, and give a polynomial-space algorithm for the resulting problem. In a quest for more efficient fragments, we identify several subclasses of sets with cardinality constraints whose satisfiability is NP-hard. Finally, we identify a class of constraints that has polynomial-time satisfiability and entailment problems and can serve as a foundation for efficient program analysis.Comment: 20 pages. 12 figure

The non-unique Universe

The purpose of this paper is to elucidate, by means of concepts and theorems drawn from mathematical logic, the conditions under which the existence of a multiverse is a logical necessity in mathematical physics, and the implications of Godel's incompleteness theorem for theories of everything. Three conclusions are obtained in the final section: (i) the theory of the structure of our universe might be an undecidable theory, and this constitutes a potential epistemological limit for mathematical physics, but because such a theory must be complete, there is no ontological barrier to the existence of a final theory of everything; (ii) in terms of mathematical logic, there are two different types of multiverse: classes of non-isomorphic but elementarily equivalent models, and classes of model which are both non-isomorphic and elementarily inequivalent; (iii) for a hypothetical theory of everything to have only one possible model, and to thereby negate the possible existence of a multiverse, that theory must be such that it admits only a finite model

Topology of two-connected graphs and homology of spaces of knots

We propose a new method of computing cohomology groups of spaces of knots in $\R^n$, $n \ge 3$, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order $\le 3.$ As a byproduct we define the higher indices, which invariants of knots in $\R^3$ define at arbitrary singular knots. More generally, for any finite-order cohomology class of the space of knots we define its principal symbol, which lies in a cohomology group of a certain finite-dimensional configuration space and characterizes our class modulo the classes of smaller filtration

Some new directions in infinite-combinatorial topology

We give a light introduction to selection principles in topology, a young subfield of infinite-combinatorial topology. Emphasis is put on the modern approach to the problems it deals with. Recent results are described, and open problems are stated. Some results which do not appear elsewhere are also included, with proofs.Comment: Small update