A Note on the Seven Bridges of Königsberg Problem

In this paper we account for the formalization of the seven bridges of Königsberg puzzle. The problem originally posed and solved by Euler in 1735 is historically notable for having laid the foundations of graph theory, cf. [7]. Our formalization utilizes a simple set-theoretical graph representation with four distinct sets for the graph’s vertices and another seven sets that represent the edges (bridges). The work appends the article by Nakamura and Rudnicki [10] by introducing the classic example of a graph that does not contain an Eulerian path. This theorem is item #54 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Czesław Byliński and Piotr Rudnicki. The correspondence between monotonic many sorted signatures and well-founded graphs. Part I. Formalized Mathematics, 5(4):577– 582, 1996.Gary Chartrand. Introductory Graph Theory. New York: Dover, 1985.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Krzysztof Hryniewiecki. Graphs. Formalized Mathematics, 2(3):365–370, 1991.Yatsuka Nakamura and Piotr Rudnicki. Euler circuits and paths. Formalized Mathematics, 6(3):417–425, 1997.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990. Received June 16, 201

The End of Mystery

Tim travels back in time and tries to kill his grandfather before his father was born. Tim fails. But why? Lewis's response was to cite "coincidences": Tim is the unlucky subject of gun jammings, banana peels, sudden changes of heart, and so on. A number of challenges have been raised against Lewis's response. The latest of these focuses on explanation. This paper diagnoses the source of this new disgruntlement and offers an alternative explanation for Tim's failure, one that Lewis would not have liked. The explanation is an obvious one but controversial, so it is defended against all the objections that can be mustered

Generalisation : graphs and colourings

The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe

Mathematical explanation and indispensability

This paper discusses Baker’s Enhanced Indispensability Argument (EIA) for mathematical realism on the basis of the indispensable role mathematics plays in scientific explanations of physical facts, along with various responses to it. I argue that there is an analogue of causal explanation for mathematics which, of several basic types of explanation, holds the most promise for use in the EIA. I consider a plausible case where mathematics plays an explanatory role in this sense, but argue that such use still does not support realism about mathematical objects.; Este artículo analiza la versión ‘mejorada’ del argumento de la indispensabilidad (EIA) de Baker y varias de las respuestas que ha recibido. Este argumento pretende establecer el realismo matemático en virtud del papel indispensable que las matemáticas juegan en las explicaciones científicas de los hechos físicos. Argumento que hay un análogo de explicación causal para las matemáticas que, de los varios sentidos básicos de explicación, parece el más adecuado para el EIA. Analizo un caso plausible en el que las matemáticas juegan un papel explicativo en este sentido, pero concluyo que este uso no es suficiente para establecer el realismo acerca de los objetos matemáticos

Mathematical explanation and indispensability

This paper discusses Baker’s Enhanced Indispensability Argument (EIA) for mathematical realism on the basis of the indispensable role mathematics plays in scientific explanations of physical facts, along with various responses to it. I argue that there is an analogue of causal explanation for mathematics which, of several basic types of explanation, holds the most promise for use in the EIA. I consider a plausible case where mathematics plays an explanatory role in this sense, but argue that such use still does not support realism about mathematical objects.; Este artículo analiza la versión ‘mejorada’ del argumento de la indispensabilidad (EIA) de Baker y varias de las respuestas que ha recibido. Este argumento pretende establecer el realismo matemático en virtud del papel indispensable que las matemáticas juegan en las explicaciones científicas de los hechos físicos. Argumento que hay un análogo de explicación causal para las matemáticas que, de los varios sentidos básicos de explicación, parece el más adecuado para el EIA. Analizo un caso plausible en el que las matemáticas juegan un papel explicativo en este sentido, pero concluyo que este uso no es suficiente para establecer el realismo acerca de los objetos matemáticos

The degree-number of vertices problem in Manhattan networks

Generally speaking, the aim of this work is to study the problem (Delta,N) (or the degree-number of vertices problem) for the case of a Manhattan digraph. A digraph is a network formed by vertices and directed edges called arcs (in the case of graphs the edges have no direction). The diameter of a graph is the minimum distance that exists between two of the farthest vertices. In the diameter of a digraph we must take into account that arcs have direction. A double-step digraph consists of N vertices and a set of arcs of the form (i,i+a) and (i,i+b), with a and b positive integers called 'steps', that is, there are connections from vertex i to vertices i+a and i+b (operations are modulo N). This digraph is denoted by G(N;a,b). A double-step graph G(N;+-a,+-b) consists of N vertices, but the edges are of the form (i,i+-a) and (i,i+-b), with steps a and b (positive integers), therefore, there are connections from vertex i to vertices i+a, i-a, i+b and i-b (mod N). In a Manhattan digraph, the arcs have directions like the ones of the streets and avenues of Manhattan (or l'Eixample in Barcelona), that is, if an arc goes to the right, the 'next one' goes to the left and if an arc goes upwards, the 'next one' goes downwards. The (Delta,N) problem consists in finding the minimum diameter of a graph or digraph given the number of vertices N and the maximum degree Delta. As this problem has been solved for the case of double-step graphs G(N;+-a,+-b), we expand these graphs transforming every vertex into a directed cycle of order 4 and every edge into two arcs in opposite directions, so that we obtain a Manhattan digraph M. In this work we find the relation between the steps of the double-step graph G(N;+-a,+-b) and the ones of the Manhattan digraph M. Moreover, we made a program that calculates the diameter of the so-called New Amsterdam digraph NA, related to the Manhattan digraph M, from the parameters of the original graph G(N;+-a,+-b).Català: En termes generals, l’objectiu d’aquest treball és estudiar el problema (o problema grau-nombre de vèrtexs) per al cas del digraf Manhattan. Un digraf és una xarxa constituïda per vèrtexs i per arestes dirigides anomenades arcs (en el cas de grafs, les arestes no tenen direcció). El diàmetre d’un graf és la mínima distància possible que hi ha entre dos dels vèrtexs més allunyats entre si. En el diàmetre d’un digraf hem de tenir en compte que els arcs tenen direcció. Un digraf de doble pas consta de vèrtexs i un conjunt d'arcs de la forma i , amb i enters positius anomenats “passos", és a dir, que existeixen enllaços des del vèrtex cap els vèrtexs i (les operacions s'han d'entendre sempre mòdul ). Aquest digraf es denota . Un graf de doble pas també consta de vèrtexs, però les arestes són de la forma i , amb passos i (enters positius), per tant, existeixen enllaços des del vèrtex cap els vèrtexs i (mod ) . En un digraf Manhattan els arcs tenen les direccions com les dels carrers i les avingudes de Manhattan (o de l'Eixample de Barcelona), és a dir, si un arc va cap a la dreta, el "següent" va cap a l'esquerra i si un arc va cap a dalt, el "següent" va cap a baix. El problema consisteix a trobar el diàmetre mínim d'un graf o digraf fixats el nombre de vèrtexs i el grau . Com que aquest problema ha estat resolt per al cas de grafs de doble pas , hem expandit aquests grafs transformant cada vèrtex en un cicle dirigit de 4 vèrtexs i cada aresta en dos arcs de sentits oposats, de manera que obtenim un digraf Manhattan . En aquest treball trobem la relació entre els passos del graf de doble pas i els del digraf Manhattan . A més, hem fet un programa que calcula el diàmetre del digraf anomenat New Amsterdam , que està relacionat amb el Manhattan , a partir dels paràmetres del graf original

The end of mystery

Tim travels back in time and tries to kill his grandfather before his father was born. Tim fails. But why? Lewis’s response was to cite “coincidences”: Tim is the unlucky subject of gun jammings, banana peels, sudden changes of heart, and so on. A number of challenges have been raised against Lewis’s response. The latest of these focuses on explanation. This paper diagnoses the source of this new disgruntlement and offers an alternative explanation for Tim’s failure, one that Lewis would not have liked. The explanation is an obvious one but controversial, so it is defended against all the objections that can be mustered

Mathematical Explanations and Mathematical Applications

One of the key questions in the philosophy of mathematics is the role and status of mathematical applications in the natural sciences. The importance of mathematics for science is indisputable, but philosophers have disagreed on what the relation between mathematical theories and scientific theories are. This chapter presents these topics through a distinction between mathematical applications and mathematical explanations. Particularly important is the question whether mathematical applications are ever indispensable. If so, it has often been argued, such applications should count as proper mathematical explanations. Following Quine, many philosophers have also contended that if there are indispensable mathematical applications in the natural sciences, then the mathematical objects posited in those applications have an independent existence like the scientific objects. Thus the question of mathematical explanations and applications has an important relevance for the ontology of mathematics.Peer reviewe