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Faith Believes, Hope Expects: The Impact of Calvin's Theology on the Mathematics of Chance 信心相信，希望期待：加尔文神学对偶然数学的影响
arXiv - MATH - History and Overview Pub Date : 2024-07-18 DOI: arxiv-2407.13312
Timothy C. Johnson
{"title":"Faith Believes, Hope Expects: The Impact of Calvin's Theology on the Mathematics of Chance","authors":"Timothy C. Johnson","doi":"arxiv-2407.13312","DOIUrl":"https://doi.org/arxiv-2407.13312","url":null,"abstract":"This paper attributes the sudden emergence of mathematical probability and\u0000statistics in the second half of the seventeenth century to Calvin's Reformed\u0000theology. Calvin accommodated Epicurean chance with Stoic determinism and\u0000synthesised emph{phronesis/prudentia}, founded personal experience and\u0000employed to deal with emph{tyche/fortuna}, and emph{episteme/scientia},\u0000universal knowledge. This meant that matters of chance, which had previously\u0000been considered too particular for mathematical treatment, became part of\u0000emph{episteme/scientia}. Clear evidence of the significance of Calvin in\u0000mathematics is in the facts that Huygens considered using the word 'hope' to\u0000describe mathematical expectation and French mathematics still uses\u0000emph{esp'erance} for mathematical expectation. Calvin asserted that Hope\u0000represented a universal, objective and indubitable idea making it\u0000characteristic of mathematics. The argument is built on a review of how the\u0000ideas of Hope, Faith and Prudence have evolved in European thought that\u0000highlights Calvin's innovations. The conclusion identifies contemporary issues\u0000in the application of mathematics in society that are illuminated in light of\u0000Calvin's doctrine.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141742631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Golden ratio in graph theory: A survey 图论中的黄金分割率：调查
arXiv - MATH - History and Overview Pub Date : 2024-07-09 DOI: arxiv-2407.15860
Saeid Alikhani, Nima Ghanbari
{"title":"Golden ratio in graph theory: A survey","authors":"Saeid Alikhani, Nima Ghanbari","doi":"arxiv-2407.15860","DOIUrl":"https://doi.org/arxiv-2407.15860","url":null,"abstract":"Much has been written about the golden ratio $phi=frac{1+sqrt{5}}{2}$ and\u0000this strange number appears mysteriously in many mathematical calculations. In\u0000this article, we review the appearance of this number in the graph theory. More\u0000precisely, we review the relevance of this number in topics such as the number\u0000of spanning trees, topological indices, energy, chromatic roots, domination\u0000roots and the number of domatic partitions of graphs.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141774626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Algorithmic methods of finite discrete structures. Automorphism of Nonseparable Graphs 有限离散结构的算法方法。不可分割图的自动形态
arXiv - MATH - History and Overview Pub Date : 2024-07-02 DOI: arxiv-2407.12045
Sergey Kurapov, Maxim Davidovsky
{"title":"Algorithmic methods of finite discrete structures. Automorphism of Nonseparable Graphs","authors":"Sergey Kurapov, Maxim Davidovsky","doi":"arxiv-2407.12045","DOIUrl":"https://doi.org/arxiv-2407.12045","url":null,"abstract":"The monography examines the problem of constructing a group of automorphisms\u0000of a graph. A graph automorphism is a mapping of a set of vertices onto itself\u0000that preserves adjacency. The set of such automorphisms forms a vertex group of\u0000a graph or simply a graph group. The basis for constructing a group of graph\u0000automorphisms is the concept of orbit. The construction of an orbit is closely\u0000related to the quantitative assessment of a vertex or edge of a graph, called\u0000weight. To determine the weight of an element, graph invariants built on the\u0000spectrum of edge cuts and the spectrum of edge cycles are used. The weight of\u0000the graph elements allows identifying generating cycles and forming orbits.\u0000Examples are given of constructing a group of automorphisms for some types of\u0000graphs.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141742633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fitting an Elephant with Four non-Zero Parameters 用四个非零参数拟合大象
arXiv - MATH - History and Overview Pub Date : 2024-07-02 DOI: arxiv-2407.07909
Dian Jin, Junze Yuan
{"title":"Fitting an Elephant with Four non-Zero Parameters","authors":"Dian Jin, Junze Yuan","doi":"arxiv-2407.07909","DOIUrl":"https://doi.org/arxiv-2407.07909","url":null,"abstract":"In 1953, Enrico Fermi criticized Dyson's model by quoting Johnny von Neumann:\u0000\"With four parameters I can fit an elephant, and with five I can make him\u0000wiggle his trunk.\" So far, there have been several attempts to fit an elephant\u0000using four parameters, but as the problem has not been well-defined, the\u0000current methods do not completely satisfy the requirements. This paper defines\u0000the problem and presents an attempt.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"2011 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Moonshot for AI Oracles in the Sciences 科学界人工智能神谕的登月计划
arXiv - MATH - History and Overview Pub Date : 2024-06-25 DOI: arxiv-2406.17836
Bryan Kaiser, Tailin Wu, Maike Sonnewald, Colin Thackray, Skylar Callis
{"title":"A Moonshot for AI Oracles in the Sciences","authors":"Bryan Kaiser, Tailin Wu, Maike Sonnewald, Colin Thackray, Skylar Callis","doi":"arxiv-2406.17836","DOIUrl":"https://doi.org/arxiv-2406.17836","url":null,"abstract":"Nobel laureate Philip Anderson and Elihu Abrahams once stated that, \"even if\u0000machines did contribute to normal science, we see no mechanism by which they\u0000could create a Kuhnian revolution and thereby establish a new physical law.\" In\u0000this Perspective, we draw upon insights from the philosophies of science and\u0000artificial intelligence (AI) to propose necessary conditions of precisely such\u0000a mechanism for generating revolutionary mathematical theories. Recent\u0000advancements in AI suggest that satisfying the proposed necessary conditions by\u0000machines may be plausible; thus, our proposed necessary conditions also define\u0000a moonshot challenge. We also propose a heuristic definition of the\u0000intelligibility of mathematical theories to accelerate the development of\u0000machine theorists.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Topology Scavenger Hunt to Introduce Topological Data Analysis 介绍拓扑数据分析的拓扑寻宝游戏
arXiv - MATH - History and Overview Pub Date : 2024-06-21 DOI: arxiv-2406.15580
Lori Ziegelmeier
{"title":"A Topology Scavenger Hunt to Introduce Topological Data Analysis","authors":"Lori Ziegelmeier","doi":"arxiv-2406.15580","DOIUrl":"https://doi.org/arxiv-2406.15580","url":null,"abstract":"Topology at the undergraduate level is often a theoretical mathematics\u0000course, introducing concepts from point-set topology or possibly algebraic\u0000topology. However, the last two decades have seen an explosion of growth in\u0000applied topology and topological data analysis, which are topics that can be\u0000presented in an accessible way to undergraduate students and can encourage\u0000exciting projects. For the past several years, the Topology course at\u0000Macalester College has included content from point-set and algebraic topology,\u0000as well as applied topology, culminating in a project chosen by the students.\u0000In the course, students work through a topology scavenger hunt as an activity\u0000to introduce the ideas and software behind some of the primary tools in\u0000topological data analysis, namely, persistent homology and mapper. This\u0000scavenger hunt includes a variety of point clouds of varying dimensions, such\u0000as an annulus in 2D, a bouquet of loops in 3D, a sphere in 4D, and a torus in\u0000400D. The students' goal is to analyze each point cloud with a variety of\u0000software to infer the topological structure. After completing this activity,\u0000students are able to extend the ideas learned in the scavenger hunt to an\u0000open-ended capstone project. Examples of past projects include: using\u0000persistence to explore the relationship between country development and\u0000geography, to analyze congressional voting patterns, and to classify genres of\u0000a large corpus of texts by combining with tools from natural language\u0000processing and machine learning.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"96 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Mathematical Entities: Corpora and Benchmarks 数学实体：语料库和基准
arXiv - MATH - History and Overview Pub Date : 2024-06-17 DOI: arxiv-2406.11577
Jacob Collard, Valeria de Paiva, Eswaran Subrahmanian
{"title":"Mathematical Entities: Corpora and Benchmarks","authors":"Jacob Collard, Valeria de Paiva, Eswaran Subrahmanian","doi":"arxiv-2406.11577","DOIUrl":"https://doi.org/arxiv-2406.11577","url":null,"abstract":"Mathematics is a highly specialized domain with its own unique set of\u0000challenges. Despite this, there has been relatively little research on natural\u0000language processing for mathematical texts, and there are few mathematical\u0000language resources aimed at NLP. In this paper, we aim to provide annotated\u0000corpora that can be used to study the language of mathematics in different\u0000contexts, ranging from fundamental concepts found in textbooks to advanced\u0000research mathematics. We preprocess the corpora with a neural parsing model and\u0000some manual intervention to provide part-of-speech tags, lemmas, and dependency\u0000trees. In total, we provide 182397 sentences across three corpora. We then aim\u0000to test and evaluate several noteworthy natural language processing models\u0000using these corpora, to show how well they can adapt to the domain of\u0000mathematics and provide useful tools for exploring mathematical language. We\u0000evaluate several neural and symbolic models against benchmarks that we extract\u0000from the corpus metadata to show that terminology extraction and definition\u0000extraction do not easily generalize to mathematics, and that additional work is\u0000needed to achieve good performance on these metrics. Finally, we provide a\u0000learning assistant that grants access to the content of these corpora in a\u0000context-sensitive manner, utilizing text search and entity linking. Though our\u0000corpora and benchmarks provide useful metrics for evaluating mathematical\u0000language processing, further work is necessary to adapt models to mathematics\u0000in order to provide more effective learning assistants and apply NLP methods to\u0000different mathematical domains.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"139 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
From Concrete to Abstract in Indian Mathematics 印度数学从具体到抽象
arXiv - MATH - History and Overview Pub Date : 2024-06-14 DOI: arxiv-2406.10147
Jaidev Dasgupta
{"title":"From Concrete to Abstract in Indian Mathematics","authors":"Jaidev Dasgupta","doi":"arxiv-2406.10147","DOIUrl":"https://doi.org/arxiv-2406.10147","url":null,"abstract":"Despite the extensive amount of scholarly work done on Indian mathematics in\u0000the last 200 years, the conditions under which it originated and evolved is\u0000still not clear. Often, one reads the ancient texts with the present concepts\u0000and methods in mind. The fact of absence of script over a long stretch of\u0000Indian history in ancient times also gets overlooked in such readings. The\u0000purpose of this article is to explore the journey of mathematics by examining\u0000what the ancient texts tell us about the nature of mathematics in their times.\u0000What one finds from the investigation of arithmetic, geometry and algebra is\u0000that while it was concrete and context bound, rooted in solving practical\u0000problems in ancient times, Indian mathematics transitioned to context free,\u0000abstract stage with the advent of algebra supported by writing.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"173 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The work of Walter Bergweiler in value distribution of meromorphic functions 沃尔特-伯格韦勒在微函数值分布方面的工作
arXiv - MATH - History and Overview Pub Date : 2024-06-14 DOI: arxiv-2406.09992
Alexandre Eremenko
{"title":"The work of Walter Bergweiler in value distribution of meromorphic functions","authors":"Alexandre Eremenko","doi":"arxiv-2406.09992","DOIUrl":"https://doi.org/arxiv-2406.09992","url":null,"abstract":"This is a colloquium talk in CAU, Kiel delivered on June 7, 2024 on the\u0000occasion of Walter Bergweiler's retirement. Walter's work on meromorphic\u0000functions consists of two parts: generalizations of Picard's theorem to\u0000differential polynomials, and the applications of the rescaling principle known\u0000as the Bloch Principle. Since the talk was aimed at the general audience, a\u0000brief introduction to Nevanlinna theory is included.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A non-circular concept of number inspired by Gottlob Frege's definition 受戈特洛布-弗雷格定义启发的非循环数概念
arXiv - MATH - History and Overview Pub Date : 2024-06-13 DOI: arxiv-2406.08715
Marco Aurélio Spohn
{"title":"A non-circular concept of number inspired by Gottlob Frege's definition","authors":"Marco Aurélio Spohn","doi":"arxiv-2406.08715","DOIUrl":"https://doi.org/arxiv-2406.08715","url":null,"abstract":"Gottlob Frege ingeniously presented a purely logical definition of the\u0000concept of number. However, one can claim that his definition is, in some way,\u0000circular, as it relies on the concept of one-to-one relation. The concept of\u0000number only makes sense when it presents the property of projection/reflection\u0000or binding. When we consider a number as an abstraction of objects, whatever\u0000they may be, saying that a number that belongs to the concept F is the same as\u0000that which belongs to the concept G means there is a projection/reflection, or\u0000binding, between the objects in F and the objects in G. We present a definition\u0000based on both equivalent approaches. First, we introduce the definition based\u0000on the relations of projection and reflection; then, we present the definition\u0000based on the relation of binding.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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