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Mathematics has become inescapable in modern, digitized societies: there is hardly any area of life left that isn't affected by it, and we as mathematicians play a central role in this. Our actions affect what others, in particular our students, decide to do with mathematics, and how mathematics affects the world, for better or worse. In return, the study of ethics in mathematics (EiM) has become increasingly important, even though it is still unknown to many. This exposition tries to change that, by motivating ethics in mathematics as an interesting, tractable, non-trivial, well-defined and good research area for mathematicians to consider., Comment: 10 pages, 1 figure. This article is a preliminary version of an article that will be published in the Notices of the AMS. It is currently under review

The resources compiled in this document provide an approach to embed and teach Ethics in Mathematics at the undergraduate level. We provide mathematical exercises and homework problems that teach students ethical awareness and transferable skills, for many of the standard courses in the first and second years of a university degree in mathematics or related courses with significant mathematical content (e.g., physics, engineering, computer science, economics, etc). In addition to the exercises, this document also contains a list of projects, essay topics, and handouts for use as final projects and in seminars. This is a living document, and additional contributions are welcome., Comment: 95 pages, 2 figures. This is the first version, and we intend to make revisions. Comments and feedback are welcome - please get in touch with us

We extend Langdon Winner's idea that artifacts have politics into the realm of mathematics. To do so, we first provide a list of examples showing the existence of mathematical artifacts that have politics. In the second step, we provide an argument that shows that all mathematical artifacts have politics. We conclude by showing the implications for embedding ethics into mathematical curricula. We show how acknowledging that mathematical artifacts have politics can help mathematicians design better exercises for their mathematics students., Comment: 23 pages; keywords: artifacts have politics, embedded ethics, mathematics in society, ethics in mathematics, mathematics education

This manifesto has been written as a practical tool and aid for anyone carrying out, managing or influencing mathematical work. It provides insight into how to undertake and develop mathematically-powered products and services in a safe and responsible way. Rather than give a framework of objectives to achieve, we instead introduce a process that can be integrated into the common ways in which mathematical products or services are created, from start to finish. This process helps address the various issues and problems that can arise for the product, the developers, the institution, and for wider society. To do this, we break down the typical procedure of mathematical development into 10 key stages; our "10 pillars for responsible development" which follow a somewhat chronological ordering of the steps, and associated challenges, that frequently occur in mathematical work. Together these 10 pillars cover issues of the entire lifecycle of a mathematical product or service, including the preparatory work required to responsibly start a project, central questions of good technical mathematics and data science, and issues of communication, deployment and follow-up maintenance specifically related to mathematical systems. This manifesto, and the pillars within it, are the culmination of 7 years of work done by us as part of the Cambridge University Ethics in Mathematics Project. These are all tried-and-tested ideas, that we have presented and used in both academic and industrial environments. In our work, we have directly seen that mathematics can be an incredible tool for good in society, but also that without careful consideration it can cause immense harm. We hope that following this manifesto will empower its readers to reduce the risk of undesirable and unwanted consequences of their mathematical work., Comment: 33 pages, 1 figure. This is the first version, and we intend to make revisions. Comments and feedback are welcome - please get in touch with us

An argument is presented that mathematics teachers would benefit from an ethics framework that allows them to reflect upon their position regarding the teaching of ethics within mathematics. We present such a framework, adapted from Chiodo and Bursill-Hall's (2018) levels of ethical engagement for mathematicians, with discussion of its applications and implications. Awareness of ethical and social responsibility amongst mathematicians and mathematics educators is a newly established field, and yet it is crucially important, if we are to use mathematics for good and the avoidance of harm - in other words to use mathematics for justice, belonging, peace, care, education, lawmaking, building, creativity, equity, connection, fairness, rulemaking and order. Here, we present a way to extend ideas from ethics in the field of mathematics for mathematicians to the field of mathematics education for mathematics teachers, aiming to contribute to this transdisciplinary work and to advance the field further. We also explore links to ideas of equity, and silence and voice, drawing on the consequences of a deeper examination of the entanglements of mathematical-ethical dilemmas for mathematics teachers both inside and outside the classroom., Comment: 36 pages, submitted

When conducting COVID-19-related research, mathematicians may find themselves in public service roles. They might engage in policy discussions, collaborate with politicians and other non-scientists, and use the predictive power of mathematics to shape substantial decisions. We aim to promote good practice for ethical mathematical modelling in these difficult times. We begin with some observations about mathematical modellers, then derive eight questions that all mathematicians should constantly ask themselves, in order to develop their mathematics responsibly and ethically., Comment: 6 pages

Building on Chiodo and Bursill-Hall (2018), we outline that mathematicians can demonstrate what we term the "four levels of ethical engagement.", Comment: 5 pages

While the consequences of mathematically-based software, algorithms and strategies have become ever wider and better appreciated, ethical reflection on mathematics has remained primitive. We review the somewhat disconnected suggestions of commentators in recent decades with a view to piecing together a coherent approach to ethics in mathematics. Calls for a Hippocratic Oath for mathematicians are examined and it is concluded that while lessons can be learned from the medical profession, the relation of mathematicians to those affected by their work is significantly different. There is something to be learned also from the codes of conduct of cognate but professionalised quantitative disciplines such as engineering and accountancy, as well as from legal principles bearing on professional work. We conclude with recommendations that professional societies in mathematics should sponsor an (international) code of ethics, institutional mission statements for mathematicians and syllabuses for incorporation into mathematics degrees., Comment: 20 pages. Submitted to Science and Engineering Ethics

In [3] is was shown that for any group $G$ whose rank (i.e., minimal number of generators) is at most 3, and any finite index subgroup $H\leq G$ with index $[G:H]\geq rank(G)$, one can always find a left-right transversal of $H$ which generates $G$. In this paper we extend this result to groups of rank at most 4. We also extend this to groups $G$ of arbitrary (finite) rank $r$ provided all the non-trivial divisors of $[G:core_G(H)]$ are at least $2r-1$. Finally, we extend this to groups $G$ of arbitrary (finite) rank provided $H$ is malnormal in $G$., Comment: 36 pages. This is the first version, comments and suggestions are welcome

We give a complete survey of a construction by Boone and Collins for embedding any finitely presented group into one with $8$ generators and $26$ relations. We show that this embedding preserves the set of orders of torsion elements, and in particular torsion-freeness. We combine this with the independent results of Belegradek and Chiodo to prove that there is an $8$-generator $26$-relator universal finitely presented torsion-free group (one into which all finitely presented torsion-free groups embed)., Comment: 14 pages. First version. Comments welcome

For a set $X\subseteq \mathbb{N}$, we define the $X$-torsion of a group $G$ to be all elements $g\in G$ with $g^{n}=e$ for some $n\in X$. With $X$ recursively enumerable, we give two independent proofs (group-theoretic, and model-theoretic) that there exists a universal finitely presented $X$-torsion-free group; one which contains all finitely presented $X$-torsion-free groups. We also show that, if $X$ is recursively enumerable, then the set of finite presentations of $X$-torsion-free groups is $\Pi_{2}^{0}$-complete in Kleene's arithmetic hierarchy., Comment: 10 pages. This is the first version, comments are welcome

We show that a construction by Aanderaa and Cohen used in their proof of the Higman Embedding Theorem preserves torsion length. We give a new construction showing that every finitely presented group is the quotient of some $C'(1/6)$ finitely presented group by the subgroup generated by its torsion elements. We use these results to show there is a finitely presented group with infinite torsion length which is $C'(1/6)$, and thus word-hyperbolic and virtually torsion-free., Comment: 17 pages, version submitted for publication. This version: new title, expanded introduction, new section on torsion in word-hyperbolic groups, minor improvements

We explore transversals of finite index subgroups of finitely generated groups. We show that when $H$ is a subgroup of a rank $n$ group $G$ and $H$ has index at least $n$ in $G$ then we can construct a left transversal for $H$ which contains a generating set of size $n$ for $G$, and that the construction is algorithmic when $G$ is finitely presented. We also show that, in the case where $G$ has rank $n \leq3$, there is a simultaneous left-right transversal for $H$ which contains a generating set of size $n$ for $G$. We finish by showing that if $H$ is a subgroup of a rank $n$ group $G$ with index less than $3 \cdot 2^{n-1}$, and $H$ contains no primitive elements of $G$, then $H$ is normal in $G$ and $G/H \cong C_{2}^{n}$., Comment: 15 pages, 6 figures. This is the version submitted for publication

Let H, K be subgroups of G. We investigate the intersection properties of left and right cosets of these subgroups., Comment: 4 pages

We construct a 2-generator recursively presented group with infinite torsion length. We also explore the construction in the context of solvable and word-hyperbolic groups., Comment: 10 pages. This is the version submitted for publication

We introduce the subgroup identification problem, and show that there is a finitely presented group G for which it is unsolvable, and that it is uniformly solvable in the class of finitely presented locally Hopfian groups. This is done as an investigation into the difference between strong and weak effective coherence for finitely presented groups., Comment: 11 pages. This is the version submitted for publication

We give a uniform construction that, on input of a recursive presentation $P$ of a group, outputs a recursive presentation of a torsion-free group, isomorphic to $P$ whenever $P$ is itself torsion-free. We use this to re-obtain a known result, the existence of a universal finitely presented torsion-free group; one into which all finitely presented torsion-free groups embed. We apply our techniques to show that recognising embeddability of finitely presented groups is $\Pi^{0}_{2}$-hard, $\Sigma^{0}_{2}$-hard, and lies in $\Sigma^{0}_{3}$. We also show that the sets of orders of torsion elements of finitely presented groups are precisely the $\Sigma^{0}_{2}$ sets which are closed under taking factors., Comment: 11 pages. This is the version submitted for publication

We say a group is finitely annihilated if it is the set-theoretic union of all its proper normal finite index subgroups. We investigate this new property, and observe that it is independent of several other well known group properties. For finitely generated groups, we show that in many cases it is equivalent to having non-cyclic abelianisation, and at the same time construct an explicit infinite family of counterexamples to this. We show for finitely presented groups that this property is neither Markov nor co-Markov. In the context of our work we show that the weight of a non-perfect finite group, or a non-perfect finitely generated solvable group, is the same as the weight of its abelianisation. We generalise a theorem of Brodie-Chamberlain-Kappe on finite coverings of groups, and finish with some generalisations and variations of our new definition., Comment: 13 pages. This is the version submitted for publication

It is well known that the triviality problem for finitely presented groups is unsolvable; we ask the question of whether there exists a general procedure to produce a non-trivial element from a finite presentation of a non-trivial group. If not, then this would resolve an open problem by J. Wiegold: `Is every finitely generated perfect group the normal closure of one element?' We prove a weakened version of our question: there is no general procedure to pick a non-trivial generator from a finite presentation of a non-trivial group. We also show there is neither a general procedure to decompose a finite presentation of a non-trivial free product into two non-trivial finitely presented factors, nor one to construct an embedding from one finitely presented group into another in which it embeds. We apply our results to show that a construction by Stallings on splitting groups with more than one end can never be made algorithmic, nor can the process of splitting connect sums of non-simply connected closed 4-manifolds., Comment: 16 pages, revised version, minor corrections made, additional results on algorithms to construct embeddings and enumerate subgroups. v3: minor changes to typesetting and theorem numberings, minor corrections made