Related papers: Mathematical Proofs 101: How Proofs Should Be Read…
This is a tract on the art and practice of mathematical writing. Not only does the book cover basic principles of grammar, syntax, and usage, but it takes into account developments of the last twenty years that have been inspired by the…
This is a reflection on the author's experience in teaching logic at the graduate level in a computer science department. The main lesson is that model building and the process of modelling must be placed at the centre stage of logic…
Mathematical proofs are both paradigms of certainty and some of the most explicitly-justified arguments that we have in the cultural record. Their very explicitness, however, leads to a paradox, because the probability of error grows…
Teaching precise mathematical reasoning can be very hard. It is very easy for a student to make a subtle mistake in a proof which invalidates it, but it is often hard for the teacher to pinpoint and explain the problem in the (often…
The book was written on the basis of materials that we presented at several faculties, either as lectures or as part of auditory exercises. Aware that there are more books and textbooks in the area in which the topics covered by this book…
Proof scores can be regarded as outlines of the formal verification of system properties. They have been historically used by the OBJ family of specification languages. The main advantage of proof scores is that they follow the same syntax…
We introduce a novel task consisting in assigning a proof to a given mathematical statement. The task is designed to improve the processing of research-level mathematical texts. Applying Natural Language Processing (NLP) tools to research…
In mathematical proof education, there remains a need for interventions that help students learn to write mathematical proofs. Research has shown that timely feedback can be very helpful to students learning new skills. While for many years…
This scoping review examines the use of student explanation strategies in postsecondary mathematics and statistics education. We analyzed 46 peer-reviewed articles published between 2014 and 2024, categorizing student explanations into…
"Systems that Explain Themselves" appears a provocative wording, in particular in the context of mathematics education -- it is as provocative as the idea of building educational software upon technology from computer theorem proving. In…
The generally accepted wisdom in computational circles is that pure proof verification is a solved problem and that the computationally hard elements and fertile areas of study lie in proof discovery. This wisdom presumably does hold for…
Despite significant developments in Proof Theory, surprisingly little attention has been devoted to the concept of proof verifier. In particular, the mathematical community may be interested in studying different types of proof verifiers…
Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of…
We present a research mathematician's perspective on current developments around in K-12 mathematics. We share activities, and highlight the different ways in which students' reasoning can progress, such as amount of abstraction,…
The growing need for trustworthy machine learning has led to the blossom of interpretability research. Numerous explanation methods have been developed to serve this purpose. However, these methods are deficiently and inappropriately…
How students use mathematics in their physics classes has been studied extensively in the physics education literature. In addition to specific mathematical methods in specific physics contexts, possible effects of more general "cultural"…
In this paper, I explore what mathematical research can tell us about ourselves, and our role in the world, using examples from my own experience. The paper is a sequel to my piece "Mathematics is a Quest for Truth", published in the…
Deep metric learning papers from the past four years have consistently claimed great advances in accuracy, often more than doubling the performance of decade-old methods. In this paper, we take a closer look at the field to see if this is…
This essay examines how automation has reconfigured mathematical proof and labor, and what might happen in the future. It discusses practical standards of proof, distinguishes between prominent forms of automation in research, provides…
To assess the ability of current AI systems to correctly answer research-level mathematics questions, we share a set of ten math questions which have arisen naturally in the research process of the authors. The questions had not been shared…