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A suite of impressive scientific discoveries have been driven by recent advances in artificial intelligence. These almost all result from training flexible algorithms to solve difficult optimization problems specified in advance by teams of…
While bibliometrics are widely used for research evaluation purposes, a common theoretical framework for conceptually understanding, empirically studying, and effectively teaching its usage is lacking. In this paper, we outline such a…
Creating learning models that can exhibit sophisticated reasoning skills is one of the greatest challenges in deep learning research, and mathematics is rapidly becoming one of the target domains for assessing scientific progress in this…
When we want to predict the future, we compute it from what we know about the present. Specifically, we take a mathematical representation of observed reality, plug it into some dynamical equations, and then map the time-evolved result back…
Using machine learning in high-stakes applications often requires predictions to be accompanied by explanations comprehensible to the domain user, who has ultimate responsibility for decisions and outcomes. Recently, a new framework for…
We consider learning mathematics through action research, hacking, discovery, inquiry, learning-by-doing as opposed to the instruct and perform, industrial model of the 19th century. A learning model based on self-awareness, types,…
Understanding and creating mathematics using natural mathematical language - the mixture of symbolic and natural language used by humans - is a challenging and important problem for driving progress in machine learning. As a step in this…
Breakthrough discoveries and inventions involve unexpected combinations of contents including problems, methods, and natural entities, and also diverse contexts such as journals, subfields, and conferences. Drawing on data from tens of…
Nobel laureate Philip Anderson and Elihu Abrahams once stated that, "even if machines did contribute to normal science, we see no mechanism by which they could create a Kuhnian revolution and thereby establish a new physical law." In this…
We argue that it is neither necessary nor sufficient for a mathematical proof to have epistemic value that it be "correct", in the sense of formalizable in a formal proof system. We then present a view on the relationship between…
Artificial intelligence (AI) is reshaping how research is conceived, conducted, and communicated across fields from chemistry to biomedicine. This commentary examines how AI is transforming the research workflow. AI systems now help…
The testimony and practice of notable mathematicians indicate that there is an important phenomenological and epistemological difference between superficial and deep analogies in mathematics. In this paper, we offer a descriptive theory of…
A recent trend in mathematical modeling is to publish the computer code together with the research findings. Here we explore the formal question, whether and in which sense a computer implementation is distinct from the mathematical model.…
Proof by induction plays a central role in formal verification. However, its automation remains as a formidable challenge in Computer Science. To solve inductive problems, human engineers often have to provide auxiliary lemmas manually. We…
While proof is a central component of postsecondary mathematical study, proof construction has historically posed significant difficulties for students who intend to earn mathematics degrees at the undergraduate level. This work is…
Evidence shows that in a significant number of cases the current methods of research do not allow for reproducible and falsifiable procedures of scientific investigation. As a consequence, the majority of critical decisions at all levels,…
Autoformalisation, the task of expressing informal mathematical statements in formal language, is often viewed as a direct translation process. This, however, disregards a critical preceding step: conjecturing. Many mathematical problems…
Artificial intelligence (AI) has acquired notorious relevance in modern computing as it effectively solves complex tasks traditionally done by humans. AI provides methods to represent and infer knowledge, efficiently manipulate texts and…
We currently witness the spectacular success of artificial intelligence in both science and public life. However, the development of a rigorous mathematical foundation is still at an early stage. In this survey article, which is based on an…
Pre-college mathematics modeling instruction often frames mathematics as being separated from reasoning about the real world -- and commonly treats reasoning mathematically and reasoning about the real-world context as separate stages of a…