Related papers: Varieties of mathematical understanding
Recent advances in computing have changed not only the nature of mathematical computation, but mathematical proof and inquiry itself. While artificial intelligence and formalized mathematics have been the major topics of this conversation,…
This overview article highlights the critical role of mathematics in artificial intelligence (AI), emphasizing that mathematics provides tools to better understand and enhance AI systems. Conversely, AI raises new problems and drives the…
We argue how AI can assist mathematics in three ways: theorem-proving, conjecture formulation, and language processing. Inspired by initial experiments in geometry and theoretical physics in 2017, we summarize how this emerging field has…
This is an explanation and defense of "mathematical conceptualism" for a general mathematical and philosophical audience. I make a case that it is cogent, rigorous, attractive, and better suited to ordinary mathematical practice than all…
Symbolic calculators like Mathematica are becoming more commonplace among upper level physics students. The presence of such a powerful calculator can couple strongly to the type of mathematical reasoning students employ. It does not merely…
After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics.
The complexity of quantum computation remains poorly understood. While physicists attempt to find ways to create quantum computers, we still do not have much evidence one way or the other as to how useful these machines will be. The tools…
As new technologies move to the fore, our understanding of the world may seem to have shrunk in comparison, for despite new developments in research, much of it is reduced or rather, abstracted for marketability. Thus, the purpose of this…
Quantum computers take advantage of interfering quantum alternatives in order to handle problems that might be too time consuming with algorithms based on classical logic. Developing quantum computers requires new ways of thinking beyond…
Artificial intelligence (AI) is the name popularly given to a broad spectrum of computer tools designed to perform increasingly complex cognitive tasks, including many that used to solely be the province of humans. As these tools become…
The interpretation of quantum mechanics is an area of increasing interest to many working physicists. In particular, interest has come from those involved in quantum computing and information theory, as there has always been a strong…
Computational thinking is a new problem soling method named for its extensive use of computer science techniques. It synthesizes critical thinking and existing knowledge and applies them in solving complex technological problems. The term…
Readability assessment is the task of evaluating the reading difficulty of a given piece of text. Although research on computational approaches to readability assessment is now two decades old, there is not much work on synthesizing this…
In their study of physics beyond the first year of University -- termed upper-division in the US, many of students' primary learning opportunities come from working long, complex back-of-the-book style problems, and from trying to develop…
Quantum computing has recently appeared in the headlines of many scientific and popular publications. In the context of quantitative finance, we provide here an overview of its potential.
In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear…
Mathematics is the language of science. Fluent and productive use of mathematics requires one to understand the meaning embodied in mathematical symbols, operators, syntax, etc., which can be a difficult task. For instance, in algebraic…
There are inherent limits in classical computation for it to serve as an adequate model of human cognition. In particular, non-commutativity, while ubiquitous in physics and psychology, cannot be sufficiently handled. We propose that we…
The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be…
During the past decades, quantum mechanical methods have undergone an amazing transition from pioneering investigations of experts into a wide range of practical applications, made by a vast community of researchers. First principles…