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Computing is an indispensable component of nearly all technologies and is ubiquitous for vast segments of society. It is also essential to discoveries and innovations in most disciplines. However, while past grand challenges in science have…
Compared with constraint satisfaction problems, counting problems have received less attention. In this paper, we survey research works on the problems of counting the number of solutions to constraints. The constraints may take various…
Mathematical reasoning---a core ability within human intelligence---presents some unique challenges as a domain: we do not come to understand and solve mathematical problems primarily on the back of experience and evidence, but on the basis…
The purpose of this note is to present a formulation of a given nonlinear ordinary differential equation into an equivalent system of linear ordinary differential equations. It is evident that the easiness of a such procedure would be able…
Implicit Computational Complexity makes two aspects implicit, by manipulating programming languages rather than models of com-putation, and by internalizing the bounds rather than using external measure. We survey how automata theory…
Questions concerning origin of mathematical knowledge and roles of language and intuition (imagery) in mathematical thoughts are long standing and widely debated. By introspection, mathematicians usually have some beliefs regarding these…
In this paper we give a mathematical model for a game that we call picture cube puzzle and investigate its properties. The central question is the number of moves required to solve the puzzle. A mathematical discussion is followed by the…
Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by having it follow from…
In problem solving, understanding the problem that one seeks to solve is an essential initial step. In this paper, we propose computational methods for facilitating problem understanding through the task of recognizing the unknown in…
Writing assignments in any mathematics course always present several challenges, particularly in lower-level classes where the students are not expecting to write more than a few words at a time. Developed based on strategies from several…
With the great success in simulating many intelligent behaviors using computing devices, there has been an ongoing debate whether all conscious activities are computational processes. In this paper, the answer to this question is shown to…
The usual aim of spreadsheet audit is to verify correctness. There are two problems with this: first, it is often difficult to tell whether the spreadsheets in question are correct, and second, even if they are, they may still give the…
The problem of advancing coordinatization of mathematics is considered. The need to develop a theory for measuring value and complexity of mathematical implications and proofs is discussed including motivations, benefits and implementation…
The two envelopes paradox is discussed. By calculating the conditional probability, we arrive at a conditional expectations which differs from existing results.
New understandings of the functioning of human brains engaged in mathematics raise interesting questions for mathematics educators. Novel lines of research are suggested by neuroscientific findings, and new light is shed on some…
Mathematical information is essential for technical work, but its creation, interpretation, and search are challenging. To help address these challenges, researchers have developed multimodal search engines and mathematical question…
A definition of what counts as an explanation of mathematical statement, and when one explanation is better than another, is given. Since all mathematical facts must be true in all causal models, and hence known by an agent, mathematical…
It is argued that the problem of interpreting quantum mechanics, and the philosophical problem of consciousness, both have their roots in the same set of misguided Cartesian assumptions. The confusions underlying those assumptions are…
Visual question answering is the task of returning the answer to a question about an image. A challenge is that different people often provide different answers to the same visual question. To our knowledge, this is the first work that aims…
This paper explores the relationship of artificial intelligence to the task of resolving open questions in mathematics. We first present an updated version of a traditional argument that limitative results from computability and complexity…