What is good mathematics?

Many mathematicians find mathematics aesthetically beautiful and even comparable to art forms such as music or painting. On the other hand, every year a great number of school students leave mathematics with total disillusionment and bitterness, without ever witnessing any beauty in it. In this work, we give some strategies to teach mathematics, especially to gifted students and to instill a love of mathematics in them. We will describe an integrated approach to teaching math...

In this note some philosophical thoughts and observations about mathematics are expressed, arranged as challenges to some common claims.

Epistemological and sociological questions about ownership of mathematical and scientific discoveries.

We claim that human mathematics is only a limited part of the consequences of the chosen basic axioms. Properly human mathematics varies with time but appears to have universal features which we try to analyze. In particular the functioning of the human brain privileges concept naming and short formulations. This leads to organizing mathematical knowledge structurally. We consider briefly the problem of non-mathematical sciences.

Mathematicians occasionally discover interesting truths even when they are playing with mathematical ideas with no thoughts about possible consequences of their actions. This paper describes two specific instances of this phenomenon. The discussion touches upon the theme of the unreasonable effectiveness of mathematics as well as the ethical obligations of mathematicians.

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...

This talk presents foundations of mathematics as a historically variable set of principles appealing to various modes of human intuition and devoid of any prescriptive/prohibitive power. At each turn of history, foundations crystallize the accepted norms of interpersonal and intergenerational transfer and justification of mathematical knowledge.

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 paper explores another class of tools for advancing mathematics research: databases of mathematical objects that enable semantic search. In addition to defining and exploring examples of these tools, we illustrate a particular line of ...

I describe some deep-seated problems in higher mathematical education, and give some ideas for their solution -- I advocate a move away from the traditional introduction of mathematics through calculus, and towards computation and discrete mathematics.

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 problems. Examples of mathematical considerations for such a theory are given. Arguments supporting such an advance related to applications in mathematical research guidance, publication standards and education are given.