The article considers structuralism as a philosophy of mathematics, as based on the commonly accepted explicit mathematical concept of a structure. Such a structure consists of a set with specified functions and relations satisfying specified axioms, which describe the type of the structure. Examples of such structures such as groups and spaces, are described. The viewpoint is now dominant in organizing much of mathematics, but does not cover all mathematics, in particular most applications. It does not explain why certain structures are dominant, not why the same mathematical structure can have so many different and protean realizations. ‘structure’ is just one part of the full situation, which must somehow connect the ideal structures with their varied examples.

Very nice philosophy paper by one of the progenitors of category theory on structure. The idea to show a correspondence between Bourbaki and category theory seems like a nice grad school project.

Ever feel like the mathematics you’re learning doesn’t make any sense to you? Good. In a way, it would’ve been worse if you thought it did make sense.

Cantor’s Diagonal Argument proves that there are an uncountable number of real numbers. But what about any interval of real numbers? Are those sets uncountable as well, no matter how small the interval?

Mathematics is based on a foundation of axioms, or assumptions. One of the most important and widely-used set of axioms is called Zermelo-Fraenkel set theory with the Axiom of Choice, or ZFC. These axioms define what a set is, which are fundamental objects in mathematics. And the Axiom of Choice is arguably one of the most important and interesting axioms of ZFC. But what does it really say? And how is it used? This video dives deep into the formal definition of the Axiom of Choice, as well as its important equivalences which have their own fascinating applications in various branches of mathematics. Furthermore, we look into the controversy behind AC, and why it has garnered much discussion throughout its mathematical history.

CyberCat is a network of researchers established in 2022 with the common goal to develop leading-edge knowledge in categorical cybernetics and applied category theory, and to show its economic usefulness. The Institute for Categorical Cybernetics was incorporated as a non-profit in 2024 to provide an institutional backbone for the network’s activities. Its purpose is to support CyberCat’s whole innovation pipeline from basic research to spinning off high-tech ventures. At the basic research level, it coordinates research activities among its members including funding and joint implementation of research projects, as well as hosting informal and formal events. At the other end of the funnel, the Institute acts as an incubator for ventures with the express goal to discover and commercialize economically useful technologies.

via: https://folk.computer/newsletters/2025-08

This is how we do math in the 21st century.

So when you see a no-go theorem that’s being given a very broad interpretation, you may do well to ask whether there is, after all, a way to get around the theorem, by achieving what the theorem is informally understood to preclude without doing what the theorem formally precludes.

A method of writing proofs is described that makes it harder to prove things that are not true. The method, based on hierarchical structuring, is simple and practical. The author’s twenty years of experience writing such proofs is discussed.

DARPA is soliciting innovative research proposals in the area of DARPA Mathematical Challenges, with the goal of dramatically revolutionizing mathematics and thereby strengthening the scientific and technological capabilities of DoD. To do so, the agency has identified twenty-three mathematical challenges, listed below, which were announced at DARPA Tech 2007.

A very nice hack for getting at the physics of the climate using some nice linear algebra techniques is presented. My first time seeing an ill-conditioned problem in the wild.

So, OK, why should you believe P≠NP? Here’s why: Because, like any other successful scientific hypothesis, the P≠NP hypothesis has passed severe tests that it had no good reason to pass were it false.

These lecture notes are based on the material I used to teach the Domain Theory (TypeSIG) course at the University of Edinburgh in 2024.