It’s been an interesting few weeks for counterexamples. This post is basically my perspective of what has been going on in the world of formalization, AI tools and, in particular, counterexamples.

Crazy shit, dog.

Our vision is to use mathematics to understand the nature of computation, and to turn that understanding into the next generation of programming languages.

We see the mathematical foundations of computation and programming as inextricably linked. We study one so as to develop the other. This reflects the symbiotic relationship between mathematics, programming, and the design of programming languages — any attempt to sever this connection will diminish each component.

Mathematicians manipulate sets with confidence almost every day, rarely making mistakes. Few of us, however, could accurately quote what are often referred to as “the” axioms of set theory. This suggests that we all carry around with us, perhaps subconsciously, a reliable body of operating principles for manipulating sets. What if we were to take some of those principles and adopt them as our axioms instead? The message of this article is that this can be done, in a simple, practical way (due to Lawvere). The resulting axioms are ten thoroughly mundane statements about sets.

This is an expository article for a general mathematical readership.

A nice read, although it definitely gets a bit difficult towards the end.

via: https://topos.institute/blog/2025-07-13-liberating-synthetic-quasi-coherence-from-forcing/

via: https://buttondown.com/hillelwayne/archive/logic-for-programmers-extra-credits/

Partial orderings are a useful concept that would have taken too many pages in the book, so I’m putting it here.

via: https://buttondown.com/hillelwayne/archive/logic-for-programmers-extra-credits/

credit: ~ac2 Melville was more into philosophy than math, and it shows with mid math references. McCarthy has him beat.

Whatever happens, the real danger is not that AI will become too good, but that it will become too evil—most likely because of the oligarchs, corporations and governments behind it. I wish I had emphasized that point, which is always on my mind.

I think I succeeded in making another point, which is that life will not become pointless simply because some other entity gets better than humans at something and knocks us off our throne. To think that the meaning of life resides in our superiority is a childish attitude.

It seems to me that we’re now over the top of this particular rollercoaster, and it will keep accelerating until we reach the bottom, wherever that might be. I don’t know whether to hope or dread that solutions to P versus NP and all our other great problems will be included in the ride—that our role, as human mathematicians, will be reduced to (at most) deciding which questions we find interesting and then understanding AI models’ answers to those questions.

But maybe that won’t happen. Maybe the new AI mathematicians will soon hit a wall, because they lack the uncomputable quantum gravity microtubules of Penrose and Hameroff, or some other magic human ingredient. The fantastical thing is that, one way or the other, we’re going to find out empirically before very long.

Melville was more into philosophy than math, and it shows with mid math references. McCarthy has him beat.

A formal type of proof most frequently encountered in elementary geometry courses in which known or derived statements are written in the left column, and the reason that each statement is known or valid is written next to it in the right column. The proof then proceeds from the known facts to the theorem to be demonstrated. […] However, this form of proof is virtually never used by practicing mathematicians because its confining and verbose format render it of very limited utility to any but the most simple of theorems.

Hmm, maybe this is why people don’t like Lean LOL.

The Arithmetic Game is a fast-paced speed drill where you are given two minutes to solve as many arithmetic problems as you can.

Welcome to the website for LGBTQ+ mathematicians and their allies. This arose from a need for recognition and community for Gender and Sexual Minority mathematicians, and we hope that this will be a resource for our community. Check back as our organization grows and evolves!

Confused by other academics telling you about their h-index? Then this handy guide may be just what you need!

This is an xkcd strip in article form. Really goes to show how arbitrary the h-index is.

see: https://adamtownsend.com/heighthunt/

This is how experienced people understand systems: they continuously form hypotheses about how the system works, and then they deliberately place themselves in situations which have a chance of invalidating their hypotheses.

I describe tutoring as “giving students an opportunity to reevaluate their assumptions, by asking the right questions to expose the critical mechanisms from their invidial perspectives.”

That’s how tutoring works, but it’s also how to debug broken code, how to learn how a network is put together, and how to learn to drive a race car. That is how to learn.

Words to live by.

Forester is a tool for scientific and mathematical hypertext authoring. It serves as your lab notebook, journal, blackboard, and lecture notes. Maintained by Jon Sterling and Kento Okura. Inspired by Zettelkasten and evergreen notes concepts.

Lecturer of computer science and math at Hendrix College.

Comprehensive collection of free mathematics resources covering general, number theory, algebra, analysis, geometry, topology, combinatorics, probability and more

Comprehensive collection of free mathematics resources covering general, number theory, algebra, analysis, geometry, topology, combinatorics, probability and more