Jay Daigle is a professor of mathematics at The George Washington University in Washington, D.C. In addition to his research in number theory, he brings a mathematical style to thinking about philosophy, politics, social dynamics, and everyday life.
I've been working my way through my reading list, and it's really interesting to see how quickly some books age. Right now I'm reading The Art of the Steal and Enlightenment 2.0, and both books had fascinating failures to predict that I wanted to comment on.
We continue our exploration of what numbers are, and where mathematicians keep finding weird ones. In the first three parts we extended the natural numbers in two ways: algebraically and analytically. Those approaches gave overlapping but distinct sets of numbers. This week we combine them to get the complex numbers, and see some hints of why the complex numbers are so useful—and so…
This week we continue our exploration of what numbers are, and where mathematicians keep finding weird ones. Last time we defined the real numbers, but it took a lot of work. Now we'll see how truly strange they are. They're so strange that it's tempting to avoid them and stick with something simpler. But the real numbers do a much better job of describing modeling the parts of the world we care…
Adam Mastroianni wrote a very interesting essay on his substack about how the difference between teaching and grading, and how much he dislikes the latter and all the problems with it. And I don't really disagree, but I don't agree either. So here's my response. Grading sucks, but it's important.
This week we continue our exploration of what numbers are, and where mathematicians keep finding weird ones. We start by asking for the area of a circle, get exhausted by Archimedes's method for finding the answer, and take a tour through the idea of limits to construct the complete field of real numbers. We resolve one of the oldest mathematical flame war topics on the internet, and finish by…
Mathematicians deal with lots of different kinds of "numbers". But where do they come from? In this series we'll see where different types of exotic numbers came from, and what reasonable questions we need them to answer. In Part 1, we're starting off with the simplest types of numbers the algebraic numbers. We'll see how we could have invented square roots and weirder things on our own, just by…
I realized you can "prove" the Fundamental Theorem of Calculus by using Euler's method for solving differential equations. It's a fun way to motivate the integral, but not one that I'm actually going to use in class.
ChatGPT is cool, but doesn't seem useful yet for doing serious intellectual work. But is it useful for more routine stuff? I wanted to see if I could use ChatGPT to write test questions for my calculus courses. I'm experimenting with using ChatGPT to write test questions. My verdict: not completely useless!
Modern AI chatbots like ChatGPT are impressive, but they work in very specific and limited ways. They produce surprisingly human-like text—as long as the human isn't paying attention. And that tells us a lot about what we can expect this technology to do for us.
This is the third part of a three-part series on hypothesis testing. Hypothesis testing is central to the way we do science, but it has major flaws that have encouraged widespread shoddy research. In this essay we consider methods that can help us draw better conclusions, and avoid the pitfalls of hypothesis testing. We start with some smaller and more conservative ideas, which basically involve…