Nihal Uppugunduri's WebsiteWhat language features are (minimally) necessary in order to implement full refactorability of symbols? (Think “Rename Symbol” in your favorite IDE.) For example, when I used Python instead of Java, I had discovered that refactoring interfaces was much harder. Is there a deeper explanation for this?
Consider the following thought experiment: we setup an experiment consisting initially of objects and known forces, and we run Newton’s laws to predict the motion of the objects. We then run the experiment, and the observed results do not match the predicted calculations. Can we just attribute the discrepancy to a new force that we haven’t yet studied, and still satisfy Newton’s laws with this just-defined unknown force? Can we explain all discrepancies with Newton’s laws in this way? Then, are the laws really saying anything materially? Can we think of forces as convenient mathematical devices to represent “contributions of mass times acceleration”?
In other words, taken away from the specific laws of gravity and the other forces, are Newton’s laws themselves falsifiable?
In this post, we investigate a curious question: say we are given 
I first learned about semilinear maps through one of my mathematical explorations. In this article, we introduce them and prove some basic results about them. We assume some knowledge of abstract algebra and related subjects (vector spaces, modules, universal algebra, category theory, etc.)
(I was still learning about the concept alongside writing this article, so if there are any inaccuracies here, please let me know!)
I have been realizing recently that many of my math ideas in their “ultimate” forms seem highly suited for formulations in model theory. However, when first trying to study some of the resources I found online for model theory, I didn’t even understand that well what the subject was about or why certain definitions were made, let alone appreciate the overall power of the subject. I think there should be a different approach to teaching model theory, incorporating better motivation and more fully reflecting the incredible power and scope of the subject.
We continue the discussion from this post.
We continue the discussion from the previous post in the series.
Topologies and measure theory’s sigma-algebras look superficially similar, but the differences in their defining axioms lead to differences in their resulting study. In this post, we write this out to understand this better.
A previous discussion on a more intuitive variant of a UFD led naturally to discussing signs on commutative monoids, where instead of thinking of signs as additive inverses in a ring, we generalized to defining a more abstract concept of sign. Upon further thought, we can probably generalize this definition even further, from a commutative monoid to a most general kind of algebra — namely, a magma. In this post, we take this approach, which has a “reverse mathematics” benefit of seeing the most general settings in which certain theorems hold and certain questions can be formulated.
In this post, we continue the exploration from the previous in the series.
