Previously on this blog, we’ve covered two major kinds of algebraic objects: the vector space and the group. There are at least two more fundamental algebraic objects every mathematician should something know about. The first, and the focus of this primer, is the ring. The second, which we’ve mentioned briefly in passing on this blog, is the field.

For a list of all the posts on Category Theory, see the Main Content page. It is time for us to formally define what a category is, to see a wealth of examples. In our next post we’ll see how the definitions laid out here translate to programming constructs. As we’ve said in our soft motivational post on categories, the point of category theory is to organize mathematical structures across various disciplines into a unified language.

Perhaps primarily due to the prominence of monads in the Haskell programming language, programmers are often curious about category theory. Proponents of Haskell and other functional languages can put category-theoretic concepts on a pedestal or in a mexican restaurant, and their benefits can seem as mysterious as they are magical. For instance, the most common use of a monad in Haskell is to simulate the mutation of immutable data.

Probabilistic arguments are a key tool for the analysis of algorithms in machine learning theory and probability theory.

Update: the mistakes made in the code posted here are fixed and explained in a subsequent post (one minor code bug was fixed here, and a less minor conceptual bug is fixed in the linked post). In our last post in this series on topology, we defined the homology group.

In this post we will assume the reader has a passing familiarity with some of the basic concepts of functional programming (the map, fold, and filter functions). We introduce these topics in our Racket primer, but the average reader will understand the majority of this primer without expertise in functional programming. Follow-ups to this post can be found in the Computational Category Theory section of the Main Content page.

This series on topology has been long and hard, but we’re are quickly approaching the topics where we can actually write programs.

One of the main areas of difficulty in elementary probability, and one that requires the highest levels of scrutiny and rigor, is conditional probability. The ideas are simple enough: that we assign probabilities relative to the occurrence of some event. But shrewd applications of conditional probability (and in particular, efficient ways to compute conditional probability) are key to successful applications of this subject.

In this final post on the basic four methods of proof (but perhaps not our last post on proof methods), we consider the proof by induction. Proving Statements About All Natural Numbers Induction comes in many flavors, but the goal never changes. We use induction when we want to prove something is true about all natural numbers. These statements will look something like this: For all natural numbers n, $ 1 + 2 + \dots + n = n(n+1)/2$.

The Problem with Cropping Every programmer or graphic designer with some web development experience can attest to the fact that finding good images that have an exactly specified size is a pain. Since the dimensions of the sought picture are usually inflexible, an uncomfortable compromise can come in the form of cropping a large image down to size or scaling the image to have appropriate dimensions. Both of these solutions are undesirable.

In this post we’ll expand our toolbox of proof techniques by adding the proof by contradiction. We’ll also expand on our knowledge of functions on sets, and tackle our first nontrivial theorem: that there is more than one kind of infinity. Impossibility and an Example Proof by Contradiction Many of the most impressive results in all of mathematics are proofs of impossibility. We see these in lots of different fields.