Tag: General and Terminology

Recently, I have been thinking about the unreasonable effectiveness of definitions in mathematics. Without making any additional assumptions or knowing any additional information about a situation, by simply making judicious definitions and studying their theory, we can get a lot farther than we could without those definitions. (As you might have noticed, “unreasonable” here is a nod to Eugene Wigner’s classic piece.) Upon further thought, it seems somewhat baffling that this works so well, especially in certain cases.

Many authors consider “mathematical logic and set theory” to be one subject, dealing with the study of concepts that are deemed fundamental to math. However, I disagree with this, for a number of reasons I will outline now.

Ostensibly, at the end of the day, math should be just about what can be proven right from wrong. Thus, we may reason that we shouldn’t care about someone’s credentials in order to cite them — we can just read their proof and check the validity of their words ourselves.

But despite this, I have often found myself evaluating the authoritativeness of authors through things like credentials when trying to pick resources to learn from. Why do I end up doing this? Are there in fact aspects of doing math for which credentials become important? For these aspects, can we devise new methods to reduce dependence on credentials, or even eliminate it, in favor of judging just based on content?

I have recently found a curious phenomenon in which teaching math can actually lead to new mathematical ideas, and as a result possibly new research. In this post, I’ll discuss some ways this can happen.

Especially in comparatively recent times, researchers in the subject of universal algebra have sought to replace that name, leading people to choose an alternative term instead. In this post, I discuss my thoughts on this matter.

Given the proliferation of computers in so many aspects of our lives, it has always surprised me that verification of submitted math research is not yet totally automated. In practice, requirements like the conferral of doctoral degrees or the referral process for arXiv are still used today in order to judge whether a proposed research contribution is valid or not. This is despite the fact that the subject of math itself is supposed to be only dependent upon formal logic for validity. The idea of automating the verification of math is not a new one, and since the twentieth century there have been some interesting proposals and thoughts towards making it happen.

Math, or rather our knowledge of it, is continually expanding in both volume and diversity of subfields. However, to better understand a general, high-level organization of it, let’s take a snapshot of current mathematical knowledge. If we roughly partitioned this corpus into its main branches, based on topical similarity and general amount of (current) content, what would they be?

There are many examples of confusing terminology in math, along with multiple reasons why they are confusing. Some terms are ambiguous, meaning that they are reused for different concepts. Other terms are “almost ambiguous,” meaning that they are not word-for-word reused, but have such similar names that it would be easy to be confused. These scenarios go from worst to best — the latter case is at least more manageable, if still not desirable.