Nihal Uppugunduri's WebsiteWe want to define topological groups as having a compatibility axiom that intuitively says that the “group operation is continuous.” But the group operation is a multivariable function, so how would we define this? And what about for general topological algebras? (Here, “algebra” is meant in the universal algebra sense, not in the “vector space with bilinear multiplication” sense.)
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.
Sometimes, it can be overwhelming to even think about how to approach a subject. Where do you start?
Here, I list some pointers and resources that have helped me most effectively learn math. I also point out some resources which I may not have used myself, but that other people I know have had positive experiences with.
There is an important concept in math called an isomorphism. This is typically introduced when you study a subject called abstract algebra, although it applies beyond abstract algebra too. While its definition generally requires some technical background, in this post we attempt to provide a non-technical introduction.
When we learn math, a typical approach is the following. We discuss some motivation, based on what we have already done. This motivation leads, in a possibly informal way, to a formal definition. (It could also be a set of formal axioms, but here we encompass that, as well as any other type of formal “framing,” under the term “definition.”) We then work with our definition using formal logic to produce theorems.
This can be a great approach. However, there are important caveats that we must keep in mind, which I call attention to in this post.
Assume we are developing the curriculum for an extracurricular math program for an advanced student. How could we introduce formal reasoning and the axiomatic method, two cornerstones of the practice of math today, as early as possible? And specifically, what are the implications for this from the philosophical discussion of the formal foundations of mathematics, in my post On the Circularity of Mathematics?
It is a well-known, but uneasy, admission that the foundations of mathematics seem to be inherently circular. Indeed, we may for example list out the axioms for the positive integers, say using the Peano formulation, but in writing down this list we already use numbers: identifying the first axiom, then the second axiom, and so on. Even if we argue that we need only formally specify a set of axioms rather than a list, this still makes the axiomatic formulation of sets circular, whether with Zermelo-Fraenkel or another system.