Nihal Uppugunduri's WebsiteThere 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.
