The foundation of math
The concept of “identity” is central to any math we do…
Mathematics is about way more than numbers, it’s fundamentally about relationships. The most powerful way that it studies relationships is by starting with simple objects, assuming simple rules on how to work with those objects, and then seeing what rich structures come out of it.
As an example, think of the simple things you do in your car: push your right foot forward and backwards, rotate your arms every now and then, and rotate your head a bit. Combine those simple things in a certain way and you can find yourself traversing the country and seeing beautiful sights, hearing beautiful sounds, meeting great people. All from a relatively simple set of things and a simple set of rules.
Given that mathematics studies relationships, and only sometimes puts labels like numbers to those relationships, there’s a strong argument we can make that science needs mathematics to be able to probe, describe, and communicate the relationships it wants to study. Relationship are, after all, the fundamental object of study across science, medicine, and engineering. Each of these disciplines takes a very different approach to studying and verifying relationships, but they’re all very similar nonetheless.
Working at the interface of neuroscience, neuromedicine, and engineering is difficult, for several reasons. One of the most challenging parts is in communication and language. There is such a diverse group of people
Abstract mathematics is, I’ve found, critical to overcoming these challenges. While I tried my best to insert the language of abstract math into depression DBS research, those efforts didn’t lead me to think abstract math was critical. It was in organizing my project in my own head that I found abstract math the most useful.
Right now I’m spending a bit of time learning some fundamental abstract algebra to better understand the art of taking axioms to larger scale structure. The goal isn’t to be rigorous, but to have a language that can unify multiple approaches to studying the same underlying system.
As a final parting thought I want to highlight one of the first examples provided in the ‘Basics of Groups’ chapter from the book mentioned above: is it possible for a group to have two identity elements? I’m also using this as a test of this blog’s MathJAX capabilities…
Suppose we have a group $G$ with operator $\times$. Suppose there are two identity elements in $<G,\times>$, $e_1$ and $e_2$. Let’s apply the identity definition to these two elements and see what we get.
$e_1 \times e_2 = e_2$ since $e_1$ is an identity element. But also $e_1 \times e_2 = e_1$ since $e_2$ is an identity element. $e_1 \times e_2$ can only be equal to one, consistent thing. So we conclude that $e_1 = e_2$.
So, we can have two different identity elements in the group, as long as both elements are identical to each other. In other words, we can’t have two distinct identity elements for a group and operator combination. How cool is that.
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