Math

Today, we're going to venture into existentialist territory: what does it mean to be a mathematician? "Somebody who does math," would be the obvious response, and that's not incorrect, just imprecise. As someone who's wanting a Philosopher's Doctorate in the field, I think it makes some sense to come up with something more concrete.

Obviously, Mathematics is a sufficiently large field that one person would be pretty hard pressed to understand all of it; even at what I'll call the "Masters" level (meaning, no further formal instruction is necessary; if you need more info, you know where to find it and can understand it when you look it up). Still, as with any field, there are some pillars upon which pretty much everything else rests and one really should grasp those before billing oneself as an expert.

It varies greatly by school but, from looking at the Qualifying Exams from various universities, it seems that the following areas are universally accepted as pillars:

• Real Analysis
• Complex Analysis
• Linear Algebra

The next tier are widely accepted, but some schools seem less concerned:

• Abstract Algebra
• Measure Theory (or, its restricted cousin, Probability Theory)
• Geometry
• Topology

Beyond that, there doesn't seem to be much agreement. I suppose I should note that Calculus is so foundational that it doesn't even get mentioned. If you don't know Calculus, there isn't much hope of grasping the rest of this stuff.

So, how does my knowledge base stand up? Real Analysis, Linear Algebra, Measure Theory and Probability Theory are no problem. I consider those pretty solid areas for me. Topology is slightly less so, but I can hang in there.

Complex Analysis is a complete blind spot. While statisticians really have no use for that silly i number, I really should address that. Fortunately, I'm told by reliable sources that it's actually fairly straightforward material. I'll probably tackle it on my own at some point.

Abstract Algebra and Geometry are areas that I seem to be OK with, but I know I'm really not. I know enough that when they come up in the context of the topics I do know, I don't get thrown. Still, if I had to actually prove a grad-level result in either area, I'm sure I'd be stumped. Unlike Complex Analysis (which is correctly regarded as undergrad material), I'm not so sure I can address those weaknesses on my own. As such, I'm signing up for an Algebra course next fall. Not really sure if I'll get a chance to pursue Geometry at UMSL but, if such an opportunity presents itself, I'll take it.

What does this have to do with my thesis? Absolutely nothing. Call me super old school, but I think some things are just worth knowing, even if you never use them.