The n-Category Café

It’s interesting that you’d promptly glom on to the most controversial of Chris Hillman’s pages, less than an hour after I advertised them.

I don’t agree with the dark tone of his assessment of Wikipedia. I think the Wikipedia is a wonderful thing. I think it will keep getting better. I use it a lot — and I correct errors in it whenever I find them. But, I think it’s worthwhile warning gullible youths that the Wikipedia is fallible. And, I think there could be reforms of Wikipedia policy that could make it more reliable over time.

I also find the assessment of Wikipedia too pessimistic, although there’s a lot of truth to it. I agree completely with what Chris Hillman considers Wikipedia’s best use: a starting point from which to find further information.

However, I use Wikipedia primarily to investigate unfamiliar terms in pure math, which is less prone than physics (and most other topics) to financially motivated manipulation and other problems Chris Hillman cites.

Just now I finished writing week248 of This Week’s Finds.

I was about to add a link to Wikipedia on the topic of magnetic reconnection, a fascinating topological phenomenon shown vividly in this movie — dramatized by a flash of white light behind the Earth. Then I noticed that the otherwise reasonable-sounding Wikipedia article ended like this:

Magnetic reconnection is the phenomena that occurs in the flow of magnetic energy of the human body that follows a “Chiropractic Treatment” which changes the justa position” of one spinal segment to another. The resultant changes in the flow of magnetic energy alters the cellular’s DNA responses to the cellular environment at the speed of light.

Whoops! I deleted it, but decided to add a different link. The process was good for me, since this other link delved far deeper into the topology of magnetic reconnection.

The standard course offerings, at least in math, are not especially standard. For example, here is a partial list of topics that might be covered in a “standard” undergraduate real analysis course:

Basic set theory, cardinality
Construction of the real numbers
Point-set topology of metric spaces
Point-set topology of general topological spaces
Rigorous development of differential calculus in one dimension
Rigorous development of Riemann integration
Classical analysis of the elementary functions
Differential and integral calculus in n dimensions
Calculus on submanifolds of Rⁿ
Lebesgue integration
Basic Fourier analysis
Hilbert spaces and Banach spaces

No one-year course can cover all of these topics in any reasonable way, nor does anyone I know try to. But everyone author or teacher makes a different selection. In my experience, this causes much less difficulty than one might expect when students get to first-year graduate courses. This is in part because the most central topics do get covered in any reasonable undergraduate course, and no one minds that some of the less central or more advanced topics get covered twice for many students. Learning the same material again from a different perspective is good for you.

Based on those observations, here’s one suggestion for tdstephens3: find out what other textbooks might also be used the classes you’re taking or have taken. (One way to do this is find the corresponding graduate texts and see if they list, in the introduction or preface, which undergraduate texts include the necessary background to read them.) Read those books, and do the problems, both from chapters on topics your class doesn’t cover, but also for another perspective on the topics you’ve already seen.