Here, Gregory Reese points out a reference to another useful course that might have made my undergraduate life a bit easier were there an Internet to speak of in the early 1990s. (These were primitive days, before Google, before Alta Vista, and when we actually put up with xdvi readers and couldn’t have imagined pdf with its tendency to work and user interface that looks like any thought at all was put into it).

In this case Reese is pointing out the Free Harvard course in Abstract Algebra. Abstract Algebra — it gets called just “Algebra” later on, when we’re not worried that undergraduates will think it’s the thing they did in middle school — is kind of what you get by taking the next set in abstracting middle- and high-school algebra.

One of the things that makes algebra a subject important enough to revolutionize thought and to get into middle- and high-school curriculums is the idea that we can do work with a number — add to it, multiply it, divide by it, raise it to powers, take its logarithm, or so — without necessarily having to know what the number is.

In abstract algebra, we consider the things that we do with numbers, in arithmetic — things like adding them, multiplying them, factoring them — and ask, can we do these things with stuff that isn’t numbers? If we put some thought into what these things are, and what we mean by addition and multiplication and such, it turns out we often can. Abstract Algebra is one of the courses that starts on this trail of doing things that look like arithmetic on things which are not numbers.

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