Tangentially related to the article, but for many years I struggled with understanding what algebra was an more importantly why it was of interest. On one hand you had groups and rings that sort of seemed related, but they weren't algebra's. Then there were special techniques like linear algebra and I could see that they were all sort of similar but why they were of interest and all considered to be part of the same subject escaped me. Sure they were interesting in a way and group theory was sort of cool but what was the point? Analysis on the other hand made sense, it was the study of how continuous things changed. As a result I though of algebra as a sort of collection of things that you got when you discretized continuous objects. Lie algebra's and groups came as a bit of a shock, but they were basically still thought of algebra as discrete.

Many of my friends raved about its beauty, but I still had no intuition as to why it was interesting. I went on a bit of a reading binge, and eventually ended up bumping into some papers of Shafarevich[0] which in turn lead me to his delightful book on algebra [1]. In it he defines algebra as the construction and study of systems of measurement. For example counting is the simplest such system and gives the natural numbers. Attempts to describe and measure the diagonal of the unit square gave irrational numbers. As we explored the world we needed to construct new systems. A more recent example is provided by quantum mechanics, where numbers are insufficient to describe our observations, however Hilbert space provides a natural setting. Attempts to describe what a simultaneous measurement of two quantities is find a natural description as commuting operators. He provides many other examples in his very readable book. For whatever reason this definition of algebra resonated with me and if this was a Zen koan I would say I was enlightened.

This is really just a very long winded way of saying that I think Shafarevich's book in which he defines algebra as the study of measurement is lovely. You should read it if such things interest you.

[0]https://en.wikipedia.org/wiki/Igor_Shafarevich [1] https://encrypted.google.com/books/about/Basic_Notions_of_Al...