[JHonaker]

That was a great article.

If I'd read this when I was first learning measure theory, I would have had a much easier time. In fact, it took me an embarrassingly long time to realize that sigma algebras were just the "nice sets and subsets" of things that we can extend measures from finite additivity to countable additivity.

I used to think mathematical objects were somehow "inherent". I was always amazed that people had discovered and proved so many interesting things about them. Once I realized they were often just defined to be the thing that has the property we want to prove something about, it got a lot less mysterious.

Note, I'm not saying we just stop there, or that this is somehow bad. The next obvious step taken by mathematicians is to start removing bits of the objects they study, and try to figure out what's still provable until we get to categories, logic, and start arguing about things like the axiom of choice.

[bdjsiqoocwk]

> Once I realized they were often just defined to be (...)

Right, that's why I often flippantly say that maths is the study of all ideas which are interesting. We could also think about other things, but they haven't been considered interesting for one reason or another.

[nyc111]

“Once I realized they were often just defined…”

I think definition is fundamental in nature because we need to define a unit in order to measure.

[JHonaker]

Yea, the more I learn about things, the more I realize that everything is defined relative to something else. Even measurements are defined relative to some standard (that beautifully manicured kilogram ball, the speed of light in a vacuum, your ruler's hand width or foot length, etc.) That's both very satisfying and extremely frustrating.

Programming languages can't escape it either; see the tautologies at the top of your favorite programming language (metaclasses, metaobject protocols, etc).