/sci/ - Science & Math

>>9744966
>I haven't seen this approach taken in any mathematics text. What makes you think it's really the most common one?
It's my main annoyance with most textbooks I read.

I specifically mentioned linear algebra as an example since many introductory linear algebra textbooks have a progression along the lines of: systems of equations, some visual stuff on [math]\mathbb{R}^2[/math] and [math]\mathbb{R}^3[/math], norms, dot/cross products, [math]\mathbb{R}^n[/math] for finite [math]n[/math] and linear transformations as matrices, real vector spaces, etc... and they never cover vector spaces over a field. Of course, there are many great linear algebra texts that don't do this (like Hoffman and Kunze). I am trying to say that given the choice I would prefer Hoffman and Kunze any day.

>And I don't quite get the meaning behind the phrase "real world examples".
As in applications to science and other fields. Though I guess my annoyance extends texts that focus too much about 'nice' special cases when the general theory has more potential (eg. intro algebra texts).

>Are you sure you're talking about mathematics? Fields outside of logic don't tend to write anything in formal logic.
I don't mean it has to be in formal logic but the more formal it is the better. To be honest, many of the texts I've liked reading aren't that formal but they are always very clear, concise, and move swiftly.

See pic related, the formalism makes it super easy to understand what is and is not being said as well as how to prove such a statement (or it's negation). It also makes it trivial to see the relationship to topology. Most importantly, it does this in half a page without monologuing about some uninteresting example that takes just as much work to understand as the definition yet is nice enough to create potential for misconceptions.