I'm the opposite (which, I admit, is backwards for a lot of engineers.) I need to have a high-level view to provide a context for the details. It doesn't help me to spend a week on defining and proving integrals without first understanding that the goal is to find the area of a rectangle or circle, or to track planetary motion. Once I see the problem, I can think about how we arrived at the solution, rather than present it as a solution in search of a problem. Newton wanted to describe planetary motion, so he invented calculus (Leibniz too, etc.) to solve it, not the other way around.
> It doesn't help me to spend a week on defining and proving integrals without first understanding that the goal is to find the area of a rectangle or circle
I will only discuss this point (but for your other points similar arguments can be made). So you want to measure some n-dimensional volume? This is a highly non-trivial problem, since it is quite possible to find subsets of R^n where you can't assign volumes in a sensible way:
If you don't care about this subtlety you end up in such a paradox.
So you think hard about how you can define "measureable" in an accurate sense and come up with the definition of a sigma algebra and think about how you can even define measures (say of volume) on a sigma algebra. Because these can be ugly to describe, you will immediately think about how one can describe the sigma algebra and the measure in a more easy way. So you come up with generators and note that the measure is defined uniquely by its value on the generator.
Doing this process on the real line using the Borel sigma algebra/measure is not that complicated (so you solved the problem for the real case). From here it is only a small step to the Lebesgue-Stieltjes integral in R^1 that is really worth doing.
But now you remember that you wanted to compute higher-dimensional volumes. So you ask how can you recycle what you've already done? The answer is obvious: You define product sigma algebras and product measures.
As you have seen (Banach-Tarski_paradox) all this is necessary just to even define volumes. I really know no easier way that is both useful in practise (at least be able to obtain the volume of spheres (i.e. not just "simple objects as simplices")) and avoids the problem that was laid out by Banach and Tarski-
I don't like this attitude at all. Euclid, Archimedes, Newton, Leibniz, Riemann, and many others did a fine job correctly calculating areas, volumes, and so on without need of measure theory.
Why is it so important to avoid the problem laid out by Banach-Tarski on the first go through the concept? We, as a species, did not even realize such a problem existed until about 1920, and yet we still managed to build stuff based on what we knew.
> I don't like this attitude at all. Euclid, Archimedes, Newton, Leibniz, Riemann, and many others did a fine job correctly calculating areas, volumes, and so on without need of measure theory.
The history of mathematics is a history of ever-increasing standards for proofs. "Proofs" that, say, Euler wrote down that were perfectly valid for the standards of their time would probably not accepted in todays standard. I personally also think it's plausible that today's proof standard in "typical" math papers will not be acceptable in 100 years anymore, when they perhaps will additionally have to be computer-checkable (and computer-checked).
