90% of the time I use the java standard lib. Sometimes I need something almost the same, but a bit different. The difference between a middling developer and a good one is what happens in that 10% of the time.

The same applies to proofs. What happens when you need something that's almost like the theorem, but not exactly? Can you tweak the proof to apply to your case, or alternately recognize that it can't be fixed and you need to do something different?

Likewise, one can certainly apply a lot of mathematical techniques in a lot of situations, without proving new theorems. So there are edge cases... not the end of the world. Deal with it when it comes up.

My problem with (most) maths texts isn't the proofs in and of themselves, it's just that they make so many assumptions about what you already know, and don't always state those assumptions clearly. And then too many maths texts, to me, fail to include enough expository text to explain the math and provide more of an intuition around what's going on.

That said, I also agree with you that more examples in terms of applications would be nice. Learning really abstract stuff in isolation is OK, but it's always nice to have a few (or more than a few) examples of how to apply the technique to something concrete.

Just skimming that part for the word "prerequisites" or "background" will usually find it.

Most actually-existing books simply write that the student should be "mathematically mature", by which they mean that the student should know all of multivariable and vector calculus, most of differential equations, much of optimization, and some amount of analysis. And also be able to do proofs, and also have good intuition.

In short, most "applied" math textbooks aim themselves at Master's degree students or beginning PhD students in math itself, to teach material which can actually be taught to anyone with a bachelor's in science or engineering (except possibly computer scientists, whose continuous maths degrade because we never use them).

Ironically, the first math textbook I've ever picked up that didn't assume far too much background was my real analysis textbook, because analysis teachers assume that they are teaching the gateway course to "real math" and have to educate little babbies who only just got done with calculus.

If you want to learn something difficult that's above your level, and learning that something includes learning a prerequisite subject that's also above your level, you simply start at the node in the DAG that's at your level and work through the prerequisites if you're serious.

But yes, there is no substitute for learning all the things you need.

Exactly. And a surprisingly large number of books aren't well-written, exactly because they don't include that bit.

For most well-studied topics, finding the books that are good enough is not that difficult.

The ability to determine when an approximation is good enough comes from a solid understanding of the underlying theory. For example, we understand that special relativity doesn't invalidate classical mechanics, because, in the limit, as speeds and energies become arbitrarily small, their predictions coincide. But, if all you have is a bunch of computational recipes, even the tiniest unexpected thing renders you unable to solve problems.

Anyway, I'm not saying that examples of applications are a bad thing. But proofs are indispensable. Even for “intuitive” people, proofs are necessary confirmation that their calculations will match their intuitions.

There is no proof of special relativity, we believe it because it makes experimentally verified predictions. Your example of limiting cases of special relativity is how I wish statistics texts were taught - it isn't based on the "proof" of special relativity. Of course I know when special relativity is applicable and when it isn't (when beta = v / c is close to one, special relativistic effects are important, and gamma = 1 / sqrt(1 - beta*beta) indicates how good the approximation is).

But I do agree that proofs are useful and needed, I think that complicated proofs could easily be placed in appendices and looked at after understanding why the theorem is relevant. Of course this is just my personal preference.

Right, that's because special relativity is a scientific theory. A scientific theory has two components: a mathematical theory (in which you can perform a priori physically meaningless calculations) and a physical interpretation (which turns the results of such calculations into predictions).

However, statistics isn't a scientific theory. It's just math, and like all math, it's about itself and nothing else. It doesn't make sense to “experimentally confirm a mathematical theory”, because, without a physical interpretation, math doesn't make any predictions about the real world.

> I think that complicated proofs could easily be placed in appendices and looked at after understanding why the theorem is relevant.

Then you're looking for books on applications. That's fine. But a book whose subject matter is a mathematical theory (it even has “foundations” in the title!) can't relegate proofs to appendices.