An N-dimensional vector space is not like R^3. A monad is not like a burrito, and >>= is not like chipotle forgetting to put chicken in your burrito, opening it up, adding chicken and wrapping in a new tortilla.
Math is a new thing rather than the same old thing with new syntax. It needs to be understood on it's own terms. If you are serious you'll learn it.
> In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.
This book: > If one generates random points in d-dimensional space using a Gaussian to generate coordinates, the distance between all pairs of points will be essentially the same when d is large. The reason is that the square of the distance between two points y and z ...
When anyone tells me that X is the ONLY way to do it, almost exclusively I have found them wrong - beyond data science. Formalism is a means of communication, not the end. You can always communicate ideas without formalism.
FYI, I am a Ph.D Computer Science and a practicing Data Scientist for many years now.
The problem is detailed in this wonderful essay by Paul Lockdhart
https://www.maa.org/external_archive/devlin/LockhartsLament....
Here's a simple concept: all polynomials with coefficients in a number system (field) have a solution (possibly in a larger field).
Do you know how hard it is to convey what exactly this means and what the consequences are? I like Lockhart's essay. It resonates with me. However, I don't see his ideas being useful in terms of learning advanced topics. At some point one has to get their hands dirty and slog through the material. Intuition will come with experience.
I never said it is easy.
> Intuition will come with experience.
And this experience that some people have already reached should be shared, not limited to a select coterie of people.
Except, I can hardly see that. At least, I see you trying - most people do not make an attempt at all.
FYI, I am a Computer Scientist.
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.
The problem is not math. The problem is the way math is explained. Most books do not go into the intuition and are very dry. People generally shy away due to the dryness of the material, not the content.