Atiscant

I suggest you start out with a course or book on formal logic in computer science. In particular doing proofs in intuitionistic logic. You can use a simple proof assistant to verify your proofs and it will help you really understand how proofs are done and how they are just a kind of programs in a special programming language called mathematics (given some assumptions). Proofs in higher level books are all informal, so you can’t do them programmatically in most cases, but it really helps with understanding how proofs works. After that, follow what subjects finds your fancy.

I don’t think there is a definitive book which represent exactly what I have in mind. It would depend a lot on your background. If you have a mathematical background with some formal logic, you could read the introduction to the Homotopy Type Theory Book, but it is aiming at something different. This video and accompanying paper could also help https://math.andrej.com/2016/10/10/five-stages-of-accepting-.... If you have no logic background, it would require a bit more.

What is your background and interest? I could perhaps help if you want to learn something like this

(Constructive) Mathematics from the group up using intuitionism, formal logic, Curry-Howard and Type theory.

One way to think about it is that you can extend your testing to prove the correctness of your program up to some notion of what “correct” means. You can reason about your program in your program and that allows you to capture another class of errors than regular static typing does. In principle, a strong enough dependent type systems allows you to encode mathematics, and you can then prove thing about your program in the same language as your are writing it. As for practical examples, you should look at Idris and some of the talks given about it. They do very fancy things while still being somewhat performant. Besides errors and testing, having dependent types also allows your IDE to reason about your program and can give much stronger hints and code completion.

But I agree that it is often a hard sell for normal developers. Given the choice, I would happily write large code bases in a dependent language, but I am also very biased having worked a lot in them.

If you care about the question more than from a purely curiosity standpoint, there is a difference tween “flesh” and “body” in (at least some) of Christianity. The “Lord of Spirits” (an orthodox podcast) has done an in-depth discussion on both the body of God and the bodies of humans, angels and demons. Just as an example, when Adam and Eve was expelled from the garden they were covered in “garments of flesh/skin” depending on the translation.

I do not want to start a discussion on Christianity, just to point to that there is a more subtle understanding in the church whether or not you believe in it.

I've added an e-mail to my profile.

Khan Academy is not bad if you like their style of presentation, but as another comment noted, the number of exercises is too few. Unfortunately, I have yet to find a good recourse with focus on exercises, so I tend to create them myself for courses I teach. If you (or any other for that matter) are interested, I can upload a work sheet somewhere.

There properly are tons of sites providing work sheets, but the focus then tends to be on K-12 education and they are often to simple IMO.

For university the importing thing is often manipulating say fractions rather then "calculating" with fractions.

Tangentially, would there be any interest in a service providing this? Pay say 2$ or something and get 50 exercises in the basics with an eye towards university with sample solution? That space is properly swamped but maybe there is a niche? I often think that there is room for textbooks of this short but then again, that market is also swamped with big publishers.

The single most important thing IMHO is to get the basics right

For context, I have a PhD in theoretical computer science and a master in mathematics. I've been teaching mathematics for several years ranging from high school to university.

The biggest problem for students, especially at the university level, is that they don't have the basics down. If you are not fluid in fractions and manipulating equations, you can't focus on the important part of learning induction proofs. This is just one example, but the return on investment on getting the basics down is huge.

For each new subject, you need to take the prerequisites serious. If you can't read a new definition and focus only on the new part, you should go back. The is of course very time consuming if you are missing a lot of prerequisites but if you want to get good, this is the way.

A well planned program will do this for you to some degree but the reason most people struggle is that they don't take prerequisites seriously or simple haven't learned them probably. You should be able to solve exercises in each prerequisite subject to a fairly good degree before tackling the next.

For anybody starting or doing university level mathematics I suggest your start refreshing pre-uni. Make sure you have fractions, linear and quadratic equations, exponentiation and logorithms and basic properties of functions are really down. If you have time, it can be very beneficial for pure mathematics to start out with some formal logic and natural deduction to really understand the way mathematicians reason in proofs and definitions.

Finally, ask all the stupid questions. Nobody, except yourself maybe, care if you misunderstood a basic point and got something wrong because of it. If "you just don't get" something, there is a high chance you missed something. If you don't understand the answer, same thing applies. Mathematics should be, if done properly, straight forward.

The part about replacing sets with categories is inaccurate. Some people are talking about categorical foundations, but as far as I know, most work is aimed at HoTT and/or (infinity-)groupoids. Thid is of course not divorced from category theory but the focus is important.