How to begin self study of Mathematics?

Question

I'm aware that this question has been asked several times, but I have specific questions hence why I'm asking again.

I began to appreciate the beauty of mathematics when I glossed over the Fundamental theorem of calculus while taking a Calculus II course. The more I come across mathematical literature the more interested I become in learning it. I'd like to add that I enjoy self studying many different subjects (Physics, Chemistry, History, etc.) and it never feels forced. Its something that is genuinely enjoyable. However as much as mathematics interests me I find it hard to motivate myself to learn by reading books. My guess it that more so than many other subjects mathematical writing is incredibly dense.

Typical answers to this question usually result in people providing a list of topics to study, and perhaps a list of books to complement that. So I know what my entry point to this journey should be. Given my background, I would begin by learning Calculus with theory. After reading various similar questions these were the three names I heard the most; Spivak, Apostol, and Courant.

Spivak's writing was by far the most pleasant and I was able to acquire a copy of this book from a friend. However I also found that I felt that a lot information was omitted from each chapter and the reader was expected to fill in the gaps to be able to do the exercises. This is as instructive as it is time consuming. I'm willing to invest the time to self study, but the ratio of extra understanding gained of the subject to the extra time spent to understand just doesn't seem very efficient. Just my opinion. At a later point I may decide to go through Spivak after learning more about mathematics.

Apostol's book seemed right up my alley, and I found that I wanted to keep reading it (I found a preview of the first chapter online). Many people said of the three Apostol was the most dry. That may be true but I felt it had a good balance of being succinct, explaining the subject manner in a rigorous manner, while still making the subject matter feel intuitive. Unfortunately I could not find a friend with a copy of this book, the book is ridiculously expensive and there was no good copy available online.

Courant is the one I attempted to go through most recently. This was the book recommended for studying Applied Math and after going through the first chapter I can see why. I personally didn't like this book very much just because of how verbose it was. What took a whole paragraph could have been shortened to a sentence or two and I found it to be a chore to get through.

So my list of questions becomes the following:

1. How do you develop and maintain the motivation to self study mathematics?
2. If I were to read a book on Algorithms it becomes immediately applicable because I can just start by implementing that Algorithm in a language. Physics is immediately relevant being that physics is a natural science; I can read, do practice problems and even create physics simulations using code. But what approach should I take for math? I was planning to go about it by, reading, doing problems, and repeat the process, but I feel this is just setting myself up to lose interest in the subject.
3. Lastly, what book should I start with? Alternatives to the books I mentioned would be great. A cross between Spivak and Apostol would make for the perfect introduction to calculus I think.

Answer 1

For these types of questions, I always appreciate when more than one answer is provided. I'll try to provide a general answer and point out where each question is considered throughout the following.

(Some background/personal experience) I'm from a small town but was allowed to take university courses throughout high school. After high school I wanted to go to a top university but, it didn't work out and I continued my education at the local university. Now, this education was very different from private university education and I didn't want to miss out so I began self-studying. My perspective has changed so much between the time I began and as I'm typing this.

(1) Developing the motivation to begin self-studying is really the most challenging part of the whole process. But it sounds like you already have the motivation or, why would you be asking for references for what to read? You are already interested in learning and that is the one aspect that no one can teach you. How do you pick up a book and read it everyday to learn the material? Well, that's a different story entirely. As you said, math books are incredibly dense. So let me tell you some things I have learned about myself and explain them also.

i) Keep a stable lifestyle - I used to have a weird sleeping schedule going to highschool, university, and I was getting paid to play video games. On some nights I would get two hours of sleep, and the next day I would take a nap and wake up at 2 a.m. after seven hours of midday sleep. But I could never consistently play well with this schedule; it was difficult to maintain technical skill with so much change. Reading math textbooks is, unfortunately, a technical skill. It is harder to do it when you aren't consistent.

ii) Vary what you read - This could be from change of books to change of subject. As you mention, Spivak's text is great but it is not a treatise; Apostol's book is drier than Spivak's but I assume more complete. I personally like both of these authors. Spivak usually gives a very detailed account of practical information; he may shy away from saying an explicit result from a more general field if it does not pertain to the use of that information in either a working knowledge or the remainder of the text. Apostol tends to cover a subject very completely in his work with the use of material that may not be of interest to you. Not only are there differences in the author, which will depend on your taste and interests but, I've noticed lately that the more abstract the material I am reading the more strain I have mentally. Maybe I can give a good 45 minutes to reading about locally ringed spaces but then I start to get headaches. (It's very consistent too, with material that takes a lot of effort to wrap my head around. After I get the headache, any hope of understanding more is lost and I essentially have to stop for a few hours). After this point, maybe it is more valuable to do computations.

iii) Ask yourself why you are interested in the subject - This could be a very useful reflection. For example, if it is to understand mathematics in more advanced physics then maybe it would be useful to go on walks and think about how the math relates to real world objects around you. This may also drive you in a direction that will increase your motivation and self-study. If it is for learning in its own right (which is really cool) then really anything is good to study but, you will find something more to your taste as you further develop your skills. This could be, as in my experience, finding distaste in measure theory because looking at the language is displeasing and very technical or enjoying algebra because it is very clean. Consider these questions since they are the best indicator as to what you should study.

(2) Reading, doing problems, repeat probably isn't what you'll like the most. There is a ton of pedagogy as to what is the best way to learn and it is a very active debate in most education systems. After reading your post I went through almost 15 other self-learning, soft-question posts and they all put emphasis on doing problems. At the basic calculus level, the difference between theory and problems is large. I'm pretty sure I made it through four calculus courses without learning any theory and I "did very well" in all of them. As I see it, the path has been paved already. Putting more emphasis on theory is wrong because then what problems can you solve? Putting more emphasis on problems is wrong because then what do you actually know? So, I would recommend going through the theory and then doing any problems that enforce the computation. Once you're comfortable with the computation go back and look for "how does the theory influence the computation?" or "how does the computation describe the theory?". For instance, the intermediate value theorem is a theorem in its own right but, what computations can you do from learning this theorem? Or, integration is explicitly defined as a limiting process of sums so why do we immediately ignore the construction of integration when we are taught it? I learned "how to integrate" years before I learned that I was integrating.

Working on problems also varies widely on the textbook. Some books use exercises to further develop the theory. These are good exercises because they give you valuable tools that could be used later in the book. Some exercises are just neat and are not very useful for learning the theory but are useful for understanding the theory. These questions you should skip but, you should find your own interesting questions in place of them and do your best to answer them yourself (this is very difficult but very rewarding). And as mentioned above, some exercises are just to become computers and, after enough exercises of this sort, are better left for computers.

Well, I am avoiding answering (3) because I think this will be something you should try to answer or hopefully someone more versed in the material will answer for me. Also, I tend to find a lot of treasures on the internet. Hopefully this helps and I will happily reply to any further questions.

Answer 2

It is not as completely detailed with the Eoin's answer, but here is what I believe.

1. Get a notebook and as you are reading, write down the important definitions and theorems, along with the proofs. You are more likely to remember things when you write them down.
2. There will come a point where you struggle to understand a definition or theorem statement/proof. Don't give up. Everybody goes through this in mathematics. Try to "simulate" or come with examples of what they mean. I remember myself struggling to understand what the $\delta-\epsilon$ definition of a limit was. The way I helped myself understand it was simulating a computer code demonstrating the statement. My code takes in a function $f(x)$, a point of interest $a$, and limit $L,$ and a fixed $\epsilon>0.$ In the code, I established what I thought the $\delta$ should be in terms of the general $\epsilon.$ I then generated a large number of random values of $x$ such that $|x-a|< \delta,$ and checked to see if $|f(x)-L|<\epsilon$ for each of those $x$ values. The code returns True if all of those values satisfied the condition, and False if at least one of those values did not. By doing this repeatedly on different functions, that's how I came to understand and check if my $\delta-\epsilon$ proofs were correct or not. The point I'm making here is make up examples for yourself as much as possible to see the idea in action, and of course write them down. This will help you understand and remember the tougher ideas.
3. Sometimes authors in proofs make some kind of statement that comes out of nowhere. They might say something like $a \in A$ and not justify it. In a case like this, provide the justification as to why $a$ belongs to the set $A$ as you write the proof.
4. As mentioned in Eoin's answer, do problems given in the book at the end of each chapter. You can really internalize the ideas when you do so.
5. Practice writing/communicating mathematics clearly and concisely. To succeed in a rigorous field like this one (or any field), you must develop the crucial skill of explaining your reasoning (or the concepts in general) to other people. Professor Francis Su has a nice guideline that is focused on how to write well. https://www.math.hmc.edu/~su/math131/good-math-writing.pdf