I think that, as long as you are able to generate interest in the fundamentals, you can go over certain areas again and again. Sometimes you may want to just meditate upon one little section at a time. At other times you may want to take many notes like a monk, work through exercises with pencil and notebooks.
Sometimes you may work with a computer, and at other times you may want to keep it off.
I think you will be studying mathematics mostly because there is not much else of interest that is so affordable and accessible.
Music. You can't just pluck a drum kit out of Library Genesis, and even if someone gave you a drum kit, you would have nowhere to practice without disturbing the peace. Schopenhauer played the flute each afternoon. Maybe going over mathematical ideas is a kind of music to us.
I used to love listening to music every day - under the spell of Dionysus. I was always drunk and singing.
That was the way Mr Hyde lived. Now Dr Jekyll is a real stick in the mud who lives like a monk. He (I) don't really like engaging in conversations with other people. I don't want to have any "fun". Nothing leads anywhere. It's all more trouble than it's worth. Negative energy here, I'll be the first to admit it. I write openly like this to you because I know you can handle it, and I trust you would never try to "cure me" of my miserable outlook. In fact, your "negative" outlook on life is actually kind of refreshing.
Thinking about fundamental mathematical ideas gives me something to focus on. Over the winter I have been focused on the theory of limits, sequences, partial sums, infinite series.
I think you may also benefit by developing a private, highly personal curriculum.
Maybe you are a bitter young man who does not like people. This will not hamper your pursuit of mathematical contemplation. In fact, you can enter a realm that is millions of miles away from popular culture and its concerns.
Personal education is now very affordable as long as you are willing to forsaken the institutions of learning. I study math the way I did drugs: in solitude.
You see, I have become grouchy and temperamental - really unsuited for polite society. It's all I can do to remain civil. I am lucky I do not have to report to an office. There would surely be trouble for me.
The reason I think studying some fundamental mathematics would suit you has to do with your temperament. Let's face it. We don't like people. We don't like snide remarks or smirking faces. We just want to be left alone in our misery.
I have some email to respond to ... an old friend who claims to be very happily married - and, to boot, a very enthusiastic and satisfied employee. I have to use self restraint when responding to the email. He questions me as to why I am so inaccessible by telephone. I have one of these tracfones that I only turn on when I need to make a call. I am inaccessible. I don't like to shoot the shiit on the telephone.
"How ya doing? Are you sure you don't sneak a drink every now and then? Blah, blah, blah ..."
You see, I have a mean streak, and I have to restrain that part of me when I am talking to others. I have a suspicion that, without realizing it, people who claim to be happy are actually cruel, dishonest, or just high on herb. In writing, I am less spontaneous and more thoughtful. I tend to show more respect in writing. For whatever reason, I don't like talking on the phone.
I could respond that I too am happy, and I do feel thankfulness when inhaling tobacco smoke in the sunshine despite the freezing cold. I know I can head indoors out of the cold! How miserable and wretched I would be to face that cold without shelter. I would surely seek some comfort in hard alcohol to warm my insides. Without books and pencils and computers, as a naked ape, I curse existence for making us so vulnerable and defenseless against the elements!
So, I guess the reason we are able to communicate so well is that we both look at life as it is, acknowledging our existential predicament, whereas, when interacting with others, they seem to live with the delusion of having their SH-IT together, that all is well in the universe.
I tell you again and again, I am not fit for polite society.
Now I will respond to some email. I will try to be gentle.
Fortunately I don't have many people who keep in touch with me, so I have to respect someone who keeps in touch at least a couple times per year. He is glad to see I am keeping my mother company and no longer wandering around aimlessly and drunk.
Anyway, back to your question. When you are alone in solitude, if you can become interested in what you are investigating, you might find a little peace of mind. It will not be the kind of "fun" that is promoted by the culture of good looks, but you just might find yourself almost "content", which is a close cousin to what we call happiness.
You have to be attune to your own moods. When I am in the mood for something formal, I will want to work with pencil and notebook along with a textbook. If I am working through exercises that are more computational in nature, I might want to see if I can work through some of the exercises with Sage or SymPy.
I know that it may sound as though I am some kind of quack when I suggest working alone, but I find I am most calm and interested when I allow myself to maintain the Beginner's Mind, where I am almost like a child in his own little world of make-believe.
A big obsticle to maintaining the Beginner's Mind when studying mathematics is the level of formality which mathematics has reached. The theory has been developed already.
If you look into the theory of limits, I would suggest not racing to get to the calculus, since calculus appears to be a formalization of certain notation which, at its core, is really shorthand abbreviation for things like Reimann sums and the limits of sequences of partial sums. Then when and if you choose to study calculus, the notation will be more intuitive. Unfortunately, after learning calculus, it seems more like magic because you compute and calculate using differentiation and integration, and all the work with dividing into intervals and taking limits seems to vanish (but it is now hiding in the calculus). This is why it is more difficult to maintain the Beginner's Mind ...
I promote solitary study because I think it is possible to attain that state of innocence much more naturally when you are alone. It is very comprable to how I would enjoy playing on the drums when I was alone, but did not like keeping time so as to "play with others". I'm not a musician. Likewise, I am not a mathematician.
I suppose my engagement with mathematics is, in some ways, similar to my engagement with music. It can be satisfying, but it can also be frustrating. When I talk about it or write about it, I feel I am "full of shiit".
And here is where being an official miserabilist may come into play. It's not something we have to announce to the world. Some people announce that they have gone on a "vacation" via airplane. And me? I spent the weekend, as usual, working through some mathematics exercises and performing some simple calcultions with for loops on a computer algebra system.
You see, as soon as you discuss your private "mathematical activities" with others, you expose yourself to mockery by who Schopenhauer called "the blockheads". Not only that, but, if you are like me, you will sell yourself short, playing the baffoon like a protagonist in one of Dostoyevsky's old novels.
While there may be some who support and encourage your continued interest, in a world of bank accounts and diapers, working through mathematics textbooks as one's main activity will most likley be looked upon as weird to say the least.
You really have to have reached a point of mutual disdain as far as public opinion goes.
It's ok to be a little bitter and cranky. If you find some kind of solace studying mathematics, knowing all the while that you are only scratching the surface, then this is nothing short of a miracle. It is like a hidden treasure within your own mind, and the only thing that is required in order for you to receive this treasure is UNDISTURBED LEISURE.
Ironically, the very thing that most people are running from - solitude, being alone - is exactly what is required in order for you to be able to discover the Beginner's Mind in privacy.
This is how I am getting through a life not worth living, Holden. Alcohol does not suit me. It leads to sobbing, at best, and bizarre psychotic episodes, at its worse.
So, even if this senseless pursuit of mathematical maturity is of no apparent benefit to me or to society, as our missing Raul de Paraguay reminded me a few weeks ago, I am not harming anyone with this obsession, not even harming myself.
Yes, by all means, Holden, I think this may be a way for you also to get through this life that is not worth living.
I hope we both develop mathematical maturity over time, and, if not, then at least we might become more efficient at calculating and computing. We can become like the depressed robot in Hitchhikers' Guide to the Galaxy.
Schopenhauer saw the drudgery inherent in arithmetic. Granted, he was absolutely right about this. The brain is just an appendage to the body. We've already figured out how unfortunate all of us are. Just in being born, we are now subject to the burden of our own existence.
Why learn the langage of mathematics? Um, because it is interesting, and while it appears difficult, it is not so difficult if one takes baby steps [the Beginner's Mind].
It is a lifelong pursuit. I can only ramble on and on like a madman about it because I am such an outsider. I don't do LaTex or concern myself with interacting with actual mathematicians.
OK. Back to my own little world ...