Sorry about your employment circumstances. And thank you for not begrudging me the current leisure afforded to me which enables me to freely explore mathematics at the slow pace it deserves.
You know, it is good to think of programming as a branch of applied mathematics. I will not push it on you, but do not think it is something that is beyond your grasp. The key for me is to know just enough C++ and Python (and a little Fortran and Basic) to be able to track down, adjust, and use small command line programs that automate some calculations that are tedious. There are plenty of posts here already about Open Source computer algebra systems such as Sage and the Python phenomena like SymPy, numpy, et al.
In that documentary you posted about HP Lovecraft, they mention all the letters to a certain handful of people he wrote. Evidently, he put a great deal of time into such letters.
One of the commenters irked me when they said that Lovecraft not being employed (with a job or career) made him a "failure as a human being". I find that kind of mindset very closed-minded.
So, I appreciate your not despising me for having a temperament and character not conducive to the servility demanded by most "jobs and careers". It helps to recall Uncle Avenir in Solzhenitsyn's "In the First Circle" :
Uncle Avenir says that if you have a position to hold down, you have to truckle … and you have to be dishonest.
“I could not even stand being a librarian, let alone a teacher.”
Innokenty asks, “What’s so hard about a librarian’s job?”
Uncle Avenir replies, “Just go and try it. You have to trash good books and praise bad ones. You have to mislead undeveloped minds.”
I will continue this correspondence as long as we are both in a position to exchange ideas.
This morning I found myself following your inclination to "plug in numerical values" into theorems, at least initially, in order to first gain an intuitive conviction before attempting to express some kind of proof in the terms of symbolic algebra.
I am tempted to type up the whole thing, but it was awkward enough on paper, and I confess that this message board is not "mathese friendly".
We have to be patient with ourselves when writing mathematical statements, using the old school notation. I will keep such "proof attempts" at a minimum.
I just wanted to mention that as I sat there first thing in the morning, I allowed the mind to drift in a natural manner. The exercise asks us to prove the theorem that "if a is odd, then a*(a^2 - 1) is divisible by 24."
Like I said, my sleepy brain began plugging in numbers on its own, without any conscious effort on my part. It noted that when a is odd, then a^2 is odd. How? By plugging in 1, 3, 5, 7, etc ...
A few numbers suffice to convince it. Then it sees that a^2 - 1 must be even, and that an odd number times an even number is always even. It drifts into plugging numbers in: 2*3 = 6, 3*4 = 12, 5*6 = 30, and so on.
The leap into actually writing down some kind of proof in algebraic terms is simply a matter of describing "odd number" in symbols rather than actual numbers.
a = 2n + 1 is our hypothesis.
All follows in a similarly mechanical manner after this. This becomes the seed, for we also "plug in" (2*n + 1) the same way we were plugging in the numbers. Observe how we replace "a" with "2*n + 1" (as well as factoring "a^2 - 1" into "(a + 1) * (a - 1)" :
a*(a^2 - 1) = a * (a + 1) * (a - 1) = (2*n + 1) * (2*n + 2) * (2*n)
= (2*n) * (2) * (n + 1) * (2*n + 1) = 4*n * (n + 1) * (2*n + 1)
We see that 2 * 2 = 4 is a factor of a*(a^2 - 1).
From here we would also show that 6 is a factor. Since we already know that 2 is a factor, this amounts to showing that 3 is a factor.
Trust me, I had to find the precise way to go about this elsewhere - not between my ears.
I'm not going to type up the rest, but it involves looking at both cases, when a certain number is odd and a certain number is even ...
I think that your intentions to return to some high school mathematics is commendable. As you know, I am not fond of "hot shots" and "whiz kids". One of the reasons I respect Alexander Stepanov is because of the way he encourages us all to return to the fundamentals, to keep going back to the basics.
Look at what I am doing. I feel weak in the area of "pure mathematics", and I realized that I was so out of it in 1984 that I missed an educational opportunity. The textbook I had ignored (as it mystified me) turns out to approach things in a formal, precise, and rigorous manner - a manner which has never come natural to me. It is almost painful to think that way, for me.
As Schopenhauer would say, it does not "appeal" to me.
And yet, I stubbornly want to develop my understanding in this area, even if only a little.
So, I too am working with my old high school text. This time - and this makes all the difference in the world - I have a Solution Key. This time I have TIME to THINK, TIME to REFLECT.
I don't think it is ever a waste of time. Invest in inexpensive notebooks and try to find texts with solutions available. I have found I am much more likely to work through exercises if I know I can not only check the results, but see full solutions worked out with some detail. Then I add more detail when working through the problems.
Consider these to be your "maths diaries", or, if you are like me and prefer to minimize the "significance" of the notebooks, just call them your SCRATCHPADS.
I think the more engrossed you become with reacquainting yourself with this realm, the less the demons at the office will be able to torment you, for you will develop an inner life where they have no power to harm you in any way.
Peace Holden. Take care.
I wish I could tell you to quit your job, but it is probably best to live a double life. Maybe your private studies will help compensate for your workday misery.
Also, it may help if you approach mathematics as I do; that is, as more of a "spiritual endeavor" than an academic one. A commitment to this kind of quest may nurture patience, humility, and an appreciation for solitude where the inner life of the mind becomes your center, not the image you produce in the minds of others, which, as Schopenhauer reminds us, is secondary. What is in your own consciousness is primary. That is Holden's world, not what goes on in the heads of your work associates.
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Important: Modern psychology (and medical psychiatry) have pathologized sadness. In Carl Jung's private diaries, which were highly guarded by his family after his death, make clear that sadness is quite appropriate amidst the "swamp of misery" [his exact words] that is life.
Hence, needless to say, joyfulness is not a requirement. The pursuit of happiness is not a realistic goal, and certainly not our purpose here. Will studying mathematics make you "happy" or "content". Probably not; but becoming more intimate with your own thought processes is at the very least an interesting distraction from the redundant demands of the will-to-live.
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