Do you think privacy may encourage intellectual honesty in one’s pursuit of mathematical maturity? Could a delicious secrecy about where one is at concerning one’s own understanding eliminate the unnecessary psychological anguish that many associate with “higher mathematics”?
I remember, in 1994, how much I would appreciate when the calculus professor (from India) would share with us his “algebraic and trigonometric” insights. To me, these were even more valuable than the teaching of the actual “calculus”.
Why? Well, because these were fundamental mind tools we were working with. It is similar to the way Integral Calculus is a fundamental mind tool for solving differential equations.
When I say, studying mathematics is NEVER a waste of time, I am attempting to transcend a tendency (in me?) to berate myself should I find myself still experiencing AHA moments of catharsis when considering something like function transformations, how the domain and range relate to the graph.
This is why someone my age may prefer solitary study. There is a certain psychological liberation when one nurtures intellectual honesty in the learning process, since one can let their own confusion become their best guide. As soon as something seems “counter-intuitive”, that is an internal recognition where the root of the confusion might exist.
When the graph shifts left or right, the domain changes, so in order for the outputs to remain consistent, the INVERSE of the transformation of the domain must be performed on the function.
The reason I mention this is because after getting through the first chapter of a textbook on differential equations, filling an entire notebook working through exercises, I decided to make a commitment to myself to relearn some fundamentals again, to slow the process down, and to devote several months to reviewing algebra, trigonometry, and integral calculus.
The thing is, this morning I suddenly suspected that this might be very boring for me. Wasn’t this a waste of time?
But then, while begrudgingly working through some exercises, I found that certain transformations on functions felt counter-intuitive to me. I was kind of amazed by this fact, so, after clearing up the confusion, I realized that returning to some fundamentals as a kind of strengthening my foundations may be seen as a “time/energy investment” that will surely enhance each phase as I forge ahead.
What role does one’s personal psychology play in one’s quest for mathematical maturity? First of all, I think the metric of maturity is defined on an individual basis and should not involve society as a whole. This is where privacy and that delicious sense of secrecy I mentioned comes into play.
Maybe students just getting out of high school should be encouraged to loaf around and study at their own pace for 10 years before committing themselves to a major course of study. This way they can take their education into their own hands and brush up on things that may not have been explained well … or maybe they were just too distracted to pay any attention.
I am interested in the role my own psychology plays. While I do intend to challenge myself, I also want to be able to appreciate reality, and not to berate myself for constantly reviewing preliminaries.
Can this “secretly studying” scenario be compared to other private pursuits involving inner growth? Learning and maturity are things that happen “inside the skin”.
Is it possible that exams, tests, grades, diplomas, job-titles, i.e. “systematic education” is a case of Nothing that is so, is so?
Maybe some things can’t be taught. They can only be learned.
Of course, there will be those who claim that just being in a position to study mathematics is tainted with the guilt of privilege, and yet how many find themselves wasting away in a redundant job or languishing in poverty, or worse still, in a jail cell.
I am not suggesting that studying mathematics, at whatever level, is a means to escaping poverty or securing economic security. No, far from it. I am suggesting it as a way to make the most of one’s poverty by attaining a life of the mind, where the most essential ingredient is leisure.
In modern mass-industrial society, those with the most leisure are the very rich and the somewhat poor. When industrialization and computerization has automated many jobs, there will be more and more superfluous members of the population whose main activity will be the securing of, the enjoyment of, and the recovering from intoxicants whose main function is to enable the user to endure its own existence.
Is self-education a social privilege?
If early education has lasting value, maybe it is simply the value as a cultural artifact, that one will have things to think about even if one finds oneself in the growing army of mind-bodies who have become apparently useless to society.
How shall the over-educated, underemployed and long-term unemployed masses continue to develop their mental lives outside the prohibitively expensive buildings of the Church of Reason if not via their own secretive self-education?
You know what, while it is interesting to talk about the masses, it is unrealistic to think in terms of mass society. That’s why, for the time being, I will only consider my own psychology. I will proceed as though learning things for the first time. Also, I have found that including a verbal explanation (to myself) of each step may convince myself (whatever that is) that I have a grasp of the concepts. Proceeding in this calm manner, from the ground up, may help me to develop a certain approach to problem solving that can be carried with me into whatever areas I choose to explore (or re-explore) in this manner.
Maybe I am just looking for a different style of education than what I experienced as a formal student. I am not promoting this style for “mass consumption”. I’m just keeping records of my learning experiences. In “planning for future studies”, I am only thinking of a few key subjects, and I foresee the study of these key subjects as lasting the rest of my life.
Isn’t it peculiar that a realistic assessment of the proper time I want to devote to these studies is in direct conflict with the pace at a university?
This makes me wonder how much intuitive understanding is actually being retained after the exams are “aced”.
Maybe I am not aiming to be a “mathematician” and will be satisfied as a kind of “mathematical technician”.
I feel the same way about computer programming.
I never had any ambitions to be a “software engineer”.
I’m just a curious chimpanzee who can read some code, and I appreciate seeing mathematical algorithms come to life in computer programs, especially computer algebra systems.
Maybe I will never possess the “mathematical maturity” to really appreciate proofs, and maybe I will never have the expertise to manage a very sophisticated piece of software.
And yet, what I do hope to attain is enough emotional maturity to be able to appreciate whatever I am able to grasp, and to stay curious.
I don’t demand too much of myself, and yet I feel I am eons ahead of those who would damn me to Hell, screaming, “Get a job you a-s-s-hole!”
related: How to Attain a Studious Life