A question for you: if I am taking account of my feelings(both good and bad),the sensations that arise in my chest from time to time & also the flood of emotions that I invariably experience while studying mathematics,would you then say that this is how math could be done, phenomenologically (Husserlian - Transcendental Phenomenology).
One more thing:sometimes I am busy doing something,it could be studying math and all of a sudden I am forced to leave it-thing in itself presents itself to me in the flesh.
Since throughout the day I am invariably always either working on a math problem, trying to implement a mathematical idea into a computer program, or satisfying basic primitive needs (which demand constant attention, as any living creature can attest), I frequently find emotions rising up to the surface out of nowhere, echoes coming from past experiences, images of those who may have abused me, or shame about some of my own behavior - guilt, fear, anxiety. These feelings are the realm of experience itself.
I firmly believe that whenever we seriously put our minds to studying anything that life will get in the way. Those who write texts rarely mention this. I suppose the material must be presented in as terse a manner as possible.
Your question about how math could be done phenomenologically may be a kind of seed question. I suppose it would be a rather complicated project of individualized description of internal (subjective) states, quite impossible to communicate, but not impossible to think about.
I'm afraid that our deepest reflections are not communicable, and we will have to give up on any hopes of describing in any meaningful way the flood of anxieties taking place beneath the surface of the brain's command center.
Often all one can do is take deep breaths to try to get a grip on a given situation and interpret what our own internal guidance system is shouting at us. We are hardwired to be afraid and anxious, maybe as a consequence of having evolved from fish, who knows. Part of our brain sees no apparent need to learn to understand the abstract ideas of mathematics, and perhaps this "beast within," the creaturely presence, this Thing-in-Itself, our own animal body as Thingly Presence, is trying to demand that the Intellect return to its subservient role, deferring to the authority of the Body, the stomache, that is, primary primitive needs.
Maybe studying math is quite unnatural, further alienating our consciousness from the evolutionary project of biological survival.
In moments such as these, the Will asserts its Natural Authority, literally shutting down any of our "higher interests".
It is unfortunate that more people do not explore these interesting obstacles.
Thanks for digging around into deeper territory, bringing into the light the darker primal forces at play. We can't just assume the life of the mind can go about its plans. Life always gets in the way. This is hardly mentioned. In other words, in the mathematics textbooks, there is no attempt to convey the actual human experience.
I wish I could articulate these kinds of things in my written solutions to exercises, but the space is precious on the paper, and I want to keep things organized for future reference.
Holden, we are a condemned species. We have language but language is not sufficient to communicate raw experience.