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I'm going to write down some pretty scattered thoughts on phenomenology here instead of my main geminispace blog because I'm partly trying to talk out some things floating around in my brain.
So one of the things in Husserlian phenomenology that gets a bad rap is "intuition of essences" and I sorta get why. It sounds very bullshitty when you first hear the term, but only because the particular words being used are in a very different sense than you might normally think.
Even some philosophy texts will refer to this as a "navel-gazing" process to understand the true nature of an object, but nothing could be further from the intent. Intuition, to Husserl and to mathematicians around this time, meant mental action of reasoning. It wasn't "gut instinct" the way we tend to use intuition or intuitive these days. The use of essence is a little more unfortunate because an object doesn't have a singular essence, which in english at least sounds counter-intuitive. I like Richard Tieszen's use of "invariant" instead of essence. Because, basically, phenomenology is ultimately about the extraction of communicable, "objective", knowledge out of purely subjective experience. Essences/invariants are properties of a thing that allow it to be categorized, described, understood.
This makes a lot of sense from a math perspective, I think, because an "essence" of the natural numbers is its nature as a group and the description of it as a set with operations that follow certain laws. This essence/invariant is something that then can be applied in other situations, related to other things: rotations of objects, permutations, changes in phase for electromagnetic fields, &c.
So, basically, I think a more accessible version of describing "intuition of essences" is as the process of abstraction by deciding what the properties to be abstracted are. In that case, Husserl's description of "free variation in fantasy" is better understood as how you figure out which properties are and aren't essential to the act of abstraction.
From this light, Husserlian phenomenology starts to sound a lot more grounded in everyday practices which is as it should be for a philosophy trying to formally investigate lived experience.