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Hey, so this is going to be a post that I'll probably update now and then but I wanted to take a first stab at it today. We're going to give some background and then books that I recommend.
Edmund Husserl, in the early 20th century, helped found an entire movement of philosophy that doesn't fit neatly into the continental/analytic divide: phenomenology. To talk about Husserl's influence is kind of like the old joke about The Velvet Underground---only a hundred philosophers really studied him but each of them started an entire system.
These days I've found you mostly get a game of telephone on what Husserl's phenomenology was: that it's like a Berkelian idealism or platonism, that it involves introspection to learn about the universe, that it involves mystical acts of intuition, or that it was supplanted by Heidegger's much formulation superior phenomenology. The latter is a belief I see repeated a lot and that, in my opinion, is almost completely attributable to Hubert Dreyfus's strange love of that most infamous Nazi.
So let's set a little bit of the stage: phenomenology is more of a meta-philosophy, a kind of foundation _for doing_ philosophy. It is, as the name might imply, about "the phenomenon" as aan object. But before we talk about what Husserl means by "phenomena", let's talk about the kind of problem that was bothering him. Husserl was a mathematician first and foremost. I can't stress enough how important that is to _everything_ and if you keep in mind how mathematicians approach problems and abstraction there's a lot that makes more sense about his formulation. Husserl was working at a time when set theory was still new and exciting and a lot of people were wondering about what on earth the foundations of mathematics could be. He started a lot of his work even before Russell found his paradox! His habilitation was on the foundations of arithmetic, particularly trying to understand if it was possible to analyze human thought and understanding of numbers and come away with some lessons about what numbers _must be_ from a logical standpoint.
Even before Frege called his habilitation a disease to philosophy and logic (no, really), Husserl had come to the conclusion that he was bordering on "pyschologism" in his previous work. Psychologism is a word you'll get used to hearing in phenomenology. It is the great enemy of early Husserl, the dragon he attempts to slay in _The Logical Investigations._ Psychologism is the idea that things like logic are merely artifacts of human thinking, an outgrowth of how our minds work. Their basis is to be found in the mind and the mind alone. Logical reasoning is thus _identical_ with the process of coming to a belief in the mind of a person. Not related, not correlated, but the same thing. Part of Husserl's attack on this is that there's no a priori reason to believe all our minds operate exactly the same way and have the exact same representations of logical ideas. In fact, it almost certainly couldn't be so because we rather obviously are capable of thinking differently than each other. And yet! We are capable of communicating logical ideas to each other, of explaining our reasoning, of holding it seperate from the individual in a form that can be checked and discussed. Surely those traits imply that there are rules here that are not entirely arbitrary.
Now, is the conclusion then that we're dealing with entities fixed by the gods? Things that have an absolute necessity and must be said to exist in the exact same way that the very surface you're reading this sentence on must exist? No! We can eschew the dichotomy between naive platonism and psychologism. Things like logical systems, like mathematical objects, have the interesting property in that they "exist" in the sense of being things we can communicate about and have rules & properties. They are just as much _phenomena_ as anything else you can experience and direct your consciousness towards. I use the phrase "direct your consciousness" because that's one of the key points here: "all consciousness is consciousness of something" is the general idea here. Anything you think, you're thinking about _something_.
It'd be reasonable to say "uhh, so?" at that observation because the point isn't entirely obvious in really anyone's writing on this topic: the fact that thinking is a two-pole activity of _me_ thinking about _another thing_ means that if we want to understand the rules of how something like logic works, to investigate a meta-logic of how logics work and are constructed, we need to consider the _subjective_ aspect of them. The fact that all experiences---even of abstract things like mathematical objects---are experienced by a person about a thing actually has a lot of implications for the philosophy of math and science. As an example, it leads to the insight of Herman Weyl that I always reference: that the subjective is the absolute and that objective descriptions are a kind of abstraction that always removes detail. Maybe that doesn't seem profound at first but it turns a lot of the implicit philosophy of science we instill in STEM education completely on its head. All of my posts on the book Feelings of Being are about putting the subjective-pole back into our study of psychology and how it changes our understanding of what even _is_ a mental illness. Although that's all I want to say on that, here, because I don't want to distract from the central point: the foundations of the sciences in general.
So while over the course of Husserl's life his _massive_ amount of writing on phenomenology it grew into a large system that underwent many revisions, the central aim was always the same: to understand the rules by which epistemology and reasoning are even possible, using the most sure and absolute data that we have---our own experiences of actions of observation and reasoning. This is not a naive idealism but a recognition that "the objective view" is already a step too far removed from the actions we undertake in science. Another way of putting it, I think, is that to know whether one is abstracting out the _right_ parts of subjective experience one needs to first understand the subjective experience in its fullness.
Okay, so all of that preface out of the way: what's the point of studying this largely out of fashion 20th century philosophic movement? A few reasons
Now we're at the point that was the original reason why I even wrote this post in the first place: where do you even start? I'll share books I think are good for getting into this morass in a rough order of accessibility
Well what started as a quick thing I was going to write before starting some other work for the day has now ballooned into nearly 2k words. Well, in any case I hope this is a helpful guide if you've ever found any of the stuff I've talked about interesting. I'd love to talk more about any of these things with someone!