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I was wondering if you could cover some historical points. During the first half of the semester, my professor lectured about Descartes, then moved onto existentialism and structuralism in the second half. How did we go from Descartes' (and Kant's) skeptic/solipsist situations and move on to a point where people were already in a world, like in existentialist or structuralist thought? Had the thinkers of the latter movements "solved" skepticism? Or did they presuppose certain conditions that allowed them to "ignore" skepticism? Or something else maybe?
Sup bro? So, the short answer to your question is "yes". All of those things happened, and several more besides.
But first let's back up and figure out exactly what Descartes did - what situations did he put us in? He's definitely most famous for two lasting legacies: the radical doubt he attempted, and the cogito he formulated to get out of it. First he went, "How the fuck can we know anything at all? Can we be certain of anything?" and then he was like, "Well, someone has to exist to do the doubting, so obviously I can be certain of me. I can be certain I have ideas." After that, he did a bunch of other stuff concerning the idea of God and how ideas worked and whatnot. Generally philosophers agree that he didn't succeed in getting all the way out of the everything-hole he had dug himself - in fact, his failure was so bad we gave it a name. We call it the "Cartesian circle" - he's arguing for himself based on an idea of himself, and that his ideas are outside him based on God, and the truth of God based on something in his ideas. We disagree over how radical a failure that is - no matter what, though, the first time you read it you're like, "Jesus, Descartes, is this fucking amateur hour?"
The point is, Descartes said that we can only be sure of things definitely, clearly present to us, so close we can't possibly fuck it up, and then he failed to show that anything except the self met that criterion. So now he's just a self, standing around, not sure if his ideas are outside himself or all internal. Plus it’s raining and he doesn't have an umbrella, so he's getting soaked. He just looks so sad. What an asshole. His failure to get out of the hole leaves us in a position where we can't be sure of shit (skepticism) except maybe ourselves (solipsism).
I can see why you'd be confused - your professor spent half a semester explaining what some notable philosophy bloggers can explain in two paragraphs (whatever, man, we all have talents) and then all of a sudden he's talking about bros who were like, "We are different from the world, and alienated from it, and we have to forge our own relation to it!" And you were like, "Woah, the what? So there is a world now? Did I miss a meeting or a memo or something? Seriously, I have got to get a better secretary." No worries, bro, I've got your back. But definitely, you should consider firing what's-her-name.
Then the early existentialists came along and said, "Wait just a fucking minute. The world isn't in the mind - the world isn't just some part of consciousness that we organize with everything else. The mind is radically different from the world, and in fact can never fully be united to the world.” Except they need a way out of the Cartesian hole - we definitely have ideas, but how do we know those ideas exist beyond us?
Meet Husserl. Husserl taught, among other bros, Heidegger and Levinas; he was kind of a big deal, and a bunch of existentialists take their cues from him, even if they disagree with him on other key points. He definitely laid the foundation for an escape from the Cartesian hole. How do we know that the ideas we have go beyond us? Because some ideas can't be contained by us. Take redness. Red isn't just a collection of some things that share a property, or an image that's just occasionally given to you. If you brought me all the red shit there is and laid it before me and said, "There! That's 'red'." I'd be like, "No, that's everything that has the property of being red. But what if I copied those two things and glued them together? NEW RED THING, asshole." And that's true of any universal property at all. So if some demon or something is feeding us 'blue', he has to invent blue for us to be fed; blue isn't just a part of some ideas. Blue exists outside of us. No matter how many ideas of blue we have, we can't exhaust the color – so in some way, blue itself is outside the mind.
Husserl didn't think that only properties exist outside of us, he just thought that was the case most obvious, first rung on the stepladder out of the hole. He says the same thing about individual things, too. We're watching a movie we can't be sure about, but we're definitely watching it. And thinking about the movie is different than watching the movie - the movie can't just be a bunch of ideas that resemble a movie. We have ideas that resemble a movie, and they're different from the movie itself. We have ideas, visual images of things like bananas, but to get to those we have to ignore actual bananas for a second. So we can't create these things in our mind, and they're not just pictures we have; our mind has to grasp at something for us to have pictures.
Now, Husserl doesn't give himself much room to develop an ontology of actual ideas; it could still be that actual bananas are 'illusions' or something, whatever. But whatever 'illusion' means, it's still something outside the mind. Husserl isn't saying anything more about what it means to be outside the mind, just that it means something. There's something else out there, a world, that we're actually experiencing, and in some sense the way we experience it is true. After all, what would it mean for a banana to be an illusion? Our mind definitely grasps something outside of itself; we can argue over whether that something is a bundle of properties or a composite object, or an idea of God, whatever; it might even be a "lie" from a demon. What makes any of those ontologies preferable to the other? What makes the thing the mind grasps as "banana" any less outside the mind or real to it in any of those frameworks? They all grant that there is something outside the mind; they just disagree over what.
And that's enough for existentialists and structuralists to fucking go to work. Even individual existentialists paint different pictures of what the world really is; the point is, it's definitely outside, and it's definitely something different than consciousness or the self, and we definitely have some fucking trouble dealing with that.
Structuralism went a step further and said that not only is the world different from our ideas, it confines them. Look, when we think about love, how much is that thinking is influenced by Romeo and Juliet? Something like all of it. We have these notions, these structures handed down via a cultural history, which is why the Eastern conception of love is so different. Now, how the fuck could all our internal ideas be constrained by structures if those structures were just other internal ideas, hmm? So these things must be outside our mind, otherwise they couldn't impose on it.
So, yeah. That's pretty much how existentialism and structuralism broke out of Cartesian skepticism. It's not perfect - all they did was say, "even if the world is an illusion, it's still forced on us from outside." And those sort of "even if" arguments comprise the bulk of the arguments against Cartesian skepticism; they aren't limited to the two groups you mentioned.
John Locke said, "Look, if pain is just part of a dream, it's still a pretty fucking painful dream - the sensation is real as shit, as we ought to act like it. If I electrocuted you, you wouldn't doubt it hurt while you writhed around - you'd genuinely be like, 'Don't taze me, bro!' Pretending pain doesn't hurt gets us precisely nofuckingwhere." Even if the world is an illusion, we should treat the sensations as real, since it's all we've got.
Hume was like, "Even if everything is an illusion, you can't help but act otherwise. You form beliefs based on sensations of pain and pleasure, and patterns we call 'cause and effect' in the world. If the world is an illusion, we've all bought right into it, and we can't help that. So, whatever bro."
The point is, Cartesian skepticism says little about how we live our lives one way or another; we pretty much just go about our business, bro. Meanwhile the shit we do experience, 'real' or not, is real to us, so we can do philosophy about it. The only problem for existentialists/structuralists was whether it's all inside the mind or not; once they got shit outside, they got what they needed.
You can read the original project of Descartes' in Meditations on First Philosophy, also available in the public domain.
Husserl lays out his solution to the Cartesian problem - part of what he calls 'phenomenology' - in several works, mostly in his Ideas: General Introduction to Pure Phenomenology.He later clarified and simplified in the context of this problem in Cartesian Meditations: An Introduction to Phenomenology.
I don't remember where I first encountered them, but Professor Paul Vincent Spade has made excellent class notes for his class on Being and Nothingness publicly available. They lay out much of the Husserlian background for existentialism.