Let's call a mental object anything that you're aware of, anything that is in your conscious mind. A sense perception, a feeling, a thought, a concept, anything that is under the spotlight of your consciousness at a given moment.
Tautologically, if you have any access to something in the world, it must be a mental object; it must enter your perception.
So how do you even know there is an external world? Why isn't it "all in your mind"? What's even the difference between a totally imaginary world playing out in your consciousness, and a "real world" that's "out there"?
Without being knowledgeable about the history of philosophy and the many people who have attempted to answer this question, I think there's a pretty straightforward way to solve this problem.
One of the things your mind can do is ask the question "same or different?" about two mental objects.
"Same or different?" can apply on multiple meta-levels. You can say "do these propositions contradict?" You can say "do these motives conflict?" You can say "does my observation match my prediction?"
The "evidence" for this proposition is just that you can try it out in your own mind and see if it works. It's also analytically necessary that this be true if we believe that people have goals.
Any kind of optimizing behavior can be viewed as optimizing to minimize a distance or difference between the desired and actual situation. Which means you need to be able to detect differences. If we are goal-achieving machines, we must also be difference-detecting machines.
Moreover, there are some computational models that fit psychophysics data quite well, like Waltz Filtering, where you can view the process of visually parsing a line drawing, as an optimization problem where we try to minimize the number of inconsistencies in interpretations of the drawing as representing a 3-d object. (Unreconcilable inconsistencies lead to impossible figures, where we "see" the figure as representing a 3-d form locally, but then can't extend that same interpretation consistently to the whole figure.)
Psychophysics data also tells us that our sensory perceptions are keyed to differences, not absolute magnitudes; far more people have relative pitch than absolute pitch, for instance, and our color perceptions are relative to background and lighting, not absolute. Detecting "same or different?" seems to be more "primitive" or "fundamental" an operation in the brain than detecting "how much?" In many cases, at least, it seems that we have "difference detectors" and construct absolute measurements out of those, rather than having "absolute magnitude detectors" and computing differences by subtracting them.
"A" and "Not A" is a flat contradiction -- an impossibility. The mind boggles. It cannot be.
But not all differences, of course, are impossibilities. It's possible to notice that a dress has fabric in two different colors, and this doesn't slow us down a bit.
What happens is a kind of "going meta", I think. You say "oh, no problem: it's A here, and not-A there." Distinction by dividing up the mental world. "Hey, A and not-A, you can share." Now there is no contradiction and everything is hunky-dory.
Or, you can explain away one half of the contradiction: "oh, I only believed not-A because I was misled by such-and-such; now I can safely discard it as a mistake." Again, no problem.
Or, you can reframe A and not-A so they are both parts of a whole, or not really opposites after all. There are a lot of things you can do.
Essentially what you're doing is handling an ontological crisis. You resolve an apparent contradiction by adding some complexity to your mental world, such that both apparently contradicting mental objects are compatible and explainable (or explain-away-able) by your new, expanded view of the world. It's the process of noticing that the blind men were seeing different parts of the elephant.
There seems to be a contradiction, but really, if you shake it all around, if you learn more, if you do some trial and error, you can get into a new configuration where the knot untangles and it all makes sense.
This is what "thinking" is, I believe. Messing around with your mental objects until apparent contradictions resolve.
And this tells us what it means to believe in an external "world." It means that you believe that your space of mental objects will come to include things that it does not yet include, but which recontextualize today's apparent contradictions so they make sense.
It means "we will understand it better by and by." Everything has an explanation; everything came from the same world. If we took a wide enough view, everything would make sense.
It is a kind of faith, but a very minimal sort -- the faith that you live in an intelligible universe, that ultimately you can make more and more sense of things. Or the stance of trying to see things from the perspective of how they would look once you had made sense of them.
Solving Problems Is What Brains Like Doing
Lulie Tanett likes to write about how "reason is fun" and "problems are good" -- that literally humans enjoy the process of problem-solving.
If you think about it, most of our "play" is puzzle-solving -- trying to achieve an objective despite an apparent obstacle, or trying to make sense of something initially confusing. Videogames are puzzles even when they're not "puzzle games". Sports are puzzles. Even reading or watching a work of fiction, or listening to music, gets much of its "fun" from a dance between predictability and surprise (and ultimate resolution of the apparent mystery or discordance.) Many of the things we do for no reason other than enjoyment are problem-solving activities.
We have an instinctive desire to tug on problems in attempts to solve them. That doesn't mean all problems are perceived as unpleasant. Sometimes the "problem-solving" happens faster than we can be aware of it; sometimes the process itself is pleasurable. Only sometimes do we have a negative feeling around the problem, and it's not merely because the problem exists.
Suffering = Problems Metastasizing
Just having a problem -- your conscious mind includes both "A" and "Not A" -- doesn't necessarily cause suffering. But if you then go "there IS a contradiction" and "but there CAN'T be a contradiction", then things start to get worse. Or "I have mixed feelings" but "but I MUSTN'T have mixed feelings", or "this is hard" and "but it SHOULDN'T be hard", and so on. Somehow a problem can result in, not an attempt at resolution, but more problems! More contradictory beliefs, which vibrate against each other, and make more and more friction in your mind.
I think this is usually what's going on when something really bothers us. Have you ever noticed that sometimes a big life problem stops being upsetting, not when it goes away, but when you find something you can do about it? You flip out of "bemoaning and denying that the issue is there at all", focus instead on some constructive activity toward solving the issue, and suddenly it gets easier? Because now all you have to cope with is the issue itself, not the exhausting "is it real or isn't it", "should I admit it hurts or tough it out", internal debates. You now have one less problem to solve.
If you have problems about problems, issues about issues, etc, it gets much harder to deal with them. These are psychological truisms: negative emotions are worse if you're ashamed of feeling them; abuse is harder to recover from if everyone around you insists it's not real; dealing with misfortune is harder if you're still in denial about it.
Having a "meta-problem" means having an internal contradiction around the concept of the problem itself. "This problem exists" and "this problem doesn't exist" battle in your head. Trying to believe in a contradiction, trying to do the literally impossible, maintaining the conviction that you should be able to believe contradictions or do the impossible -- all of these are ways of making meta-problems out of problems.
The Buddhist concept of tanha is usually translated as "craving" or "desire", but it literally means "fuel" and is associated with clinging or persistence, trying to make mental states stick around. If I were to try to map it to this model, I'd say that tanha is the mental motion of returning to a contradiction or a knot in the mind, and trying to will it to not be a knot, thereby creating a bigger meta-knot. It's what you do when you remind yourself of a frustration in a way that makes it more and more frustrating. You keep the contradiction bouncing back and forth, louder and louder -- it IS, but it SHOULDN'T BE, and moreover nothing should BE HOW IT SHOULDN'T, but some people told me I should ACCEPT THINGS AS THEY ARE, but I DON'T LIKE THAT...
That "outward spiral" makes it harder and harder to resolve the problem at the root of the whole thing, because every time you notice it, it activates all the other meta-problems. I think this is the structural underpinning of what we experience as negative emotions, "touchy subjects", and "sore spots." They're wounds that defend themselves from healing.
Ultimately, if we believe in a world, we must believe that this process too comes from the world, that there is a reason why problems sometimes grow their own defenses. But it's tricky and deserves exploration at length. You're looking at a process that doesn't "want" to be looked at.