This is interesting! I like how you differentiate by "perception," and I want to pursue that. However, first I have to ask you to clear up how you're using some terms for me. The OP asks how we define "knowledge" and "belief"; your post defines "truth" and "belief", and goes on to define belief as "something that we know without (or perhaps even in spite of) perception." So I must ask, similarly as to Orion, what do you mean by "to know" here?
My apologies, I was using words interchangeably. To make it less confusing, I define it like this:
Knowledge = Information that has been duly perceived by the holder(s) of said information.
Belief = Information that has not been duly perceived by the holder(s) of said information.
Perception, of course, is the interpretation of stimuli by our brains. The term "duly" here is meant to refer to more scientific means of affirmation; or, put differently, that the perception is repeatable/verifiable. For example, the knowledge that my phone is black. I can look at my phone as many times as I want to confirm this. Others can look at my phone to confirm this as well. However, take the case of a ghost sighting. Such a perception typically occurs "out of the corner of one's eye," and cannot be perceived a second time despite people's best efforts.
I like that line, but must ask again: would you say "Fifteen hundred years ago, everybody knew that the Earth was the center of the universe" or "Fifteen hundred years ago, everybody believed that the Earth was the center of the universe"? Or does it not make a real difference?
I would say that they
did "know." However,
now we "know" that those statements are not true. My definition of knowledge allows for said knowledge to be incorrect. Knowledge, as I define it, is mutable; a concept separate from truth (which I, henceforth, define as the immutable realities/occurances that exist regardless of our perception of them).
The reason I make this distinction is because we don't operate in especially different ways than we did all those years ago. Just as we "know" based on our current perceptions, they "knew" based on theirs. They looked out over the horizon, had no means to perceive a curve, and knew the world was flat. They looked at the stars, perceived that they moved around us, and knew that the earth was the center of the universe. The only difference is that we now have the means to perceive what they could not. Someone perceived points of light in the night sky moving backwards, and therefore "knew" that something in their thinking was wrong.
So who is to say that, centuries from now, the things that we so adamantly regard as "true" based on our "knowledge" won't later be thought of as incorrect by the minds of our ancestors? Sure, you could assume a plateau in our potential capacity to perceive (and that's definitely possible), but such a plateau would also be based on our current capacity. I'm sure that hundreds of years ago, they could not have fathomed the strides that that modern world has taken. They likely thought that our abilities as humans would plateau well before the point that we've reached due to their own limits (speculation, admittedly, but I still very much doubt that they were able to envision our current situation). The same could easily be true of us far into the future; or perhaps even near into the future, if the rate of technology continues to increase as exponentially as it currently does.
Fascinating video that you (indirectly) pointed me to! Also relevant to what we are discussing here. In this video, the host points out that math is not empirical, or in your terms "perceivable", yet math makes claims to truth! You suggest that it does this through matching up with other things that we do perceive, but that is not technically how we "prove" things in mathematics. "Pure" or abstract mathematics is not concerned with what can be perceived in the physical universe, and even applied mathematics might be considered "quasi-empirical" at most. This suggests that if we accept mathematical "truths," we need a different model of "Truth" (or knowledge).
Wikipedia - I used this article to supplement what was presented in the video. It also includes arguments for mathematical empiricism, which the video (and I) did not address.
Indeed, I may have been hasty in my use of mathematics to support that I do not promote solipsism. It's a fascinating idea to think that these concepts, lauded as truth, are things that are imperceptible in and of themselves. Truly, mathematics often prove themselves in hindsight. The rules are determined, tested out in (ideally) real world application, and then verified after the fact should the results come out as planned. Yet, that only works in instances where a real world application is feasible, which is certainly not the case for a lot of upper-level mathematics. Then comes the thought that some of these decisions seem rather arbitrary. Calculus was formed after finding that Algebra wasn't able to answer all mathematical questions, for example. Physics, meanwhile, seems constantly threatened with being rewritten.
Mathematics, I suppose, would just be another example of our capacity to perceive, then. Just as back in the days when we thought the earth was the center of the universe, we accept our current understanding of mathematics until such a time as we discover something that requires the rules be rewritten to match it.
This is even more interesting than what I understood you to be saying above- if I understand you correctly here, you're saying that knowledge is perception, regardless of whether that perception is "correct" or corresponds to some immutable truth or "what really happened." In that case, it would be very different from Orion's definition of knowledge as relating to facts and thus correct. Am I on the right track?
In a nutshell, yes, that's exactly what I'm saying. Again, we operate on the assumption that our knowledge is the truth, no matter the era. It's only later that we discover if our past knowledge matched with truth; and even then, it may still only be probable that our perceptions are truth.
Going back to my phone example, my phone is black. I "know" this, because I've looked at my phone numerous times. As such, without even looking at my phone, I can be reasonably assured that it remains black. Yet, despite the extremely high likelihood of "my phone is black" being true (to the point where I can operate under the assumption of that knowledge), it's always possible that something happened to change the color of my phone.
Now, again, I want to make it clear that I'm not trying to promote solipsism. We can operate under the assumption of our knowledge (and have been doing so since antiquity). Not to mention that, in order to perceive something; something must, therefore, exist to be perceived. I merely feel it appropriate to distinguish knowledge and belief from truth via the concept of perception, since that's precisely how we operate. After all, the only way for us to determine truth is to utilize our capacity to perceive. We cannot intrinsically "know," we must first perceive. Which makes perception an irremovable part of the process.
To end this post with another example, let's go back to color. Not our ability to perceive color, however; but rather, that of the mantis shrimp. Sight is, obviously, a large part of how we perceive the world. We have three photoreceptors (red, green, and blue) that allow us to visually perceive our surroundings. Logically, we base a lot of our knowledge on that ability, to the point where we convert wavelengths beyond our ability to perceive (ultraviolet, infrared, etc.) to formats that we can.
But, as many are aware, not all animals share the same number of photo receptors. Some have different numbers of photoreceptors. Dogs, for instance, only have blue and green. Many birds have a fourth photoreceptor that lets them perceive ultraviolet waves. Some creatures, like Octopuses, don't have color photoreceptors, but instead have those that let them see polarized light. Then you have the mantis shrimp. They have sixteen photoreceptors to our three, over five times as many. Their eyes allow them to see ultraviolet, our visible spectrum, "depth," and polarized light (including circular polarized light, something that only they are known to perceive naturally).
To me, this is fascinating. Since we attempt to determine truth from knowledge, and our knowledge is based on perception, our search for the truth is limited by our ability to perceive (as I've said before). So what, then, if our limits were different? How much else would we "know"? How different would our body of knowledge be if we had the eyes of a mantis shrimp? It's things like that which lead me to define knowledge and belief as inseparable from perception.
EDIT: Well, this post ended up being longer than I expected, lol