OK, so let's talk about physics. The first thing you have to understand is that physics is a model of reality. We can't trust our senses to tell us about reality; we have to make tools to look at things closely enough to figure out what's really going on. Of course, in the end, you have to trust your senses about what those tools tell you; the philosophy of it is that you ultimately have to accept that:
1. I'm (you're) real.
2. I'm (you're) the same me (you) that I (you) was (were) an instant, or ten minutes, or a lifetime ago.
3. My (your) senses send impulses to my (your) brain that are an accurate representation of some "reality" out there.
4. All of you (us) are real just like me (you).
You can't prove any of this. You either accept it, or you don't. If you do, then you're agreeing with the majority, and you're "sane;" otherwise, you're either playing philosophical games, or you're truly "insane."
Given that we accept this is real, then we look for how we can describe it. There's natural language, of course, and for most people, this is plenty. But if you want to know why things are the way they are, and in fact if you really actually want to know how they are, then you have to do more than just describe them, which we call "qualification;" you have to describe them in detail, which we call "quantification." In other words, "there's some rocks" is a qualitative description; "there are eighteen rocks located as follows, of the following sizes and compositions and masses," is a quantitative description. The first is good enough for natural language; the second is only barely good enough for the simplest kind of physics. Physicists get really, really precise about how they describe things, and to do that, they use math.
Now, math is a language for describing things quantitatively. That's what it's made to do. But you always have to remember, it's a language, and it describes; never confuse it with the reality being described.
As it turns out, we know enough to make really incredibly detailed descriptions. So detailed, we can describe things that we can't actually sense directly with our own senses. We can measure those things, and we can describe them, but we can't see them. So how do we know they're right?
The answer is, reality appears to be consistent. In other words, our universe appears to be a place where, although random things can happen, not just anything can happen. Only certain sorts of random things can. For example, if you get out of bed and walk to the store and buy some brewskis and come home and sit on the couch and drink one, you're still you. You don't turn into a penguin when you walk around the corner, and you don't cease to exist when you sit down on the couch. And this implies some things about the nature of our universe- and those things add up to consistency. Rocks don't just disappear, or appear out of nowhere. The planet beneath our feet is there all the time, and holds us to itself.
Physicists have a way of describing consistency like this. They call it, "symmetry." Physicists believe that there are certain things that remain the same no matter what you do. Those things are called "physical laws." They are descriptions of how objects behave, put in ways that are pretty much always true, and if they aren't under some certain set of circumstances, there's a way to describe how those circumstances can be translated into ones that make sense and show that in fact, it really was true, it just didn't look true. These symmetries are mathematical in nature, but you can describe them in natural language, with a bit of quantification thrown in.
Because of these symmetries, it is possible to state these laws of physical reality in mathematics; math is a language that's made to be consistent just the way the world is. There are ways to check if math is consistent, and those ways tell whether the math is useful for describing the world. And the beauty of math is, it can describe things that are impossible, but only if you start with things that are impossible; if you start with real world things, then as long as your math obeys the rules of consistency, it will continue to describe reality no matter how complicated it gets.
So that's how we can trust math to tell us about reality, and that's how we can trust the big pieces of reality that we can see to tell us about the little or huge ones we can't. That consistency is a key idea. It's the basis of something called the "Scientific Method," which uses that consistency to let us make these mathematical theories that accurately describe our world.
So, physics is written in math. And the rules of consistency of that math are as close as we can make them to the rules of consistency of the real world as we can make them; and we go looking for little teeny faults in the math, by testing it against the real world, all the time. If we find something wrong, then we figure out what we did wrong and fix it, and then our math is a better description.
But you always have to remember that the math is just a description; it's the old thing about the map, not the territory. You can look at the map all you want, but until you've walked it you really don't quite know what's there. And when you have a bunch of math that describes stuff you can't ever directly sense for yourself, then you have to just trust the math, and look for ways to check it.
Now, the math talks about probabilities when it talks about quanta. It talks about waves- or, rather, the equations we use to talk about those phenomena are the types of equations we use to talk about waves we can see. It talks about particles- again, rather, the equations we use are the types we use to talk about solid objects with apparently definite locations and other definite characteristics, boundaries, qualities, and so forth, that we can hold in our hands, as well as many we can't. But that's not to say there really are waves, or particles. All we know is that if we use this particular math right here to describe this aspect of reality, when we make predictions with it, those predictions come true. And we don't predict precisely what will be precisely where precisely when- we predict probabilities that this thing will be there then. That's all the math can tell us. If we try to use path that doesn't include those probabilities, it doesn't predict what's going to happen. So we can tell that math that doesn't make its predictions in those terms is wrong, because it doesn't agree with the real world. We can tell that math that doesn't talk about waves is wrong, because the predictions don't come out right. We can tell that math that doesn't talk about particles is wrong, because when we look, the particles are there. But even though we know it's right, we also know it's not reality- it's just a description of reality.
So how come we know when it's wrong or right? Because if it's right, it works. It describes what we see, and predicts what we will see. If we make something using it, that something works- it does what we expect. If it's wrong, it doesn't work. It's just that simple. We keep checking it against the real world.
We also know that we can't ever "look for ourselves" at quanta. In books, you see pictures of little balls floating around, generally lit by diffuse light from over your left shoulder. Quanta aren't little balls, and they aren't lit by anything. There isn't anything to see them with but other quanta, that are around the same size they are. They don't have precise edges, and we can't say they're "right there-" we can only say where they probably are. And we only know that much by bouncing another quantum off them, and then capturing it in one of our machines and measuring it. So all we will ever know about them is the math that works. That's as close as we can get- and that math only talks about probability, not about certainty.
Next, I'll talk about "interpretations" of quantum mechanics- how we describe that math in natural language, and try to understand what it means in terms we can understand.
