We'd all agree, I think, that there is a world around us, one in which we live. But a question we don't usually ask is why we think there's a world around us. Why think that there is any world outside your mind? This may seem like a rather odd question to ask; but it turns out that how we answer it has ramifications for many other fields of thought. Different answers will affect, for instance, how you view the results of scientific investigation; they will change what you count as a good or bad argument for God's existence (or God's nonexistence); they will shift your views about how to approach testimony and hearsay. And this is because your answer to the question of how you know that there is a world outside your mind will influence what you think we can definitely know about that world. One of the important features of early modern philosophy is that this question was explicitly asked, and received a wide variety of answers.
Lady Mary Shepherd approaches the question by splitting it into three different questions. When we ask how we know that there is a world outside our minds, we are really asking three things (Shepherd is following David Hume here):
(1) How do we know that anything continues to exist when we aren't perceiving it?
(2) How do we know that there is anything external to us?
(3) How do we know that there is anything that doesn't depend on our minds?
Each of these questions implicitly makes a distinction: continuous/interrupted; external/internal; independent/dependent. The world outside our mind is continuous (it doesn't just pop into existence when we sense it); it is external (it isn't just part of our minds); and it is independent (it exists on its own).
We can start with the first question. Why do we think anything continues to exist when we aren't perceiving it? I look at a mountain, look away, and then look at it again. I'm pretty certain that it's the same mountain, and that it was still there even when I wasn't looking at it. I cannot currently see the wall behind me; I'm pretty sure it's there, continuing to exist without my sensing it. I'm pretty sure that Toronto continues to exist, despite the fact that I'm not there; I'm pretty sure the stars exist, even though it's daytime and I cannot see them. We expect the world to be there even when we aren't sensing it. But this means that our notion of the external world isn't derived entirely from our senses. We cannot perceive that anything continues to exist when it is unperceived! We cannot sense that the things we sense still exist when they are not sensed! What else needs to be added in order to get this conclusion?
When Hume answered the question, his answer, in Treatise 1.4.2, was complicated -- notoriously complicated. But Shepherd's answer will actually be straightforward. And this is because both Hume and Shepherd recognize that this question depends on our view of causation -- and they have very different accounts of causation. Shepherd's account of causation is reason-based: causal reasoning is rigorous reasoning. Indeed, Shepherd goes so far as to say that all mathematical reasoning is one kind of causal reasoning. Because this account of causal reasoning makes causal reasoning much, much stronger than Hume's account does, Shepherd's answer to the first question is massively more simple than Hume's. (Fortunately for us!) Strikingly simple, in fact. The basic idea is that
(1) Our senses make "irregular calls" for things. (For instance, I can arbitrarily look behind me to see the wall.)
(2) On the basis of this we find that things are very often "ready to appear".
(3) The only way this readiness to appear can be explained is if either the world is set up so that things are automatically created every time we sense them (they are "created purposely, ready to appear") or the things continue to exist even when not sensed (they "continue to exist, ready to appear"), so that things are consistently there when our senses call on them.
(4) We have so many different perceptions that we recognize there must be many different things with uninterrupted existence capable of causing them, because otherwise they would begin to exist without a cause.
(5) Therefore, for every kind of existence exhibiting this consistent readiness to appear, there must be something with the continuing capability of appearing in this way.
In other words, Shepherd essentially is identifying a well-known effect that we experience (ready appearance at irregular calls of sensation) and giving a cause for it. There has to be a cause for it, because these appearances begin to exist, and what begins to exist must have a cause. Likewise, there must be something that exists that explains the consistency of these appearances, because otherwise there is something about the beginning-to-exist of these appearances that has no cause. (The reasoning behind this is somewhat more complicated, but we don't need to go into details here.) From this we see at once that there are things that appear to us in sensation that must continue to exist even when we aren't sensing them.
It is important to notice two things about this, I think.
(1) This is very common-sensical, and deliberately so. After all, what will people usually do when you challenge them to prove that things continue to exist even when they don't sense them? Very often their inclination is to try to prove it by making what Shepherd calls "irregular calls of the senses". If you want to make sure that something is continuing to exist, or is continuing to exist in (at least roughly) the same way when you aren't looking, one natural way to do it is to look at irregular intervals to see if you can catch it out, surprise it, so to speak, show that it's not ready to appear. This is actually quite important. If Hume's account of our belief in continuous existence is right, for instance, this method of "irregular calls" proves nothing; the readiness of things to appear is simply left a mystery and Hume's account of causation makes causal inference weak enough that we can't really prove anything about the underlying causes. But in Shepherd's account this is entirely good reasoning. She recognizes, of course, that it's still a bit loose; we can make our reasoning much more sophisticated and systematic. But the kind of reasoning is good reasoning, and more sophisticated and systematic versions of this simple, ordinary inference don't make it good -- they just make it better.
(2) On its own this actually doesn't get us much. It tells us that there's a lot that corresponds to our sensations and there continues to be a lot corresponding to our sensations even when we don't have those sensations. But we don't have a full account of the external world here: continuity is not enough. It's still possible, for instance, that all the continuing causes of our sensations are really in our mind; it's still possible that they are in the mind of someone else, like God; it's still possible that they are all really caused by us. Many possibilities still exist because we don't know yet whether the world is wholly internal to our minds or at least partly external to them or whether it is dependent on us or independent of us. Likewise, we don't know its full nature, beyond the fact that it somehow corresponds to sensation and keeps existing. We only have one step here.
But we do have a step, and what is significant is that, if Shepherd is right, we know with absolute certainty that there is a world, something that continually exists so that we have the sensations that we do. Getting more than this requires answering the other questions.