Metaphysics takes its name from the work of Aristotle, the famous Greek philosopher, and literally means "after the physics"; legend has it that the Alexandrian librarians christened the writings thus because they followed his Physical Treaties. Since then metaphysics has come to be split into two sub-fields:
Ontology is the study of existence. It asks what there is, what it means to exist and what kind of things there are. These questions are important because we have all (probably) experienced occasions when we were sure something was there but found out later that it was just a dream, a trick of the eye or an over-active imagination. In short, things aren't always what they seem to be, so it makes sense to ask what reality really is.
Cosmology is the study of the nature of the universe (or cosmos, as the name suggests). It asks questions about what is possible, such as time travel and parallel or alternate universes. These ideas are of interest because we may be able to find some reasons why a suggestion from science fiction will or won't work before we spend our time studying the practicalities of flux capacitors.
Some historical perspectives:
There has been a good deal of controversy over the years with regard to metaphysics and its importance. When Aristotle was talking of reality he said that if there were no other thing making up our universe beyond what we often call the natural then natural science would be able to study it and hence be what he called the "first philosophy", or "first science". On the other hand, if there is something else, above and beyond nature, then its study must come before natural science.
What kinds of things could we mean by this "something else"? An obvious answer is of course God, but other philosophers have proposed ideas other than theological ones. Dualists, for example, suspect that the universe may be composed of two substances: mind and matter (and hence dual). Descartes was a famous philosopher who made this claim. Monists, conversely, suggested that in fact there is only one substance making up things (and hence mono). Berkeley, for instance, insisted that there were only ideas, whereas materialists consider that everything is composed of matter in some way. In both cases, the two substances of the Dualists are reduced to one.
Another issue to look at is the existence of things like numbers or symphonies. Is mathematics, say, invented or discovered? We sometimes want to say the latter because it seems that mathematics just had to be the way it is, but if that's the case then did it already exist as it is? We can also consider, as Popper did, the difference between the different things we might want to say exist: a symphony, for example, can only come about if it was written, stored in some form and played by an orchestra; nevertheless, it doesn't seem accurate to say that it is the score and the musicians. What can we say about the existence of such things and what can we remark about the way they are?
Once we've thought about what exists we can move on to asking questions about it. Can we separate substance from property? In other words, can we distinguish between that something is and what we can say about it? If so, what properties does matter have? By posing this problem long before our modern scientific approach could help them, some of the Greek philosophers were able to come up with the early atomic theory; Lucretius suggested that atoms made up our universe and proceeded to wonder what this meant for further investigation and whether he could explain what was already seen using his idea. Berkeley, for his part, tried to explain how it was that some things appeared to be solidly real if they were ultimately only composed of ideas in some way.
Other questions arose from Dualist ideas: if mind and matter are separate, how does the one influence the other? If there is really only one substance, how is it that our minds appear often to be distinct from our bodies? Perhaps there are different ways of thinking about our minds and our bodies that employ different concepts and hence lead to a confusion?
Another concept analysed by metaphysics was causality, the idea that every event has a cause. Is it possible to justify this except to say that so far we know of no exception? Aristotle looked into the problem and it is still considered today.
Some philosophers were hostile to metaphysics. Hume famously declared:
"If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames; for it can contain nothing but sophistry and illusion."
To be fair to him, though, we should note that he had his own ideas about how we might decide what was worth considering and what wasn't, and some of the writing around in his day was obtuse enough to make his remark quite accurate.
The idea that only information gained from mathematics or the sciences was of any value was taken up by the group of philosophers centred in the Austrian capital and called the Vienna Circle. Some of their ideas would become known as positivism and suggested that metaphysical speculation would, along with theology and superstition, be replaced by positive scientific understanding. The rejection was especially strong in Comte and the early Carnap, who could find no use for it at all except to hold back the advance of science. Wittgenstein, too, had little time for metaphysics throughout the better part of his writings
Kant also argued that metaphysics could not help us, but for different reasons. He used the term to mean our attempts to study what lay beyond the natural or apparent world and suggested that even though this realm may exist, we can say nothing about what transcends our abilities.
In recent times the rise of scientism - the idea that only science can tell us anything about the universe - has led to a decline in metaphysics in some opinions and the use of the word in a negative sense; "more metaphysical clap-trap, Hugo" is a popular refrain in my locale, for example.
Why study metaphysics?
If metaphysics is out of fashion for many people, why should we waste our time looking into it? Well, some philosophers think that metaphysical choices or assumptions come before anything else we do. Suppose, for instance, we agree that the natural world is "all there is" and proceed to use science to find things out. If science is subsequently successful, is it because there really is only the natural world, or for some other reason? Can we justify our assumption in the first place or only later on by pointing to its success?
Many of the interesting areas in modern philosophy that we'll come on to later in the series are very much concerned with metaphysics, like the problem of realism or the philosophy of mathematics and mind. Generally speaking metaphysics is involved in questions where many avenues of philosophy meet, such as talking about truth or cognitive abilities. Before we even ask these things, though, are we not presuming that other people exist to answer them? If so, we are using ideas about what there is and what it's like - metaphysics.