Once more unto the breach, dear friends, once more.
This is a regular topic. What follows is a re-write of stuff from three years ago.
Speaking very roughly, just to get started, realism holds that ...stuff... is independent of what we say about it; anti-realism, that it isn't.
"Stuff", because the content makes a difference.
For instance, if our topic is aesthetics, then aesthetic anti-realism is the view that beauty is in the eyes of the beholder; but an aesthetic realist might hold that beauty and ugly are a part of whatever it is we are beholding.The realist says that something is either beautiful, or it isn't, while the anti-realist perhaps says that being beautiful is an attitude we take towards the item.
A further example. An ethical realist might say good and bad are as much aspects of the world as matter and volume; while an ethical anti-realist might say that no observation of the world will reveal good or bad, because they are not 'out there' to be found.
Stealing blatantly from my Rutledge Encyclopaedia of Philosophy, a realist would generally hold to a set of beliefs that includes: that correspondence to the facts is what makes a statement true; that there may be truths we do not recognise as such, do not believe and do not know; that the Law of excluded middle holds for things in the world; and that the meaning of a sentence may be found by specifying it's truth-conditions.
An ant-realist may in contrast hold that truth is to be understood in sophisticated epistemic terms, perhaps as what a "well-conducted investigation" might lead us to believe; that there can be no unknown truths; that we need include "unknown" as well as true and false in our logical systems; and that the meaning of a sentence is to be found in what it might assert.
Going back to the main point I'd like to make here, one can be a realist in one area and an anti-realistin another. SO for my part, I've argued against typical examples of anti-realism such as pragmatic theory, logical positivism, transcendental idealism and Berkeley's form of idealism. I have however also defended a constructivist view of mathematics, an anti-realist position; and off-handedly rejected realism in ethics and aesthetics.
It is important to note that there is a difference in
logic sitting behind the distinction between realism and anti-realism. Realists supose that a proposition is either true or it is false, and that there are no alternatives. Their attitude towards truth is binary. On the other hand, anti-realists are happy to admit at least a third possibility, that a proposition might be neither true nor false, but have some third value. Anti-realism became more prominent towards the end of last century with the development of formal paraconsistent and many-valued logics.
I think a large part of the difference between realism and anti-realism can be explained by making use of Anscombe's notion of direction of fit. This is the difference between the list you take with you to remind yourself of what you want to buy and the list the register produces listing the things you actually purchased. The intent of the first list is to collect the things listed; of the second, to list the things collected. The first seeks to make the world fit the list, the second, to make the list to fit the world. So perhaps anti-realism applies to ethics and aesthetics because we seek to make the world as we say, while realism applies to ontology and epistemology because we seek to make what we say fit the world.
This by way of background.