In analytic philosophy, anti-realism is an epistemological position first articulated by British philosopher Michael Dummett which encompasses many varieties such as metaphysical, mathematical, semantic, scientific, moral and epistemic. The term was coined as an argument against a form of realism Dummett saw as 'colorless reductionism'.
In anti-realism, the truth of a statement rests on its demonstrability through internal logic mechanisms, such as the context principle or intuitionistic logic, in direct opposition to the realist notion that the truth of a statement rests on its correspondence to an external, independent reality. In anti-realism, this external reality is hypothetical and is not assumed.
Because it encompasses statements containing abstract ideal objects (i.e. mathematical objects), anti-realism may apply to a wide range of philosophic topics, from material objects to the theoretical entities of science, mathematical statement, mental states, events and processes, the past and the future.
See also: Philosophical skepticism
One kind of metaphysical anti-realism maintains a skepticism about the physical world, arguing either: 1) that nothing exists outside the mind, or 2) that we would have no access to a mind-independent reality, even if it exists. The latter case often takes the form of a denial of the idea that we can have 'unconceptualised' experiences (see Myth of the Given). Conversely, most realists (specifically, indirect realists) hold that perceptions or sense data are caused by mind-independent objects. But this introduces the possibility of another kind of skepticism: since our understanding of causality is that the same effect can be produced by multiple causes, there is a lack of determinacy about what one is really perceiving, as in the brain in a vat scenario. The main alternative to this sort of metaphysical anti-realism is metaphysical realism.
On a more abstract level, model-theoretic anti-realist arguments hold that a given set of symbols in a theory can be mapped onto any number of sets of real-world objects—each set being a "model" of the theory—provided the relationship between the objects is the same (compare with symbol grounding.)
In ancient Greek philosophy, nominalist (anti-realist) doctrines about universals were proposed by the Stoics, especially Chrysippus. In early modern philosophy, conceptualist anti-realist doctrines about universals were proposed by thinkers like René Descartes, John Locke, Baruch Spinoza, Gottfried Wilhelm Leibniz, George Berkeley, and David Hume. In late modern philosophy, anti-realist doctrines about knowledge were proposed by the German idealist Georg Wilhelm Friedrich Hegel. Hegel was a proponent of what is now called inferentialism: he believed that the ground for the axioms and the foundation for the validity of the inferences are the right consequences and that the axioms do not explain the consequence. Kant and Hegel held conceptualist views about universals. In contemporary philosophy, anti-realism was revived in the form of empirio-criticism, logical positivism, semantic anti-realism and scientific instrumentalism (see below).
Main article: Mathematical anti-realism
See also: Post rem structuralism
In the philosophy of mathematics, realism is the claim that mathematical entities such as 'number' have an observer-independent existence. Empiricism, which associates numbers with concrete physical objects, and Platonism, in which numbers are abstract, non-physical entities, are the preeminent forms of mathematical realism.
The "epistemic argument" against Platonism has been made by Paul Benacerraf and Hartry Field. Platonism posits that mathematical objects are abstract entities. By general agreement, abstract entities cannot interact causally with physical entities ("the truth-values of our mathematical assertions depend on facts involving platonic entities that reside in a realm outside of space-time") Whilst our knowledge of physical objects is based on our ability to perceive them, and therefore to causally interact with them, there is no parallel account of how mathematicians come to have knowledge of abstract objects.
Field developed his views into fictionalism. Benacerraf also developed the philosophy of mathematical structuralism, according to which there are no mathematical objects. Nonetheless, some versions of structuralism are compatible with some versions of realism.
Anti-realist arguments hinge on the idea that a satisfactory, naturalistic account of thought processes can be given for mathematical reasoning. One line of defense is to maintain that this is false, so that mathematical reasoning uses some special intuition that involves contact with the Platonic realm, as in the argument given by Sir Roger Penrose.
Another line of defense is to maintain that abstract objects are relevant to mathematical reasoning in a way that is non causal, and not analogous to perception. This argument is developed by Jerrold Katz in his 2000 book Realistic Rationalism. In this book, he put forward a position called realistic rationalism, which combines metaphysical realism and rationalism.
A more radical defense is to deny the separation of physical world and the platonic world, i.e. the mathematical universe hypothesis (a variety of mathematicism). In that case, a mathematician's knowledge of mathematics is one mathematical object making contact with another.
Main article: Semantic anti-realism (epistemology)
The term "anti-realism" was introduced by Michael Dummett in his 1982 paper "Realism" in order to re-examine a number of classical philosophical disputes, involving such doctrines as nominalism, Platonic realism, idealism and phenomenalism. The novelty of Dummett's approach consisted in portraying these disputes as analogous to the dispute between intuitionism and Platonism in the philosophy of mathematics.
According to intuitionists (anti-realists with respect to mathematical objects), the truth of a mathematical statement consists in our ability to prove it. According to Platonic realists, the truth of a statement is proven in its correspondence to objective reality. Thus, intuitionists are ready to accept a statement of the form "P or Q" as true only if we can prove P or if we can prove Q. In particular, we cannot in general claim that "P or not P" is true (the law of excluded middle), since in some cases we may not be able to prove the statement "P" nor prove the statement "not P". Similarly, intuitionists object to the existence property for classical logic, where one can prove , without being able to produce any term of which holds.
Dummett argues that this notion of truth lies at the bottom of various classical forms of anti-realism, and uses it to re-interpret phenomenalism, claiming that it need not take the form of reductionism.
Dummett's writings on anti-realism draw heavily on the later writings of Ludwig Wittgenstein, concerning meaning and rule following, and can be seen as an attempt to integrate central ideas from the Philosophical Investigations into the constructive tradition of analytic philosophy deriving from Gottlob Frege.
In philosophy of science, anti-realism applies chiefly to claims about the non-reality of "unobservable" entities such as electrons or genes, which are not detectable with human senses.
One prominent variety of scientific anti-realism is instrumentalism, which takes a purely agnostic view towards the existence of unobservable entities, in which the unobservable entity X serves as an instrument to aid in the success of theory Y and does not require proof for the existence or non-existence of X.
In the philosophy of ethics, moral anti-realism (or moral irrealism) is a meta-ethical doctrine that there are no objective moral values or normative facts. It is usually defined in opposition to moral realism, which holds that there are objective moral values, which any moral claim are either true or false. Specifically the moral anti-realist is committed to denying one of the following three statements: 
Different version of moral anti-realism deny different statements: specifically non-cognitivism denies the first claim, arguing that moral statements have no meaning or truth content, error theory denies the second claim, arguing that all moral statements are false,  and ethical subjectivism denies the third claim, arguing that the truth of moral statements is mind dependent.
Examples of anti-realist moral theories might be:
There is a debate as to whether Moral relativism is actually an anti-realist position because, while many versions deny the metaphysical thesis, some do not, as one could imagine a system of morality which requires you to obey the written laws in your country.  Such a system would be a version of moral relativism, as different individuals would be required to follow different laws, but the moral facts are physical facts about the world, not mental facts, so they are metaphysically ordinary. According to Richard Joyce in the Stanford Encyclopedia of Philosophy "Moral relativism is sometimes thought of as a version of anti-realism, but (short of stipulating usage) there is no basis for this classification; it is better to say that some versions of relativism may be anti-realist and others may be realist." 
See also: Metaepistemology
Much like moral anti-realism, epistemic anti-realism asserts the nonexistence of facts about certain normative facts, but in the domain of epistemology rather than that in ethics. Thus, the two are now sometimes grouped together as "metanormative anti-realism". Prominent defenders of epistemic anti-realism include Hartry Field, Simon Blackburn, Matthew Chrisman, and Allan Gibbard, among others.
No single description is likely to capture all realist views, but a reasonably accurate rule is to understand moral realism as the conjunction of three theses: The semantic thesis: The primary semantic role of moral predicates (such as "right" and "wrong") is to refer to moral properties (such as rightness and wrongness), so that moral statements (such as "honesty is good" and "slavery is unjust") purport to represent moral facts, and express propositions that are true or false (or approximately true, largely false and so on). The alethic thesis: Some moral propositions are in fact true. The metaphysical thesis: Moral propositions are true when actions and other objects of moral assessment have the relevant moral properties (so that the relevant moral facts obtain), where these facts and properties are robust: their metaphysical status, whatever it is, is not relevantly different from that of (certain types of ordinary non-moral facts and properties).
This one is used to designate that family of ethical positions in which it is supposed that moral judgements do not possess truth-value and hence can not be known. An example of a non-cognitivist position is emotivism; that is, the claim that moral judgements are merely expressions of emotion.
The moral error theorist thinks that although our moral judgments aim at the truth, they systematically fail to secure it. The moral error theorist stands to morality as the atheist stands to religion.
A subjectivist ethical theorist is a theory according to which moral judgements about men or their actions are judgements about the way people react to these men and actions - that is, the way they think or feel about them.
In all cases, it may be that what determines the difference in the relevant contexts is something “mind-dependent”—in which case it would be anti-realist relativism—but it need not be; perhaps what determines the relevant difference is an entirely mind-independent affair, making for an objectivist (and potentially realist) relativism.
Moral relativism is sometimes thought of as a version of anti-realism, but (short of stipulating usage) there is no basis for this classification; it is better to say that some versions of relativism may be anti-realist and others may be realist.