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). Mathematical anti-realism 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. Counterarguments 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