When Lakoff claims the mind is "embodied", he is arguing that almost all of human cognition, up through the most abstract reasoning, depends on and makes use of such concrete and "low-level" facilities as the sensorimotor system and the emotions. Therefore embodiment is a rejection not only of dualism vis-a-vis mind and matter, but also of claims that human reason can be basically understood without reference to the underlying "implementation details". (Thus Lakoff would strongly reject a number of formulations of the Strong AI[?] position.)
Lakoff offers three complementary but distinct sorts of arguments in favor of embodiment. First, using evidence from neuroscience and neural network simulations, he argues that certain concepts, such as color and spatial relation concepts (e.g. "red" or "over"), can be almost entirely understood through the examination of how processes of perception or motor control work.
Second, based on cognitive linguistics' analysis of figurative language, he argues that the reasoning we use for such abstract topics as warfare, economics, or morality is somehow rooted in the reasoning we use for such mundane topics as spatial relationships. (See conceptual metaphor.)
Finally, based on research cognitive psychology and some investigations in the philosophy of language, he argues that very few of the categories used by humans are actually of the black and white type amenable to analysis in terms of necessary and sufficient conditions. On the contrary, most categories are supposed to be much more complicated and messy, just like our bodies.
Lakoff argues that the best way to understand what mathematical and philosophical ideas are really about is to consider them in light of the structure of the embodied mind. This has generated some controversy, especially in the philosophy of mathematics. It is as yet unclear whether philosophers not so mathematically inclined are terribly interested in or bothered by Lakoff.
As an example of a controvertial Lakovian idea in this vein is that, when considering the significance of mathematics, we should remain agnostic about whether math is some how wrapped up with the very nature of the universe. "Mathematics may or may not be out there in the world," Lakoff told the AAAS in early 2001, "but there's no way that we scientifically could possibly tell." This claim bothers a number of people, some because they think there really is a way we could "tell", others, presumably, because it implies that mathematics involves a good deal less certainty than one might expect.
Lakoff has publicly expressed both ideas about the conceptual structures that he views as central to understanding the political process and some of his particular political views. He almost always discusses the latter in terms of the former.
Moral Politics gives book-length consideration to the conceptual metaphors that Lakoff sees as present in the minds of American "liberals" and "conservatives". Lakoff makes an attempt to keep his personal views confined to one particular section near the book's close. It is not entirely clear whether this work is more relevant to cognitive science or to political analysis.
Lakoff has distributed some much briefer political analyses via the Internet. One article distributed this way is Metaphor and War: The Metaphor System Used to Justify War in the Gulf, in which Lakoff argues that the particular conceptual metaphors used by the Bush administration to justify American involvement in the Gulf ended up either obscuring reality, or putting a handy conservative spin on the facts. Presumibly it is contributions such as this that have helped endear Lakoff to some who otherwise wouldn't care less about theories of the mind.