The significance of Gödel’s Incompleteness Theorems is that it not only proves the system Principia Mathematica to be incomplete, but any possible formal system that can be constructed. You could revise your system T to include T2, which allows G to be proven, but then T2 would itself be susceptible to its own G2, and so on to infinity. In a sense, this seems entirely reasonable, just as in physics there can be no perpetual motion machines, in logic there can be no complete, self-contained formal axiomatic system, there must a “push” from the outside. In a fell swoop, Gödel had demolished the great project of the 19th century. Mathematics, it seems, cannot account for mathematics



Philosophy, by extension, is concerned with many of these same problems (as many mathematicians were involved in both fields). Where epistemology had been the central focus of Western philosophy since Descartes, logic and meaning became the axis of philosophy in the 20th century. Along with this program came an intense self-awareness and scrutiny of language and other systems of meaning. Essentially, the question of philosophy shifted from “What do we mean?” to “How do we mean?”. Quintessential to this line of philosophy is Ludwig Wittgenstein, a pupil of Bertrand Russell, who also came from a background in mathematics and engineering. Wittgenstein’s project was the analysis of language and meaning, about which he developed two different, conflicting theories throughout his life. First, he developed a “picture” theory of meaning, in which words are seen as depicting objects in the world, with language as a whole creating a picture of reality. He later abandoned this idea for a theory of meaning as use, by which he meant that words themselves are arbitrary, and in order to understand what a word means, one must look at how it is used, and how it fits into the total background of culture. “A language is a way of life”, by which he meant that two speakers of a language are able to communicate because the speaker and the listener both share a similar mapping of words to concepts dictated to them by the culture at large. Every word has meaning by virtue of the meaning of every other word, hence language as a whole is not a picture that stands for reality, but a system of agreed upon symbols that stands in for reality. Hence his famous aphorism “The meaning is the use”



Meanwhile, the world of art was dealing with these problems in its own way. In many respects, the project of modernism in art is analogous to the project of mathematics, in that both are methods of modelling reality that were struggling with the question of how to model themselves. Clement Greenberg, one of the standard chroniclers of modernism, devised a narrative of the progression of modern painting as the incremental flattening of the canvas, of art becoming more aware of itself, shifting its goals from creating an illusion of space as the Renaissance masters did, to exploring painting as an object in its own right, without the need to depict anything. Central to this narrative of modernism is the dialectic. Owing its origins to Plato, and its formalization to Hegel, the dialectic is a method for arriving at knowledge of the world via a series of dialogues, in which one party posits a point, then another posits a counterpoint, addressing the weaknesses of the former, and from the interaction of these two comes a synthesis, which in turn becomes a new point, and the cycle continues. It is by this method that we successively hone in on truth, dispelling our misconceptions and illusions about the world with each pass. This was the modus operandi of modernism in art, each successive school was responding to the statements of the previous, incrementally extricating itself from renaissance illusionism, further abstracting and purifying itself, and becoming the ground for the schools to follow.