Einstein's view on how apriori knowledge is possible

This article is a 250-word summary of Einstein's address to the Prussian Academy of Sciences titled 'Geometry and Experience' that I wrote several months back in college. There are three ways (or maybe more) to look at what the article does. It addresses the conflict between Euclidean Geometry and the discovery that the structure of space is not euclidean. Secondly, it addresses the question : How is math so certain and absolute, while the sciences are constantly finding faults in their theories and correcting them. Is math always right ? How is the nature of math and science different ? Thirdly, it summarises Einstein's view on the apriori-posteriori analytic-synthetic distinction, where he rules out the apriori-synthetic possibility. All of these are similar if not identical problems. Rather than rewriting a more-or-less same argument at length in a post, I am posting the paper itself.

This paper summarizes Einstein’s argument for how a priori knowledge is possible.

Einstein argues against Kant’s view of synthetic a priori knowledge of the world. He argues for two separate possibilities : analytic a priori, and synthetic posteriori knowledge.

Einstein’s analytic apriori view talks about meaningless symbols manipulated according to a certain set of rules. The axiomatic system by which we do mathematics, Einstein says, assumes the axioms to be vacuously true. These axioms don’t speak anything about the structure of reality, but are like the pieces in a chess game manipulated by certain formal rules. Because all rules of this formal system are rigidly and certainly laid out, there is no possibility of uncertainty as in the empirical world, just like there is no uncertainty about possible states of a chess game.

Einstein’s synthetic posteriori view explains how these vacuously true analytic statements come to model reality. The axioms of the formal system are picked from the empirical world through induction. Only after a certain time, we detach these axioms from the empirical world and give them a vacuous truth value. Thereafter, even if the world changes, the axioms continue to be true in a vacuous sense. For example, Euclidean geometry was proven to not be universally applicable in Physics, but Euclidean geometry in itself remains ‘true’ in the vacuous sense, because it falls upon Physics to decide which formal system to use. Therefore, we simply replace one formal system with another, or make necessary changes to our prior system for it to model reality. For some mysterious reason, these changes are rare, therefore formal systems like math predict the world quite accurately.

Einstein takes a formalist view of analytic a priori knowledge. These formal statements aren’t ‘true’ or ‘false’, merely meaningless, and something outside the formal system attributes meaning to the statements, a meaning that could be changed because it isn’t dictated by the rules of the system.