## Foundational crisis

The **foundational crisis of mathematics** arose at the end of the 19th century and the beginning of the 20th century with the discovery of several paradoxes or counter-intuitive results.

The first one was the proof that the parallel postulate cannot be proved. This results from a construction of a Non-Euclidean geometry inside Euclidean geometry, whose inconsistency would imply the inconsistency of Euclidean geometry. A well known paradox is Russell’s paradox, which shows that the phrase “the set of all sets that do not contain themselves” is self-contradictory. Other philosophical problems were the proof of the existence of mathematical objects that cannot be computed or explicitly described, and the proof of the existence of theorems of arithmetic that cannot be proved with Peano arithmetic.

Several schools of philosophy of mathematics were challenged with these problems in the 20th century, and are described below.

These problems were also studied by mathematicians, and this led to establish mathematical logic as a new area of mathematics, consisting of providing mathematical definitions to logics (sets of inference rules), mathematical and logical theories, theorems, and proofs, and of using mathematical methods to prove theorems about these concepts.

This led to unexpected results, such as Gödel’s incompleteness theorems, which, roughly speaking, assert that, if a theory contains the standard arithmetic, it cannot be used to prove that it itself is not self-contradictory; and, if it is not self-contradictory, there are theorems that cannot be proved inside the theory, but are nevertheless true in some technical sense.

Zermelo–Fraenkel set theory with the axiom of choice (ZFC) is a logical theory established by Ernst Zermelo and Abraham Fraenkel. It became the standard foundation of modern mathematics, and, unless the contrary is explicitly specified, it is used in all modern mathematical texts, generally implicitly.

Simultaneously, the axiomatic method became a de facto standard: the proof of a theorem must result from explicit axioms and previously proved theorems by the application of clearly defined inference rules. The axioms need not correspond to some reality. Nevertheless, it is an open philosophical problem to explain why the axiom systems that lead to rich and useful theories are those resulting from abstraction from the physical reality or other mathematical theory.

In summary, the foundational crisis is essentially resolved, and this opens new philosophical problems. In particular, it cannot be proved that the new foundation (ZFC) is not self-contradictory. It is a general consensus that, if this would happen, the problem could be solved by a mild modification of ZFC.

https://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis

Personal Note:

This is why Logic if vital to everything. Logic would have “purified” the Math and cleaned it to correct for error. Logic is mandatory prior to beginning Mathematics.