In 1931, at just 25 years old, Austrian logician Kurt Gödel shattered the foundations of mathematics. In 300 BCE, the philosopher Euclid developed proofs, which are logical arguments mathematicians use to justify how a statement about numbers is true. Axioms, which are irrefutable, are basic statements that are used to form the foundations of these theorems.
If you can overcome the war flashbacks from algebra class, we can ground these concepts with a simple example. Let’s start by proving that even integer squared is also even. This proof requires a few basic axioms, such as the closure of integers (adding or multiplying integers gives you an integer), associative multiplication ((a*b)*c = a*(b*c)), and the definition of even (an integer n is even if and only if n = 2k, where k is some other integer). By establishing what an even integer is, squaring both sides (n2 =(2k)2) will result in an integer value, therefore n2 must be even. Now, we have established that an even integer square gives you an even result with the help of a simple proof.
The basis of mathematics rested on the fact that axioms were indisputable and that they could construct complex, mathematical truths. Before, mathematical statements could only describe numbers, but in 1931, Gödel formulated logical statements into numerical codes. He assigned statements about axioms to unique numbers he invented in order to easily manipulate the logic contained in the axioms to learn more about what the system represented. Gödel’s numbers can be arranged in a unique way to form the first self–referential statement, describing, in itself, an intrinsic truth.
A profound paradoxical statement inspired Gödel’s breakthrough conclusion: “this statement is false.” Because the theorem puts forth self–referential statements, if the statement were false, it must be true. However, if the statement is true, it directly contradicts itself. By applying self–referential logic, Gödel introduced a new criterion of mathematical statements — provable and unprovable given a set of fixed axioms. He concluded that the only resolution is that the statement cannot be proved. His most revolutionary idea is that even the “most true” mathematical statement asserts its own unprovability and that every axiomatic system has its own unprovable true statement. In short, no complete, perfect, mathematical system can exist.
Mathematicians reacted to the newly uncovered gaping hole at the heart of their field with mixed emotions. The repercussions of Gödel’s discovery were felt across disciplines, from physics to computer science. Alan Turing applied Gödel’s principle to his “Turing machine” — an idealized computer that is supposed to read and write bit by bit. He showed that an algorithm on the computer cannot determine if the calculations can be completed in a finite amount of time. This applies to material science as well. The “spectral gap” between energy levels of electrons determines a metal’s basic properties. Lowering materials to extremely low temperatures reduces the gap, creating superconductors. By creating a model with an infinite 2D crystal lattice of metal atoms, researchers have found that the quantum states of the atoms in the lattice embody a Turing machine. Therefore, it is impossible to know if the computation ends.
“In short, no complete, perfect, mathematical system can exist.”
Although Gödel’s breakthrough was used to make sense of physical phenomena in other fields, it also crushed the dreams of generations of mathematicians hoping to discover the “ultimate truth.” Gödel’s shocking theorems unveiled that mathematics is paradoxical and imperfect.