Are mathematical proofs reliable?

Are mathematical proofs reliable?

Mathematical proofs are often regarded as the epitome of rigor and precision. Are mathematical proofs as solid as we are led to believe?

Early mathematical proofs are indeed the epitome of beauty and rigor. But more recent proofs often drag on hundreds of pages, or build on hundreds of earlier papers. Few have the patience or knowledge to go over all the relevant material to verify the correctness of the whole procedure.

Occasionally, someone does find a mistake somewhere. Zhang Yitang, while a student, did find a bug in his supervisor’s paper. He couldn’t find a job after graduation.

A paper, when new, may face some scrutiny. If the paper and its author gain prominence, fewer and fewer people will risk time, effort, reputation, and their careers to examine the potential problems in the paper. 

Let me use an example from economics literature, which I am more familiar with. In the initial Arrow Debreu paper, they acknowledged one of their assumptions was “clearly unrealistic”. As they became more and more prominent, fewer and fewer people mention the theory is built on unrealistic assumptions. Later versions of general equilibrium models, such as the one getting Nobel prize in 2025 “for the theory of sustained growth through creative destruction”, rarely concede their models were built on “clearly unrealistic” assumptions. Since general equilibrium theory is not consistent with physical laws, it has to be built on unrealistic assumptions. As the foundation of economic theory is built on equilibrium theory, few can survive questioning the general equilibrium theory, no matter how absurd it is. 

Is mathematics any better? Human nature is universal, for economists and mathematicians.