How do we know that math is correct?

The correctness of mathematics is established through a combination of logical reasoning, rigorous proof, and empirical verification in certain cases. Here are some key aspects that contribute to the assurance of the correctness of mathematics:

1. Axiomatic Foundations: Mathematics is typically built upon axioms, which are self-evident truths or statements accepted without proof. From these axioms, mathematicians derive theorems and propositions through deductive reasoning.

2. Logical Reasoning: Mathematical proofs are constructed using strict logical reasoning. Each step in a proof must be logically justified and follow from previously established statements or axioms. If a logical flaw is found in a proof, it indicates an error that needs to be corrected.

3. Peer Review: Mathematical work undergoes rigorous peer review by other mathematicians. Before a theorem or proof is widely accepted, it is scrutinized by experts in the field. If a flaw or error is discovered, it can lead to revisions or corrections.

4. Consistency: Mathematics aims to be consistent, meaning that its results and principles do not contradict each other. Any internal contradictions would indicate a problem in the mathematical framework. The pursuit of consistency helps ensure the reliability of mathematical structures.

5. Applicability: In many cases, mathematical concepts are applied to real-world problems. If the mathematical models consistently provide accurate predictions or solutions in practical situations, it lends additional support to the validity of the underlying mathematics.

6. Historical Success: The success of mathematics in explaining natural phenomena, predicting outcomes, and solving practical problems throughout history supports its correctness. The fact that mathematical theories and methods have consistently produced reliable results over time contributes to confidence in their correctness.

7. Formalization and Rigor: Mathematicians often formalize their work using symbolic notation and rigorous definitions. This helps eliminate ambiguity and ensures that concepts are precisely defined, reducing the likelihood of misinterpretation or error.

It's important to note that while mathematics strives for certainty, absolute certainty is elusive. Gödel's incompleteness theorems, for example, demonstrate that in any consistent mathematical system, there are true statements that cannot be proven within that system. Despite these theoretical limitations, mathematics remains a highly reliable and powerful tool for describing and understanding the structure of the world.