Formal vs Informal Proofs: What Builds Trust?

Formal proofs are fully machine-checkable derivations from axioms, while informal proofs are human-readable arguments that rely on shared mathematical conventions. A proof is trustworthy when its steps are verifiable by an independent party; it is useful when it conveys understanding and can be reused. Both forms serve distinct purposes, and the choice between them depends on the audience, the stakes, and the mathematical context.

Background: The Evolution of Mathematical Proof

For most of history, mathematics was communicated through informal proofs. Euclid’s Elements (c. 300 BCE) set a standard for deductive reasoning, but even his arguments relied on geometric intuition and implicit assumptions. The 19th century brought a push for rigor: mathematicians like Cauchy and Weierstrass formalized calculus to eliminate paradoxes, and later Hilbert and Russell attempted to ground all of mathematics in formal systems. In the 20th century, computers enabled the first fully formal proofs, with the Automath project in the 1960s and modern proof assistants like Coq, Isabelle, and Lean. Today, formal proofs are used in software verification, cryptography, and advanced mathematics, while informal proofs remain the norm in research and education.

What Is a Formal Proof?

A formal proof is a sequence of statements in a precise, machine-readable language, each derived from previous ones using a fixed set of inference rules. The key features are:

• Explicit axioms: Every assumption is stated, and nothing is taken for granted.
• Mechanical checking: A computer program (proof assistant) verifies each step, leaving no room for human error or hidden gaps.
• Absolute certainty (within the system): If the axioms are consistent, a formally verified theorem is true in that framework.

For example, the proof of the four-color theorem (first proved with computer assistance in 1976) was later fully formalized in Coq in 2005. The formal proof is thousands of lines long, but every step is checkable.

What Is an Informal Proof?

An informal proof is a narrative argument written for human readers. It uses natural language, diagrams, and standard mathematical notation. It relies on the reader’s understanding of definitions, known theorems, and common proof techniques. Features include:

• Abstraction: It omits tedious details that are considered obvious or routine.
• Intuition: It often explains the idea behind a proof, not just the mechanical steps.
• Flexibility: It can adapt to the audience’s level and the context.

Most published mathematics, from textbooks to research articles, uses informal proofs. They are the primary vehicle for mathematical communication.

Trustworthiness: Why Formal Proofs Are Considered More Reliable

Formal proofs eliminate two major sources of error: gaps in reasoning and mistakes in the application of rules. In an informal proof, a step like “clearly, this function is continuous” might hide a subtle issue that the author overlooked. Formal verification forces every such claim to be justified. This is why formal proofs are used in safety-critical systems, such as verifying the correctness of computer chips or cryptographic protocols. For instance, the seL4 microkernel (a small operating system kernel) was formally verified in Isabelle/HOL, guaranteeing that it behaves exactly as specified. No human review could provide that level of assurance.

Usefulness: Why Informal Proofs Remain Essential

Despite the reliability of formal proofs, informal proofs dominate mathematics. The reasons are practical:

• Readability: A formal proof can be thousands of lines long and almost impossible for a human to follow. An informal proof conveys the key ideas in a few pages.
• Insight: Informal proofs often include the motivation, the strategy, and the connections to other results, which are crucial for learning and for generating new mathematics.
• Speed: Writing a formal proof can take months or years, while an informal proof can be written in days.
• Flexibility: Informal proofs can be modified, generalized, or applied to new contexts without redoing the entire derivation.

For example, Andrew Wiles’s proof of Fermat’s Last Theorem was an informal proof (though extremely detailed). It was accepted by the mathematical community after peer review, even though it was not fully formalized. The proof’s usefulness lay in its explanation of deep connections between number theory and geometry, not just in its conclusion.

How to Choose: When to Use Which

The choice between formal and informal proofs depends on the goal:

• For teaching and learning: Informal proofs are essential. They build intuition and teach problem-solving skills.
• For research mathematics: Informal proofs are the standard, but formal proofs are increasingly used to verify complex or controversial results (e.g., the Kepler conjecture, formalized in 2014).
• For software and hardware verification: Formal proofs are mandatory in high-assurance settings where a bug could be catastrophic.
• For automated reasoning and AI: Formal proofs are necessary because they are machine-readable, enabling automated theorem provers and proof assistants.

In practice, many mathematicians use a hybrid approach: they develop an informal proof, then formalize it in a proof assistant to check its correctness.

Common Problems and Misconceptions

One common misconception is that informal proofs are not rigorous. In reality, a well-written informal proof is rigorous if it provides enough detail for a knowledgeable reader to fill in the gaps. The difference is one of degree, not of kind. Another issue is that formal proofs are often so long that they obscure the underlying ideas. The four-color theorem’s formal proof is a prime example: it is correct, but it does not explain why the theorem is true in a way that aids human understanding. Additionally, formal proofs are only as reliable as the axioms and the proof checker itself. If the axiom system is inconsistent, any statement can be proved, and if the checker has a bug, it might accept invalid proofs (though this is rare).

Costs and Alternatives

The cost of formalization is significant: it requires specialized software, training, and a huge time investment. For instance, formalizing a single theorem like the Pythagorean theorem can take hours; formalizing a major result like the classification of finite simple groups would be a massive project. Alternatives to full formalization include:

• Semi-formal proofs: Using formal notation but not machine-checking every step.
• Computer-assisted proofs: Using software for calculations or case analysis, but not for the entire logical derivation.
• Peer review: The traditional method of relying on expert scrutiny to catch errors.

These alternatives offer a balance between reliability and practicality.

Frequently Asked Questions

Can an informal proof be wrong?

Yes, informal proofs can contain errors, even in published works. Sometimes the errors are minor and easily fixed; other times they are fatal. The history of mathematics includes famous examples of incorrect proofs that were accepted for years before being challenged. Formal proofs are designed to eliminate this risk.

Why don’t mathematicians use formal proofs all the time?

Because formal proofs are extremely time-consuming to write and often unreadable for humans. They are not practical for everyday mathematical work, where the goal is to communicate ideas and build on existing knowledge. Formalization is reserved for situations where absolute certainty is required.

Are all formal proofs computer-generated?

No, a formal proof can be written by hand, but it is so tedious that in practice almost all formal proofs are created and checked with proof assistants. The computer does not generate the proof automatically; it verifies each step the human provides.

Does a formal proof guarantee that a theorem is true in the real world?

No, a formal proof guarantees truth only within the chosen axiom system. If the axioms do not match reality (for example, if they include the axiom of choice, which some mathematicians reject), the theorem may not apply to real-world situations. Also, the proof assumes the consistency of the axiom system, which cannot be proven within the system itself.

Can informal proofs be converted into formal proofs?

Yes, any valid informal proof can, in principle, be formalized. However, the process is not automatic; it requires translating the human reasoning into the formal language and filling in all the gaps. This is a major undertaking, which is why only a small fraction of mathematical results have been formalized.

Which is more important for a proof: trustworthiness or usefulness?

Both are important, but they serve different purposes. Trustworthiness is crucial when the proof is used as a foundation for further work or in safety-critical applications. Usefulness is crucial for human understanding and for advancing mathematical knowledge. A proof that is trustworthy but useless (e.g., a formal proof that no one can read) has limited value, and a proof that is useful but untrustworthy (e.g., an informal proof with a gap) is dangerous.

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