Gains:
- Ability to use artificial intelligence to find the idea and method of proof (direct, contradiction, inductive, contrapositive) and self-check the validity of each logical step
- Ability to identify evidentiary gaps, implicit assumptions and unjustified leaps behind expressions such as 'clearly', 'without prejudice to generality'
- Ability to distinguish between fluency and validity by looking for counterexamples before relying on proof without being sure of the truth of a claim.
Mathematical proof is the precise derivation of a claim in logical steps from accepted axioms and previously proven theorems. Proof is the most rigorous product of mathematics: a single invalid logical transition, an omission or implicit assumption that we call a "gap", refutes the entire proof. Artificial intelligence is very skilled at producing convincing-looking text for proof — and that's exactly why it's dangerous. A text that seems convincing is not a valid proof. In this unit you will learn how to use AI as a proof drafting partner and how to inspect each logical step.
First two definitions. A proof sketch is a summary that gives the main idea and skeleton of a proof, but does not fill in every detail. A proof gap is a leap in which the proof says "here it follows" but does not actually justify it. The biggest risk when working with AI is the gaps covered by persuasive sentences: the text is fluid, full of conjunctions like “therefore” and “obviously,” but with leaps in between that are not actually proven.
Strengths and weaknesses of AI in proof
AI does two things well in proof: (1) evoke the standard idea of proof of a known theorem, (2) suggest what method (induction, contradiction, direct, contrapositive) might be appropriate for a proof. Its weakness is this: ensuring that each step of an original or subtle proof is actually valid. AI can produce “fallacious proofs” that appear true but are actually false — for example, it can skip the basic case in an induction step, or it can say “without breaking generality” but make an assumption that actually breaks generality.
So the golden rule in proof: use AI to find and outline the idea of the proof; Check the validity of each logical step yourself. Before "accepting" a proof, make sure that each "therefore" is actually valid.
Step by step: checking a proof
1. Clarify the claim and assumptions. What is being proven? Under what assumptions? If these are vague, the proof is also vague.
2. Know the proof method. Directly, by contradiction, inductively, contrapositively? Know the structural requirements of the method (e.g., in induction, base case + induction step is essential).
3. Question every “therefore”. At each logical transition, "does this really follow from the previous steps?" ask. The most insidious gaps hide behind the expressions "obviously", "it is easily seen", "without losing generality".
4. Look for implicit assumptions. Does the proof rely on an unspoken assumption? For example, it may be silently accepted that a number is positive or a function is continuous.
5. Try a counterexample. If the claim is false, a counterexample demolishes it. Before accepting the proof, test that the claim is actually true in simple special cases.
6. Consult a procurement authority. Compare the standard proof for known theorems with a reliable source (textbook, peer-reviewed source).
Hint: The phrase "without loss of generality" in the proof is a double-edged sword. Sometimes it is actually valid (if there is symmetry), sometimes it is a hidden error. AI uses this expression a lot. Justify yourself each time that "generality is not really broken"; Don't take the AI's word for it.
Proof methods and pitfalls
proof method
Structure
The most common AI trap
direct
Assumption → ... → Conclusion
skipping a step in between
contradiction
Assume the opposite → find contradiction
The contradiction is not real
induction
Base case + step
Forgetting the basic situation
contrapositive
¬Conclusion → ¬Assumption
false negation
Counterexample (rebuttal)
single counterexample
Counterexample is invalid
three mini cases
Case 1 — Incomplete base case. A teacher had the AI prove the formula "1 + 2 + ... + n = n(n+1)/2" by induction. The AI wrote the induction step correctly but never checked the base case (n=1). The teacher asks "where is the base case?" he asked; AI added. Without the ground state, induction is invalid; A 30-second check saved the proof.
Case 2 — Secret division by zero. One student saw a ridiculous “proof” like “a = b for every a, b” and asked the AI “where is the mistake here?” he asked. YZ correctly showed that the proof divides by (a − b) in one step, and under the assumption a = b, this is division by zero. Here the AI was successful as an auditor; but the student still verified this step with his own hand.
Case 3 — Convincing false evidence. An engineering student had an AI prove an inequality. The text was fluent and convincing, but when taking square roots in one step, it ignored the possibility of both positive and negative roots and took only the positive. The student found this gap when he questioned every step. The proof became valid when an additional condition (positivity of the variables) was added.
Four copyable templates
1) Requesting a proof draft (idea):
Which METHOD would be appropriate to prove the following claim (direct, contradiction, inductive, contrapositive)? Just give the MAIN IDEA and the skeleton of the proof, do not write the full proof. Claim: [here]
2) Step by step, reasoned proof:
Prove the following claim with [method]: [claim]. Write down which axiom/theorem/definition you rely on for each step. DO NOT use expressions such as "clearly" or "easily"; Fully justify each transition. If induction, show the base case and the induction step separately.
3) Proof loophole hunt:
Check out the proof below. JUST look for logical gaps, implicit assumptions, and unjustified leaps. Check whether each "therefore" actually follows from the previous steps. Write down each gap you find with which step it is in. Proof: [here]
4) Search for counterexample:
I want to test whether the following claim is TRUE: [claim].Test it in simple special cases first; try to find a COUNTEREXAMPLE. If you find a counterexample, show it; If you can't find it, list the situations you tried (but this is not proof, just looking for evidence).
Weak prompt / Strong prompt
Weak: "Prove that √2 is irrational."
Result: The standard proof comes, but a step (e.g. "then p is even") may have been skipped without justification and you won't notice.
Strong: "Prove BY CONTRADICTION that √2 is irrational. Write down which assumption you used at each step; also justify intermediate claims such as 'If p² is even, then p is even'. Finally, show clearly where exactly the contradiction arises."
Result: Every intermediate claim is justified, the source of the contradiction is clear, no gaps are left.
Common mistakes
- Confusing fluency with validity. A persuasive text is not a valid proof; Every step must be supervised.
- Skipping the ground state in induction. AI often forgets the base case; The induction step alone is not enough.
- To accept "without losing generality" without question. This statement may be a latent error; Justify it every time.
- Not seeing implicit assumptions. Assumptions such as positivity, continuity, nonzero, etc. can silently leak into the proof.
- Trusting the proof without trying a counterexample. If the claim is false, the proof is also false; Test the truth of the claim in simple cases first.
Caution: AI can produce “proof” even for a claim that is actually false — because it produces text, it does not guarantee logical validity. If you are unsure of the accuracy of a claim, look for a counterexample first. The "proof" of a false claim necessarily contains a loophole; Your job is to find that gap.
In summary
Proof is the most rigorous product of mathematics, and AI can produce convincing but invalid “proofs.” Use AI to find the proof idea and method; Check the validity of each logical step yourself. Look for key cases, implicit assumptions, and loopholes behind phrases like “clearly” and “without prejudice.” If you are unsure of the truth of a claim, try a counterexample before trusting the proof. Fluency is not validity.
Application task
Choose a standard theorem (e.g. "the sum of two even numbers is even" or "√2 is irrational"). Have the AI prove it step by step with the 2nd template. Then give the same proof as the 3rd template again for the gap hunt — let him check his own proof. Then manually query each "therefore": is there a base case, is there an implicit assumption, is each transition justified? Find and note at least one potential gap or improvement point.
checklist
- [ ] I clarified the claim and assumptions.
- [ ] I got to know the proof method and its structural requirements.
- [ ] I verified that each "therefore" follows from the previous steps.
- [ ] I did a base case / implicit assumption check.
- [ ] I tested the claim in simple cases and looked for counterexamples.
- [ ] I compared the standard proof for known theorems with the reliable source.