Which of the following constitutes a mathematical proof that all multiples of 4 are even?
AChecking that 4, 8, 12, 16, and 20 are all even
BA survey showing 95% of mathematicians believe it
CThe argument: if n = 4k, then n = 2(2k), which is 2 times an integer, hence even
DA computer program that checks the first million multiples of 4
Option C is a deductive proof: it starts from the definition (n = 4k), performs valid algebra (4k = 2·2k), and arrives at the conclusion by definition (2 times an integer is even). It covers all multiples of 4 at once, not just specific ones. Options A and D check examples, which is testing, not proving. Option B is an appeal to authority.
Question 2 True / False
A mathematical theorem that was proven 200 years ago could be overturned by new evidence discovered today.
TTrue
FFalse
Answer: False
Mathematical proof is deductive: if the axioms are accepted and the logical steps are valid, the conclusion follows necessarily and permanently. A proven theorem cannot be 'overturned by evidence' the way a scientific theory can be revised by new data. The only way a proven theorem fails is if an error is found in the proof itself — which is a correction, not new evidence.
Question 3 Short Answer
Explain why Goldbach's Conjecture (every even number greater than 2 is the sum of two primes) is not considered proven, despite being verified for all even numbers up to 4 × 10¹⁸.
Think about your answer, then reveal below.
Model answer: Verification of specific cases, no matter how many, is not a proof. A proof must show the statement follows logically for ALL even numbers greater than 2 — infinitely many cases. The 4 × 10¹⁸ verified cases show the conjecture is very likely true, but it is possible (however unlikely) that some larger number is a counterexample. Only a deductive argument covering all cases would constitute proof.
This highlights the fundamental gap between evidence and proof in mathematics. Scientific disciplines accept strong evidence as provisional truth. Mathematics requires logical certainty. Goldbach's Conjecture is one of the oldest unsolved problems precisely because the gap between 'checked extremely many cases' and 'proven for all cases' is the hardest part of mathematics.