In mathematics, proofs must be conducted rigorously and logically based on axioms, definitions, theorems, or other previously agreed-upon or proven knowledge. However, there are several common misconceptions in mathematical proofs that many people often make. Two of these are the use of empirical arguments and circular reasoning.
An empirical argument in mathematics is a statement that involves observation or specific examples to conclude a theorem or a more general statement. Although these examples may support the hypothesis, such arguments cannot be considered definitive proof since they do not cover all possible cases. In fact, mathematics requires proofs that are general and encompass all possible scenarios.
Some common examples of empirical arguments in everyday life include the following:
- To verify the statement, “All students in the Mathematical Proof and Continuity course come from Java Island,” a researcher asks three students from the class. All three confirm that they are from Java Island. The researcher then immediately concludes that the statement is true. This conclusion would only be valid if the class consisted solely of these three students. If there are other students, the conclusion becomes invalid due to an unlawful generalization.
- A tourist visits City A for five days and observes that the weather is always sunny. The tourist then concludes, “The weather in City A is always sunny.” This conclusion is based on only five days of observation, even though the weather can vary throughout the year.
- A customer buys three electronic products from Store B, and all of them work well. The customer then concludes, “All electronic products sold at Store B are of high quality.” However, without further testing or a larger dataset, this conclusion may be incorrect.
- A teacher observes that three students who frequently sit in the front row always get high scores on exams. The teacher then concludes, “All students who sit in the front row always score high.” However, many other factors influence academic performance.
- Someone observes that two of their vegetarian friends appear healthy and fit. They then conclude, “All vegetarians must be healthy and fit.” This conclusion ignores other factors such as genetics, a complete diet, and overall lifestyle.
- A person moves to a new neighborhood and does not witness any crime for the first two weeks. They then conclude, “This neighborhood must be safe and crime-free.” However, this conclusion is based on a short observation period and may not reflect long-term conditions or unseen incidents.
Empirical arguments are also frequently presented as formal proof in mathematical contexts. These arguments are typically based on observations from specific examples and then generalizing a universal truth from them. Below are some examples of empirical arguments in mathematical proofs:
- A student verifies that several odd numbers less than 8 are prime numbers: 3, 5, and 7. They then conclude that all odd numbers are prime. However, further observation disproves this conclusion—9 and 15 are odd numbers but are not prime.
- A student observes a pattern in a series of numbers. They compute that $2^2 + 1 = 5$, $4^2 + 1 = 17$, and $6^2 + 1 = 37.$ These results are all prime numbers. The student then concludes that $(2k)^2 + 1 = 4k^2 + 1$ is always prime for every positive integer $k$. However, further observation disproves this claim—for instance, $8^2 + 1 = 65$ is not a prime number because $5$ divides $65.$
- A student is asked to prove that the sum of the interior angles in any triangle equals $180^\circ.$ They draw two types of triangles: a right triangle with angles $90^\circ -45^\circ -45^\circ$ and an equilateral triangle with each angle measuring $60^\circ.$ The student then argues that since the sum of the three angles in both cases is $180^\circ,$ the statement must be true. However, since they only used two specific types of triangles, their conclusion cannot be generalized to all triangles.
Another common misconception is circular reasoning, also known in Latin as circulus in probando. Circular reasoning is a logical fallacy that occurs when someone assumes the statement they are trying to prove as part of the proof itself. As a flawed form of argumentation, circular reasoning is an error where an argument is made by assuming that what needs to be proven is already true.
People who believe the assumptions they use in circular reasoning often speak with confidence. This confidence can make their arguments sound more convincing to others, even though they are not logically valid. Moreover, if the person presenting the circular reasoning has authority or credibility in a particular field, others may be more inclined to accept the argument without questioning its logic. For instance, a mathematics professor or a government official may be more easily trusted, even if their argument is circular. Additionally, persuasive language and strong rhetoric can obscure logical weaknesses in an argument. If the presenter uses technical terms or complex language, others may feel overwhelmed and unable to challenge the argument, ultimately failing to recognize the logical flaw.
The general form of circular reasoning is as follows:
- $X$ is true because $Y.$
- $Y$ is true because $X.$
Examples of circular reasoning in everyday life include:
- “Killing is wrong because killing is immoral.” This statement assumes that immoral acts are wrong without providing an independent reason why killing is wrong.
- “This policy is the best because it was designed by experts who know what is best.” This argument assumes that experts are always right without providing additional evidence for why the policy is the best.
- “This product is very popular because many people buy it.” Then, why do many people buy it? “Because the product is popular.” This argument assumes that the product’s popularity is the reason people buy it while also using its sales as proof of its popularity.
- “Michael’s house is next to Floryna’s house.” Then, where is Floryna’s house? “Next to Michael’s house.” This argument is often used as a joke by comedians because it sounds amusing.
How about circular reasoning in mathematics? As one of the most common misconceptions, even mathematics teachers sometimes prove mathematical statements using circular reasoning. This happens when they struggle to distinguish between the statements assumed to be true and the statements that need to be proven in a mathematical proof. Below are some examples:
- Given the statement, “The sum of the interior angles of any triangle is $180^\circ,$” someone provides the following proof:
“The sum of the interior angles of any triangle is $180^\circ$ because a triangle has three angles, each measuring $180^\circ \div 3 = 60^\circ.$ Therefore, we get $60^\circ + 60^\circ + 60^\circ = 180^\circ$.” - Given the statement, “The square root of a positive number is always positive,” someone provides the following proof:
“Since the square root of a number is a positive number that, when squared, results in a positive number, the square root of a positive number must always be positive.” - Given the statement, “If $f: A \to B$ and $g: B \to C$ are surjective functions, then the composition $g \circ f$ is also surjective,” someone provides the following proof:
“Suppose $g \circ f$ is surjective. This means that for every $c \in C$, there exists some $a \in A$ such that $g(f(a)) = c$. Thus, $g \circ f$ is surjective.”
Now, try to come up with at least three examples of circular reasoning in mathematics. Discuss with your peers if needed. The goal is to help you recognize that these examples involve circular reasoning and to increase your awareness of the importance of logical and valid arguments in mathematical proofs.
When writing a proof for a mathematical statement, you are essentially reporting your findings. Once you have found the correct method to prove a statement, you must communicate it to others so they understand it. This report does not include the process of discovering the method, such as ideas and trial-and-error, but rather focuses on the final result of your findings. Such a proof is known as a formal proof, which is widely regarded as valid and acceptable by mathematicians.