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Misrepresents an opponent’s position to make it easier to attack or refute. Example: Person A claims that we should have more math classes. Person B argues that Person A believes students should only study math.

Believing that past random events affect the probability of future random events. Example: She rolled six sixes in a row on the dice, so she is less likely to roll a six next time.

Assumes what’s true for individual parts must also be true for the whole. Example: Every part of this mathematical model is simple, so the entire model must be simple.

Asserts that a proposition is true because many or most people believe it. Example: Most people believe that prime numbers hold mystical significance, so it must be true.

Tries to shift the obligation to prove an argument onto the opponent. Example: Prove that this mathematical theorem is not important for our field of study.

Claims that a proposition is true because it hasn’t been proven false, or vice versa. Example: No one has ever proven that the Riemann Hypothesis is false, so it must be true.

Applies a general rule too broadly without considering possible exceptions. Example: Since mathematical truths are universal, any mathematical solution works under all conditions.

This fallacy occurs when an argument assumes that a specific consequence necessarily implies the condition that leads to it. Example: If it rains, the ground gets wet. The ground is wet, therefore it has rained. (But the ground could be wet for other reasons, like a sprinkler.)

Makes an appeal to purity as a way to dismiss relevant criticisms or flaws of an argument. Example: No good mathematician would ever make such an obvious mistake. Therefore, anyone who makes such a mistake is not a good mathematician.

Drawing an equivalence between two things that are not comparable in the relevant aspects. Example: Saying calculus homework is just as unacceptable as stealing, because they both cause students distress.

This fallacy arises when an argument presents a situation as having only two exclusive options, one of which is often presented as the inevitable conclusion. Example: If we do not prove the conjecture to be true, it must be false.

Introduces an irrelevant topic to divert attention from the original issue. Example: During a debate on number theory, someone insists on discussing unrelated mathematical history to distract from the mathematical argument at hand.

An argument where the conclusion is assumed in the premises. Often involves using a word's or phrase's connotation to prove itself. Example: We must encourage our youth to practice higher-level mathematics to improve math skills because higher math improves problem-solving abilities.

Makes a broad conclusion based on a small or unrepresentative sample. Example: Two students in the class solved the problem easily, so the problem must be easy for everyone.

This fallacy exploits the ambiguity of a word or phrase that has multiple meanings to draw a misleading conclusion. Example: A mathematical ‘function’ has a specific meaning. But in the context of a formal event, arguing that there ‘function’ implies calculations is a fallacy.

Assumes that since one event followed another, the first event must have caused the second. Example: I used a new calculator and got a high score on my math test, so the new calculator must have caused my high score.

A conclusion or statement that does not logically follow from the previous argument or statement. Example: She’s a mathematician, so she will definitely fix your computer.

Relies on the status of an authority figure or institution instead of a logical argument or concrete evidence. Example: The famous mathematician claimed this equation is unsolvable, so it must be true.

This fallacy involves the assumption that rejecting the antecedent of a conditional statement also allows one to reject the consequent. Example: If I am a mathematician, then I am good at numbers. I am not a mathematician, hence I am not good at numbers. (One might not be a mathematician but still good with numbers.)

Argues that a relatively small step will lead to a chain of related events resulting in some significant effect. Example: If we allow calculators in exams, eventually students won't learn math at all.

Assumes what’s true for the whole must also be true for the individual parts. Example: This algorithm is very complex, so every step of the algorithm must also be complex.

This fallacy occurs when the conclusion of an argument is assumed in one of the premises. Example: The theory must be true because the evidence supports the theory, and the evidence is valid because the theory says it is.

Cherry-picking data clusters to suit an argument, or finding patterns to fit a presumption. Example: Highlighting statistical areas where a certain mathematical theory appears true while ignoring the areas where it does not.

Assumes that if two things are similar in some ways, they are similar in all ways. Example: The human brain is like a computer. Since computers can compute infinitely large numbers, so can the human brain.