A mathematical proof does relate to our ordinary dictionary meaning of “truth”, but it has many more elements to it. The main idea behind the proof is the idea of logic. Math is a science and there is nothing fictional in the logic used to solve problems. Proofs are a way of using that logic to create a path through the maze often presented by mathematical concepts. Because math is so concrete and isn’t influenced by outside factors we can rely on some basic rules and concepts to help navigate the solution.
Non-proof based mathematics lacks the deductive reasoning that proof-based mathematics offers. It is much like observation-based mathematics – the answer may be right but there exists no understanding of how or why. The ancient
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A typical student should be required to use the proof-based method until they have mastered mathematics and can intuitively (based on their background knowledge) solve problems without all the steps. Like learning to drive a car, a person must pay attention to every detail until those details become an automatic thought and there is no need to keep a mental checklist every time they get behind the wheel.