Robert B. Reisel

Even if they plan to become theoretical, as opposed to experimental, scientists, science students must spend a significant amount of time studying how to conduct laboratory work. They must comprehend both the methods used in experiments and the significance of the findings. In science, experiments are used to determine the viability of theories. A novel concept must be abandoned if it fails to materialize in a test setting. If it succeeds, it is at least provisionally approved. Therefore, in science, laboratory experiments serve as the yardsticks for the approval or disapproval of findings. Mathematics is unique. It is not intended to imply that experiments are not relevant to the topic. Mathematicians often form conjectures as a result of numerical computations and the examination of particular and simplified examples, but a conjecture is only accepted as a theorem after a proof has been created. In other words, just as laboratory experiments are to science, proofs are to mathematics. Students of mathematics must consequently learn what constitutes a valid proof and how to build one. How does this work? By doing, just like everything else. Students in mathematics are required to attempt to verify their conclusions before having the work of more seasoned mathematicians critiqued. Both correct and bad proofs must be thoroughly analyzed, and they must learn to appreciate good writing. Naturally, they must begin with simple proofs and progress to more challenging ones.