In 1905 or the year referred to as annus mirabilis, Albert Einstein published four papers that propelled the development of modern physics. His famous equation that beautifully establishes equivalence between two variables and one constant— E (energy) = m (mass) c^2 (speed of light)— also comes from one of the papers. Instead of discussing the result of his paper, it might be more helpful to talk about a way of thinking that Einstein celebrates in his papers. In Einstein’s research, he conducts thought experiments instead of physical experiments. For example, in his paper, “On the Electrodynamics of Moving Bodies”, he invites the audience to follow along with thought experiments that yield a new understanding of relativity. The setting up of a thought experiment is a constructive process, in which he combines some natural instincts or maybe some unexpected discoveries to intentionally design a typical situation leading to the anticipated conclusion.

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Sometimes, instead of solving the problem in a conventional way (such as physical experiments), jumping out of the box to think constructively or imaginatively might reduce the complexity and lead to a simplistic result (remember, MATES subjects are empowered by and appreciate simplicity). This way of thinking is equivalently applicable to other fields of study, such as mathematics. Perhaps all groundbreaking math proofs could be considered construction. In Andrew Wiles’s proof for Fermat’s Last Theorem, his work bridges together two separate threads of research and creates a contradiction.

 

Although the examples I gave involve the most intelligent brains applying this way of thinking to solve grand problems, constructive thinking is quite applicable in our daily problem-solving. In my summer research, we studied a classical problem, the Traveling Salesman Problem, which asks an optimal route for a traveling salesman to travel around a network of cities visiting each city exactly once and returning to the start. We would like to give a proof of the nonexistence of a general algorithm that computes the optimal route of every traveling salesman problem. To solve the problem, we applied constructive thinking: we resolved the problem by simply finding low-order contradictions for both oriented and unoriented scenarios. If the problem was not being solved in this constructive way but by pattern-seeking, it might be exponentially difficult or laborious to tackle.

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If you are interested in reading my research, I am open to criticism or suggestions. :)