In 2013, an enigmatic user by the name Cleo created an account on the Mathematics Stack Exchange. This website—one with math only really intelligible by high-degree-holding mathematical academics—hosts a myriad of problems for fellow mathematicians to scramble to solve. Cleo, a seemingly unremarkable and unknown user back in 2013, rampaged the website for nearly 2 years, solving numerous extremely difficult problems with zero explanation on how the problem solving was done. This lack of explanation caused a frenzy on the Stack Exchange—how did Cleo get these answers, and are these answers even valid without proper explanation and justification?
I would argue that yes, these answers are valid. Sure, Cleo is not a saint: there are obviously some steps that are purposefully omitted from her answers and I believe that she is acting a fool just to rile up the masses on the website. However, that is not entirely what I will be discussing here; I want to discuss how some called her correct answer invalid due to a lack of explanation.
Mathematics has always required some explanation. You couldn’t make a statement past the most simplest of axioms without someone asking how you did it. My earliest years of education have required me to answer “why” to even the most simple of addition and word problems. I think this makes sense for education and teaching purposes; it is impossible to know if a child just used random fundamental arithmetic to achieve an answer or if they actually understood why they applied said arithmetic. In more recent years, less mathematics courses have required me to display the steps by which I achieve my answer. Calculus has thrown numerous theorems and facts at the class with no further explanation or proof. Work being shown is a very minor portion of any test. The axioms for each calculus course have become more and more abstract: I doubt most calculus 1 students can prove the chain rule on their own, they just accept it.
Outside of an educational setting, there is no reason why it is necessary to show your work. Recently, a friend proposed a math problem to me with nothing but the answer being required to be submitted. After a few hours of python programming, I was able to generate an answer by brute force. My friend disputed this answer, claiming that I “cheated” and it is not how the problem was intended to be solved. However, if no proof is explicitly required in the problem, I disagree. All work being done in the world using mathematics doesn’t require an explanation; it requires an answer, a definitive number value so that whatever physical job is being done can be done. Complicated fluid dynamics are never done with pen and paper proof-based mathematics, but are done with supercomputers simulating the mathematics for you. In any practical sense, no answer that works requires a justification or proof.
Proofs can be beautiful and impressive. Wiles’ proof for Fermat’s Last Theorem is one of the largest mathematical achievements of the last century. However, I think it is imperative to separate this abstract mathematics from practical mathematics. Cleo solving a very complex integral in a few short hours is impressive from the practical world view, the world view which gets things done. An answer is an answer, after all. Nonetheless, I do see why Stack Exchange users may have been tearing her to shreds. That website focuses on the abstract, the solving problems for the sake of solving problems rather than for practicality.
My point here is that I believe practicality is often more important than the abstract side of problems. If a problem is solved in any way, that solution can be used for all situations where said problem applies. The practical solution is one where all real life abstractions of the problem are solved. If one’s employment required crunching numbers all day, no one cares how the numbers are crunched as long as they are accurate. Abstractions can be fun and expand the bounds of the field, but are frankly unimportant for the operation of mankind day-to-day. My friend’s problem was just required to be solved, and I did exactly that. The firm administering the problem would see no difference between someone that solved it by hand and solved it with a computer. Practical solutions to problems should be fair-game and not looked down upon in fields which do not require strict proofs.
And yes, I did just write this because I want my friend to accept my answer as valid.