Do you think like an artist? Perhaps you think more like an engineer? Do you make broad strokes in your arguments- sometimes glossing over the finer details? We know what this sounds like,
“Who cares if the constant is wrong? It’s just a constant.”
Or perhaps you are very concerned about the precision of your work. You can complete computations correctly the first time and have confidence in your answer. Does hyperbole bother you when used in scientific contexts? Is it more important that there are a LOT of chickens on that farm or is it important that there are 12,393 chickens?
I think mathematicians tend to think like artists. This is why we believe our craft to be creative and challenging. We consider a proof like an artist considers a canvas. Bob Rauschenberg will make rules for himself about how he is going to create a piece of art. …then breaks them. Haven’t we done that from time to time? We say, “let’s prove by contradiction” then we start our proof and realize that, in the end, we proved it directly. Our artwork is sometimes hard to understand and is more beautiful the longer the audience studies the craft- just like fine art is. Mathematicians are the abstract artists of analytical thinking. Our most beautiful theorems can require years of training before they can be appreciated. We don’t just care about the finished product (the theorem) but we want to analyze the brush strokes of the proof. As though we were talking about an oil painting we ask,
“How did they layer the colors of the argument to provide a beautiful conclusion?”
Despite my waxing poetical treatment here, the line between the artistic whimsy and mathematical rigor is crisp.
Mathematicians must also think like engineers. We must be very concerned about precision and accuracy. Because we can’t BS our way through a proof. Our colleagues will see straight through us. Perhaps this is why I can’t sell my abstract paintings for millions. The trained fine art audience can see through my flimsy attempts at their art. We must care that there are 12,393 chickens on the farm. It may be important to our argument that the number of chickens isn’t prime. We must swap between rigor and whimsy to succeed. I think this is part of what draws people into mathematics. I don’t know about you, but I can’t think of too many other fields were individual creativity is valued as highly as precision and accuracy.