Consider the following possibly familiar situation. Let us assume that you, the reader, are a musical novice, and you therefore know nothing about jazz. You and some friends go to a jazz club, however, and there’s a trio playing (piano, bass, drums). Throughout each tune the trio weaves its way through esoteric chord changes, constantly adapting to one another’s improvised flourishes and syncopations. You, the novice, of course have no idea what’s going on, but assuming the musicians are at least decent the following two statements could most likely be made. First, you’re probably enjoying the music, at least somewhat. And second, you can appreciate and admire the talent of the musicians, the hard work that they’ve put in to accomplish what they’re doing in front of you, and you might even respect and look up to the musicians themselves for their devotion to the craft.
Now consider the following possibly less familiar situation. Suppose instead of seeing professional musicians “do what they do”, you were seeing professional mathematicians “do what they do”. Now, this is a somewhat hard example because one rarely gets to observe a mathematician do his/her work in real time, but fortunately we don’t even need to concoct an example of this. We merely need to reflect on what the general public’s reaction to the existence of a professional mathematician at all would be. I believe, in general and with sufficient observational evidence to back it up, that the response of the general populace to the realization that there are professional mathematicians is “WHY?!”.
This is not a “why?” as in “why do we need people to do math?”, because I think almost everyone believes that “math is important” is a true statement (to some extent), and therefore that people understand that mathematicians should exist. Instead, this “why?!” is more along the lines of “why would anyone CHOOSE to be a professional mathematician?”.
Note how this is in stark contrast to the reaction of seeing a professional musician. No one questions why anyone would become a professional musician, or painter, or architect. Conversely (almost) everyone wonders why anyone would WANT to do math. But why is this? After all, musicians, painters, architects, and mathematicians all a) deal with esoteric concepts that take years to fully understand and b) spend years working in solitude on these ideas. So the answer can’t be that “math is hard” or that “mathematicians are weird”. I would argue that being good at math is just as “hard” as being good at music, painting, and architecture (in the sense that they all take comparable amounts of time, energy, and focus to master), and that professional mathematicians are just as “weird” as their counterparts in other fields (by “weird” I actually just mean passionate and enthusiastic).
So then why is there such a “mathematical double standard”? I believe the answer is easy. Subjects like music, painting, writing, and architecture, are all COOL. People generally know that the core ideas of these crafts are interesting, beautiful, and worthy of committing one’s life to the study of. Thus, when a non-musician meets a professional musician, they can at least think “hey, I have no ability to understand what you really do and think about, but I think that what you do and think about is COOL and interesting, so I’ll respect and admire you for devoting your life to it”. Let’s compare that to the thought that often happens when a non-mathematician meets a mathematician: “Hey, I have no ability to understand what you really do or think about, and I also hate math a lot, so I have no idea why you’d EVER want to purposefully devote time to the subject, and therefore you’re weird and I’m confused by your existence”.
Now I’m not saying that we need to make mathematicians replace the jocks as the cool kids in high school. Instead what I’m saying is that we need to make math itself COOL in the following particular sense: we need to make people understand that the core ideas and thought processes in math are cool and interesting and worthy of purposefully devoting one’s time to. THIS is the only way people will start to change their opinions about math, even if they don’t decide to pursue math seriously, just as I wouldn’t want everyone who has ever gone to a jazz club to attempt to become a professional musician. Luckily, math really IS cool and interesting! Just usually not the math that is actually in the curriculum. And THIS is why we need the actual core ideas and thought processes of math—abstraction, proof, rigor—to be made more apparent.