His point here is that being a programmer is better than being a doctor or a lawyer because you can begin contributing at an earlier age.
Here's the "arrogant" part: for that point to be valid, one has to assume that the future contributions of being a doctor aren't worth as much as the contributions to software one made as a teenager. From my point of view, that is simply not the case.
That is something that was not said or implied on the text in question. As a matter of fact, the point given across has very little to do with how helpful either one of us perceives a programmer's and doctor's "contributions" over the course of their careers, but how there is no age limitation which stops a person from "helping" others and how the "value" of your service is determined by talent, discipline, and effort while not necessarily being attached to a title.
Here's a thought, I come a family of doctors, lawyers, and architects (in order of occurrence within the family), and I've achieved the realization that thinking the "importance" of a person is defined by their profession is incredibly misguided. Being a doctor does not mean that you'll save a life, and being a programmer does not mean that you wont. Actually, there are as many type of doctors as you have organs, diseases, and appendages, as many lawyers as there are clauses in a EULA, and as many programmers as programming language books... and all of them are just people whose helpfulness will be defined by what they do with their life's and not because one's a doctor and the other a circus clown.
I'm not trying to be mean spirited or demean the work of doctors or lawyers (or programmers for that matter), but you've stated that the article's writer sounded arrogant because for you his opinion was only valid if it followed a specific train of thought that lead to a direct comparison of two stereotypes - the lifesaving doctor and the cave-dwelling programmer - and I believe that to be wrong. I know I'm being overly dramatic with my examples, but let me finish with one small nugget: Who's "contributions" are worth more? Neil deGrasse Tyson's or Dr. Robert Rey's?
Here's the "arrogant" part: for that point to be valid, one has to assume that the future contributions of being a doctor aren't worth as much as the contributions to software one made as a teenager. From my point of view, that is simply not the case.