> Do they explain anywhere why we consider radicals special and ask these questions about them in the first place? Nobody ever explains that. To my programmer brain,
But regardless of whether you're a programmer, mathematician, machinist, carpenter, or just a kid playing with legos, there's always a good time to be had in the following way: first you look at the most complex problems that you can manage to solve with simple tools; then you ask if your simple tools are indeed the simplest; and then if multiple roughly equivalently simple things are looking tied in this game you've invented then now you get the joy of endlessly arguing about what is most "natural" or "beautiful" or what "simple" even means really. Even when this game seems pretty dumb and arbitrary, you're probably learning a lot, because even when you can't yet define what you mean by "simple" or "natural" or "pretty" it's often still a useful search heuristic.
What can you do with a lot of time and a compass and a ruler? Yes but do we need the "rule" part or only the straight-edge? What can we make with only SKI combinators? Yes but how awkward, I rather prefer BCKW. Who's up for a round of code-golf with weird tiny esolangs? Can we make a zero instruction-set computer? What's the smallest number of tools required to put an engine together? Yes but is it theoretically possible that we might need a different number of different tools to take one apart? Sure but does that really count as a separate tool? And so it goes.. aren't we having fun yet??
But regardless of whether you're a programmer, mathematician, machinist, carpenter, or just a kid playing with legos, there's always a good time to be had in the following way: first you look at the most complex problems that you can manage to solve with simple tools; then you ask if your simple tools are indeed the simplest; and then if multiple roughly equivalently simple things are looking tied in this game you've invented then now you get the joy of endlessly arguing about what is most "natural" or "beautiful" or what "simple" even means really. Even when this game seems pretty dumb and arbitrary, you're probably learning a lot, because even when you can't yet define what you mean by "simple" or "natural" or "pretty" it's often still a useful search heuristic.
What can you do with a lot of time and a compass and a ruler? Yes but do we need the "rule" part or only the straight-edge? What can we make with only SKI combinators? Yes but how awkward, I rather prefer BCKW. Who's up for a round of code-golf with weird tiny esolangs? Can we make a zero instruction-set computer? What's the smallest number of tools required to put an engine together? Yes but is it theoretically possible that we might need a different number of different tools to take one apart? Sure but does that really count as a separate tool? And so it goes.. aren't we having fun yet??