For human mathematicians and scientists, elegance may be not just an aesthetic preference but a practical heuristic.
A human exploring the immense space of possible proofs "by hand" (by brain?) takes a long time to work through any given approach. That means going down a blind alley is very costly, and the need for heuristics is high; approaches that can be explored quickly and discarded right away if they don't pan out.
What makes a proof "elegant?" I'm not a mathematician (my dad was), but I would guess there's a strong correlation with simplicity; ease of understanding; an outcome that seems to flow naturally and inevitably from the premises; "one weird trick" that, once grasped, makes the rest just fall into place. All of those seem to me like fair proxies for "quicker to explore."
But if the cost of exploration falls, then the need for heuristics is reduced, and it becomes more feasible to take a brute-force approach where you just let the machines bash their way through. Our instincts rebel at the results because we're used to a world where cognition is a scarce resource and it feels wasteful.
There's a whole lot more I want to add about parallels in software and the benefits and drawbacks of abstractions, but I think I better stop or I'll be at this all day. :)
...Okay, just one more thing. I can stop whenever I want to, I swear.
I suspect elegant solutions offer one other benefit, which is reuse. If you make a conceptual breakthrough -- a new way of framing the problem that makes the solution easy -- you can often apply that same framing to many other problems too. It's another way to economize on scarce cognition, seeking solutions that provide more bang for your cognitive buck.
Bash-your-way-through approaches are much less likely to produce such force multipliers. What you see is what you get.
For human mathematicians and scientists, elegance may be not just an aesthetic preference but a practical heuristic.
A human exploring the immense space of possible proofs "by hand" (by brain?) takes a long time to work through any given approach. That means going down a blind alley is very costly, and the need for heuristics is high; approaches that can be explored quickly and discarded right away if they don't pan out.
What makes a proof "elegant?" I'm not a mathematician (my dad was), but I would guess there's a strong correlation with simplicity; ease of understanding; an outcome that seems to flow naturally and inevitably from the premises; "one weird trick" that, once grasped, makes the rest just fall into place. All of those seem to me like fair proxies for "quicker to explore."
But if the cost of exploration falls, then the need for heuristics is reduced, and it becomes more feasible to take a brute-force approach where you just let the machines bash their way through. Our instincts rebel at the results because we're used to a world where cognition is a scarce resource and it feels wasteful.
There's a whole lot more I want to add about parallels in software and the benefits and drawbacks of abstractions, but I think I better stop or I'll be at this all day. :)
...Okay, just one more thing. I can stop whenever I want to, I swear.
I suspect elegant solutions offer one other benefit, which is reuse. If you make a conceptual breakthrough -- a new way of framing the problem that makes the solution easy -- you can often apply that same framing to many other problems too. It's another way to economize on scarce cognition, seeking solutions that provide more bang for your cognitive buck.
Bash-your-way-through approaches are much less likely to produce such force multipliers. What you see is what you get.