Would it be correct to consider things like proof by contradiction and proof by refutation as falling on the generation side, as they both rely on successfully generating a counterexample?
Completely separately, I want to make an analogy to notation in the form of the pi vs tau debate. Short background for those who don’t want to wade through the link (though I recommend it, it is good fun): pi, the circle constant, is defined as the ratio between the diameter of a circle and its circumference; tau is defined as the ratio between the radius of a circle and its circumference. Since the diameter is twice the radius, tau is literally just 2pi, but our relentless habit of pulling out or reducing away that 2 in equations makes everything a little harder and less clear than it should be.
The bit which relates to this post is that it turns out that pi is the number of measurement. If we were to encounter a circle in the wild, we could characterize the circle with a length of string by measuring the circle around (the circumference) and at its widest point (the diameter). We cannot measure the radius directly. By contrast, tau is the number of construction: if we were to draw a circle in the wild, the simplest way is to take two sticks secured together at an angle (giving the radius between the two points), hold one point stationary and sweep the other around it one full turn (the circumference).
Measurement is a physical verification process, so I analogize pi to verification. Construction is the physical generation process, so I analogize tau to generation.
I’m on the tau side in the debate, because cost of adjustment is small and the clarity gains are large. This seems to imply that tau, as notation, captures the circle more completely than pi does. My current feeling, based on a wildly unjustified intuitive leap, is that this implies generation would be the more powerful method, and therefore it would be “easier” to solve problems within its scope.
Would it be correct to consider things like proof by contradiction and proof by refutation as falling on the generation side, as they both rely on successfully generating a counterexample?
Completely separately, I want to make an analogy to notation in the form of the pi vs tau debate. Short background for those who don’t want to wade through the link (though I recommend it, it is good fun): pi, the circle constant, is defined as the ratio between the diameter of a circle and its circumference; tau is defined as the ratio between the radius of a circle and its circumference. Since the diameter is twice the radius, tau is literally just 2pi, but our relentless habit of pulling out or reducing away that 2 in equations makes everything a little harder and less clear than it should be.
The bit which relates to this post is that it turns out that pi is the number of measurement. If we were to encounter a circle in the wild, we could characterize the circle with a length of string by measuring the circle around (the circumference) and at its widest point (the diameter). We cannot measure the radius directly. By contrast, tau is the number of construction: if we were to draw a circle in the wild, the simplest way is to take two sticks secured together at an angle (giving the radius between the two points), hold one point stationary and sweep the other around it one full turn (the circumference).
Measurement is a physical verification process, so I analogize pi to verification. Construction is the physical generation process, so I analogize tau to generation.
I’m on the tau side in the debate, because cost of adjustment is small and the clarity gains are large. This seems to imply that tau, as notation, captures the circle more completely than pi does. My current feeling, based on a wildly unjustified intuitive leap, is that this implies generation would be the more powerful method, and therefore it would be “easier” to solve problems within its scope.