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A 19 century topologist would explain this by dimension count. By Riemann-Hurwitz, a surface of genus $g$ covering the sphere with 2 sheets has 2g+2 ramification points which gives 2g-1 free complex parameters, since 3 ramification points can be fixed. On the other hand, according to Riemann, when $g>1$ the space of Riemann surfaces of genus g depends on $3g-3$ parameters which is strictly more than $2g-1$ when $g>2$. I suppose this is how this fact was discovered.