I know that angles can't be trisected. But suppose that I know for sure that unless I manage to find a way to trisect angles in the next ten years, the human race will be destroyed. What should I do? Surely I should try to trisect, hoping that there are mistakes in the impossibility proofs. And surely here at least is a place where I can do what I should. So it is possible to try to do what one knows to be impossible.
5 comments:
If you know angles cannot be trisected in the same way that you know the human race will be destroyed if you fail, would you put in comparable effort to find a "way out" so to speak? Or is it because the trisected angle seems simpler and less time-consuming that you might decide to work only or mostly on that?
Good question. I was assuming you're more sure that the human race will be destroyed iff you fail.
Otherwise, yeah, it's a pragmatic judgment as to your talents. I'm a mathematician, so angle trisection is probably the better thing to try. If you're good at negotiating with terrorists, maybe that's the better option.
Alex,
I don't think you can try to do what you know to be impossible. The impossible state of affairs (trisecting angles) includes many other possible states of affairs (e.g., drawing lines). You can try one of those. And you can do so while wishing that doing so would lead to a trisected an angle. But that isn't trying to do what you know to be impossible. Rather, it is trying to do something you know to be possible, while wishing you were doing something known to be impossible.
I'm probably splitting hairs here. But I'd like to know what you think.
Consider two scenarios. In one, you're asked to construct a 20 degree angle with ruler and compass, which you know is impossible, and in the other you're asked to construct a 30 degree angle with ruler and compass, which you have a justified false belief is impossible.
Whether your justified false belief in impossibility is knowledge doesn't affect whether you can try. So either you can try in both or in neither.
Now imagine that in the 30 degree case you do various sorts of things that would count as trying if you believed the task was possible. And you succeed! Then it seems clear that the right way to describe what happened was this: "You succeeded at constructing a 30 degree angle." But you can't succeed without trying. So you tried to construct a 30 degree angle.
So in the 30 degree case you can try. Thus, likewise, in the 20 degree case you can try.
My daughter found it necessary several years ago to prove to me that there CAN be such a thing as a square circle.
http://philochristos.blogspot.com/2007/09/square-circles.html
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