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Alicorn comments on Procedural Knowledge Gaps - Less Wrong
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While I wouldn't prefer a fast-food-place plastic spoon, I don't think it would be in danger of melting in this specific case. Boiling water is a fixed temperature and it will stay that temperature until the water is all boiled off, if I understand it correctly; and the spoon doesn't spend much time pressed against the pot itself, since the idea is to smoosh a vegetable between spoon and pot.
The pot itself can't get hotter than boiling either, as long as there's a bunch of water in it. (This, btw, is how rice cookers detect when the rice is done.)
The inside of the pot can't get significantly hotter than ... right, he water just turns gas phase more rapidly.
It won't melt, but depending on the type of plastic it might become too soft and flexible to be useful for vegetable smooshing. From experience, some types of plastic spoons become too soft to even support their own weight when placed in boiling water.