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Apples and oranges when you jump from algorithm design to software in general.

An algorithm has mathematical underpinnings and can be created entirely on paper first. Turning ideas into applications however is drastically enhanced by prototyping extensively.

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Agreed. If you have the wrong algorithm, it doesn't matter how well you implement it: it'll still give you the wrong answer. But no matter how right your algorithm is, if your I/O doesn't work properly, you'll still get the wrong answer. And that's just for things involving a single algorithm; if you're talking about the larger piece of software as a whole (since a lot of programming is not algorithms, algorithms is just one of the harder parts), testing and prototyping and all that becomes even more important.

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