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Comment by crispyambulance | Hacker News Reader

    > So, why would we want to write mathematics any differently?

I think you're stretching the unit-test-to-mathematics analogy a little too far.

I haven't looked at the book in question, but generally, you don't just "know if" your solution is correct, you prove it, or at least consider examples that can demonstrate if your solution works in some conceivable cases. That's how "unit tests" work in mathematics.

If the book has armed you with enough knowledge to work though a problem to the end, it has done it's job. The real test will come as you build on that knowledge later on in the book or in real-life applications.

Moreover, it's not like this stuff is obscure knowledge. You can find help online or by looking at other books or connecting with others.

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As I read it, the point is not about using unit tests to verify correctness, but as an interactive means of learning the problem inside and out prior to finding a complete solution. The trick is in learning what you don't know, and one can do that faster with regular feedback. There does seem to be a delicate balance between letting a student "stew" for a bit to really learn how to find and be confident in solutions, and preventing the student from heading down a wrong course. Books can never provide real feedback, I suppose, but it's a nice option to have solutions available even if students must be trusted to independently exhaust the available resources for attaining them first. Often having the solution is at least helpful as a "sanity check" when it comes to mathematics - I feel it is all too easy for beginners to make proofs on hidden and untenable assumptions and not necessarily realize it.

The point isn’t we need to do formal verification to test correctness. It’s that you probably wouldn’t write a program first pass and assume it is but free. So, why would you assume beginners that don’t know how to write proofs would write correct proofs without feedback to check?

Absolute beginners may not able to write an actual proof that their answer is correct, but at least they can verify that it satisfies the problem (depending on the actual problem).

In any case, since the study of mathematics is cumulative. Even if the student can't prove or verify that their answer to a specific problem is correct they will eventually reach a point, often within the same problem-set, where inability to solve a problem will force them to encounter their gaps/misconceptions.

The most important thing in a mathematics text, as with any other text, is lucidity of writing and the preparation of the student.

The issue isn’t just that the person can’t write a valid proof, but that the person thinks they have a valid proof, but it is actually invalid and they don’t have feedback to know it.

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