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Comment by bennofs | Hacker News Reader

Transforming a statement using a series of equavilence transforms into an always true statement isn't a valid proof? After all, you've shown that the original statement is the same as a true statement. Isn't that a valid proof as well? It is often easier to go in that direction, since you have a clear starting point. In the other direction, you'd have to guess from where to start.

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1. Many, many theorems are not full equivalences. Students who write proofs bottom-up are, ime, MUCH more likely to mistake implications for equivalences when using theorems/lemmas that are not equivalence conditions.

2. Writing proofs upside-down is bad form. Sometimes it doesn't matter -- e.g., for very short proofs or for proofs that are just a lot of symbol pushing. But for proofs that benefit from intuition, writing the proof in the correct order (from assumptions to conclusion) is infinitely easier to follow.

3. Even when all the steps are honest equivalences, it's a matter of convention and familiarity. Yes, community norms are sometimes arbitrary and silly. And there are all sorts of exceptions to those norms. But I don't walk around my neighborhood naked or in a speedo, and I don't write relatively short direct derivations upside down ;-)

4. This is not unique to mathematics. Legal texts, philosophical texts, and even business plans tend to follow the same structure when presenting arguments.

> In the other direction, you'd have to guess from where to start.

But a proof is NOT a brain dump of your process! It's an exposition of your final result.

It's fine to write it that way as long as you justify that the steps are equivalent and thereby showing you know what you are doing. If you don't write something like 'by equivalence' then the assumption should be that you don't know what you are doing. Or, when there comes a time where there is actually a proof or step of a proof where there is no equivalence, mark it wrong. Don't patronize your students but do write better problems, perhaps.

>It's fine to write it that way as long as you justify that the steps are equivalent and thereby showing you know what you are doing.

Have you heard of cargo culting? Some students will just learn that writing "by equivalence" sometimes gets them most of the points. Correct proofs don't necessarily indicate understanding. That's not patronizing, it's just true.

But that's really beside the point.

The primary problem with these proofs is not that they are logically wrong. Re-read #2 and #3 above.

These proofs are usually difficult to read and understand, in part but not exclusively because they break with traditional style. And not just mathematical style (#4).

Don't be the guy who complains that he doesn't need to follow style guides or write maintainable code because "my code doesn't have bugs".

I don't want a brain dump of the student's thought process. I want a clear exposition.

Sometimes upside down proofs provide a clear exposition; in #3 I mention these exceptions! But more often, the proof is upside down because the student is narrating their problem solving process instead of communicating a valid argument.

Unreadable code is technical debt.

Unreadable documentation will lose you customers.

Unreadable proofs are wrong.

Developers who write easy-to-understand code and clear documentation are strictly superior, all else equal, to developers who write in an idiosyncratic and difficult-to-follow style.

Math is no different, and one goal of proof-based courses is to teach students how to communicate technical content clearly.

> Don't patronize your students but do write better problems, perhaps.

At this level, the problem is usually just a bog standard statement of a theorem in established mathematical prose.

Some number of students new to proof writing will write upside down proofs regardless of what you ask them to prove.

The wording of the question does not prompt this behavior.

A student writing mathematical proofs is not a developer writing client code.

It's a metaphor.

One goal of proof-based courses is to teach students how to communicate technical content clearly.

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