There are several difficulties.
- When creating the presentation, the correct level of detail is tricky. With too much detail, the proof takes too much time and people may be lost to boredom. With too little detail, the steps of the proof involve too-great a jump. This is very difficult to judge.
- What may be an easy step in the careful thought of a quiet room is not so easy when you are occupied by the process of presentation.
- What may be easy after having gone over this (and other) proofs is not so easy to follow in the first pass by a viewer.
These problems seem only correctable by process of repeated test-and-revise.
- When presenting the proof, simply speaking with sufficient precision is substantially harder than in normal conversation (where precision is not so critical). Practice can help here.
- When presenting the proof, going at the right pace for understanding is difficult. When we use a blackboard/whiteboard, a natural reasonable pace is imposed by the process of writing. Unfortunately, writing doesn’t scale well to large audiences for vision reasons, losing this natural pacing mechanism.
- It is difficult to entertain with a proofÃ¢â‚¬â€there is nothing particularly funny about it. This particularly matters for a large audience which tends to naturally develop an expectation of being entertained.
Given all these difficulties, it is very tempting to avoid presenting proofs. Avoiding the proof in any serious detail is fairly reasonable in a conference presentation—the time is too short and the people viewing are too heavily overloaded to follow the logic well. The “right” level of detail is often the theorem statement.
Nevertheless, avoidance is not always possible because the proof is one of the more powerful mechanisms we have for doing research.