Go up to B Proving Propositions
Go forward to B.2 Preliminaries
A proof can be given on different levels of detail; on the lowest level, the individual reasoning steps are so small that their correct application can be even automatically verified by a computer. However, such proofs become very large and are rarely written by humans.
On a higher level, the reasoning steps are larger and less self-evident; still the structure of a (less detailed) high-level proof matches the structure of the (more detailed) low-level proof as it is determined by the proof rules. If some opponent questions our high-level proof, we can correspondingly refine it by decomposing a large reasoning step into some smaller steps. This process can be repeated until we reach the lowest level (which is automatically checkable) or our opponent is satisfied. Even if we are not writing all details in a high-level proof, the skeleton provided by the proof rules help us to maintain its power of persuasion and justify our confidence in its correctness. This is the kind of proofs that humans typically write in scientific works.
To choose the right size of reasoning steps is a tradeoff between demonstrating the "key ideas" of a proof (by skipping details that the reader is supposed to be able to fill in herself) and giving convincing arguments (by providing sufficient details that enable the reader to understand the line of reasoning). However, while we may easily bore our audience with too many details, we may even more easily lose its understanding by giving too few. Also for the sake of correctness, it is better to elaborate more details than we may be originally inclined: many textbook proofs are wrong6, because the author tried to save some work: she used reasoning steps that were so large that she could ultimately not grasp them herself, or she took wrong shortcuts in the application of proof rules.
If a proof is not understood, the proof uses too large reasoning steps, has bad style, or is simply wrong (the boundaries are fluid). This is in general not the fault of the audience; the presenter has the duty to adapt her presentation to the available knowledge, to make a clear argument and to refine it down to a satisfying level of detail. Science is not theology: no one is obliged to believe our claim unless we give a compelling argument.