It has come to my attention that Mathematicians don't like to ask questions. The main reason seems to be that they are afraid of looking stupid. It has annoyed me for quite some time and it has several severe disadvantages for the field of Mathematics: * Students don't ask questions in class because it's very important to appear as though you have completely grasped and understood everything immediately. As a result, students learn less and get educated more slowly. Because in the end, it won't help you to only pretend you understood math. * Many fellow PhD students do not use their real name on pages like math.se and mathoverflow because they fear intellectual persecution if people were to associate their name with completely stupid questions. As a result, it's harder to get in touch and find out who is working on similar problems. Anonymity is fine, I am a big fan. However - if it's out of shame for intellectual curiosity that you remain anonymous, then something in that society needs changing. I believe that these are symptoms of a severe sickness that has spread in Mathematics and which is causing a chronical occlusion to our metaphorical vascular system of information flow.
The heart of the ProblemI wanted to write a post where I simply complain loudly and encourage my fellow Mathematicians to ask more questions, but somehow I believe that my above words have expressed my feelings about that to a sufficient degree. Instead, I began wondering about the cause of this illness that pains me to see in so many of my good friends and colleagues. The root of our problem is simple. It is devious, horrifying and simple: There is one aspect of formal Mathematics that is not well-defined, not covered by our otherwise clear rules and terminology. I inquire of you:
When talking or writing about a mathematical topic, which are the statements that may be considered trivial?This is a very important question, because you can absolutely not reduce every reasoning you perform to the Zermelo-Fraenkel set theory axioms. On the other hand, by definition of Mathematics, they are the only statements that are truly "trivial" in the sense that they afford (or rather, do not even allow) any further explanation. If you call any other statement trivial, which Mathematicians do all the time, implicitly or explicitly - then this is an expression of helplessness towards the sheer impossibility of reducing everything to the axioms. It's a plea to the audience to perform the remaining reductions in their head. Hence, Definition. A mathematical statement is called trivial if your audience is deeply familiar with it or can verify its correctness immediately. Clearly, if every statement which is implicity or explicitly tagged "trivial" was indeed trivial according to this definition, then by definition any person in the audience would only ask completely legitimate questions. I am a bit subtle here - I regard implications as mathematical statements, therefore an implication can also be trivial (or not). To say that $(2x^2=8\land x>0)$ implies $x=2$ might be trivial to you, but depending on the audience I might have to use finer steps because this implication is not trivial to them. So, this is the problem. The meaning of trivial depends on your audience. If a person in your audience can not verify the statement immediately, then this can have various reasons. Mathematicians seem to be a very hateful crowd, for it would seem that most of the time, people assume that the reason must be a lack of mathematical capability. This is a very narrow and patronizing assumption. Also, it obstructs progress, which is the worst kind of attitude one can have in science.