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 Problem

I 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.

### The solution to the problem

To find a solution to the problem, we have to consider two cases.

#### Spoken Mathematics

When Mathematics is presented by a speaker, it mostly amounts to a social paradigm shift. I think it would be best if asking for more detail in an argument would be taken for what it is, namely scientific curiosity. People should not be afraid of asking questions and it should be commonplace that there is lively discussion. It nurishes understanding instead of stifling it with silence. I try to live this belief in my daily life, and I am writing this blog post - not much more I can do.

#### Written Mathematics

When Mathematics is written in an article, book or thesis, I can't even picture a truly satisfying solution. I could bore you with my dreams of computer-verifyable and still perfectly human-readable proofs which can cascade into different levels of detail, all in a world where digital documents are as commonplace as paper is today. However, you might think I am mad.

## 5 Replies to “Ask Questions!”

1. Phille says:
I think the heart of the problem is that there is a cult of genius in (pure) mathematics. Many of the big shots in mathematics are extreme persons with extreme gifts and the general feeling is that a few geniuses at the top of the mathematical food chain are responsible for almost all mathematical progress. The ordinary professors are just good enough to work out some details or special cases. This situation may be similar in other sciences but I think it is much more harshly felt in pure mathematics, where money generally isn't a big motivation and it is really hard to kid yourself into thinking your work advances mathematics in any way. Even a nobel laureate cannot conduct experiments much faster than a phd-student, but in pure mathematics most phd-projects could probably be done in a matter of days by the big expert in the field. So a mathematician basically has a choice, he can hope to be a genius one day, or he can accept his work is probably completely irrelevant. That's a tough choice, no wonder people try to fake understanding everything as long as possible, crippling their mathematical progress in the process. So I state the hypothesis: The sooner you realise that you won't amount to anything in mathematics, the more likely you are to amount to anything in mathematics. My own latest feeble attempt at mathematical immortality has been foiled a few weeks back: I noticed that I would turn 31 this year, a prime number, and I realised it was the 11th prime number, which is a prime number again, namely the fifth prime, again a prime number … and so on all the way down to the last turtle. Turns out this sequence of primeth primes has been known for decades ...
2. I had a whole detailed answer laid out in my head, but then I realized that it all comes down to this: If what you say is true, wouldn't that mean that pure mathematics has navigated itself into a terribly narrow corner? I woudn't call myself a pure mathematician, so I do not really know how narrow that corner really is. But I always hoped it wasn't quite as narrow as it seemed from this distance. Because if it is, then I would expect that there is serious doubt about its merit, even from a mathematician's point of view. What good is an area of Mathematics if it's results (and its beauty) is hidden to all but 5 genius mathematicians in the world?
3. Phille says:
Well, understanding and appreciating a result is very different from coming up with it. And of course my view on pure mathematics may be extreme, because I did set theory, which is arguably the most extreme form of pure mathematics. But in set theory I had the distinctive feeling, that its corner was narrow indeed. I also had the impression that many young researchers went off on a tangent, tackling less principled questions, to avoid competing with the established big guys. The sentiment that "ordinary" professors can only fill in the details for the geniuses was actually stated by my professor … Maybe someday there will be areas of mathematics that have been cut off from research, because there is nobody alive who still understands the most advanced results.
4. Man, that sure sounds like a sad story. On the bright side however, every generation of Mathematicians has the chance to reshape the landscape of mathematics - theoretically, at least. There is no actual need to get stuck in narrow corners.