Ask yourself dumb questions and answer them (2020)

I think most people commenting in this thread are reading the headline and/or reading the article as outside the context of mathematics. This is Terry Tao's blog and not a VC's microblog. The very first block of text is a quote from "I want to be a mathematician". He is purely talking about how to approach mathematics at a level sufficiently close to the frontier. What he is saying is that when you are taught something like optimization theory, the definition of convexity is presented to you in the first lecture. A "stupid" question to ask is what makes convex functions special. After thinking a lot about this question, I've come to the conclusion that convex functions are functions that are trivial to optimize but the converse is not true. Not all function that are trivial to optimize are convex functions. There are then plenty of questions like which other classes are easy to optimize. In Terry's words, "the answers to these questions will occasionally lead to a surprising conclusion, but more often will simply tell you why the conventional wisdom is there in the first place, which is well worth knowing.". Another example I ran into was the notion of metric spaces and completeness. It is "stupid" to ask what happens if you get rid of completeness. Stupid because you were just taught completeness for a reason. Trying to understand gives you a whole new appreciation than being presented the formal definition refined over many years.

One of the things that separates ordinary people from smarter people is the topic of this article, the ability to imagine new concepts, questions, ideas. Colloquially we call this <i>creativity</i>, and it stems from a large degree of playfulness and enjoyment of the subject at hand.<p>What separates geniuses from the crazies is the ability to test those ideas and the humility to dismiss the ones that do not work. It’s that combination that makes a great thinker.

Related tangentially perhaps, but I find this to be a particularly helpful application of things like ChatGPT. Instead of asking it to give me the answer, I tell it what I’m trying to do and ask it to ask me questions about it. It’s great for gathering thoughts for writing, and I’ve been using it a little to try to help me think through software problems more quickly.<p>It’s very good at asking “dumb” (ew) or at least simple, building-block questions, and it almost always ends up asking me about something I hadn’t considered that ends up contributing to what I’d have done without the machine. It’s also kind of fun.

&gt; It’s also acceptable, when listening to a seminar, to ask “dumb” but constructive questions to help clarify some basic issue in the talk.<p>I think this is one of the reasons I hit a wall in my study of mathematics forty-five years ago. In the master&#x27;s program I was enrolled in, the students never asked questions during class. The teachers would probably have welcomed questions, but the culture among the graduate students was never to show ignorance by asking. The result was that I often got lost in the lectures, was unable to catch up by studying the material on my own outside of class, and never became a mathematician.

Even more productive: ask yourself, is this really the best way to accomplish this? Why are we doing it this way? Is the standard solution bound by some constraint that was valid in its time but can now be discharged?<p>My canonical example for this is standard mathematical notation, which was invented before computers and optimized for quill pens and chalk boards. How much leverage could you get by discarding it and starting over with something a computer can more easily process, like, say, s-expressions?<p>Or... most programming languages use ascii for their standard syntax but nowadays we have unicode. How much leverage can you get by using more non-ascii characters, like, say, «balanced quotation marks»?

Once we had a very smart young guy on work. But he always asked things, even if he knew the answer. It was like a sickness, people refused to work with him, because of his many also stupid questions. So I told him he can ask me anything, but he allways has to present at least two ideas to solve the problem. He always came with a possible solution and if I asked him witch solution then the better would be, he always picked the right one. After a very short time he asked much less and it was much more calm to work with him.

I have found that many people don&#x27;t respond well to one asking dumb questions. It can quickly marginalize you in a room of people trying to be the smartest in the room.

&gt; <i>if a simple result is usually proven by method X, you can ask whether it can be proven by method Y instead</i><p>in particular, see <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Mathematics_Made_Difficult" rel="nofollow noreferrer">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Mathematics_Made_Difficult</a><p>(one of the distinguishing features of a good hack is that it leaves the Sherpas behind and takes a different —usually more direct— route to the solution than the siege approach that Conway&#x27;s-Law-compliant enterprise team(s) would take)

Great read. As a past adjunct, you can learn this through your students. No question is a dumb question. I eventually got to a point where I couldn&#x27;t wait for the &quot;dumbest&quot; question of the session so that we could explore it as a class.

Another useful technique in understanding a topic is to find is to simple (but nontrivial) examples and make sure one can solve them fully and easily.<p>Then slowly add complexity from that example to gain insight to more challenging cases.

Could someone recommend a good maths research book that covers the whole messy discovery process, rather than just the end result?<p>I don&#x27;t mind in which field, and it&#x27;s fine if it&#x27;s quite formal (not looking for pop science). But it would be good if it is relatively self-contained (doesn&#x27;t require too much prior knowledge, it establishes some foundations and builds from them).<p>I&#x27;m a fan of On Numbers and Games by Conway for instance, and the more literary Surreal Numbers by Knuth (written earlier but inspired by Conway&#x27;s research before publication).

I&#x27;d go further to say that if you don&#x27;t ask to yourself a single question while you&#x27;re reading a substantial part of a book (a section, a chapter), then you don&#x27;t really internalize what you think you&#x27;re learning.<p>The questions don&#x27;t need to be answered perfectly, nor they need to be &quot;good&quot; questions. The questions can be general and simple, like:<p>- Does it have to be this way? Why?<p>- How is it related&#x2F;connected to what I&#x27;ve learned previously (last chapter, last year, last decade)? Do I need to unlearn something or adjust my previous mental model?

Answering popular questions like the &quot;contrarian question&quot; Thiel poses is a good exercise. Flesh out the answers to a degree you would feel good presenting it in an interview.

To me the existence of this post is particularly relevant given the contemporary context: <a href="https:&#x2F;&#x2F;twitter.com&#x2F;StefanFSchubert&#x2F;status&#x2F;1738327056239169618" rel="nofollow noreferrer">https:&#x2F;&#x2F;twitter.com&#x2F;StefanFSchubert&#x2F;status&#x2F;17383270562391696...</a>

I need to involve students in carbon footprint activity, suggest an activity involving mathematical formula, simple for students

Is this different from Rubber Duck Debugging?

Very good thought process here… I always try to steel man oppositional arguments of things that I may think are true