> “Incidentally, am I alone in finding the expression ‘it turns out’ to be incredibly useful? It allows you to make swift, succinct, and authoritative connections between otherwise randomly unconnected statements without the trouble of explaining what your source or authority actually is. It’s great. It’s hugely better than its predecessors ‘I read somewhere that...’ or the craven ‘they say that...’ because it suggests not only that whatever flimsy bit of urban mythology you are passing on is actually based on brand new, ground breaking research, but that it’s research in which you yourself were intimately involved. But again, with no actual authority anywhere in sight.”
When I was doing my physics undergrad, my friends and I would always joke about how some professors (and some papers!) would use "it trivially follows..." in the same way. When an approximation or step in a proof was particularly hairy or weird, that phrase kept cropping up -- and why not? It handwaves away the step while simultaneously discouraging any pesky questions (if it's trivial, surely only a weaker student would ask about it).
Honorable mentions are: "hence" and "the remainder is left as an exercise for the reader."
A professor is giving a lecture, scribbling some formulas on a blackboard and at some point saying "it's obvious that ...". Then he pauses, stares at the blackboard, says "one moment please" and runs out of the door. Five minutes later he comes back in, with a wad of scratch paper in his hand, picks up the chalk and says "it is indeed obvious that ..." and continues the lecture.
I remember taking a set of courses in Mechatronics years after graduation. For the uninitiated, Mechatronics is a combination of Electrical Engineering, Mechanical Engineering and Software/Firmware to design & analyze software-controlled electromechanical systems. Most of the students had been in industry for decades and even I was almost 10 years out of college by that point. The Feedback Controls instructor, a brand-new Mathematics PhD, was fond of answering questions with a bewildered look at the board and the question, "well, isn't it obvious?" (probably not, if an engineer who hasn't solved a differential equation in 30 years is asking you). He was obviously very bright, but could not teach the subject to save his life.
Thankfully, the guy we had teaching the labs was an engineer who had been building control systems his entire career and could answer the theory questions as well as the practical aspects, or no one would have passed that class!
Hence has its place, as do the holes left in theorems that our betters leave for us. These gaps leave ups to leaps of faith at our beginnings and thorough proofs as we mature. When I was a green tutor I would attempt to make every single last step as obvious as possible, but I later recognized the value of making the the pupil do the work. It builds their confidence and character to have faith in themselves or, if failing, their ability to find the answer that they could not bridge themself.
As long as you guide them doing so and help them out if they get stuck - which is your job as a tutor. No one is helped if the gaps make it impossible for students to understand the concepts.
> These gaps leave ups to leaps of faith at our beginnings and thorough proofs as we mature.
As a student who had been in such a situation, I can say that both of those seriously impacted my and my colleagues' understanding of the subject.
I've had professors who didn't explain their reasoning for certain steps at all and others who "explained" it by simply reading through the mathematical proof. That was about as helpful as getting thrown into vi without any help documents vs getting thrown into vi with just a copy of the full source code next to you: Neither helps you understand how to actually work with vi.
I believe the goal of, say, a mathematical lecture should be to explain the basic idea of a mathematical tool and the reasoning why it is there and how one can use it. I believe if students have understood that, they will have less need to dive into some hairy details of the actual proofs as well.
As long as you guide them doing so and help them out if they get stuck - which is your job as a tutor. No one is helped if the gaps make it impossible for students to understand the concepts.
Apparently Apple Store "Geniuses" have a manual that instructs them to use this phrasing when discussing the problem with hardware a customer has brought in for repair.
Not sure why it works, but I've borrowed this phrasing with some success in my work and personal life.
Speaking of getting mileage out of a phrase, a related thing was discussed two days ago[0] - "anticipatory procrastination". I loved the following passage from the linked paper:
"For an early example, consider Fermat’s 1637 statement, “I have a truly marvellous proof of this proposition which this margin is too narrow to contain.” This statement incorporated a landmark proof technique. The ingenuity of this technique allowed Fermat to delegate his proof to 400 years of his students while keeping his name attached to the theorem."
In all honesty the "technique" is simply an appeal to authority (not necessarily a fallacy it turns out * ). The technique isn't that "he said it" it's that "he said it". E.g. if I scribble some nonsense on the side of a paper almost nobody will pay attention one way or another.
Look no further than religion to see how apparently banal things get passed on for millennia, and even more of them are attached to those same religious figures simply because they are "the authority".
That is a great point. I watched a documentary on the person who finally solved the proof (twice!) and I don’t remember his name, but I do recognize Fermat’s theorem.
I like the fact that people notice and call attention to these subtle devices. I use them myself when writing[1] and they're useful tools, but we should always be aware when we are basically clubbing our reader over the head with our own take on things.
I would love to see a compendium of these sorts of subtly domineering turns of phrase.
[1]: it turns out I used "it turns out" in the last thing I wrote that was at the top of HN, heh.
> They learn to trust the writers who use the phrase, in large part because they come to associate it with that feeling of the author’s own dispassionate surprise: “I, too, once believed X,” the author says, “but whaddya know, X turns out to be false.”
The show ended before a large swath of HN's demographic was born. I watched it as it aired and barely remember any of it. I wouldn't know that was a Seinfeld reference if no one said anything.
> “Incidentally, am I alone in finding the expression ‘it turns out’ to be incredibly useful? It allows you to make swift, succinct, and authoritative connections between otherwise randomly unconnected statements without the trouble of explaining what your source or authority actually is. It’s great. It’s hugely better than its predecessors ‘I read somewhere that...’ or the craven ‘they say that...’ because it suggests not only that whatever flimsy bit of urban mythology you are passing on is actually based on brand new, ground breaking research, but that it’s research in which you yourself were intimately involved. But again, with no actual authority anywhere in sight.”