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This is a very basic fact from...

It then clearly follows that...

Obviously, we have...

The proof is trivial...

I could add plenty of other phrases to this list that mathematicians are prone to use when trying to communicate that a particular concept is so simple that they refuse to discuss the details. I can't think of a time when words like these helped me. Usually they just make me realize how lost I am, and how much more the professor/author knows than I do. Even if I agree that the fact is basic, or the proof is trivial, it didn't help me to hear that it was easy or trivial, because I already thought that.

To get to my question, consider the following amendments:

This is a fact from...

It then follows that...

We have...

The proof is left as an exercise.

Is there ever a context in which it is more helpful for students to hear the first set of sentences as compared to this second set? Are words like "easy," or "trivially," ever constructive? In what setting? As math educators, should we work to weed them out of our written/spoken vocabulary?

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Something related was discussed on MathOverflow mathoverflow.net/questions/16193/… –  quid Apr 22 at 18:00
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Yes, they are helpful. For example, when there are multiple inequalities that could apply, I normally opt to use sharper bounds, however, it might be enough to apply something as rough as union bound. It's also a short way of letting the audience know that nothing unexpected happens. In other words, such a statement "prunes the search tree" and might lead sometimes to great speed-ups. Nevertheless, one should not use such phrases if they are not warranted, e.g. it only seems it's obvious, but was not properly checked. –  dtldarek Apr 22 at 19:32
    
Also related: Mathematics StackExchange and Academia StackExchange –  Joel Reyes Noche Apr 23 at 2:10
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This page is about the computer engineering use of "trivial", which is similar to the mathematician's use; you might find it amusing. fishbowl.pastiche.org/2007/07/17/… For me personally, "clearly" usually indicates where I'm about to make an error. :-) –  Eric Lippert Apr 23 at 13:01
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I'd add that I agree with your statement that either the characterization of a theorem as trivial is insulting to the reader if the theorem is not understood, and redundant if it is. I edit programming books as a hobby and often discourage writers from saying things like "It is easy to do X in C#"; the reader who already finds it easy likely does not have to read the book! –  Eric Lippert Apr 24 at 6:33

11 Answers 11

I can think of a few instances where it might be useful:

  1. To situate the current piece of the concept among others coming up
  2. To call-back to something earlier in the course that really should be easy to them at this point
  3. To intentionally make fun of something you know they thought was stupid (e.g. high school? if you're teaching a proofs course)

And, examples of each of these:

  1. To start off this proof, we prove this easy (or easier) theorem... (again, the goal is to tell them that something harder is coming soon)
  2. Obviously, we have 2n+1 is odd, by the definition of odd. (again, later in the course--not if you've just introduced quantifiers)
  3. And now, to finish off this proof, we do a trivial computation of this obnoxious integral. You all learned this in high school, right? (here, the goal would be for people to roll their eyes)

I do agree with you on the general idea though. We use these terms far too often when we're explaining things, and they are almost always not a good idea.

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On point (2), it can sometimes be outright confusing if an author doesn't acknowledge that an obvious step is obvious. I'm left worrying that I don't understand something, and it's more complicated than I think. –  Jack M Apr 22 at 18:03
    
Almost always being, "at all but finitely many times" or "at all but countably many times"? :D –  M. Vinay 9 hours ago

The main point here is that these words/expressions should not be used as substitute for an argument.

They obviously have some negative effects:

  • You evaluate your students by them and not in the positive sense. If you say "obviously", then it has a message that "it better be obvious". And if it is for someone not clear, then it is an evaluation that this person is not up to the course. This can turn into humiliation of your audience (or reader, if it is in notes) quickly.
  • If you turn out to be wrong, the you loose your face quickly. This is a consequence of my previous point: you humiliated your audience, and it turned out that you did it wrongly. You are not a person to be taken seriously.

You should, however, use these words in context, explanation. For example, to stress that though a calculation looks complicated, it is actually a simple idea what is behind.

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Yes! One of my old math teachers always banned us from saying "Basically..." when we were explaining our solutions up at the board. It was deemed "a put-down word", because it implied that if you didn't understand, you were stupid. –  SimonT Apr 23 at 0:11
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@SimonT well I wouldn't classify "basically" as one of those words. That can be used when you've just explained a complicated series of steps that can be summarized more concisely. –  David Z Apr 23 at 2:05
    
+1 for "should not be used as substitute for an argument". –  Nico Burns Apr 30 at 22:48

In addition to what @andras-batkai said, seeing words like 'obviously', 'clearly', and so on, in assignments or texts raises red flags and scepticism with me. (Recall that, for thousands of years, it was obvious that the earth was flat. Also, obviously a set contains all of its elements.)

As both a former student of formal logic and an occasional tutor, this is a pet peeve of mine. I feel these words should not be in educational or instructional texts (manuals, course notes, troubleshooting):

  • They don't improve comprehension. If something is obvious, by its nature, it doesn't need to be pointed out.

  • They put the burden of mathematical rigour on the reader, rather than the author. This makes it more likely for hidden assumptions to leak into proofs.

  • They sweep complexity under the rug. If it really is obvious, wouldn't the explanation fit into a footnote?

  • They discourage discussion. If the assumption does turn out to be wrong, students will be more likely to assume they misunderstand than they are to ask about it.

I did notice that this practice seems to be going out of style. At least in institutions I've worked with, lecturers are advised to take teaching classes, and it looks like this topic is given some attention.

P.S. The author experimented with the word 'obviously' in providing IT support to lecturers. The experiment was not well received.

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They aren't any four-letter words in the set, so they are allowed in polite company. But use them sparingly, and when you really mean it. When you say something is "simple," you should make sure it really is simple for most of your audience. Be careful, what is trivial to you (presumably the world expert on the topic you are writing about) can very well be a profound mistery to the average reader.

If in doubt, err on the side of explaining (a bit) too much.

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Have you deliberately combined "mystery to the average reader" with "misery to the average reader"? –  Matt F. Apr 22 at 19:54
    
@MattF., no. Chalk it up to "fortunate typo" ;-) –  vonbrand Apr 22 at 20:04

Yes, they are useful, but they can be over-used, or used when not true.

If you state "it is a basic fact from...", and the reader does not see why it follows, then they know that they're not following your argument as you intended it to be followed.

If you merely state "it is a fact from...", and the reader does not see why it follows, then they might think that it's a deep result and that you intend either that they should accept it on trust, or that you're presenting a lemma to be proved later.

Compare:

"It is a very basic fact from distributivity/associativity/commutativity that $(1-x)(1+x) = 1 - x^2$" -- the reader should be able to immediately visualize the proof.

"It is a very basic fact from arithmetic that $(1-x)\sum\limits_{i=0}^{n-1}x^i = 1 - x^n$" -- the reader can immediately see how a proof for a fixed $n$ might be carried out, and you're telling them that the calculations do indeed work out to save them the bother of writing it. The detail around the implied "for all $n$" is being ignored, which is a little shady. Really you want a proof that retains the summations and therefore involves distributivity over the summation symbol rather than just distributivity over binary addition, and that's still within the reach of the reader.

"It is a fact from the Axiom of Choice that every set has a well-order" -- the reader might well know or be able to construct a proof, but they're not expected to instantly produce it. It is not a "very basic" consequence except perhaps to an audience of skilled set theoreticians who are indeed all expected to trot out that proof.

"It is a fact from the Axiom Choice that a sphere can be decomposed into finitely many pieces, which themselves can be rotated and translated to produce two spheres of equal size to the first. This is the so-called 'Banach-Tarski paradox'" -- Information presented for interest and not proved. The student probably thinks "okaay, I can't even imagine how to prove that", but the proof might come in a sufficiently specialized undergraduate course.

There is a similar difference between "the proof is trivial" and "the proof is left as an exercise". If someone attempts the exercise and finds themself part way into a proof that isn't trivial, then if you've said it's trivial then they know there's a better proof they've missed. If you haven't they don't.

"Clearly follows" and "obviously" are similar, although I think they're frequently mis-used for things that objectively are neither clear nor obvious to part of the audience. It's just that the speaker thinks they ought to be clear or obvious and therefore doesn't care to spend time and screen real-estate on them.

There's another such phrase, "it follows immediately", which asserts that no new gizmos need to be introduced to the proof. To stretch the use of the term, it may turn out that "immediate" actually means a few lines of multiplying out brackets, but that's OK.

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There's two separate issues here. The first is whether there's important work which the word clear/trivial/obvious is doing when used in mathematical explanations. As explained by Jack M. and Steve Jessop and several of the MO posts, the answer to that question is yes: it indicates to the reader/listener that there's a very short argument that's being skipped over rather than something deep. The second issue is whether "clear/trivial/obvious" are good choices of words to use for the concept. I think in the context of teaching the answer is no. (In a research context they're fine, because all readers will know that they're technical terms of art that don't actually mean what clear/trivial/obvious mean in everyday speech.) However, I don't know what would be a better choice of word which would be less likely to be misunderstood. Has anyone found a word choice that works better with students?

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I would argue that they are.

When one reads a word such as "clearly", it is a sign that an argument has been omitted, and that said argument should be relatively easy to find. I think it engages the reader and makes them "participate" with the material more.

I wouldn't worry about making people feel bad when they don't get something - everyone eventually has to learn to get over that feeling (and we all experience it, time and time again). Students who are assertive will ask you if they didn't understand something you said or wrote.

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There is a difference here. When stating that something "clearly" holds, then is is also stated that it is expected of you to get it, and if you don't get it, you are not up to the course. This would not be a big problem if it would be objective. But it is highly subjective what is "obviuous" and what is not. Independently of the mathematical ability. –  András Bátkai Apr 23 at 11:35
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Feeling bad for not getting something shouldn't be accompanied by having to put up with someone telling you (written or otherwise) that it is obvious/clear/easy to see. Moreover, like András said, "relatively easy to find" is also subjetive. –  Mark Fantini Apr 23 at 21:06
    
I think that worrying about how people feel when they hear words like "easy" is an issue that really shouldn't be ignored. It is obvious to any instructor who cares about understanding that a student who feels like everyone else finds something easy (even if they don't) will be unable to fully engage with the material. I take your point about it signalling a missing argument, but I think it is important to contrast the difficulty constructing an argument with the ease of presenting it. What is clear or obvious in presenting an argument is often difficult to come up with in the first place –  jbaldus Apr 25 at 2:16

I would say that "trivial" is okay, the others are probably less so. While "trivial" has connotations of "exceptionally easy", in math speak we often use it to roughly mean "an exceptionally simple statement, requiring little complexity to prove or define". I'd probably shy away from "the proof is trivial" a little more, but even so, if it's a tiny 3 line lemma, even if it's not easy and involves a huge leap of logic, it's still "trivial" by my metric.

I view "trivial" like "simple", and "simple doesn't mean easy" is practically a catch phrase of mine at this point. Things can be simple in that they only rely on two or three axiomatic facts, but actually thinking of which two or three axioms to use isn't always easy. It may pay off to remind students of this fact. "It's not exceptionally complex, but can be hard to understand" goes a long way.

Basic is the next I'd be okay with, but only in limited cases. I'd pretty much solely use "basic" to refer to things like so-called "basic facts" (2+2=4; you can't divide by zero, etc), or more generally literal axioms of the system you're working in.

"Obviously" and "clearly" I'd avoid at all costs, unless maybe you're saying "2+2=4". Though I've been known to use "obviously" in humor, especially with contradictions ("And, well, obviously 0=4 so..."). It just alienates people who didn't get the intuition. Unfortunately, these two words are almost the proof-writer's version of "um...". People don't notice they're saying them, they're filler words. I always try to proof read them out of anything I write, it actually tends to make things more readable in addition to not alienating the students who don't understand a concept.

Perhaps one tiny caveat is that sometimes look more complex than they are, and it can be useful to reassure students that this gnarly thing isn't really that scary. Newton-Raphson comes to mind as something that made my brain overflow when I first saw the formal definition of it in a textbook we were using. Being assured that, yes, it really is just this simple iterative process was helpful in making it less scary. See also: functions with 25 Greek letters where 24 of them are stupid constants.

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Functions with 25 Greek letters are highly suspicious. –  user11235 Apr 30 at 19:50
    
@user11235 Deliberate absurdity, I know the Greek alphabet only has 24 letters :p –  Jsor Apr 30 at 19:53

Obviously, and its ilk, do not contain much information. They can be replaced by referring to the tool that is used to prove or notice the obvious fact.

For example: By triangle inequality, by definition (of something specific where necessary), by calculation, by integrating by parts.

Also: By lengthy calculation, by very clever choice of test function, and so on.

Here the statement communicates something about the difficulty and the tools used, thus making it easier to immediately verify something, or alternatively telling why one should not be discouraged at not immediately seeing it.

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I think the most important thing to keep in mind is your audience. Obviously if you were teaching a calculus class you wouldn't have to remind them on how to add fractions. Well, in this case the correct word might be "hopefully". Hopefully you don't have to remind them, but I've had calculus students that couldn't add 2 fractions to save their lives. So one should really think about one's audience first.

However, the use of such phrases builds confidence if used appropriately. I remember when I was a student and my professors would use "trivial" or "obvious" ...sometimes I would agree and other times I would disagree. When I agreed, I felt that all my studying paid off. The problem is when the student disagrees. In that case, it's up to both the student and the professor to bridge the gap. This is why it's important for both students and teachers to ask questions.

This is just one facet of the question that I feel wasn't mentioned.

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I don't have the rep to comment, but @Carlos - if you use hopefully there's no clarity of meaning between a) I hope I don't have to remind you of the standard method used to add fractions and b) I hope the standard method of adding fractions, which you're familiar with, is true in most instances.. –  portll Apr 23 at 21:38

These words can be quite helpful, because it is important to know how difficult a skipped pproof is. The problem is that these words have contradictory meaning, they depend on context and are often used out of laziness.

Since it can be quite frustrating to get stuck on an abused "trivial", the trust of students and readers in this kind of word can be low, so it is almost always better to be more precise, either by giving more details on the difficulty:

"The fully-written proof of this lemma would take half a page of intermediate difficulty among the proofs written in this book."

"The reader can verify the base case of the induction (same difficulty as the one-star exercices in this book)."

or be more honest on the sense of trivial:

"The proof of this result is elementary, uninspiring and long, so we skip it."

"The proof of this result is so well-known that I expect my readers to know it."

"I am so scared about boring anyone that I would rather risk losing half the audience."

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