When is something "obvious"?

Question

I try to be a good student but I often find it hard to know when something is "obvious" and when it isn't. Obviously (excuse the pun) I understand that it is specific to the level at which the writer is pitching the statement. My teacher is fond of telling a story that goes along the lines of

A famous maths professor was giving a lecture during which he said "it is obvious that..." and then he paused at length in thought, and then excused himself from the lecture temporarily. Upon his return some fifteen minutes later he said "Yes, it is obvious that...." and continued the lecture.

My teacher's point is that this only comes with a certain mathematical maturity and even eludes the best mathematicians at times.

I would like to know :

1. Are there any ways to develop a better sense of this, or does it just come with time and practice ?

2. Is this quote a true quote ? If so, who is it attributable to and if not is it a mathematical urban legend or just something that my teacher likely made up ?

Answer 1

Like Florian, I really like Gowers' definition of obvious. Of course this is a very personal definition. A proof that instantly springs to mind for one person may not spring to mind for another. I am not really sure what there is to say at this level of generality beyond that.

Really phrases like "it is obvious that..." and "clearly..." are bad habits. In a mathematical argument they are the places you should look at first for possible errors.

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Perhaps another story will be illuminating: a professor of mine once made an assertion in lecture that I didn't quite see instantly. I asked him "is that obvious?" and he replied "yes." I asked him "is it obvious that that's obvious?" and, after a short pause, he replied "no."

Answer 2

I really like the following definition (here given by fields medalist Timothy Gowers, and he credits his former colleague):

A statement is obvious if a proof instantly springs to mind.

However, for many mathematicians and teachers the meaning of "obvious" unfortunately is much broader.

Answer 3

I had an economics professor who thought one result was obvious and spent over 3 hours explaining the "obvious" result to a fellow mathematician. After 3 hours, he realized it might not have been so obvious and found it to be, in fact, false, and published 3 papers from the counter-examples.

Answer 4

I believe the famous mathematician was G. H. Hardy, and I am sure he had a different view as to what is "obvious" compared to lesser mortals (like myself).

I would say that "obvious" should only ever be applied to things that the speaker believes that his listener should find "simple" ..

Answer 5

The level of triviality of a mathematical is relative to the observer. That is to say, a mathematician who has worked in geometry his entire life would find a certain set of geometric statements $P$ obvious, whereas an undergraduate student learning from his/her first geometry textbook will likely only consider some subset $M \subset P$ of the statements that the mathematician does as obvious.

A good illustration of this would to consider the following statement,

Project the edges of a regular icosahedron $I$ inscribed in $\mathbb{S}^2$ onto $\mathbb{S}^2$ from the center of $I$. Then $\mathbb{S}^2$ is obviously divided into 20 equilateral triangles of area $4\pi / 20 =0.628...$

Embedded in the statement is quite a few trivial facts that an undergraduate student completely new to geometry (maybe this particular example would only be frustrating to a high school student) may not be aware of, and so upon reading this statement for the first time may be frustrated by the use of "obvious".

My answer to the question is that as you navigate your way through learning more and more mathematics over time, you are expanding the cardinality of your set $M$. There does not seem to be a way to intentionally increase your knowledge of trivial or obvious statements while not learning the other non-trivial parts of mathematics that those trivial statements hint towards.

Answer 6

That quote is often told of G. H. Hardy. You will find it in Lion hunting & other mathematical pursuits, by Ralph P. Boas Jr:

The story is told of G. H. Hardy (and of other people) that during a lecture he said ‘It is obvious… Is it obvious?’ left the room, and returned fifteen minutes later, saying ‘Yes, it's obvious.’ I was present once when Rogosinski asked Hardy whether the story were true. Hardy would admit only that he might have said ‘It is obvious… Is it obvious?’ (brief pause) ‘Yes, it's obvious.’

Answer 7

I consider this in the same vein as "it is easy to see that..." , which is probably encountered more often than "it is obvious that..." and there is a good discussion about that here.

Answer 8

A statement attributed to Littlewood in a footnote of Cassels' book "Geometry of numbers" says "Every equality is trivial when it is known"

Answer 9

Mathematical statements are only evaluated by individuals. Since individuals differ in mathematical ability, the answer is that "something" is never obvious to everyone or to yourself. The crux of the joke is that it was only obvious to the professor after reflection, which is deliberate irony since there would be no point in reflecting if something were explicitly obvious. Hence, the point is that if even the expert professor had to make sure it was obvious, then students should only check axioms even more diligently, no matter how hallowed the axiom.

Answer 10

I have never experienced a non obvious "obviously" in a textbook in mathematics. If it wasn't obvious at first sight it became obvious after some investigations.

To me, what is really obvious is a claim that is true and can be easy proved, but that might not be that obvious from the wrong perspective.

Anybody can fail to see what is obvious and anybody with appropriate knowledge can agree after some thinking. So there is no way to develop a sense of it, but having knowledge about the subject.