Why should we prove obvious things?

Question

Obviously, there are obvious things in mathematics. Why we should prove them?

• Prove that $\lim\limits_{n\to\infty}\dfrac{1}{n}=0$?
• Prove that $f(x)=x$ is continuous on $\mathbb{R}$?
• $\dotsc$

Just to list few examples.

Answer 1

Because sometimes, things that should be "obvious" turn out to be completely false. Here are some examples:

1. Switching doors in the Monty Hall problem "obviously" should not affect the outcome.
2. Since Gabriel's horn has finite volume, then it "obviously" has finite surface area.
3. "Obviously" we cannot decompose sphere into a finite number of disjoint subsets and reconstruct them into two copies of the original sphere.
4. Since the Weierstrass function is everywhere continuous, then "obviously" it must have at least a few differentiable points.

Of course, mathematics has shown that switching doors is to the player's advantage, that Gabriel's horn actually has infinite surface area, that you can indeed get two copies of the original sphere (see Banach-Tarski paradox), and that the Weierstrass function is everywhere continuous but nowhere differentiable. The point being, there are many things out there which are "obvious" but actually turn out to be entirely counterintuitive and opposite what we would otherwise expect. This is the point of rigor: to double check and make sure our intuition is indeed correct, because it isn't always.

Answer 2

I think the answer has four parts.

1. If you ask a random person at Walmart what $$\lim_{n\to \infty} \frac{1}{n}$$ is, then you might not get much. If you tell them that it is $0$, then they probably won't think that it is obvious.

   Conversely, if you go to a high level research talk, you will hear "It is obvious that ..." or "It is clear that ..." a lot. And you might not think that it is very obvious.

   The point is: Whether something is obvious or not is relative to the person.

2. Mathematics is built around proving things. There is a justification for everything. This is the very nature of mathematics. We start with some axioms and then we prove everything. So when you ask, "Why should we prove something?", the answer always contains: "Because we are doing mathematics."

3. You don't need much experience teaching mathematics before you meet a student who is confused about losing points on an exam because of lack of justification. Often the student will respond that they just thought that it was obvious. When you press them a bit harder it becomes clear that they, in fact, have no idea how to justify what they did. Whether or not the student arrived at the correct answer is irrelevant; the point is: if something is truly obvious, then it shouldn't be hard to prove it.

4. If you want to get good at proving difficult things, why not get experience with proving things by starting to focus on simple or "obvious" things? I think that the experience gained from proving even simple propositions is valuable later in your career as a mathematician.

Answer 3

The main procedural reason is to show that your axioms correctly capture what you want them to capture: that is to say they are both "correct" and sufficient.

If it turned out that under our axioms $\lim\limits_{n\to\infty}\dfrac{1}{n}\neq0$ then we would probably choose a different definition of $\lim$ (or a different name for it), since it would not be describing anything that we'd like to call a "limit". It would not be "correct". Of course that wouldn't be a problem in some unusual topology, since by calling it "unusual" we mean that we don't expect it to behave the same as the usual one, so limits might be different. There may be a fine line between a result that's counter-intuitive but that we stand by our system anyway, and a result that causes us to conclude that our definitions or axioms aren't as useful as we thought they were.

Consider that Euclid tried and failed to prove the "obvious" parallel postulate. Fast forward 2000 years or so, and it's finally proved not to be a theorem of Euclid's other axioms. His first four axioms were not sufficient to describe what was "obvious". Furthermore, non-Euclidean geometries (in which the postulate is not true) are interesting and useful.

It is valuable to know whether or not "obvious" things are provable from your axioms.

When learning mathematics, it's useful to prove "obvious" results in addition to "non-obvious" ones because:

• you "know" they're true before you start, which can save some frustration
• the ease or difficulty of proving the obvious teaches you something interesting about the area you're working in
• you train yourself to reason only using formal axioms, not by assuming any old "obvious" things you like, no matter how tempting they are
• similarly you train yourself to accept from others only things that are proven, no matter how true they look
• probably other benefits.

Then when something is stated as "obvious", or you want to state it so yourself, you quickly either prove it to yourself, or at least satisfy yourself that a proof is possible and you could write it out if really needed, or else you question the "obvious". It might turn out to be false (in which case you've avoided an error) or it might turn out to require quite a difficult proof (not so obvious after all despite your intuition being correct). Normally you would want to restrict the use of the word "obvious" to things where the first proof your reader would think of works (and hence anyone can easily prove them if they bother to write it out), not to things where your intuition is correct but the proof is tricky.

Answer 4

I always say that the most difficult exercise in my undergrad studies was the first question in linear algebra. We were taught about the axioms of a field. Then we had to prove the following thing:

For every $x$, $x+0=x$.

The catch is that the axioms we were given stated $0+x=x$. So we had to use the axiom of commutativity first, then we could conclude that.

Why was that so difficult? Because of two reasons. The first is that you had to understand not just what you should prove, but also why you should prove it. The second reason is that you had to come up with a proof which was not the single word "obviously" or "trivially".

So why do we have to prove trivial things?

1. Because it's good practice. It's a great practice for understanding why, how and what to do when you're writing a proof. And the good thing about trivial things is that they are trivial and you know they're true, so you don't have to bust your hump in order to prove something which might not be true after all.

2. Because it teaches you to sit down and prove everything. Later on in your studies, you might have to prove more complicated things, and sometimes things look obviously true, but since you don't spend time proving them, you will take them for granted, only to waste precious time before realizing they are false.

   If you sit down to prove everything, you'll learn to do that later on in your work, and avoid wasting time on false assumptions, like I've done recently. Several times. (Yes, do as I say, not as I do!)

3. Because trivial things are generally easy to prove, and it makes sure that you understand the process of finding the proof by verifying the definition. To show that $f(x)=x$ is continuous is easy. Given $\varepsilon>0$ take $\delta=\varepsilon$ and you have that that if $|x_0-x|<\delta$, then $|x_0-x|<\varepsilon$.

   By doing so, you review the definition of continuity, you understand it better. Using this understanding it is easier to tackle more difficult questions.

4. Because kihon is as important, if not more important than advanced ideas. Let me digress and tell you a short story about my past (it's a true story). Some decade ago or so I used to practice ninjutsu for about a year and a half. It was great, I loved it. We had a great group, and a great sensei, whom despite not seeing for the better part of ten years now, I would still rally to his call immediately.

   We would often do advanced techniques involving throwing, or evading weapons, or using weapons, or whatever. But he would constantly remind us that kihon is the most important part. Kihon, in Japanese, is basics. In the context of ninjutsu it means that you have to know how to punch properly, how to kick properly, and how to fall properly. If you know that, then you have a much better chance of winning a fight (and in ninjutsu, generally, there's no scoring or rules, the winner is the guy who can walk away).

   What does that mean? It means that the guy who spent a month punching fifty thousand punches, will never punch improperly, and in a fight he will have a better chance to survive than someone who give half-assed punches, but can make a really mean throw.

   So what does all that have to do with mathematics? Kihon. In the context of mathematics kihon means three things. It means being able to understand a definition, it means being able to understand the problem that you have to prove, and it means being able to write a proof.

   If you try to jump, your kihon is weak. And it will haunt you. Trust me on that. You will get stuck later on, and it will trouble you. But if you sit down to write a proof why $x+0=x$, then you understand the definitions of a field (i.e. the axioms), and you understand how to read the problem (i.e. why do we have to prove something here), and you understand how to write a proof (i.e. well, really just how to write a proof).

   These skills, the mathematical kihon, will make you a mathematical ninja at some point. And the better your kihon is, the better you will be.

So sit down to write proofs for trivial things, but remember that as the levels go by, you can allow yourself leeway. When you've mastered one level, it's okay to "trivialize" certain proofs; but from time to time it's also good to repeat them.

Answer 5

Things that are obvious to one person are not necessarily obvious to another. Futhermore they dispel (most) skeptics. Just thinking they are true does not mean they are true. For example, before I entered university, I was under the impression that there were twice the number of elements in $\mathbb{Z}$ than in $\mathbb{N}$. I would've called this obvious, but after learning more about it, it is no longer "obvious", as $|\mathbb{N}|=|\mathbb{Z}|$.

If I tell you that $|\mathbb{N}|=|\mathbb{Z}|$, you might not believe me, but if I proved it to you- showed you without a shadow of doubt that my assertion was true, then you would believe me. So, it is a sort of argument to show the reader that a claim is true.

Answer 6

There was just an answer I provided a couple of minutes ago that was wrong. The question was:

$$\lim_{n\to\infty}(1+2^{-n})^{2^{n+2}}$$

What I thought:

$$\left(1+\frac1{2^{\infty}}\right)^{2^{\infty}}=\left(1+0\right)^\infty=1$$

But the answer was actually $e^4$. Even the computer made a mistake (when $n$ got too high). Sometimes, something that seems obvious may be wrong.

Answer 7

Because obviously a continuous function must be piecewise monotone. And therefore differentiable at all but at most countably many points.

Ampère "proved" this result in 1806 and it was considered a theorem for quite a while.

Then Riemann came up with an example of a function than when integrated produces a function (i.e. $x \mapsto \int_a^x f(x)\,dx$) which is not differentiable on a dense set of points.

The final nail in the coffin came in 1872 when Weierstrass published his proof that $x \mapsto \sum_{n=1}^\infty b^n \cos(a^nx)$, $0<b<1$, $ab > 1+3\pi/2$ is continuous (because the series is uniformly convergent) and nowhere differentiable.

The history of real analysis putting the results of calculus on a solid footing is peppered with plenty of such stories. Things that sound intuitively obvious but are in fact wrong.

Analysis by Its History (Undergraduate Texts in Mathematics) by Ernst Hairer and Gerhard Wanner is likely to contain more examples, but that's the most famous one that I can think of off the top of my head.

Answer 8

My old real analysis professor once made the following statement:

Mathematical proofs come in three types: proving the obviously true to be true, proving the not-so-obviously true to be true, and proving the obviously false to be true.

He goes on to give an example of the third kind of proof with the following theorem:

Theorem. Let $I$ be an interval and $f : I \to \mathbb{R}$ strictly monotone. Then the inverse function $f^{-1} : f (I) \to I$ is continuous.

The point is that some "obvious" things are not so obvious at all, and the truth may turn out to be the complete opposite of what you expected.

Answer 9

Just an addition to the many good answers here: Every axiom is obvious, and every rule of inference is obvious. (Otherwise, they wouldn't be very good axioms.) Thus, every single proof in existence is a long string of trivialities. Thus if you agree that two trivialities make a triviality, then everything in math that is provable is also trivial, and a proof is merely a mechanism for showing how trivial the result is. This may seem like an argument to absurdity, but in fact it is the case that every proof is "trivial", once you understand the steps. That's what the proof is there for.

So, to answer the original question, we prove things in order to show how obvious they are, not the other way around (which is what occurs in the minds of people that would call the fact "obvious"). Everything is obvious under the right mindset; a proof just shows people the right mindset to use, after which they too will find the result obvious. That's how mathematical knowledge is transferred.

Answer 10

Because your first example is not even true without suitable conditions.

The limit of 1/n as the integer n increases forever depends on your number system. In the standard reals the limit is indeed zero. However in the reals augmented with infinitesimals (as nonstandard reals) the limit does not exist because every positive infinitesimal is strictly less than every 1/n when n is a positive integer, and there is no greatest positive infinitesimal (since 2x > x when x is positive).

Anyone claiming it is "obvious" that this limit is 0 is implicitly claiming it is "obvious" that infinitesimals don't exist. Why should that be obvious?

Setting infinitesimals aside, consider the standard complex numbers. Suitably defined, it is indeed the case that lim 1/n exists there and is 0. But how do you define "limit" so as to make this true? Could it not happen that the limit does not exist because the set of complex numbers closer to 0 than any given 1/n cannot be linearly ordered?

One can rule out this possibility with a suitable definition of limit, but "obvious" is no less appropriate for the complex numbers than for the reals augmented with infinitesimals.

Answer 11

Here's an obvious one:

There are half as many points on a number line between 0 and 1 as there are between 0 and 2.

(Of course, as "obvious" as that may seem, it's not true. In fact, there are the same number of points between 0 and 1 as there are between 0 and 2. But proving that in a way that would convince a skeptic may be a bit challenging.)

That said, the main reason for proving obvious things is that proofs are the fundamental building blocks of mathematics. If something is true, a mathematician should be able to prove it. If something cannot be proven, that will (or should) stick in the mathematician's craw. It is not enough to appear true, it must be proven true.

Two examples that spring to my mind are:

• the four-color map problem, and
• Fermat's last theorem

Both of these were widely regarded to be true (no counterexamples could be found), but the proofs were elusive for decades or centuries.

The long and short of it: No proof, no truth – no matter how obvious something may seem.

Answer 12

Someone called Jerry Bona once pointed out the following.

The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?

The point is that all three statements are equivalent, but the obviousness of the statements vary.

The Axiom of Choice states that if you have some infinite sets then you can take an element from each set and make a new set out of these elements. This is obviously true.

The Well-Ordering Principle states that every set has a well-ordering (so a total order where every subset has a least element). This is clearly false! (It is far too strong.)

Zorn's Lemma states that if you have a partially ordered set $P$ which has the property that every chain has an upper bound in $P$. Then the set $P$ contains at least one maximal element. I have only recently understood what this actually means.

Informally, the Axiom of Choice is this: If you are in an old fashioned sweetie shop, with some infinitely big sweetie jars, then you can take one sweetie from each jar and put them all in another jar. Just take the top sweetie, you say? Well, that requires that the sweeties in each jar are well-ordered...

Answer 13

In the 19th century it took a while to clarify the notion of continuity of a function. Ampère actually published a "proof" that any continuous function is differentiable, period. Even Cauchy seems to have felt that a continuous function should have at most a finite number of points of discontinuity (Boyer in his book erroneously reverses the roles of Ampère and Cauchy). For a more detailed discussion see this article. Such a "theorem" must have seemed intuitively "obvious" at the time.

Answer 14

The obvious is hard to prove and often wrong.

Answer 15

While enough counterexamples have been covered and the subjectivity of obvious has been noted, there is another subjective aspect of "it's obvious" that has to do with the student's own mathematical education. I would like to add this one little thing to the answers here: The advances in one's education proceed on many fronts, but a step forward occurs when the student recognizes that something that used to seem obvious is in fact more complicated.

A friend of mine in graduate school once got a fortune cookie, "Never try to prove what no one doubts." Oh, did that get a laugh! But it is not such a bad principle. A good teacher ought to be able to create such doubt in a student, at the appropriate time. One of the earliest doubts I recall, certainly from high school but maybe earlier, is whether or which things could be proved at all. It was a naive view and included nonmathematical things. Perhaps my interest was idiosyncratic and one of the reasons I came to like mathematics.

I alluded above that timing can be important. There is inevitably an order in which topics are discussed in education. I don't think it helps much to prove properties of arithmetic before the student notices that there are properties of arithmetic. In the case of arithmetic, we usually wait to long and the moment when the student is curious about the properties passes. By college they are as familiar as the fact that the sky is blue. I don't think it does much good to simply assert that obvious is subjective, that if it is obvious then you should be able to prove it easily, or give them counterexamples that they cannot understand. You have to show them things they can understand. If they are ready, a counterexample can be just the thing.

On the other hand, that obvious things are easily proved is not clear to me. It seems pretty obvious that there is an empty set. Perhaps it can be proved based other axioms. And perhaps, someday, we can have a system of nonobvious axioms to prove all the obvious things. :) Going further down this road to absurdity won't be useful, so let me turn back. The problem I see is that if the student does not see the need or the usefulness in proving, then the student is free to disregard the teacher's authority or the accusation implicit in the charge, if it's obvious then it's easy. Students have been made to do a lot of boring busy work throughout high school. If doing the proofs seem that way to them, then the teacher is losing their attention and possibly their respect.

My purpose, prompted by the education tag, was simply to add the dimension the development of human understanding to the answers, which so far have mainly addressed mathematical aspects of the question.

Answer 16

There are at least two reasons:

1. If you wish to improve or refresh your knowledge of how the definitions work.

2. If you wish to improve or refresh your knowledge of useful general theorems proven by others, that have the obvious result as a special case. For example, the second statement trivially follows from (a) that all polynomials are continuous, or (b) that all differentiable functions are continuous, etc. etc. ... I wonder what is the most complex theorem that it trivially follows from?

Answer 17

If it's obvious then the proof should be trivial and pose no inconvenience. If you find it difficult to prove then perhaps it's not so obvious.

Answer 18

In my opinion, only axioms should be treated as obvious, above all while a theory is being explained to others. I think it is simply immoral for a mathematician who is writing a proof of a proposition in a book not to give every smallest detail in the chain of logical inferences, skipping the task of making the subject perfectly clear through a lazy abuse of the magic words trivial, obvious, exercise. If one wants to write mathematics for others, he must be as clear and rigorous as possible, at least while developing theory and examples. If instead he's too lazy to bear such a sacred task, then he should not even begin to write mathematics, because the essence of mathematics is in its logical clarity, and not in a pseudomystic vagueness - alas, many important books of great mathematicians are "written" in this terrible style! -.