# What really is mathematical rigor? How can I be more rigorous?

I'm an undergraduate mathematics student who has received some constructive feedback from two instructors at the end of my exams. Namely, that I am a bit hand-wavey and not always very rigorous. While I greatly appreciate this feedback since I intend to apply to graduate school, I worry because when I hand in assignment, I generally feel the proofs are rigorous already, and I'm not sure what I'm missing. Obviously I will be engaging in a dialogue with my professors about what I can do to be more rigorous, but I'm hoping the experts on this forum can provide some examples or distinctions between rigorous arguments and arguments that are close, but maybe gloss over important details so I can more clearly see the difference.

It is worth noting that I have taken a course in mathematical reasoning and logic and did very well in it. Something has happened in the last year or so to reduce the quality of my arguments, or rather, the expectation has gone up and my level of rigor has not matched it.

I think another aspect of what makes this a troublesome problem for me is that i typically do really well on homework problem sets. 90% consistently, sometimes with little mistakes. I feel like I am getting mixed signals from some of my classes when I am consistently told I am doing well, but am given this advice. All of this is said without spite or malice, I just sometimes feel confused about my own level of understanding.

So if anyone has any general advice, or useful examples of arguments that look rigorous but need to be patched up, that would be greatly appreciated. I want to get a sense for what a really solid proof looks like compared to a less polished argument that looks passable to someone still with naivete in them.

Edit: Here is an example of one such question in which I struggled for a long while to produce the proof posted, and even then I was missing something to be fully rigorous. Homology groups of orientable surfaces.

• Maybe you could share with us (part of) the proofs that were the subject of this feedback? As it is, I feel the question could be closed as too broad. – zarathustra May 15 '15 at 19:34
• This most recent comment came from a final exam. I can link you another thread where I struggled with rigor. – Alfred Yerger May 15 '15 at 19:35
• Just a little bit thought. As an undergrad math major, I am not smart (in fact I'm not even close to it). So I always write my proof in hardcore detail and that never got me into trouble. I guess if someone who has little math background can follow your proof and be convinced then you're good to go. – 3x89g2 May 15 '15 at 19:35
• By the way by saying "has little math background", I mean someone who has taken several college math classes including introductory real analysis. – 3x89g2 May 15 '15 at 19:36
• change to physics, we don't care about rigour! :) – tired May 19 '15 at 18:04

You should be able to delineate the precise mathematical theorems that allow you to make each step in a proof. For example, if you have $(x,y) \in \mathbb{R}^2$ and you write: let $r,\theta$ satisfy $x = r\cos \theta,y=r\sin \theta$ with $r\geq 0$ and $2\pi > \theta \geq 0$, you are using a theorem that says that:

Proposition. For all $x,y \in \mathbb{R}$, there exists $r \in \mathbb{R}_{\geq 0}$ and $\theta \in \mathbb{R}$ such that $x=r\cos \theta$ and $y = r \sin \theta$ and $2\pi > \theta \geq 0$.

Make sure you can explicitly write out the theorems that allow you to make the steps you are making.

The other thing is that you need to develop an intuition for what the instructor (reader, etc.) expects you to take for granted. You may need to write some follow-up lemmas if there are steps in your proof which invoke theorems that you really shouldn't be taking for granted.

• So I should take great care to challenge my own assumptions. Make sure I am assuming nothing more than what is given. I will make sure to pay attention more closely to this in the future. – Alfred Yerger May 15 '15 at 19:46
• @AlfredYerger, more than that, be aware of the theorems that justify each inference. If you can't identify the precise statement of the theorem that allows you to make an inference, you may need to divide that inference up into multiple steps. – goblin May 15 '15 at 19:47
• So for every statement which is not a purely arithmetic operation, I should be able to cite some result which I am given or have already shown to justify it. – Alfred Yerger May 15 '15 at 19:48
• @AlfredYerger, not quite. You should be able to state a result that justifies the step. Whether or not you actually need to prove that result is a complicated matter: obviously, if the result is given in the lecture notes, you do not have to prove it, but even if its not explicitly given, it is often okay to take results for granted, as long as they're well-known or "standard" results. If they're specialized results, you should offer a lemma/proof pair in favor. – goblin May 15 '15 at 19:51
• @AlfredYerger, possibly. Another good trick is to avoid vague references: instead of "this implies the statement under question," write "this implies $(*)$" and go back and put a $(*)$ before the statement under question. – goblin May 15 '15 at 19:53

I can relate. Throughout the years, many of my exams and homework assignments had deductions for not being rigorous enough.

The trick to mathematical rigour is managing the leaps that you make from statement to statement. The size of the leaps is inversely proportional to the rigour.

Let me mention some heuristics that tend to work well for me:

• If you spend a long time thinking on a step, also spend a long time on writing it down — apparently it isn't clear at first sight.
• When using theorems relevant to the subject matter and the level of the proof, mention them. If necessary, explicitly verify that the hypotheses are met.
• Try to verify how your proof works in a small or otherwise manageable example.
• Whenever you give in to the temptation of using "obviously", "trivially", "clearly" and their brethren, or just leaving an intermediary step out, pay extra attention to make sure that the leap you're making is indeed small enough to be disposed of using these terms.

That should hopefully be of some assistance in assessing your own rigour.

Two obvious pointers remain:

• Practice. You will get there.

As a side remark, I've always found writing answers on Maths.SE a good means to combine these.

• This summer I am doing a reading course with one of the instructors who said this, so I fully expect to improve a lot by the end. I'm just looking for some advice to work with until then, and this is definitely what I had in mind. In another comment another user and I talked past each other a little bit, which makes me thnk that my writing isn't always clear. I think I fall into some of these traps. Especially wih phrases like "obviously, etc." – Alfred Yerger May 15 '15 at 20:12
• "The trick to mathematical rigour is managing the leaps that you make from statement to statement." Sooooo true. I like to conceptualize rigour in terms of formalizability: the greater the computational difficulty of removing the gaps and omissions in a proof, the less rigorous it is. – goblin May 15 '15 at 20:43

A proof of a proposition is rigorous if it convinces the reader that the proposition is true beyond a reasonable doubt. In math, as in everything else, what constitutes "reasonable doubt" is flexible. It depends on the cultural context:

It depends on the situation at hand:

• What's rigorous in a published paper, meant to convince experts of a new result, may not be rigorous in a homework assignment, meant to verify that a student really knows what they're talking about.
• What's rigorous when teaching middle school students, who are often very willing to rely on intuition, may not be rigorous when teaching graduate students, who know from bitter experience how intuition can lead them astray.

Knowing what level of rigor to use in a given context is a social skill, and it can only be learned through social interaction.

To get acculturated to the standard of rigor in your courses, look at proofs written by other people—your classmates, your teachers, your textbooks' authors. Look at places that teachers have flagged as gaps in students' work, and see how other people bridged those gaps. As you suggested, ask your teachers about how you can strengthen your arguments in places where they're too weak for the course.

I think it's normal, in an undergraduate course, for a proof to get a good grade but also a note that it ought to be more rigorous. As an analogy, if I were marking an essay for a planetary geology class written in English by a student who was still learning English, I would flag awkward phrasing and grammatical errors to help the student improve their writing, but I wouldn't punish the student grade-wise for them. In an essay for a English writing class, of course, it might be a different story. I think most math teachers feel that students should be nudged toward an appropriate standard of rigor to help them improve their writing, but not punished harshly for straying from it, except of course in a class devoted to teaching a certain standard of rigor.

TL;DR. The bad news is that rigor, like all human social standards, is somewhat vague and arbitrary, and thus frustrating to learn. The good news is that you're a human, so you'll learn it. Good luck!

p.s. If you enjoyed this answer, you might also enjoy Eugenia Cheng's "Mathematics, morally," which touches on many related topics.

• "Sufficient unto the day is the rigour thereof" (T.W.Korner in The Pleasures of Counting). – David May 21 '15 at 3:56
• I think tl:dr's are best placed at the top of a post so someone doesn't have to scroll through the whole post to get the gist of what you're saying. – Alex Ortiz Sep 16 '16 at 1:38

For rigour you must Either quote an authority for each step yoiu make, or prove the step by whatever intermediate steps are needed: For an example of rigour look at Euclid's "elements of Geometry" where the first thing is a definitionof point, line, plane surface, then Euclid's basic tool, the congruence of triangles is proved, by methods which would ring true to a modern day topologist: the triangle congruences: three sides ewual to three sides (SSS) then two sides and the included angle, (SAS) and two sides and the corresponding angle ASA) are all in detail proved, with the fourth; Right angle, hypoitenuse and one side, then subsequent proofs use these to reason the new deeper property being canvassed. Hope this helps/ Bill Greig (BSc St And)