# Rigor in Mathematics

Since starting my undergrad studies (in Maths -- !) I have struggled to grasp the concept of "rigor". My "proofs" tend to have the right framework but somehow they are said to "lack rigor", this has cost me many marks and is very frustrating... Any advice?

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Perhaps you should provide a concrete example of a proof you wrote which you feel "lacks rigor." I feel like this will help make your problem clearer and help people help you. – Deven Ware Aug 27 '11 at 22:05
Every step you make, ask yourself, "What is it exactly that I'm taking for granted here?" and "If I was tutoring someone who didn't understand this, how should I explain why this is justified?" – anon Aug 27 '11 at 22:19
Yes, without an example, we can't really give advice. As for not having any examples, where are you getting docked - only on tests? If on homework, post one of your homework answers. – Thomas Andrews Aug 27 '11 at 22:20
It's certainly a fine line between rigor and pedantry. For now, err on the side of being pedantic. One thing to keep in mind are that words have specific meanings in math - don't refer to a "number," refer to a "real number," an "integer," etc. Even if the context is clear, the word "number" is (rarely) used alone - it is too general a term. Mathematicians don't use words that they haven't previously defined, so if you start using casual terms, you are bound to lose rigor. – Thomas Andrews Aug 27 '11 at 23:15
Another way to learn how to right proofs with the appropriate amount of rigor, is to read good mathematics! So if you are doing calculus, put down Stewart and start reading Apostol. Then, in the exercises at the end of the chapter, try to imitate what he did earlier. – Ragib Zaman Aug 28 '11 at 1:50

I think one important response to this is that it is not only about mathematics, but about language, and implied context. That is, in ordinary use of language, there is usually so much shared, understood context that what we say can be very imprecise, just a hint, in effect referring to a huge body of prior understanding. Thus, horrific grammar, completely ambiguous antecedents of pronouns, lack of verbs, and such things are effectively harmless, because the current message need impart so little information.

In fact, ordinary language is mostly "failing"... by a strict standard... but, in fact, it does not fail, because the actual context is so rich. (Think of explaining the details of a very cultural-specific tradition to someone from a wildly different culture.)

In contrast, in the very stark context of beginning university mathematics, almost nothing is "understood" or taken for granted. The implied context has almost nothing in it, in comparison to ordinary language. Still, this is not about mathematics itself... When we finally look specifically at technical-mathematical issues, in addition to all the previous, we are trying to capture things near the edge of human experience (and intuition). I'd claim it's not only about "logic" or "rigor" (meaning conformity to standards, which may change depending on context), but as much as anything about the crazy level of precision necessary in situations where there's essentially a null shared context.

To overcome lack of practice in operating in such a situation is as difficult for reasons of language as reasons of mathematics, I think.

(Indeed, in "ordinary" situations, people who give needless details, or insist on having irrelevant, tiny details, are considered to lack common sense...)

Thinking a little more broadly about precision in more ordinary language may be useful. It's not only about mathematics.

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Thanks, Paul. Indeed, as I commented above, the varying levels of rigor expected from different modules is very confusing... And sometimes I can't help feeling that the marker is slightly pedantic :P – Grigory Aug 27 '11 at 22:48
It's possible that some of your graders are being a bit too stringent - not all mathematicians are pedantic, but the field certainly self-selects for the trait. – Thomas Andrews Aug 27 '11 at 23:17
As for different modules requiring different levels of rigor, that's the nature of the beast. Foundations of logic and math are, for example, necessarily more pedantic because you have to take special care to know what your assumptions are. Also, as a beginner, you are taught to err on the side of rigor, while later courses will not require the same rigor because you will be assumed to know how to "flesh out" certain parts of an argument. – Thomas Andrews Aug 27 '11 at 23:20

As Deven Ware notes, it's hard to say what the issue might be without seeing some examples. However, perhaps it might help you to specifically practice writing rigorous proofs from first principles.

For example, try to prove that $1 + 1 = 2$ based on the Peano axioms of arithmetic. (It's not very hard.) At every step of the proof, note exactly which axiom or definition you've used. Don't skip any steps.

Of course, the dirty little secret is that very few proofs actually written by working mathematicians are completely rigorous in this sense — they often omit "trivial" steps or rely implicitly on unstated "well-known" lemmas. However, to be acceptable, a proof does need to be at least rigorous enough that a reasonably competent mathematician can see how to fill in the gaps. Practicing rigorous proofs helps you see where the gaps in your proofs are, and to be sure that you won't leave any gaps you don't know how to plug if needed.

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Thanks, Ilmari, I'll try that. – Grigory Aug 27 '11 at 22:49

I had the same problem making the "transition" from computational mathematics to proof-oriented classes. Two books that I can definitely recommend on the subject are "Mathematical Proofs" by Chartrand et al. and "How to Prove It" by Velleman. Carefully following the examples and working through the exercises really helped me to gain an understanding of the difference between knowing that a proposition is true and providing a "rigorous" proof of it.

As far as check-points for creating a proof, I prefer to sketch out a plan and then try "attack" it myself, looking for places where the logic is not clear or where the statements are ambigous, and then try to fix each of those holes one by one.

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Thanks, Robert. I will definitely check them out! :) – Grigory Aug 27 '11 at 22:33
@Grigory Good Luck!!! – user12998 Aug 27 '11 at 22:35