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?
Rigor in Mathematics
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15Perhaps 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. – 2011-08-27
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0@Deven: Thanks for the suggestion. Unfortunately I don't have any at hand... Just wondering if anyone else has difficulty in their high-school (hand-waving) to university (rigor) transition and if yes, perhaps they have been enlightened since then and would not mind sharing...? Perhaps check-points when writing a proof? – 2011-08-27
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8Every 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?" – 2011-08-27
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2Yes, 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. – 2011-08-27
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0@anon: Thanks, good suggestion. – 2011-08-27
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0@Thomas: Yes, mainly on tests. I'm usually fine with hw because I have a lot of time to do them so I can write everything down. My proofs then become really long, but they cover everything inc. a lot of stuff that are not necessary to prove/state. The thing is I don't know what bits are required and which aren't... I would have posted an example but I left my folder in uni where I am not at right now... Sorry :S – 2011-08-27
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0@Grigory expanding anon's suggestion, IMO, you should consider the scope of the proof in each step you take, is every argument you include in that step is enough to make it solid, if it isn't, what's missing from it? (Question your arguements constantly to see they are bullet proof) I had quite a handful of agony since I came from computational mathematics to calculus where on the contrary had to substantiate every step of the proof. Good luck! – 2011-08-27
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0Different levels of rigor are required at different levels of study. Since you are just starting to prove things, you may be taking too many things for granted in your arguments and glossing over what seem to be "obvious" things, but technically are not based on the prior results and assumptions in your course. A lot of "obvious" things are only obvious because you've been introduced to them without proof! – 2011-08-27
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0@Cu7l4ss and tomcuchta: Thanks guys. To add to the confusion, even within the course different modules tend to require very different levels of "rigor". Pure modules are very strict on that, whereas applied modules are very loose. It is very confusing! – 2011-08-27
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4It'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. – 2011-08-27
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1Another 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. – 2011-08-28