Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 0 down vote favorite

Many people feel that a combinatorical proof of a particular algebraic identity is not as rigorous as an analytic one. However, I'm not quite sure why people feel this way. So, I am curious as to what people on this site think of this style of proof. Is it rigorous enough? If it's rigorous, do they lack motivation?

share|improve this question
10  
I’ve never encountered such a prejudice; there is no inherent lack of rigor in combinatorial arguments. In my experience combinatorial proofs are frequently preferred, on the grounds that they offer more insight. – Brian M. Scott Mar 28 '12 at 5:39
Like @Brian said. You could provide references for, or examples of, the alleged prejudice. – Did Mar 28 '12 at 5:49
1  
If you are talking about Isaac Solomon, he just likes to joke that way. He does know that combinatorics is a rigorous subject. – Daniel Montealegre Mar 28 '12 at 5:51
1  
I am quoting one of the offered reasons for closure: "This question is not a good fit to our Q&A format. We expect answers to generally involve facts, references, or specific expertise; this question will likely solicit opinion, debate, arguments, polling, or extended discussion." Perfect fit - click... – Alex B. Mar 28 '12 at 6:01
1  
@Alex: If the premise on which it’s based were correct, it would be a natural and reasonable question, deserving an answer. The OP seems to have believed the premise and asked the question in good faith; I see no reason to close it simply because the premise is in fact false. – Brian M. Scott Mar 28 '12 at 20:21
show 5 more comments

closed as not constructive by Alex B., Did, Hans Lundmark, Qiaochu Yuan Mar 28 '12 at 20:35

As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or specific expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, see the FAQ for guidance.

1 Answer

up vote 1 down vote

I think that this question is based on a misapprehension: in several decades of doing and teaching mathematics I’ve never encountered a prejudice against combinatorial proofs of algebraic identities. On the contrary, I’ve found that they are often preferred, on the grounds that they offer more insight $-$ more sense of real understanding of why the result is true $-$ than purely computational arguments.

There is certainly no basis for such a preference: there is no inherent lack of rigor in combinatorial arguments. It’s true that by their nature they can often be stated somewhat informally and still be understood; this is in large part precisely because they tend to have a clear underlying idea, instead of being the result of a comparatively unmotivated string of computations. But this does not mean that they must be expressed informally: it is always possible to dot the i’s and cross the t’s just as meticulously as one would in any other argument. In practice there may be less need to do so, however, since $-$ the basic idea being relatively clear $-$ the reader who wishes to do so is likely to be able to fill in those details.

share|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.