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