This quote is fascinating to me:

    How much should we worry about this potential omission?
    It depends on your point of view. If you look upon the
    result as a mathematical proof classifying all
    minimally rigid sphere clusters, then the proof has a
    big gap. But, as I noted in my American Scientist
    column, mathematical rigor was not the highest priority
    in this work and was already jeopardized by the use of
    a numerical algorithm (Newton’s method) that is not
    guaranteed to converge in all cases. The original
    motivation for both the Harvard and Yale projects came
    from chemistry and physics, not from geometry and graph
    theory.

So chemistry and physics are using mathematical results, but not strictly proven mathematical results. Does anyone know how common this is? I wonder if we'll see more of it as we accumulate more unproved-but-so-far-so-good mathematical conjectures.