On the beach I finally got a chance to start (if not finish) Nahin’s When Least is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible. So far it is an engrossing survey of work on maximum/minimum problems that leads one gently into the mathematical intricacies of the history without being overly technical.
One of the first examples that Nahin discusses is Torricelli’s Funnel (also known as Gabriel’s Horn). One can think of it as the result of rotating the "first quadrant branch" (positive x, positive y) of the hyperbola xy = 1 around the x-axis. If we consider its surface, then we can show that the total surface is greater than any finite number. But if we consider its volume, then we can show that it is finite. For this case, the volume is pi. While I had come across this example before in Mancosu’s book, Nahin raises a paradox which I think should be more widely known for those working on mereology and "intuitions" in philosophy. As Nahin puts it, if the surface is infinite, then we cannot paint the funnel with a bucket of paint, no matter how large the bucket is. But if the volume is finite, we can paint the inside of the funnel by filling the funnel with paint! Do we have a proof that this figure is impossible?
No! Nahin points out that we are tacitly working with two different notions of paint: "mathematical paint" and "real paint". If we consider real paint, it is composed of molecules of some finite size. So, if we pour the real paint into the funnel, then it will not coat the inside of the funnel because at some distance its molecules will no longer fit. But if we consider "mathematical paint", then a finite amount can coat the outside of the funnel, or the inside of the funnel, because a finite volume can be spread out to any degree over a surface if we place no limit on the thickness of the coating. This elegant example shows just how liable we are to make mistakes when we consider these sorts of mereological questions if we fail to pay attention to the mathematical subtleties.
2 comments:
In fact, it seems that there could have been no privileged position in this debate until the end of the 19th century. The claim that paint droplet can (or cannot) be modeled as a continuum is a substantial physical claim -- and I can't think of any evidence to support either position that doesn't involve 20th century physics!
On Nahin: I like the separation of "real" and "mathematical paint," but wouldn't go so far as to separate them once and for all. There may yet be a role for empirical facts (e.g., about "real paint") when engaging in mathematical reasoning -- I once posted a couple of neat examples. I'd rather say the paradox of Gabriel's Horn shows that importing physical reasoning into mathematics can only be done with great care...
Thanks Brian. Given your post, you might like Nahin's book, as it is (so far at least) structured around the isoperimetric problem from pre-calculus through calculus.
What about the converse problem, that is, importing mathematical reasoning into physics? This seems to have created just as many problems, especially when the mathematical concepts have been confused in a hidden way!
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