The Smallest Uninteresting Number and Fuzzy Logic
I’ve tried to think of something interesting about the number 2013 and haven’t come up with anything. This reminds me of the interesting number paradox.
Theorem: All positive integers are interesting.
Proof: Let n be the smallest uninteresting positive integer. Then n is interesting by virtue of being the smallest such number.
The interesting number paradox is semi-serious, and so is the resolution I propose below. Both are jokes, but they touch on some serious ideas.
“Interestingness” is not an all-or-nothing property. Some numbers are more interesting than others, so perhaps we should use fuzzy logic to quantify how interesting a number is, say on a scale from 0 to 1.
For a given ε > 0, define as interesting the set of numbers whose interestingness is greater than ε. Suppose the interestingness of numbers trails off after some point. (Otherwise, if the interestingness dropped sharply, the first number after the drop would be interesting.) The largest interesting number then is barely interesting. The number one larger than a barely interesting number is even less interesting. So the proof of the interesting number paradox doesn’t apply in the continuous setting.
On a more serious note, many paradoxes in mathematics can be resolved by replacing a binary criterion with a continuous one.
For example, the sum of a trillion continuous functions is continuous, but the infinite sum of continuous functions may not be. How can that be? The problem is that we’re viewing continuity as an all-or-nothing property. If you have a series of continuous functions that converges to a discontinuous limit, the degree of continuity must be degrading. The partial sum after some large number of terms is continuous, but not very continuous. The modulus of continuity of each partial sum is finite, but is getting larger, and is infinite in the limit.
Classical statistics is filled with yes-no concepts that make more sense when replaced with continuous measures. For example, instead of asking whether an estimator is biased, it’s more practical to ask how biased it is.
Computer science is often concerned with whether something can be computed (i.e. exactly). But sometimes it’s more important to ask how well something can be computed. Many things that cannot be computed in theory can be computed well enough in practice.
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