THere [sic] can even be uncertainty in mathematics. For example, mathematicians in the 1700’s kept finding paradoxes in mathematics, which you would have thought was well-defined. For example, what is the answer to this infinite sum: 1+ (-1) + 1 + (-1) …? If we group them in pairs, then the first pair =>0, so the sum is: 0+0+0… = 0. But if we skip the first term and group it in pairs, we get 1 + 0+0+0… = 1. So which is it?
Mathematicians call these “ill-posed” problems and argue that ambiguity in posing the question causes the ambiguity in the result. If we replace the numbers with variables, do some algebra on the sum, we find the answer. It’s not 0 and it’s not 1, it’s 1/2. By the 1800’s a whole field of convergence criteria for infinite sums was well-developed, and the field of “number theory” extended these results for non-integers etc. The point is that a topic we thought we had mastered in first grade–the number line–turned out to be full of subtleties and complications.
Nearly every statement of Sheldon here is wrong. And not just wrong -- wildly wrong, as in "I have absolutely no idea of what I'm talking about" wrong.
1. Uncertainty in mathematics has nothing to do with the kinds of "infinite sums" Sheldon cites. "Uncertainty" can refer to, for example, the theory of fuzzy sets, or the theory of undecidability. Neither involves infinite sums like 1 + (-1) + 1 + (-1) ... .
2. Ill-posed problems have nothing to do with the kind of infinite series Sheldon cites. An ill-posed problem is one where the solution depends strongly on initial conditions. The problem with the infinite series is solely one of giving a rigorous interpretation of the symbol "...", which was achieved using the theory of limits.
3. The claim about replacing the numbers with "variables" and doing "algebra" is incorrect. For example if you replace 1 by "x" then the expression x + (-x) + x + (-x) + ... suffers from exactly the same sort of imprecision as the original. To get the 1/2 that Sheldon cites, one needs to replace the original sum with 1/x - 1/x^2 + 1/x^3 - ..., then sum the series (using the definition of limit from analysis, not algebra) to get x/(1+x) in a certain range of convergence that does not include x=1, and then make the substitution x = 1.
4. Number theory has virtually nothing to do with infinite sums of the kind Sheldon cites -- it is the study of properties of integers -- and has nothing to do with extending results on infinite series to "non-integers etc."
It takes real talent to be this clueless.
