If there’s one consistent aspect of creationism, it’s that people lacking understanding and training are put forth as experts. Here we have yet another example, from the creationist blog Uncommon Descent. There physicist Rob Sheldon is quoted as saying
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.
It’s “ill posed” in the sense that it’s unclear what is being asked in “1 – 1 + 1 – 1 + …”
From wikipedia:
“The mathematical term well-posed problem stems from a definition given by Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that:
1.) a solution exists,
2.) the solution is unique,
3.) the solution’s behavior changes continuously with the initial conditions.”
The given problem fails on the first two points, not the third.
After that I agree it’s just gobbledygook. His problem is so ill posed that we cannot even talk about a solution, so it’s nonsense to claim that the solution is 0, 1, or 1/2 until you define exactly what you mean.
“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.”
This is actually a point I agree with. I’ve heard university professors discuss whether they should teach the definition of a real number to first year maths students. We may think we have a reasonable idea what real numbers are, but in fact we mostly just understand rational numbers. There ARE a lot of subtleties and complications.
You may think you understand what an “ill-posed problem” means for mathematicians by quoting Wikipedia, but you don’t.
No actual mathematician would agree that 1 + (-1) + 1 + (-1) + … is “ill-posed” in the sense understood by mathematicians.
Here, check this out.
We may think we have a reasonable idea what real numbers are, but in fact we mostly just understand rational numbers.
Speak for yourself, please.
robert79 preceding the italicized sentence with this: “I’ve heard university professors discuss whether they should teach the definition of a real number to first year maths students.” If he had gone on from there to say “We may think we have a reasonable idea what a real number is, but in fact we mostly just understand what a rational number is”, he’d have been on firmer ground.
The point is that what we really have “a reasonable idea of” is not “what real numbers are” in a sense that would require us to know what a real number is; what we do really have is “a reasonable idea of” what the system of real numbers is. There are many different constructions of “the real number system”: in one of them, “a real number” is a Dedekind cut in the rational numbers; in another, “a real number” is an equivalence class of Cauchy sequences of rational numbers, where the equivalence relation is “modulo null sequence”; in yet another, “a real number” is an “infinite decimal expansion that is not eventually all 0s”. As constructed (in naive set theory, say) these “real number systems” are entirely distinct and disjoint. But when each of them is equipped with an appropriate small collection of definitions (arithmetic operations of addition and multiplication, a subset of “positive real number”, a topology or a limit operation on sequences) that can (more or less easily) be proved to satisfy a small number of properties (addition and multiplication make the system a field, the set of positive real numbers induces a total order on the system, each Cauchy sequence of real numbers in the system converges to a unique real number in the system, and the Archimedean Principle), it turns out that these distinct “real number systems” are naturally isomorphic; and, further, that any proposed “real number system”, i.e., something which satisfies those properties (in short, a complete, Archimedean ordered field…am I forgetting anything?) is isomorphic to each of them. So, our “reasonable idea” of what the real number system is (and should be) is a model of the axioms which I have enumerated as properties that (for instance) the given systems each have; and the reasonableness of this idea follows from the theorem (which I asserted, and obviously did not prove) that the given axioms are “categorical” in the sense that any two models of those axioms are canonically isomorphic. The axioms don’t give us a “reasonable idea” of what a “real number” is, because a real number belongs to some specific model. BUT IT DOESN’T MATTER. The system properties are important (for analysis, algebra, what have you).
Obviously you (Jeffrey) know this (or a version of what I just wrote with any technical errors kindly corrected); perhaps robert97 does too, but if he does, I hope he will appreciate why I’m beating him over the head with it, for the sake of (possible) other readers who are less well-informed.
As to whether “university professors […] should teach the definition of a real number to first year maths students”, well, that depends on the “first year maths students”. Just 10 years before you (Jeffrey), as a freshman at the same university, I took my first real mathematics course (in high school, thanks to the upwardly mobile parents of a school friend, I got to leave school early and take the Rapid Transit across town to Case Institute of Technology, where a special section of freshman calculus was taught to a group of 7 or 8 high school students—the rest of them all from tony East Side suburbs—by Richard Vesley out of a standard Thomas-style textbook), “Advanced Calculus”, with Dieudonné’s Foundations of Analysis as the textbook. The (future) engineers in the class did not enjoy it. It was a revelation to me! Dieudonné used the axiomatic method, followed by a proof that the construction by Cauchy sequences modulo null sequences is a model of the axioms. I think he stated (but did not prove) categoricity. I am fairly sure that (at whatever cost to his self-esteem as an established metamathematician) Vesley would never have dared introduce the real number system like that at Case Institute, nor was there any such material in the textbook used there (of course the Least Upper Bound property was discussed, but proved? are you kidding??) and I don’t think any model of the real number system more sophisticated than “infinite decimals with appropriate fiddling” was in the book.
But “first year maths students” isn’t well-defined, so in a way this whole discussion is … ill-posed.