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.
Sheldon’s confusing comments about the “paradox” in summing 1-1+1-1…
ReplyDeletebrought back memories of a discussion from a graduate course in which Knopp’s Theory and Application of Infinite Series was used as a reference.
In the edition I have, in the first pages of Chapter XIII Knopp points out Euler’s use of the geometric series evaluated at x = -1 to assign a value of 1/2 to the sum (and his assignment of 1/3 to it when x = -2). He also mentions that not all mathematicians were comfortable with this, and says (page 458)
t is true that mot mathematicians of those times held themselves all of from such results in instinctive mistrust, and recognized only those which are true in the present-day sense. But they had no clear insight into the reasons why one type of result should be admitted, and not the other.
So already we see that the thinking on these issues was far more nuanced than “If we replace the numbers with variables, do some algebra on the sum, we find the answer.”
Knopp continues
Here we have no space to enter into the very instructive discussions on the point among the mathematicians of the 17th and 18th centuries. We must be content with stating, e.g. as regards infinite series, that Euler always let these stand when they occurred naturally by expanding an analytical expression which itself possessed a definite value. This value was then in every case regarded as the sum of the series.
Knopp continues and points out that there was no justification for this approach, and mentions that there was no reason why the same series could not arise from the expansion of multiple analytical expression for a different value of x. Again, it seems Sheldon’s description of the state of affairs is far too simple; whether because of lack of familiarity or by intention I don’t know.
At the risk of running on too long with this, here is one last bit from Knopp’s introductory section.
Euler’s principle is therefore insecure in any case, and it was only Euler’s unusual instinct
for what is mathematically correct which in general saved him from false conclusions in spite of his copious use which he made of divergent series of this type.
It seems Sheldon was trying to say that the mathematicians of the day were simply making things up and being sloppy with their work, which could not be further from the truth. And, like you, I have no idea how or why he tries to make the link to number theory.
I think the criticism about "ill-posed" is unfair. He's clearly using that term in the ordinary sense, not in that specialized sense. But yes the rest of this is baloney.
ReplyDeleteYour objection is evidently invalid, as the quote from Sheldon himself
ReplyDelete"Mathematicians call these “ill-posed” problems"
shows he wasn't using it in the "ordinary sense".
Coincidentally, I just saw on another blog a comment from a historian who linked to his own blog essay on mistaken views atheists (citing "new atheists" such as Hitchens) have about religious suppression of ancient pagan writings (such as the works of Democritus). He was able to cite a long list of misconceptions or factual errors with a lot of passion, similar to this post.
ReplyDeleteMy point is that perhaps both some creationists and some atheists are guilty of biased simplifications of complex subjects in order to support their own views. I happen to think that is more prevalent among creationists, but they probably believe the opposite. Given this possibility, maybe it would be better to ascribe such errors to the individuals who make them, without characterizing those individuals as either creationists or atheists, to avoid the implication that all creationists or atheists make the same errors (even if they do).
Aside from that minor qualm, the post was interesting and taught me some things that I did not know.
The issue for me is that this is typical behavior for creationists.
ReplyDeleteThey have very few academically-respectable people among their ranks, and the few that they have are usually not that great at what they do. Nevertheless, they crave the respect of the educated, and so one of their typical ploys is to take some creationist professor who is rather mediocre (or worse) in his/her profession, and elevate them to exalted status. Here we have a guy who can't even get mathematics at the level of 1st year university right; yet he is quoted as some kind of expert.
Jim V
ReplyDeleteI'd like to see that post.
For now, here is the burn order from the Codex Theodosianus for the works of the Greek Philosopher Porphyry of Tyre (He wrote a book "Agianst the Christians" no longer extant):
CTh.16.5.66pr.
Idem aa. Leontio praefecto Urbi. Damnato portentuosae superstitionis auctore nestorio nota congrui nominis eius inuratur gregalibus, ne christianorum appellatione abutantur: sed quemadmodum arriani lege divae memoriae Constantini ob similitudinem impietatis porfyriani a Porfyrio nuncupantur, sic ubique participes nefariae sectae nestorii simoniani vocentur, ut, cuius scelus sunt in deserendo deo imitati, eius vocabulum iure videantur esse sortiti. (435 aug. 3).
He's lumped together with Nestorius in order to smear Nestorius as a non-Christian.
ReplyDeleteJeffrey, don't be so quick to say "It takes real talent to be this clueless."
The only one of your four points that you were right about was 4. As we assume, Number Theory is about the relationships of the natural numbers. But series of integers can be classed as number theory as well, and many series can be infinite in length.
https://en.wikipedia.org/wiki/Category:Number_theory
And Sheldon did use scare quotes around "number theory". You would have to ask him why.
Your point number 2 does not tell us which equation is right. And how is the definition of “ ... “ significant?
1 +(-1) + 1 +(-1) + 1 +(-1) + 1 +(-1) + 1 +(-1) + 1 +(-1) + 1 ... = S
(1 +(-1)) + (1 +(-1)) + (1 +(-1)) + (1 +(-1)) + (1 +(-1)) ... => 0
1 +((-1) + 1) +((-1) + 1) +((-1) + 1) + ((-1) + 1) +((-1) + 1) ... => 1
1 -S = S => S = 1/2
Your point number 1. is just arguing the definition of uncertainty. Just look at the context and go with that. Maybe some other word would better express what Sheldon is trying to say. But, never argue definitions, it is pointless.
Your point number 3 is not in the ballpark, and I can tell you that there is nothing that I can say to convince you of that. So, I will let Dr. James Grime explain it to you, as you may take his word over mine.
https://www.youtube.com/watch?v=PCu_BNNI5x4
Ivan, what is your level of mathematical training?
ReplyDeleteWell, you didn't answer, Ivan, so I'll try.
ReplyDeleteYour claim about point #3 is incorrect. The assertion that 1+ (-1) + 1 + (-1) + ... = 1/2 is not correct, because in the ordinary understanding about infinite series, the series simply does not converge. (If you do not know what it means for a series to converge, consult any book about calculus, and look up "limit of an infinite series".)
There is an alternate notion of summation, called Cesaro summation. (There is a Wikipedia article on it.) In Cesaro summation it is sometimes possible to assign values to expressions like 1+ (-1) + 1 + (-1) + ..., even though the corresponding infinite series do not converge, by taking the limit of the arithmetic means of partial sums. The Cesaro sum of 1+ (-1) + 1 + (-1) + ... is 1/2.
Finally, while I like the effort of James Grime and his "Numberphile" videos (and indeed my work was briefly featured in a recent video by him), his enthusiasm sometimes gets the better of him and leads him to make claims that are strictly incorrect.
Your other points are similarly based on misunderstandings.
It's late at night and maybe I'm making a stupid error, but isn't the Cesaro sum of that sequence 0? The sum of the first n terms is either 0 or 1 (depending on whether n is even or odd). Divide by n and you get 0 or 1/n. This converges to 0.
ReplyDeleteOK, I see where I went wrong. You take the means of the partial sums (not the terms). The Cesaro sum is indeed ½. Sorry about that.
ReplyDelete