The Paradox of 1 – 1 + 1 – 1 + 1 – 1 + …

[bhobba]

In the latest edition of Scientific American, they had an opinion piece titled The Paradox of 1 – 1 + 1 – 1 + 1 – 1 + …. For those with access:

https://www.scientificamerican.com/article/1-the-paradox-of-1-1-1-1-1-1/

Scientific American, the once-venerable publication I eagerly awaited every edition of in my misspent youth, has let its standards slip somewhat. The opinion piece is a case in point. I still subscribe to and recommend it because it is very cheap digitally and keeps up (mostly) with the latest developments, but this one is misleading.

However, understanding why is very instructive. It will take us to the very foundations of mathematics.

You can purchase books and take entire university courses on the foundations of mathematics. An interesting debate between Wittgenstein and Turing (plus a few others) took place in 1939:



It has been published as a book:

https://www.amazon.com.au/Lectures-Foundations-Mathematics-Ludwig-Wittgenstein/dp/0226904261

I will only skim that issue here. Suffice it to say the formalist view is nowadays the most prominent:

https://www.math.kent.edu/~edd/UmeaII.pdf

Of course, one needs to assume some things, and in math, what is nearly always assumed is the so-called Zermelo-Fraenkel axioms.

https://people.math.ethz.ch/~halorenz/4students/LogikGT/Ch13.pdf

And a guy you have probably heard of, Kurt Godel, discovered some surprising results in 1931 that had profound implications for what mathematics is.

Every Tom, Dick and Harry is taught a programming language like Python at school these days; it is seen to be not quite as mysterious as it once was being related to the so-called Halting Problem (which was discovered by Turing):



Naturally, Turing and Wittgenstein had their views on it:

https://discovery.ucl.ac.uk/10093229/1/Thesis Final3-compressed.pdf

This is just to set the stage. Given a sequence S = a1 + a2 + a3 + a4 ......, we ask its sum. We know math is a formalism, and the answer is a freely chosen definition. This naturally is done with tact, taking into account other ideas, notably those of what is called real analysis, and the definition is if Sn = a1 + a2 + a3 + a4 ......, then S is limit n → ∞ Sn. Such is familiar to most who have completed high school or at least the first year of university. The definition for S = 1 - 1 + 1 - 1 ..... is meaningless. Normally, teachers stop there, and any dissent leads to chalk flying past the dissenter's ears.

Already, the alert reader may have spotted something. The definition was made to make sense within real analysis. However, a similar game can be played with complex numbers to give us complex analysis. As good formalists, we are at liberty to see what happens if complex analysis is used instead. The problem is easily reformulated as S(x) = 1 + x + x^2 + x^3 ..... with x = -1. If |x| < 1, the same definition used in real analysis is also used to get the answer. Nothing special here - it fails at x = -1. Except in complex analysis, there is this sneaky process called analytic continuation:

.

Now, the cat is among the pigeons. S(x) can be extended to the whole complex plane, and, low and behold, at x = -1, S(x) = 1/2. Indeed, using this trick, all sorts of methods have been devised to sum series that can't be summed using the real analysis definition. I could give a link to a number of these. Still, a well-known mathematical physicist/applied mathematician, Carl Bender, has done a whole series of lectures exploring this and all sorts of other strange stuff, as well as interesting Feynman anecdotes:



The critical point is that this is not just a mathematical curiosity - it has many real-world applications. One method, Borel Summation, has an interesting story. It was discovered by the famous mathematician Borel when he was young, well before he was famous. His summation method gave the 'right' answer for many classical divergent series. He decided to make a pilgrimage to Stockholm to see Mittag-Leffler, the recognised lord of complex analysis. Mittag-Leffler listened politely to what Borel had to say, and then, placing his hand upon the complete works by Weierstrass, his teacher, he said in Latin, 'The Master forbids it'.

Returning to the original article, we see that the definition of an infinite sum is flexible; there are many (as the article points out). None are better or worse than any other. However, they are all variations on a theme - using complex analysis instead of real analysis and analytic continuation. The choice of which to use depends on what one wants to accomplish.

The Scientific American article explains this but in less detail. I hope I have shown the conclusion is somewhat dubious: Grandi’s series does not sum to anything, but if it did, it would sum to ½. Grandi's sum is based on whatever definition we choose, freely chosen depending on what we, as a mathematician, wish to accomplish. In essence, mathematics is about formal concepts, with the concept chosen to suit the situation.

Dedicated to Fresh_42, who I think would enjoy the ideas in this post.

Thanks
Bill

 

[jedishrfu]

Numberphile on YouTube has three videos on the topic:





and

 

[bhobba]

The last video has taken one of my interests, 1 + 2 + 3 + 4....... = -1/12, much further than my on-and-off meanderings. It, too, succumbs to the general methods using Analytic Continuation. There is also another summation definition called Ramanujan Summation, which also works (although Hardy, who wrote the textbook on Divergent Series cautions against it):

https://en.wikipedia.org/wiki/Ramanujan_summation

Terry Tao, as alluded to in the last video, examines the connection in much more detail (from the references in the Wikipedia article):

https://terrytao.wordpress.com/2010...tion-and-real-variable-analytic-continuation/

And finally, we have the paper alluded to in the video:

https://arxiv.org/abs/2401.10981

Also, regarding .99999999999......, that leads to the theory of infinitesimals, about which I wrote some insight articles. Working in the hyperrationals (not hyperreals, which is an extension of hyperrationals), .99999999.... is not equal to 1 but is, in fact, infinitesimally close to it. In fact, this can be used to define real numbers from the hyperrationals, but that would take us way too far from the intent of this thread.

Thanks
Bill

 

[jedishrfu]

The hyperreal work has allowed people to rewrite Calculus books in which we can treat dy and dx values as ordinary numbers that can be divided, meaning that dy/dx is a ratio of infinitesimals with all the same arithmetic properties as other numbers.

It eliminates the need for epsilon-delta limit proofs.

Keisler's Calculus Book:

“Elementary Calculus: An Infinitesimal Approach” by H. Jerome Keisler. This textbook uses hyperreal numbers to simplify the teaching of calculus, particularly by eliminating the need for epsilon-delta limit proofs. Instead of the traditional approach, Keisler’s book introduces calculus through infinitesimals, making concepts like derivatives and integrals more intuitive for beginners.

 

[PeroK]

Keisler's Calculus Book:

“Elementary Calculus: An Infinitesimal Approach” by H. Jerome Keisler. This textbook uses hyperreal numbers to simplify the teaching of calculus, particularly by eliminating the need for epsilon-delta limit proofs. Instead of the traditional approach, Keisler’s book introduces calculus through infinitesimals, making concepts like derivatives and integrals more intuitive for beginners.

That, however, leaves the issue of mapping hyperreal infinitesimals to physical spacetime. If, indeed, you want to apply calculus to physical problems.

And, of course, if you've avoided real analysis, then you are committed to hyperreal vector calculus and hypercomplex analysis. And, it leaves you in a weak position to study Hilbert spaces and analytic topology generally.

 

[jedishrfu]

I think the point of the book was to remove the fear of limit proofs in the minds of students back to a later chapter so you could learn calculus more as an extension of algebra.

Later you can get into the reason and seriousness of epsilon-delta proofs.