Reading Isaac Asimov’s essay collection “The Roving Mind”, I was surprised to see some rather harsh comments aimed at Sherlock Holmes author Arthur Conan Doyle. Asimov didn’t have any issues with the quality of the fiction, but was pointing out some scientific errors in Doyle’s writing. Some of his most severe comments were aimed at the fact that the evil Professor Moriarty was presented as a mathematical genius, and said to have written “a treatise on the binomial theorem”. Here are Asimov’s comments on that credential:

*Moriarty was 21 years old in 1865 (it is estimated), but forty years earlier than that the Norwegian mathematician Niels Henrik Abel had fully worked out the last detail of the mathematical subject known as “the binomial theorem,” leaving Moriarty nothing to do on the matter. It was completely solved and has not advanced beyond Abel to this day.*

Asimov, Isaac. The Roving Mind (p. 141). Prometheus Books - A. Kindle Edition.

Is this really true? Did Asimov land a slam dunk against poor Doyle, forever disproving his mathematical competence, and casting doubt on the talents of evil genius Moriarty? Well, let’s take a closer look.

First, let’s refresh our memories on the Binomial Theorem. Most simply viewed, this is a theorem that talks about the coefficients of the terms when you expand the expression (x + y) taken to the nth power. For example, x + y to the 1st power has the exponents 1 1 , since there is 1 x and 1 y. If you square x + y, you get x^2 + 2xy + y^2, so the exponents are 1 2 1. Continuing further, if you cube it the exponents are 1 3 3 1. If you write out these exponents in rows, each one staggered so the middle terms appear between two numbers above, you get the famous construct known as Pascal’s Triangle. Aside from the diagonal sets of 1s going down the left and right edges, each number in the triangle is the sum of the two numbers above. The 2 in the second row is the sum of the 2 1s above it, the 3s in the 3rd row are each the sum of a 1 and 2 above, etc. The binomial theorem basically states that Pascal’s Triangle correctly represents the coefficients for any exponential power of (x+y).

I have, as usual, oversimplified a bit here. If you write out a few examples, you’ll quickly see that in the form I just stated, the theorem seems to just fall out naturally from the way algebra works. It’s pretty easy to prove by simple induction. And in fact, limited cases have been understood since the 4th century A.D. But once you allow non-integer or irrational exponents, the theorem becomes a lot more complex. As Asimov stated, Niels Abel is credited with proving a generalized version of the theorem in the 1820s, as part of an amazing streak of mathematical contributions before his untimely death at the age of 27.

But does this mean that Asimov is correct, and Moriarty could not have demonstrated his mathematical talents with a treatise on this theorem? The most ironic aspect of Asimov’s comments, I think, is the lack of self-reflection. After all, what are Asimov’s essay collections? Essentially he is writing about well-established areas of science and mathematics, to improve their understanding by the general public. Nearly every one of his essays covers facts that were discovered and/or proven many years before he wrote. Indeed, if I comment indirectly about this exact work by writing “Asimov was a great essayist, and I enjoyed his commentary in ‘The Roving Mind’ about the binomial theorem”, would that invalidate my own credentials, as I can’t possibly praise someone’s expository writing about a well-established theorem?

Here’s one way to brush aside that quibble: perhaps Asimov was implicitly considering popular essays about science and math, like his books or like the Math Mutation podcast, to be intellectually trivial exercises, not indicating any particular intelligence by the author. I’ll refrain from further comment in that domain from a motive of self-preservation. But even assuming we accept that judgement for now, I think Asimov made a deeper error here.

In mathematics, you are never really “done”. There are nearly always ways to improve or generalize a theorem. Can we apply it to vectors, matrixes, or fields? Can we apply it to general types of functions other than the algebraic expressions Abel originally considered? Can we find analogs of the theorem that apply to powers of more complex expressions, or in higher-dimensional spaces? While Abel may have definitively proven the theorem as originally stated, that was far from the end of possible work in related areas. A little web searching, in fact, uncovers a 2011 paper by modern-era mathematician David Goss called “The Ongoing Binomial Revolution”, which discusses several major 20th century mathematical results that descend from the Binomial Theorem. I’d like to provide more details, but I’m afraid most of the paper is a bit over my head. However, it ends with a telling comment: “Future research should lead to a deeper understanding of these recent offshoots of the Binomial Theorem as well as add many, as yet undiscovered, new ones.”

It looks like Asimov himself, while a very intelligent man and a great writer, held a somewhat naive view of mathematics. It’s a common mistake, often made by people in applied fields who have always consumed math as well-established, fully presented theorems and formulas. We need to recognize that a huge part of the genius of mathematics is the idea of abstracting and generalizing previous observations, and this work never really stops. When a theorem is proven, that usually leads to even more opportunities for offshoots, generalizations, and other further expansion of knowledge. Thus, I think we have to conclude that Isaac Asimov was wrong, and it was perfectly reasonable for an academic to write meaningfully about the Binomial Theorem many years after Abel, like Goss in his 2011 paper. Holmes was right to fear Moriarty, as there is nothing more dangerous than an archvillian who understands mathematics.

And this has been your math mutation for today.

References:

- https://www.amazon.com/dp/B003ZSISM6/
- https://arxiv.org/pdf/1105.3513.pdf
- http://kobotis.net/math/MathematicalWorlds/Fall2016/131/Biography/Abel.pdf
- https://biography.yourdictionary.com/niels-abel
- https://en.wikipedia.org/wiki/The_Seven-Per-Cent_Solution
- https://en.wikipedia.org/wiki/Binomial_theorem
- https://en.wikipedia.org/wiki/A_Treatise_on_the_Binomial_Theorem