Month: February 2021

Over the past few days, the mathematics world has been abuzz over the news that Sir Michael Atiyah, the famous Fields Medalist and Abel Prize winner, claims to have solved the Riemann hypothesis.

If his proof turns out to be correct, this would be one of the most important mathematical achievements in many years. In fact, this would be one of the biggest results in mathematics, comparable to the proof of Fermat’s Last Theorem from 1994 and the proof of the Poincare Conjecture from 2002.

Besides being one of the great unsolved problems in mathematics and therefore garnishing glory for the person who solves it, the Riemann hypothesis is one of the Clay Mathematics Institute’s “Million Dollar Problems.” A solution would certainly yield a pretty profitable haul: one million dollars.

The Riemann hypothesis has to do with the distribution of the prime numbers, those integers that can be divided only by themselves and one, like 3, 5, 7, 11 and so on. We know from the Greeks that there are infinitely many primes. What we don’t know is how they are distributed within the integers.

The problem originated in estimating the so-called “prime pi” function, an equation to find the number of primes less than a given number. But its modern reformulation, by German mathematician Bernhard Riemann in 1858, has to do with the location of the zeros of what is now known as the Riemann zeta function.

The technical statement of the Riemann hypothesis is “the zeros of the Riemann zeta function which lie in the critical strip must lie on the critical line.” Even understanding that statement involves graduate-level mathematics courses in complex analysis.

Most mathematicians believe that the Riemann hypothesis is indeed true. Calculations so far have not yielded any misbehaving zeros that do not lie in the critical line. However, there are infinitely many of these zeros to check, and so a computer calculation will not verify all that much. Only an abstract proof will do.

If, in fact, the Riemann hypothesis were not true, then mathematicians’ current thinking about the distribution of the prime numbers would be way off, and we would need to seriously rethink the primes.

The Riemann hypothesis has been examined for over a century and a half by some of the greatest names in mathematics and is not the sort of problem that an inexperienced math student can play around with in his or her spare time. Attempts at verifying it involve many very deep tools from complex analysis and are usually very serious ones done by some of the best names in mathematics.

Atiyah gave a lecture in Germany on Sept. 25 in which he presented an outline of his approach to verify the Riemann hypothesis. This outline is often the first announcement of the solution but should not be taken that the problem has been solved – far from it. For mathematicians like me, the “proof is in the pudding,” and there are many steps that need to be taken before the community will pronounce Atiyah’s solution as correct. First, he will have to circulate a manuscript detailing his solution. Then, there is the painstaking task of verifying his proof. This could take quite a lot of time, maybe months or even years.

Is Atiyah’s attempt at the Riemann hypothesis serious? Perhaps. His reputation is stellar, and he is certainly capable enough to pull it off. On the other hand, there have been several other serious attempts at this problem that did not pan out. At some point, Atiyah will need to circulate a manuscript that experts can check with a fine-tooth comb.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to William Ross

Credit: Getty Images

Better living through set-builder notation

As a publicly mathematical person, one of the matters upon which I am called to adjudicate is what I think of as “viral order of operations questions” with a math problem along the lines of “48÷2(9+3) = ?”

In the past, I used to tell people who asked me about one of those questions something like, “I think the correct answer is __, but it’s better to write the expression unambiguously.” But recently I decided I need to put my foot down. I will no longer give an answer to those questions. The only way to win is not to play.

An ambiguous sequence of digits and mathematical operation symbols is not interesting to mathematicians. Most of us learn something about the order of operations fairly early on in our mathematical educations. We might learn PEMDAS or BIDMAS, or “Please excuse my dear aunt Sally.” All of these expressions tell us that we’re supposed to take care of expressions inside parentheses or brackets first, followed by exponential expressions, followed by multiplication and division, then addition and subtraction. It’s good that we have a system, but mathematicians and other people who use mathematical expressions regularly would never write down something like “48÷2(9+3) = ?” because its potential for causing confusion is too great.

In general, mathematicians strive to reduce ambiguity when possible. A mathematician would write (48÷2)(9+3) or 48÷(2(9+3)), depending on which one they meant. Viral order of operations problems are unappealing. Just toss in a few more parentheses to clarify your meaning and move on. There are cat pictures to scroll through, for goodness’ sake!

In fact, I think if mathematicians had their way, they would get rid of easily-fixed ambiguous order of operations problems altogether, and I don’t think they’d stop there. The English language often leaves room for ambiguity, and I think mathematical notation could help us make some improvements.

I remember chuckling to myself when I saw the phrase “I like to play board games and read a book while taking a bath.” It conjures up an image of a very exciting game of Monopoly where Baltic Avenue is replaced by the Baltic Sea. Set-builder notation could resolve that titillating ambiguity there while simultaneously freeing me of a tiny shred of joy in this world of woe. Mathematicians use curly brackets to indicate things that are in one set and can be treated as one object. A person who enjoys two separate activities, one of which is playing board games and one of which is reading a book while taking a bath, could write “I like to {play board games} and {read a book while taking a bath}.” A person with a much more interesting bathtime routine than mine could write “I like to {play board games and read a book} while taking a bath.”

Curly brackets take care of written English, but we communicate through speech as well. The late Danish pianist and comedian Victor Borge had a routine about phonetic punctuation. He assigned sounds to some common punctuation marks and inserted them into sentences.

He didn’t include a sound for curly brackets, and I’m not sure the best option. Perhaps a “zzp” sound would work, but I’m open to other suggestions. In the meantime, there is a precedent for air quotes, and perhaps we could add extend that to air parentheses.

When I started thinking about using brackets and air-parentheses in English writing and speech, I wondered if I was just reinventing sentence diagramming. I don’t know how many people learned to diagram a sentence in school, but a sentence diagram is a graphical representation of a sentence that shows how each word and phrase functions in the sentence. The diagram for the sentence “I like to {play board games} and {read a book while taking a bath}” is different from the diagram for “I like to {play board games and read a book} while taking a bath.” It’s been a while since I did any sentence diagramming, so I beg for lenience from any grammar teachers reading, but these were the diagrams I came up with. The placement of the “while” clause is the only difference between the two diagrams.

Two possible diagrams for the sentence “I like to play board games and read a book while taking a bath.” Credit: Evelyn Lamb

Diagramming the sentence does remove the ambiguity, and it gives even more information than the set-builder notation I’m suggesting, but it comes at a cost of both space and effort. It’s much more practical to throw a few brackets into English prose than to draw every sentence as a complex, multi-storied building.

Set-builder brackets are just the tip of the iceberg when it comes to resolving linguistic ambiguity through mathematical notation. After we’ve mastered those, we can consider incorporating union and disjoint union symbols and the distinction between or and xor into our speech and writing. But that will have to wait until after we’ve relaxed with an extra-realistic game of Battleship in the bathtub.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Evelyn Lamb