When mathematics loses its shine, and then recovers it
A curious email hit my inbox this week. A friend, on behalf of his friend, sent me an outline of a mathematical proof. I routinely get these emails, but they usually originate from students looking to validate their solutions to a problem set. This was different: the proof in question is of the Twin Prime conjecture; one of the great unresolved mysteries of mathematics. The author was seeking a sanity check on his approach. If his argument proves valid, my friend’s friend will have his name etched in mathematical folklore. I may even make the footnote as his great enabler.
My rush to excitement was checked with a feeling of overwhelming scepticism. I struggled to accept that mathematical greatness had landed at my fingertips. My reaction speaks volumes about my relationship with mathematics.
The Twin Prime conjecture states that there are infinitely many prime numbers spaced two apart. Examples of Twin Prime pairs include {5,7}, {29,31} and {41,43}. Euclid proved long ago that primes themselves are in infinite supply; the Twin Prime conjecture pushes their existence a step further. It has been the subject of study for over 150 years. In 2014, mathematicians rejoiced when Yitang Zhang showed that there are infinitely many prime pairs to within 70 million. That may be worth a second read; it says that primes do indeed crop up in pairs indefinitely, but the spacing between them may be as large as 70 million.
The fact that an upper bound of any kind could be established was cause for celebration. So you can only imagine how hard the mathematicians partied a year later when the bound was reduced to 246. Some of the world’s best number theorists are racing to bring this bound down to 2, which would render the Twin Prime conjecture true.
The Sieve — surely too simple (source)
And now, sitting in my inbox, is a 2-page summary from a novice mathematician (a programmer with a basic grasp of undergraduate mathematics) who claims to have won the race. He has shared only an outline, lest the glory of his fully fledged proof be stolen from him. At first blush, it seems to be no more than a clever application of the Sieve of Eratosthenes. The author acknowledges the apparent simplicity of his approach —yet he is hopeful that it is novel enough to have eluded the great mathematical minds all these years.
I believe in miracles, but I am not naïve enough in thinking they reside in simplistic mathematical proofs.
I cannot fault the guy for trying — I’ve been there, and my own attempt failed at hello. It is the dreamers among us that shoot for the stars. The uncomfortable reality, however, is that the few who get there only do so through considerable effort and sacrifice.
Andrew Wiles dared to dream when he stumbled upon Fermat’s Last Theorem (which was inappropriately named since it was a mere conjecture at the time) as a child. Wiles devoted his life in pursuit of a proof that had escaped mathematicians for over 350 years. Wiles knows something of sacrifice, describing his work as an obsession and locking himself away for seven years to fulfil his ambition. He is among the lucky ones; I am reminded of Wiles’s legacy every time I walk past the mathematical institute that bears his name in Oxford. But far more have faltered.
For every Andrew Wiles there are a great many mathematical minds yet to have a building named after them.
The rewards of a successful proof may be life-changing, but when the odds of success are so low our curiosity often gives way to defeatism. My primary reason for not pursuing new mathematical proofs is also my most poignant: they are beyond me. My doctorate exemplified the excruciating effort required to squeeze out drops of new mathematical truths. No longer are these discoveries clothed in elegant 5-line proofs. While professional mathematicians delight in furthering the frontiers of knowledge, they would also confess to the brute-force, honest effort that it demands.
To all but the obsessive few, mathematics loses its shine when you plumb its greatest depths; so much so that I could not bring myself to properly review the 2-page proof outline.
Not all was lost. The episode was a natural talking point over lunch with a mathematician friend (the best kind). As I reflected on my cold rejection of mathematical voyage, my friend reminded me of the strange case of Thomas Royen.
Royen hit the headlines in 2014 when he proved another highly sought after conjecture, the Gaussian Correlation Inequality (GCI). Royen was retired, and not particularly renowned in his field. He presented a short proof of the GCI — too short for others to take it seriously (mathematical proofs tend to span several tens of pages; Wiles’s proof reached over a hundred). Royen’s cause was helped none when he submitted his proof to an obscure journal, as a Word document no less (an act that invites derision from mathematicians, who expect arguments to be neatly packaged in Latex).
Thomas Royen — the man who gives us reason to believe again (source)
And yet despite all this, Royen’s proof was legitimate. His 6 pages brought down the conjecture, ending a search that others had been immersed in for decades with an argument he recognises is ‘totally different and comparatively short’.
Lesson learned: perhaps mathematical miracles do still happen. We just need to believe that little bit more.
Royen concludes with an acknowledgement to Donald Richards, who helped him along the way. Perhaps that’s the prize I have awaiting me if I support the aspiring solver of the Twin Prime conjecture.
Royen’s example has reawakened a part of me that believes mathematics can be elegant and surprising, even at the highest levels. At the very least, it has tipped the balance of Pascal’s Wager — my evenings will be spent combing through the two-page outline. I remain doubtful, but my cynicism has receded and I am not quite ready to relinquish my romance with mathematics altogether.
I am a research mathematician turned educator working at the nexus of mathematics, education and innovation.
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If you liked this article you might want to check out my following pieces:
The time I ‘nearly’ solved the Twin Prime Conjecture_My first authentic mathematical experience_hackernoon.com
I no longer understand my PhD dissertation (and what this means for Mathematics Education)_Earlier this week I read through my PhD dissertation. My research was in an area of Pure Mathematics called Functional…_medium.com
How my supervisor saved my PhD — the Goldilocks principle for mathematics_Mathematics is all in the asking_mystudentvoices.com