Machine Generates Brand New Mathematics

By Peter Dockrill, Science Alert

February 8, 2021 – His name was Srinivasa Ramanujan, and he had a unique gift for dreaming up mathematics of a kind few, if any, had ever contemplated.

Attributing his skills to a divine goddess, the Indian mathematician introduced thousands of mathematical ideas and equations to the world, and was especially known for devising conjectures: mathematical propositions not yet proven to be true (in which case they become classified as theorems [when proven]).

Such an ability – crafting mathematical statements that are both informed and yet uncertain – is rare, and relatively few mathematicians make their name on the basis of such output, let alone theorists with little in the way of formal training.

But now, a new algorithmic invention developed by researchers in Israel could help us automate the discovery of mathematical conjectures like those Ramanujan once pioneered.

Named after Ramanujan – who died in India at the age of 32 – the 'Ramanujan Machin  is a computerised system capable of self-generating conjectures involving mathematical constants: strange numbers like π and e that seem to crop up all over the place, even if entirely by coincidence.

"Fundamental mathematical constants such as e and π are ubiquitous in diverse fields of science, from abstract mathematics and geometry to physics, biology and chemistry," researchers from Technion – Israel Institute of Technology explain in a newly published study detailing the system.

"Nevertheless, for centuries new mathematical formulas relating fundamental constants have been scarce and usually discovered sporadically."

The Ramanujan Machine might speed things up a little on that front. A system of algorithms powered by a community of cloud-connected computers, it's capable of producing conjectures and discovering mathematical formulas for fundamental constants that stand to reveal the underlying structure of the constants.

So far, the algorithmic machine has generated conjectures that were easily provable, while discovering new fractional ways to calculate constants like π, and also coming up with conjectures that are yet to be proven.

"The computer doesn't care if proving the formula is easy or difficult, and doesn't base the new results on any prior mathematical knowledge, but only on the numbers in mathematical constants,"explains senior author and physicist Ido Kaminer.

"It's important to point out that the algorithm itself is incapable of proving the conjectures it found – at this point, the task is left to be resolved by human mathematicians."

The researchers observe there are limitations to what the Ramanujan Machine can produce; notably, in some instances, what appear to be previously unknown conjectures generated by the algorithms may be "merely mathematical coincidences that break down once enough digits are calculated".

So far, however, there are reasons to get excited about what these algorithms are enabling – especially the discovery of a new algebraic structure concealed within Catalan's constant, which hints the machine might be capable of generating actual breakthroughs the math world has never seen before.

"We believe and hope that proofs of new computer-generated conjectures on fundamental constants will help to create mathematical knowledge," the researchers explain.

If you like the idea and want to get involved, there are several perks to unlock if you join the Ramanujan Machine's community. Lend your computer's processing power, and you might get a conjecture named after you.

Formulas and algorithms themselves are also up for naming rights, depending on your aptitude for mathematical proofs or developing code.

The findings are reported in Nature.

Scientists Invent a Machine That Generates Mathematics We've Never Seen Before (sciencealert.com)