Of all the scientific disciplines, pure math, also known as theoretical math, is arguably one of the most cerebral.
While chemists tend to make discoveries in the lab, mathematicians are more likely to jump out of bed in the middle of the night to hastily jot down formulae they conceived in their dreams. Unsurprisingly, that is a problem when it comes to the pace of discovery.
Waiting around for a once-in-a-generation mind to uncover formulae for essential mathematical concepts — fundamental constants like Pi, for example, which, by the way, describe most of our physical world — can make progress in mathematics staggered and uneven, to say the least.
“There are many known ubiquitous constants, and while they’ve been studied for years, there is still a lot more to be discovered that can potentially shed lights on new frontiers in mathematics and beyond,” Yaron Hadad, a co-author on the study and independent mathematician, explains to Inverse.
But now we may be able to bypass mathematicians’ seemingly super-human intuition altogether. A team of physicists from Israel has designed a new way to achieve these breakthroughs using an algorithm-generating invention they call the Ramanujan Machine. With this machine, scientists can automatically examine mathematical patterns and discover new formulas for existing mathematical constants and dramatically speeding up discovery.
Essentially, physicists may have made mathematicians obsolete.
Why it matters — Finding a way to regulate the pace of new math and formulae could not only explain the world around us, but also fuel new discoveries not just in pure mathematics, but physics, computer science, and chemistry as well. Without this math, for example, we would not know what we do about black holes, cell division, or quantum computing.
The research was published Wednesday in the journal Nature.
The background — In its simplest definition, math has two main components:
- Conjectures: The assertion of mathematical ideas
- Proofs: The formal ways of confirming conjectures
The connection between the two is made up of equations and arbitrary case studies designed to prove beyond doubt that purely theoretical ideas are true.
Most mathematicians could work tirelessly for decades to uncover these connections using the fabric of the universe to guide them — if they manage at all. But there are a handful of mathematicians who, through history, have asserted new conjectures without showing any work to support their ideas — only to be proven correct later on.
Perhaps no mathematician is as well-known for this innate ability as the new invention’s namesake, Srinivasa Ramanujan, whom Dev Patel portrayed in the 2015 film The Man Who Knew Infinity.
Ramanujan had little formal math training, but his talents were so profound he ended up at the University of Cambridge in 1913, where he worked on prime numbers, infinite series, and number theory. Ramanujan would often credit a Hindu goddess with gifting him mathematical revelations in his sleep, without being able to initially provide any formal work or proofs.
Ultimately, what separates mathematicians like Ramanujan from those who slug it out daily in the numerical trenches is a refined ability to identify patterns in data, argue the authors of this new paper.
Lucky for us, that’s something algorithms are pretty good at too.
“This back-and-forth dynamics between algorithms and mathematicians is the essence of what we believe can be achieved.”
What they did — To imitate the intuitive power of a mathematician’s mind, the researchers designed two algorithms to look at data — without any prior knowledge of its mathematical structure — and pull out patterns. The algorithms then begin conjecturing formulae to fit these data, and continually optimize their assertions until they are essentially fool-proof.
From there, it’s up to human mathematicians to confirm the new formulae with rigorous proofs to show they apply to all possible scenarios stemming from these data, and not just cherry-picked ones.
What they discovered — At first, the machine began spitting out previously proven formulae of fundamental constants from different areas of math, like combinatorics, chaos theory, and continued fractions. This was a good sign the machine seemed to be working as expected. But then it started doing something interesting — churning out new, unproven formulae.
The authors shared their initial results on the preprint server, arXiv, and many of these new formulae have already been proven by the mathematical community.
“We consider the discoveries and the way they are discovered as an approach to inspiring more people about mathematics,” Hadad says.
He tells Inverse they are especially excited about attracting a younger crowd to math. One of the most exciting community proofs thus far was done by a 15-year-old, Hadad says.
What’s next — Right now, this invention is limited in that it looks only at a narrow range of mathematical constants, but in the future, the researchers hope the machine could be trained to look at constants beyond pure math, including those integral to physics, biology, and chemistry.
Algorithms like these could play a big role in unearthing new and important formulae, but mathematicians won’t be out of a job anytime soon. In fact, they will be essential for confirming and advancing these discoveries, write the authors.
“This back-and-forth dynamics between algorithms and mathematicians is the essence of what we believe can be achieved with automatically generated conjectures of fundamental constants,” they write.
(Inverse reached out the American Institute of Mathematics for comment on this story, but they did not respond.)
Abstract: 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. Nevertheless, for centuries new mathematical formulas relating fundamental constants have been scarce and usually discovered sporadically. Such discoveries are often considered an act of mathematical ingenuity or profound intuition by great mathematicians such as Gauss and Ramanujan. Here we propose a systematic approach that leverages algorithms to discover mathematical formulas for fundamental constants and helps to reveal the underlying structure of the constants. We call this approach ‘the Ramanujan Machine’. Our algorithms find dozens of well-known formulas as well as previously unknown ones, such as continued fraction representations of π, e, Catalan’s constant, and values of the Riemann zeta function. Several conjectures found by our algorithms were (in retrospect) simple to prove, whereas others remain as yet unproved. We present two algorithms that proved useful in finding conjectures: a variant of the meet-in-the-middle algorithm and a gradient descent optimization algorithm tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values; consequently, they conjecture formulas without providing proofs or requiring prior knowledge of the underlying mathematical structure, making this methodology complementary to automated theorem proving. Our approach is especially attractive when applied to discover formulas for fundamental constants for which no mathematical structure is known, because it reverses the conventional usage of sequential logic in formal proofs. Instead, our work supports a different conceptual framework for research: computer algorithms use numerical data to unveil mathematical structures, thus trying to replace the mathematical intuition of great mathematicians and providing leads to further mathematical research.
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