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Ramanujan Machine Automatically Generates Conjectures for Fundamental Constants:
A team of researchers at the Israel Institute of Technology has built what they describe as a Ramanujan machine—a device that automatically generates conjectures (mathematical statements that are proposed as true statements) for fundamental constants. They have written a paper describing their device and have uploaded it to the arXiv preprint server. They have also created a webpage for people who wish to allow the network to use their computer's process cycles, suggest a proof or develop code toward new mathematical structures.
The Ramanujan machine is named for famed Indian mathematician Srinivasa Ramanujan, a self-taught mathematician who grew up in India and was "discovered" by fellow mathematician G.H. Hardy. After moving to England, he became a fixture at Cambridge, where he shook up the math world with his unorthodox mathematics—instead of pounding away at math proofs, he obtained results to famous problems through intuition and then let others find the proofs for them. Because of this, he was sometimes described as a conjecture machine, pulling formulas out of thin air as if they received from a higher being—sometimes in dreams. In this new effort, the researchers in Israel have sought to replicate this approach using computing power.
The Ramanujan machine is more of a concept than an actual machine—it exists as a network of computers running algorithms dedicated to finding conjectures about fundamental constants in the form of continued fractions—these are defined as fractions of infinite length where the denominator is a certain quantity plus a fraction, where a latter fraction has a similar denominator, etc.) The purpose of the machine is to come up with conjectures (in the form of mathematical formulas) that humans can analyze, and hopefully prove to be true mathematically. The team that created the machine is hoping that their idea will inspire future generations of mathematicians—to that end, they note that any new algorithms, proofs or conjectures developed by a participant will be named after them. The researchers note that their machine has already discovered dozens of new conjectures.
The Abstract and full paper are available on arXiv.org.
From the abstract (with formulas adjusted so they could be displayed here):
Fundamental mathematical constants like 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. In this paper we propose a novel and systematic approach that leverages algorithms for deriving new mathematical formulas for fundamental constants and help reveal their underlying structure. Our algorithms find dozens of well-known as well as previously unknown continued fraction representations of π, e, and the Riemann zeta function values. Two new conjectures produced by our algorithm, along with many others, are:
e = (3 + (-1 / (4 + (-2 / (5 + (-3 / (6 + (-4 / (7 + ... ) ) ) ) ) ) ) ) )
4 / (π - 2) = 3 + (1·3) / (5 + (2·4) / (7 + (3·5) / (9 + (4·6) / (11 + ...) ) ) )
We present two algorithms that proved useful in finding new results: a variant of the Meet-In-The-Middle (MITM) algorithm and a Gradient Descent (GD) tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values and thus find new conjecture formulas without providing proofs and without requiring prior knowledge on any mathematical structure. This approach is especially attractive for fundamental constants for which no mathematical structure is known, as it reverses the conventional approach of sequential logic in formal proofs. Instead, our work presents a new conceptual approach for research: computer algorithms utilizing numerical data to unveil new internal structures and conjectures, thus playing the role of mathematical intuition of great mathematicians of the past, providing leads to new mathematical research.
(Score: 2) by GreatAuntAnesthesia on Wednesday July 17 2019, @09:34AM (8 children)
So we could end up with mathematical proofs (and by extension algorithms, engineering etc built upon the maths) without actually understanding the fundamentals of how or why they work?
"So this new invention can cure cancer / fly people to Mars / provide cheap, clean power / automatically punch telemarketers in the face."
"Cool, how does it work?"
"uuuhhhhhh...."
(Score: 1) by khallow on Wednesday July 17 2019, @10:16AM
(Score: 3, Insightful) by takyon on Wednesday July 17 2019, @10:36AM (4 children)
Computer Generated Math Proof is Largest Ever at 200 Terabytes [soylentnews.org]
That ship has sailed.
Once we get AI agents or artilects working on software, science, and engineering problems, technology could transition quickly into "magic".
[SIG] 10/28/2017: Soylent Upgrade v14 [soylentnews.org]
(Score: 2) by Freeman on Wednesday July 17 2019, @04:13PM (2 children)
42
Joshua 1:9 "Be strong and of a good courage; be not afraid, neither be thou dismayed: for the Lord thy God is with thee"
(Score: 2) by takyon on Wednesday July 17 2019, @04:18PM
420 yottabytes
[SIG] 10/28/2017: Soylent Upgrade v14 [soylentnews.org]
(Score: 2) by maxwell demon on Wednesday July 17 2019, @08:05PM
If they ever try Ramanujan on the constant 42, watch out for yellow space ships!
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2) by maxwell demon on Wednesday July 17 2019, @06:04PM
So you say, our technology may soon be sufficiently advanced?
The Tao of math: The numbers you can count are not the real numbers.
(Score: 5, Interesting) by AthanasiusKircher on Wednesday July 17 2019, @02:53PM (1 child)
There is absolutely nothing new about this. Engineers have practical approximation equations they use in all sorts of circumstances. Some of them are derived with simplifying assumptions in mind, but others are strictly empirical in nature -- derived from data fitting rough curves. But they are used because they work in practical circumstances.
Moreover, this sort of modeling is a foundation of modern science. If you accept the "Scientific Revolution" as valid, you also must accept this, as it was a cornerstone of the shift that happened in the 1600s.
Prior to that time, the authority was Aristotle, who was always concerned about explanations, which to Aristotle and other ancient Greeks, were often framed in terms of "causes" (though that term "cause" has a broader meaning to the Greeks than the English word usually does).
The Scientific Revolution is often portrayed as some sort of growth in empiricism (though scholars had been doing empirical experiments for many centuries before) or denouncing of religion (though ignorant religious explanations had been doubted in some cases for centuries, though in other cases they continued to be believed well into the 20th century by large numbers of scientists). These really weren't the main shifts, though.
But the true break of the Scientific Revolution was with Aristotle. First, there was the rise of Mechanistic explanations of reality, championed by philosophers such as Descartes, who postulated that the traditional boundaries of Aristotelian physics were no longer useful. Instead, everything (except the conscious mind, for Descartes) could and should be explained by more fundamental physical processes common to everything. Terrestrial and celestial matter should follow the same laws on a microscopic scale. Living and inorganic matter should follow the same laws on a microscopic scale. Nobody had any clue at that time about how to explain all of this (though Descartes took a stab at it), but the fundamental shift diverged from Aristotle.
Terrestrial matter, for example, in Aristotelian physics could only move when caused to move. Otherwise it would come to a halt. There needed to be an explanation of "how or why" the matter came to be in motion. Galileo's experiments on inertia showed that maybe this view was wrong, and Descartes followed up with this. (Eventually, it would be formulated into Newton's First Law.) Meanwhile, celestial matter apparently stayed in motion forever, and there were various explanations about why this should be so, too. Descartes had little idea of how biological organisms could work under the same principles as inorganic matter, too, but he postulated it nonetheless -- the "how and why" could be sorted out later, but we should work from an assumption that there's a governing model for all matter. In traditional physics, animal behavior was governed by "causes" about the motivation of animals. Descartes instead postulated it could all be broken down into basic physics. The need for "cause" as an explanatory tool dissipated.
This break from Aristotle was truly fundamental (and primarily "philosophical" from our modern perspective, rather than "scientific"), but if you look at people who first report an intellectual "revolution" happening in the 1600s when they discussed it a century later, this was where the "revolution" happened for them in their thought.
The real test of this came when Newton proposed his theory of Universal Gravitation, which was a primarily mathematical theory, proved by geometry, that claimed to explain the physics of both terrestrial and celestial matter under one set of rules. But in the process, it required something incredibly weird to the empirical and scientific mindset of the day: invisible forces acting at a distance. Newton had no explanation for these "forces." Such forces in the 1600s were associated with the occult -- like magnetism (a "force" at that time which encompassed not just actual magnetic rocks, but other supposed "forces" exerted remotely by one thing upon others, causing changes in behavior, emotional changes, etc.), they were "spooky" stuff that legitimate scientists lumped in with quackery.
Yet Newton persevered. His mathematical model worked. He could not explain the "fundamentals of how or why" it worked, though. He had no Aristotelian "cause." He just knew his theoretical mathematical model fit the data. Many scientists of the day were skeptical, and his spooky "action at a distance" weirdness was derided as unscientific if not downright mystical. In the later editions of his Principia Newton explicitly added further discussion of this issue, and put down the doctrine that would truly come to characterize the difference between Aristotle and modern science.
Modern science would no longer require explanations of "cause" (effectively chains of "hows" and "whys") to support a theory -- instead, empiricism should move toward a view that mathematical models can be effective and "scientific" even without a thorough explanation of how they work. Unfortunately, we've seen some of the legacy of that perspective in recent decades as studies misuse statistics and produce a lot of crap, leading to modern cries of "correlation does not imply causation"! Which is also true. But it's the abuse of math and the abuse of the language of cause (which Newton cautioned against) that has led to bad mathematical modeling and the "replication crisis" in modern science.
Newton's point is subtle, but it's very important to understand: mathematical models are simply models. They don't tell us the "cause" of anything, and we may not understand how or why they work in all of their details. But they don't have to explain everything in order to be a valid part of scientific methodology and inquiry.
(Score: 4, Interesting) by AthanasiusKircher on Wednesday July 17 2019, @03:07PM
After submitting this wall of text, I note that I mostly was addressing parent's implication that it could be problematic for to build practical engineering or algorithms on mathematical models that aren't understood.
I forgot to also address the "pure math" proof aspect, though I don't think that needs further explanation at all. Math has always been primarily a practical endeavor for the vast majority of people. Formal "proof" to justify the "hows and whys," though important for Euclid and geometry, was mostly an afterthought for practical math. The point is that you had a method that worked. Math that could do something. Explaining "why" it happens was irrelevant to the vast majority of people who have used math. The obsession with proof in all mathematical fields is a mostly recent development, growing greatly in the past couple centuries.
Regardless, Ramanujan is a great name for this present development, as he himself practiced a sort of experimental mathematics [stephenwolfram.com] and was often less concerned with rigorous proof.