Machines Are Inventing New Math We've Never Seen (vice.com) 44
An anonymous reader quotes a report from Motherboard: [A] group of researchers from the Technion in Israel and Google in Tel Aviv presented an automated conjecturing system that they call the Ramanujan Machine, named after the mathematician Srinivasa Ramanujan, who developed thousands of innovative formulas in number theory with almost no formal training. The software system has already conjectured several original and important formulas for universal constants that show up in mathematics. The work was published last week in Nature.
One of the formulas created by the Machine can be used to compute the value of a universal constant called Catalan's number more efficiently than any previous human-discovered formulas. But the Ramanujan Machine is imagined not to take over mathematics, so much as provide a sort of feeding line for existing mathematicians. As the researchers explain in the paper, the entire discipline of mathematics can be broken down into two processes, crudely speaking: conjecturing things and proving things. Given more conjectures, there is more grist for the mill of the mathematical mind, more for mathematicians to prove and explain. That's not to say their system is unambitious. As the researchers put it, the Ramanujan Machine is "trying to replace the mathematical intuition of great mathematicians and providing leads to further mathematical research." In particular, the researchers' system produces conjectures for the value of universal constants (like pi), written in terms of elegant formulas called continued fractions. Continued fractions are essentially fractions, but more dizzying. The denominator in a continued fraction includes a sum of two terms, the second of which is itself a fraction, whose denominator itself contains a fraction, and so on, out to infinity.
The Ramanujan Machine is built off of two primary algorithms. These find continued fraction expressions that, with a high degree of confidence, seem to equal universal constants. That confidence is important, as otherwise, the conjectures would be easily discarded and provide little value. Each conjecture takes the form of an equation. The idea is that the quantity on the left side of the equals sign, a formula involving a universal constant, should be equal to the quantity on the right, a continued fraction. To get to these conjectures, the algorithm picks arbitrary universal constants for the left side and arbitrary continued fractions for the right, and then computes each side separately to a certain precision. If the two sides appear to align, the quantities are calculated to higher precision to make sure their alignment is not a coincidence of imprecision. Critically, formulas already exist to compute the value of universal constants like pi to an arbitrary precision, so that the only obstacle to verifying the sides match is computing time.
One of the formulas created by the Machine can be used to compute the value of a universal constant called Catalan's number more efficiently than any previous human-discovered formulas. But the Ramanujan Machine is imagined not to take over mathematics, so much as provide a sort of feeding line for existing mathematicians. As the researchers explain in the paper, the entire discipline of mathematics can be broken down into two processes, crudely speaking: conjecturing things and proving things. Given more conjectures, there is more grist for the mill of the mathematical mind, more for mathematicians to prove and explain. That's not to say their system is unambitious. As the researchers put it, the Ramanujan Machine is "trying to replace the mathematical intuition of great mathematicians and providing leads to further mathematical research." In particular, the researchers' system produces conjectures for the value of universal constants (like pi), written in terms of elegant formulas called continued fractions. Continued fractions are essentially fractions, but more dizzying. The denominator in a continued fraction includes a sum of two terms, the second of which is itself a fraction, whose denominator itself contains a fraction, and so on, out to infinity.
The Ramanujan Machine is built off of two primary algorithms. These find continued fraction expressions that, with a high degree of confidence, seem to equal universal constants. That confidence is important, as otherwise, the conjectures would be easily discarded and provide little value. Each conjecture takes the form of an equation. The idea is that the quantity on the left side of the equals sign, a formula involving a universal constant, should be equal to the quantity on the right, a continued fraction. To get to these conjectures, the algorithm picks arbitrary universal constants for the left side and arbitrary continued fractions for the right, and then computes each side separately to a certain precision. If the two sides appear to align, the quantities are calculated to higher precision to make sure their alignment is not a coincidence of imprecision. Critically, formulas already exist to compute the value of universal constants like pi to an arbitrary precision, so that the only obstacle to verifying the sides match is computing time.
Sure, why not? (Score:1)
Math itself is just logic.
As long as you can provably say that whatever function you're performing is logically equivalent to and compatible with the other math you're going to interact with... then sure, you can make whatever new mathematics functions you want.
They're all tools - and that's why we developed computers, so we can automate the math beyond what we were limited to on blackboards.
Ryan Fenton
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They've made small steps toward guessing at Taylor series.
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Math itself is just logic
Gödel might have disagreed with this, although there are plenty of neo-logicists who would buy it, or something in the same neighbourhood.
Best wishes,
Bob
Re:Sure, why not? (Score:5, Interesting)
"Math itself is just logic." I think what you mean is that mathematical proofs are just proofs in a logic. There are more logics than you'd care to know about. Math, as an endeavor of mathematicians, is much more than blindly following a series of logical rules. In fact, the proof itself is merely the end product of a fair amount of mental blood and sweat.
Put another way, if math were just logic, we could program computers to turn out mathematical proofs of increasing complexity and length. Those are proofs simply because the rules were followed. And then you'd still run into Goedel's theorems: any sufficiently powerful formal system (sufficient means able to code arithmetic) will have true but unprovable statements where unprovable means unprovable in the system.
So now your computer must generate increasingly powerful formal systems as well as produce proofs in those systems. And it isn't clear which more powerful system you need to prove the true but unprovable statement of the old system.
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Directions of thought (Score:2)
Not quite Mathematics is following a direction of thought. Two of those directions are already quite known: the Greek and the Arabic math.
The ancient Greek were not mathematicians, but more like builders. Greek math is about building. It only expresses lengths and surfaces, and whole numbers. For Greek math, you need spatial awareness
Arabic math was more general and abstract. For Arabic math, you need abstract thinking abilities.
The funny thing is that this makes some problems easier in one form than the ot
They called them "Machines" (Score:2)
Re:They called them "Machines" (Score:5, Insightful)
Critically, formulas already exist to compute the value of universal constants like pi to an arbitrary precision, so that the only obstacle to verifying the sides match is computing time.
They just need to add "coin" to it and they'll get countless yahoos giving them more computing time then they could ever hope to muster or even feasibly use.
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Consider the poker programmers who enumerated every way to manipulate sets of ranks within suits using and/or/xor/not so that they had efficient functions for pair, trips, quads, and full houses detection, all relying on some underlying co
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INCOMING MESSAGE (Score:2)
This is the voice of world control [youtu.be]...
This isn't really special (Score:5, Informative)
The are randomly trying new constants to see if they are equal to old constants, and this kind of thing is great for brute forcing to find interesting things. But there is nothing really intelligent about it. Not much more then throwing assortments of constants in a for loop then finding associations and then seeing if there is math gold in it, good luck.
Re: This isn't really special (Score:2)
And yet, the movements of many organisms (including humans), under certain conditions can be described in terms of a random walk (aka a levy flight). Is this intelligence?
I'm of the view that intelligence is tailored to the environment an organism exists within (actually its better thought of as a single animal-environment system). I'm not sure how we define the environment in the context of a computer (some kind of computational space, perhaps?), but in this view, I think something the what the researchers
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This isn't random walk, it's not even genetic search
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What is perhaps new here is automating the conjecture as well as the proof. Normally in a theorem prover the human has to give it a goal, otherwise you can trivially churn out a literally infinite number of provably true things that are not interesting at all. So they are linking this to a conjecture-generator that first uses heuristics to identify "interesting" things that might be true (conjectures). Then I
All operators in an 8 term equation (Score:1)
From the equations kept, lets add another 8 more items to the equation and recompute, if it gets closer to the constant, keep it
rinse, repeat, rinse, repeat
The question is does it find any equations which are as simple as or even simpler than the existing best known equation?
Then we
Just more proof that math is not science (Score:3)
Millions of monkey typewriter squads discover new math.
Re:Just more proof that math is not science (Score:4, Insightful)
Of course math isn't science.
Math discovers interesting relationships between abstract concepts, and then proves those relationships always hold.
Science makes an observation about how some real-life objects might behave, then tries it out several times to see if it seems to pretty much work on the ones we tried.*
* Alternatively, faux science starts with a statement that the sponsor wishes to support, and tries to design experiments and analysis which will appear support that statement,.while borrowing scientific lingo.
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Who has ever said that math is "science"?
The application of math is science.
Math is language.
Same that application of a language writing system is a text (poetry, narrative, etc.); and a language is not a text by itself.
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It was the best of times, it was the blurst of times.
Stealing Grad Students Jobs! (Score:1)
inventing (Score:4, Funny)
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I also had to repeat diff. eq....
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Can you invent new math
I think it will be difficult to improve upon Tom Lehrer [youtube.com].
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In my words: There is "Math" and there is "math."
"Math" is God, the reason for existence, and the rules that everything follows. "Math" must be discovered or "divined."
"math" is the language that mankind uses to describe "Math."--Physics adheres to "Math" but is described using "math."
So, there can be new "math." There cannot be new "Math."
Saving paper (Score:5, Funny)
As the researchers explain in the paper, the entire discipline of mathematics can be broken down into two processes, crudely speaking: conjecturing things and proving things.
Actually, this system both creates the conjectures and finds the associated proofs. However, in an effort to go green and save trees, it just prints the following message in the margin of each printout:
"I have discovered a truly marvelous proof of this, which this margin is too narrow to contain"
Re:Saving paper (Score:5, Funny)
I thought the printout would simply be a statement of the conjecture and the text, "The proof is left as an exercise for the reader".
this book is 50 years old (Score:2)
the US and Russian govts built super brian computers to control warfare. when the computers found each other, they started sharing and learning new math not yet invented by man.... mankind doesn't do so well in this movie.
here's the trailer: https://www.youtube.com/watch?... [youtube.com]
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Surely you meant "Batman Brian" [nocookie.net]. Peter is Superman.
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New Math We've Never Seen? (Score:1)
Is there such thing as new math we have seen or old math we've never seen?
car analogy incoming! (Score:2)
It's used but New To You math.
Often, journalists and science don't mix (Score:5, Informative)
Hard to tell from the vapid article, but I imagine the actual paper does have some real contribution to make.
But the "this is a huge breakthrough! machines inventing new math!" headlines are absurdly overblown. There have been plenty of other algorithms useful for conjecturing formulas and relationships for particular constants, like the Ferguson-Forcade algorithm (1979), LLL lattice basis reduction ('82), and PSLQ ('92). See wikipedia [wikipedia.org]. That has resulted in lots of little curiosities and some handfuls of actually useful formulas. The algorithm in the paper may come up with some more of both.
But with all the hype in the headline here, you'd think they'd come up with an artificial general intelligence which was solving Millennium problems.
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I fear that is too optimistic. The overview of the code [github.com] strongly suggests that it's not even at the level of generating high-precision estimates of random continued fractions and then using an integer relation algorithm to relate them to the constants. Instead it just builds a hash table of derived constants and searches that hash table.
Sadly, that seems to put it in the same class as most of the Nature papers that I hear about nowadays: im
So in a nutshell (Score:2)
It's a machine that creates and solves math homework.
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No, it just creates it. They're recruiting volunteers to solve it.
Bistromathics (Score:2)
It is an open project - link from paper (Score:3)
The actual paper looks interesting and can be viewed with an enhanced pdf viewer which I haven't seen before.
full article [nature.com].
The paper contains a link: www.ramanujanmachine.com [ramanujanmachine.com]
Below is the text containing the link. (RF means regular function, and they are searching for polynomial continued fractions that match arbitrary universal constants some of which have no formulation at all, which is cool.)
In contrast to the method we present, many known RFs for funda-mental constants were discovered by conventional proofs, that is, sequential logical steps derived from known properties23. In our work, we aim to reverse this process, finding new RFs for the fundamental constants using their numerical data alone, without any prior knowl-edge about their mathematical structure (Fig. 1). Each RF may enable reverse-engineering of the mathematical structure that produces it. In certain cases, where the proof uses new techniques, it may also provide insight into the field. Our approach could be especially valuable when applied for empirical constants, such as the Feigenbaum constant from chaos theory (Table 1), which are derived numerically from simulations and have no analytic representation.
Given the success of our approach to finding new RFs for fundamental constants, there are additional avenues for more advanced algorithms and future research. Inspired by worldwide collaborative efforts in mathematics such as the Great Internet Mersenne Prime Search (GIMPS; https://www.mersenne.org/ [mersenne.org]), we launched the initiative http://www./ [www.] RamanujanMachine.com, dedicated to finding new RFs for fundamental constants. The general community can donate computational time to find RFs, propose mathematical proofs for conjectured RFs, or suggest new algorithms for finding them (Supplementary Information section B). Since its inception53, the Ramanujan Machine initiative has already yielded fruit, and several of the conjectures posed by our algorithms have already been proved
Not "New Math" (Score:2)
Keynesian multiplier (Score:2)
It's entirely possible that voting machines employ a Keynesian Multiplier and a Potemkin Village Filter. How's that for some new math?
So what? (Score:1)
There are an infinite number of possible theorems under most axiom systems. Only a small finite number are interesting.
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There are an infinite number of possible theorems under most axiom systems. Only a small finite number are interesting
FTFY: /. posts under most axiom systems. At most a small finite number are interesting
There are an infinite number of possible
1,1,2,5,14,... (Score:1)