New machine math? Elegant patterns and helplessness
According to some experts, machines can invent or, if you like, discover completely new mathematics that we humans have never seen or thought of. Others argue that machines do not invent anything on their own, they can only represent the formulas we know in a different way, and they cannot cope with some mathematical problems at all.
Recently, a group of scientists from the Technion Institute in Israel and Google presented automated system for generating theoremswhich they called the Ramanujan machine after the mathematician Srinivasi Ramanujanawho developed thousands of groundbreaking formulas in number theory with little or no formal education. The system developed by the researchers turned a number of original and important formulas into universal constants that appear in mathematics. A paper on this topic has been published in the journal Nature.
One of the machine-generated formulas can be used to calculate the value of a universal constant called Catalan number, more efficient than using previously known human-discovered formulas. However, scientists claim that Ramanujan's car it is not meant to take math away from people, but rather to offer help to mathematicians. However, this does not mean that their system is devoid of ambition. As they write, the Machine "attempts to emulate the mathematical intuition of the great mathematicians and to provide hints for further mathematical quests."
The system makes assumptions about the values of universal constants (such as) written as elegant formulas called continued fractions or continued fractions (1). This is the name of the method of expressing a real number as a fraction in a special form or the limit of such fractions. A continued fraction can be finite or have infinitely many quotients.i/bi; fraction Ak/Bk obtained by discarding the partial fractions in the continued fraction, starting from the (k + 1)th, is called the kth reduct and can be calculated by the formulas:-1=1,A0=b0, AT-1=0,V0=1, Ak=bkAto-1+akAto-2, ATk=bkBto-1+akBto-2; if the sequence of reducts converges to a finite limit, then the continued fraction is called convergent, otherwise it is divergent; A continued fraction is called an arithmetic ifi=1, b0 completed, bi (i>0) – natural; arithmetic continued fraction converges; every real number expands to a continued arithmetic fraction, which is finite only for rational numbers.
1. An example of writing Pi as a continued fraction
Ramanujan Machine Algorithm selects any universal constants for the left side and any continued fractions for the right side, and then calculates each side separately with some precision. If both sides appear to overlap, the quantities are calculated with more precision to ensure that the match is not a match or inaccuracy. Importantly, there are already formulas that allow you to calculate the value of universal constants, for example, with any precision, so the only obstacle in checking page conformity is the calculation time.
Before implementing such algorithms, mathematicians had to use an existing one. mathematical knowledge i theoremsmake such an assumption. Thanks to the automatic guesses generated by algorithms, mathematicians can use them to recreate hidden theorems or more "elegant" results.
The most notable discovery of researchers is not so much new knowledge as a new assumption of surprising importance. This allows calculation of the Catalan constant, a universal constant whose value is needed in many mathematical problems. Expressing it as a continued fraction in a newly discovered assumption allows for the fastest calculations to date, defeating earlier formulas that took longer to process in a computer. This seems to mark a new point of progress for computer science since when computers first beat chess players.
What AI can't handle
Machine algorithms As you can see, they do some things in an innovative and efficient way. Faced with other problems, they are helpless. A group of researchers at the University of Waterloo in Canada discovered a class of problems using machine learning. The discovery is connected with a paradox described in the middle of the last century by the Austrian mathematician Kurt Gödel.
Mathematician Shai Ben-David and his team presented a machine learning model called maximum prediction (EMX) in a publication in the journal Nature. It would seem that a simple task turned out to be impossible for artificial intelligence. Problem posed by the team Shai Ben David comes down to predicting the most profitable advertising campaign, focused on the readers who visit the site most frequently. The number of possibilities is so great that the neural network is not able to find a function that will correctly predict the behavior of website users, having only a small sample of data at its disposal.
It turned out that some of the problems posed by neural networks are equivalent to the continuum hypothesis posed by Georg Cantor. The German mathematician proved that the cardinality of the set of natural numbers is less than the cardinality of the set of real numbers. Then he asked a question that he could not answer. Namely, he wondered whether there is an infinite set whose cardinality is less than the cardinality set of real numbersbut more power set of natural numbers.
Austrian mathematician of the XNUMXth century. Kurt Gödel proved that the continuum hypothesis is undecidable in the current mathematical system. Now it turns out that mathematicians designing neural networks have faced a similar problem.
So, although invisible to us, as we see, it is helpless in the face of fundamental limitations. Scientists wonder if with problems of this class, such as infinite sets, for example.
