Can Deep Learning-Based AI Unlock the Mysteries of Mathematics?

Computers originated from mathematics, so as computer artificial intelligence technology continues to update and evolve in recent years, we can’t help but ask: can computers also help us “evolve” ancient mathematics? In fact, mathematics is a great testing ground for artificial intelligence capabilities. If we consider each mathematical theorem as an independent node, you will find that there are countless links between nodes, forming a vast and complex network. In fact, this network is so complex that today’s mathematicians can hardly master every node in it. However, processing complex networks is precisely the strong point of computers, so the remaining problem is how to convert existing mathematical theories created by human wisdom into calculable data and let computers handle it to discover new theorems.

In fact, we find that in this mathematical network, there is a “distance” between theorems, with some theorems being very “close” and others being very “far” apart. You discover that theorems that are close to each other can be easily proven or are similar or even isomorphic. Whether this distance between theorems is due to “fate” or “historical coincidence” is unknown. Looking back at history, you will find that many times, two seemingly unrelated mathematical subfields eventually connect miraculously as human understanding of mathematical knowledge deepens. For example, Andrew Wiles used elliptic curves and modular forms to prove the famous Fermat’s Last Theorem in number theory. Another example is Shinichi Mochizuki’s proof of the ABC conjecture using inter-universal Teichmüller theory (although nobody understands it yet…). Therefore, some people think that if we can quantify the distance between theorems and input this network into a deep learning network, we might be able to create a primitive “artificial mathematician”. Of course, to succeed, we still need to overcome the difficulty of expressing the final result in a simple or natural way and letting the machine continue to learn from its own creations.

In fact, many previous artificial intelligence projects have encountered this problem. For example, when MIT developed the “MACSYMA” computer algebra system based on LISP, one of the important problems to be solved was how to determine whether the current operation is heading in the correct (simplified) direction during calculation. The artificial intelligence of that era mostly relied on tree structure search, while mathematical operations produce a large number of branches at each step, and exploring them one by one would waste a lot of time. To improve speed, they proposed a hierarchical structure, using heuristic algorithms combined with known mathematical knowledge to optimize the overall operation. In fact, this method has a similar idea to the current AlphaGo’s use of Monte Carlo tree search.

Another project that cannot be ignored is a program developed by Professor Douglas of Stanford University, which can “invent” mathematical theorems on its own. At the beginning, he defined a set of symbols and a set of “interesting” relationships between symbols, and then input these two sets of data into the computer. After that, his program did not intervene anymore, but let the computer discover the relationships on its own. Finally, he found that the computer can re-invent how to count, how to do addition, how to do multiplication, how to identify prime numbers, and even re-”discover” the Goldbach conjecture through these two sets of data. These are, of course, knowledge that humans have known for thousands of years, but achieving such results is still exciting. However, what puzzled him was that after this, the computer could not continue to create useful knowledge, and the system could not use the theories it derived to produce new theories. So the limited mathematical knowledge derived by the computer may be because the input data already contained these theorems, and the artificial intelligence he created could not create new knowledge on its own.

Back to the present, deep learning can enable us to utilize existing hardware resources more effectively. However, based on the current computer system framework, even if we successfully input all mathematical theories into a neural network, the best result estimated is that we can analyze existing human mathematical knowledge and supplement some missing “links” between theorems. Current artificial intelligence still does not have the ability to create new mathematical fields like genius mathematicians or penetrate the mysteries of mathematics.