Can Deep Learning AI Unlock the Mysteries of Math?

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 perfect testing ground for artificial intelligence capabilities. If we consider each mathematical theorem as an independent node, we will find that there are countless links between nodes, forming a vast and complex network. In fact, this network is so complex that current mathematicians find it difficult to master every node in the network. 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 then hand it over to computers for processing and discovery of new theorems.

In fact, we find that in this mathematical network, there is a concept of “distance” between theorems, with some theorems being very “close” and others being very “far” apart. We 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, we find that many 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 Fermat’s Last Theorem, a famous conjecture in number theory. Another example is Shinichi Mochizuki’s proof of the ABC conjecture using the Inter-universal Teichmüller Theory (although no one can understand 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 basic “artificial mathematician”. Of course, to succeed, we still need to overcome the difficulty of expressing the final results in a simple or natural way and allowing the machine to continue learning 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 key problems to be solved was how to determine whether the current operation was heading in the correct (simplified) direction during calculation. The artificial intelligence of that era relied heavily on tree-structured searches, but mathematical operations at each step would produce a large number of branches, and exploring them one by one would be extremely time-consuming. 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 principle to the Monte Carlo tree search used by AlphaGo.

Another notable project is a program developed by Professor Douglas of Stanford University, which can “invent” mathematical theorems on its own. Initially, he defined a set of symbols and a set of “interesting” relationships between them, and then input these two sets of data into the computer. After that, his program did not intervene further, but instead allowed the computer to discover the relationships on its own. Finally, he found that the computer could re-invent how to count, how to add, how to multiply, how to identify prime numbers, and even re-”discover” the Goldbach conjecture. Although these are all knowledge that humans have known for thousands of years, 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 had deduced to produce new theories. Therefore, the limited mathematical knowledge deduced by the computer may be due to the fact that the input data already contained these theorems, and the artificial intelligence he created could not create new knowledge.

Back to the present, deep learning enables us to make more effective use of existing hardware resources. However, based on the current computer system framework, even if we successfully input all mathematical theories into a neural network, the best result we can expect is to analyze existing human mathematical knowledge and fill in some missing links between theorems. Current artificial intelligence still lacks the ability to create new mathematical fields like a genius mathematician or to penetrate the mysteries of mathematics.