The question of whether mathematics is invented or discovered is a longstanding open debate. From the way results are deduced and obtained as corollaries from axioms and theorems, it appears that mathematics is discovered. From what mathematicians choose to study or the names they give to things, it appears that some part of it is created or invented. The binary is not at all clear.
A more serious and less-discussed discussion is that of whether machines are capable of doing mathematics the way mathematicians do. The question is about the possibility of artificial but intelligent mathematicians. This should not be taken to mean that we aspire to know if machines can do complex manipulations or calculations. Rather, this should be taken to mean if machines are capable of inventing or creating new mathematics. Surprisingly enough, denial is not an acceptable option anymore.
According to a research article titled “Advancing mathematics by guiding human intuition with AI” published with open access in the journal ‘Nature’ on the first day of this very month, authored by fourteen contemporary scholars, there exist “examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning, thereby demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems.” The authors write in the abstract of the paper, “We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures.”
This seems to have serious implications, to say the least. The question of whether mathematics is discovered or invented is a bygone tale. The hot topic is if mathematicians are required or needed to discover or invent, whatever it be, mathematics. This is a bolt from the blue. The question of whether machines can reproduce more of their kind is one, and now, the question of if machines can invent theorems, proofs and theories is another, and almost settled.
Though it is not rocket science to understand that computers and machines can come very handy when it comes to doing drudgery and monotonous calculations, the fact remains that, it was not clear until very recently that artificial intelligence can be immensely helpful in something more meaningful and more important, and that is the psychological mystery called “intuition”. Amazingly enough, there is something non-routine and something unexpected that the findings published in the ‘Nature’ journal revealed. In the words of the authors, “Our contribution shows how mature machine learning methodologies can be adapted and integrated into existing mathematical workflows to achieve novel results.” They further proclaim that “AI can also be used to assist in the discovery of theorems and conjectures at the forefront of mathematical research. This extends work using supervised learning to find patterns by focusing on enabling mathematicians to understand the learned functions and derive useful mathematical insight.” There are a few ways to look at it. One is a positive perspective. And one is a very enervating one. If Artificial Intelligence is potent enough to discover theorems or to make conjectures, it will simplify the work of mathematicians. However, there is another danger. That one thing humans almost always took pride in is the notion that creativity is not possible for machines and that it is the monopoly of the flesh-and-blood creatures only. This notion appears to be shattered. The supremacy of humans over machines appears to be a misnomer. A machine, or more specifically artificial intelligence, is doing much more than what it was expected to do. It is leaving us high and dry, and it is making us rack our brains regarding our understanding of its power. Our understanding of how much we know about AI is put into question again and again. In the words of Eliezer Yudkowsky, “By far the greatest danger of Artificial Intelligence is that people conclude too early that they understand it.”
The danger is that the advent and advancement of AI in mathematics – with the possible potential of discovering theorems, patterns and conjectures – is making inroads, making us doubt ourselves and making us feel inferior. However, the fact remains that we may be filled with a similar complex of being inferior when we draw a parallel between ourselves and a beautiful object such as a flower or a rainbow. AI is just one among a plethora of things that make us self-doubt and that make us think about our position in the world. In the words of Alan Kay, “Some people worry that artificial intelligence will make us feel inferior, but then, anybody in his right mind should have an inferiority complex every time he looks at a flower.”
—The writer is Assistant Professor at Government Degree College Sopore. [email protected]