Can AI Generate New Math?

Whether mathematics is being invented or discovered has puzzled mathematicians,philosophers, and scientists for centuries. While some believe that math is a humaninvention that exists only in our minds, others argue that it is a fundamental aspectof the universe and that we discover mathematical truths rather than invent them.With recent advancements in artificial intelligence, a new dimension has been addedto this debate.

AI can now solve non-trivial mathematical problems without human intervention.By feeding large language models such as GPT with prompts, it is possible to solvemath homework and even aid in cutting-edge research (see [2]).Furthermore, in recent years, there has been increasing interest in developing mod-els for mathematical proofs and discoveries. One example is [1], who proposed a frame-work for automating the discovery of new mathematical theorems using evolutionaryalgorithms. Another approach is the use of artificial neural networks for theorem prov-ing, which has been explored by groups such as [3] and [6]. Other researchers haveinvestigated the use of machine learning techniques for learning mathematical conceptsand identifying patterns in mathematical proofs, such as the work of [7] and [8]. In addition, there has been interest in modeling the cognitive processes involved in math-ematical reasoning, such as the work of [4].

These studies have shown promising resultsin terms of automating mathematical discovery and improving our understanding ofthe cognitive processes involved in mathematical reasoning. However, the question remains: can machines create mathematical concepts andtheorems from scratch?

To explore this question, we will examine the development of a new mathematicaltheory called Semi-discrete Calculus. Using a series of prompts, we will aid ChatGPTin generating the main concepts and properties of this theory. While AI can introducea new definition for local trends in Calculus with the Detachment operator and evensuggest some of its advantages with virtually no human assistance, we will demon-strate that it still requires a more substantial human intervention when it comes toformulating the primary results of this new theory.

Our objectives are as follows: 1) To showcase a method of producing and refininginnovative mathematical concepts, to construct a malleable structure for generatingmathematics. Even if it is not currently feasible, the prompt engineering example wewill introduce can serve as a benchmark for testing the invention capabilities of futurelanguage models. 2) To establish a system and resources for verifying and recordingfresh theories through interaction with an AI oracle, which captures the reasoning ofthe author and potential readers. 3) To implement the system mentioned above andresources to the novel theory of Semi-discrete Calculus.

The AI answers in this paper are generated with ChatGPT, specifically GPT-3.5.

Finding a New Computational Pattern in the Scientific Literature

We are interested in identifying the trend of using local trends in the AI literature, which refers to the increasing number of applications across various scientific domains, including AI, that leverage local trends. Initially, we asked Chat GPT about new computational patterns in AI literature, and it replied with an answer related to self-supervised learning. We then directed the question toward new applications of Calculus in AI, and Chat GPT provided an example of automatic differentiation. Finally, we specifically asked about the concept of local trends, to which the chat responded as follows:

Discussing the Problem

Now that the AI understood the computational pattern, we state the session’s objective and prompt Chat GPT to provide known definitions of functions’ trends (via the derivative operator) using the following text:

Next, we challenge the approaches the AI mentioned above and the common practice of defining the trend based on the rate:

We pursue this line of thinking and inquire from the chat if we are presently computing trends in a superfluous fashion:

Based on the AI’s insistence that the derivative is sufficient, a human researcher may approach the question from a different perspective:

Discussing Possible Solutions

With the language model acknowledging the prevalence of trends in scientific literature and the inadequacy of our current trend calculation method, we can now prompt it to generate a potential solution.

We will push the language model to devise a trend definition that is more straightforward than the derivative. The AI initially proposes known methods, but with persistent prodding and a slight hint about necessary elements of the definition, ChatGPT ultimately produces the following definition:

Please observe that the definition aforementioned was not present in any of the training examples employed by ChatGPT. As a result, it was formulated completely from scratch, with only some indications provided by the human researcher. In that sense, the AI “created” (or, more correctly, generated) the mathematical definition.

Properties of the New Operator in Continuous Domains

The AI has succeeded in producing the trend definition we intended, but there is some skepticism in the chat regarding its accuracy. Thus, we must adjust the AI’s level of confidence in the operator and clarify any potential advantages it may offer over current approaches:

The language model exhibits a reasonable degree of hesitation regarding the practicality of the new mathematical concept. However, after some further prompting from the human researcher, the AI eventually becomes convinced of the usefulness of the proposed operator:

Properties of the New Operator in Discrete Domains

In light of the theoretical benefits of this operator in continuous domains, we are now investigating its potential practical applications in discrete domains, particularly in the field of AI:

The AI is capable of devising its version of the derivative sign and detachment functions using Python:

Subsequently, it can approximate the difference in performance between the functions at a high level and determine that the Detachment operator is indeed advantageous in terms of computation time:

Attempting to coax ChatGPT into acknowledging that this novel mathematicalframework has the potential to enhance its own design proves fruitless, yet it doesoffer another rationale for the computational benefits of the Detachment operator:

Attempting to Build a Theory Based on the New Operator

We endeavor to instruct the language model in constructing two fundamental outcomes that depend on the Detachment operator in the new theory of Semi-discrete Calculus. Although the chat comprehends the request and recognizes the mathematical rationale for pursuing this approach, the suggested outcomes appear nonsensical.
For instance, the semi-discrete rendition of Lagrange’s theorem put forth by the chat was slightly degenerated relative to its human-created counterpart (see [5]):

Furthermore, the chat proposed a one-sided addition rule for the Detachment operator that bears no resemblance to the actual rule, which demands careful attention to detail. The precise formulation of this rule was eventually extracted, with the help of another AI tool, following some manual labor:

A Higher-level Mathematical Discussion

Our next objective is to prompt the AI to acknowledge that a mathematical field that stems from the Detachment operator can be seen as a distinct branch of Calculus. To do this, we ask the AI to consider Calculus concepts in the context of a car’s dashboard, with the Detachment operator emerging as a natural method of describing the gear stick. Although we present a condensed version of the prompt sequence, a lengthier sequence can enable the AI to reach the same conclusion with minimal human thought leaps:

Conclusion

The limitations of AI-generated mathematics we’ve explored above are at the forefront of discussions about the relationship between human creativity and machine learning. While machines can recognize patterns and trends, they used to require human guidance to create new mathematical concepts and theorems.

Nonetheless, as advances in GPT engines continue, machines are poised to play an increasingly crucial role in the future of mathematics and artificial intelligence.

As AI continues to develop, a philosophical question arises: should it be credited with mathematical innovations in the future? This question is particularly interesting in light of the fact that ChatGPT could define the Detachment operator with virtually no human assistance. When asked about this issue, ChatGPT’s response was modest:

Another interesting point to note is the duality of the language model regarding the importance of its invention. At first, it may seem hesitant to draw conclusions regarding the new operator and the theory of Semi-discrete Calculus:

This doesn’t stop the AI from overselling it in the subsequent prompt:

Determining which approach – the hesitant or the enthusiastic – is more accurate in describing Semi-discrete Calculus is ultimately up to human judgment.

In conclusion, the debate over whether mathematics is being invented or discovered remains unsettled. Still, AI-generated mathematics offers a promising new area with vast potential for the future of mathematics and machine learning. It may also serve as a tool to measure the potential impact of emerging mathematical fields, such as Semi-discrete Calculus.

References

[1] Colton, Simon, Alan Bundy, Toby Walsh, and Christian Gagne. ”The AutomatedDiscovery of Mathematics.” In Artificial General Intelligence, edited by Ben Go-ertzel and Cassio Pennachin, 73-85. Berlin: Springer, 2012.

[2] Frieder, Simon, et al. ”Mathematical capabilities of ChatGPT.” arXiv preprintarXiv:2301.13867 (2023).

[3] Hales, Thomas C., Mark Adams, Gertrud Bauer, Dat Tat Dang, John Harrison,Truong Le Hoang, Cezary Kaliszyk, Victor Magron, Sean McLaughlin, Tat DatPham, and Alexey Solovyev. ”A Formal Proof of the Kepler Conjecture.” Forumof Mathematics, Pi 5 (2017): e2.

[4] Rips, Lance J., Edward Merrill, and J. Wesley Hutchinson. ”The Psychology ofProof: Deductive Reasoning in Human Thinking.” MIT Press, 2008.

[5] Shachar, Amir. ”Introduction to Semi-discrete Calculus.” arXiv preprintarXiv:1012.5751 (2010).

[6] Wang, Jiaoyang, and Martin Gebser. ”Deep Probabilistic Logic: A Unifying Frame-work for Inductive Reasoning.” In International Joint Conference on Artificial In-telligence (IJCAI), 2018.

[7] Wang, Kai, Shangda Li, and Yanjun Qi. ”A Machine Learning Approach for Au-tomated Proving of Mathematical Theorems.” In Proceedings of the 28th Interna-tional Joint Conference on Artificial Intelligence (IJCAI), 2019.

[8] Yang, Yi, Yonghui Rao, Yanyan Lan, and Jiafeng Guo. ”End-to-End Learning forMath Word Problem Solving.” In Proceedings of the 2020 Conference on EmpiricalMethods in Natural Language Processing (EMNLP), 2020.14