Mathematical methods and human thought in the age of AI

Abstract. Artificial intelligence (AI) is the name popularly given to a broad spectrum of computer tools designed to perform increasingly complex cognitive tasks, including many that used to solely be the province of humans. As these tools become exponentially sophisticated and pervasive, the justifications for their rapid development and integration into society are frequently called into question, particularly as they consume finite resources and pose existential risks to the livelihoods of those skilled individuals they appear to replace. In this paper, we consider the rapidly evolving impact of AI to the traditional questions of philosophy with an emphasis on its application in mathematics and on the broader real-world outcomes of its more general use. We assert that artificial intelligence is a natural evolution of human tools developed throughout history to facilitate the creation, organization, and dissemination of ideas, and argue that it is paramount that the development and application of AI remain fundamentally human-centered. With an eye toward innovating solutions to meet human needs, enhancing the human quality of life and expanding the capacity for human thought and understanding, we propose a pathway to integrating AI into our most challenging and intellectually rigorous fields to the benefit of all humankind.

The authors of this paper come from academic domains that are frequently viewed as polar opposites: mathematics and the study of art. But we both have found it beneficial to incorporate several AI tools into our disparate areas of research on a day-to-day basis, and found a surprising amount of common ground regarding the very messy, but universal, philosophical questions that real-world AI use poses. Using mathematics as a model, we will consider the benefits, risks, ethics and outcomes of incorporating AI into routine workflows and then expand these observations to broader real-world use. Despite the risks that these new, and not necessarily morally neutral technologies present, we argue twofold that AI tools should be developed, implemented, and applied both within mathematics and in other domains: they have the potential to radically augment our natural human abilities and they are capable of expanding what is possible beyond what we humans could do individually or within the limits of our own collective capacity. Drawing from our own experiences with these tools, we particularly examine the human/AI interface and offer suggestions on the evolution of these technologies in ways that offer more benefits than harms to humanity and value the unique contributions of human thought and action in concert with the new modalities that future AI development promises.

The “prisoner’s dilemma” of such competition has pressured many individuals and organizations to experimentally adopt these tools as hastily as possible, at the expense of more deliberate evaluation of the economic, social, or moral costs and benefits of such an adoption – or, more fundamentally, why we should be developing such technologies in the first place.

Modern AI. But modern AI can automate large portions of the creative process itself, enabling the mass-generation of intellectual products, such as artwork, mathematical proofs, or scientific or philosophical theories, with far less human oversight than was previously required2. This has created an unprecedented decoupling between the outward form of such products, and the values and thought processes used to create these products. A diffusion model may now create an aesthetically pleasing landscape, for instance, which was not directly inspired by any particular location in the physical world, though countless images of actual landscapes (as well as many images completely unrelated to landscapes) were certainly used to train the outputs of that model; the aesthetic response of the image thus becomes decoupled from the original sources of such aesthetics.

Frontier AI models can now solve increasingly complicated mathematical problems, with proofs that can be independently verified, without directly reproducing the problem-solving practices of human mathematicians (such as testing out special cases, and then generalizing from those examples), though its training data would include proofs generated in such a traditional fashion; and so mathematicians will increasingly encounter situations in which the ability to prove theorems is decoupled from the reasoning processes needed to discover and understand such proofs.

Additionally, it is noteworthy that modern AI tools do not pursue or intuit “truth” through manifestation in the physical world, or comprehension of the immutable nature of our reality’s physical laws; instead, these models rely heavily on human-generated data, often without attribution, as well as significant amounts of human feedback to iteratively improve itself. Models cannot be built to be less reliant on human intellectual labor without a serious risk of contaminating our collective body of information with AI-generated information. There is a clear limit to how much AI can be used to generate “new information” in a domain before AI collapse [43] becomes a serious problem. Without a sufficient amount of genuine content, AI becomes ungrounded from reality, caught up in a mode of thought that has no connection to the real world and significantly hampers the meaningful interactions at the human/AI interface. Mathematics, with its formal verification process, may have a tolerance for higher levels of AI contamination than other domains; but as we have seen, it is not completely immune to this danger.

But while technique is certainly an essential component of each of these disciplines, it does not capture the full experience of how mathematics, science, and the arts are conducted in practice, and provides little guidance on such practical questions as how to motivate the next generation of students, or what directions of curiosity-driven research to pursue. So, one could instead retreat to a radically different position, in which one ascribes an ineffable special status to human intellect or human creativity, permanently distinguishing any activity exercising these human traits such talents at a fundamental level from any artificial replication of that activity, regardless of how accurately the latter could replicate or surpass the former at a technical level. In this framework, Artificial Intelligence will forever be “No True Scotsman”: lacking true “soul” or “understanding”. With the long familiarity with our own species, we are used to humans being unreliable, “spiky” in their abilities, and sometimes lucking into successfully achieving a task through random word association and rote memorization; but when AI tools exhibit similar behaviors, one can be inclined to judge them far more harshly, for instance attributing such failings to their inherent nature as “stochastic parrots”. But perhaps this position is simply denying an uncomfortable truth: that some portion of our vaunted human capabilities are in fact not that much more sophisticated in nature than the AI algorithms we have now designed to mimic them. And as AI performance continues to advance, such a human-chauvinistic viewpoint risks degenerating into an increasingly untenable “god of the gaps” philosophy, in which an ever-shrinking list of qualities are touted as indicators of essential human achievement that AI is still not yet able to replicate.

A third option, particularly favored by some enthusiasts of these technologies, is to hold that all human cognitive abilities will soon be completely superseded by their AI equivalents, rendering philosophical discussions about the value of human contributions and concerns to mathematics, science and the arts increasingly moot. In the more extreme versions of this position, the very exercise of human intellect is viewed as an undesirable and tedious activity, which ought to be replaced by automation as quickly as possible, in order to free up time and mental space for more leisurely or hedonistic pursuits. Obviously, an implementation of this philosophy would carry many risks, such as the degradation of human abilities to the point where our species will become collectively unable to monitor, control, or even understand the actions of that increasingly powerful AIs that we will have delegated our civilization to23.

6.4. A Copernican view. One possibility is to embrace a cognitive analogue of the Copernican revolution in astronomy. In antiquity, the dominant models of cosmology (insofar as the universe was viewed in mechanistic terms) were geocentric in nature, in which the Earth had a privileged ontological status as the immobile center of the universe, fundamentally distinct in nature from the heavens above or the underworld beneath. However, multiple advances in astronomy and physics dismantled this view, successively demonstrating over the centuries that the Earth was in fact in motion around its axis, and in orbit around the Sun, with the Sun itself orbiting the center of our galaxy, which in turn was part of an expanding universe that lacks any notion of a spatial center. Indeed, it became extremely fruitful to adopt a completely opposing philosophical viewpoint, now known as the Copernican principle: that the Earth was just one planet among countless others in the universe, receiving no preferential treatment whatsoever from the fundamental laws of nature.

At first glance, this view feels quite threatening to humanity’s emotional attachment to our home planet, but ultimately there is no fundamental contradiction between the universe’s disinterest in the planet Earth, and our own strong investment in it; we can still quite justifiability prioritize issues specific to planet Earth over those on other planets, while simultaneously accepting that these other planets exist and would be of comparable importance to their own inhabitants. Similar revolutions can be seen in the historical development of other sciences, for instance in the Darwinian revolution dethroning the unique status of humans among other constantly evolving species, or the dethroning of the privileged role of Euclidean geometry as a source of synthetic a priori truth in mathematics.

Until recently, our species has similarly embraced an intellectual analogue of the geocentric model, in which human intelligence stood at the center of the cognitive universe, thus affording it a special philosophical status. But now we are discovering (or creating) other “planets” of intelligence comparable in many ways to our own, while simultaneously being quite distinct in many aspects. Instead of denying the existence or importance of these planets, or arguing over which of these planets deserves to be the “center”, one can instead accept that both human and artificial intelligences exist in the same ontological category, though with many distinctive differences and complementarities. While our interests and attachments will still largely be tied to the human intellectual sphere, its relationship with other forms of intelligence can be explored, both for practical purposes of more efficiently achieving various real-world objectives, as well as for more philosophical reasons, such as achieving an external perspective on human cognition that was previously difficult to attain.

7. Conclusion

The unstructured, chaotic, and widespread release of AI technology into the world has already dramatically shifted social, intellectual, and economic spheres in ways that are as alarming as they are beneficial. While unquestionably, some kind of collective effort by humanity is needed, whether through regulation, market pressure, or by some as-yet defined force; we have decidedly not yet reached a tipping point from which we cannot extricate ourselves from the high economic and social cost of these new technologies. Approaches to integrating AI into the field of mathematics have just as rapidly demonstrated the promising benefits that AI can bring to academic research, scientific progress, and to humanity at large. The largely objective and verifiable nature of mathematical research presents a unique opportunity to experiment with these new technologies and study the resulting impacts in ways that do not present an ethical or existential risk to the individual or broader society. From the application of AI to mathematics, we are able to explore the pressing philosophical and moral questions of broader global AI use. Further, we can extrapolate potential pathways to relieve the tensions at the AI/human interface and suggest new paradigms of cooperative AI/human thought that respect the unique and valuable qualities that each modality brings to the metaphorical table. Though we will never get the genie back in the bottle, we are optimistic that, as our understandings and action rapidly advance, we can yet clear the smoke away and look toward a bright, if somewhat uncertain, future.