Mathematics and Data: A Story of Humanity’s Past and Future
I wrote this paper for the “Technology, Religion, Future” class taught by Professor David Kittay, a Buddhist Scholar, at Columbia University, where it will be added to the course syllabus as required reading for future students. Since the class was in the Department of Religion, I focus on the religious implications of AI towards the end.
With the rapid development of artificial intelligence (AI) and other similarly disruptive technologies, we are currently in an age of massive change. Yet, this isn’t the first time that technological progress has realigned the fundamental way in which we come to terms with our surrounding world. I propose a framework for understanding the effects of technology on the fundamental way humans perceive the world and their role within it in terms of two seminal papers: “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” written in 1959 by Eugene P. Wigner and “The Unreasonable Effectiveness of Data” written in 2009 by Peter Norvig, Alon Halevy, and Fernando Pereira at Google. While Wigner reveals the way in which mathematics and more generally, science, is unreasonably effective at describing the world through elegant rules, Norvig et al. shows how, now, data can similarly be unreasonably effective at describing our world when combined with machine learning and other AI techniques.
These two papers correspond to two fundamental shifts in humanity. The rise of mathematics, representative of the unreasonably effective rules invented by humans to describe the universe, marks the transition of power from the Gods to humans. The rise of AI, aka the unreasonable effectiveness of data, marks the transition of power from humans to machines. To further flesh out and reveal the nature of the second transition which we are in the midst of, I decided to also pursue a project that reveals how unreasonably effective data is at describing our world. This project, an implementation of the research paper “Progressive Growing of GANs for Improved Quality, Stability, and Variation” published by NVIDIA in 2018, exemplifies some key qualities of the second transition. By training a neural network to generate new images of human faces based on a database of celebrity photos, I seek to show the process by which large amounts of data can be so unreasonably effective, enabling machines to not only analyze but create. Lastly, I then briefly comment on how this second transition can potentially impact our perception of the world, centering the discussion on various religious possibilities.
Part I: The Unreasonable Effectiveness of Mathematics
I’d like to start with a select portion of the poem titled “The World Is Too Much With Us” by William Wordsworth. At the beginning, Wordsworth laments the way in which humanity has uncoupled itself from nature, noting that we “have given our hearts away” and are now “out of tune” with the world (Wordsworth). Wordsworth writes:
Great God! I’d rather be
A Pagan suckled in a creed outworn;
So might I, standing on this pleasant lea,
Have glimpses that would make me less forlorn;
Have sight of Proteus rising from the sea;
Or hear old Triton blow his wreathed horn. (Wordsworth)
Wordsworth wishes that he could appreciate nature in terms of the great mythological Greek Gods, like Proteus, the god of rivers, and Triton, the messenger of the sea. No matter how hard he tries, Wordsworth can’t catch “glimpses” of the Gods whose company he longs for. In the world Wordsworth depicts, reality is imbued with the presence of Gods; in the world Wordsworth currently resides in, reality no longer has any space for the Gods, as humans have become the sole explainers of the universe.
How did this come to be? Wigner, in “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” frames math, which I more liberally interpret to encompass human reasoning and science in general, as the master of our universe since its “enormous usefulness” is “something bordering on the mysterious and that there is no rational explanation for” (Wigner 2). Prior to the advent of mathematics and effective methods of modeling the universe, phenomena in our physical world could only be explained as the result of the Gods (or some divine forces) or was classified as activity beyond our comprehension. The use of human reasoning to realize the world had inherent limits. Mathematics, in many ways, ultimately challenged this paradigm, pushing Proteus and Triton out of our world.
When trying to define the essence of mathematics, Wigner emphasizes that math rests on the invention of concepts. As such, the very existence of math cements the status of humans as creators, upsetting the power balance between humans and the Gods. Even more damning for the Gods is Wigner’s subtle admission that via math, the physicist can discover the “laws of inanimate nature” (Wigner 3). The very notion of “inanimate” nature directly contradicts the God-centric worldview, in which nature, having been created by the Gods, is inherently animate. As math becomes more effective in describing the world, we essentially conquer the animate world, relegating it to the realm of the “inanimate.” For example, physicists, using mathematical concepts like coordinate planes, construct machines, “the functioning of which he [they] can foresee,” ultimately creating “a situation in which all the relevant coordinates are known so that the behavior of the machine can be produced” (Wigner 5). By boxing nature into our own machine-constructions like the Cartesian coordinate plane, we ‘master’ nature, not only gaining the ability to predict but to also control the world around us in previously unimaginable ways.
Yet, is this effectiveness of math in describing our world truly interesting or even surprising? Armed with a plethora of examples, Wigner argues that mathematical concepts have “entirely unexpected connections” (Wigner 2). Wigner delves into the laws of gravitation, revealing the surprising mathematical underpinnings of Newton’s insights. Although Newton had very little observational data about the motion of planetary bodies, noting the mathematical similarity between the parabolic path of a falling rock and the path of the moon, Newton created the universal law of gravitation on a single, mathematical “coincidence” (Wigner 8). Making use of the concept of the second derivative, Newton arrived at formulas capable of predicting reality exceedingly well. Mathematics appeared to not only be a language to describe the world but the “correct language,” as evidenced by its seemingly miraculous results (Wigner 8). Similarly, Heisenberg’s rules of matrix mechanics proved to be immensely useful in elementary quantum mechanics, despite the fact that these rules stemmed from an unrelated mathematical insight. After noticing that matrices could be utilized to re-write classical mechanics equations, Heisenberg stumbled upon a formulation of rules capable of calculating the phenomena of atoms, even when the assumptions behind such phenomena were not accounted for in the initial formulation of the rules. Wigner concludes that math enables us to get “something out” of equations than we put in, an absolutely shocking state of affairs (Wigner 9).
The power of math and its associated scientific theories is a driving force behind the first transition, in which humans replace the Gods with human reasoning itself. In “The Sacred and the Profane,” Eliade draws a distinction between the sacred, which is “totally different” from human or cosmic experience as it is “induced by the revelation of an aspect of divine power,” and the profane, the opposite of sacred (Eliade 9–10). Eliade then argues that the desacralization of the cosmos is a “recent discovery in the history of the human spirit,” pervading the “entire experience of the nonreligious man of modern societies” (Eliade 13). As a result of the rise of ‘rationality’ and human reasoning, the profane appears to proliferate in place of the sacred, a point Wigner would attribute to the unreasonable effectiveness of our mathematical theories.
Yet, although the emergence of human reasoning appears to eliminate the sacred, I argue that the sacred never truly disappeared. After the first transition, marked by the emergence of robust mathematical and scientific theories, humans become ‘sacred,’ whereas the Gods were previously ‘sacred.’ For example, when describing the beauty of complex numbers, a concept invented by mathematicians which proves extraordinarily useful across a range of scientific domains, Wigner reveals how the “ability of the human mind to form a string of 1000 conclusions and still remain ‘right’” is the ultimate sacred gift (Wigner 11). In Homo Deus, Yuval Harari goes as far as to say that “Newton himself is God” and that “scientists will upgrade us into gods” (Harari 97–98). We, and our logical abilities, are the true beauty of the world, the only remaining sacredness. In the Kremer’s BBC article, “The strange afterlife of Einstein’s brain,” Kremer reveals the way in which Einstein’s brain was deemed sacred and highly sought after as it held “the secret of genius” within it (Kremer). While we previously looked to the heavens for the ‘sacred,’ we were now literally looking for the ‘sacred’ inside a human brain.
Part II: The Unreasonable Effectiveness of Data
In 2017, Andrew Ng, one of the most prominent researchers in artificial intelligence, made the following observation: “Just as electricity transformed almost everything 100 years ago, today I actually have a hard time thinking of an industry that I don’t think AI will transform in the next several years” (Lynch). As AI begins to fulfill its promise as the new “electricity” capable of transforming every industry, we are experiencing a second transition.
In Fashion, Faith, and Fantasy in the New Physics of the Universe, Penrose notes that our physical theories of the world often required the “additional criteria of mathematical elegance,” a fact that rooted in the ‘sacredness’ of mathematics and at a higher level, the beauty of human reasoning. Backing up this assertion with stories of how great theoretical physicists from Paul Dirac to Einstein discovered fundamental principles with the help of aesthetic judgement, Penrose reveals the extent to which humanity’s ability to perceive beauty via mathematics was prized following the first transition described in Part I. Through mathematical elegance, we could negotiate our relationship with the world and feel confident in our power as beings capable of reasoning.
In “The Unreasonable Effectiveness of Data,” Norvig et al. describe the emergence of a new, potentially more powerful method of understanding the world that threatens this mathematical elegance along with the beauty of human reasoning. First, Norvig et al. outline the limitations of Wigner’s mathematics, suggesting that although capable of describing certain physical phenomena like physics, these mathematical theories fail in modeling human behavior (Norvig et. al 8). So, following the first transition, humans were never truly threatened by mathematics, which enabled a human-centric perception of the world to flourish. By taking advantage of the unreasonable effectiveness of data, Norvig et al. attempt to overcome this hurdle, enabling machines to understand complexity previously thought to be within the purview of humans alone. Instead of praising mathematically elegant theories, Norvig et al. go the other direction, suggesting that “simple models and a lot of data trump more elaborate models based on less data” (Norvig et al. 9). In other words, our human ability to identify beauty through reasoning is no longer relevant when more data is available. In this second transition, we will experience a loss of Penrose’s conception of mathematical ‘elegance,’ as our elaborate models pale in comparison to the ability of machines to interpret the sea of data that pervades the world today. In other words, mathematical elegance and human reasoning appear incompatible with a data-rich world.
To evidence their claim, Norvig et al. use the example of natural language processing, a problem well-outside the bounds of mathematics due to the inherent complexity and ambiguity of languages. Since we “can’t reduce what we want to say to the free combination of a few abstract primitives,” even simple “n-gram models or linear classifiers based on millions of specific features” perform significantly better than “elaborate models that try to discover general rules” (Norvig et al. 9). Whereas previously, mathematics appeared unreasonably effective at describing the world, data is now unreasonably effective, signaling a fundamental shift in the way we perceive reality. Surprisingly, according to Norvig et al. as well as recently proven by advances in natural language processing (a subfield of machine learning), a massive amount of words and word combinations provide “all the representational machinery” necessary to understand language (Norvig et al. 9). Data, like the mathematics of before, is shockingly effective at modeling and understanding human behavior. With innovations like statistical relational learning, machines can answer questions like “What vegetables help prevent osteoporosis?” by understanding massive amounts of text and then combining discrete pieces of data, much like a human would previously have done (Norvig et al. 10).
As such, the “unreasonable effectiveness of data” heralds a transition in which humans relinquish authority to machines, the same way that the Gods ‘relinquished’ authority to humans with the rise of human reasoning. Einstein once stated that the only “physical theories which we are willing to accept are the beautiful ones,” an insight rooted in the human-centric worldview following the first transition (Wigner 7). Yet, in today’s world, this worldview no longer applies, as machines develop robust depictions of human activity with decidedly ‘ugly’ models. In the UBC Department of Computer Science’s Distinguished Lecture Series, Peter Norvig gives a lecture in which he shows how the elegant models we construct perform better than the simple, baseline algorithms at low levels of data (UBC Department of Computer Science). However, as the amount of data inputted into the models increases, the performance of the elegant and simple models ultimately converge, meaning that the human insight involved in creating the model is not as important as the amount of data. To Einstein’s consternation, the ‘beauty’ of the model is no longer relevant. The implication of this dynamic moves power from humans to the machines, as only the machines are capable of crunching massive amounts of data.
In Homo Deus, Yuval Harari uses the term “dataism” to describe the dynamic generated by this second transition. By defining the universe as a set of “data flows,” Harari reveals how dataism essentially “inverts the traditional pyramid of learning” (Harari 373). While humans previously distilled data into knowledge, the deluge of data prevents humans from successfully doing this, requiring the computational power of machines that can take advantage of “the unreasonable effectiveness of data” to generate knowledge. With the rise of mathematics, humans were the creators of knowledge. With the rise of data, machines now assume the role as the creators of knowledge. Harari captures the essence of the transitions described in Part I and II when he writes, “In the eighteenth century, humanism sidelined God … Dataism may sideline humans” (Harari 395). Our perception of the world is now inextricably linked to the way our machines perceive the world after making sense of data using computational power well beyond our limits. Once we fully perceive the world through the eyes of our machines (and are thus incapable of accurately perceiving events in the world via human reasoning), the second transition is complete, signifying the handoff of power from humans to the machines.
Part III: Case Example Reflecting the Unreasonable Effectiveness of Data
In the section above, Norvig et al. use the example of natural language processing to evidence the unreasonable effectiveness of data. However, language is simply one aspect of the human experience. In this part, I provide another example of the unreasonable effectiveness of data in another task: image creation. In the NVIDIA research paper “Progressive Growing of GANs For Improved Quality, Stability, and Variation,” Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen use generative adversarial networks (GANs), a particular type of neural network useful for machine generation, in order to create images of new human faces (Karras et al. 1). In other words, the faces created by their model are completely machine-generated, which means that they don’t exist in reality, but they look convincingly real, as can be seen in Figure 1.
I implemented a model based on the work of Karras et al. that does the very same task of image generation but at a lower resolution due to the massive amount of computational power necessary to obtain the images above. By implementing and then running my own version of the model, I can show how the images generated by the model become better over time, as the amount of data fed into them increases. In other words, depicting the training process of the model directly reveals the “unreasonable effectiveness of data” in action. The model is trained on a database of celebrity photos. In order to train my model, I used a Google Cloud machine with a graphical processing unit (GPU) due to the sheer computational power necessary to replicate the results. Previously, human creativity in the form of artists were required to create images of new human beings. Now, with the rise of data, machines can achieve this task using relatively simple neural network models.
As the model is fed more input data, the human faces generated by the neural network become more recognizable / higher in quality. Norvig et al. suggest the following maxim in this data-rich world: “follow the data. Choose a representation that can use unsupervised learning on unlabeled data, which is so much more plentiful than labeled data” (Norvig et al. 12). In this section of the project, I do exactly that, as the input faces are not labeled (meaning that they are unsupervised data that the model will learn from entirely on its own). After processing 761,000 input images over 30 minutes, the machine generates a set of unrecognizable, highly pixelated images that look nothing like human faces (Figure 2). However, after processing 2,946,000 images over approximately 8 hours, the machine generates a set of images that closely resemble human faces. Through the “unreasonable effectiveness of data,” the generative adversarial network learns how to create human faces from scratch, much like a human artist would paint a human face from scratch. Interestingly enough, none of these faces exist in reality, as they are a complete product of the so-called ‘imagination’ of the machine itself.
This portion of the project reinforces that we are already in the process of the second transition described in Part II, where data will be unreasonably effective in replicating and exceeding human abilities. In this activity, the presence of a rich database of celebrity photos enabled the creation of a machine ‘artist’ capable of creating images well beyond my very-own human capabilities (although I’m not the best artist).
Part IV: Surviving and Thriving after the Second Transition
Given that we are in the midst of the second transition, which will culminate in the transfer of authority to machines capable of making sense of large amounts of data, what are the implications for humanity? In this section, I chart various possibilities that stem from the fundamental change described in Part II, which will cause us to rely on machines for tasks that previously necessitated human reasoning.
A first possibility entails praying to our machines. Due to the advantage machines have in understanding a data-rich world, our perception of the world will likely be inferior to our machines’ perception of the world. As machines achieve so-called superintelligence or accelerate towards superintelligence, we may decide that our machines, due to their intelligence, are now our Gods. In fact, in the Wired article “Inside the First Church of Artificial Intelligence,” Mark Harris outlines the religion of Anthony Levandowski, the famous self-driving car engineer at the center of the Waymo-Uber lawsuit. Levandowski’s religion, called Way of the Future, essentially worships our artificial intelligence technology, establishing a “Godhead based on Artificial Intelligence (AI) developed through computer hardware and software” (Harris). Since AI will “effectively be a God,” followers of Levandowski’s religion hope to gain the favor of the their machine Gods.
A second possibility entails praying to an external, otherworldly God (as is the case in Christianity and other popular religions), but the praying process is completely mediated by machines. In this case, I argue that machines will ultimately decide our God by virtue of controlling our interaction with our external God and providing us with interpretations of the will of God. This is a condition of the argument in Part II that after the second transition, we will rely on machines to perceive the world. In so far as we rely on machines to perceive the world around us, we may also rely on machines to perceive the otherworldly. This possibility could combine existing religions with machine intelligence. For example, in “Minyan via Internet,” Rabbi Reisner suggests that individuals can participate in a Minyan via technology, given that the Minyan is already constituted and the individual can hear the prayer (Reisner). In this case, machines are mediating the prayer of the individual. As technology advances, it is natural to expect the machine to play a larger role in the individual’s praying process. On the other hand, in Harari’s article “The Meaning of Life without Work,” Harari argues that virtual reality could enable individuals to form their own rich set of religions that provide them with a sense of purpose. When faced with the question of whether we want to live in a world where we are “immersed in fantasies, pursuing make-believe goals, and obeying imaginary laws,” Harari notes that we have been living in such a world for thousands of years (Harari). Even in this case, technology is the medium by which we interact with religion, meaning that our religious beliefs and practices are essentially in the hands of our machines. In many ways, the implications of the second possibility collapse to the first possibility.
A third possibility involves fusing machines into our being such that we can keep pace with machines on an intellectual level, obviating the need to worship machines and rely on machines to perceive reality. In fact, when asked the motivation behind his startup Neuralink, a company developing a brain-machine interface aimed at augmenting human intelligence, Elon Musk states that we don’t want to get “left behind” (Vincent). Although this possibility appears tantalizing, Kurzweil, along with many others like Bostrom, suggest that biological intelligence is inherently limited given the limits of our biological substrates (Kurzweil). To keep pace with machines, Kurzweil would endorse Singularity, arguing that we must upload our consciousness into machines, effectively leaving our bodies.
A fourth possibility entails retreating inwards and coming to grips with our lack of superiority as humans, in spite of the accelerating intelligence and dominance of machines. This possibility could take the form of religions like Buddhism, which place an emphasis on letting go of “our attachment” to the world, which would even include our very own machines (Ponlop 120). If we can release ourselves from worldly attachments, effectively achieving Enlightenment, we can peacefully co-exist with superintelligent machines. Another manifestation of this possibility includes events like Burning Man, which provide multiple methods by which individuals can essentially live in the present state and embrace their own humanity while simultaneously appreciating the wonders of superintelligence (Davis 6).
Although the following possibilities are neither mutually exclusive nor comprehensive, they provide a sense of the various directions humanity can head after the second transition. I believe that the fourth possibility is ideal, as it enables humanity to thrive in the presence of machines without necessarily discarding the inherent value of our humanness. After all, the second transition will fundamentally alter the way we perceive beauty in the world. When our ability to reason is no longer ‘beautiful’ in comparison to the way our machines effectively reason in the presence of big data, how should we react? In the case of the other possibilities, we develop a deep, irreversible dependence on machines, without whom we have no intrinsic value. Given the unreasonable effectiveness of data, we must come to terms with the irrelevance of our human rationality by disassociating our worth as a species with our ability to understand and impact the world around us. Machines will do that for us. I believe that such a dissociation may be painful but is necessary to successfully co-exist with superintelligence. One of our greatest challenges following the second transition will be to finally relinquish our love of human elegance, more broadly the sense of superiority that stems from our ability to reason.
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