From Self-Referential Paradoxes to Intelligence
Our Self-Exploratory Journey to Externalize Some of the Best in Us
Introduction
As Large Language Models (LLMs) like GPT-4 ascend to prominence, it’s fascinating to reflect on how our next words are essentially statistical patterns that machines can emulate. This notion mirrors the contemplations of early 20th-century mathematicians who questioned whether their creative process was merely an application of logical rules. This striking parallel carries an interesting twist: it has broadened from a specialized group of mathematicians to encompass us all. In this article, we strive to illuminate the extraordinary journey that has brought us to this critical juncture in history and delve into the untapped possibilities that the future holds.
At its core, this journey is about our self-exploration, isolating and examining aspects of ourselves, such as logical reasoning. Interestingly, this introspection has transformed logical reasoning from a linguistic and cognitive skill into a system of symbolic manipulation that can be embedded in ‘machines.’
However, when we prompt these logic machines to reference themselves — mirroring our own introspection — we inevitably encounter the machines’ limitations. These limitations are manifested in self-referential paradoxes. A prime example is the Liar’s Paradox, where the statement “this statement is false” results in a logical contradiction, paradoxically appearing to be both true and false simultaneously. Such paradoxes serve as a potent reminder that we transcend our inventions.
The birth of the computer was an unexpected side effect of wrestling with self-referential paradoxes. It’s a double whammy! While we acknowledge superiority over machines, we also found exponential growth in our capabilities as computer programmers.
Our newfound abilities as computer programmers culminated in the development of AI and LLMs that seem to outperform us, stirring memories of early twentieth-century mathematicians. This forces us to ponder: Are we in a recursive cycle where our inventions confront their self-references, revealing another facet of human transcendence over machines and leading to the invention of new machines? Ultimately, we might find that the machines are simply a series of deepening mirrors of ourselves — a testament to our creativity and the ongoing journey of self-exploration.
The Abstract Realm and This Tangible World
In the intellectual epicenter of ancient Athens, a spirited rivalry unfolded between two eminent schools of thought — the Lyceum and the Academy — founded by Aristotle and Plato, respectively. These philosophical institutions stood as strongholds of intellectual discourse, each representing distinct philosophical traditions and approaches to knowledge. The following hypothetical dialogue between Plato and Aristotle encapsulates the essence of their contrasting views on the nature of a circle.
Plato starts, “Aristotle, think of a circle as a perfect entity existing in an abstract realm. The circles we see in our world are just imperfect copies of this perfect form. Our job in seeking knowledge should be about thinking over these perfect forms. That’s where our human intellect comes in, our ability to dream up and consider perfection.”
Aristotle, listening attentively, counters, “Master Plato, while I respect your viewpoint, I disagree. Circles are echoed as a potentiality in objects in the physical world, and while they may be imperfect, they, such as wheels, can serve practical purposes. Our pursuit of knowledge should be about studying the natural world through observation and harnessing our ingenuity in our machines.”
This imagined exchange between Plato and Aristotle encapsulates the divergence of their philosophical viewpoints. This vibrant interchange, this discourse straddling the abstract and the tangible, the dominance of human intellect in the philosophical realm, and the impressive proficiency of machine ingenuity in our world have formed a fertile ground where our greatest advancements have taken root.
Distilling the Logical Mind
In recognition of the paramount importance of scientific exploration and discovery, Aristotle went a step further than his counterpart at the Academy. Within the hallowed halls of the Lyceum, he established a dedicated laboratory, a sanctuary where the wonders of experimentation and empirical investigation flourished.
In the botanical section of his laboratory, Aristotle cultivated an impressive collection of plant specimens from various corners of Greece. Notably, his analysis of the opium poppy provided insights into the potent effects of certain plant compounds on the human body.
Adjacent to the botanical section, Aristotle’s laboratory housed an area solely dedicated to the study of animals. He unraveled the anatomy, behavior, and interactions through dissections and observations. His study of marine life included a groundbreaking classification system for fish species, serving as a foundation for subsequent taxonomical studies.
High above the laboratory, Aristotle’s astronomical observatory adorned the Lyceum. Equipped with sophisticated instruments of his time, he observed the celestial bodies that adorned the night sky. Aristotle’s accurate recording of lunar eclipses and recognition of the Earth’s spherical shape laid the groundwork for future astronomical advancements.
Recognizing the limitations of deriving logical principles from scientific observations, Aristotle embraced self-reference in his exploration of logical reasoning. Despite Aristotle’s approach to self-inspecting logical reasoning differing from scientific inquiry, he employed the same scientific discipline: direct, isolate, and inspect. He introduced the concept of syllogism, using the mortality of Socrates as an example. Employing symbolic notation, he demonstrated the logical structure: A (All men) are B (mortal), C (Socrates) is A (a man), and therefore C (Socrates) is B (mortal). Through this illustration, Aristotle prompted his students to discern the universal pattern represented by the formal abstractions A, B, and C, extending beyond Socrates’ specific instance.
Self-references emerged as Aristotle employed logical principles to analyze and comprehend logic itself. This approach minimized ambiguity and subjectivity inherent in natural language. Students realized the refinement of their thinking and reasoning through this formal discipline, empowering them to excel in scientific debates, legal discourses, and everyday disputes with others.
Thus, the journey of formal logic began— and it was just the beginning.
As we will delve into further in the next section, the pivotal capacity for self-referential analysis of our reasoning and cognitive processes played a crucial role in the evolution of more abstract forms of reasoning. This detachment allowed us to construct symbolic representations of logic, establishing the foundation for the advent of computing technology.
Note: This story is a hypothetical exploration of Aristotle’s journey, combining historical context with imaginative storytelling. The examples provided are based on Aristotle’s contributions to studying plants, animals, and astronomy in ancient Greece (Herman, 2013) (Rubenstien, 2003).
Demonstrating the Realm: Masterstroke of Logic
The Academy in Athens bore an inscription at its entrance: “Let no one enter here who is ignorant of geometry,” testifying to the paramount importance attributed to this field of study. However, Euclid brought classical geometry to its zenith within the ancient city of Alexandria, Egypt. In his influential work, “The Elements,” Euclid employed logical reasoning to transform a jumble of knowledge about geometry into a cohesive system.
In “The Elements,” Euclid fastidiously ordered the study of geometry, commencing with a collection of fundamental presumptions termed axioms. These axioms, such as the existence of a straight line and the equality of angles, formed the bedrock upon which all future reasoning would be constructed.
Starting these axioms and definitions, Euclid crafted a logical scaffold to derive numerous theorems. Euclid’s mastery of logical reasoning enabled him to construct sequential arguments, providing infallible proof for each theorem.
While Euclid’s organization of “The Elements” provided a roadmap for readers through the labyrinth of the geometric realm, he was not intrigued by logical reasoning itself. In contrast, Aristotle’s intellectual heirs embarked on diverse journeys, concentrating on the language of formal logic itself to uncover its innate power. This shift of focus nearly dismantled the abstract realm, representing a departure from the philosophical leanings of Plato and Euclid.
Logic’s Leap: Triumph and Despair
In the 17th century, the renowned philosopher and mathematician Gottfried Wilhelm Leibniz envisioned a universal language of reasoning that would reduce logical arguments to mathematical calculations.
Building upon Leibniz’s ideas, George Boole propelled the mathematical approach to logic in the 19th century. His work, now known as Boolean algebra, treated logical expressions as algebraic structures. This perspective enabled the systematic manipulation of logical statements, forming the bedrock of digital logic design, a fundamental component of modern computers.
However, the path toward progress was not without its challenges. In the early 20th century, Gottlob Frege, a German philosopher, logician, and mathematician, embarked on an ambitious project. He aimed to demonstrate that all mathematics could be derived from logical axioms and rules, an endeavor known as logicism. Frege’s work introduced powerful symbolism that represented logical reasoning as unambiguous strings of symbols. This laid the groundwork for modern first-order logic, encompassing elementary logic such as (and), (or), (not), (implies), (for all), and (there exists). For example, the following reasoning All dogs are mammals; therefore, the tail of a dog is the tail of a mammal, which can be represented as
in modern first-order logic, where D, M, T denotes “Is a dog,” “is a mammal,” and “is the tail of.”
While a human reader might struggle to understand the symbolic representation of reasoning, it appears perfectly suited for mechanistic processes.
Yet, Frege encountered a formidable obstacle in his pursuit — a self-referential paradox that threatened the foundations of his logical system. This paradox, later known as Russell’s paradox, was discovered by Bertrand Russell. It exposed a flaw in the logic fabric, demonstrating how self-references could lead to contradictions. Russell’s paradox involved a set of all sets that do not contain themselves, which both had to contain itself and not contain itself, creating a contradiction. This revelation signaled the need for a more rigorous approach to logic, urging scholars to seek a solid and coherent foundation.
Barber’s paradox emerged as a more digestible alternative to Russell’s, presenting a deceptively simple yet deeply perplexing scenario: a barber shaves everyone and only those who do not shave themselves. This premise leads to an engrossing quandary: does the barber shave himself? If he does, he violates his own rule, for he shouldn’t. Yet, if he doesn’t, he contradicts his rule, for he should. This paradox encapsulates the deep-rooted challenge that self-references pose to logical reasoning.
In the aftermath of the paradox that unsettled the logical foundations of mathematics, the intellectual descendants of Aristotle remained undeterred. The next phase of this journey saw a shift in perspective, ushering in a novel way of viewing mathematics.
Redefining Mathematics: The Game Without Players
Bertrand Russell, the logician who uncovered the paradox that challenged Frege’s ambitious project, proposed a radical idea. He suggested that mathematics could be perceived as a kind of game, a system of rules and procedures devoid of inherent meaning (Russell, 1903). This perspective, known as formalism, treated mathematics as a purely syntactical discipline, focusing on manipulating symbols according to formal rules (Detlefsen, 2005). In this view, mathematical symbols didn’t need to signify anything real or tangible. They were merely game pieces, akin to the pieces on a chessboard. The rules of the game, the axioms, and the theorems of mathematics, called formal systems, dictated how these pieces could be moved and manipulated. The game’s objective was to construct valid proofs, sequences of moves that led from a set of axioms to a desired conclusion (Detlefsen, 2005).
Russell wittily observed that the entities we grapple with in mathematics — such as points, lines, and triangles in Euclidean geometry — can be arbitrarily chosen. They could represent tables, chairs, apples, and oranges (Russell, 1919).
Within this framework, any mathematical intuitions about numbers or geometry are viewed as forbidden territories, serving as a precautionary measure to avoid potential pitfalls associated with self-referential paradoxes. Russell’s monumental work, “Principia Mathematica” (Russell & Whitehead, 1910–1913), which he co-authored with Alfred North Whitehead, adhered to this principle. This seminal text systematically constructed arithmetic, a fundamental mathematics branch solely based on logic. It dedicated a substantial number of pages to demonstrate seemingly simple truths, with an astonishing 362 pages solely dedicated to proving the statement that 1+1 equals 2 (Urquhart, 2015).
A German mathematician, David Hilbert, extended Russell’s idea of formalism even further. He proposed that mathematics was a game that didn’t require players (Hilbert, 1925). In other words, the process of mathematical reasoning could be entirely automated, executed by a machine following a set of predetermined rules (Hilbert & Ackermann, 1928). This was part of Hilbert’s program, which also encompassed the decision problem, a question about the existence of a universal method to solve all mathematical problems (Hilbert & Ackermann, 1928).
Hilbert’s vision was profound. It suggested that the power of formal logic could be taken to its ultimate extreme. We could create machines that could carry out our reasoning for us, machines that could play the game of mathematics without any human intervention (Davis, 2000).
However, should Hilbert’s premise hold, the nature of mathematical truth would emerge as a critical question. Moreover, if machines showcased their proficiency in executing complex reasoning tasks, the necessity of human intuition and creativity within mathematics could be questioned.
The next chapter in this narrative introduces two significant figures: Kurt Gödel and Alan Turing, each embodying different philosophical persuasions. Through Gödel’s revolutionary incompleteness theorems and Turing’s significant contributions to computation, the dialogue within mathematics experienced a monumental shift, marking one of the most pivotal turning points in human history.
The Revenge of Plato
As the game of mathematics unfolded, a new player entered the scene, one who would challenge the very rules of the game. Kurt Gödel, a mathematician, and logician of the 20th century, would bring a twist to the tale that no one saw coming.
Gödel was a Platonist, a believer in the existence of abstract mathematical objects. This view starkly contrasted the formalist perspective championed by Russell and Hilbert. Gödel’s work would bring what some might call the revenge of Plato.
In 1931, Gödel published his incompleteness theorems, stating that in any sufficiently powerful formal mathematical system, there would always be statements that couldn’t be proven or disproven within the system itself.
Gödel’s theorems underscored the unattainability of the mathematical formalist’s dream — the aspiration to distill all mathematics into a game of symbol manipulation within a formal system. These theorems revealed that there would invariably be truths beyond the confines of the game’s rules, supporting the independent existence of the realm.
Intriguingly, Gödel’s proof of his incompleteness theorems employed a form of self-reference, a concept central to Russell’s paradox. Gödel constructed a statement declaring, “This statement cannot be proven.” It couldn't be proven if the statement were true, just as it claimed. However, if it were false, it could be proven, contradicting its own assertion. Gödel leveraged this paradox to expose the inherent limitations of a formal system. Ironically, despite the formalists’ rigorous attempts to avoid them, formal systems could not evade the mesmerizing trap of self-references.
Gödel’s method was indeed ingenious. He employed a technique now known as “Gödel numbering” to encode mathematical statements about numbers as numbers themselves. This approach enabled number theory, a mathematical branch of studying natural numbers, to examine and reason about its own statements. It was a remarkable self-reference, purposely crafted to create a self-referential paradox.
Yet, the question of the decision problem still lingered. It was a question about the existence of a universal method — an algorithm — capable of deciding the truth or falsity of any mathematical statement provable or disprovable within a formal system. At this pivotal point, a new player, Alan Turing, an English mathematician, and logician, entered the fray.
The Birth of the Computer
Turing aimed to demonstrate that such a universal method could not exist. To accomplish this, he needed a clear-cut definition of an algorithm, a concept somewhat nebulous at the time. Turing’s brilliance emerged in his realization that an algorithm was essentially a set of instructions followed by a “computer” — a human role utilizing pen and paper in his time. This insight led him to imagine a theoretical machine, now known as a Turing machine, which was simple enough to simulate the operation of any algorithm. The widely held belief is that everything in the universe is computable if, and only if, it can be reduced to the operations of a Turing machine. This concept is referred to as the Church-Turing Hypothesis.
With the concept of a Turing machine established, Turing moved forward to demonstrate the nonexistence of Hilbert’s universal proof mechanism.
Taking inspiration from Gödel’s numbering scheme and the resulting self-referential paradox, Turing devised a similar strategy by numerically encoding a machine as a number that can serve as an input that other machines, including itself, can read.
Having established the necessary foundation, Turing proceeded to present the Halting Problem: Is it possible for a halting oracle to exist, one that could take two inputs — a machine and an input — and determine whether the machine would halt or loop indefinitely on the given input? Ingeniously, Turing circumvented the need to elaborate on the mechanics of such an extraordinarily potent machine. He showed that the mere proposition of its existence would lead to a paradox.
Turing assumed there was a special machine M, that does exactly the opposite of what the halting oracle does. If the halting oracle decides a machine halts when taking itself as the input, M will loop forever. On the other hand, if the halting oracle decides a machine loop forever, M will halt in this case. This way, you can say M purposely sabotages the halting oracle. Now the question is: does the special machine M halt when it takes itself as input?
- If M halts on M, M will loop forever by the definition of M: A contradiction.
- If M loops forever on M, M will halt. Another contradiction
This self-referential paradox is illustrated below:
Therefore, the halting oracle cannot exist. The halting problem cannot be solved, or technically speaking, it is undecidable.
Returning to Hilbert’s decision problem: if a universal proof mechanism existed, we could convert it into a halting oracle by recasting any query as a mathematical statement. However, the halting oracle and the universal proof mechanism don't exist.
Conversely, if the halting oracle did exist, we could easily create a universal proof mechanism by following these steps: first, write a program to endlessly search for all possible proofs of a mathematical statement and have it halt upon finding one; next, ask the halting oracle if the search program will halt. Thus, the halting problem's unsolvability directly implies the problem’s unsolvability and vice versa.
Gödel unveiled the existence of mathematical truths that transcend formal systems. Moreover, Turing asserted that no universal method could answer any mathematical questions within these systems. These revelations seemed to reinforce human supremacy, maintaining the realm of mathematics as a palpable reality, consistent with Platonism.
However, the narrative veered sharply just when mathematical Platonism appeared to be on the cusp of victory. Alan Turing found himself unexpectedly morphing the perceived setback of mathematical formalism into a showcase of human ingenuity. While the Platonic realm might linger somewhere, humanity stood prepared to usher it into our tangible world.
The Rise of the Programmer
As part of his strategy to demonstrate the unsolvability of the halting problem, Turing conceptualized a groundbreaking entity known as the universal machine. This machine, in essence, could interpret and simulate any encoded Turing machine number, allowing it to replicate its behavior. This abstract machine was integral to Turing’s proof, as he needed machine M to emulate the operations of the elusive halting oracle.
Turing’s concept of the machine became modern computers in an astounding development. These computers, acting as physical embodiments of the universal machine, function based on stored programs, effectively equating numbers with computer programs. This unanticipated consequence, originating from Turing’s proof, turned the challenge of mathematical formalism into a triumph of human ingenuity. The self-referential paradox, devised as a firm rebuttal to the aspirations of mathematical formalism and yet ironically built upon its craft, emerged as one of humanity’s most noteworthy innovations.
The birth of computers marked the rise of programmers, who were pivotal in creating effective programs. The need for such human creativity was underscored by the reality that randomly generated programs couldn’t depend on an illusory halting oracle for verification. This highlighted the invaluable role of human ingenuity in the sphere of computing. However, is this entirely true? As the narrative of computing evolved, we found ourselves circling back to a familiar theme: self-reference.
The Hard Problem of Educated Guesses
The emergence of the computer provided more than just a new technological asset; it also offered a novel perspective on the decision problem that had indirectly sparked its inception.
Gӧdel wrote about it in his 1956 letter to John von Neumann (1903–1957, mathematician, physicist, known for von Neumann architecture, the blueprint of all modern computers) :
If there actually were a machine with [running time] ∼ kN (or even only with ∼kN²), this would have consequences of the greatest magnitude. That is to say, it would clearly indicate that, despite the unsolvability of the Entscheidungsproblem, the mental effort of the mathematician in the case of yes-or-no questions could be completely replaced by machines. One would indeed have to simply select an N so large that if the machine yields no result, there would then also be no reason to think further about the problem (Aaronson, 2016).
This letter predates the establishment of computational complexity theory as a discipline. According to this theory, there are two major categories of running time: polynomial and exponential. For a given problem size N, the running time of a polynomial algorithm grows proportionally to the power of N, like 2N or 3N². On the other hand, exponential algorithms see their running time expand as an exponential function of N, such as 2^N or 3^N. Problems requiring exponential time are considered “infeasible.” They could leave a human, even one utilizing a computer that employs all the resources in the universe, waiting for an answer for a period comparable to the universe's age.
Looking at Gödel’s letter with a computational complexity theory lens, we realize that choosing a big enough ’N’ for the decision problem isn’t the main issue. The real question hinges on Gödel’s starting assumption — can we solve the decision problem in polynomial time? We can refer to this question as the computational decision problem. The twist is that it is one of our most intriguing open problems.
If a mathematical proof exists, it can be verified by a program operating in polynomial time. However, it’s important to note that a program’s ability to ‘find’ this proof doesn’t necessarily mean it will also function within the same time frame. The complexity class for problems that polynomial-time programs can solve is P, while NP (Non-deterministic Polynomial Time) refers to the class of problems where solutions can be ‘guessed’ and then verified within polynomial time. The computational decision problem falls into this latter category. All problems that can be resolved in polynomial time comprise the complexity class P, a subset of NP.
The complexity class assigned to a problem, such as P or NP, is determined by the fastest known program capable of solving it. If a faster solution is found, the problem’s class may change. Whether P equals NP is one of our era’s great unsolved conundrums, an issue Gödel poignantly highlighted.
If P were to equal NP, it would mean any problem whose solution can be quickly verified (a characteristic of NP problems) could also be solved quickly. This would give rise to the existence of an efficient solver for NP-complete problems – a class of problems to which all NP problems can be reduced. Such a solver would act as a kind of ‘oracle’, capable of decisively solving complex computational problems in a feasible timeframe.
However, the prevailing consensus leans towards P not equalling NP, suggesting that the computational decision problem, along with educated guesswork in general inherently presents substantial challenges to resolve.
So, how does a mathematical impossibility become infeasibly hard work from a computational standpoint? The secret’s in the differences between math and computer science. Math doesn’t get the idea of ‘making things easier to compute’ and assumes guessing randomly or doing exhaustive searches. Computer science, though, while it can’t solve the problem in a strict math sense, can make educated guesses or do smarter searches.
So, this correlation between mathematics and computer science implies there’s no easy way to make educated guesses to prove mathematical statements. However, if that’s true, how have we made such huge strides in math, not just in one person’s lifetime, but across many generations? Even more important, how can we make machines do what we humans have been doing?
Math Genius and the Wise Minds
French mathematician and physicist Henri Poincaré noted,
Logic, which alone can give certainty, is the instrument of demonstration; intuition is the instrument of invention.
Echoing Poincaré, Hungarian mathematician George Pólya pointed out,
The result of the mathematician’s creative work is demonstrative reasoning, a proof, but the proof is discovered by plausible reasoning, by guessing.
To delve deeper, consider how humans make educated guesses. Tracking back to the root of logical reasoning, Aristotle’s syllogisms ensure the truth of their conclusions given their premises. However, in the face of uncertain or incomplete information, we must draw probable, albeit not certain, conclusions. For example:
- Most dogs have four legs.
- Fido is a dog. Therefore.
- Fido probably has four legs.
The truth of the premises doesn’t guarantee the conclusion, as Fido might be one of the few dogs with fewer or more legs. However, Fido does likely have four legs.
While this style of reasoning may seem ‘weak’ in conventional logical terms, it nevertheless demonstrates potent probabilistic power. It finds applications in a vast array of human endeavors. Detectives use it to pursue justice, piecing together clues to solve crimes. Scientists rely on it when formulating and rigorously testing hypotheses. We all harness its strength in making everyday decisions, like whether to bring an umbrella based on the day’s forecast. The acclaimed physicist and probability theorist Edwin T. Jaynes once stated, ‘Probability theory is an extension of logic.’ This form of probabilistic reasoning, fundamental to our daily lives, forensic investigations, and scientific research, is also the tool that mathematicians deftly use. It helps them make informed ‘guesses’ about which conjectures to propose and which theorems to select to construct their proof steps.
Our initial question can be reframed as: how do we capture our collective use of weak syllogism and probabilistic reasoning? How do we distill the guessing mind, much like the logical mind?
Perhaps, not surprisingly, at this point in our journey, we circle back to our double-edged tool: self-reference. However, there’s an intriguing twist: the ingenuity of machines now becomes entangled with the notion of human supremacy.
Distilling the Guessing Mind
Theoretically, we could construct a universal set of rules for probabilistic reasoning, leading us to a one-size-fits-all way of making educated guesses for any question. But, like in Hilbert’s program, an old mathematical endeavor, we hit a roadblock when we need self-referentially to make educated guesses about our guesses. This is like a dog chasing its tail — an endless loop nearly impossible to work out.
This is where Artificial Intelligence (AI) enters the picture, providing a unique form of self-reference. Using AI programs such as Large Language Models (LLMs), we apply probabilistic reasoning to our own process of making educated guesses, essentially performing probabilistic reasoning on our own probabilistic reasoning (Lu, 2023). These models, acting as stand-ins for human reasoning, are trained to emulate our cognitive processes by interpreting the data we generate about our behaviors and actions.
While our typical information gathering tends to be more passive, we assume a much more proactive role within AI. Humans actively generate and provide the data that AI systems use. For instance, when an LLM predicts the next word in a sentence, it relies on patterns we’ve taught it, using vast amounts of data. We essentially shape LLMs into artificial reasoning agents, enabling them to make educated guesses that mirror our collective thought processes.
The distinctiveness of AI and LLMs resides in their ability to replicate human-like guessing by emulating our collective reasoning. Hence, our role extends beyond mere observation and decision-making to include the active construction of artificial systems that reason similarly to us. In this regard, AI is a reflective surface for the human ‘guessing mind,’ mirroring our collective thought patterns and reasoning capabilities.
Conclusion
Our journey has always been a voyage of self-exploration. We began by probing the depths of formal logic, moved on to uncover the mysteries of symbolic logic, and then stepped outside ourselves to embed our logical thinking in computing machines. It was a significant shift, taking us from self-contemplation of our thought processes to letting machines reflect on workings, revealing their hiccups along the way. Encountering these challenges sparked a self-discovery of sorts; we realized we could stretch our minds to program these machines, defining a new role for ourselves in this grand adventure — that of computer programmers.
We’ve also seen the interaction between the Academy and the Lyceum, the realm of ideas and the tangible world. Without the thinkers who carried on Aristotle’s legacy, pushing the boundaries of logic and grappling with its limits, we wouldn’t have been able to breathe life into our abstract ideas through our inventions or unlock our hidden potential. On the other hand, without those who believe in a Platonic world of ideas and our capacity to conquer it, we wouldn’t have been able to elevate our inventions beyond their initial creation. These perspectives have been crucial in getting us where we are today.
Will the pattern and the interplay repeat itself? Our journey did not conclude at the doorstep of externalizing logical reasoning. We moved ahead, embracing another compelling facet of human nature — probabilistic reasoning. This mode of thought, characterized by making educated guesses from probable outcomes, manifests some of our finest attributes: consistency, open-mindedness, and a non-ideological approach, all underpinned by common sense. We use this ‘best-guess’ logic not only on an individual level but also to comprehend collective human behavior, essentially applying the concept of educated guessing to our collective thought processes.
This is mirrored in what large language models (LLMs) do. An LLM is making large-scale educated guesses about the next word, drawing from an extensive pool of data collected from us. It’s making educated guesses in our images.
This raises the question: can they go to the next level of self-reference, become observers, and self-reflect like us? This is reminiscent of Hilbert’s logical machine. Similarly, will they encounter self-referential paradoxes? As we continue to develop AI, will we discover new roles that transcend mere programming the machines?
Perhaps in our continued exploration of AI, we’re learning more about ourselves than we realize. Turing certainly thought so when he said:
The whole thinking process is still rather mysterious to us, but I believe that the attempt to make a thinking machine will help us greatly in finding out how we think ourselves.
As humanity advances, we become inseparable from our inventions. The open question is: what else in the best in us should we externalize in our next inventions?
References
CP Lu (2023). Beyond the Bayesian vs. Frequentist Debate in the Advent of Large Language Models
CP Lu (2020a). AI since Aristotle: Part 1 Logic, Intuition and Paradox.
CP Lu (2020b). AI since Aristotle: Part 2 The Limit of Logic and The Rise of the Computer.
CP Lu (2020c). AI since Aristotle: Part 3 Intuition, Complexity and the Last Paradox.
Aaronson, S. (2016). P=?NP. In Open Problems in Mathematics. Springer.
Herman, A. (2013). The Cave and the Light: Plato Versus Aristotle, and the Struggle for the Soul of Western Civilization. Random House.
Rubenstein, R. E. (2003). Aristotle’s Children: How Christians, Muslims, and Jews Rediscovered Ancient Wisdom and Illuminated the Middle Ages. Harcourt.
Urquhart, A. (2016). RUSSELL AND GÖDEL. [Online] Available at: https://doi.org/10.1017/bsl.2016.35
Pierpont, J. (1928). Mathematical rigor, past and present. [Online] Available at: https://doi.org/10.1090/s0002-9904-1928-04507-x
Burman, J. T. (2016). Piaget’s neo-Gödelian turn: Between biology and logic, origins of the New Theory. [Online] Available at: https://doi.org/10.1177/0959354316672595
Westland, J. C. (2004). The IS Core XII: Authority, Dogma, and Positive Science in Information Systems Research. [Online] Available at: https://doi.org/10.17705/1cais.01312
Davis, M. (2000). Engines of Logic: Mathematicians and the Origin of the Computer. W. W. Norton & Company.
Detlefsen, M. (2005). Formalism. In S. Shapiro (Ed.), The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press.
Hilbert, D. (1925). On the Infinite. In P. Benacerraf and H. Putnam (Eds.), Philosophy of Mathematics: Selected Readings. Cambridge University Press.
Hilbert, D., & Ackermann, W. (1928). Principles of Mathematical Logic. Chelsea.
Polya, G. (1954). Induction and Analogy in Mathematics.
Polya, G. (1954). Mathematics and Plausible Reasoning. Martino Publishing.
Polya, G. (1973). How to Solve It (Second ed.). Princeton University Press.
Russell, B. (1903). The Principles of Mathematics. Cambridge University Press.
Russell, B. (1919). Introduction to Mathematical Philosophy. George Allen and Unwin.
Russell, B., & Whitehead, A. N. (1910–1913). Principia Mathematica. Cambridge University Press.
Tymoczko, T. (Ed.). (1986). New Directions in the Philosophy of Mathematics. Birkhäuser.
Urquhart, A. (2015). The Logic of Principia Mathematica. Review of Symbolic Logic, 8(4), 759–785.
