The Missed Opportunities in Ancient Thought
Messy notes, but this is a science journal and science can be messy. To truly grasp how our current understanding of science and mathematics has evolved, we need to dig deep into the intersections between philosophy, language, and societal structures that both fueled and hindered progress. Many pivotal thinkers, whose theories aligned seamlessly with modern science, were often ignored or misinterpreted, not due to the validity of their ideas but due to the social, political, and even religious contexts of their time.
Take for example Heraclitus—often overshadowed by Plato and Aristotle—who famously posited that "everything flows" (Panta Rhei), essentially suggesting that the universe is in a state of constant change. While his views weren’t entirely dismissed, they didn’t fit neatly into the more rigid, deterministic frameworks of later thinkers like Aristotle, who sought to categorize the world in fixed terms.
Heraclitus’s ideas actually have striking parallels with modern quantum mechanics, particularly the concept of wave-particle duality and the indeterminacy embedded in the fabric of reality. The ancient belief in constant flux aligns more closely with what we now understand about subatomic particles, which are never fully "static" but exist in probabilities and shifting states. Yet, Heraclitus’s ideas were largely ignored in favor of the Aristotelian view that dominated Western thought for centuries—a view that sought to impose order on the natural world, and in doing so, perhaps limited the scope of inquiry.
Another missed opportunity lies in the Atomism of Democritus and Leucippus, whose theories about the fundamental, indivisible particles of the universe were dismissed by the more influential philosophical schools of the time. Their ideas, though primitive by modern standards, laid the groundwork for atomic theory, which resurfaced in the 19th century. The dismissals they faced can be attributed, in part, to the dominance of Platonic idealism, which sought metaphysical explanations for the nature of reality, rather than mechanistic ones.
Take the example of Leibniz’s monads. Leibniz envisioned monads as fundamental, indivisible units of reality that don’t interact causally in the traditional sense. Instead, these monads reflect the entire universe in themselves, operating in a kind of pre-established harmony. To his contemporaries, the notion of non-causal interaction was metaphysical and speculative. But this idea, dismissed or underappreciated in its time, resonates strikingly with modern quantum mechanics, specifically quantum entanglement—where particles remain connected across vast distances without any direct interaction. In this context, Leibniz’s monads weren’t just a philosophical thought experiment—they foreshadowed the concept of entangled states that are foundational to quantum physics.
However, the friction arises because scientific paradigms are not static. The language in which they are expressed evolves, often making previous insights incomprehensible or irrelevant to the frameworks of the time. Wittgenstein’s view that language is action-bound means that the scientific community of Leibniz’s day couldn’t fully grasp the implications of his monads because their "language game" didn’t yet include the conceptual tools necessary to make sense of non-causal interaction.
Fast forward to today’s discussions on quantum entanglement—we have the benefit of quantum field theory and probabilistic mathematics, but the core of the problem remains the same: our ability to understand and explain these phenomena is still bound by the limits of language and current mathematical frameworks. The friction Wittgenstein points to is alive and well: science, as a construct, evolves, but not without the struggle of reinterpreting the very terms and frameworks we’ve built around it. Returning to the concept of non-causal interaction, this friction isn’t just a relic of the past—it’s ongoing. In quantum entanglement, we see this non-causal interaction play out, where two particles, regardless of distance, remain connected in such a way that the state of one instantly influences the state of the other. This mirrors Leibniz’s vision of monads existing in a pre-established harmony. Here, though, the concept is not merely metaphysical—it’s empirically observed. Yet, even in today’s scientific community, entanglement stretches the limits of our understanding, revealing how difficult it is for the "language" of classical physics to fully explain or accommodate quantum phenomena.
Yet, the concept of non-causal interaction—whether in Leibniz’s monads or today’s understanding of quantum entanglement—requires us to rethink the way we engage with scientific progress. It’s not just about the ideas; it’s about the language we use to express them. As Wittgenstein teaches us, the words we choose, the frameworks we use, shape not only our understanding but the very reality we are capable of observing. It’s the friction between these evolving language games that drives progress. The key is being able to recognize when the game has changed—and being bold enough to redefine the rules.
Now, how does Évariste Galois fit into this narrative of friction and the evolution of knowledge? Galois, who developed group theory, wasn’t just offering a new mathematical tool—he was disrupting the entire structure of how we understand symmetries in mathematics. His insights into how algebraic solutions could be organized into groups of permutations were largely dismissed at the time, not because they were wrong, but because the mathematical community lacked the framework to recognize their potential. Galois was dealing in a language game that hadn’t yet evolved enough to accommodate his ideas.
What Galois’s group theory eventually provided was a way to describe symmetry in mathematical systems—a concept that has direct applications in quantum mechanics, particularly in the study of particle interactions and conservation laws. His work is essential to modern physics, underpinning much of the mathematical architecture that describes the behavior of fundamental particles. Like Wittgenstein’s language games, Galois’s group theory created a new "grammar" for understanding the relationships between complex systems. But just like with Leibniz’s monads, the initial rejection of Galois’s work highlights how society’s "forms of life" can be blind to the significance of disruptive, forward-thinking ideas.
Philosophies and theories that didn’t align with the dominant cultural or political ideologies were frequently suppressed. During the Middle Ages, for example, much of the ancient Greek knowledge—including mathematics and natural philosophy—was preserved and expanded upon in the Islamic world.
Scholars like Alhazen (Ibn al-Haytham) developed theories of optics and scientific methods that directly influenced the Renaissance,
yet their contributions were often marginalized in Western narratives of scientific progress due to the Eurocentrism of the time. It’s worth considering how these "language games" evolve and propagate out, creating friction as they do so. In the same way that Alhazen’s work on optics was dismissed in Europe before being rediscovered during the Renaissance, or how Sophie Germain’s contributions to elasticity theory were overlooked because she was a woman in a male-dominated field, scientific progress is as much about navigating these social and linguistic barriers as it is about empirical discovery.
Similarly, Sophie Germain, a self-taught mathematician who made significant contributions to number theory and elasticity, had her work often dismissed or ignored because of her gender. Her correspondence with Carl Friedrich Gauss highlights her intellectual rigor, but much of her work wasn’t fully appreciated until long after her death. The social structures of her time, which marginalized women, played a role in sidelining her contributions.
As we move into the postmodern context, Wittgenstein’s concept of "language games" becomes highly relevant.
Scientific terms, much like philosophical ones, are deeply contextual. The shift from classical mechanics to quantum mechanics didn’t just introduce new theories—it introduced a new vocabulary, a new way of speaking about reality. Terms like "uncertainty," "probability wave," and "superposition" were not just descriptors of new phenomena; they were linguistic tools that reshaped the framework through which we understood the universe.
Thomas Kuhn’s idea of "paradigm shifts" in science further emphasizes how language and societal structures influence which theories gain traction. It’s not necessarily the best ideas that rise to prominence, but those that align with the current scientific "language game." When quantum mechanics first emerged, it was met with resistance from classical physicists who were entrenched in deterministic thinking. It wasn’t until the scientific community began to adjust its linguistic framework that quantum theory gained widespread acceptance.
In our current postmodern landscape, we’ve become more aware of these dismissals, sidelined theories, and forgotten philosophers. Yet, even now, the "language games" continue to limit how far we can go. Our obsession with measurable outcomes and technological applications often pushes aside philosophical musings or mathematical theories that don’t have immediate, tangible benefits.
Category theory, for example, is an abstract branch of mathematics that deals with the relationships between different mathematical structures. It’s been called "generalized abstract nonsense" by some due to its perceived lack of practical applications. Yet, in the realm of theoretical computer science and quantum computing, category theory is providing profound insights into the nature of computation and the fabric of logic itself. This is a perfect example of how society often dismisses deep, abstract ideas until they find a direct application, at which point they are re-evaluated and celebrated.
The history of science and mathematics is full of ideas that were dismissed, ignored, or forgotten, not because they were incorrect, but because they didn’t fit the dominant narrative or the language of the time. As we move forward, especially in this postmodern era, we must become more aware of the language games we play. Our current understanding of science and society is built on layers of knowledge, some of which have been consciously discarded or repressed.
Reclaiming these forgotten threads, whether from philosophers like Heraclitus or mathematicians like Galois, isn’t just about revisiting the past—it’s about recognizing the limitations of our current frameworks and pushing beyond them. If we can become more aware of how society shapes what we consider "valid" knowledge, we can start to question and evolve those structures, leading to new, deeper understandings of both science and the world around us.
At the heart of Gödel’s Incompleteness Theorems is a profound insight into the limits of formal systems: any sufficiently powerful formal system is incomplete—it contains truths that cannot be proven within the system itself. This created a rupture in the mathematical world, where many had believed that, through formal logic and rules, we could describe all truths in mathematics. Gödel shattered this dream, showing that no matter how comprehensive or consistent a system of rules might seem, there will always be statements that are true but unprovable.
This presents an interesting friction: Gödel’s work shows that formal systems (including mathematics) are inherently limited, while Wittgenstein would argue that our understanding of these limitations is shaped by the linguistic context in which they emerge.
Gödel’s work has deep implications for how we think about scientific theories. Many scientists (and perhaps society at large) operate under the assumption that there is a "complete" and discoverable set of truths about the universe—a final theory that will explain everything. Gödel’s theorems remind us that this quest may be fundamentally flawed. Any formal system we devise to describe the universe will have inherent limitations. This doesn’t just apply to mathematics—it applies to any framework we use to understand reality.
This resonates with quantum mechanics and relativity, where the friction between different theories suggests that a complete, unified theory may forever elude us. Gödel’s incompleteness forces us to reckon with the idea that science, like mathematics, will never be "complete." There will always be phenomena that are true but cannot be encapsulated in any theory we devise. Here is where our work on XAWAT shines: challenging the rigid, mainstream scientific narratives by embracing the idea that the current structures of knowledge may not be sufficient to explain reality in its entirety.
In parallel, Wittgenstein’s concept of language games illustrates the limits of how we communicate and understand science. Scientific terms and concepts, like mathematical symbols, are only meaningful within their specific contexts. The language we use in quantum mechanics, for example, is full of metaphors and constructs (like "particles" and "waves") that may not fully encapsulate the true nature of reality. Wittgenstein would argue that these terms gain their meaning not from some inherent truth, but from the way they are used within the scientific community. The friction arises when new discoveries or insights (like those in quantum physics) demand a shift in language, but the old "language games" resist change.
Think of how non-causal interaction (as seen in quantum entanglement) strains the language of classical physics. We use terms like "action at a distance" to describe entanglement, but these are relics of an older "game" that doesn’t quite fit. Leibniz’s monads, much like Gödel’s theorems, point to the limits of current linguistic and conceptual frameworks.
In exploring Gödel and Wittgenstein, we confront the reality that our scientific and mathematical systems are inherently incomplete, and the language we use to describe them is both a tool and a limitation. XAWAT’s vision aligns with this understanding: by constantly questioning and pushing the boundaries of scientific language and narrative, we allow for the emergence of new truths—truths that may otherwise be hidden within the cracks of our formal systems.
As we continue to explore non-causal interactions and push the envelope of scientific discourse, the interplay between Gödel’s incompleteness and Wittgenstein’s language games becomes even more relevant.
We are not just dealing with the limits of what we know, but the limits of how we know and express it.
The real challenge is to embrace the incompleteness of our systems and evolve the language to better capture the complexities of the universe—something that XAWAT seems primed to do.
