A Simple History of Philosophy: Structuralism in Contemporary Philosophy and Plato — Examples from Greek Philosophy and Mathematics

A Simple History of Philosophy: Structuralism in Contemporary Philosophy and Plato — Examples from Greek Philosophy and Mathematics

• Plato’s Dilemma: Understanding Contemporary Philosophy Through Examples

Looking across the history of Western philosophy and, especially, ancient Greek philosophy offers good examples for grasping contemporary philosophy.
Take Plato’s theory of Forms (Ideas). By asking why such a theory was needed in the first place, we can sketch both the historical problems the Western tradition carried and the rough way contemporary philosophy came to address them.
As a way to study contemporary philosophy, it often helps not only to follow the timeline from ancient to modern philosophy, but also to attend to the Western psyche—its deep-seated mental structures and how they changed. With that in mind, I will offer a clear, accessible explanation.

• Ancient Greek Philosophy

Early Greek philosophy was “natural philosophy” (proto–natural science), a quest for the archē—the origin or essence—of all things.
This “essentialism” is a basso continuo running through the entire history of Western thought. Essentialism is also a kind of “truth-ism”: the basic theme is that there is essence or truth.

Beyond Ionian natural philosophy, the Greek polis and democracy fostered a culture and institutions of public speech and debate. Natural science seems friendly to rationality and logic; but politics, law, and administration are equally bound up with reason and argument. The two kinds of rationality diverge in important ways, and that divergence shaped Greek temperament and social life.

The difference later appears as the clash between philosophers and sophists and becomes one cause of Socrates’ execution.
The Greeks had the terms kosmos (order), nomos (social norms or law), and logos (reason; the root of today’s “logic”). They also had rhētōrikē (rhetoric), the root of “rhetoric”, and “sophists”, the root of “sophisticate”.

Philosophy–natural science–mathematics on the one hand, and politics–institutions–law on the other, both wear rational, logical faces, yet they can conflict. Both may aim at kosmos (order), but the former is logical and philosopherly, the latter rhetorical and sophist-like. The first pursues non-contradiction and coherence; the second tends toward populist, even sophistical, political pragmatism and usefulness. The first is truth-seeking and scholarly; the second can be mass-oriented, populist, propagandistic.

Though they may seem like oil and water, in modernity both come to emphasize “logic”—as if logic itself were a kind of justice.

• A Brief Sketch of Western Philosophy

From early on, there seems to have been a sense that the world runs by principles or laws governing nature, society, and human beings. This sense is refined in Socrates, Plato, Aristotle, and the Stoics of the Hellenistic era.

In the sweep of Western philosophy, Plato’s signal move was to split the sensible world and the world of Forms. That division became the backbone of the entire tradition. Much of Western philosophy then tries to justify reconciling what ought not be divided. Whether they truly “ought not be divided” is uncertain, but the obsession to reunite them is one strand in the tradition’s history.

In the end, around phenomenology and existentialism—just before contemporary philosophy proper—the sentiment becomes: “It’s fine if those realms remain distinct.” Hegel, just prior, offered a grand system that reunifies the sensible and the ideal, which one might see as a culminating shape of early modern philosophy.

Aristotle, often called the father of all disciplines, surveyed the knowledge of his day and added much of his own. He is also called the father of logic; his Organon underwrote the syllogistic logic of the medievals. Later, the Stoics pioneered a kind of propositional/logical calculus—limited and incomplete, yet characteristic in treating logic as formal, almost algorithmic.

Presupposing that the world is a kosmos, they envisioned reality as ordered, harmonious, mechanical, consistent—a system or structure. With deft approaches, one can extract laws from that whole and express them formally, in straight-lined language (symbols). To think of the world as machine/system/structure is not only a modern Western move; indeed, Judaism and Christianity (and Islam) as we know them were shaped under Hellenistic influence, so even the “world created by God” was imagined as cosmic—ordered.

Accordingly, Greece early on forged prototypes of modern mathematics: Pythagoreans, Euclid, Diophantus, Archimedes—names emblematic of number theory, geometry, algebra, and analysis. Alongside great names and schools there must have been many anonymous contributors.

Computers and machines are not solely modern. The Antikythera mechanism was an ancient astronomical device. In the Middle Ages, theologians even imagined machines to compute theological conclusions. In the early modern era, Pascal built the earliest surviving numerical calculator; Leibniz improved it and envisioned the conceptual prototype of today’s computers.

As logic and mathematics advanced in the modern period, the Platonic habit of dividing worlds reappeared—at least twice in different guises.

• It Was Humans—not God—Who Made Irrational Numbers

Ancient Greece offers famous stories about the real numbers. The Pythagoreans, who held that number is the essence of reality, are said to have killed the discoverer of irrational numbers. Zeno’s paradoxes are also well known—Achilles never catching the tortoise, the arrow at motionless rest—paradoxes of infinite division.

Kronecker, himself Jewish, famously said, “God made the integers; all else is the work of man.” In contemporary mathematics, one might go further: “Even the natural numbers are human-made,” constructed via the Peano axioms.

Across fields that bear the adjective “modern” or “contemporary”—from formalist/utilitarian mathematics to structuralist thought to contemporary psychology and the human sciences—the emphasis shifts from discovery to invention, from “finding” to “making”.

In natural science we separate theory from experiment. Theory is something we make; experiment/observation may be called “discovering”. In analysis, Cauchy and Weierstrass grounded calculus by handling infinity via real numbers and continuity. Intuitively, the continuity of the reals seems obvious, yet classical handling of infinity begets paradoxes like Zeno’s (e.g., point-sets of zero area never summing to length or area).

The picture of the real line as unbroken feels natural and self-evident; but modernity says: it is not self-evident—we decide it to be so. One expression: “There are no cuts (gaps) in the real line.” Slice the line with a “knife” wherever you wish; you always hit some number—sometimes rational, sometimes irrational. In contrast, the rationals alone allow a cut that hits no number—hence the need to forbid such gaps by making irrationals to fill them. This is the spirit of Dedekind cuts.

If irrationals “already existed”, that would be Platonic realism. But modern mathematics brackets such ontological claims and simply posits irrationals as human work—Kronecker’s “work of man”. Equivalently, we can enforce completeness by the Cauchy-sequence view: “Every Cauchy sequence converges.” Completeness means: no holes.

Another equivalent: the monotone convergence property, often cast as Weierstrass’s theorem here—“Every increasing sequence bounded above has a least upper bound.” In modern style: “We decide that every increasing sequence with an upper bound has a supremum.”

These equivalent formulations rely on several ingredients: a linearly ordered set; taking the rationals as a base field (a field under +, −, ×, ÷); and the Archimedean property (no infinitesimals, no infinite elements, and density: between any two numbers there is a third). Add completeness—again, a decision—and we obtain the continuous real line.

Thus irrationals and the continuity of the reals are constructed by combining rules—operational, constructive, structural, formal. This contrasts with a Platonic or Euclidean intuition that reals “are just continuous by nature.” Classical mathematics may get by with that, but the rigor of modern mathematics demands the constructive/formal approach.

This is a fine example of structuralism: by stipulating rules—density, completeness, order, field structure—we build irrationals and continuity. Whether reals “exist” in a realm of essences does not matter; axioms (rules) and the totality of what follows from them define a mathematical field. A machine can carry out such formal manipulations; computer or AI may even be more accurate than us.

Philosophically, whether or not the Platonic world of Forms exists, we can make the reals and do mathematics. Plato prized mathematics—“Let no one ignorant of geometry enter”—and yet, two millennia later, mathematics resolved his philosophy not by refuting it, but by relativizing it—showing philosophy and mathematics can proceed independently.

Continuity can be reinforced further: beyond set-theoretic constructions, we may impose topological rules (“connected open sets”, etc.) to recover smoother intuitions. With the rise of modern mathematics and philosophy—especially structuralism—anything labeled “modern” tends to become artificial, operational, constructive, structural, formal.

Physics is often said to depend on mathematics, but more precisely, it splits into theory and experiment. Experiment observes, measures, verifies; theory is made. A theory should cohere with existing data; if it explains them without contradiction and predicts new phenomena, we test those predictions. This looping is true in early modern science as well. The difference is that, under Plato’s influence, early modernity sought to unify theory and nature, almost obsessively—as if under Socrates’ curse or Plato’s compulsion, perhaps abetted by religious habits. Contemporary thought releases the pressure to fuse the two.

• Phenomenology as an “Enfant Terrible”

Another thread is phenomenology. Husserl—student of Weierstrass and Kronecker—began in the foundations of mathematics, then shifted to philosophy to seek the foundations of all phenomena.

He proceeds with strict logic: within us, things appear. When attention turns to them, they present themselves; when it doesn’t, they still hover as background. Call the totality of such presentations “phenomena”, and analyze their structure. Husserl frankly admits: from the presence of phenomena we cannot infer with certainty an outside world or “things in themselves.” Like Kant, he acknowledges the limit.

So he proposes the epoché—to bracket the question of external existence and study only the appearing itself. Influenced by psychology and psychoanalysis, he invents a new method of philosophizing: hence not an “-ism” but phenomenology. Heidegger then radicalizes the question of why and how things appear within our being; Sartre seasons it with the harsh mid-20th-century mood into an existentialism that captivated France.

This, too, moves away from Plato’s split into ideal essence and shadowy particulars—by dwelling on the side of appearance without forcing a reunion. In a sense, contemporary philosophy arose both as a critique of modernity and from fields adjacent to it: mathematics, linguistics, cultural anthropology. Structuralism is less a direct offspring than a foster child—brilliant, rebellious, not bound by blood.

The 20th century saw magnificent inventions and discoveries—but also wars, massacres, ideological absolutes, and persecutions. The death toll dwarfed all that came before; for Jews, fully one-half to one-third of the population was murdered.

• Conclusion

Greek philosophy was embedded in, and deeply influenced, Western intellectual history. We often hear that Christianity caused a “dark Middle Ages,” that Aristotle re-entered via Islam, and that this spurred the Renaissance and modernity—but Christianity itself is saturated with Greek philosophy; Judaism and Islam also absorbed Hellenic influence. The New Testament was compiled from Greek texts. Different canons across Judaism, Catholicism, Protestantism, and Islam reflect different editorial choices. Jews lived widely across the Mediterranean after the Babylonian Exile and under Hellenistic and Roman rule; Greek thought left deep marks.

In Greek antiquity, “philosophy” encompassed all scholarly pursuit, including mathematics and logic. Logic, rhetoric, dialectic, and proof burgeoned. Terms like kosmos, nomos, logos, rhetoric, dialectic, and demonstration imply a world with order, system, structure, and coherence—laws to be uncovered and described linearly in language and symbols. That spirit birthed Western modern science and technology, and from that golden age came contemporary civilization.

Yet the ultimate outcome—structuralism and post-structuralism—converged on something close to Mahāyāna Buddhism. After French theory and the post–Cold War turn to neoliberal globalization, we arrived at a present where such models strain sustainability—fueling environmental harm, resource waste, and social anxiety, prompting conservative reactions.

Amid this, Japan—often said to be “left behind” by neoliberal globalization—has oddly been reassessed. Philosophically, two reasons stand out: (1) despite Westernization, Japan—as an island nation with a relatively homogeneous people—has preserved old artifacts and forms, some say even traces from Jōmon or earlier; modernization elsewhere often erased such strata. (2) Japan has mechanisms to preserve the old—akin to Britain’s islanded conservations—helped by scant experience of foreign conquest. Islands, peninsulas, and mountain fastnesses tend to conserve antiquity.

Uniquely, Japan is virtually the only Mahayana nation-state. Mahayana’s kinship with contemporary philosophy means that, from the outset—since the first transmissions from the Korean peninsula—script and Mahayana doctrine were entwined with state formation. In other words, Japan has, from the beginning, combined the oldest with the newest: prehistoric artifacts and spirit alongside the cutting edge of Western thought.

With many virtues and vices, Japan’s path may still be to “learn the new by warming the old” (onko chishin)—to survive by preserving and renewing.