- 2025年8月18日
A Simple History of Philosophy: Structuralism in Modern Philosophy, Plato, and Examples from Greek Philosophy and Mathematics
A Simple History of Philosophy: Structuralism in Modern Philosophy, Plato, and Examples from Greek Philosophy and Mathematics
Plato’s Conflict: Modern Philosophy Through an Example
When we look at the history of Western philosophy, particularly ancient Greek philosophy, we find good examples for understanding modern philosophy. A key one is Plato’s theory of Forms (or Ideas).
By considering why a theory like the theory of Forms was necessary, we can grasp, in broad strokes, the history of Western philosophy, the problems it has faced, and the methods modern philosophy uses to solve them. To study modern philosophy, it is often easier to understand not only the history from ancient to modern philosophy but also the deep psychology and mental structures of Western people and their changes over time. I would like to explain this in an easy-to-understand manner.
Ancient Greek Philosophy
A primary characteristic of early ancient Greek philosophy was its focus on natural philosophy (akin to natural science), seeking the origin of all things. This “origin” could also be called “essence.”
This “essentialism” is like a continuous bass note flowing through the entire history of Western thought. “Essentialism” is also a form of “truth-ism”—the fundamental theme of Western philosophy is that such things as essence and truth exist.
Beyond Ionian natural philosophy, there was a culture and system of debate born from the polis and democracy. While natural science aligns well with rationality and logic, social sciences like politics, law, and administration are also deeply connected to them. There seems to be a slight difference between the rationality and logic of these two fields, which in any case, appears to have become a defining factor in the Greek temperament and social nature. This difference later manifested as a conflict between philosophers and Sophists, leading to Socrates’s execution.
In Greece, there were the words “kosmos,” “nomos,” and “logos.” Kosmos means order, nomos means social norms, and logos is something like human reason. Logos is the origin of our modern word “logic.” There was also “rhētorikē,” meaning something like rhetoric, which is the origin of our modern “rhetoric.” Additionally, there were the Sophists, from whom the modern word “sophisticated” derives.
Philosophy, natural science, and mathematics on one hand, and politics, institutions, and law on the other, both appear rational and logical, yet they sometimes have conflicting differences. While both may aim for kosmos (order), the former is “logical” and philosophical, while the latter is “rhetorical” and Sophistic. The former seeks non-contradiction and coherence, while the latter has a populist, sophistical political nature and a pragmatic pursuit of practical benefits. The former is academic and research-oriented in its pursuit of truth, while the latter caters to the masses, being populist and propagandistic. Although they seem like oil and water, it is interesting that in modern times, both emphasize logic. It seems logic is seen as a form of justice.
A Brief History of Western Philosophy
From the beginning, there must have been a sense, which was gradually cultivated, that there are principles and laws in the world, and that nature, society, and humans move according to them. This was refined by Socrates, Plato, Aristotle, and the Stoics of the Hellenistic period.
From the perspective of the entire history of Western philosophy, Plato’s achievement was to separate the real world from the world of Forms. This became the backbone of all Western philosophy. A main theme of Western philosophy then became devising theories to unify what should not be separate. Whether they “should not be separate” is actually uncertain, but a kind of obsession with this idea marks one aspect of the history of Western philosophy.
To state the conclusion, around the time of modern philosophy, or slightly before with phenomenology and existentialism, the feeling became “it’s okay for them to be separate.” A little before that, Hegel’s German Idealism, which perfected a theory unifying the real world and the world of Forms, may have been one of the culminations of modern philosophy.
Aristotle is called the “father of all sciences.” A true intellectual, he compiled the knowledge of his predecessors and contemporaries while adding his own unique contributions. He is also known as the father of logic, with his work, the Organon. Syllogistic logic seems to have been central during the Middle Ages, and Aristotle is synonymous with the syllogism. Later, during the Hellenistic period, the Stoics brought developments to logic that were a precursor to modern symbolic logic, though they were somewhat limited and incomplete. However, the idea that logic could be performed through a formal algorithm like calculation was characteristic.
The premise for this is that the world is a kosmos. The world is designed as an ordered, harmonious, mechanical, coherent, consistent, and non-contradictory system or structure. Through a good approach, one can discover its laws. And the idea that it can be expressed formally and linearly with words (symbols) through a clever approach was nurtured.
Thinking of the world as a machine, system, or structure is not exclusive to modern Western thought. It is not found in the Semitic Bible either. Rather, Judaism and Christianity were influenced by Hellenism—that is, Greek civilization—to become what they are today. It cannot be attributed to Rome, nor to the Celts or Germanic peoples. Its influence from the Orient is unknown to me. Greece, alongside India and China, is truly special. Karl Jaspers, a psychiatrist-turned-philosopher, named the period from around 800 BC to 200 AD, which brought about the evolution of human intellect, the “Axial Age,” and these three regions are particularly significant.
The worldviews of Christianity, Judaism, and Islam—a world created by God—are also influenced by Greek thought. This means that the world created by God is a kosmos.
Thus, Greece quickly laid the foundations for modern mathematics. The Pythagorean school, Euclid, Diophantus, Archimedes—they are all representative figures of number theory, geometry, algebra, and analysis in mathematics. Besides these famous organizations and individuals, there must have been many anonymous contributors.
Computers and machines are not unique to the modern era. The Antikythera mechanism was a machine for astronomical observation, proving that complex machines were being built in antiquity. In the Middle Ages, a theologian conceived of a calculating machine to derive theological conclusions. In the modern era, Pascal created the oldest surviving calculator, a numerical calculating machine, and Leibniz improved upon it while also conceiving the prototype of the modern computer.
At the intersection of modern logic and mathematics, Plato’s idea of separating the two worlds emerged. And not just one, but at least two such ideas were born.
Humans, Not God, Created Irrational Numbers
In ancient Greece, there are several famous stories about real numbers. A well-known legend is that the Pythagorean school, which claimed that the essence and origin of the world, what constitutes it, is number, killed the person who discovered a number that was not rational—an irrational number.
Zeno’s paradoxes are also famous. Many have heard of the infinite division stories, such as Achilles being unable to catch a tortoise or a moving arrow being stationary.
It is famously said by Leopold Kronecker, who was Jewish, “God made the natural numbers; all else is the work of man.” Incidentally, in modern mathematics, the view is that “humans created natural numbers and all other numbers.” Natural numbers are constructed using the Peano axioms.
When the word “modern” is attached to fields like modern mathematics (formalism, axiomatism), modern thought (structuralism), and modern humanities and social sciences, the perspective of “creation” is necessary. It is not “discovery” but “invention.”
In natural science, there is a division between theoretical and experimental/empirical fields. Theories are created. For experiments, verification, observation, and measurement, one might use the word “discover.”
In Kronecker’s time, Cauchy and Weierstrass established the foundations of analysis by using the concepts of real numbers and the continuity of functions to solve problems in analysis, in other words, to manipulate infinity. In the classical view, the continuity of real numbers is self-evident. However, dealing with infinity classically leads to strange results, including Zeno’s paradoxes. For example, no matter how many points with no area are connected, they cannot form a length. Similarly, no matter how many points with no area are laid out, they cannot form a surface.
The classical geometric idea that the number line, representing real numbers, has no gaps is intuitive, natural, and considered self-evident.
However, the modern approach is not to see it as self-evident but to decide that it is so. One expression for this is that there are no “break points” on the number line. This means that if you try to cut the number line into two with a blade, wherever you cut, you will always hit some number, which could be rational or irrational. In fact, a number line consisting only of rational numbers can be split into two half-lines without hitting any number. To prohibit this, something was created to fill the gaps in the rational number line, and it was named the irrational number. This is one expression of Dedekind’s axiom defining the continuity of real numbers.
If irrational numbers existed from the beginning, they would be real and essential. But if we ignore the idea of their pre-existence and simply decide that there is something called an irrational number, it becomes “the work of man,” as Kronecker said. This means that humans created irrational numbers to fill the gaps in the rational number line.
Emphasizing this perspective of creation, the concept of completeness—that “every Cauchy sequence converges”—also works. The convergence of a Cauchy sequence is called “completeness.” In other words, the term “completeness” carries the nuance of making a line complete and without gaps by creating and filling them with irrational numbers. Incidentally, “to converge” means to have a limit. For a sequence of rational numbers that gets progressively narrower, it is not possible to state definitively, from a classical, intuitive, or natural sense (whether innate or instilled by education), that there are no gaps. If it cannot be stated definitively, then we can simply decide it to be so. In other words, the existence of a limit for a Cauchy sequence is not something to be figured out, but something humans decide exists. This is equivalent to saying that humans create irrational numbers so that Cauchy sequences can converge.
Let’s look at the Weierstrass theorem: “A monotonically increasing sequence that is bounded above has a supremum (least upper bound).” Like the Dedekind cut, this is an endpoint problem. It is also an expression using sequences, like a Cauchy sequence. It is useful to rephrase this to understand modern mathematics and structuralism, which may have been established with modern mathematics as a prototype.
A clearer way to put it is: “We decide that a monotonically increasing sequence that is bounded above has a supremum.” It is not so from the beginning; we don’t know, but humans decide it to be so, creating and placing a supremum point.
I have presented this in three different ways, and they break down into several components. One is that it is an ordered set. If we use rational numbers as a known base to define or create irrational numbers, then since rational numbers form a field where addition, subtraction, multiplication, and division are possible, it is the field of rational numbers. These also have the Archimedean property. The Archimedean property has several meanings, including “there are no infinitesimal numbers,” “there are no infinite numbers,” and “it is dense.” “Dense” simply means that between any two numbers, there is always a third, different number. On top of these properties, we add the property of being “complete.” To emphasize the sense of human intention, one could write “it is defined as complete.”
The existence of irrational numbers, or the continuity of real numbers, is artificially, operationally, and constructively created by combining several such rules. This is different from the Platonic, idealist view that the continuity of real numbers is naturally determined, or the naive image of real numbers and their continuity found in Euclid, such as “a point is that which has no part” and “a line is breadthless length.” However, without such an operational, constructive, and formal approach, modern mathematics, which pursues rigor, would be unusable, unlike classical mathematics. Not only modern mathematics but anything with “modern” attached to it tends to become artificial, operational, constructive, and formal.
This is a good example of structuralism. By establishing rules like “it is dense,” “it is complete,” “it has an order,” and “it is a field,” we construct irrational numbers or make real numbers continuous. It doesn’t matter whether real numbers exist as essential entities in the world of Forms, as in classical mathematics, or not. These rules are called axioms. A field of mathematics is formed from the entirety of what can be derived by combining these axioms. This process can be automated by a machine or a computing device through formal operations, without human involvement. In simple terms, a computer or AI would suffice. In fact, a computer might be more accurate than a human using their brain.
Philosophically speaking, regardless of the existence of Plato’s world of Forms, whether essence is real or not, humans can create real numbers and do mathematics. Plato valued mathematics so much that he allegedly hung a sign at the entrance to his Academy that read, “Let no one ignorant of geometry enter.” It’s not meant to be ironic, but Plato might have complex feelings knowing that 2000 years after his death, mathematics would resolve his philosophy—not by negating it, but by relativizing, neutralizing, or making it independent. His philosophy feels overly serious, like a top student’s, or rather rigid. Having his master Socrates killed in such a way, perhaps he wanted to cling to something like essence or truth. Or perhaps he became stubborn and overcomplicated things. Consequently, the history of Western thought influenced by Plato has become extremely earnest, elitist, and meritocratic.
It is possible to add more to reinforce the continuity of real numbers. For instance, so far we have used set theory, but we could also add rules from topology to express continuity. To add a smoother, more intuitive image, we could bring in and add the topological concept of a “connected open set” as a rule.
With the emergence of modern mathematics and modern philosophy, or more broadly, structuralism, anything labeled “modern” has changed in nature from its classical counterpart. In physics, mathematics is said to be important, but it is divided into theoretical and experimental branches. The experimental side discovers new facts or confirms existing theories through observation, measurement, and verification. It is the department that gathers data from nature, though it is not directly related to the reality of nature itself. On the other hand, the theoretical side creates theories. As concluded from what I wrote earlier, a theory is fine as long as it is self-contained. Whether there is an actual nature that realizes that theory is sometimes a separate issue from the theory’s brilliance. However, physicists are natural scientists, not mathematicians, so they design their theories to be consistent with data already gathered from nature. If a theory is well-made, explains existing data without contradiction, and has the potential to reveal new natural phenomena and gather new data from nature, then to perform a stress test on its validity as an explanatory system for natural phenomena, they try to verify it by gathering the data predicted by the theory from nature and phenomena. Natural science is a repetition of this process. This cycle itself is not unique to modern natural science; modern science from the early modern period did the same. However, until the modern era, under Plato’s influence, there was an attempt to see these two—theory and reality—as one. Was it the curse of Socrates or the obsession of Plato? There was an almost obsessive effort to unite them. The influence of religions like Christianity, Judaism, and Islam might also be a factor. Since the modern era, it has simply become unnecessary to strive to unite the two.
Phenomenology, the “Enfant Terrible”
Another stream is phenomenology. Husserl, a mathematician who was a student of Weierstrass and Kronecker, initially researched the foundations of mathematics. But perhaps thinking that if he were to study foundations, the subject didn’t have to be mathematics, he moved to the field of philosophy to explore the foundations of all things.
Here, he thought very logically. Within our inner world—be it the mind, the head, or the spirit—various things appear or “are present” (phenomena). Let’s tentatively call the totality of what is present when we direct our consciousness to it and feel its reality (noesis, noema), and what floats in the background but becomes present when we direct our consciousness to it, a “phenomenon.” Husserl clearly acknowledged that just because there are presences and phenomena, it is fundamentally uncertain whether an external world exists, or if there is something in that world that is the source of these presences and phenomena. Up to this point, it is similar to Kant, who considered the “thing-in-itself” and the dilemma of being unable to reach it.
Husserl proposed that we should stop thinking about the uncertain external world whose existence is unknown (suspend judgment = epoché) and focus only on the mechanisms of phenomena and presences, as that is what constitutes a rigorous science. This was likely influenced by the psychological trends of the era, or perhaps psychoanalytic influences. Since science is the spirit of method, he devised a new methodology for philosophizing. That is why it is called phenomenology, not an “-ism.”
The exploration of why various things are present and appear in our inner world, and what the relationships are between these appearing things or with oneself, was advanced by Heidegger. In France, Sartre added the bleak atmosphere of the mid-20th century, seasoned it with his style of existentialism, and popularized it. This also moved the direction of philosophy toward separating Plato’s world of Forms and its shadow, the objects before our eyes.
However, while modern philosophy was born from a critique of modern philosophy (in the sense of the early modern period), the aspect of it being born from within modern philosophy is not very strong. In the sense that structuralism was born from fields like mathematics, linguistics, and cultural anthropology—which studied the relationship between Western civilization and “savage,” primitive cultures—it is more like an adopted child than a biological one. Perhaps like a rebellious adopted child, not related by blood to its foster parents.
The 20th century was a century of both greatness and misery, with wonderful inventions, discoveries, growth, and progress, as well as wars, massacres, ideological absolutism, and persecution. The number of people killed in the 20th century was orders of magnitude greater than the number of people killed in all of previous human history. For example, between one-half and one-third of the entire Jewish population was killed. It was chaos. Of course, many other peoples, cultures, and human groups also perished or disappeared during the process of Westernization.
Conclusion
Greek philosophy was embedded in the history of Western thought and influenced it. It is said that the Middle Ages were “dark” because of Christianity, and that the Renaissance and the modern era began with the re-importation of Aristotle from the Islamic world. However, Christianity itself was heavily influenced by Greek philosophy. Furthermore, Judaism and Islam were also influenced by Greek civilization. I had thought that for Judaism, after the Babylonian Captivity, there was a slight gap in Hellenistic influence until around the time of the Books of the Maccabees (which are not in the Hebrew Bible for some sects), but it seems that considerable materials remain. In fact, the New Testament is supposed to be a compilation of books and letters written in Greek. The canons of the Bible also differ among Judaism, Catholic and Protestant Christianity, and Islam, with different documents being adopted during their compilation.
Jews were occupied by the Greeks during the Hellenistic period, were incorporated into the Roman Empire which had adopted Greek civilization, and even after the Babylonian Captivity, they did not live only in Israel but were widely spread throughout the Mediterranean world, receiving ideological influence.
The word “philosophy” today refers to a specific discipline, but in the Greek era, it encompassed all academic fields. Naturally, this included mathematics and logic. Logic, oratory, dialectics, and proof developed overwhelmingly in Greek philosophy. I don’t know the Greek words for cosmos, nomos, logos, rhetoric, dialectics, and proof, but they imply that the world has order, a system, a structure, and coherence, and that by using the right approach, one can extract laws, explain the world, make arguments, and describe it linearly using language and symbols. This spirit gave birth to modern Western science and technology. From this golden age of humanity, created by Westerners, modern civilization was born.
However, the final conclusions that emerged from it were modern philosophy, structuralism, and post-structuralism, which convergently evolved to the same place as Mahayana Buddhism. After the era of modern French thought, the world, with the end of the Cold War, entered the present through neoliberalism and globalism. But neoliberalism and globalism have instead advanced unsustainable environmental destruction, resource depletion, and social unrest, causing a conservative backlash today.
Amidst this, Japan, which was left behind by neoliberalism and globalism, has been receiving some positive evaluation recently. There are likely various reasons for this, but from a historical and philosophical perspective, I believe there are two main points.
The first is that despite being Westernized, Japan has preserved its old culture and artifacts, which overlap with its identity as a nation and an ethnic group due to being an island country with a relatively homogeneous population. Some theories suggest that Japan retains elements from the Jomon period or even the earlier Stone Age. Generally, modernization tends to lead to the loss of a country’s old traditions. The fact that they remain is a unique aspect of Japan.
The second is that there is a mechanism for preserving old things. The United Kingdom, another island nation, has a similar tendency, so there may be geographical and geopolitical factors at play. It is also significant that Japan has rarely been invaded or ruled by other peoples. One theory suggests that old things tend to survive in islands, peninsulas, and mountainous regions. In addition to this, Japan has the unique characteristic of being almost the only Mahayana Buddhist country in the world. Since Mahayana Buddhism shares the same ideas as modern philosophy, and since Chinese characters and Buddhism were first transmitted from the Korean peninsula at the beginning of its history, Japan is a country whose origins are intertwined with both written language and Mahayana Buddhism. In other words, it is a special country that, from the very beginning of its history, has lived at the forefront of Western thought while preserving prehistoric artifacts, spirituality, and its imperial dynasty.
These various characteristics have both good and bad aspects, but I hope Japan will continue to survive and endure with the spirit of onko-chishin—learning from the past to gain new knowledge—which is part of its identity.