- 2025年11月16日
A Provisional Map of the Disciplines:
Mathematics, Logic, Relations, Natural Language, and the Humanities
(Putting Contemporary Philosophy Aside for the Moment)
A Provisional Map of the Disciplines:
Mathematics, Logic, Relations, Natural Language, and the Humanities
(Putting Contemporary Philosophy Aside for the Moment)
The aim of this essay is to propose that we could breathe new life into the entire world of scholarship—including the humanities—by integrating mathematics more deeply into it.
A second aim is to sketch, from the perspective of mathematical foundations (set theory, category theory, type theory, etc.), a rough map of “what mathematics is / what scholarship is / what modern literacy is,” and then connect that map with the perspectives of contemporary philosophy and Mahāyāna Buddhism.
Mathematics as a Good Illustration of Structuralism, but…
Structuralism is more difficult to grasp than realism (especially naïve realism).
Modern mathematics provides extremely good examples for explaining structuralism, and so it is often used to illustrate structuralist ideas.
Mathematics, structure, and relations are themselves related concepts (to put it in a somewhat awkward way that sounds like the “Shinjirō-style” phrasing currently popular in Japan). We can think of mathematics as a discipline that studies structures, but when we ask “What is mathematics?” or “What is structure?”, the notion of relation becomes indispensable.
This is, in a sense, a relationship of mutual dependence.
- On the one hand, we can use the concept of relations to build or define mathematics and structures.
- On the other hand, we can also use mathematics and structures to define or analyze relations.
Even if we do not go so far as full-fledged logic, anyone who studies set theory will usually find, in the opening chapters of a textbook, both the definition of relations and a brief introduction to logic. The way relations are defined, however, depends on what kind of “cut” we take from the standpoint of mathematical foundations—i.e., which framework we use to define or express mathematics itself.
From Mathematical Foundations: What Is Mathematics? And What Is Scholarship?
In a loose way of speaking, we might say that mathematics is the study of structures.
When we use the word “structuralism,” we have also tended to use the word “structure” in this somewhat fuzzy way.
In what follows, I will leave those notions deliberately vague, and offer a rough map of the whole of scholarship—including the humanities—centered around mathematics. It is only a map, so I will not go into detailed explanations of every point.
I have explained set theory and category theory elsewhere (to the extent that my memory serves), so if needed you can read my earlier writings, ask an AI, or simply search online.
Perspectives on Mathematics and the Building Blocks That Form Mathematics
Anything can be viewed, discussed, and thought about from multiple angles—through different perspectives or “cuts.”
Pursued further, this quickly becomes a post-structuralist method.
Yamamoto Shichihei coined the term “multiple-opposition object grasping” (複対立対象把握), and that way of seeing things allows, for instance, the practical execution of Derridean deconstruction in contemporary philosophy.
With that in mind, let me list some of the components of mathematics, and the different “cuts” on mathematics, from the standpoint of mathematical foundations:
- Set theory
- Logic
- Category theory
- Type theory
- Proof theory
- Model theory
- Computability theory (≒ theoretical computer science?)
- Reverse mathematics
The items above are parts of mathematics, or tools for explaining what mathematics is, or even for constructing mathematics itself.
- In many classical areas of mathematics, for example, you can describe everything in terms of set theory + logic.
- Category theory, taken by itself, is also such a “part.” Naive category theory alone is not sufficient to build or explain all of mathematics. For that, we need category theory + logic, or else we construct categories in which logic is internalized (e.g., toposes).
If we do that—building a category that carries its own internal logic—then we can encode logic inside category theory itself, and even reconstruct set theory within categorical terms.
However, the axioms of set theory (the ZFC system) and the kind of set theory / logic arising in topos theory are not identical. In a topos or categorical setting, the internal logic is typically intuitionistic logic, so there is a certain gap between that and “classical” mathematics built directly on ZFC-based set theory.
I will leave such fine points aside here (though some might object that they are not “fine” at all).
- Intuitionistic logic evokes Brouwer.
- Type theory calls to mind Bertrand Russell.
- Proof theory brings us back to Hilbert’s program.
For readers who know a rough outline of the history of mathematics, this is quite an interesting constellation.
Likewise, set theory recalls Cantor. Structuralism and category theory call to mind Bourbaki, Grothendieck (and Yoneda). So mathematical foundations is a field that is not only about foundations, but also quite enjoyable from a history-of-mathematics perspective.
These topics are familiar not only to lovers of pure mathematics, but also to people working pragmatically in information science and the IT industry—programmers, system engineers, project managers, and so on.
Mathematics Also Turns Contradiction into an Object of Study
The humanities have long described humans and societies full of contradictions as raw material. But when it comes to how to treat those contradictions formally, mathematical logic has, in some respects, moved further ahead.
One might feel tempted to tease the humanities a little for being somewhat negligent here: the items listed above under mathematical foundations mostly deal with non-contradictory structures.
Yet if scholarship is to be a form of inquiry into human intellectual curiosity, then we must also be able to deal with entities that include contradictions.
Ideally, the humanities would study such things in a systematic way, but that does not seem to have been done extensively. Ironically—or perhaps naturally—it is mathematics that has taken the lead in systematically studying them.
For example:
- Paraconsistent logic (矛盾許容論理, paraconsistent logic)
- Dialetheism (矛盾真理論, dialetheism)
These are attempts to study systems that do not collapse even in the presence of contradictions.
The stability of a system is important. Humans are full of contradictions, and societies are full of contradictions, yet they somehow manage to hang together and function (at least sometimes).
- Paraconsistent logic studies systems that do not explode even when contradictions like “P and not-P” occur.
- Dialetheism explores the idea that there can be true contradictions, and studies systems where “thesis and antithesis” can both hold without the system breaking down.
Logos, Pathos, and Mythos; Philosophers (Logos) and Sophists (Rhetoric);
Apollo and Dionysus (Bacchus); Order (Monophony and Symphony) and Carnival (Polyphony)
Mathematics has succeeded in systematizing non-contradictory structures, and furthermore, it now studies the systematization of frameworks that can even accommodate contradiction.
In recent times, some areas of the social sciences—such as certain parts of economics and sociology—have also been increasingly mathematized or “scientized.”
However, even if we do mathematics, symbols and logical notation alone are not enough. Mathematics by itself is not currently in a position to study every domain of the world from every perspective.
There are things that mathematics cannot adequately express. In broad terms, there are non-mathematical dimensions, such as:
- natural language,
- the humanistic,
- meaning,
- value,
- emotion,
- will or motivation,
- sensory experience, and so on.
If we caricature a bit:
- Mathematics is logos-like, classically philosophical, Apollonian, and oriented toward order (monophony, symphony).
- In contrast, the non-mathematical side is pathos (from the same root as “passion”), mythos (story), sophistic rhetoric, Dionysian/Bacchic, and carnivalesque (polyphony).
Strictly speaking, the humanities ought to treat both contradiction and non-contradiction, just as mathematics does. But to put it somewhat cynically, the humanities sometimes look as though they have neglected the systematic study of non-contradictory structures.
This is not to disparage or devalue the humanities. On the contrary, I am speaking here from the conviction that the humanities are critically important.
What Is Needed for Truly Comprehensive Scholarship?
Purely mathematical approaches are not enough.
The world is not made solely of “mathematical stuff,” and a purely mathematical perspective cannot exhaustively describe everything. Therefore we also need research into non-mathematical phenomena, and we need scholarship with a wider horizon that includes mathematics as one part.
Fortunately, mathematics itself is being vigorously developed.
Logic has been refined by mathematicians and has become a full-fledged subfield of mathematics.
Within mathematics, the theory of relations is not left as something vague or fuzzy: relations are defined with precision. Moreover:
- We can use the concept of relation to construct mathematics, structures, and logic;
- Conversely, we can use mathematics, structures, and logic to derive and analyze relations.
Mathematics and the mathematical sciences, the quantitative and formal disciplines, including logic, are being developed energetically, systematically, and with deepening foundations. They provide robust tools and expressive forms that clearly “exist” as powerful instruments of thought.
Given that such good tools already exist, we should use them and, in a sense, embed them inside the broader enterprise of scholarship (especially the humanities) to enrich and renovate it.
Just as fields that were once counted as “humanities,” such as economics, are now often seen as closer to the sciences, so too should other “humanities” fields incorporate mathematics.
In other words: people in the humanities should study mathematics—
or be required to study it—so that mathematics becomes a universal tool of thought available to everyone.
I have been working to spread contemporary philosophy, but if mathematics looks too difficult or inaccessible—if the barrier to starting and continuing mathematical study feels too high—then an indirect route is possible:
- First, study contemporary philosophy (structuralism and post-structuralism).
- If even that is hard, one can study Mahāyāna Buddhism instead.
By studying śūnyatā (空) and Madhyamaka (中観) in Mahāyāna Buddhism, one can acquire a way of thinking that overlaps significantly with contemporary philosophy and modern mathematical thinking.
To study contemporary philosophy or the Threefold Truth (三諦論) in Mahāyāna Buddhism (which systematizes emptiness theory and Madhyamaka) is already to adopt a structuralist perspective. And from the outset, these frameworks treat both non-contradictory and contradictory phenomena without discrimination, as part of their domain.
In Closing
Contemporary philosophy, I would say, is a “finished” discipline.
Here, by “finished” I mean “brought to a certain completion.”
Likewise, modern mathematical foundations are, in a sense, also completed. Yet I would hesitate to say they are entirely finished, because:
- they still have the power to produce new components and new cuts on mathematics,
- they are in fact producing them, and
- whenever we feel some area is missing—say, the need to study systems that tolerate contradiction—there is enough vitality in mathematics to open up such new lines of research.
Mahāyāna Buddhism, in a sense, has been “finished from the beginning.”
It was complete at the moment it came into being—or rather, when the Buddha attained awakening and began to teach, the core was already there, even if expressed in a somewhat rough form.
Later figures such as Nāgārjuna and Tiantai Zhiyi systematized and refined this core, but they were essentially organizing and polishing what was already present. In that sense, Buddhism can be seen as a discipline that was complete from the moment of its birth. (There is a strong line of thought that says Buddhism is not so much a “religion” as it is a “philosophy.” Of course, Buddhism is diverse, so any simple statement has its limits.)
In any case, societies—and the world at large—need basic literacies shared by everyone.
- Language is one such literacy;
- traditional reading, writing, and arithmetic are another;
- as in the terakoya of Edo-period Japan, a minimum of ethics and morals may also be necessary;
- without financial literacy, one cannot function in society;
- today we also need literacy in computers, communication, the internet, and ICT;
- literacy in English, as a global lingua franca, is likely helpful as well.
To this list, I would add literacy in contemporary philosophy—which could equally be literacy in modern mathematical foundations, or literacy in (Mahāyāna) Buddhism, because at a deep level these can serve similar roles.
By “literacy in contemporary philosophy,” I mainly mean literacy in structuralism and post-structuralism. The latter is relatively easy to pick up; structuralism is somewhat more difficult to approach, but relying solely on (naïve) realism has many disadvantages.
Whether in terakoya-style basic education, compulsory schooling, or in the general-education curricula of higher-education institutions, it would be better if structuralist ways of thinking were treated as something like a compulsory component—so that people at least have a chance to encounter them. That, I believe, would make our shared intellectual life richer.