Recently, I have been researching the history of mathematical symbols.
At first, it was just a small question: why does mathematics need so many symbols? Why couldn’t we simply keep using natural language to describe quantities and relationships? Why did we have to invent notations like +, -, and =?
But the further I followed the question, the more I realized that what symbols truly changed was not merely the “writing” of mathematics, but our ability to express mathematical relationships. Before the equals sign, people could of course understand that two quantities were equal, but they had to rely on language to express that relationship. Once = appeared, “equality” was compressed into a simple form that could be used repeatedly, combined with other symbols, and incorporated into increasingly complex reasoning. Mathematics was gradually acquiring a formal language of its own through this process.
And this led me to a strange question:
If mathematics is a language, then who is it communicating with?
What is even stranger is that this language was created by humans, yet it can describe with remarkable precision a world that was not created by us.
And so, a somewhat absurd idea occurred to me:
Mathematics is the language with which humans converse with God.
I have no idea whether this idea is actually true. So rather than rushing to a conclusion, I would rather treat it as a hypothesis for now and follow it through, to see what it can explain and where it runs into problems.
A Strange Idea That Suddenly Appeared#
Looking back at the history of mathematical symbols, it is easy to understand the development of mathematics as a continuous expansion of our ability to calculate: numbers became more sophisticated, algorithms became more complex, and humans became capable of solving more and more problems.
But the history of symbols offers another way of looking at it.
Humans had been able to calculate for a very long time, yet it took much longer to gradually develop a formal language capable of directly expressing relationships. A person could know that two quantities were equal without having a simple symbol with which to fix that relationship; they could know that one number was greater than another without having a standardized notation that allowed the relationship to become part of a more complex mathematical expression.
Once symbols appeared, something fundamental changed. When a relationship was fixed into a particular form, it was no longer merely an understanding in the mind or a sentence in natural language. It became something that could be manipulated, combined, and reasoned with.
So from this perspective, what is really worth paying attention to in the history of mathematical symbols may not be “who invented which symbol,” but the gradual emergence of a new human ability:
Relationships that could previously only be understood became forms that could be manipulated.
And once we begin to see mathematics as a formal language of this kind, a larger question naturally appears.
Language is usually used for communication.
So who is on the other end of mathematics?
Language Is Not Only Expression, but Communication#
If we understand “language” in a broader sense, it does not necessarily exist only between human beings.
Programming languages are an interesting example. A programmer encodes their intentions into a form that a computer can process; the computer runs the program according to its rules and returns a result; the programmer observes the result, modifies the program, and runs it again. Real software development is therefore not simply “human → computer,” but a continuous cycle:
Human → Program → Computer → Result → Human.
What matters here is not merely the “command” itself, but the fact that both sides can exchange information through a set of commonly followed rules.
The emergence of AI makes this relationship even more apparent. Humans no longer always need to translate their intentions in advance into a form that a machine can execute. We can increasingly communicate with AI directly through natural language, then continue questioning, correcting, or even restructuring our questions based on the results we receive. Language is therefore not merely a tool for transmitting information; it can also become a bridge for continuous interaction between two systems.
If we understand language as a rule system that allows information to move back and forth in this way, mathematics suddenly becomes very interesting.
Because the “other side” that mathematics faces is not a computer made by humans.
It is the world itself.
So Who Is Mathematics Communicating With?#
The symbols of mathematics were, of course, created by humans.
1, +, =, π, ∫, and the rules of notation built around them all belong to human civilization. We could even redesign the entire system of symbols and use completely different notations to express the same mathematical relationships.
But the things those symbols point to do not necessarily belong to us.
For example:
We created the letters, numbers, and equals sign, but we did not create the motion of physical objects itself. What we are really trying to express is a relationship in nature that exists regardless of which symbols we choose to represent it.
This reveals something very unusual about mathematics: the form comes from humans, while what the form attempts to express does not.
The difference becomes even clearer if we bring art into the comparison.
Art also requires form, and it also uses symbols created by humans, but its forms are more directly grounded in human experience and perception. Painting depends on vision, music on hearing, literature on language; an artist organizes their experiences, feelings, or understanding into some form, and then another person’s perception allows that form to enter their own experiential world.
It can be roughly represented as:
Human experience → Form → Human perception
The natural sciences, however, confront a different kind of object: an objective world that exists independently of human consciousness. Of course, we can only come to know it through perception, instruments, and experiments, but the way we know the world is not the same thing as the world itself.
And mathematics happens to stand right between these two sides.
Its symbols, forms, and modes of expression come from humans, yet they can enter the natural sciences and be used to describe a world that was not created by us.
This may be precisely what is strangest about mathematics.
Why Are Art and Mathematics Different?#
The difference between art and the natural sciences cannot simply be reduced to one being “subjective” and the other “objective.” Art can certainly represent the real world, and the natural sciences cannot escape human perception and cognition. What truly differs is their respective points of departure.
The form of art ultimately has to return to human perception in order to complete the act of expression. A painting, a melody, or a piece of writing can function as an expression because humans possess the corresponding perceptual and cognitive abilities. Art can depict a real tree, but it can also depict dreams, imagination, or things that have never existed in reality, because what it ultimately faces is the human world of experience.
The natural sciences, by contrast, take as their object a world that does not depend on human consciousness.
Here, two questions that are often mixed together need to be distinguished:
How do we come to know the world?
and:
Does the world itself depend on our knowing it?
The first question of course cannot be separated from human beings. Whether we observe a star with the naked eye or obtain data through telescopes, microscopes, and particle detectors, we ultimately have to transform that information into concepts that human beings can understand before it can become what we call knowledge.
But this does not imply that the world depends on our knowledge of it. A star does not temporarily stop nuclear fusion simply because humans have not yet observed it; a chemical reaction does not wait for us to develop a theory before it begins to occur.
So, human perception is the gateway through which we come to know the world, but it is not a condition for the world’s existence.
And mathematics happens to appear at this intersection: it takes human symbols and cognition as its form, yet it can point toward an objective world that exists independently of us.
If art is more concerned with organizing forms within the human world of experience, while the natural sciences are concerned with understanding a world independent of us, then mathematics seems to possess both characteristics.
At this point, the statement “mathematics is a language” begins to take on another meaning.
It is not merely a language through which humans transmit knowledge to one another; it may also be a language through which humans establish connections with the world.
“Knowing the World” and “The World Itself”#
Once we separate “knowing the world” from “the world itself,” the previous line of thought can move one step further.
We can never step completely outside our own perception and cognition and observe the world from beyond it. Whether we see a planet with our own eyes or obtain a set of measurements through an instrument, we ultimately have to pass that information through human perception, understanding, and organization before it can become concepts, models, and theories.
So the process by which science comes to know the world can be roughly represented as:
Objective world → Perception and experiment → Concepts and models → Theory → New predictions
Notice that the arrows here describe a process of knowing, not a relationship of existence.
What is truly interesting is that once mathematics enters this process, it does more than help us organize things we have already observed.
A mathematical model can organize our understanding of the world into a precise form, and that form can then generate new predictions; those predictions can in turn be taken back to reality and tested through experiment.
A cycle emerges:
World → Observation → Mathematical model → Prediction → World.
If a prediction fails, we have to go back and examine our understanding; if a prediction receives support from reality, we can use it as the basis for asking new questions.
Mathematics is therefore not merely something left behind after we come to know the world. It also participates in the process through which we come to know it.
And this makes me wonder whether “description” is really the most accurate word.
The Strange “Dual Identity” of Mathematics#
From this perspective, mathematics seems to possess a rather strange dual identity.
As symbols and forms, it clearly belongs to human civilization. We create notations, define modes of expression, establish definitions and axioms, and organize these forms through reasoning; yet as a tool of the natural sciences, mathematics can express relationships in the objective world and generate predictions that can be tested against reality.
So mathematics belongs, on the one hand, to the human world of cognition and formal structures, while on the other hand it can continually reach toward the structures of the objective world.
This brings us to a classic question:
If mathematics is merely a set of forms created by humans, why can it describe so effectively a world that was not created by humans?
Of course, we could say that humans continually seek mathematical tools that are useful for describing reality, while structures that appear to have no practical application remain within pure mathematics. There is nothing wrong with this explanation, but it still leaves us with a more puzzling question: why are there so many abstract structures that can later establish such stable and profound connections with the real world?
Some mathematical theories initially appeared to have no clear practical application, yet later became important tools for describing natural phenomena.
I do not know what this ultimately means.
But at least by this point, I have begun to understand why I think of mathematics as a kind of “language.”
It seems to do more than record what we already know. It allows us to organize our understanding of the world into a sufficiently precise form, and then use that form to ask the world new questions.
Mathematics as a Language for Communicating with the Physical World#
If we understand “communication” as an ongoing process of information moving back and forth between two systems, then mathematics and the natural sciences present a rather interesting structure:
Human → Mathematics → World → Observation → Human.
We propose a model, and the model generates predictions; reality produces results, and we modify the model based on those results; the new model then generates new predictions.
Of course, the world does not actually “understand” our mathematics. It simply continues to operate according to its own laws. What is truly remarkable is that the forms created by humans can nevertheless establish stable connections with those laws, and that these connections must repeatedly withstand the test of reality.
So rather than saying that mathematics is “the language for describing the world,” I now prefer to think of it as:
Mathematics is the language with which humans formally communicate with the physical world.
This kind of “communication” does not mean commanding the world to change anything. It means that humans use mathematics to express their understanding of relationships in the world, and then use observation and experiment to test whether that understanding holds. We can modify our equations, but we cannot modify the laws of the universe; what we can continually change is our own expression and understanding.
Perhaps this is what makes this kind of “communication” so unusual: language is created by humans, but the world to which it points is not determined by human will.
Where Does the Idea “Mathematics Is the Language with Which Humans Converse with God” Come From?#
At this point, the strange sentence with which we began can finally reappear.
If mathematics can serve as a language through which humans formally communicate with the objective world, then why call that world “God”?
Because in this hypothesis, I am only borrowing the word temporarily, using it as a metaphor in a thought experiment.
Here, “God” is not a religious entity whose existence has been established. It refers instead to a world that exists independently of human consciousness, does not change according to our wishes, and yet continually responds to our mathematical models through reality.
We propose an equation, and it does not directly answer “correct” or “incorrect”; but an experiment can tell us whether the relationship expressed by that equation corresponds to reality. We predict a phenomenon that has not yet been observed, and if it is later observed, we seem to have received an answer; if it is not observed, that too is an answer. It simply tells us:
Your understanding of me is not accurate enough.
If we call this relationship a “conversation,” then mathematics is the language with which humans formally converse with the physical world; and if we temporarily call that world, which exists independently of us and continually responds to our questions, “God,” then:
Mathematics is the language with which humans converse with God.
This is, of course, only a metaphor. It is not a proof of the existence or nonexistence of God, nor is it a religious proposition.
But a good metaphor can sometimes make something familiar suddenly appear strange. We use mathematics every day, yet we rarely stop to ask: Why can these symbols created by humans establish such stable connections with a world that was not created by humans?
But the Hypothesis Immediately Runs into a Problem#
If the story ended here, it would be almost too perfect.
Because mathematics immediately raises a difficult question:
What about pure mathematics?
Number theory, set theory, abstract algebra, topology, and a vast amount of highly abstract mathematics do not necessarily begin by describing any specific physical object. Mathematicians can study an abstract structure simply because it has meaning within mathematics itself, rather than because it immediately explains some experiment.
So who, exactly, are they “conversing” with?
If mathematics is merely the language between humans and the physical world, where should pure mathematics fit?
At this point, the original hypothesis develops a crack.
Perhaps mathematics is not merely about describing the world. Perhaps it also contains a relatively independent abstract domain in which humans can explore relationships between structures, and those structures may sometimes establish connections with the real world while at other times showing no apparent connection at all.
If so, “mathematics is the language with which humans converse with God” can explain only part of mathematics. It cannot serve as a complete definition of what mathematics is.
And beyond that, we encounter an even older question:
Are mathematical structures created by humans, or discovered by humans?
I still do not have an answer.
But I increasingly feel that this crack is not a failure of the hypothesis. In fact, it may be the most valuable part of it. If a hypothesis can only continually search for evidence supporting itself, while never allowing itself to encounter counterexamples, it can very easily become a belief rather than a question genuinely worth pursuing.
So perhaps what is really worth keeping is not the conclusion that “mathematics is the language with which humans converse with God,” but the fact that it forces us to reconsider a more fundamental question:
What role does mathematics play between humans and the world?
A Question Without an Answer—For Now#
So in the end, I still do not know what mathematics really is.
“Mathematics is the language with which humans converse with God” may be an illuminating way of understanding it, or perhaps ultimately just a beautiful metaphor. It can help us understand why mathematics can become such an important language of the natural sciences, but it is clearly not enough to explain mathematics in its entirety, especially the pure mathematical structures that do not directly point toward the physical world.
Yet after following this path all the way from the study of mathematical symbols, I seem to have become increasingly clear about something: what truly interests me is no longer any particular symbol itself.
What truly interests me is:
Why can abstract forms created by humans reach the world so effectively—a world that exists independently of human consciousness?
We created numbers, symbols, and equations. We also created definitions, axioms, and proofs. Yet once these forms are organized together, they can help us ask questions about the world and allow those questions to be tested against reality itself. We reason on paper while the world continues to operate in reality; we make predictions and the world produces results; when those results do not match our expectations, we have to revise our understanding.
It really does resemble a long conversation.
Except that the other side of the conversation never accommodates us. It does not accept a theory simply because it is beautiful, nor does it change its laws because we believe in a particular answer. What we can change is only our language and our understanding of the world.
Perhaps that is why I thought of the word “God.”
If it is merely a metaphor, then perhaps it does not point to some deity whose existence can be proved or disproved, but to that which always stands beyond us, exists independently of our will, and yet continually responds to our questions through reality.
Then:
Mathematics is the language with which humans converse with God.
As for how correct this statement really is, I do not know yet. Perhaps it breaks down in the face of pure mathematics. Perhaps mathematics has another side that we have not yet seen. Perhaps we will eventually need an entirely different framework to understand it.
But at least for now, this question fascinates me more than the question of “why mathematics can calculate an answer.”
Because an answer may be merely a result. What is truly astonishing is that humans can create an abstract language, write down their understanding of relationships and structures, and then repeatedly take those forms back to the real world.
For thousands of years, the world has never changed its language to accommodate us.
It is we who have always changed.
And this conversation seems far from over.