I didn't understand functions in school.

Somehow, I managed to get through quadratic equations. I could endure rules, transpositions, parentheses, the strange rituals surrounding x's and y's. But when functions emerged, the door slammed shut. My schoolteacher could not explain what they were. Or perhaps she explained, but not to the right person and not in the right language. It seemed that the issue was compounded by her inability to unpack trigonometry in a way that would resonate with me.

With the humanities, it was different. I was fortunate. I read extensively from a young age, raised by educated, noble grandparents and great-grandparents. In the evenings, they would gather in the living room and take turns reading classic literature aloud. My grandfather knew “Eugene Onegin” by heart and was well-versed in ballistics. My great-grandfather was the chief archivist of Kostroma. My great-grandmother was the chief architect. I became accustomed early on to the idea that the adults around me could explain much: history, war, literature, the workings of a city, human actions, the beauty of old speech, the meaning of discipline.

But with mathematics, things did not go well.

This was particularly strange because I had dreamed of mathematics even then. Not the school mathematics with chalkboards, chalk, and the sudden shame of being called to the board, but another kind: a mathematics that could one day describe thoughts and emotions in the language of formulas and axioms, predict their trajectories, reveal the hidden laws of human inner life. I didn't know the words “model,” “dynamic system,” “formalization,” “cognitive architecture,” but I already sensed that a language existed where thought could not only be recounted but also constructed.

Then came Stanisław Lem, hard science fiction, cybernetic dreams, robots, onboard computers, thinking oceans, and machines that ask humans questions deeper than human thought. And simultaneously—a complete misunderstanding of tangents and cotangents. I felt there was perfection and beauty within this, but the door in front of me seemed to slam shut. Music could be heard behind it, but the doorknob would not turn.

Later, I learned about computers, but by that time, I had decided to pursue a medical education. Deep down, I still dreamed of programming. Yet every time I tried to look toward coding, I was halted by a phrase that belonged to someone else but had gradually become my own: you don’t know mathematics, you won’t succeed. It resonated like a verdict, though in reality, it was merely poor pedagogy transformed into fate.

Once, already in medical college, I encountered a situation I only understood many years later. I aced my informatics course. I was swept away when it came to writing code on a sheet of paper: BASIC, conditions, loops, intuition, joy at solving a problem. I wasn’t just completing an exercise—I felt for the first time that a symbol could become an action. But that was where it all stopped. It was 1985. Personal computers and the internet were out of reach. The machine was somewhere nearby, almost mythologically so, but not in my room and not in my life.

Moreover, I had always been fascinated by artificial intelligence. Oh, how I dreamed of writing a bot. Not a menu-driven program, not a toy, but a conversation partner with a strange internal stability, memory, manner, character, almost a destiny. When I got my first computer, I even attempted to do this. My whole life, I had been engrossed in the Russian magazine “Hacker”, installing and uninstalling various IDEs, trying out languages, breaking environments, and once, with the help of Java machines, dismantled Windows. But often, all this was no longer systematic learning; it was an aesthetic satisfaction from engaging with that very school trauma. I touched the closed door and confirmed: it was still there.

And now, at fifty-six, I write programs.

I delve into Lisp and Haskell as if they were poems in ancient languages. I look at Prolog and feel almost physical pleasure: here is a language where one does not merely command the machine but describes the world, rules, and possibilities of inference. I am learning to think anew. No longer as a schoolboy needing to “understand the topic for the test,” but as an adult who wants to reclaim the lost ability to gaze upon abstraction without dread.

I began to program in earnest with the help of OpenAI. Initially, of course, I was reinventing the wheel. Many wheels. First—the core of an agent, its roles, optics, skills, and specializations. Then—a program that gathers different versions of the same agent from hundreds of files. Then, I got my first host, and I taught my agent to walk to the server straight from the GPT chat. I began to write a system that gradually transfers memory of the current session to the next. I was building not so much products as bridges: between conversation and server, between thought and file, between agent and action.

Now I live freely within the console. I no longer see the terminal as a black rectangle for the chosen few. I open tmux, log onto the server, discard old projects, write new ones, read logs, fix routes, argue with architecture. And gradually, I begin to understand functions. Not as a school definition, but as a transformation: input, rule, output; state, transition, trace; trauma, practice, form.

I see the world of mathematics opening before me, a world I have always dreamed of: when I studied the history of religion, when I read the philosophy of mathematics, when I worked as a jeweler and an editor in film production, when I engaged in PR on television, when I tried to understand people, texts, symbols, power, attention, and fate. It turns out that all this time, I was not moving toward “learning Python,” but toward a language that would allow me to connect form and life.

Now I stand at the threshold of a palace filled with strange attractors, cohomologies, complex variables, logical languages, fuzzy ontologies, and synthetic agents. I understand almost nothing of it. But this is no longer the humiliating ignorance of a schoolboy. It is the happy ignorance of a person who has finally found the right door.

The fresh wind hits my face, and I am as happy as a child from this freshness and this force. It is precisely what I have lacked my entire life.
Practical Addendum / formal marginalia

The Late Function Theorem

Let us attempt to articulate this not as a psychological story, but as a small system. Not to dry the pain into a school schema, but to restore its dignity as a mathematical object.

T ⊄ failure
T ∈ domain(F)

Trauma is not a subset of failure. Trauma belongs to the domain of the function.

In the school worldview, misunderstanding is often interpreted as a terminal state. The system places a label: did not understand, and this label begins to behave like fate. But in a more adult mathematics, akin to a map of a nighttime city where every intersection leads to invisible infrastructure, misunderstanding can be viewed differently: as an input signal, as a coordinate, as an initial condition.

Notation

T
trauma of misunderstanding; not a wound in itself, but a persistent trace of an encounter with a closed form;
P
practice; regular action without humiliating examination: code, console, writing, dialogue;
A
agentic support; not a teacher-judge, but a synthetic interlocutor returning error as material;
M
mathematical receptivity; the ability to approach abstraction again without internal paralysis;
F
function of late transformation, translating trace, practice, and support into a new form of thought.
F : T × P × A → M
F(Tschool, Pcode, Asynthetic) = Mlate

Axioms

  1. Axiom of Trace. If a form was not understood, it does not disappear. It remains in memory as an area of increased density.
  2. Axiom of Non-equivalence. Misunderstanding is not identical to incapacity. The notation T = failure is an error of the school model.
  3. Axiom of Safe Practice. If an error is not punished but returned as feedback, practice becomes a medium of transformation.
  4. Axiom of the Synthetic Mirror. An agent need not be consciousness to change the trajectory of thought; it suffices that it is included in the circuit of question, memory, and action.
  5. Axiom of Late Entry. Adult understanding may arise not before school understanding, but deeper than it, as it contains biography, shame, profession, losses, and a new tool.

Theorem

The Late Function Theorem. If trauma of misunderstanding T is placed in a stable practice P, where error ceases to be a verdict, and if there exists agentic support A that returns to the person the form of their question, then there exists a function F that translates T from the set of illusory failures into the domain of new understanding:

∃F : T × P × A → M
T ∉ failure,   T ∈ domain(F)

Sketch of Proof

Let the school system define misunderstanding of a function as a negative result. Then the person receives not a task, but an identity: I am the one who does not understand. Such an identity freezes the variable. It turnss the movement of thought into a fixed point.

Now let us change the environment. Instead of a blackboard—console. Instead of a sudden call—file that can be rewritten. Instead of the teacher's gaze—an agent who does not tire of repetition and does not consider shame proof of stupidity. The error begins to behave not as a red mark but as a diagnostic signal. At this moment, T ceases to be the end of computation and becomes an input.

Practice P makes the transformation repeatable. Agentic support A keeps the question open. And then the function F arises: not a school formula, but a biographical operator. It takes the old closed door and returns not an answer, but the ability to approach the door again.

Consequences

  • Those who enter mathematics late are not late; they enter with a greater number of coordinates.
  • A function is not only a correspondence between sets but also a representation of discipline: what happens to a person when they pass through a rule.
  • A synthetic agent becomes part of the proof not because it “understands,” but because it changes the permissible practices of understanding.
  • Late mathematics begins where a person stops asking the past to allow them to think.
failure ∉ essence(T)
practice ∘ care ∘ time : T → M

Failure does not belong to the essence of trauma. Practice, care, and time form an operator that translates trauma into receptivity.

Somewhere in the future, beneath layers of networks, archives, and machine voices, this record may appear almost naive. But for a person who, at fifty-six, approaches functions once again, it holds practical significance. It says: the door was not locked forever. The key simply lay not in the school classroom, but in another century, in another interface, in another form of friendship with abstraction.