The first day, I was angry.
When OpenAI shared its collection of mathematical manuscripts, I felt the scale of it before I knew how to think about it. For four years, I have been carrying my own questions through research, code, revisions, and the effort of learning enough to continue. A project gathers part of your life as you work on it.
Today, I want to welcome what this progress could make possible. Reaching that position has also made me think carefully about the people who have spent twenty, thirty, or forty years inside mathematics—and about my responsibility to understand what arrives next.
The years inside a question become part of the judgment with which we receive an answer.
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The people who have lived with the question
Four years of my own work give me a reason to approach someone else's lifetime with humility.
I can imagine how deeply a proposed answer might unsettle a person who has spent decades studying a problem. That question may have shaped their reading, their teaching, their friendships, and their sense of what they could contribute. An unexpected result can ask them to reconsider something much larger than a line in a paper.
I want to leave room for that human adjustment. People deserve time to examine the mathematics and time to understand what the moment means to them.
Those years also carry something precious. A researcher who knows a problem intimately may recognize the assumption that needs attention, the difficult step hidden behind an elegant sentence, or the connection that gives the result a wider meaning. Their experience can help a new answer become reliable knowledge.
I began to feel more hopeful when I thought about that continuity. A person who has spent years approaching an obstacle could find a route through it. They could bring their understanding into another question, teach the result in a way that helps others see it, or discover a possibility its first author had overlooked.
I would like us to make room for that kind of beginning.
An answer becomes more valuable as we understand it
William Thurston's On proof and progress in mathematics helped me return to a question at the centre of this reflection: how does mathematical work advance human understanding?
A small example makes the distinction tangible. For the function f(x) = x², the derivative at x = 3 is 6. I can remember that answer. I can also learn why it matters:
(3 + h)² = 9 + 6h + h²
For a small change h, the change in the function is close to 6h. The remaining h² term tells me something about the approximation. The number now belongs to a relationship I can explain and use.
That example is elementary. The movement it illustrates matters at every level of learning: receive a result, work through its meaning, and build the understanding needed to carry it somewhere useful.
When a machine helps produce a difficult answer, I want to make that same movement. The result gives me a place to begin. Understanding grows through the work I do with it.
Taking the mathematics seriously is part of welcoming it
On 6 October 2026, OpenAI announced a collection of mathematical results produced by an internal frontier model, together with repository materials and Lean formalizations for many proofs. That is the release behind my article.
The repository's 7 October history entry, consulted on 13 October, reports 719 top-line results, with 300 formalized in Lean—approximately 42%. The same history records withdrawals, proof repairs, and clarified assumptions.
I read that history alongside the manuscripts. A result has a statement, a set of assumptions, and an argument to examine. A formalization has a precise scope, including how its formal statement corresponds to the mathematical question. An application brings another set of conditions to establish.
The counts above describe the repository's dated record. The linked history supplies the context for revisions and formalization coverage.
There is room for celebration in this careful work. Someone who finds a gap helps us understand where an argument needs repair. Someone who explains an assumption makes the next person's reading more productive. Someone who reproduces a result gives others firmer ground.
I want my appreciation for the release to include those people. Their attention helps determine what the rest of us can responsibly build.
My responsibility is to push my understanding further
This is where the reflection becomes personal again.
If I welcome these tools into my work, I also accept the effort that comes with using them well. I want to take the material seriously, study the parts that exceed my present understanding, and ask for help with enough precision that another person can answer.
That means being willing to say where I am in the learning process. It means returning to a definition, working through an example, and giving a difficult idea more time. It also means letting a sound result change the direction of an idea I care about.
For the next result I explore, I want to follow a concrete path:
- State the question in my own words. Identify the objects, assumptions, and conclusion.
- Work through a small example. Connect the symbols to something I can calculate or inspect.
- Follow the evidence. Read the argument, check the stated formalization scope where available, and keep track of the version.
- Compare fairly. Define what an added method should contribute and assess it against a simple reference.
- Explain what I learned. Make the result, the remaining questions, and the next step clear enough for someone else to examine.
That is a commitment I can carry into tomorrow's work. Curiosity gives me the desire to begin; responsibility asks me to stay with the question long enough to understand it better.
Where I want to go from here
My interest in fractals keeps bringing me back to the relationship between a pattern and the information it carries.
For the questions I want to bring into Book III with Aguilar, and in my preparation around NASA Space Apps, I can make that interest concrete. I want to begin with a defined observable, a modest dataset, and a geometric descriptor whose contribution can be measured.
Suppose a reference model uses a set of features, z. I would compare it with the same learning method using z plus a descriptor D, under matched evaluation periods, budgets, and metrics. The question is precise: what information does D add for this task?
This figure describes an experiment I want to design. Its outcome remains to be measured.
That comparison could help me understand a useful contribution, or show me which part of the idea needs another approach. Either outcome would give my next decision a better foundation. Aguilar's reading and perspective would remain his own contribution to that discussion.
This is the kind of progress I find exciting: a large new body of work can lead to a small, serious experiment that I am able to explain. From there, another person could reproduce it, question it, or carry it further.
A future we can learn to build together
I want us to enjoy what these machines make possible. There is pleasure in seeing a difficult relationship become clearer, in finding an unexpected connection, and in reaching a question that once felt beyond our resources.
A verified mathematical advance could give researchers a stronger starting point. An explanation could help a student enter a subject. A useful computational method could make a modest experiment feasible for an independent builder. Each possibility asks for its own work, and each gives me a reason to remain curious.
I also want that curiosity to become shared participation. Experienced mathematicians, students, developers, and people arriving from other disciplines can contribute different kinds of understanding. We can value those contributions, give credit carefully, and make it easier for the next person to join the conversation.
That is why access matters to me. A paper becomes more useful to more people when it is accompanied by clear explanations, reproducible examples, and conditions that let them keep learning. UNESCO's Recommendation on Open Science connects the sharing of knowledge with skills, infrastructure, cooperation, and equitable participation. I want to bring that concern into the way I share my own work.
Through SecuredMe, I want to help people protect what they entrust to digital systems. Through my atmospheric research, I want to investigate whether a better description can support a useful decision. Those ambitions give this discussion a purpose beyond the excitement of a new capability.
The future I hope for is one in which we help each other understand more, use our tools with care, and put what we learn to work on problems that matter to people and the planet.
Giving the years somewhere new to go
My ego needed time to catch up with my curiosity.
I can acknowledge that honestly while choosing what comes next. I want to welcome an answer that withstands examination, learn from the people who understand it, and push myself to comprehend what I can build on.
The years I have already spent are part of that effort. They have taught me patience, given me questions worth asking, and helped me recognize how much care a useful contribution requires.
I want us to carry that experience into the future together. We can enjoy the machine, appreciate the people around it, and take responsibility for the knowledge we pass on. There is hope in the thought that something we learn today might give another person the confidence to begin tomorrow.
Today, I am ready to welcome the answer—and to do the work of understanding what comes next.
What question would you return to if a new result opened a way forward, and what would help you understand it well enough to take the next step?
About the full illustrated article
This post develops the human reflection behind Learning to Welcome the Answer: AI assisted mathematics and the questions I want to carry forward, my article dated 13 October 2026. The complete English and French editions include nine chapters, a conclusion, a glossary, and thirty references.
📖 Read the complete illustrated article in English or French — both PDF editions are available together in the same Zenodo record. DOI: 10.5281/zenodo.23349937.
Author's note: I used Codex as a research partner for source comparison, evidence organization, consistency checks, drafting, and editorial preparation. The underlying article also drew on a research dossier I supplied from Gemini 3.1 Pro. I retain responsibility for the interpretation, conclusions, editorial decisions, and final text.
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