What Would Count as Proof?

I have a bachelor’s degree and a master’s in mathematics. I can list the qualifications without hesitation.

I have more trouble saying that I am a mathematician.

Suppose I am reading a proof and cannot follow a step. I go back to a definition, try an example, discover something I had forgotten or never properly understood. There is work to do, and I know roughly how to begin.

But before I begin, another thought arrives. Someone with my education should already know this.

Now I am thinking about the degrees, the years, and what I should have become by the end of them. The missing step begins to look like the place where the whole account of me comes apart. Nobody needs to accuse me. Nobody even needs to be there.

Say I do the work. I return to the definition, find the assumption I missed, and the step makes sense. The question that stopped me has an answer.

Would that persuade me to use the name?

I can already imagine finding the next thing I do not understand. There is enough mathematics for me to keep doing this indefinitely. I know enough of the subject to supply questions that would defeat me, and the more I learn, the more places I can find to look.

That is the part I keep passing over when I tell myself I need to know more. I could learn the thing that embarrassed me and still leave the judgment untouched. Whatever I understood would join the things I should have known already. Whatever remained would become the reason I was still not ready.

Then I compare myself with someone whose work I admire. They see a connection I would have missed, or move easily through something I find difficult. There is something there I want to learn. Almost immediately, I am using the difference to decide whether what I know counts at all.

Would I do that to another person? If they could explain the argument, would I refuse to count their understanding because they had needed to return to a definition? Would I expect them to be as good as the strongest mathematician I could find before allowing them the word?

I would have difficulty defending that standard. I have been much less reluctant to live under it.

So I need to ask what I have been trying to obtain. Another degree would give me more study. A result could show that I had understood something well enough to contribute. Neither would prevent someone, including me, from finding the next question I could not answer.

That seems to be the promise I keep hearing inside the name. I want to say it without risking that a future moment of confusion will make it sound dishonest.

I still want to understand the missing mathematics. But I would like to be able to call myself a mathematician without turning the next thing I have to learn into evidence that I lied.