What Would Count as Proof?

I have a bachelor’s degree and a master’s in mathematics. Before those, there was the high-school diploma. I can list the qualifications without hesitation.

I have more trouble saying that I am a mathematician.

There are too many things I still do not understand. Subjects I could study for years. Arguments I cannot yet follow. Questions for which I would have to begin much earlier than I feel someone with my education should need to begin. The qualifications do not disappear when I think about those things. They simply stop feeling like an answer.

I know I studied mathematics. I know I earned the degrees. What I cannot quite settle is how much knowledge the word mathematician asks me to possess before I can use it without feeling that I have overstated something.

The same hesitation follows me into AI, software engineering, and electrical engineering. I have evidence that I can do the work. Between me and myself, the doubt remains.

Nobody needs to accuse me. Nobody even needs to be there.

So what would I need to have? Another qualification? A particular result? The ability to answer a question I cannot answer now?

What would count as proof, and would I accept it if I had it?

What I can say the diploma proves#

Perhaps part of the difficulty is that I do not connect my diplomas very strongly with the knowledge I have.

I do not mean that the degrees were meaningless. They took study and work. They record something I completed. But when I ask myself whether I understand mathematics, I am asking about what I can do with an idea now. Can I explain why an argument holds? Can I reconstruct the reasoning? Can I recognize when an assumption matters instead of merely remembering that it was on the page?

A qualification cannot answer each of those questions for me.

There is a reasonable concern here, and I do not want reassurance to erase it. Passing an assessment and understanding a subject deeply are not identical. Knowing a result and knowing how to arrive at it are different achievements. There are gaps worth finding, including gaps in things I have already studied. If a piece of work depends on one of them, the degree does not do the missing work on my behalf.

But there is another step I seem to take almost immediately.

I find something I do not understand, and the question becomes whether I deserve the name at all.

That is much more than the discovery tells me. The gap might be serious. It might limit what I can responsibly attempt or explain. It does not, by itself, tell me what to make of everything I do understand. Yet the missing knowledge feels more revealing than the knowledge I have. It seems to show the truth about me, while the qualifications remain something I would have to explain.

I can be cautious about what a diploma proves without treating it as though it proves nothing.

I understand that sentence more easily than I feel it.

How much does the name promise?#

Saying that I studied mathematics describes something that happened. Saying that I am a mathematician feels like making a claim about whatever might happen next.

Someone could ask me something.

Even alone, I can imagine the question I would not know how to answer. I do not need an actual person to ask it. I already know enough about the size of the subject to supply questions that would defeat me.

Perhaps this is where the name becomes difficult. I hear more inside it than I could possibly demonstrate in a sentence. It seems to include familiarity with subjects I have not studied, ease with ideas I still find difficult, and a kind of intellectual readiness I do not consistently feel.

What exactly am I claiming when I say it?

That I have an education in the field? That I work with its ideas? That I can contribute something? Or that I should be able to meet any part of it without being lost?

These claims ask for different evidence. If I leave them mixed together, I can meet one and still feel dishonest because I have not met another.

I do not need to settle a universal definition of who gets to be called a mathematician. People use the word in different ways. But I need to understand what I require of myself when I hear it. Otherwise, I am trying to satisfy a condition I have never stated clearly enough to examine.

The people I compare myself with#

I compare myself with people at the top of the fields I care about.

There is much to learn from doing that. Their work makes possibilities visible. An argument can show me a level of understanding I want to develop. Something they make can change what I think is worth attempting. I do not want to stop looking closely at excellent work just because the comparison can be uncomfortable.

But being inspired by someone’s understanding and using it to decide whether my own understanding counts are different uses of the same encounter.

If someone knows much more mathematics than I do, that difference may be real. I do not need to protect myself by pretending otherwise. The question is what follows from it. Does their greater ability tell me where I could learn, or does it mean I am pretending to belong to the field?

Somewhere in the comparison, good enough to call myself this becomes difficult to separate from as good as the people I most admire.

Then even substantial knowledge can feel insufficient. I can acknowledge that I know something and immediately place it beside someone who knows it more deeply. The comparison is accurate in one respect and still unfair in what I ask it to decide. Another person’s excellence cannot tell me, on its own, whether I am misrepresenting my own ability.

There is also a question I find uncomfortable: would I apply the same requirement to someone else?

Would I hear about their education, look at what they can do, and then refuse to count it because a much stronger person exists? Would I expect them to reach the top of a field before they were allowed to say they worked in it?

I need to answer that honestly. If my standard changes when the person is me, calling it rigor does not settle whether it is fair.

The feeling survives a change of subject#

Mathematics gives this difficulty its clearest form for me, but changing fields does not remove it.

I feel it in AI, in software engineering, and in electrical engineering. There is evidence that I can do these things, and there is still a private reluctance to let that evidence become a description of me.

The evidence has limits, of course. Being able to do one kind of work does not establish competence in every other kind. A successful implementation does not answer every theoretical question. Understanding a method does not mean knowing how to build every system that might use it. I would be making a mistake if I treated one success as permission to claim everything nearby.

But I can make the opposite mistake too. I can ask a particular achievement to establish so much that it is guaranteed to seem insufficient.

Then I am left with an odd arrangement. I can say that I did the work. I can accept that the evidence exists. But saying what that suggests about my ability feels like a step I am not entitled to take.

Perhaps I distrust that step because I do not want to become someone whose confidence travels much farther than his understanding. I care about the difference. I want the words I use about myself to be accurate.

Accuracy should also make room for what is there.

If I acknowledge only my limitations, I have not necessarily described myself more honestly. I have described one part in detail and left the other part out.

The gap deserves attention. What else does it deserve?#

I do not want to solve this by declaring every doubt an imposter feeling. Sometimes I am unprepared. Sometimes there is something I should understand before I continue. Feeling better about myself would not repair that.

What I want is to be more exact about the demand a gap makes.

Suppose I cannot explain a step in a proof. I may need to return to a definition, work through an example, or discover an assumption I missed. There is a question I can pursue. I can eventually check whether I understand the step better.

If I turn the same difficulty into “Am I really a mathematician?”, the task changes. Understanding the step may no longer be enough. There will be another proof, another subject, another person whose knowledge makes mine look small. I have attached a much larger uncertainty to something that initially asked me to learn a particular thing.

That example is the distinction I want to keep in view. A real limitation can require work without requiring a verdict on everything I have become.

The struggle is that knowing this does not automatically change how the limitation feels. I can explain why the larger conclusion does not follow and still find it convincing when it concerns me. The doubt persists beside the explanation.

This is why another person listing my qualifications back to me would not quite answer the problem. I have not forgotten them. I am having trouble deciding what I am allowed to conclude from them.

Evidence I could actually use#

Perhaps the question needs to become smaller before I can answer it honestly.

What am I claiming I can do?

For a particular mathematical argument, I can ask whether I understand the definitions, whether the steps follow, and whether I can account for the assumptions. Someone else can examine the reasoning and find what I missed. For a piece of software, I can ask what it is supposed to do, how it behaves, and what has actually been checked. The evidence will be different because the claims are different.

These questions do not ask me to feel impressive. They ask me to make an account of the work that can be examined and corrected.

They also leave room for an honest limit. I can know enough for one task and need help with another. I can have a substantial education and still need to study something fundamental. I can be capable of learning a missing idea without pretending I already know it.

None of that guarantees that I will feel comfortable with the name. But it gives me something more useful than asking whether I am enough for an entire field at once.

There are three questions here that I keep bringing too close together:

What can I do?

What do I claim I can do?

Do I feel like the person those answers describe?

The first two can be made precise. The third may take longer. I do not want to force agreement by inflating what I know. I also do not want the feeling to have the final word about evidence it has not answered.

What would I let count?#

I still hesitate.

Writing this does not give me the missing mathematics. It does not make me as good as the people I compare myself with, and it does not remove the distance between being able to do something and feeling entitled to say so.

But it makes me less certain that another qualification would settle the question. If I cannot say what evidence I would accept, I cannot reasonably expect the next achievement to persuade me.

I want to keep studying. There are things I do not understand because I have not learned them, things I need to revisit, and things difficult enough that understanding them would be meaningful work. I want to give those questions the attention they deserve.

I also want to be able to recognize what I have already learned without feeling that recognition is a promise never to be confused again.

My degrees cannot answer every question someone could ask me. The work I can do cannot make me capable of every task. I can accept those limits. What I am trying to understand is why the limits so often feel like the whole description.

For now, I can say something smaller than “I have finally proved myself.”

I studied mathematics. I earned the degrees. There are things I understand and things I can do because of that work. There is also a great deal I do not know.

I am trying to let the last sentence stand without making it erase the others.

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