Christopher Thomas Ryan recently shared with me an essay titled Reverse Perspective: On Subjectivity in Mathematics. His central metaphor stayed with me. Much of mathematics education is presented in linear perspective: a problem points toward one correct answer, one proof, or one canonical method. Reverse-perspective mathematics opens in the other direction. A mathematical idea becomes a starting point from which many questions, representations, models, and interpretations can emerge.
Ryan connects this distinction to the arrival of powerful AI systems. If AI becomes extraordinarily good at problems with objectively verifiable answers, perhaps people can spend more time using mathematics creatively: posing problems, describing the world, and developing mathematical taste.
I found this picture both hopeful and useful. But my experience of mathematical research led me to modify it in one important way.
The proof is objective. The path is not.
A finished proof is objective in an important sense. Once the assumptions and definitions are fixed, the argument can be checked. But the process of discovering that proof feels very different.
In research, we rarely formulate the right question and then simply solve it. The question itself changes as we try to solve it. A failed proof attempt reveals that an assumption matters more than expected. A counterexample exposes a distinction we had not noticed. A construction repeatedly produces a quantity, and eventually we realize that this quantity - not the one we originally cared about - is the real structure of the problem.
The proof may be objective. The path to it is not.
Research therefore looks less like a one-way map from question to solution and more like a cycle:
Convergence and divergence are not separate phases. They continually generate one another.
Mathematical taste is formed through struggle
There is a great deal of subjective judgment inside problem solving. We often think: this bound looks unnatural; this assumption should not be necessary; this proof is correct but does not explain the phenomenon; there should be a more canonical representation; these two objects feel as if they should be manifestations of the same structure.
These are not formal deductions. They are expressions of mathematical taste.
But taste does not exist independently of problem solving. It is formed through the work itself: trying representations that fail, encountering obstructions, noticing which quantities repeatedly appear, seeing which assumptions resist removal, and learning which difficulties are merely technical and which are structural.
Even after a theorem has been proved, correctness is not the only criterion by which we judge the result. Two proofs can both be valid, yet one may be more satisfying because it reveals the relevant structure, identifies the right invariant, generalizes naturally, or makes the conclusion feel inevitable. Those judgments are also part of mathematics.
The discarded paths matter
This is where the role of AI becomes especially interesting.
I am uneasy with a clean division of labor in which humans pose interesting questions and AI solves them. Our ability to ask good questions may come precisely from struggling with solutions. If an AI returns a correct proof immediately, I may be able to verify every line while understanding less than I would have after trying and failing myself.
Failed attempts teach us why a natural approach does not work, which assumption is doing the real work, why a particular construction is necessary, and why a theorem has exactly the form that it does. A proof is not only a certificate of truth. For a researcher, searching for a proof is also a way of interrogating the mathematical object.
The discarded paths matter.
AI as a higher-bandwidth mathematical collaborator
The important question, then, is not simply whether AI can solve mathematical problems better than we can. It is whether AI keeps us intellectually engaged with the process through which mathematical understanding is formed.
If AI becomes a black box that transforms conjectures into proofs, something important may be lost. The loss would not come from giving up the manual production of proofs. It would come from losing contact with the failed attempts, alternative representations, and structural questions through which understanding develops.
There is, however, another possibility. AI may let us interact with mathematical space at much higher bandwidth. We may be able to test more ideas, encounter more failed approaches, generate counterexamples, compare representations, and ask "why?" much more aggressively. Used in this way, AI could deepen mathematical understanding rather than diminish it.
This suggests a criterion for mathematical AI that goes beyond correctness: does the system help us see why an argument works, why nearby arguments fail, and which new questions become visible as a result?
Reverse perspective is already inside research
Mature mathematical research has perhaps never been purely linear perspective or purely reverse perspective. Solving a problem changes our understanding. That understanding changes the questions we ask. Those questions send us back into exploration.
In that sense, reverse perspective is already present inside problem solving itself.
AI does not have to remove the struggle through which mathematical taste is formed. It could instead make that struggle richer, faster, and more expansive - provided that we design our interaction with AI around exploration and understanding, not only around the production of correct proofs.