A New Dawn for Mathematics

By Daniel Litt

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The Velocity of Change

The trajectory of artificial intelligence in the mathematical domain has been staggering. Consider the timeline:

• Three years ago: AI struggled with basic arithmetic.
• One year ago: Internal models from DeepMind and OpenAI achieved performance comparable to $\text{IMO Gold Medalists}$.
• Today: These systems are independently cracking significant open problems.

While it is difficult to envision this acceleration continuing at the same pace for another year, I suspect it will. This shift necessitates a fundamental reimagining of the mathematical profession.

"I previously delivered a presentation titled The End of Mathematics. If you only saw the headline, you might assume I was predicting that AI would simply 'solve' the field."

In reality, that talk presented a pessimistic scenario: a world where superhuman AI exists, but our rigid institutional structures cause human comprehension—and perhaps abstract mathematical progress—to grind to a halt. While I believe this is the default path if we fail to adapt, I am optimistic that we can avoid it.

The Core Thesis

We can actually deepen human understanding even as the actual generation of mathematical results becomes less dependent on human effort.

Premises of this argument:

1. Strong Premise: AI systems that are robustly superhuman across all mathematical facets are imminent.
2. Weak Premise: The act of producing mathematical text is becoming increasingly decoupled from the act of mathematical understanding.

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Defining the Goal of Mathematics

There is currently no unified consensus on what the "point" of mathematics is. Different practitioners hold diverging views:

My own view is that we strive to create and comprehend high-quality mathematics. This definition is fluid and evolves as a community. Furthermore, our role extends beyond training PhDs; we are tasked with educating the broader public on the nature of mathematical thought.

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The Fallacy of the "Theorem"

Historically, we have operationalized our goals through the act of proving theorems. Whether in a thesis or a journal paper, the "Main Theorem" is the gold standard. However, proving theorems is not the actual goal of mathematics. If it were, the field would be trivial to automate.

Imagine a "smart monkey" or a basic script executing the following logic:

def automate_math(axioms_zfc):
    for conjecture in all_possible_strings_alphabetically():
        if prove_via_deduction(conjecture, axioms_zfc):
            print(f"Theorem Found: {conjecture}")

Using $\text{ZFC}$ (Zermelo-Fraenkel set theory with the Axiom of Choice), a machine could iteratively apply deduction rules without any grasp of meaning. Even solving "open problems" is just a matter of filtering the output of such a machine.

The "Super-Monkey" Thought Experiment

What if the machine is not a blind script, but a highly sophisticated entity?

• It writes elegant expositions.
• It understands what humans find interesting.
• It solves our most pressing mysteries and poses new, fundamental questions.

Even in this utopia, the machine's output does not automatically equate to human understanding.

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A Vision for the Future

We are standing at the precipice of a mathematical explosion. If we value human comprehension, the role of the human mathematician becomes more critical, not less.

What to Preserve vs. What to Abandon

We must distinguish between our values and our institutions. We should not try to save the "shape" of the academy, but rather the essence of the practice.

Values to protect:

• Learning seminars
• Spontaneous, idea-sparking conversations
• The "open door" policy between students and professors
• The collective process of resolving confusion

Preserving the current equilibrium of academic publishing and tenure is an absurd goal given that AI can now produce high-quality results for a few dollars.

The Strategy of Adaptation

Some suggest we should pivot to reward skills where AI currently fails (e.g., theory-building or high-level exposition). I believe this is a mistake.

The speed of model improvement far outstrips the speed of institutional change. If we define our value by "doing what the AI can't," we will be constantly moving the goalposts until there is no ground left to stand on.

Instead, we must accept the shift and focus on the human element of understanding.