Redefining Mathematical Value: Beyond the Proof

Originally a guest contribution by Grant Sanderson on Terence Tao's blog.

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The Proxy Problem in Mathematics

For a long time, the mathematical community has operated under a specific set of assumptions. The ultimate goal of the field is to expand human understanding, but because "understanding" is difficult to quantify, we have used certain shortcuts.

"Solving problems and generating proofs have always served as proxies for the true goal of mathematicians, which is to further human understanding."

However, we are entering an era where $\text{Proof} \neq \text{Understanding}$. When machines can generate a formal proof without any accompanying human insight, the value of the proof as a proxy for understanding collapses.

The Crisis of Perception

If the world sees that AI can produce proofs, and the academic community continues to reward only proofs, the public will naturally conclude that mathematicians are obsolete. To prevent this, the community must only value the result $\rightarrow$ celebrate the process of insight.

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A Proposal for "Motivated Explanations"

I suggest that we formally recognize and reward "motivated explanations." This is not merely "popularization" for the masses; rather, it is any work—regardless of how technical—that answers the fundamental question:

"How would one actually think of this idea?"

Goals for the Community

• Establish a clear definition of a "motivated explanation."
• Grant academic credit for these explanations equivalent to that given to new proofs.
• Shift the internal reward structure to reflect the true value of human intuition.

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Proof vs. Motivated Explanation

To understand this shift, we must contrast the two approaches. While a proof is a rigid chain of logic, a motivated explanation is a narrative of discovery.

Comparison Matrix

The Logic of Discovery

In a proof, you cannot start with a mistake. In a motivated explanation, starting with a "simple-but-wrong" idea is often the most effective way to teach. This is what Michael Nielsen calls "discovery fiction."

The "Discovery Fiction" Algorithm

If we were to represent the process of a motivated explanation in pseudo-code, it might look like this:

def motivated_explanation(concept):
    while not human_understands(concept):
        idea = propose_intuitive_approximation()
        failure_point = find_contradiction(idea)
        refine_idea(failure_point)
        # The goal is to make the user feel: 
        # "I could have thought of this myself!"
    return "Aha! moment"

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Measuring the "Squishy" Metrics

One major hurdle is that proofs are easy to grade: they are either right or wrong. Human understanding is "squishier." However, avoiding subjective metrics means avoiding the human element of mathematics.

I use a specific, actionable guideline to test my own work: $\text{Guideline} = \text{"Does this feel like the viewer could have discovered it themselves?"}$

While not a binary $\text{Yes/No}$ check, it provides a verifiable standard for whether an explanation is complete.

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Real-World Exemplars

We don't need to invent this from scratch; it already exists in high-quality exposition. A prime example is Part IV of the Princeton Companion to Mathematics.

In this volume, experts (such as Andrew Granville or David Ben-Zvi) provide introductions to complex fields. These aren't just lists of theorems; they capture the intuition and motivation typically reserved for private conversations at a chalkboard.

Visual Context

The following images illustrate the intersection of intuition and formal math:

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Final Thoughts

As someone whose career exists outside the traditional academic tenure track, I have no personal "skin in the game" regarding how credit is distributed. However, if we want the world to understand what mathematicians actually contribute, we must stop treating clarification and exposition as second-class citizens. We must celebrate the motivation as much as the proof.