Peter Thiel’s Zero to One distinguishes between two types of progress: going from “1 to n” (horizontal progress through copying or incremental improvement) versus going from “0 to 1” (vertical progress through creating something entirely new). While Thiel wrote about startups and business, this framework offers a fascinating lens for understanding mathematical research.
The Mathematics of 1-to-n vs 0-to-1
In mathematical research, 1-to-n progress might look like:
- Proving the next case of a known theorem
- Extending existing results to slightly more general settings
- Applying established techniques to new but similar problems
- Incremental improvements to known bounds or constants
0-to-1 progress, by contrast, involves:
- Developing entirely new proof techniques (like Grothendieck’s schemes in algebraic geometry)
- Creating new mathematical frameworks (non-Euclidean geometry, category theory)
- Finding unexpected connections between distant fields
- Paradigm shifts that change how we think about entire areas of mathematics
The Hidden Value in 1-to-n Work
Here’s where mathematics differs from Thiel’s business context: incremental mathematical progress can be profoundly valuable, even when it looks routine. What appears to be simple “1-to-n” work, proving the next case, extending a theorem slightly, can reveal deep structural insights that weren’t visible before.
Some of the most important mathematical ideas emerge at the “end of research trees,” when pushing an incremental approach to its limits reveals something unexpected that opens entirely new directions. The challenge is that these gems are often buried in the vast literature of incremental work, making them hard to discover and build upon.
Finding Mathematical “Secrets”
If 0-to-1 breakthroughs come from having unique insights, what Thiel calls “secrets,” how do mathematicians discover them?
Cross-pollination between fields remains one of the most fertile approaches. Many breakthroughs come from applying techniques from one area to problems in another. But this requires unusual breadth of knowledge and willingness to venture outside established specialties.
Studying failure patterns can be revealing. When many capable researchers have tried similar approaches and hit walls, that’s often where secrets hide. The key is recognizing why standard approaches fail and what that reveals about underlying structure.
Reading old papers with fresh eyes connects to Goethe’s insight: “Everything has been thought before, but the difficulty is to think of it again.” Techniques that seemed limited in their original context can become powerful when combined with modern tools.
Following the “why” questions rather than just proving that something works often leads to more general principles with broader applications.
The Recognition Problem
There’s a temporal challenge here: it’s often easier to recognize truly valuable 1-to-n work only in retrospect, after other researchers have built upon it in unexpected ways. When incremental work first appears, its ultimate significance may not be apparent.
This creates a kind of mathematical myopia. The most obviously revolutionary 0-to-1 work gets immediate attention, while important incremental contributions that could seed future breakthroughs get lost in the noise of published research.
The Collaborative Reality
Unlike in business, where you might want to avoid competition entirely, mathematics is fundamentally collaborative and cumulative. Even the most revolutionary 0-to-1 work builds on previous foundations. Mathematical truth exists independently of human discovery, so “secrets” are more about finding the right approach or perspective than having proprietary insights.
This suggests that the most valuable mathematical research often combines both approaches: using genuinely new methods (0-to-1 thinking) to make progress on established problems, or finding unexpected connections between different areas that have been developed incrementally.
Looking Forward
The 0-to-1 framework helps us think more clearly about the different types of mathematical progress and their relative values. But it also highlights a challenge: in our rush to celebrate the obviously revolutionary, we might be overlooking incremental work that contains the seeds of future breakthroughs.
Perhaps what mathematics needs is better ways to identify which 1-to-n work contains hidden 0-to-1 potential, before it gets buried in the ever-expanding literature. The next breakthrough might be waiting in a paper that looks routine today but contains exactly the insight someone needs tomorrow.