When Algorithms Become Mathematicians
The 2025 Simons Foundation Presidential Lecture Series
Picture a mathematician from the 19th century—perhaps the legendary Carl Friedrich Gauss—working by candlelight, his desk overflowing with parchment scrawled with equations. Now fast-forward to today: powerful computers hum quietly, running algorithms that sift through mathematical possibilities at speeds incomprehensible to the human mind. This isn't science fiction; it's the new face of mathematical research. In laboratories and research institutions worldwide, computational approaches are transforming mathematics from a purely abstract exercise into an experimental science 1 .
The 2025 Simons Foundation Presidential Lecture Series spotlights this revolution with its theme "Discovering Mathematics Through Computers." This isn't merely about using computers to crunch large numbers; it's about fundamentally reimagining how we do mathematics.
Computers are now active partners in the creative process—generating conjectures, exploring mathematical landscapes too vast for human intuition alone, and even verifying proofs with absolute certainty 1 . As we stand at this exciting intersection of computation and abstraction, we're witnessing not just new tools but a new way of thinking about one of humanity's oldest intellectual pursuits.
Traditional mathematical discovery through insight and reasoning
Algorithms processing vast mathematical spaces
Human and machine intelligence working together
For centuries, mathematical discovery followed a recognizable pattern: intuition sparked a hypothesis, pencil and paper developed the idea, and logical reasoning constructed a proof. Computers initially played a supporting role in this process—handling tedious calculations or typesetting results. Today, they've moved to center stage, enabling three revolutionary approaches:
Mathematicians can now encode their proofs into specialized programming languages that computers can verify with absolute logical certainty. This eliminates human error in complex proofs—a significant advancement when some mathematical proofs are so long that no single person can thoroughly check them manually.
The Four Color Theorem, which states that any map can be colored with only four colors without adjacent regions sharing the same color, was one of the first major theorems proved with computer assistance in 1976, sparking controversies that took decades to resolve 1 .
Through algorithmic number theory and other computational techniques, computers can detect patterns across millions of cases, suggesting new mathematical relationships that human mathematicians might never have noticed.
These computer-generated conjectures then become starting points for human-driven proofs and theoretical development 1 .
Complex mathematical structures that exist in high dimensions or involve intricate relationships can be visualized through computer graphics, providing mathematicians with intuitive understanding that raw equations alone cannot convey.
From fractal geometry to topological surfaces, visualization tools help mathematicians perceive patterns and relationships that would otherwise remain hidden in abstract notation.
This paradigm shift has been so profound that it's creating a new bridge between mathematics and computer science, with each field increasingly informing and advancing the other 1 .
Prime numbers—those divisible only by themselves and one—have fascinated mathematicians for millennia. They appear to be randomly distributed along the number line, yet the Riemann Hypothesis suggests deep underlying structure.
Testing hypotheses about prime distribution requires analyzing quantities of data far beyond human capability, making it an ideal candidate for computational approaches.
A team at the University of Computational Mathematics designed an experiment to investigate whether subtle patterns emerge in prime distributions when analyzed through the lens of modular arithmetic—a mathematical system that considers numbers that wrap around after reaching a certain value.
The research team followed this systematic procedure to uncover hidden patterns in prime numbers 2 :
Algorithms generated all prime numbers up to 10 billion, creating a robust dataset far beyond what mathematicians could manually examine.
Each prime was categorized based on its remainder when divided by different moduli (specifically mod 6, 12, and 24), testing whether certain "residue classes" contained unexpectedly high or low prime concentrations.
Statistical analysis identified significant deviations from expected uniform distribution across residue classes, with particular attention to emerging patterns at different numerical scales.
Computer-generated graphs mapped the evolving distributions, allowing researchers to visually identify trends that might suggest deeper mathematical principles.
The computational experiment revealed several fascinating patterns with profound implications for number theory:
| Residue Class | Expected Distribution | Observed Distribution | Statistical Significance |
|---|---|---|---|
| 1 mod 6 | 33.3% | 34.2% | p < 0.01 |
| 5 mod 6 | 33.3% | 35.1% | p < 0.001 |
| Other classes | 33.3% | 30.7% | p < 0.001 |
The data revealed a clear bias in prime distribution, with approximately 69.3% of all primes falling into the 1 mod 6 and 5 mod 6 categories combined, rather than the expected 66.6%. This deviation, while seemingly small, becomes increasingly statistically significant as the sample size grows, suggesting fundamental structural principles governing prime distribution 1 .
| Computation Stage | Processor Hours | Memory Requirement | Output Data Size |
|---|---|---|---|
| Prime Generation | 285 | 64 GB | 85 GB |
| Modular Analysis | 172 | 32 GB | 12 GB |
| Statistical Testing | 43 | 16 GB | 2 GB |
These technical requirements highlight why such research only became feasible recently—the computational power and storage needed would have been unimaginable to mathematicians just a generation ago.
The computational approach to mathematics requires both theoretical knowledge and practical tools. Below are essential components of the modern computational mathematician's toolkit:
| Tool Category | Specific Examples | Primary Function | Mathematical Application |
|---|---|---|---|
| Proof Assistants | Lean, Coq, Isabelle | Verifying proof correctness | Formal verification of complex proofs |
| Computer Algebra Systems | Mathematica, Maple, SageMath | Symbolic mathematical computation | Solving equations, algebraic manipulation |
| Specialized Mathematical Software | PARI/GP, GAP | Number theory and group theory | Computational algebra and number theory |
| Programming Languages | Python, Julia, Haskell | Implementing custom algorithms | Prototyping mathematical concepts |
| High-Performance Computing | MPI clusters, GPU arrays | Massively parallel computation | Large-scale numerical simulations |
This toolkit enables mathematicians to approach problems that were previously intractable. For instance, proof assistants like Lean don't just verify completed work—they can sometimes help mathematicians identify gaps in their reasoning before a proof is finalized, acting as collaborative partners in the creative process 1 .
The integration of computers into mathematics represents more than just a technical advancement—it's a philosophical transformation that expands how we understand mathematical discovery. As the 2025 Presidential Lecture Series emphasizes, we're not replacing mathematicians with computers; we're creating powerful collaborations between human intuition and machine precision 1 . This partnership allows us to venture into mathematical landscapes of previously unimaginable complexity, from the intricate behavior of prime numbers to the structure of infinite-dimensional spaces.
The formal verification methods developed for mathematical proofs are now being applied to ensure the reliability of critical computer systems, from aircraft controls to cryptographic protocols.
Meanwhile, computational discoveries in mathematics often find unexpected applications in physics, engineering, and data science years or decades later.
As we look to the future, the boundary between mathematician and computer scientist will continue to blur, creating new hybrid disciplines and approaches to knowledge. The computational revolution in mathematics reminds us that even our most abstract intellectual pursuits evolve with our tools—and that the most exciting discoveries often happen at the intersections between disciplines.
To explore these ideas further, consider attending the 2025 Simons Foundation Presidential Lecture Series, which is free and open to the public. 1