The Riemann Hypothesis aligns with the model that primes are pseudo-random. If proven false, it would imply a deep, undiscovered pattern in their distribution. This 'secret patent' to the primes would shatter the foundations of cryptography, as any hidden structure could lead to an exploit.
Terence Tao argues against hyper-optimizing one's time. Serendipitous interactions—like bumping into someone in a hallway or browsing a physical journal—spark new ideas. Over-scheduling and efficiency tools eliminate these random encounters, potentially stifling the unexpected connections that lead to breakthroughs.
The classic scientific model involved devising a theory and then collecting data to test it. The modern paradigm, driven by big data, often reverses this. Progress now frequently comes from analyzing massive datasets first to discover patterns, and only then forming hypotheses to explain them.
Astronomy has always been a data-bottlenecked field, forcing practitioners to become world-class at "squeezing every last possible drop of information" from limited, noisy datasets. This specific skill of finding weak signals is directly transferable and highly valued in quantitative finance.
Unlike other sciences, mathematics has historically lacked a strong experimental branch. AI changes this by enabling large-scale studies—for example, testing a thousand different problem-solving approaches on a thousand problems. This creates a new, data-driven methodology for a field that has been almost entirely theoretical.
We have formal languages like Lean for deductive proofs, which AI can be trained on. The next frontier is developing a language to capture mathematical *strategy*—how to assess a conjecture's plausibility or choose a promising path. This would help automate the intuitive, creative part of mathematical discovery.
Historically, generating a good hypothesis was the most prestigious part of science. Now, AI can produce theories at near-zero cost, overwhelming traditional validation systems like peer review. The new grand challenge is developing scalable methods to verify and filter this flood of AI-generated ideas.
Great ideas like deep learning were not immediately recognized. Their value emerged over time as others built upon them. This suggests an idea's fruitfulness is a product of its context and cultural adoption, not just its isolated brilliance, making it difficult for an AI to evaluate its ultimate impact.
True intelligence is adaptive and builds upon partial progress. Terence Tao notes current AIs demonstrate "cleverness" by using trial-and-error at massive scale. They can't yet grab a 'handhold,' stay there, and pull others up—a cumulative process that defines collaborative human intelligence.
Copernicus's simpler heliocentric model was less accurate than the highly-tweaked Ptolemaic system. This shows that progress isn't linear accuracy; a new, conceptually superior framework might perform worse at first. It requires further refinement, as Kepler provided for Copernicus, to realize its full potential.
Darwin communicated his theory in plain, persuasive English, accelerating its acceptance. In contrast, Newton wrote in Latin and was secretive, slowing his ideas' spread. This highlights that exposition and narrative are critical, non-technical skills for driving scientific progress and convincing others to invest in a new idea.
Kepler's method of testing numerous, often strange, hypotheses against Tycho Brahe's precise data mirrors how AIs can generate and verify countless ideas. This uncovers empirical regularities that can later fuel deeper theoretical understanding, much like Newton's laws explained Kepler's findings.
Mathematician Terence Tao finds AI doesn't speed up his core problem-solving but makes his papers "richer" by adding complex plots and deeper literature searches. Tasks that were previously infeasible are now easy. AI expands the scope and quality of work rather than just shortening the timeline for existing tasks.
AIs excel at exploring millions of problems at a surface level (breadth), a scale humans cannot match. Human experts provide the depth needed to tackle the difficult "islands" AIs identify. Science must shift from its current depth-focused model to one that first uses AI to map entire fields and clear away low-hanging fruit.