Do We Need New Math to Understand the Universe? With Terence Tao

The Institute for Pure and Applied Mathematics (IPAM) serves as a hub for bringing together pure mathematicians, applied mathematicians, scientists, and industry professionals. This interdisciplinary approach allows for the identification and resolution of mathematical obstacles in emerging technologies, leading to significant advancements. For example, early workshops on AI and self-driving cars, and a collaboration that led to MRI scans 10 times faster than traditional methods, now used in modern machines.

IPAM brings together pure and applied mathematicians, scientists, and industry to work on topics where math is needed. They held early workshops on AI, deep fakes, and self-driving cars. A collaboration led to MRI scans 10 times faster, with their algorithms now used in modern MRI machines.

Pure mathematics is curiosity-driven, exploring abstract patterns without immediate practical application, while applied mathematics focuses on developing tools and equations valuable to scientists and engineers. Despite their different motivations, they share a symbiotic relationship where pure math often provides the foundational concepts that applied math later leverages for real-world problems. This exchange can also flow in reverse, with physical observations inspiring new pure mathematical conjectures.

Pure math is curiosity-driven, exploring abstract patterns like digits of pi. Applied math develops software or equations for practical value to scientists, like modeling climate. Physicists' observations, such as universal distributions like the Gaussian curve, can inspire mathematicians to find explanations.

The Collatz conjecture, also known as the hailstone conjecture, is a deceptively simple problem that has stumped mathematicians for over a century. It proposes that any positive integer, when subjected to a rule (divide by two if even, multiply by three and add one if odd), will eventually reach the 1-4-2 loop. Despite extensive computer testing up to quadrillions, no counterexample has been found, yet a formal proof for all numbers remains elusive, illustrating how simple operations can lead to immense complexity and 'chaos.'

The Collatz conjecture involves a simple rule: if even, divide by 2; if odd, multiply by 3 and add 1. It's conjectured that all numbers eventually reach the 1-4-2 loop. Computers have tested numbers up to quadrillions, but no proof exists for all infinite cases. This is an example of chaos from simple operations.

Abstract mathematical concepts, developed purely for intellectual curiosity, frequently find profound and unexpected applications in the physical sciences, often decades later. The most famous example is non-Euclidean geometries, which mathematicians explored for their own sake but later became the precise language Einstein needed to formulate General Relativity and describe curved spacetime.

Eugene Wigner called this the 'unreasonable effectiveness of mathematics.' Non-Euclidean geometries, developed by mathematicians like Riemann for curved space, were later found by Einstein to be almost exactly the language needed for general relativity.

While changing the numerical base system (e.g., from base 10 to base 2 or 60) doesn't alter fundamental mathematical truths (like A+B=B+A), it can influence the speed or direction of discovery. Different bases might highlight or obscure certain patterns, potentially slowing down or accelerating the development of specific theories. Historically, humans have used various bases (Babylonian base 60, remnants of base 12 and 20), and computers now predominantly use binary (base 2) for efficiency.

Babylonians used base 60, influencing time measurement. Humans have used base 12 (dozens) and base 20 (scores). Computers use base 2 (binary). While A+B=B+A remains true regardless of base, using different bases can 'slow it down or speed it up a little bit' in terms of accessing discovery.

When tackling complex, unsolved mathematical problems, it's crucial to explore not just what works, but also the 'negative space' – what doesn't work. By actively trying to prove the opposite of a conjecture or finding counterexamples that satisfy hypotheses but not the conclusion, mathematicians can identify missing parameters or flawed assumptions. This process of mapping out pitfalls helps reveal the narrow path to a valid proof, emphasizing the value of partial progress and avoiding emotional investment in a single outcome.

It's important to prove 'negative results' by finding counterexamples that satisfy hypotheses but not the conclusion. This helps identify what's missing. Mapping out the 'negative space' reveals the narrow path to a goal. Mathematicians value partial progress and are not emotionally invested in one outcome.

While current mathematics excels at explaining most of the universe, it 'breaks down' at extreme scales and conditions, such as the early universe, the centers of black holes, and the realm of quantum gravity. To reconcile these areas, physicists believe new mathematical frameworks are necessary, potentially requiring a complete re-evaluation of our fundamental notions of space and time. Existing concepts like non-Euclidean geometry are insufficient for describing quantum spacetime, prompting theories like string theory to propose entirely new mathematical structures.

Our math breaks down in places like the early universe and black holes. The current math doesn't give sensible answers for these extreme conditions. We need a theory of quantum gravity, which requires abandoning our notions of space and time. Even non-Euclidean geometry won't be enough for quantum spacetime. String theory is a proposal, but it hasn't convincingly fit reality.