From: mk_thisisit
Professor Adam Kanigowski, winner of the European Mathematical Society’s main prize, discusses chaos theory and its implications for predicting the future, particularly in relation to quantum physics [00:00:03].
The Butterfly Effect and Predictability
The core concept of chaos theory is the “butterfly effect,” which states that a minimal disturbance from an actual state can lead to an entirely different future [00:00:30]. For example, a butterfly flapping its wings slightly changes wind force, which can have “colossal consequences” [00:00:21].
Theoretical Possibility vs. Practical Impossibility
Theoretically, if all parameters and initial conditions were known with infinite precision, the future could be predicted [00:01:35]. However, in practice, this is unattainable due to measurement errors [00:01:48]. The human eye and other measurement tools introduce inaccuracies, making it impossible to precisely determine the future state of a system [00:02:30].
The Nature of Chaos
While the term “chaos” suggests disorder, Professor Kanigowski notes that the underlying laws governing these systems are orderly [00:03:03]. Chaos emerges when small initial disturbances lead to significant divergences from predictions over time [00:03:11]. The scientific endeavor is to model initial conditions and use mathematics to get as close as possible to understanding this “written version” of the future, even if it cannot be fully known [00:04:04].
Impacts and Applications of Chaos Theory
Chaos theory has significant applications:
- Weather Prediction: The butterfly effect originated from the work of mathematician and meteorologist Edward Lorenz, who concluded that long-term weather prediction is impossible due to deterministic chaos [00:04:38]. Even a minimal disturbance like a temperature change can have colossal consequences [00:05:04]. Increasing measurement precision, for example, from an accuracy of 1 to one-millionth, can extend prediction time from one day to three days [00:09:34]. The speed at which information is lost in such systems is exponential, governed by the Lyapunov exponent [00:10:26].
- Stock Markets: While direct long-term prediction is challenging, chaos theory and probability are used to predict stock market movements for increasingly longer periods [00:09:10].
- Evolution of Species: Chaos can determine the fate of species. Minimal disturbances in initial conditions can decide whether one species survives while another does not [00:13:08].
- Celestial Mechanics (N-Body Problem): While some systems like the movement of two celestial bodies can be precisely predicted, the “n-body problem” (e.g., three or more celestial bodies like Earth, Sun, and Moon) becomes incredibly difficult to solve due to the terrifying number of variables and sensitivity to initial conditions [00:13:47]. Predicting movements for more than 2 million years is currently impossible [00:13:47].
Chaos Theory and Probability
Since exact prediction for complex systems is practically impossible due to an overwhelming number of variables, the focus shifts to predicting what will happen with a high probability [00:16:41]. This is where probability theory and dynamic systems intersect [00:16:46]. Optimization theory also plays a role, allowing for corrections over time to bring a system closer to an expected result [00:29:22].
Chaos Theory and Quantum Physics
Chaos theory has been successfully developed in the context of classical physics, specifically Newtonian physics [00:18:16]. However, its relationship with quantum physics is a more nascent field.
- Non-Deterministic Quantum World: Unlike the deterministic classical world, the quantum world is not deterministic [00:17:46]. Phenomena like a particle being in multiple places simultaneously (e.g., a photon) highlight its non-intuitive nature [00:17:52].
- Mathematical Challenges: Mathematics is “just beginning” to learn how to study phenomena from quantum physics [00:18:37].
- Quantum Ergodicity: Hypotheses like “Quantum Ergodicity” suggest that if velocity is precisely known, the position is evenly distributed, implying zero information about the position and equal probability for all possible states [00:18:54]. This concept aligns with the ambiguity principle (Heisenberg Uncertainty Principle), which states that position and velocity cannot be simultaneously predicted [00:19:03].
- The Interface: There is a significant challenge in combining classical and quantum physics mathematically [00:06:02]. This requires experts in quantum physics, probability theory, and the ability for these experts to collaborate [00:36:52]. Organizations like Microsoft Research are building teams to address these complex interdisciplinary problems [00:37:16].
- Unpredictability: At the quantum level, the world is fundamentally unpredictable, described using probability theory [00:34:53]. However, even unpredictability can often be quantified [00:37:58].
Mathematics: Discovered or Created?
Professor Kanigowski believes that good mathematics is “discovered,” implying that it exists independently and is found rather than invented [00:07:12]. He views mathematics as a “book” where “everything is written,” and mathematicians are simply uncovering its pages [00:07:38]. Concepts like differential calculus are considered discovered [00:07:24]. While some mathematics might be “created” or less interesting, the most significant work is discovered [00:46:17].
AI and the Future of Mathematical Proofs
AI models dedicated to mathematics are emerging with the ability to reason [00:24:04]. The hope is that AI could assist in proving theorems by helping with sequences of cause-and-effect reasoning and intermediate steps [00:24:34]. However, there is a question about whether AI will eventually be able to prove entire theorems independently [00:25:00].
Personal and Cultural Impact
Professor Kanigowski’s interest in chaos theory was sparked by the film The Butterfly Effect [00:26:45]. He notes that chaos theory resonates with pop culture because it reflects the human desire to understand the consequences of seemingly insignificant daily decisions [00:27:49].
For aspiring mathematicians, Professor Kanigowski advises viewing every unsuccessful attempt at problem-solving as a step forward, understanding what doesn’t work [00:49:31]. He emphasizes that the goal should be to foster curiosity about the world and a desire to discover [00:50:50].
The Limits of Prediction
Ultimately, for complex, large-scale systems with numerous variables, the future is best understood not as written and precisely predictable, but through probabilities [00:52:26]. Even with infinite precision, fundamental limits exist at the quantum level, such as the Planck constant, beyond which information cannot be obtained [00:30:42]. This inherent unpredictability ensures that human mathematical inquiry remains relevant [00:30:53].