From: mk_thisisit
The question of whether the future can be predicted was explored in a conversation with Professor Adam Kanigowski, a recipient of the European Mathematical Society’s main prize, focusing on chaos theory at the National Museum of Technology in Warsaw [00:00:00].
The Deterministic View: A Written Future?
Theoretically, if all parameters of a system were known with infinite precision, the future could be determined [10:51:00], [01:35:00]. This perspective suggests that the future is “written” [03:56:00]. If we had infinite accuracy in measuring the present state of a system, we would be able to predict its future with infinite accuracy [11:38:00].
The Butterfly Effect: Order within Deterministic Chaos
The “butterfly effect” describes how a minimal disturbance from an actual state can lead to an absolutely different future [00:30:00], [02:43:00]. For instance, a butterfly flapping its wings slightly changes wind force, which can have colossal consequences [00:21:00], [05:12:00]. This phenomenon, which gave rise to chaos theory, implies that while theoretically predictable, practical prediction is unattainable due to the need for infinite precision in initial conditions [01:48:00], [02:01:00].
Professor Kanigowski clarifies that despite its name, the butterfly effect doesn’t represent chaos but rather a profound sensitivity to initial conditions within a deterministic system [02:54:00]. The “chaos” appears when predictions diverge from the actual state over time, due to the exponential loss of information caused by measurement errors [03:11:00], [10:26:00]. This rate of information loss is described by the Lyapunov exponent [10:30:00].
Practical Limitations of Prediction
In practice, the human eye and measurement tools introduce errors, making it impossible to know the future with certainty [02:16:00]. Even with advancements in computing, the goal is to slow down the butterfly effect to enable longer prediction periods by increasing measurement precision [09:47:00].
Applications and Contexts of Chaos Theory
Chaos theory describes one variant of system evolution, distinct from systems that evolve in an orderly manner, such as the movements of planets [12:41:00].
Weather and Stock Market Prediction
Edward Lorenz, a mathematician and meteorologist, discovered the butterfly effect while attempting to predict weather for longer periods, concluding it was impossible due to deterministic chaos [04:38:00], [04:47:00]. Despite this, the goal is to predict weather and stock market movements for increasingly longer periods through enhanced measurement accuracy [00:46:00], [09:10:00].
Planetary Movements and the N-Body Problem
Even with knowledge of celestial body trajectories, predicting their movements beyond a few million years is not possible due to extreme sensitivity to initial conditions [13:38:00], [14:08:00]. The n-body problem, which has roots with Newton, illustrates this difficulty; predicting the movement of three or more bodies (like Earth, Sun, and Moon) is incredibly complex, with no general solution known for three or more bodies [14:32:00], [15:15:00].
Evolution of Species
Chaos theory also applies to the development of species. Even a minimal disturbance in initial conditions can lead to one species surviving while another does not [13:02:00], [13:10:00].
The Indeterministic Challenge: Quantum Physics
The deterministic view of the world is challenged by quantum physics, which suggests that reality is not entirely deterministic [17:22:00], [17:46:00]. The interface between classical (Newtonian) and quantum physics is an area of active mathematical research [05:32:00], [06:02:00].
While classical chaos theory is well-developed for Newtonian physics, mathematics is only beginning to learn how to study phenomena in quantum physics [00:39:00], [18:37:00].
Quantum Unic Ergodicity and the Uncertainty Principle
The ambiguity principle in quantum physics states that it’s impossible to predict both the position and velocity of a particle at the same moment [19:00:00]. A hypothesis called “Quantum unic ergodicity” further suggests that if more is known about velocity, the position becomes evenly distributed, meaning there’s zero information about it, and all possible states are equally probable [19:16:00].
The Role of Probability
Given the vast number of variables in the real world, absolute prediction is impossible [16:07:00], [52:28:00]. Instead, probability theory is used to determine what will happen with high probability [16:43:00], [16:55:00]. This approach allows for making optimal corrections along the way to stay close to an expected outcome [29:32:00].
Mathematics: Discovered or Created?
Professor Kanigowski personally believes that good mathematics is “discovered” rather than “created” [07:12:00], [07:27:00], [46:17:00]. He imagines that “somewhere there is some book which has in which everything is written” [07:38:00]. This perspective also applies to the concept of failure in problem-solving: if everything is written, then understanding why a method doesn’t work is itself progress [49:42:00].
AI and the Future of Mathematical Proofs
AI could significantly aid mathematicians by providing intermediate steps in complex proofs [24:46:00]. However, there is a concern that AI might eventually prove entire theorems, potentially marginalizing human mathematical research [25:00:00].
Personal Reflections and Curiosity
Professor Kanigowski’s interest in chaos theory stemmed from the film The Butterfly Effect [26:42:00]. The film’s concept highlights how seemingly insignificant decisions can have drastic future impacts, echoing the core idea of the butterfly effect [27:11:00]. This idea’s natural appeal to human curiosity about consequences is why it has permeated popular culture [27:49:00].
Ultimately, the goal in mathematics is to understand the world, which can be achieved by embracing curiosity and the desire to discover [50:46:00], [51:17:00]. While complete prediction of complex systems is impossible due to the sheer number of variables and the non-deterministic nature of quantum phenomena, examining the probability of events remains the practical approach [52:44:00].