From: mk_thisisit
Chaos theory, a field explored by Professor Adam Kanigowski, winner of the European Mathematical Society’s main prize, investigates systems where a minimal disturbance to the initial state can lead to a vastly different future [00:00:05], [00:00:30]. This concept is famously known as the butterfly effect [00:00:30].
The Butterfly Effect and Determinism
The butterfly effect illustrates that a slight change in a parameter, such as a butterfly flapping its wings altering wind force, can have “colossal consequences” over time [00:00:21], [00:05:12]. While this phenomenon appears chaotic, it fundamentally operates within a deterministic framework [00:04:50], [00:09:47]. The term “chaos” refers to the divergence between predicted and actual states of a system after a period, not a lack of underlying order [03:00:00], [03:06:00]. If all parameters were known with infinite precision at time zero, the future state would be entirely predictable [01:35:00].
Prediction: Theory vs. Practice
Theoretically, if all factors, circumstances, and data accompanying the present moment were known with infinite precision, predicting the future would be possible [01:16:00], [01:35:00]. However, in practice, due to inevitable measurement errors and the human eye’s accuracy limitations, it becomes unattainable [01:48:00], [02:16:00], [02:30:00]. Minimal disturbances from the actual state cause “an absolutely different future” [02:45:00].
The speed at which information is lost in chaotic systems is exponential, described by the Lyapunov exponent [10:26:00], [10:30:00]. This means even with increased measurement accuracy (e.g., one millionth accuracy), predictions can only be extended for a few days [09:37:00], [10:06:00]. The more precisely something is measured, the longer predictions are possible [09:54:00].
Historical Development and Applications
The butterfly effect originated from the work of mathematician and meteorologist Edward Lorenz, who concluded that long-term weather prediction was impossible due to deterministic chaos [04:38:00], [11:24:00].
Chaos theory finds applications in:
- Weather Prediction [00:46:00], [04:34:00]
- Movements of Stock Markets [00:48:00], [09:13:00]
- Evolution of Species where minimal disturbances in initial conditions can determine species survival [13:02:00].
- Celestial Body Movements (N-body problem): While the movement of two bodies is a solved problem, predicting the movement of three or more bodies (like Earth, Sun, Moon) remains a very difficult challenge, famously studied by Poincaré [14:43:00], [15:18:00]. Even for celestial bodies, predictions cannot extend beyond a few million years due to sensitivity to initial conditions [13:47:00].
“Chaos describes one of the possible variants of the evolution of the system” [12:47:00].
Mathematical Perspective
Mathematics provides a “deep understanding or proper understanding of the phenomena that we observe” [06:15:00]. The question of whether mathematics is “discovered” or “created” is fundamental [07:10:00]. Professor Kanigowski personally believes that “good mathematics is discovered,” citing differential calculus as an example [07:12:00], [07:24:00]. This perspective views mathematics as a hidden book where everything is written, and mathematicians are simply discovering its pages [07:38:00].
When faced with many variables, probability theory becomes crucial to predict what will happen with “high probability” [16:41:00], [16:55:00]. Some parameters in a chaotic system are more significant than others due to their associated Lyapunov exponent, which dictates the rate of exponential growth in error [04:56:00], [45:05:00].
Chaos Theory and Its Relation to Quantum Physics
Chaos theory, while well-developed for classical Newtonian physics, is a much younger field when applied to quantum physics [18:16:00], [18:34:00]. Mathematics is only beginning to learn how to study phenomena in the quantum realm [00:39:00], [18:39:00].
“We do not know what connects the classical world with the quantum world, we have no idea, but we know that it exists and we know that the quantum world is not deterministic, that it is non-intuitive…” [17:39:00]
One hypothesis in quantum mathematics is Quantum Ergodicity, which suggests that beyond the ambiguity principle (Heisenberg’s uncertainty principle), if velocity is known, the position is “evenly distributed” or has “zero information” [18:54:00], [19:03:00], [19:18:00]. Research institutions like Microsoft Research are building interdisciplinary teams to combine quantum physics with mathematical theory and probability [36:40:00], [37:17:00].
Future of Chaos Theory Research
Current research in chaos theory, particularly Professor Kanigowski’s award-winning work, focuses on “slow chaos” [19:41:00], [19:53:00]. This involves systems where information is lost at a polynomial rate, which is significantly slower than the exponential rate of deterministic chaos, allowing for more investigation with less measurement precision [19:56:00]. The aim is to maximize the probability of a future state with a minimum amount of data [21:00:00].
The development of computers with higher precision allows for longer predictive periods, but the underlying challenge remains the exponential loss of information [09:47:00], [10:26:00]. There is also discussion about the potential influence of artificial intelligence (AI) in assisting with mathematical proofs and reasoning [24:04:00].
Despite the complexities, the core idea of chaos theory—that seemingly insignificant decisions can have drastic future impacts—resonates beyond science into pop culture, media, and daily human curiosity about consequences [27:01:00], [27:49:00].