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Functions and Iterations
This set of notes is based on a course of the same name over at the amazing educational website run by the Santa Fe Institute (SFI) called Complexity Explorer, which I encourage anyone interested in applied math and interdisciplinary studies to check out.
It has a whole host of courses on all kinds of topics related to the study of complex dynamical systems, but which can also be used as stand-alone courses for learning a particular subject among the ones offered. These range from courses on differential equations and vector algebra to game theory and machine learning. It also has a great course, currently on-going, called Origins of Life which is not just any old course teaching stuff that was discovered decades or centuries ago, but is what the website calls the first in their series of “Frontiers Courses” which aren’t too advanced but still let learners explore an active area of research so that they can get a glimpse of what scientists in that domain are currently up to.
Writing up these notes took more time than they should’ve because I had to learn LaTeX from scratch in order to write down the equations in an understandable and presentable manner. Also had to spend hours looking for the right commands and getting the syntax right for the Python program at the end of this post because learning Python at the appropriate time in university would’ve been too subordinate of me (my rebellious nature is ridiculous at times, but it’s sort of an indispensable part of my personality).
Have decided to start learning programming in earnest from now on because a scientist without programming skills is as useless as an armwrestler without arms, so stay tuned for many more interesting programming-related posts here in the near future. Have also planned to go through the rest of the tutorials on the Complexity Explorer website – will post the notes for those here accordingly.
Chaos and Reductionism (Notes)
There is a spectacular course called Human Behavioral Biology (BIO150) by Robert M. Sapolsky (one of my all-time favorite people) on Youtube, which is worth watching in its entirety but which I’ve only watched selectively due to time constraints. He has two great lectures in the series on chaos and complexity which I watched in full because they align closely with my current interests. Following are a set of relatively comprehensive notes on lecture #21 of the series, entitled “Chaos and Reductionism”.
Before Reductionism
- Dark Ages began after fall of the Roman Empire (post 400 A.D.). Almost all past knowledge lost.
- 1085 A.D.: first European Christian conquest of Alhambra in Spain, leading to first major influx of knowledge into Europe since the beginning of the Dark Ages
- People started thinking transitively: a>b, b>c => a>c
- Rediscovery of syllogistic thinking à la Aristotle
- Thomas Aquinas’ 3 properties of God: God cannot sin, He cannot make a copy of Himself and He cannot make a triangle that is > 180o
Notes on (very basic) Chaos Theory
The following set of notes are based on the brilliant video series entitled “Chaos” by the math-enthusiast supreme, Jos Leys. He’s a mechanical engineer by profession but has a deep interest in math, in particular, how mathematical concepts can be visualized using modern software. His website is full of animations of well-known structures (such as the Mandelbrot set, Klein bottle, hyperbolic 3-space etc) and also numerous curiosities (such as hybrid fractal spaces, linked Klein bottles and fractals resembling alien creatures etc). Leys created another set of educational videos under the title of “Dimensions”, which I’ll go through very soon and post the notes for here.
This set of videos is a very elementary (at most, high school-level) introduction to chaos theory, a field of mathematics pioneered by the legendary Poincaré and first applied to weather forecasting by Edward Lorenz. Before Lorenz, most climate scientists believed the atmosphere could be modeled almost exactly using a set of simplified differential equations and supercomputers powerful enough to analyze the tremendous amount of data involved. To them, the hurdles in the way of accurate forecasting were merely practical (better mathematical models, greater data collection, faster computers).
But what Lorenz showed in his groundbreaking 1963 paper, entitled “Deterministic Nonperiodic Flow” was that long-term accurate weather forecasting was doomed to failure “in principle” because the Earth’s atmosphere is complex enough such that its future state (especially for large times) is highly sensitive to initial conditions. Even slight deviations (of the order of a millionth even) can lead to drastically different outcomes down the road. Therefore, since we know that climate modelling can never pin down the present state of the system to an arbitrary number of decimal places, there will always remain the possibility of the system diverging exponentially from our predictions, making long-term climate modelling impossible. The best we can do is give statistical predictions a week or two in advance.
This discovery of chaos playing an essential role in nearly all everyday phenomena rapidly took off and influenced nearly every field of science, including applied math, physics, chemistry, biology, ecology etc. It has been termed by practitioners as the greatest advance in our understanding of the natural world since relativity and quantum mechanics. This is true, because all three of these pillars of modern science completely shattered the naive conception of the world we had before and opened our eyes to a whole new realm of possibility.
Elementary definition of chaos: “sensitive dependence on initial conditions”
Stellar Magellanic Thoughts 