Category Archives: Mathematics

Evolution – Variation and Similarity

Evolutionary thinking (due to Darwin) is no doubt one of those paradigm shifting moments in scientific history, changing how we conceive of the world around us and ourselves. The idea of ‘Descent through Modification’ is now well established and accepted.

While evolution is not a disputable fact, a major source of debate a few decades ago (and even nowadays, to some extent) has been the causes for evolution. Enter a evolutionary biology class and you will see that everyone tries to explain observable traits (non-jargon way of saying phenotypes) using fitness arguments – how this or that trait was required for survival and reproductive success, and hence it is here today. These arguments stem from a view that is called the ‘Modern Synthesis’ – evolution happens primarily through natural selection, and natural selection requires a set of variants to select from, and this variation within a population is given by random genetic mutation. It is called the ‘Synthesis’ since it combined ideas from evolution and genetics to give a plausible answer to the mechanism of evolution. The whole idea of evolutionary game theory rests on this hypothesis, and so does evolutionary psychology.

However, a physicist or a mathematician or anyone else who tries to look for patterns in phenomena will tend to be exasperated by natural selection arguments for everything: in some cases, it is obvious that natural selection caused evolution, while it is not so in others. However, a knee-jerk answer to any evolutionary question by a biologist will invoke natural selection. Now, most of these answers are plausible, but that does not mean anything. For example, a crash in a predator population can easily be put down to a lack in fitness, but everyone who has studied the predator prey model will tell you that this crash comes about due to interactions between predator and prey populations, and has nothing to do with genes or natural selection.

Creating evolutionary fairy tales frees the biologist from looking at a phenomena at a deeper level, and sometimes one feels that depth is what is lacking when one reads up evolutionary biology. The oft quoted example is of the Fibonacci spirals in plants – this shows up everywhere, from shapes of galaxies to arrangement of seeds in flowers. A hardliner selectionist would tell you that this is because there were many variants of the universe and ours was the only one that managed to survive (reproduce?), and thus all such successful survivors will have Fibonacci spirals because of their ‘fitness improvement’. Now, one cannot disprove this, no doubt, but the question is whether one should accept it.

For me atleast, the answer is no – while selection of variants has its place in biology and (I sceptically say this) in other fields, it cannot explain the unity underlying phenomena: Certain things ‘just happen’ to look/behave/think similarly, and this evolution via selection cannot explain. Are there physical, chemical, informational constraints on a living being that simply does not allow certain variants? Are ‘gaps in the fossil record’ actually ‘gaps’ –  is there a step jump from one form to another? Answering these questions is way harder than coming up with ‘plausible’ selectionist arguments, and has very rarely been attempted in the history of biology. However, if evolutionary theory has to have the depth seen in physics or mathematics, such work has to inevitably happen.

The problem with nonlinearity (AKA why I cannot predict the weather)

Being from an engineering background, and having mainly engineers for friends, I normally get asked why I cannot predict tomorrow’s weather, and jibes as to how weather prediction is a pseudo-science etc etc., Thus, I decided to just rant about how life is so difficult for me.

Engineers of all kinds like to work with computationally friendly methods of analysis. One way to ensure this is to use mathematical maps that are linear in nature, and preferably orthogonal. What I mean by this is that it should be representable by a matrix, and all columns should have a zero inner product with every other column but itself. The classic example is the Discrete Fourier Transform. One of the most important properties (atleast to me!) of a linear system is that of superposition, i.e, if x and y are any two ‘signals’ or vectors, and F is a linear transform, then F(x+y) = F(x) + F(y). This property tremendously simplifies the analysis of the behavior of the transform. It is easy to identify ‘problematic’ vectors and do away with them.

For example, if im building a music system and I have a linear amplifier which I know goes nuts if I input music containing the 2 Khz frequency, I can remove that frequency in advance so that there are no problems in performance. Thus, a signal localised in a certain frequency band will not ‘leak’ to other bands. The case is not so in nonlinear systems. There is a transfer of energy from one part of the spectrum to another (eg: the Kolmogorov spectrum in turbulence), and thus there is no guarantee that your amplifier will be well behaved for all time.

This also implies that the superposition principle no longer applies. Since energy in one frequency invariably finds its way to other places, there is interaction between different frequencies and thus the resulting behavior of the system is not just the addition of the behavior of the system with the individual frequencies as inputs, i.e, F(x+y) \neq F(x) + F(y). Thus, the resulting behavior is not easy to predict in advance, and pretty much impossible if the number of interacting components is huge, like in an ecosystem or the climate. This is called emergent behavior, since it cannot be predicted by looking at the individual components themselves.

If losing superposition was a problem, the problem of chaos is as bad, if not worse. Chaos is a fancy way of saying that nonlinear systems are extremely sensitive to their inputs and their mathematical formulation. For example, if you had perfect knowledge about every quantity but not a perfect model of the phenomenon being observed, you will make errors in prediction, which are huge. Similarly, if your models were perfect, but you were not able to measure accurately enough, the same fate. In real life, both are true. We don’t understand natural phenomena well enough (Of course, dam builders will disagree), nor do we have measurements that are accurate enough. Thus, even the fact that we can say whether tomorrow will be cloudy or not with reasonable confidence is a testament to how well weathermen have learnt to live with nonlinearity.

And if all this was not enough, there is the problem of phenomena occuring at multiple scales. A typical cyclone has a horizontal extent of around 1000 km, while the convection that drives it is of the order of 1 km. There are planetary waves that have a wavelength of 10000 km, and they are dissipated by turbulence acting at the micrometer level. Any model that tries to incorporate the largest and the smallest scales will probably tell us about tomorrow’s weather sometime in the next century!!

And coming to the worst problem of all, rain.While one can say with reasonable confidence about whether it will rain or not, since that is constrained by the first law of thermodynamics and behavior of water vapor, it probably is next to impossible to predict when or how much. Quite amazingly, there still does not seem to have been found a sufficient condition for rainfall to occur: the necessary conditions are known, and still we don’t know when it will rain.

Interestingly, average behavior is more predictable, since averaging ‘smooths” out the nonlinearity in the system, and thus we are able to reasonably estimate climate, which is a long time-average of weather. The constraints of thermodynamics, which seem to be the only thing that will never be violated, are stronger as we go into longer and longer time scales.

Handling nonlinear systems is hard, but we are getting there! (In a century or so.)

Math: The geometry of Matrices

Was attending a workshop on Linear Algebra, and one of the lectures was by Dr. C. R. Pradeep. It was supposed to be on positive-definite matrices (whatever that might be!), but finding that no one really understood what a matrix was or stood for, it became a geometry class, to everyone’s benefit.

The standard way to look at a matrix is that is a set of numbers arranged in some order in between some brackets, and that they can be added with some effort and are multiplied in a completely obscure manner. This much one learns by the time one leaves Pre-University, and this does not help one bit in appreciating the whats and the whys behind the whole thing.

The best way to start off is with an example. Consider a general vector multiplying a 2\times 2 matrix:

\begin{pmatrix}1&1\\ 1&1\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix} = \begin{pmatrix}x+y\\ x+y\end{pmatrix}

If we take the general vector \begin{pmatrix}x\\ y\end{pmatrix} to be some point on the plane, then note that every point on the right hand side of the above equation has its x-coordinate equal to its y-coordinate. Therefore, if we think of this matrix as a machine that takes in all vectors in the plane and spits out some other vectors also in the plane, then things begin to look very nice.

This is because, if we start feeding this machine all the points on the plane, what we will get as a result is all the points whose coordinates are equal. From elementary geometry, this is a line passing through the origin at an angle of 45 degrees to both the coordinate axes:

p-q=0

How is it managing to do this ? Consider two special cases: x = 0, y=1 and y=0, x=1. These are the well known unit vectors on the plane, representing the y axis and the x axis respectively. Substituting these values in the equation, we see that they both are sent to the same point! Therefore, this matrix is collapsing the plane onto a single line, very much like closing a Chinese hand held paper fan. We can write any point on the plane in terms of the unit vectors, and similarly, we can write any point on the line p-q=0 using both the columns of the matrix. In this case they are the same, so it is a trivial relationship. But in general, the columns of the matrix are such that any point at the output can be written uniquely in terms of them.

Therefore, this matrix seems to be taking as its input a 2 dimensional ‘space’, i.e, the entire plane and giving back a 1 dimensional ‘space’ – a line through the origin. Another interesting thing to note is that both the points \begin{pmatrix}x\\ y\end{pmatrix} and \begin{pmatrix}y\\ x\end{pmatrix} both end up in the same point \begin{pmatrix}x+y\\ x+y\end{pmatrix}. This means given a point \begin{pmatrix}x+y\\ x+y\end{pmatrix}, we would not know which point on the plane it corresponds to, i.e, the inverse is not well-defined.  (In fact, given any point \begin{pmatrix}p\\ p\end{pmatrix} at the output there are an (uncountably) infinite number of points which could have produced that) When we say that a matrix is singular or non-invertible, this is what we mean. We normally check this by finding the determinant of the matrix (which is zero in the singular case), but this normally does not appeal much to intuition.

To generalise this, any matrix that takes a higher dimensional space to a lower dimensional space is not invertible. The natural next question to ask is that if a matrix maintains the dimension of the output space equal to the input, is it invertible ? This can in fact be proved to be true, and this can be taken as a simple interpretation of the Rank-Nullity Theorem.

Going back to our example, what if the point/vector at the input was already on the line p-q=0 ? For example, the point \begin{pmatrix}1\\ 1\end{pmatrix} would end up at \begin{pmatrix}2\\ 2\end{pmatrix}, which is the same as 2\times\begin{pmatrix}1\\ 1\end{pmatrix}. This is consistent with out Chinese fan approach, the line in the middle of the fan does not really move when it is closed (It does not get elongated either, but that is a special case of this). In Linear Algebra terminology, such points/vectors are called eigenvectors, and the value by which they are multiplied is called the eigenvalues of the matrix. In the general case, the axes do not collapse into each other, but maintain some angle between them. Even then, there will be some set of points/vectors which do not move, and these are called the eigenvectors.

Almost all the basic concepts of Linear algebra can be interpreted in this geometric manner. The heart of this whole discussion is the concept of linear transformations, which are represented by matrices for convenience and analysis.

Linear Algebra is an interesting subject, and on this is built almost all of engineering!!

Review: Godel’s Proof

KISS at its best
KISS at its best

This was something that I had to stop reading because of the exam season, and finally finished within 24 hrs of the exam getting over.

It is a rightly acknowledged classic in the field of popular mathematics (Stuff which makes abstruse mathematical ideas accessible to the (almost) general public).

Like the title says, it deals with a set of proofs that Godel published which demolished one of the most ambitious projects of modern times, to axiomatize all of mathematics.

It was believed that given a finite set of axioms and a set of rules to operate on them, one can mindlessly manipulate symbols to generate every known theorem. A simple example would be translating geometry into algebra (x^2 + y^2 = c^2 is how a circle would look in algebraic terms ). So, we can say that if  algebra is consistent, then geometry is consistent.

Hilbert developed a method to provide proofs of consistency, which was then taken over by the logicians of the time to try and provide similar proofs to certain parts and eventually all of  mathematics.

That is, until Godel came along. How he did it is explained beautifully in the book and is unecessary to reproduce here. Godel proved that any system for number theory that is consistent is incomplete, i.e,  you can produce a true theorem which cannot be proved within the system. Thus, we cannot have a formal system describing all of mathematics. We can have formal systems with finite set of axioms, but not for something like number theory, where Godel proved it was impossible to generate a finite set of axioms.

The book also has a section on the philosophical implications of this proof, which essentially says that present day A.I is a far cry from imitating the human brain, and unless we can actually describe the human brain as a logical system with a finite set of axioms.

There are many approaches to imitate human intelligence, like neural networks, but they run on a system with a very limited set of axioms and rules of inference commonly known as a digital computer. Maybe we need some new kind of ‘computer’. Researchers have a lot of work to do!!

Simply, a must read.