linear thinking

I feel I have always been taught just enough math to get by. Since I am by training a physicist and work in the area of medical imaging, that means that I have digested some amount of math, but I have come to realize that what I know mostly amounts to tricks, and when I step back and try to see whether those tricks form a coherent whole, I am sadly disappointed.

Part of the problem may be that the same mathematical concepts crop up sometimes in two or more analogous contexts but at other times they get used in contexts that are fundamentally different. Take, for example, the supposedly simple concept of "linearity." When this term is applied to a function, e.g. height = some function of distance h(x), it very simply means that the function describes a ramp of constant slope. If you figure out the height you reach by travelling a horizontal distance x₁ and then go back to your starting position and figure out the height for the distance x₂, then the total height you achieve by travelling x₁ followed by x₂ is simply the linear sum of the two separate heights. This implies that h(x) = c₁ x + c₂, where c₁ and c₂ are constants. The word linear here ends up meaning a function of x that doesn't curve, i.e. doesn't have terms like x².

How about moving across a terrain that isn't just going up a hill with a constant slope. Let's make the height some function h₁(x) that goes up and down like a piece of rolling countryside. Now we imagine being in some airplane that is designed to change its height relative to the ground h₂(x) which could for example be the horizontal distance it travels squared, in which case h₂(x)= x². In general the total height is given by: h(x) = h₁(x) + h₂(x), which is no longer a linear function of distance. And indeed, if you repeat the same experiment by going separate distances x₁ and x₂, the heights no longer add up to the overall height you achieve by travelling x₁ + x₂.

Let's say you are able to make adjustments to your plane so that the height reached is given by: h(x) = h₁(x) + c h₂(x). Here, the plane still rises above the ground in a manner described by h₂(x), but the parameter c is a multiplier that determines the scale of this effect. The point is that this system isn't linear in how it behaves as a function of distance, but it is linear with respect to the combining of the height of the ground plus the additional height achieved by the plane.

So what, you may well ask. Well, the point is that if you know the height of the countryside, in this case h₁(x), and you know how the plane rises above this height as a function of x, given by h₂(x), then the measured height of the plane as a function of x, h(x) enables you to solve for c, because this physical system is described by a linear system of equations. The final height of the plane has to be a linear combination of the two functions of x, h₁(x) and h₂(x). It turns out to be a simple matter to figure out what proportion of each function is needed to achieve that height.

This is an example of linear regression. Solving for the magnitude of a constant slope hill is another example of linear regression, but in this case the fact that the function of x is itself linear is in fact irrelevant. What is relevant is that the height (measured relative to some arbitrary height such as the sea level) is the linear combination of a constant term and another term (i.e. function of x) which just happens to be linear.

In a follow-up post I will talk more about linear systems, before moving on to the concept of non-linearity. By that stage, what might appear here to be nitpicky hair-splitting becomes genuinely important to actually understanding what is going on.

the strange world of mathematics

Mathematicians use all sorts of evocative words and phrases to describe some of the fancy games they play with numbers and spatial constructs. A few posts back I quoted Penrose talking about fibre bundles, for example. This is supposed to bring to mind an image like a hairbrush, i.e. a bundle of fibres (the bristles). Each bristle is the same kind of object and locally the way the structure fits together looks the same all around the brush, but in fact mathematically what the bristles represent can be "twisted" like a Mobius strip. I have only the dimmest notion of what this means or what the point of it is, but I gather it has something to do with mathematical structures that have symmetries (like a bristle that looks the same if you turn it upside down) enabling the global structure of the bundle itself to change. Apart from delighting mathematicians in an abstract sense, this concept of a fibre bundle is apparently relevant to the modelling of so-called gauge symmetries in theoretical particle physics.

But before you know it, you're reading about really weird-sounding things like natural fibrations and jet prolongation functors. Even though I have no clue what is going on, I love stumbling upon articles that are replete with sentences such as: Let us also mention the jets of modules over a commutative ring. Sure, go ahead, why not mention them!

Apart from all these crazy notions from the more exotic realms of mathematics, most of us, I think it is true to say, find it hard enough keeping straight such seemingly elementary notions as the meaning of the word "nonlinear." One hears of the effects of nonlinearity in creating real world behaviours such as shock waves and turbulence but what does this actually mean? The exponential growth in a population is certainly nonlinear, yet it arises as a simple solution to a very straightforward linear differential equation. In a nutshell, the word "nonlinear" is usually only of significance when it describes the nature of a system, rather than the time dependence of what goes in or comes out of the system (which is basically without exception a nonlinear function). But more of these mundane matters next time.

thoughts on the meaning of health "insurance"

In the current health care debate going on in the US right now, one of the major issues has to do with individuals who can't get private health insurance because they have a costly existing condition. Similarly, individuals who develop a condition that requires ongoing expenditure that is covered by their current insurer are afraid to change jobs because their medical status will make it hard for them to get insurance from a new company.

There seems to be a common misconception about how insurance has to work in order for it to really work. In general, there are two different types of situation where insurance companies are known to restrict their munificence. The first is where someone pays a premium and later becomes rightly eligible for an insurance payment, but the insurance company tries to wriggle out by coming up with unjustified excuses. This, needless to say, is an unethical business practice, and ideally the company's reputation should suffer in such a way that there is in general a strong business disincentive to engage in such practices.

The second situation concerns an insurance company's decision not to insure certain individuals. I have often heard proponents of private-only health insurance saying that there should be legislation to prevent this sort of thing. But this simply denies the reality of how insurance works, where the premium charged is normally directly proportional to the estimated risk. Insurance works when the risk is reasonably low but the possible loss is high. A large group of individuals cooperate in a highly effective manner so that each person's financial exposure is restricted to the cost associated with their personal risk. If the insurance company decides to offer insurance to individuals with ten times my own personal risk, I have no problem with that because they are paying their way.

If an insurance company decided to charge the same premium to everyone, and insured all applicants, then the premiums would have to increase - I would assume substantially in the case of health care costs. Even a not-for-profit mutual society would not be able to work this way, because the premiums would be too high for some individuals. This is why some hold the opinion that the more caring, charitable aspects of a nation's medical system should be organized as part of the national taxation system. This incorporates the concept that poorer people's health care is subsidized. But it also removes the conundrum of the individual who has developed a condition that is covered while he stays in his current job, but not if he changes jobs. For, if he does change jobs, his relationship to the government as an insured tax payer doesn't change.

But a single insurance scheme has far broader implications. For, in fact, before someone is born, they have already implicitly signed up with the scheme, which then considers each individual to have the same overall risk. So that now, apart from ability to pay, each member of the population really should pay the same premium, because they are assumed to start with equal risk. Of course genetic testing potentially spoils this ideal concept. And I make no apology for the fact that idealism is involved here. But if one is to use the word "insurance" to cover both government and privately run schemes, then this is surely what is actually behind a single government run system. Ultimately, any one of us might have been born with some severe medical condition, and it is in that sense of shared humanity that we can dream of a system where someone born with such a "pre-condition" was in a sense already signed up for an insurance policy before the risk itself turned into a certitude.

lost in the equations

Mathematics is my poor excuse for disappearing off the blogger radar screen for over three months. It started with Penrose's The Road to Reality which is pretty heavy on the math, even when accompanied by the author's rather splendid - and sometimes clarifying - illustrations. Penrose has written books, such as The Emperor's New Mind, that were aimed a bit more squarely at the "general reader," which of course here really means someone who is already fairly well versed in scientific notions and is highly motivated to read more of the same, i.e. not all that general a reader after all! But Penrose's aim for The Road to Reality is quite charmingly naive. According to the Preface, one supposes that Penrose set out to take a mathematically incompetent individual who can't even deal with fractions, and to lead them steadily towards the sort of bedtime reading that enchants with sentences such as:

A bundle (or fibre bundle) B is a manifold with some structure, which is defined in terms of two other manifolds M and V, where M is called the base space (which is spacetime itself, in most physical applications), and where V is called the fibre (the internal space, in most physical applications).

So this led me to thinking about whether or not it really would be possible for someone like myself - with a moderate level of mathematical sophistication somewhere between the fraction-challenged person and Penrose himself - to:

a) actually try and stretch a bit and gain some additional mathematical ability so that
b) I could actually try to communicate the essential mathematical concepts at least to someone else with a moderate mathematical background if not to a more general audience and
c) what would it be possible to truly communicate to an intelligent but mathematically unversed audience.

Meanwhile, I started to think about the mathematical concepts that I often need to communicate as best I can to other scientists with whom I work, usually issues concerning statistical analysis of data and so on. There is a field known as "science communication," which refers to the communication of scientific ideas to the "layperson." But as a scientist, I am aware that scientists need to learn better how to communicate essential ideas and methods to other scientists, e.g. statisticians need to communicate their knowledge effectively to less mathematically inclined researchers.

Astute readers of this blog may have noticed that I often have a problem in aiming at a clearly defined audience. There is a part of me that shares Penrose's naivety in believing that extremely careful communication can bridge any gap and a marvellous flow of ideas can occur between people, irrespective of their intellectual backgrounds or inclinations. I still think that this is an ideal to strive for, but realistically a misdirected communication can so easily fall between the cracks, in my case between the "lay" and the "scientific" audiences.

So during my blog hiatus, I have been thinking about how to deal in future with my desire to communicate both to fellow scientists and also to a more general audience. Do I start a second blog? Explicitly identify my target audience for each post? I still haven't quite hit on the solution, but I thought I'd start posting again and see what happens.

giving a parabola its legs

I remember being taught in high school how to find the solutions to the equation: y(x) = ax² + bx + c = 0. The parabola in the figure clearly crosses the horizontal x-axis twice and gives us two solutions for y=0 that are in fact:

x = ( -b ± √(b² - 4ac) )/2a

But if you lift the parabola up so that its vertex is above the x-axis, there is no way this curve is ever going to cut through y=0. Or is there?

All those years ago, I was shown that the problem had to do with the square root in the expression above. A parabola that is above the x-axis corresponds to a negative value for b² - 4ac. Since we would have to take the square root of a negative number to get a solution, why not define the so-called imaginary number i=√-1 and just carry on! This is one way to motivate the invention of complex numbers, and I thought that was so cool or incomprehensible or both that I never went back to the geometrical picture of what's going on.

As many readers will know, the standard approach is to define a complex plane of numbers by adding an imaginary number axis perpendicular to the more familiar (real) number line. I have talked about the beauty of this before, here and here. A complex number is made up of a real part and an imaginary part, which is why you need a plane to describe these numbers geometrically. Now let's imagine what y=x² looks like when x is a complex number that can lie anywhere on the complex plane.

Our normal picture for y=x² is a parabola with a vertex at x=0 with arms rising majestically above the real x-axis. What about y as a function of a purely imaginary x=ai (where a is just a real distance along the imaginary axis)? Well, y = x² = a² i²= -a² (since i² = -1) so that y now takes negative values. In other words, we take a copy of our usual parabola, rotate it by 90 deg so that it is above the imaginary axis, and flip it so that it points down. This is nicely shown for a general parabola here.

For any x not on the real or imaginary axes, y is complex, which turns out not to be helpful for finding solutions to y(x)=0. But with two parabolae pointing in opposite directions, we've really got the y-axis covered. Now we can place the vertex above or below y=0 and we will have two arms ascending towards ever more positive y values, and two legs descending towards ever more negative y values. If our parabola has arms and its vertex is above y=0, all we need do is create the legs and follow them down to dig up the two complex roots to its equation.

When you think about it, the initial problem only occurs for polynomials of even degree, i.e. y=ax²+... or y=ax⁴+... because these are the ones that have arms or legs that end up pointing in the same direction. Obviously y=ax+b is going to cross the x-axis somewhere and so will y=ax³+... etc. The invention of complex numbers results in the even-degree polynomials acquiring both arms and legs so that they too are bound to cross y=0. Because we only have a single problem here - ensuring that all polynomials cover all negative and positive values of y - we don't need to invent anything more exotic than complex numbers. I love this geometrical picture, because it gives us an extremely informal demonstration of this so-called closure of the complex numbers, which is known in fact as the fundamental theorem of algebra.

brake cables and drip coffee machines

A while back, I had what might be called an epiphany of ignorance. In such a situation, what makes the sudden insight so mind-broadening is the realization that one has lived for so many years on this planet in complete ignorance of a simple aspect of everyday life. In this instance, a colleague at work said he was pretty sure that the plastic housing around bicycle brake cables have metal coiled around the inside. Being naturally argumentative, I started to disagree without having given the issue a moment's thought. Eventually I started to come around, and we eventually figured out that there has to be a push on the housing to balance the brake lever's pull on the inner wire, with confirmation from an informative website on bicycle minutiae. The author says it is all about Newton's third law ("every action has an equal and opposite reaction"). Being Newton's Ocean and all, I have given this rather too much thought, and I have come to the conclusion that it is more to do with the need to balance forces so that an object compresses rather than undergoes a bulk acceleration.

If we start by imagining a centre-pull style front brake, just pulling on a bare wire should work as long as you are sitting firmly on the seat, so that a force is also transmitted downwards to keep the front of the bicycle from lifting upwards. In principle, I suppose one could still end up lifting the front of the bike right off the road that way, so the use of a reinforced cable housing to transmit the necessary downward balancing force along the same path as the upward pull on the brakes via the inner wire makes more sense! Plus it allows the cable to bend and still transmit a differential force between the inner and outer components. It also allows for less symmetrically designed brakes such as side-pull and linear-pull systems, where the inner wire is connected to one side, and the balancing compression force from the housing is applied to the other side.

So there you have it. The other day, this same colleague confided that he sometimes pours day-old coffee into the reservoir of our communal drip-style coffee machine, in order to reheat it. This led to a discussion about whether this would gunk up the machine, which led to a debate concerning how exactly such a machine works! The issue was whether the water would boil and deposit any residue in the reservoir. Now I kind of knew that the water didn't get totally turned into steam, but how does it move up the tube to get to where it drips down onto the coffee grinds? It turns out that pockets of steam and a one-way valve ensure that the heated water moves against gravity up the tube.

Perhaps the real epiphany is how much clever stuff there is in the "simple" things around us.

the trouble with physics by lee smolin

Lee Smolin works just down the road from me. Well that's a slight exaggeration - I live and work in Toronto, while Lee Smolin works out of the Perimeter Institute in Waterloo, about 100 km west of Toronto. The PI was founded in 1999 with help from the Canadian company Research in Motion (RIM), also based in Waterloo and best known for the BlackBerry.

Smolin is one of the originators of loop quantum theory, which is basically a rival to string theory. Reading The Trouble with Physics, published in 2006, I kept wishing for more about loop quantum theory, but I'll just have to get myself a copy of Three Roads to Quantum Gravity, and hope it's not too out of date (it was published in 2001).

Early in The Trouble with Physics, Smolin lays out the five big remaining problems in physics:

1. Quantum + gravity unification
2. Quantum foundations
3. Particle + force unification
4. Freely tunable parameters (in particle physics)
5. Cosmological discrepancies (dark matter and dark energy)

If I understand correctly, the reason Smolin favours more "foundational" methods - for example, to unify the quantum world with Einstein's gravity world - has to do with a preference for "background-independent" methods. Whereas Newton's laws play out on a fixed background of Cartesian (or Euclidian) space and an equably flowing arrow of time, Einstein's general relativity is well-known to define the very space and time that the events of the universe unfold across. So theorists who continue in the spirit of Einstein are like artists who do not depend on a predefined canvas but create everything from scratch. The ultimate appears to be a theory in which not only particles and forces can be seen as emergent properties of a vacuum, but space and time can themselves emerge - possibly in some quantized state. Needless to say, theories such as loop quantum gravity are background-independent whereas quantum mechanics, the standard model of particle physics, and (super)string theories are background-dependent. Now we all know that superstring theories require extra dimensions that are supposedly curled up so we can't see them. But I guess this is like an artist having a technique that requires a canvas with various quirky features, rather than a different artist whose canvas emerges as a natural part of his or her art.

A seemingly bigger issue is the fact that there are a huge number of string theories that can apparently be devised by changing any of a large number of free parameters. The "super" in superstring means proposing that fermions and bosons have supersymmetric partners, so a selectron is a boson partner to an electron, while a gluino is a fermion partner to a gluon. None of these particles has been observed, but with all those free parameters it is easy to just propose that they are way too massive to have been created in any accelerators. Abstruse mathematical models can be proved to be consistent (although even this is pretty hard at this level) but they do not necessarily correspond to reality.

Smolin talks a lot towards the end of his book about the sociology of today's physics departments, with hiring practices that favour those who will work on the now-popular string theory approach. But I think the more fundamental issue is the one he alludes to earlier concerning the adoption of the anthropic principle. Since it has become hard to test these theories experimentally, the anthropic argument implies that a physicist should pick a model universe in which he/she could exist, and which can be rigged to not look obviously different from our world. But this is hard to do, and perhaps these physicists have become just as intrigued with worlds that are mathematically possible rather than truly anthropic. In this case, theoretical physics departments may have turned into specialized mathematics departments, and string theory is now preferred precisely because it is such a rich source of mathematical worlds, even if none of them even remotely corresponds to our world!