Newton's Ocean

linear thinking

noreply@blogger.com (NewtonsOcean) — Fri, 09 Oct 2009 23:42:00 +0000

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

noreply@blogger.com (NewtonsOcean) — Thu, 17 Sep 2009 23:30:00 +0000

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"

noreply@blogger.com (NewtonsOcean) — Fri, 28 Aug 2009 00:31:00 +0000

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

noreply@blogger.com (NewtonsOcean) — Tue, 18 Aug 2009 01:17:00 +0000

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

noreply@blogger.com (NewtonsOcean) — Sat, 09 May 2009 00:07:00 +0000

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

noreply@blogger.com (NewtonsOcean) — Sun, 26 Apr 2009 21:39:00 +0000

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

noreply@blogger.com (NewtonsOcean) — Sat, 18 Apr 2009 18:30:00 +0000

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!

the viruses and the worms

noreply@blogger.com (NewtonsOcean) — Mon, 13 Apr 2009 21:00:00 +0000

In the beginning was the hardware, although it wasn't much use without some software. And then, to spoil the Edenic harmony of Intel and Microsoft (I'm simplifying the history of computers a bit here!) there came the viruses and the worms, those evil packages of ones and zeros that every computer owner and network administrator fear.

The conficker worm has been in the news off and on since October of last year, and has of course become much talked about since it awoke from its dormant state on April 1 of this year. I was going to write about it anyway, and then I got to see a friend's PC a few days ago after it had been infected with something - although the symptoms didn't seem to fit those of any of the conficker variants. Believe it or not, this was the first time I'd actually seen a computer that was obviously infected. It was kind of spooky to see the internet browser jump to websites other than what you had selected. After installing Norton's Internet Security presumably way too late, it refused to run a full system scan and soon refused to fire up altogether.

How much is a computer virus like its biological cousin anyway? Since a biological virus is basically a computer program written in the four base code of its DNA or RNA, and it replicates itself using the cellular machinery of a host organism, the computer version is obviously quite aptly named. A big difference, of course, is that computer viruses have not been configured to mutate as they spread. (Earlier versions of conficker can apparently update to the latest version, but that is a different can of worms altogether.)

Meditating on both forms of virus makes one think about the stuff we call "information." Viewed from one angle, a biological virus seems like such a harmless assortment of letters from the genetic code of life. But of course those letters are inextricably tied to the powerful workings of life, and the information packed into a virus has the power to spread horrible diseases among humans and many animal species. Biological viruses are such curious entities because it is debated whether one can even define them as life forms or not. It is therefore hard to view them as intrinsically evil. (Then again, a single-celled bacterium does not sit around and ponder whether to wreak havoc on a population of humans either!) Computer viruses are also just pieces of information, but - being the product of intelligent design - carry the explicit malevolence of the designer. It would be a cheap shot at the ID community to make the obvious logical step concerning an assignment of good or evil to a designer of biological viruses. But it is surely an interesting starting-point in a discussion of whether the "stuff" of good and evil appears to be a basic constituent of our universe, or whether it in some sense simply emerges within the context and structuring of life, where information and knowledge can unleash the power to create and destroy.

laser-induced fusion

noreply@blogger.com (NewtonsOcean) — Fri, 03 Apr 2009 19:00:00 +0000

Well I've been busy trying to make a start on some basement renovations but I decided to come up for air and to make contact with my poor neglected blog. I just happened to notice the curious image (left) on the main page of wikipedia which led me to today's post. A few days ago, the US national lab in Livermore announced that their National Ignition Facility is now completed, tested and ready to start doing real experiments in inertial confinement fusion.

Many years ago, I remember being aware of magnetic confinement experiments using donut-shaped prototype fusion reactors. These "tokamak" reactors with electromagnetic coils wrapped around their donut shape are rather like small versions of particle accelerators, with the fusion fuel forced to circulate around the middle of the donut. Fusion holds the promise of clean energy (just like the Sun's!) since the idea is to use the same nuclear fusion reactions involving isotopes of hydrogen (deuterium and tritium) that occur in the Sun. This, of course, doesn't obviously indicate that the energy production would be clean. Deuterium and tritium combine to form helium innocuously enough. But all the unreacted tritium is radioactive, which has to be taken into account during decommissioning, but its half-life is only 12 years so it doesn't represent a long-term headache. However, the sides of the reactors themselves become radioactive due to exposure to energetic neutrons that are produced along with the helium. Nevertheless, if fusion energy became feasible via magnetic confinement, these radiation concerns are very different from today's nuclear fission reactors that create long-lived radioactive waste in direct proportion to the amount of energy created. The real issue is that no one has managed to get more energy out than they have to put in to keep the deuterium and tritium confined and hot enough to react.

But I am ashamed to admit that I was totally ignorant of the laser approach to the problem of attaining temperatures and pressures equivalent to the interior of the Sun. I feel so out of touch here it's as though I've been hiding in my basement for years not days! This laser inertial confinement method is reminiscent of the H-bomb design, whereby a fission explosion is used to implode the fusion material to create the higher temperature and pressure for the fusion reaction to proceed. Indeed, the number one motivation behind the National Ignition Facility is in fact to continue H-bomb research without doing full-scale tests. They will use 192 extremely powerful lasers which converge on a tiny pellet of fusion fuel inside the target chamber shown in the picture. The final amplified laser energy will give a total power of 500 trillion watts. This is about 1000 times the electrical power that is being used across the whole of the US at any one time, but it is only maintained for a split second. After this the optics get distorted from the heat involved in the light amplification and the system has to cool down for several hours before the lasers can fire again. Ignition is said to occur if more energy comes out than goes in.

Whereas an isolated laser firing performs a valid simulation of an H-bomb detonation, a power station would require a constant stream of fusion fuel pellets to be ignited, so that a continual stream of fusion energy could be made available to convert water into steam to run turbines for electricity generation. France has a similar project called Laser Megajoule expected to be completed next year. A large European High Power laser Energy Research facility (HiPER) aims to focus on more efficient ignition methods by using one laser pulse to provide adequate compression when combined with another laser pulse to heat the compressed fuel to reach ignition conditions.

These big science projects are all very impressive displays of technological prowess, but for many years, the promise of power stations running on nuclear fusion has remained a far-off dream. There's something comical about playing around for years with extremely expensive facilities where you always end up pumping in way more energy than you get out. No wonder we all got excited by the possibility of cold fusion!

pyramid schemes and falling dominos

noreply@blogger.com (NewtonsOcean) — Sat, 21 Mar 2009 23:36:00 +0000

So Bernie Madoff has pleaded guilty to a pretty massive investor defrauding scheme. Like Charles Ponzi, he started small, but as former chairman of NASDAQ, his fall in reputation has been rather greater. But was Madoff's truly a pyramid scheme?

A pure pyramid scheme collapses when the ever growing numbers of new recruits that are required to fuel it hits a limit imposed by finite resources. Ponzi's scheme relied on the rather ineffectual-sounding business of buying international postal reply coupons in Italy and selling them in the USA. In his case, the finite resource was the reply coupons themselves. So when a newspaper pointed out that his business would require about 6000 times the available supply of coupons, it became obvious that he must have turned to forging them.

But was Madoff's scheme - fraudulent as it evidently was - bound to collapse eventually? I have no idea, but he had been going strong for quite a while and his weakness only became exposed when the domino effect of the sub-prime mortgage fallout hit. Now don't get me wrong - fraud is fraud and I'm not trying to find excuses here. While a reasonably informed investor knows that his or her money may get eaten up by the market, any losses should all be above board and not due to a broker using your investment capital to feed someone else's fake returns, even temporarily. The funny thing is that if Madoff's scheme was primarily intended to demonstrate reliable but not unrealistic returns by temporarily borrowing from investor Peter to pay Paul during dips in the market, then he might have got away with it in perpetuity as long as there wasn't any panic-driven run on his "bank."

Again, I stress that I am not condoning his activities in any way, especially as I am too ignorant of precisely what was involved anyway. But I think it is correct to say that even conservative banks would have a hard time if everyone decided to pull out their cash at once. Since regular bank accounts are insured by the federal government, there is no reason why there would be a run on these accounts. Needless to say, this is not true of the stock market, which still has inadequate safety systems to prevent the sorts of extreme volatility that occur when people start to panic simply because they know others are panicking. The only thing we have to fear is fear itself etc.

In terms of physical systems, the stock market is like an elevator on a bungee cord. Instead of dropping down smoothly to a lower floor in response to some newly emerging reality that "the economy" needs to cool off, the market overreacts and drops down way further than required for a realistic correction. The system is mostly designed to enable microeconomic transactions to occur very quickly, with potential investors able to get their hands on up-to-date information. It is precisely these design criteria that cause problems when the mass media start to broadcast panic signals not about one particular company but about the economy as a whole.

So yes, Madoff should be made to pay for his recklessness and what was clearly totally fraudulent practice. But what would be so much more valuable than throwing all the Madoff's in jail is if we could figure out a way to keep the market as responsive as it currently is when a regular number of investors jump in or out, but add some sort of natural brake when the herd instinct starts to occur. Communist countries have traditionally enjoyed control over the mass media. How ironic to muse on the fact that the free market can get damaged so badly by the pronouncements of doom that are fed to us by the free press!

happy belated pi day

noreply@blogger.com (NewtonsOcean) — Mon, 16 Mar 2009 22:52:00 +0000

Last year, I wrote my sixth post on March 14, in order to help celebrate Pi Day. This business of celebrating a bit of math on 3.14, just because these are the first three digits of the irrational number known as pi, is a definite case of math trumping physics, since we should really have been celebrating the anniversary of Einstein's birth on the 14th.

Einstein was a pretty straightforward and rational kind of guy. The number pi is also a straightforward enough character, being the simple ratio of a circle's circumference to its diameter, but everyone knows it is irrational. Well I thought this was well-known until I browsed through the historian Fernandez-Armesto's Ideas that Changed the World recently, where he confidently tells us that pi = 25/8 or some such rational approximation to the real pi. I don't recall what he thought the number pi had done to change the world, but it wasn't anything to do with its irrationality.

A safer bet for revealing the wonders of mathematics is a book like Bridges to Infinity by Michael Guillen. I have a feeling it is out of print - my copy dates from 1983, when it was initially published. I confess I've only ever dipped into it - I don't know why I haven't read it cover to cover because it contains some nice accessible mathematical insights. For example, Guillen demonstrates how Georg Cantor could be sure that there were more irrational than whole numbers, even though - of course - there are an infinite number of the latter. Here goes:

Imagine you can list all the irrational numbers in order and you label them using whole numbers:

1 .17643...
2 .23482...
3 .62346...
etc...

Obviously these aren't the first three but you get the idea. Cantor then showed that you can make a new irrational number that is different from every single one on this list, even though it's supposed to be the complete list! You take the first digit of the first irrational number on the list and pick a different digit to start the new number, so you know it will be different from the first number. You change the second digit from the second number and so on. Clever huh?

Happy belated irrational Pi Day!

believable theories of everything

noreply@blogger.com (NewtonsOcean) — Sun, 08 Mar 2009 15:41:00 +0000

A few years ago I got about half way through The Elegant Universe by Brian Greene, which is a classic introduction to superstring theory. I'd like to think I gave up because of intellectual skepticism, but I think I just ran out of steam. But now I'm cruising through Lee Smolin's The Trouble with Physics. Both books are written by experts in "theories of everything," but they line up on opposite sides of one of the great debates. Namely, whether superstring theory is terribly clever and exciting or whether it's Not Even Wrong.

Greene is known for his ability to popularize what is in reality a ludicrously arcane mathematical theory. The motivation behind this blog is the notion that synthesizing scientific concepts and trying to present them intuitively to a lay audience is an extremely worthwhile exercise. Science becomes a beautiful aesthetic experience when something messy can be described by an elegant theory. Such a theory may perhaps still contain technically difficult mathematics, but if it proposes a model of reality that gives us insight at some intuitive level, then it can be appreciated by scientist and layperson alike. Also, it becomes intuitively believable. This follows the principle of the medieval scholar William of Ockham, who proposed that the theory that explained the most by way of the least knobs and twiddles is to be preferred. For example, proposing that the earth and other planets orbited the sun led to way fewer twiddles in describing the motion of those other planets viewed from the earth.

Then there is the aesthetic of finding a theory that provides a more unified explanation of nature. A good example is Maxwell's theory of electromagnetism. The equations can look a bit daunting to a layperson but there is certainly something very elegant here. There is a nice symmetry to the way the electric and magnetic fields enter the equations. There is incidentally a lack of symmetry in that electric charges are incorporated into the model, but the magnetic equivalent - magnetic "monopoles" - are conspicuously absent. But when we just look at the fields, Maxwell's equations predict the propagation of an electromagnetic wave at a speed that can be calculated from two properties - one electric, the other magnetic - of a vacuum. And this speed is exactly the speed of light (in a vacuum). This implies that the speed of light is fixed, regardless of the speed of any observer, and this, of course, was Einstein's starting point for his theory of special relativity.

The "theory of everything" physicists are continuing this tradition of trying to unify areas of physics. Faraday showed experimentally that electricity and magnetism were connected, and Maxwell finished the job off mathematically. This led to Einstein's special relativity, which unified space and time, and also mass and energy. Oh, and he showed even more deeply than Maxwell how electricity and magnetism are in fact the same thing, only viewed at different relative velocities. Then came Einstein's theory of general relativity, which unifies gravitation with acceleration, and hence explains the equivalence of gravitational and inertial mass, while also demonstrating that matter influences the geometrical properties of space.

Although few of us have much grasp of general relativity, we know that there is something elegant about it. But as one reaches out in an attempt to unify ever more diverse phenomena, one's theories may of course develop an alarming number of knobs and twiddles. And in fact, things have become quite messy. There are three forces that affect elementary particles, and then there is gravity, which is only noticeable when there are vast numbers of elementary particles grouped into astronomical sized objects. Not surprisingly, gravity has always been hard to connect up with the other three forces, which are part of the quantum mechanical world of subatomic particles. So let's keep gravity out of it for now. This leaves us with the Standard Model of particle physics, which is a weird combination of elegance and messiness, as some deep symmetries of nature have been "spontaneously broken" while others remain intact.

We live in a world where different kinds of matter share some properties but not others. Some carry electrical charge, and so are sensitive to electromagnetism. Some carry the nuclear charge called color, and so are sensitive to the strong nuclear force. And all carry weak isospin and hence are all affected by the weak force. In their respective domains, these charges are conserved, corresponding to underlying symmetries. But electrons are fundamentally different from the quarks that make up protons and neutrons, the different quarks have different masses, and all of this represents a breaking of symmetries.

Similarly the three forces are carried by particles that are very different from each other. The familiar photon that carries the electromagnetic force is massless, as is the gluon that carries the strong force, whereas the weak force is carried by three very massive (and shortlived) particles. And yet, the photon and weak force carriers can be shown to be beautifully related in a unified "electroweak" theory that also connects electrons to neutrinos and explains how neutrons can change "flavor" and turn into protons during radioactive decay.

Needless to say, force particles are different from matter particles. Yet superstring theory aims to connect even these aspects of reality together in an overarching theory that combines all four forces of nature and all flavors of matter together. This makes Maxwell's synthesis of electricity and magnetism seem quite modest. Modest yet elegant has been superceded by extremely ambitious and extremely messy.

This may sound like an excuse, but I think the reason I stopped reading The Elegant Universe was that I knew that Greene was having to hide all that messiness under the rug in order to convince me of the elegance. And so I couldn't even begin to judge the believability of any of it. I feel more comfortable with Smolin's skeptical approach, and I've finally caught the theory of everything bug.

from leaning towers to particle accelerators

noreply@blogger.com (NewtonsOcean) — Mon, 02 Mar 2009 00:18:00 +0000

So I got The God Particle out of the library a week or so back. At first I loved the sense of humour that spills out irresistibly as the Nobel-prize winning Leon Lederman tells his story of particle physics. Near the beginning he has this extended - very extended - conversation with Democritus' ghost about the modern-day quest to find the ultimate indivisible building blocks of the universe. Along the way the jokes keep coming until you feel that Lederman is doing more groaning than you are.

Now Democritus coined the word "atom," and it's funny because I always knew that this meant uncuttable in Greek. But it wasn't until seeing Lederman write it as a-tom that I made the connection with "tomography," which means the imaging of slices, as in computed tomography, or CT. Duh!

I confess that the book had to go back to the library before Lederman got anywhere close to the god particle, aka the Higgs boson. This is the massive weakly-interacting critter that the Large Hadron Collider will be looking for when it starts up again after its initial technical difficulties last September. But along the way, Lederman's historical tour made a stop in Pisa where Galileo is contemplating whether to throw different weights off the conveniently leaning tower of his birthplace. Everyone since Aristotle had been hoodwinked into believing that a heavier weight has a stronger purpose to fall and so will reach the ground faster. But does it? Why didn't they just do an experiment? Well, of course, air resistance may have confused the casual observer but how hard would it be to do a half decent test?

Lederman indicates that Galileo didn't even need to do a test. He only had to perform a thought experiment to convince himself of the correct answer. I had never heard of this piece of logic and I think it's ever so clever. Assume that the lighter object does fall more slowly than a heavier one. Now tie the two objects together. The lighter object should try to hold back and slow down the heavier object, yet together they form an even heavier object that should go faster. We have a contradiction that is only resolved if indeed the two objects take the same time to fall any distance.

Anyway, I'll take a shot at talking about gauge theories and spontaneous symmetry breaking another day. If only particle physics involved such simple thought experiments or such inexpensive real experiments as the physics of Galileo's day!

hidden factors

noreply@blogger.com (NewtonsOcean) — Sat, 28 Feb 2009 17:06:00 +0000

In my last post, I tried to emphasize the difference between the evolution and climate change debates. I made a joke about god being less easily measured than sea ice. I was mostly trying to emphasize that we ought to all be able to come to a basic agreement about the plain facts that are relevant to the climate change debate, so the comparison to a theological debate was a bit of a throwaway comment.

In the climate change debate, the issue is whether anthropogenic factors can explain most or all of the observed climate fluctuations. Are there hidden factors that should be considered? In the evolution debate, the issue is whether Darwinian mechanisms can explain the observed variety of species from a primordial starting point. Or is there a hidden factor such as a biblical-style creator god?

My throwaway comment made the mistake of presuming that the existence of god is a defining issue when debating evolution. This is of course not the case, given that there are loads of Christians and other theists who embrace evolution, several of whom do me the honour of regularly reading this blog. There is of course yet another debate, the most fundamental one of all, in which the question is, whether a scientific view of the world can potentially explain everything, and if not, what kind of hidden factor should one add in to the model?

Both the evolution and climate change debates revolve around a set of observations that have been made in the present and near past and include the interpretation of records that stretch way way back into geologic time. In the case of evolution, these ancient records are fossils and also the genetic record. In the case of climate change, the ancient records are the isotopic concentrations of bubbles of gas trapped at various depths in the cores of ice that are brought to the surface.

All scientists know that extrapolating even a relatively simple graph beyond the range of actual measurements can be extremely dangerous. In the case of evolution, it is not a graph that is being extrapolated but a fundamental principle of how living organisms change over time. So if there is adequate evidence that this principle operates over some observable time-scale, the simplest explanation is that this principle operated all the way back. This does not rule out the possibility that some radically different mechanism may have operated at some earlier, unobserved time. But for someone proposing another scenario, the onus is on them to find supporting evidence.

Climate depends on numerous factors. Unlike evolutionary theory, climate change science is not about a simple principle. It is all in the details: specifically, whether the recently observed changes are attributable in large part to human activity. We all know that there have been some huge climate changes that occurred on this planet way before living organisms started to pump massive amounts of oxygen into the atmosphere, let alone before humans found all that organic matter and gave it the chance to release its rightful contribution of carbon dioxide into the atmosphere. We also know that life itself is capable of having a profound influence on the composition of the atmosphere: the presence of oxygen being the obvious example.

Apart from the apparent seriousness of the climate change debate, it's a fascinating situation in terms of assessing the credibility of scientists' claims. On the one hand, the doubters do have a point: when you look back at past records of scientists' attempts to extrapolate into the future, they have often overlooked other perhaps hidden factors that become the dominant aspects of the issue over time. Some explanations we accept far more readily than others. When we are told that certain species of fish are disappearing from our oceans, nobody denies that it is due to anthropogenic overfishing. But when we see the loss in sea ice over a roughly equivalent period of time, we are understandably not so sure. We know that humans can dramatically impact global aspects of the biosphere, but we expect the earth's climate to be more resilient, especially with respect to something as ubiquitous as carbon dioxide.

Frankly, I know relatively little solid information about climate change science, and I am hoping to thoroughly educate myself in this area over the next year or so. Even when I have a certain gut belief in something, I like to approach it as a skeptic. I believe that is what science is all about. So I hope to bring some critical appraisals of climate change science over the next few months.

politically-influenced shrinkage factors

noreply@blogger.com (NewtonsOcean) — Wed, 25 Feb 2009 00:15:00 +0000

When scientists with an atheistic bent argue with fundamentalists about whether God did or did not have a hand in the manifestations of life on this planet, the trickiest aspect of the debate is that it concerns an entity that does not reveal himself directly enough to be weighed or photographed.

But the other great science-related debate of our times concerns physical quantities that can be measured, such as whether there has been any change in the extent of sea ice over the last few decades. Given the accuracy with which these quantities can be measured, the facts should be the same, whether they are perceived by left-wing "let's intervene and fix it" types or laissez-faire right-wing types.

Earlier this month, conservative columnist George F. Will let rip with some skeptical opinions concerning global warming. This included a claim that the sea ice extent has rebounded to 1979 levels. But the research group he was supposedly quoting in fact claims that the arctic sea ice is down 8% compared to the same time of the year in 1979, as visualized here.

Now my point is not in fact to criticize anyone who honestly doubts whether humans are dramatically affecting the climate on this planet. My point is that the debate should be focused on the interpretation of the facts, which is quite a tricky business, not the facts themselves, most of which are pretty hard to dispute. Once humanity figured out how to get satellites up, measuring the extent of sea ice is frankly not rocket science. And for a newspaper of the supposed quality of the Washington Post, checking a simple fact like this is a whole lot easier than checking whether Deep Throat was telling the truth during their award-winning reporting of the Watergate scandal back in the 70s!

Of course the proposed solutions to an acknowledged problem will likely be different according to one's political stripe; that's as it should be and is what makes democracy so much fun. Of course this sets up a bias in how believing or skeptical one is concerning a certain interpretation of the facts, which is regrettable but is hard to avoid. But in a "climate" (groan!) of intelligent debate, it should be in neither side's interest to cherry-pick or distort the facts, because by doing so they ought to lose credibility.

I actually only got around to seeing Al Gore's "An Inconvenient Truth" last weekend. I found it quite persuasive and Gore appears to have a lot of facts at his disposal, probably a lot more than someone like George F. Will. Well I would hope he does, since he is devoting so much of his time and energy to the climate change issue. It occurred to me that, impressive as his rhetorical analogies are, and sincere and extremely well educated about the data as I believe him to be, his one-man show ends up further polarizing the debate - at least in the US - precisely because he can be labeled as anti-Republican.

In the 2000 presidential election, Gore won the popular vote (by a 0.27% margin) but Bush won the all-important electoral vote with a confused counting of the Florida vote where he won by a controversial 0.01% margin. Because this margin was so slim, the facts themselves were subject to debate - whether the votes had been accurately counted in Florida. Compared to these numbers, an 8% decrease in sea ice is huge. In the global warming debate, the facts are the easy part.

high-speed orbits

noreply@blogger.com (NewtonsOcean) — Sun, 22 Feb 2009 21:52:00 +0000

The news a week or so ago about the high speed crash between the Iridium communications satellite and a defunct Russian military satellite got me wondering about why orbiting satellites have to go so damn fast. The artist's impressions of these things always give you the feeling that they are just drifting around up there but clearly that ain't so. So I got out my envelope and started to doodle on the back of it...

First of all let's figure out how fast someone on the equator is moving simply due to the Earth's rotation. The radius of the Earth r₀ = 6,400 km. A person on the equator covers a distance equal to the Earth's circumference = 2 π r₀ = 40,000 km in 24 hours, corresponding to a speed of 1,700 km/hr! Of course we don't notice this because everything around us is moving the same way, but it gives us a reference point to work from.

Next, let's figure out what radius (r) orbit will keep a satellite over the same patch of Earth using only the effect of gravity to keep it at the right speed. Gravity must pull on the satellite just enough to change the direction of its velocity vector v through a certain angle over time, i.e. at a certain angular velocity (ω = v/r = 360^o/day). This change in velocity corresponds to an acceleration = v ω = r ω².

We know that the acceleration due to gravity at a distance r from the center of the Earth g(r) is proportional to 1/r².

So g/g₀ = (r₀/r)², where g₀ = g(r=r₀) = 9.8 m/s² (the acceleration due to gravity that we experience at the Earth's surface).

Now we equate g = (r₀/r)² g₀ = r ω² (*)

which gives a geosynchronous orbit at a radius r = 42,000 km. A satellite in this orbit must travel at v = r ω = 11,000 km/hr.

What about a low orbit such as the unfortunate Iridium satellite was recently in? These satellites apparently orbit about 780 km above the Earth's surface, i.e. r = 6400+780=7180 km. Plugging this into equation (*) gives ω = 360^o every 100 min, i.e. a complete circuit around the Earth every 100 min. This corresponds to a velocity of 27,000 km/hr.

Of course, if you're outside the Earth's atmosphere in nice empty space, any velocity feels like you're just drifting around. Unless it's not quite empty and you hit something.

colliding submarines

noreply@blogger.com (NewtonsOcean) — Thu, 19 Feb 2009 00:10:00 +0000

So what is the actual probability of any two objects colliding at random in the Atlantic? Well it was high enough for the Titanic to come a cropper on its maiden voyage of 1912, but then again there had been warnings of large numbers of icebergs and the Titanic was steaming right through their path as they drifted down from the coast of Greenland. And, needless to say, the Titanic and the icebergs were constrained to move around on the two dimensional surface of the sea.

Now of course neither of the two crew members posted to look out for icebergs had binoculars. Similarly, nuclear subs nowadays spend a lot of time in stealth mode, which means turning off their sonar in order to be more invisible. Blind but invisible. Now take two equally blind subs and let them prowl around the bottom of the Atlantic and see what happens. By choice, they constrain themselves to the lower depths of the ocean which increases the probability of a collision. Then they have their favourite trenches to hide in and apparently use temperature gradients to help evade detection - assuming another sub has actually turned its sonar on of course!

So earlier this month it really happened. A French and British sub - both nuclear powered and loaded with nuclear weapons - went bump in the night. Imagine how that must have freaked them out! The funniest part is that in a way - assuming the crews weren't all just fast asleep at the tillers - it shows how amazingly successful both subs were being at what they were supposed to be doing.

So this has been quite the month for spectacular collisions, what with the Iridium communications satellite smashing into a defunct Russian satellite at very high speed. These satellites were in low pole-to-pole orbits. Unlike the much higher geostationary orbits above the equator, the pole-to-pole orbits look like lines of longitude and in fact the satellites crossed paths almost at right angles to each other. They were travelling about 10000 times faster than the two subs so I guess it's a good thing that - unlike the subs - they weren't armed to the teeth.

Darwin, Wallace and consciousness

noreply@blogger.com (NewtonsOcean) — Thu, 12 Feb 2009 19:10:00 +0000

There are several reasons why Charles Darwin got more credit for the theory of natural selection than did his contemporary Alfred Russel Wallace, who entertained some rather similar thoughts on the subject. I think it is well established that Darwin's original thinking did indeed precede Wallace's and I am not about to spoil our commemoration of Darwin's achievements on this bicentennial of his birthday by discussing whether Wallace deserved more of the credit.

While Wallace was certainly quite the free thinker, Darwin was intellectually the bolder of the two. For him, the mechanics of evolution were powerful enough to bring forth adaptations as striking as consciousness itself. Wallace, on the other hand, believed not only in the materialistic workings of evolutionary theory but felt the need for an additional spiritual force in order to create life in the first place and later to create forms of consciousness in animals and humans.

Today, it is probably true enough to say that people who have thought about how consciousness might be explained can be divided into three main groups. There are the die-hard evolutionists, who believe that life and consciousness evolved from the raw materials of a materialistic universe. There are those, like Wallace, who believe that evolution can explain huge physical adaptations such that fish can turn into mammals just as shallow seas can become thrown up into dramatic mountain peaks, but reckon that some extra ingredients are required to explain consciousness. Lastly, there are those who believe above all in a spiritual force and choose sources of evidence supposedly revealed to us by this spiritual force instead of the materialistic implications of evolutionary theory.

Unlike modern-day creationists, Wallace was a very independent thinker and was certainly not hung up on issues of biblical truth. But he epitomizes the many people who are quite prepared to give evolutionary processes enormous powers over the physical adaptation of life-forms but who baulk when it comes to the evolution of consciousness. While it is easy to see the irreconcilable differences between the evolutionists and the creationists, the differences as exemplified between Darwin and Wallace with respect to this problem of consciousness are surely more worthy of sustained intellectual discussion.

Humans have in the past naturally been drawn to assume that an all-intelligent force created the universe, such that thought must precede existence. Everything we know about existence itself is a conscious knowledge, hence Descartes' assertion that "I think therefore I am." But Darwin's (and Wallace's) theories lead inexorably to the notion that our own ability as a species to develop self-conscious thoughts as individuals must have been the product of evolution, although Wallace could not accept it. And this is what we observe every time a couple of bags of DNA come together to form a new human being - a potential for conscious thought develops into another self-aware individual. Here indeed the emergent property of an individual's consciousness in some sense recapitulates its initial evolutionary development as an adaptive feature of an increasingly complicated central nervous system.

We live neither in a world of mechanical determinism nor in a world where strange spiritual forces must be invoked to breathe life and consciousness into dead matter. It takes a while to get used to the reality that the forces of random change and natural selection can create something as adaptively superb and complicated as an eye. There seems to be no scientific reason not to take the leap and include fully the organ of consciousness within evolution's grasp. By all accounts, Darwin himself probably had more difficulty understanding how on earth peacocks came up with those ridiculous tail feathers than the more sensible accomplishment of consciousness.

debating evolution

noreply@blogger.com (NewtonsOcean) — Sat, 07 Feb 2009 01:26:00 +0000

As you can see, I've stuck up an icon for the Blog for Darwin carnival. Please click on it if you want to help celebrate the 200th anniversary of Darwin's birthday (12th Feb) by writing a post or just reading the resulting "blog swarm." Thanks to eTrilobite.com for making me aware of it.

The phrase "blog swarm" makes me a tiny bit nervous. It seems to imply more than just an educational opportunity. Of course, the carnival rubric makes it clear that the intelligent design proponents are not welcome, which could be criticized as a lack of open debate, but it will at least prevent the otherwise inevitable slugfest. It still amazes me that the battle-lines are drawn so sharply between the Evolutionists and the Creationists. When I first thought about writing a science blog, my initial browsing put me right off because it seemed that all science bloggers ever did was to trade insults with creationists.

Now I do realize that there are real battles to be fought over how children should be educated in public schools. But outside of this quite specific concern, I see no point in either side wasting endless hours attempting to logically argue their case, when each side comes to the debate with completely different axiomatic starting points. If the children whose education both sides claim to care about ever hear the vitriolic abuse that too many on both sides hurl at their opponents, I only hope they can learn to behave in a more civilized manner when they themselves grow up, whether they believe themselves to be made in the likeness of god or of monkeys.

There is something fascinating about how radically different are these two opposing viewpoints concerning where we all come from and where we're all headed. In so many ways, it makes almost no difference to how people objectively appear to go about their actual day-to-day lives. One might think that religious conviction would turn most adherents into monkish types who shun all earthly concerns and focus purely on some spiritual dimension. And the materialist atheists should surely succumb to despair over the meaningless of existence. The fact that most of us strive to make the most of our time on this earth fits in with Darwin's ideas but is also not incompatible with a religious search for something "higher."

I believe creationists to be tremendously misguided. I also find that many scientists exude an arrogance that leads me to have more sympathy with those who search for a greater meaning behind it all. Surely some of the hatred that passes for debate about evolution is a sign of an insecurity that exists on both sides because, let's face it, the existence of this universe, with us sitting around on this earth contemplating it, is in fact a complete and utter mystery to us all.

oncogenes

noreply@blogger.com (NewtonsOcean) — Thu, 05 Feb 2009 04:08:00 +0000

Counting down to the 200th anniversary of Darwin's birth, I'm going to keep with biology related posts for a while. I think they're more popular than quantum mechanics anyway!

When I was a graduate student in medical biophysics more than twenty years ago, there was a group of physicists like myself interested in medical imaging and then there were the cell biologists. We spoke rather different languages, but every week we would have a double seminar so that we would all be exposed to talks given by members of both groups. Having had no biology education whatsoever, I found the cell biology talks quite exciting, when they were at least slightly comprehensible to me.

One factoid that I grabbed onto was about oncogenes. A normal cell has bits of DNA that code for proteins that help regulate how much the cell manufactures other proteins, which then affects what type of cell it is. Everyone knows that cancer has to do with a breakdown of normal cellular regulation. So, not surprisingly, when these bits of DNA suffer some mutation, they can become cancer-causing oncogenes.

But the really interesting thing is that certain viruses are carrying these very same oncogenes and this explains how some viruses can induce cancer when they enter the body's cells. The origin of viruses is uncertain. Whether they were totally derived from cellular genetic material that escaped from cells or whether they coevolved as separate entities, they are always replicating inside host cells and so have a rather intimate relationship with cellular DNA. And so, at some stage, a virus can pick up an oncogene from a cell where a mutation has occurred, and can then spread it around. We normally think of viral or bacterial infections as a transmission of the viruses or bacteria themselves. But here, the virus basically picked up "cancer" some time in the past, and can do harm not only in the normal way - killing infected cells - but by destabilizing cells that take in the oncogene so that they become cancer-forming.

Now unless such a virus manages to infect a germ cell, this process is not going to directly affect our genetic lines. But for single-celled bacteria etc, it's another story. For these little creatures, the family tree can have strange horizontal paths that connect up totally different species. Bacteria grow by dividing asexually, but the viruses that infect them (called bacteriophages) can transfer DNA in a "sexual" sort of way with a virtuosity that we multi-cellular beings know nothing about. From the point of view of messing up a family tree, it would be like getting a whale to mate with a chaffinch. Then again, bacteria are pretty flexible about their group identity over time.

the first step

noreply@blogger.com (NewtonsOcean) — Sun, 01 Feb 2009 21:54:00 +0000

Have you ever wondered whether life may have actually started up from scratch more than once on this planet? There are several issues to consider. First, the conditions in primordial times were probably much more favourable to the genesis of life than at any subsequent period. Second, once life got established, with its universal currency of nucleotides encoding amino acids to produce proteins etc., any second shot at reinventing life would presumably not stand a chance unless it could integrate with what was already thriving in abundance. Mind you, some believe life started up around hot deep sea vents. Have they changed much? Maybe they have, I honestly don't know. But one can just about imagine the odd isolated colony of new life - maybe with a different genetic code - lurking in the depths. Then again, maybe the way life turned out on this planet, with the genetic code just so, was the only possible or the only really favourable one? In which case, if life started up from scratch again, we would never notice.

Almost as fascinating is to consider whether life was invented only once in those primordial days. After all, researchers have looked at genetic similarities and differences across the whole spectrum of modern-day life-forms, and have attempted to trace us all back to a last common ancestral cell. Again, since we all share the same genetic code, this is not so crazy. Now in all probability, if you went back further, you would find an increasing number of common ancestors, so this does not in itself resolve the issue.

From a scientific standpoint, it is perhaps almost irrelevant whether critical discoveries in self-replication etc can be credited to more than one molecule, experimenting in the primordial soup. After all, scientific discoveries are often made by more than one scientist at about the same time - namely, when the state of prior knowledge creates a fertile breeding ground for the next creative step. But wouldn't it be amazing if we knew that every living cell on this planet was connected by an unbroken thread of DNA replication all the way back to a single crucial innovation by some molecular configuration several billion years ago?

immortality

noreply@blogger.com (NewtonsOcean) — Sat, 31 Jan 2009 16:51:00 +0000

Most of us know why we have kids. The cells that make up our bodies are just not meant to keep going for ever. If our body cells suffer damage, it counts as wear and tear and is never a good thing, partly because the whole system of cells has evolved to work in a particular way. If a cell has new ideas, it will no longer fit in with the rest of the system. Of course, the ultimate damage is the transformation of a cell to produce an immortal cancer cell line that proceeds to take over without any concern for the rest of the system.

Our normal body cells can only divide a fixed number of times because the DNA copying machinery cannot copy all the way to the end. There is a blank piece of DNA called a telomere (labeled white in the picture) at the end of each chromosome which keeps getting shorter and once it disappears the cell normally commits suicide. Embryonic stem cells express the protein telomerase, which is able to add the uncopied end bits of the telomere back at each cell division, so that many divisions can occur during the rapid growth phase of an individual.

Bacteria reproduce asexually so their cells might be considered immortal, except for the fact that they reproduce so rapidly that a diverse range of mutations occurs within a short time and a subset of these come to dominate the new population thanks to environmental pressure. Their identity changes so much that it makes little sense to consider them immortal. Our own germ lines certainly do not maintain a specific identity over generations because they get mixed in with those of our partners.

But life itself did learn the trick of immortality. One wonders how much time was spent in the primordial soup not only figuring out how to make biomolecules that could transmit instructions for the continued building of future generations of cells and ultimately multicelled organisms, but how to keep it all going for ever and ever. Simple tricks such as telomerase had to be discovered obviously. Then there is the beauty of natural selection operating on individuals. Unlike the cells in your body that are tightly linked up to the other cells in your body, individuals really are ideal experimental units for dealing with mutations. For bacteria the cell is the organism and it's obvious to us how bacteria as a group evolve under random mutation and selective pressure from the environment. But for multicelled organisms it is also a no-brainer. If a mutation is bad, the individual doesn't make it and the species carries on. If a mutation is good, then future individuals will benefit. Meanwhile, many genes have such vital roles that any change is almost bound to be deleterious. These genes are said to be highly conserved and are found to be almost identical across many species. They of course are the immortal ones.

schrodinger's kittens - part 3

noreply@blogger.com (NewtonsOcean) — Sun, 25 Jan 2009 18:54:00 +0000

This is the final part in my series on quantum theory weirdness. With the help of the invaluable guide to these issues as set down in Schrodinger's Kittens by John Gribbin, but also making shameless use of wikipedia to fill in any gaps, I will be colouring in the spectrum that lies between two extreme positions. On one end of the spectrum we have seen that the original Copenhagen interpretation requires the quantum fuzziness inherent in probability waves and Heisenberg's uncertainty principle to be resolved (or collapsed) into a definite outcome. On the other end of the spectrum, we have the many worlds scenario, whereby the universe splits endlessly so that we only observe one scenario but the other possible outcomes are all accounted for in other universes.

Erwin Schrodinger used a thought experiment in 1935 to draw attention to the ridiculous conclusions that must be drawn from the Copenhagen interpretation. He deftly demonstrated the crux of the problem by having a quantum event, such as the radioactive decay of one atom, be detected by a standard instrument, the Geiger counter. The subatomic world of probabilities is suddenly brought into the macroscopic world by a standard detector that amplifies the energy and makes it observable. Now Schrodinger imagined placing this apparatus inside a box along with a cat. On detecting the radioactive decay, the apparatus then triggers the release of a poison that will kill the cat. According to the Copenhagenites, the cat remains in a half-dead, half-alive state until some suitably "conscious" observer looks inside the box to resolve the probabilities into certain life or death. Of course, cat lovers (myself included) are not only appalled by such a callous-sounding thought experiment, but will point out that a cat is perfectly aware enough to be the conscious observer in the first place. Others claim that quantum mechanics predicts that the uncertainty of a large object's state would only last for a miniscule split-second anyway.

I feel I have been beating/flogging a dead horse here with this Copenhagen interpretation. (OK now the horse lovers are getting upset.) But we should remember that, apart from the "conscious observer" silliness, we do still have a problem to resolve. According to quantum theory, things don't work out properly unless various aspects of subatomic particles (position, spin, whatever) are given probabilities at certain times rather than anything as definite as is implied by their final observed state. In part 1, we saw in the classic interference experiment that you simply get the wrong answer (no interference) unless you have some wavelike entity extended over space to cover the probability of a single particle passing through either slit.

One seemingly simple way to get around this is to propose that each particle does go through only one slit at a time, but coexists with a wavelike entity that guides the particle through such that interference effects are predicted as observed. This sounds like mere semantics, but it in fact corresponds to a long-lasting debate initiated by Einstein's famous dislike of quantum probabilities. The alternative was that quantum theory was incomplete, and there were hidden variables, e.g. the mathematical details of a guiding wave. Unfortunately, the mathematician von Neumann had supposedly proved that hidden variables were impossible, until an error was found in his logic and David Bohm demonstrated a theory with his so-called quantum potential acting as the guiding wave.

One thing appears to be certain (no pun intended). In order to satisfy Einstein's dislike of probabilities, everyone seems to agree that you have to deal with issues of non-locality, where things going on in two regions of space that are distant from each other appear to sort of communicate with each other at speeds faster than that of light. So on the face of it Einstein would still be distressed and indeed he was: with Podolsky and Rosen he came up with the EPR paradox. To discuss this, Gribbin brings in his hypothetical kittens of Schrodinger's original cat. For quantum weirdness extends to the idea of what is called quantum entanglement. It has been shown that if you generate two subatomic particles whose spins are dependent on each other, and you have them shoot off in opposite directions, the measurement of one of the particles will instantaneously affect the other. In terms of the original Copenhagen interpretation, this amounts to a quantum system that has got spread out over space, but is described by some probability wave that must collapse immediately everywhere. Thinking loosely here, this doesn't necessarily violate the speed of light limit. Perhaps we just have to think of a system spreading itself out over a huge region of space and staying connected in such a way that when the whole system has to change its configuration, it does so as a "holistic" system without one end having to communicate via speed-of-light messages to the other end. So where did the kittens come in? Well, just imagine the macabre situation of killing a kitten thousands of miles away by opening a box and finding its playmate alive and well.

OK, moving on from the effects of quantum entanglement on kittens... John Gribbin favours a so-called transactional interpretation first proposed by John Cramer in 1986 and based on the earlier ideas of Wheeler and Feynman. Thinking back to the particle going through the two-slit apparatus, the basic idea is that it puts out some wave, "offering" to make some transaction with a particle on the photographic screen. This other particle sends out a "confirmation" wave which travels backwards in time. The beauty of this is that it resolves the apparent paradox we saw in part 1: namely that a particle generates an interference pattern but if it is observed after passing through the slits, it fails to interfere with itself. The transactional analysis approach allows the particle to take the whole apparatus into account and then behave consistently. And what difference does it make whether to us the wave is traveling forwards of backwards in time, when - according to Einstein - to the wave itself, time is frozen still?

The amazing thing of course is how well quantum theory works in practice. Quantum electrodynamics (QED) displays agreement with experiment to within 10 parts in a billion. So, like Gribbin, I will finish this series with a quotation from the ever pragmatic Richard Feynman, one of the most famous of the developers of QED theory. Concerning the fact that no one understands quantum weirdness, he wrote: "Nobody knows how it can be like that." Of course, Gribbin finds this ironic, given Feynman's own early contribution to what Gribbin considers a promising solution, that really does appear to resolve a lot of the puzzle. I wonder if Einstein would have liked it?

anthropocentric arguments

noreply@blogger.com (NewtonsOcean) — Fri, 23 Jan 2009 02:50:00 +0000

Before we get to part 3 of my extended discussion about quantum weirdness, I want to look at the apparent anthropocentricity of the Copenhagen interpretation. Dedicated readers :-) will recall from my previous post that the Copenhagen interpretation specifies that conscious observers are required in order to collapse the fuzzy probability status of a quantum mechanical system and thus resolve it into a definite observation. It smacks of the old philosophical question: if a tree falls in a forest and no one observes it, did it really fall?

Science requires observers to remain detached from their experiments. Quantum theory developed in a way that threatened that ideal, perhaps far more dramatically than it should have. What is interesting here is that scientists were willing to embrace such an anthropocentric proposition that gave conscious observers so much power - seemingly to create observable reality out of vague fuzziness. In cosmology too, some scientists have been tempted to put humans back on center stage after Copernicus pushed us off to the side. In 1973, an astrophysicist named Brandon Carter came up with an Anthropic Principle to address the apparently careful tuning of various fundamental physical constants that has enabled intelligent life to develop on earth. This is a real game of probabilities, for what is the probability of a universe existing/forming in the first place?

We are in fact back to the creation vs evolution debate, only now we are talking about universes rather than life forms. If our universe is the one and only universe, then indeed one could argue that some creator had a specific desire for life to exist, since slightly changing the physical constants would radically alter whether atoms such as carbon could have been produced or whether stars and planets would even exist. But if many universes are formed, they will likely sample a range of different physical constants, i.e. starting ingredients. Since our universe supposedly acquired its particular combination of fundamental forces by the random process of spontaneous symmetry breaking, this can easily be thought of as a random component to an evolutionary scenario. A universe will then be selected based on issues such as whether it is stable enough to exist for a while. If a universe wants to produce sentient creatures that can be impressed by it, it will have to be a bit luckier. Our universe was simply one of those lucky ones, so - no surprise - here we are!

This single versus multiple universe assumption is echoed in the Copenhagen interpretation and one of its competitors. The Copenhagen interpretation seems to imply that you need a god observing the universe in order to collapse its total quantum mechanical probability wave into hard definite reality. Alternatively, the multiple universe idea can be extended dramatically such that instead of the quantum probability states needing to be collapsed, each state now spawns a new universe. This many-worlds interpretation was suggested by Hugh Everett in 1957. While I'm happy to believe in more than one universe, this idea seems a bit extravagant. But as we will see in my next post, there are a number of alternative ways to brush this quantum weirdness safely under the rug. But there is another aspect of the problem that is weirder still.

schrodinger's kittens - part 2

noreply@blogger.com (NewtonsOcean) — Mon, 19 Jan 2009 21:30:00 +0000

In general, the framework for describing quantum phenomena, which works unbelievably well by the way, always seems to start off with uncertainty. This is often explained in terms of observers introducing uncertainty by interacting with the very system they are trying to measure. Classically you can observe something without perturbing it by, for example, turning down your light source to an infinitesimally low power, as long as you can construct an incredibly sensitive detector. But with quantum theory comes the existence of photons with discrete quantities of energy, so that the minimum you can use to observe a subatomic particle is one photon, which will still jolt the very object you are trying to measure. In part 1, I described the double-slit experiment that seems to be mandatory when introducing quantum phenomena. The celebrated physicist Feynman often stated that any quantum weirdness was fundamentally an expression of the wave-particle enigma illustrated by this experiment. Being a no-nonsense kind of guy, he would point out that a simple measurement of the momentum as a particle either bounced through the left slit or the right slit would cause an uncertainty in the position of the slits enough to wipe out the interference effect.

This is all true enough, but others point out that the uncertainty is inherent in the system before anyone tries to observe it. In fact, it seems that the biggest issue is in trying to explain how one goes from uncertainty to a certain result. When results of experiments that are designed to probe the subatomic world impinge on us at a macroscopic level they give us definite answers. Why?

Macroscopic measurements usually entail looking at the net result of millions of individual submicroscopic events, but this does not change anything. It simply means that the probabilities for one particle become statistical distributions of lots of particles. So in the double-slit experiment, the interference pattern gets built up one photon at a time, and reveals that the probability for each photon to hit the screen at various locations was higher in some places than others. So the result that is built up from millions of photon observations still tells us that each photon was interfering with itself as its probability wave took in the presence of both slits.

However, the recording of each submicroscopic event involves the interaction of the subatomic particle of interest with a whole host of particles that form the detection equipment. The photographic film used to record the interference pattern presents itself as a macroscopic obstacle that each photon must interact with, and so each photon acts as a well-defined particle, giving up its energy at one spot on the film and not as a probability wave spread out across it. This transition from subatomic fuzzy probabilities to macroscopic concreteness is hard to grasp but is in a sense irrelevant to the practical work of physicists dealing with quantum mechanics.

But this aspect of quantum theory has nevertheless caused a lot of philosophical debate since Heisenberg first articulated his uncertainty principle in 1926. At the time he was working in Bohr's Copenhagen lab, and together they concocted what became known as the Copenhagen interpretation of quantum phenomena. Basically, the fact that experimenters observe definite states was interpreted as the power of the experimenter's own consciousness to collapse the probability wave into a definite outcome. Now this is so kooky that it ranks with the most absurd statements that have ever been made way out on the fringes of serious scientific debate. But these scientists remained right in the centre of the new science of quantum mechanics as highly respected founding members. Of course this interpretation does have a sort of pragmatic value. It could be rephrased in the following way. Since we don't have a clue what goes on between our extremely useful probability-wave model of subatomic goings-on and our final concrete results, let's keep it as a black box and note that it is as if our (conscious) observation forces the indecisive particles to make up their minds and come out like rolled dice from a tumbler each showing a definite face to the world.

But in fact, a surprising amount of philosophical debate did ensue, including an attempt by another founding father of quantum theory, Erwin Schrodinger, to highlight the absurdity of this crude attempt to deal with the leap from fuzzy small things to our hard-edged observations in the macroscopic world. In part 3, we will join him with his furry and possibly fuzzy cat.