## Archive for the ‘**mathematics**’ Category

## reading matters 1

The universe within by Neil Turok (theoretical physicist extraordinaire)

Content hints

– Massey Lectures, magic that works, the ancient Greeks, David Hume and the Scottish Enlightenment, James Clerk Maxwell, quantum mechanics, entanglement, expanding and contracting universes, the square root of minus one, mathematical science in Africa, Paul Dirac, beauty and knowledge, the vitality of uncertainty, Mary Shelley, quantum computing, digital and analogue, Richard Feynman, science and humanity, humility, education, love, collaboration, creativity and thrill-seeking.

## towards James Clerk Maxwell 6: Newton’s universal law of gravitation and G

Newton’s law of gravity goes like this:

where *F *is the force of gravitational attraction, *G i*s the constant of proportionality or gravitational constant, *m *is an entity, particle or object with a particular mass, and *r *is the distance between the centres of mass of the two entities, particles or objects.

What’s the relation between all this and Maxwell’s electromagnetic work? Good question – to me, it’s about putting physics on a mathematical footing. Newton set us on this path more than anyone. The task I’ve set myself is to understand all this from the beginning, with little or no mathematical expertise.

The law of gravity, in its un-mathematical form, says that every object of mass attracts every other massive object with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

It seems often to be put about that Newton was revolutionary because he was the first to wonder why objects fell to the ground. This is unlikely, and Newton wasn’t the first to infer an inverse square law in relation to such falling. Two Italian experimenters, Francesco Grimaldi and Giovanni Riccioli, investigated the free fall (where no force acts besides gravity) of objects between 1640 and 1650, and noted that the distance of the fall was proportional to the time taken. Galileo had previously conducted free fall experiments and found that objects fell with uniform acceleration – an acceleration that is proportional to the square of the elapsed time. Nor was he the first to find a time-squared relationship. The point of all this is that science doesn’t proceed via revolutions proceeding from one brilliant person (which shouldn’t diminish Galileo or Newton’s genius). The more you find out about it, the more incremental and fascinatingly collaborative and confirmative over time it is.

Galileo used the geometry of his time to present his time squared law, but algebraic notation, invented principally by Descartes, superseded this approach in the seventeenth century.

What about the gravitational constant? This appears to be a long and complicated mathematical story. I think it tries to answer the question – why do objects fall to Earth at such and such a rate of acceleration? But I’m not sure. The rate of acceleration would have been easy enough to measure – it’s approximately 9.8 m/sec^{2}. This rate would appear to be caused by the mass of the Earth. The Moon has a fraction of Earth’s mass, and I believe the gravitational force it exerts is approximately one sixth that of Earth. It has been measured as 1.62 m/sec² (for Mars it’s 3.71).

It’s frustratingly difficult to get an explanation online of what the gravitational constant (G) is or really means – without very quickly getting into complex (for me) mathematics. Tantalisingly, Wikipedia tells us that the aforementioned Grimaldi and Riccioli ‘made [an attempted] calculation of the gravitational constant by recording the oscillations of a pendulum’, which means nothing to me. Clearly though, there must be some relationship between G and the mass of the Earth, though how this can be ascertained via pendulums is beyond me. Anyway, on with the struggle.

We do have a number for G, or ‘Big G’, as it’s called (explanation to come), and it’s a very very small number, indicating that, considering that the multiplied masses divided by the square of the distance between them then get multiplied by G, gravitation is mostly a very small force, and only comes into play when we’re talking about Big Stuff, like stars and planets, and presumably whole galaxies. Anyway here’s the actual number:

G = 0.0000000000667408, or 6.67408 × 10^{-11}

I got the number from this useful video, though of course it’s easily available on the net. Now, my guess is that this ‘Big G’ is specific to the mass of the Earth, whereas small g is variable depending on which mass you’re referring to. In other words, G is one of the set of numbers in g. We’ll see if that’s true.

Now, looking again at the original equation, *F* stands for force, measured in newtons, *m *for mass, measured in kilograms, and and *r* for distance in metres (these are the SI units for mass and distance). The above-mentioned video ‘explains’ that the newtons on one side of the equation are not equivalent to the metres and kilograms squared on the other side, and G is introduced to somehow get newtons onto both sides of the equation. This has thrown me into confusion again. The video goes on to explain how G was used by Einstein in relativity and by Max Planck to calculate the Planck length (the smallest possible measure of length). Eek, I’m hoping I’m just experiencing the storm before the calm of comprehension.

So, to persist. This G value above isn’t, and apparently cannot be, precise. That number is ‘the average of the upper and lower limit’, so it has an uncertainty of plus or minus 0.00031 x 10^{-11,} which is apparently a seriously high level of uncertainty for physicists. The reason for this uncertainty, apparently, is that gravitational attraction is everywhere, existing between every particle of mass, so there’s a signal/noise problem in trying to isolate any two particles from all the others. It also can’t be calculated precisely through indirect relation to the other forces (electromagnetism, the strong nuclear force and the weak nuclear force), because no relationship, or compatibility, has been found between gravity and those other three forces.

The video ends frustratingly, but providing me with a touch of enlightenment. G is described as a ‘fundamental value’, which means we don’t know why it has the value it does. It is just a value ‘found experimentally’. This at least tells me it has nothing to do with the mass of the Earth, and I was quite wrong about Big G and small g – it’s the other way round, which makes sense, Big G being the *universal *gravitational constant, small g pertaining to the Earth’s gravitational force-field.

Newton himself didn’t try to measure G, but this quote from Wikipedia is sort of informative:

In the

Principia, Newton considered the possibility of measuring gravity’s strength by measuring the deflection of a pendulum in the vicinity of a large hill, but thought that the effect would be too small to be measurable. Nevertheless, he estimated the order of magnitude of the constant when he surmised that “the mean density of the earth might be five or six times as great as the density of water”

Pendulums again. I don’t quite get it, but the reference to the density of the Earth, which of course relates to its mass, means that the mass of the Earth comes back into question when considering this constant. The struggle continues.

I’ll finish by considering a famous experiment conducted in 1798 by arguably the most eccentric scientist in history, the brilliant Henry Cavendish (hugely admired, by the way, by Maxwell). I’m hoping it will further enlighten me. For Cavendish’s eccentricities, go to any online biography, but I’ll just focus here on the experiment. First, here’s a simplification of Newton’s law: **F = GMm/R ^{2}**, in which

**M**is the larger mass (e.g. the Earth), and

**m**the smaller mass, e.g a person. What Cavendish was trying to ascertain was nothing less than the mass and density of the Earth. In doing so, he came very close – within 1% – of the value for

**G.**Essentially, all that has followed are minor adjustments to that value.

The essential item in Cavendish’s experiment was a torsion balance, a wooden bar suspended horizontally at its centre by a wire or length of fibre. The experimental design was that of a colleague, John Michell, who died before carrying out the experiment. Two small lead balls were suspended, one from each end of the bar. Two larger lead balls were suspended separately at a specific distance – about 23cms – from the smaller balls. The idea was to measure the faint gravitational attraction between the smaller balls and the larger ones.

Wikipedia does a far better job than I could in explaining the process:

The two large balls were positioned on alternate sides of the horizontal wooden arm of the balance. Their mutual attraction to the small balls caused the arm to rotate, twisting the wire supporting the arm. The arm stopped rotating when it reached an angle where the twisting force of the wire balanced the combined gravitational force of attraction between the large and small lead spheres. By measuring the angle of the rod and knowing the twisting force (torque) of the wire for a given angle, Cavendish was able to determine the force between the pairs of masses. Since the gravitational force of the Earth on the small ball could be measured directly by weighing it, the ratio of the two forces allowed the density of the Earth to be calculated, using Newton’s law of gravitation.

To fully understand this, I’d have to understand more about torque, and how it’s measured. Clearly this weak interaction is too small to be measured directly – the key is in the torque. Unfortunately I’m still a way from fully comprehending this experiment, and so much else, but I will persist.

**References**

https://en.wikipedia.org/wiki/Newton’s_law_of_universal_gravitation

https://en.wikipedia.org/wiki/Gravitational_constant

https://energyeducation.ca/encyclopedia/Gravitational_constant

https://en.wikipedia.org/wiki/Cavendish_experiment

https://www.school-for-champions.com/science/gravitation_cavendish_experiment.htm#.XSrCrS3L1QI

Go to youtube for a number of useful videos on the gravitational constant

## algorithms, logarithms and biorhythms

In a recent conversation I was talking about the algorithms used by organisations like Facebook and Google to get people hooked into reading or listening to or otherwise attending to a whole host of stuff related to what they attended to yesterday. My interlocutor continued the conversation, but inadvertantly replaced the word ‘algorithm’ with ‘logarithm’ throughout, which I didn’t correct, perhaps not so much out of politeness but because, though I knew enough about the difference to use the right word, I would have difficulty in defining either term, in case he asked. As for biorhythms, I threw them in just for fun.

Mathematics is probably my weakest subject, and for that reason I’ve often been drawn to it in my dilettantish fashion, as to a kind of secret society I’ve been excluded from due to lack of the requisite. So I’ll start with my extremely limited and probably wrong working understanding of algos and logos (and bios), then provide a more informed account through a bit of research.

An *algorithm*, I think, is a largely computer-generated aggregate of a person’s data-gathering which provides some predictive power about their future data-gathering, or future interest, so that the data can be presented to them for gathering. Or something like that. A *logarithm, *I think, has something to do with exponentials. So that if something increases exponentially, it also increases logarithmically. Or logarithmickly? Or maybe a logarithm is some mathematical relation between a number and its exponent. Or something like that. A *biorhythm *is something to do with your supposed natural cycle of activity/inactivity throughout the day. It was new age stuff of the seventies, my early teen years. It seems to have died the death.

So now for the details and correctives.

An algorithm is indeed most associated with computing and data processing, though it may be used in a broader sense, mathematically or not. In fact it can be used to describe the *process *to get any result. In that sense a cookbook is a set of algorithms. But even in computing, the term is broader than my working understanding of it implied. I was thinking of *particular *algorithms, the ones used by certain web presences like Google, Facebook and Netflix to get you hooked into continuing to use them. To get you addicted, in a sense. So that’s all that needs to be said here about algorithms in general. In another post I’ll look at those particular algorithms that have worked so well for their creators (or the hirers of those creators) that they’ve become some of the richest folk in the world. That should be much more interesting.

A logarithm is also something like what I thought it was, but of course my understanding was very vague. It’s described as the inverse function to exponentiation – just as I sort of thought. For example, take the number 64, and think of it as 2 x 2 x 2 x 2 x 2 x 2, or 2^{6 }. The 6 here is the exponent (*n*), while the 2 is the base (*b*), so the general formulation is *b*^{n}. The exponent (*n*) is also called the *power*, as in two to the sixth power, 2^{6}. Now, say that you want to know what power to raise the number two to get 256. That’s to say, 2^{n} = 256, find *n*. This number that we want to find is the logarithm. So the question or equation can be restated thus:

log2(256) = *n* , find *n*

and the answer is *n *= 8. So the logarithm is the *power* that we have to raise a specified (base) number to, in order to get to another specified number. That’s all we need to know as a beginning. Obviously it gets more complicated with higher powers – but then, with higher powers, anything’s possible.

A biorhythm chart is a development of ideas concocted by one Wilhelm Fliess, a good mate of the only slightly less eccentric Sigmund Freud, back in the late 19th century. I well remember this fad gaining super-popularity in the seventies, with a paperback on the topic floating around our household. That and ‘speed reading’ were all the go. The basic hypothesis was that our lives – presumably in terms of energy, ‘clarity’ and competence – go in cycles, which can be charted as sinusoidal waves. The standard model argued for a 23-day physical cycle, a 28-day emotional cycle and a 33-day intellectual cycle. I’ve actually met someone who drew up these charts for herself back in the day. You live and learn. Needless to say, there’s no evidence whatever to back these ideas up, but they’re claimed often as ancient learning which has been eclipsed by the monstrous juggernaut of modern science, much like that other font of cyclical predictive wisdom, astrology…..

**References**

https://computer.howstuffworks.com/what-is-a-computer-algorithm.htm

Khan academy video: intro to logarithms

## towards James Clerk Maxwell: 1 – a bit about magnetism

Canto: So what do you know about magnetism?

Jacinta: Well not a lot but I’m hoping to learn a lot. Some metals – but perhaps it’s only iron – appear to be attracted by other metals – or other bits of iron – so that they’re pulled together and are hard to pull apart, depending on the strength of the magnetism, which is apparently some kind of force. And I believe it’s related to electricity.

Canto: We shall learn more together. All this enquiry stems from a perhaps vague interest in James Clerk Maxwell, who famously connected electricity and magnetism in an equation, or a series of equations, or laws, with a great deal of mathematical sophistication, which I don’t have. Maxwell is hardly a household name in the way that Newton and Einstein are, but he’s undoubtedly revered among mathematical physicists. My own interest is twofold – I’d like to understand more about physics and maths in general, and – I’m Scottish, sort of. That is, I was born there and grew up among Scottish customs, though I’ve lived in Australia since I was five, and I always like to say that I haven’t a nationalist cell in my body. I’ve never waved a flag or sung any of those naff national anthems, and I have dual British/Australian citizenship only as a matter of convenience – and I suppose the more nations I could become a citizen of, the more convenient it would be. And yet. I’ve always felt ‘something extra’ in noting the Scottish contribution to the sciences and the life of the mind. James Hutton, Charles Lyell, James Watt, Adam Ferguson, David Hume and Adam Smith are names I’ve learned with a glimmer of unwonted or irrational pride over the years, though my knowledge of their achievements is in some cases very limited. And that limitation is perhaps most extreme in the case of Maxwell.

Jacinta: So we’ll get back to him later. There are good, easily available videos on all matters scientific these days, so I’ve looked at a few on magnetism, and have learned a few things. Magnetism apparently occurs when the atoms in a block of material are all aligned in the same direction, because atoms themselves are like tiny magnets, they’re polarised with a north and south pole, which I think has something to do with ionisation, maybe. Most materials have their atoms aligned in an infinity of orientations, with a net effect of no magnetism. Don’t quote me on that. The Earth itself is a gigantic magnet with a north and south pole. If it wasn’t, then the solar wind, which is a plasma of charged particles, would strip away the ozone that protects us from UV radiation. Because that field is sucked in at the poles, we see that plasma in the northern and southern latitudes, e.g. the northern lights. We now know that magnetism is essential to our existence – light itself is just a form of electromagnetic radiation (I think). But what we first learned about this stuff was pretty meagre. There were these rocks called lodestones, actually iron ore (magnetite), which attracted iron objects – swords and other tools of the iron age. What was this invisible force? It was named magnetism, after the region of Magnesia in what’s now modern Greece, where presumably lots of these lodestones were to be found. Early discoveries about magnetism showed that it could be useful in navigation…

Canto: But that wasn’t too early – there’s something of a gap between the discussions in Aristotle and Hippocrates and the 12th century realisation that a magnetic needle could be used for navigation. At least in Europe. The Chinese were well ahead in that regard. But I should stop here and say that if we’re going to arrive at Maxwell, it’s going to be a long, though undoubtedly fascinating road, with a few detours, and sometimes we might move ahead and turn back, and we’ll meet many brilliant characters along the way. And, who knows, we may never even arrive at Maxwell, and of course we shouldn’t assume that Maxwell is at the summit of all this.

Jacinta: So the first extant treatise on magnets was the *Epistola de Magnete, *by Petrus Peregrinus, aka Pete the Pilgrim, in 1269. It was described as a letter but it contained 13 chapters of weighty reading. The first 10 chapters apparently describe the laws of magnetism, a clear indication that such laws were already known. He describes magnetic induction, how magnetism can be induced in a piece of iron, such as a needle, by a lodestone. He writes about polarity, being the first to use the term ‘pole’ in this way – in writing at least. He noted that like poles repel and unlike poles attract, and he wrote of a south pole and a north pole. That’s to say, one end of a needle points north when given its head – for example when suspended in water. He also describes the ‘dry’ pivoted compass, which was clearly well in use by that time.

Canto: What he didn’t know was *why *a needle points north – actually magnetic north, which isn’t the same as the north pole – but close enough for most navigational purposes. He didn’t know that the Earth was a magnet.

Jacinta: On compass needles, there’s a neat essay online on how compasses are made. I’m not sure about how GPS is making compasses obsolete these days, but it’s a bit of a shame if it’s true…

Canto: So the next name, apart from the others, to associate with work on magnets was William Gilbert, who published *De Magnete *in 1600. This gathered together previous knowledge on the subject along with his own experimental work. One of the important things he noted, taken from the 1581 work *The Newe Attractive*, by Robert Norman, was magnetic inclination or dip, probably first noted by the Bavarian engineer and mathematician Georg Hartmann in the mid sixteenth century. This dip from the horizontal, either upward (steepest at the south pole) or downward (north pole) is a result of the Earth’s magnetic field, which doesn’t run parallel to the surface. Inspired by Norman’s work, Gilbert conducted experiments with a model Earth he made, concluding that the Earth was a magnet, and that its core, or centre, was made of iron…

Jacinta: Just how did he he work that out? Did he think that a bar magnet passed through the centre of the Earth from north to south pole?

Canto: I don’t think so, it’s probably more like he thought of Earth as a gigantic spherical lodestone with iron at its centre. It’s understandable that he would infer iron to be inside the Earth to make it magnetic, but he was the first to give a geocentric cause for the behaviour of compass needles – others had thought the attractive force was celestial. Interestingly, Gilbert was also a Copernican, in that he thought it absurd that the stars, which he believed to be vastly distant, revolved around the Earth. So he argued that the Earth turned, a view that got Galileo into so much trouble a few decades later.

Jacinta: Useful to be a Protestant in those times. Thank Dog for Henry VIII.

Canto: He also took an interest in what was later called electricity, though he didn’t consider it connected to magnetism. He built a versorium, the first electroscope, used to detect static electric charge. It was simply a metallic needle pivoted on a pedestal, like a compass needle but not magnetised. The needle would move towards a statically charged object, such as rubbed amber. In fact, Gilbert’s experiments strove to prove that static electricity was distinct from magnetism, which was an important development in early modern science.

Jacinta: I suppose we’re going to learn exactly what ‘static’ electricity is and how it fits in the over-all picture?

Canto: We shall try, though I shudder to think about what we’re embarking on here.

Jacinta: And I shudder to think about what cannot possibly be avoided – mathematics.

Canto: Well, yes, as we enter the 17th century, we’ll be encountering some great mathematical developments – with figures like Descartes, Pascal, Fermat, Liebniz and Newton all adding their weighty contributions to Galileo’s claim that nature is a book written in the language of mathematics.

Jacinta: Shit, I’m having a hard enough time trying to understand this stuff in English.

Canto: Hopefully it’ll be a great and rewarding adventure, and on the way we’ll learn about Coulomb’s inverse-square law, which is central to electrostatics. Meanwhile, it seems not much was added to our understanding of magnetism for a couple of hundred years, until Hans Ørsted’s more or less accidental discovery in 1819 that an electric current could create a magnetic field, by noting that a compass needle moved when placed near an electrified wire. Alessandro Volta had invented the voltaic pile, or battery, twenty years earlier, leading to a pile of electrical experiments in subsequent years.

Canto: Tragic but true.

## The over-population clock

Canto: From time to time I’ve shown my students the world population clock (WPC), because I’ve brought my discourse round to it for some reason, and they’ve been mostly fascinated. And I’ve usually told them that the world’s population will level out at about 9.5 billion by mid-century, because I’ve read or heard that somewhere, or in a few places, but is that really true?

Jacinta: So you’re wanting to investigate some modelling?

Canto: Well yes maybe. I was looking at the WPC the other day, and was shocked at how births are outnumbering deaths currently. What’s actually being done to stem this tide?

Jacinta: Looking at the WPC website, there’s a lot more data there that might enlighten you and calm your fears a bit – if it can be trusted. Ok we went past 7.4 billion this year and you can see that so far there’s 70 milliom births compared to around 29 million deaths, and that looks worrying, but you need to look at long-term trends. The fact is that we’ve added a little over 40 million so far this year, with a current growth rate of about 1.13%. That figure means little by itself, but it’s important to note that it’s less than half of what the growth rate was at its peak, at 2.19% in 1963. The rate has been decelerating ever since. Of course the worry is that this deceleration may slow or stop, but there’s not much sign of that if we look at more recent trends.

Canto: Okay I’m looking at the figures now, and at current trends the projection is 10 billion by 2056, by which time the growth rate is projected to be less than 0.5%, but still a fair way from ZPG. The population, by the way, was two point something billion when I was born. That’s a mind-boggling change.

Jacinta: And yet, leaving aside the damage we’ve done and are doing to other species, we’re doing all right for *ourselves*, with humanity’s average calorie intake actually increasing over that time, if that indicates anything.

Canto: Averages can carpet over a multitude of sins.

Jacinta: Very quotable. But the most interesting factoid I’ve found here is that the current growth rate of 1.13% is well down on last year’s 1.18%, and the biggest drop in one year ever recorded. In 2010 the growth rate was 2.23%, so the deceleration is accelerating, so to speak. It’s also interesting that this deceleration correlates with increasing urbanisation. We’re now at 54.3% and rising. I know correlation isn’t causation, but it stands to reason that with movement to the city, with higher overheads in terms of housing, and with space being at a premium, but greater individual opportunities, smaller families are a better bet.

Canto: You bet, cities are homogenously heterogenous, all tending to favour smaller but more diverse families it seems to me. That’s why I’m not so concerned about the Brexit phenomenon, from a long-term perspective, though we shouldn’t be complacent about it. We need to maintain opportunities for trade and exchange, co-operative innovation, so that cities don’t evolve into pockets of isolation. Ghettoisation. Younger people get that, but the worry is that they won’t stay young, they won’t maintain that openness to a broader experience.

Jacinta: Well the whole EU thing is another can of worms, and I wonder why it is that so many Brits were so pissed off with it, or were they duped by populist nationalists, or are they genuinely suffering under European tyranny, I’m too far removed to judge.

Canto: Well, if there were too many alienating regulations, as some were suggesting, this should have and surely could have been subject to negotiation. Maybe it’s a lesson for the EU, but you’re right, we’re too far removed to sensibly comment. Just looking at the WPC now – and it’s changing all the time – it has daily birth/death rates which shows that the birth rate today far exceeds the death rate – by more than two to one. How can you possibly extrapolate that to a growth rate of only 1.13%?

Jacinta: Ah well that’s a mathematical question, and I’m no mathematician but obviously if you have a birth rate the same as the death rate you’ll have ZPG, no matter what the current population, where as if you have a disparity between births and deaths, the percentage of population increase (or decrease) will depend on the starting population and the end-population, as a factor of time – whether you measure is annually or daily or whatever.

Canto: Right so let’s practice our mathematics with a simple example and then work out a formula. Say you start with 10, that’s your start population at the beginning of the day. And 24 hours later you end up with 20. That’s a 100% growth rate? But of course that could be with 1000 additional births over the day, and 990 deaths. Or 10 more births and no deaths.

Jacinta: Right, which indicates that the total number of births and deaths is irrelevant, it’s the difference between them that counts, so to speak. So let’s call this difference d, which could be positive or negative.

Canto: But to determine whether this value is positive or negative, or what the figure is, you need to know the value of births (B) and deaths (D).

Jacinta: Right, so d = B – D. And let’s set aside for now whether it’s per diem or per annum or whatever. What we’re wanting to find out is the rate of increase, which we’ll call r. If you have a start population (S) of 10 and d is 10, then the end population (E) will be 20, giving a birth rate r of 100%, which is a doubling. I think that’s right.

Canto: So the formula will be: r* = *S – E… Fuck it, I don’t get formulae very well, let’s work from actual figures to get the formula. It’s actually useful that we’re almost exactly mid-year, and the figure for d (population growth) is currently a little under 42 million. That’s for a half-year, so I’ll project out to 83 million for 2016.

Jacinta: So d now means *annual *population growth.

Canto: right. Now if we remove this year’s growth figure from the current overall population we get as our figure for S = 7,391,500,000 and that’s an approximation, not too far off. And we can calculate E as 7,474,500, approximately.

Jacinta: But I don’t think we need to know E, we just need S and d in order to calculate r. r is given as a percentage, but as a fraction it must be d/S. And this can be worked out with any handy calculator. My calculation comes out at 6.6% growth rate.

Canto: Wrong.

Jacinta: Yes, wrong, ok, a quick confab with Dr Google provides this formula. d = ((E – S)/S).100. But we already have that? E-S is 83 million. Divided by S (7,391,500,000), and then multiplying by 100 gives a growth rate annually of 1.1229%, or 1.12% to two decimal places, which is not far off, but significantly less than, the WPC figure of 1.3%. I must have stuffed up the earlier calculation, because I think I used the same basic formula.

Canto: Excellent, so you’re right, my fears are allayed somewhat. Recent figures seem to be showing the growth rate declining faster than expected, but let’s have another look at the end of the year. Could it be that the growth figures are higher in the second half of the year, and the pundits are aware of this and make allowances for it, or are we actually ahead of the game?

Jacinta: We’ll have a look at it again at the end of the year. Remember we did a bit of rounding, but I doubt that it would’ve made that much difference.

**Some current national annual population growth rates (approx):**

Afghanistan 3.02%

Australia 1.57%

Bangladesh 1.20%

Brazil 0.91%

Canada 1.04%

China 0.52%

India 1.26%

Iran 1.27%

Germany 0.06%

Morocco 1.37%

Nigeria 2.67%

Pakistan 2.11%

South Africa 1.08%

United Kingdom 0.63%

(These are not, of course, calculated solely by births minus deaths, as migration plays a substantial role – certainly in Australia. Some surprises here. The highest growth rate on the full list of countries: Oman, 8.45%. The lowest is Andorra with -3.61%, though Syria, with -2.27% on these figures, has probably surged ahead by now).

## more on Einstein, black holes and other cosmic stuff

Jacinta: Well Canto I’d like to get back to Einstein and space and time and the cosmos, just because it’s such a fascinating field to inhabit and explore.

Canto: Rather a big one.

Jacinta: I’ve read, or heard, that Einstein’s theory, or one of them, predicted black holes, though he didn’t necessarily think that such entities really existed, but now black holes are at the centre of everything, it seems.

Canto: Including our own galaxy, and most others.

Jacinta: Yes, and there appears to be a correlation between the mass of these supermassive black holes at the centres of galaxies and the mass of the galaxies themselves, indicating that they appear to be the generators of galaxies. Can you expand on that?

Canto: Well the universe seems to be able to expand on that better than I can, but I’ll try. Black holes were first so named in the 1960s, but Einstein’s theory of general relativity recast gravity as a distortion of space and time rather than as a Newtonian force, with the distortion being caused by massive objects. The greater the mass, the greater the distortion, or the ‘geodetic effect’, as it’s called. The more massive a particular object, given a fixed radius, the greater is the velocity required for an orbiting object to escape its orbit, what we call its escape velocity. That escape velocity will of course, approacher closer and closer to the speed of light, as the object being orbited becomes more massive. So what happens when it reaches the speed of light? Then there’s no escape, and that’s where we enter black hole territory.

Jacinta: So, let me get this. Einstein’s theory is about distortions of space-time (and I’m not going to pretend that I understand this), or geodetic effects, and so it has to account for extreme geodetic effects, where the distortion is so great that nothing, not even light, can escape, and everything kind of gets sucked in… But how do these massive, or super-massive objects come into being, and won’t they eventually swallow all matter, so that all is just one ginormous black hole?

Canto: Okay I don’t really get this either but shortly after Einstein published his theory it was worked out by an ingenious astrophysicist, Karl Schwarzschild – as a result of sorting out Einstein’s complex field equations – that if matter is severely compressed it will have weird effects on gravity and energy. I was talking a minute ago about increasing the mass, but think instead of decreasing the radius while maintaining the mass as a constant…

Jacinta: The same effect?

Canto: Well, maybe, but you’ll again reach a point where there’s zero escape, so to speak. In fact, what you have is a *singularity. *Nothing can escape from the object’s surface, whether matter or radiation, but also you’ll have a kind of internal collapse, in which the forces that keep atoms and sub-atomic particles apart are overcome. It collapses into an infinitesimal point – a singularity. It was Schwarzschild too who described the ‘event horizon’, and provided a formula for it.

Jacinta: That’s a kind of boundary layer, isn’t it? A point of no return?

Canto: Yes, a spherical boundary that sort of defines the black hole.

Jacinta: So why haven’t I heard of this Schwarzschild guy?

Canto: He died in 1916, shortly after writing a paper which solved Einstein’s equations and considered the idea of ‘point mass’ – what we today would call a singularity. But both he and Einstein, together with anyone else in the know, would’ve considered this stuff entirely theoretical. It has only become significant, and very significant, in the last few decades.

Jacinta: And doesnt this pose a problem for Einstein’s theory? I recall reading that this issue of ‘point mass’, or a situation where gravity is kind of absolute, like with black holes and the big bang, or the ‘pre-big bang’ if that makes sense, is where everything breaks down, because it seems to bring in the mathematical impossibility of infinity, something that just can’t be dealt with mathematically. And Einstein wasn’t worried about it in his time because black holes were purely theoretical, and the universe was thought to be constant, not expanding or contracting, just *there.*

Canto: Well I’ve read – and I dont know if it’s true – that Einstein believed, at least for a time, that black holes couldn’t actually exist because of an upper limit imposed on the gravitational energy any mass can produce – preventing any kind of ‘infinity’ or singularity.

Jacinta: Well if that’s true he was surely wrong, as the existence of black holes has been thoroughly confirmed, as has the big bang, right?

Canto: Well of course knowledge was building about that in Einstein’s lifetime, as Edwin Hubble and others provided conclusive evidence that the universe was expanding in 1929, so if this expansion was uniform and extended back in time, it points to an early much-contracted universe, and ultimately a singularity. And in fact Einstein’s general relativity equations were telling him that the universe wasn’t static, but he chose to ignore them, apparently being influenced by the overwhelming thinking of the time – this was 1917 – and he introduced his infamous or famous cosmological constant, aka lambda.

Jacinta: And of course 1917 was an early day in the history of modern astronomy, we hardly knew anything beyond our own galaxy.

Canto: Or within it. One of the great astrophysicists of the era, Sir Arthur Eddington, believed at the time that the sun was at the centre of the universe, while admitting his calculations were ‘subject to large uncertainties’.

Jacinta: Reminds me of Lord Kelvin on the age of the Earth only a few generations before.

Canto: Yes, how quickly our best speculations can become obsolete, but that’s one of the thrills of science. And it’s worth noting that the work of Hubble and others on the expansion of the universe depended entirely on improved technology, namely the 100-inch Hooker telescope at Mount Wilson, California.

Jacinta: Just as the age of the Earth problem was solved through radiometric dating, which depended on all the early twentieth century work on molecular structure and isotopes and such.

Canto: Right, but now this lambda (λ) – which Einstein saw as a description of some binding force in gravity to counteract the expansion predicted by his equations – is very much back in the astrophysical frame. The surprising discovery made in 1998 that the universe’s expansion is accelerating rather than slowing has, for reasons I can’t possibly explain, brought Einstein’s lambda in from the cold as an explanatory factor in that discovery, which is also somehow linked to dark energy.

Jacinta: So his concept, which he simply invented as a ‘fix-it’ sort of thing, might’ve had more utility than he knew?

Canto: Well the argument goes, among some, that Einstein was a scientist of such uncanny insight that even his mistakes have proved more fruitful than others’ discoveries. Maybe that’s hero worship, maybe not.

Jacinta: So how does lambda relate to dark energy, and how does dark energy relate to dark matter, if you please?

Canto: Well the *standard model *of cosmology (which is currently under great pressure, but let’s leave that aside) has been unsuccessful in trying to iron out inconsistent observations and finding with regard to the energy density of the universe, and so dark energy and what they call cold dark matter (CDM) have been posited as intellectual placeholders, so to speak, to make the observations and equations come out right.

Jacinta: Sounds a bit dodgy.

Canto: Well, time will tell how dodgy it is but I don’t think anyone’s trying to be dodgy, there’s a great deal of intense calculation and measurement involved, with so many astrophysicists looking at the issue from many angles and with different methods. Anyway, to quickly summarise CDM and dark energy, they together make up some 96% of the mass-energy density of our universe according to the most currently accepted calculations, with dark energy accounting for some 69% and CDM accounting for about 27%.

Jacinta: Duhh, that does sound like a headachey problem for the standard model. I mean, I know I’m only a dilettanty lay-person, but a model of universal mass-energy that only accounts for about 4% of the stuff, that doesn’t sound like much of a model.

Canto: Well I can assure they’re working on it…

Jacinta: Or working to replace it.

Canto: That too, but let me try to explain the difference between CDM and dark energy. Dark energy is associated with lambda, because it’s the ‘missing energy’ that accounts for the expansion of the universe, against the binding effects of gravity. As it happens, Einstein’s cosmological constant pretty well perfectly counters this expansive energy, so that if he hadn’t added it to his equations he would’ve been found to have predicted an expanding universe decades before this was confirmed by observation. That’s why it was only in the thirties that he came to regret what he called the greatest mistake of his career. Cold dark matter, on the other hand, has been introduced to account for a range of gravitational effects which require lots more matter than we find in the observed (rather than observable) universe. These effects include the flat shapes of galaxies, gravitational lensing and the tight clustering of galaxies. It’s described as cold because its velocity is considerably less than light-speed.

Jacinta: Okay, so far so bad, but let’s get back to black holes. Why are they so central?

Canto: Well, that’s perhaps the story of supermassive black holes in particular, but I suppose I should try to tell the story of how astronomers found black holes to be real. As I’ve said, the term was first used in the sixties, 1967 to be precise, by John Wheeler, at a time when their actual existence was being considered increasingly likely, and the first more or less confirmed discovery was made in 1971 with Cygnus x-1. You can read all about it here. It’s very much a story of developing technology leading to increasingly precise observational data, largely in the detecting and measuring of X-ray emissions, stuff that was undetectable to us with just optical instruments.

Jacinta: Okay, go no further, I accept that there’s been a lot of data from a variety of sources that have pretty well thoroughly confirmed their existence, but what about these supermassive black holes? Could they actually be the creators of matter in the galaxies they’re central to? That’s what I’ve heard, but my reception was likely faulty.

Canto: Well astrophysicists have been struggling with the question of this relationship – there clearly is a relationship between supermassive black holes and their galaxies, but which came first? Now supermassive black holes can vary a lot – our own ‘local’ one is about 4 million solar masses, but we’ve discovered some with billions of solar masses. But it was found almost a decade ago that there is correlation between the mass of these beasties and the mass of the inner part of the galaxies that host them – what they call the galactic bulge. The ratio is always about 1 to 700. Obviously this is highly suggestive, but it requires more research. There are some very interesting examples of active super-feeding black holes emitting vast amounts of energy and radiation, which is both destructive and productive in a sense, creating an active galaxy. Our own Milky Way, or the black hole at its centre, is currently quiescent, which is just as well.

Jacinta: You mean if it starts suddenly feeding, we’re all gonna die?

Canto: No probably not, the hole’s effects are quite localised, relatively speaking, and we’re a long way from the centre.

Jacinta: Okay thanks for that, that’s about as much about black holes as I can stand for now.

Canto: Well I’m hoping that in some future posts we can focus on the technology, the ground-based and space-based telescopes and instruments like Hubble and Kepler and James Webb and so many others that have been enhancing our knowledge of black holes, other galaxies, exoplanets, all the stuff that makes astrophysics so rewarding these days.

Jacinta: You’re never out of work if you’re an astrophysicist nowadays, so I’ve heard. Halcyon days.

## how to debate William Lane Craig, or not – part 5, the fine-tuning argument

Dr Craig’s fifth argument is the well-known fine-tuning argument. Once again I should point out that when Dr Craig brings up these science-related topics it isn’t from a fascination with science itself – indeed Dr Craig likes to use the term ‘scientism’ when he refers to science other than when he’s using it to support his obsession. He uses science solely to mine and manipulate it to convince himself and others that there’s a warrant for a supernatural agent who has a personal love for him. So you should always consider his use of science with that in mind. And you should ask yourself, too, why is it that the physicists and cosmologists and mathematicians of the world, the people who work on a daily basis with the so-called laws of nature and the physical constraints of the universe, are by and large so completely lacking in belief in a personal deity? This is a sub-population that is more atheistic than any other sub-group on the planet. How does Dr Craig account for this? Madness, badness, indoctrination? How is it that the greatest physicist, by general acclaim, of the twentieth century, Einstein, regularly described belief in a personal god as a form of childishness? Why is it that Bertrand Russell, one of the greatest mathematicians and logicians of all time, wrote, ‘I am as firmly convinced that religions do harm as I am that they are untrue’? What is it with the Richard Feynmans, the Stephen Weinbergs, the Stephen Hawkings of this world that they’ve been so indifferent or hostile to the claims of religion? Perhaps Dr Craig should consider launching a wholesale attack on these disciplines, since they seem such a breeding ground for views so completely out of synch with his obsessions. How can they not know that all their researches and discoveries converge on the screamingly obvious fact that a loving human-focused supernatural being designed everything. What a bunch of blind fools.

The fine-tuning argument has been around for a long time despite its seeming ultra-modernity, though of course it gets updated in terms of constants and constraints. It’s of course, a rubbish argument like all the others. This universe wasn’t fine-tuned for anything. There was no tuner, as far as we know, and it would be impossible to predict what possibilities could emerge from the hugely complex and almost entirely unknown preconditions of the universe’s existence. Our universe will provide us with many many surprises long into the future, and I would not be surprised if those surprises include forms of life hitherto thought impossible, due to the ‘laws of nature’. Dr Craig claims that the various constraints and quantities that he talks about are independent of the laws of nature, which is a nonsense, as it’s only through our application of physical laws that we’ve been able to determine these quantities. So I don’t know what to make of his claim that these constraints aren’t physically necessary. The constraints exist as an essential part of the physical nature of this universe. The question of necessity or chance just doesn’t arise. These are the constraints we have to work with, and we find that, within these constraints, intelligent life is clearly possible, though perhaps very rare, though perhaps not so very rare as we once thought. I think we must all agree that we live in exciting times in the search for extra-terrestrial intelligence and extra-terrestrial life more generally. We’re homing in on the zones elsewhere that meet all the conditions for the emergence of life, and I believe we will find that life in time. Intelligent life, by our standards, will no doubt take longer.

Dr Craig says the odds of this universe being life-permitting are astronomically small. Some cosmologists agree, but they don’t then make any leaps to a supernatural cosmic designer. And I mean none of them do. It’s interesting that the cosmologist Alan Guth, to whom Dr Craig has already referred, believes that humans will one day be able to design new universes, no doubt with the help of quantum computers, and there are others who suggest that this may be how our universe came into being. All highly speculative stuff, and not particularly mainstream, but good fun, and worthy of reflection. Others, such as Stephen Hawking, have proposed a superposition of possible initial conditions for the universe which provides for an ‘inevitability’ of us finding ourselves in just this kind of life-sustaining universe at a later stage. It’s all to do with the manipulation of time-perception apparently. This hypothesis eliminates the need to posit a multiverse. There are many other hypotheses too, of course, including the multiverse, the bubble universe and others. It’s an exciting time for cosmology. Tough, but exciting, and far more interesting and rewarding than theology, I can promise you that. As students, I hope you continue to follow this stuff, for its own sake, not to mine it as confirmation for preconceived ideas.

## how to debate William Lane Craig, or not – part 4, on mathematics and gods

Now we come to the argument that God is the best explanation of the applicability of mathematics to the physical world. My intuitive response to this – and of course I’m not a mathematician – is that mathematics appears to me to be be a kind of abstraction from, that’s to say a manipulation of, a play on and further development of, the regularities that exist in the world, and that if no such regularities existed, the world wouldn’t exist. Or at least would not be in any sense describable. For example, the most basic form of regularity required would be a binary contrast, describable in mathematical or logical terms as *x *and* not x. *The real world, though , offers far more opportunities for playing on and manipulating regularities than this. So many opportunities have been found in fact, and so many beautiful theorems have been developed from them over the centuries that mathematics has often been given a mystical, miraculous status. One thinks of the Pythagoreans in ancient times, and the mathematically-obsessed philosophers of the seventeenth century, such as Descartes, Spinoza and Leibniz. However, I think it’s fair to say that, historically, when mathematics has been raised to mystical heights, great problems have ensued. So I don’t see anything particularly miraculous in the fact that a tool for understanding the regularities of the world can be developed and manipulated to underpin theories which further deepen or extend that understanding.

Eugene Wigner’s 1960 essay, ‘The unreasonable effectiveness of mathematics in the natural sciences’ is available online, and everyone should be encouraged to read it – though it doesn’t make for easy reading. I think it’s a little unfortunate that Wigner uses the word ‘miracle’ a number of times in the essay, but he certainly doesn’t refer at any time to a god. And while I would hesitate to interpret Wigner from my lay background, I’m not sure I agree with his view in the essay that, while elementary mathematical concepts derive directly from the perceived regularities of the actual world, more complex and abstract mathematical concepts don’t so derive, and yet can be applied with uncanny reliability, or if you like profitability, from our perspective, to that world, as is the case with much modern physics. If that were so, if the mathematical abstractions our minds create were completely removed from the world’s actual regularities, and yet just happened to apply to them to provide us with a richer and more developed view of our universe, then that would indeed be a ‘happy coincidence’. But abstraction doesn’t occur in a vacuum. Just as non-Euclidian geometry derives from the regularities of nature that Euclid strove to axiomise in a set of rules, and just as multi-dimensionality derives from the standard three-dimensional world of our experience, mathematical abstraction is always tied to some underlying actual regularity, however obscured by its overlay. The applicability of maths is not a happy coincidence (which isn’t to say all mathematical abstractions are applicable of course), but that is just because the world *has regularity. *Thus when we look at Dr Craig’s formal argument:

1. If God did not exist, the applicability of mathematics would be a happy coincidence.

2. The applicability of mathematics is not a happy coincidence.

3. Therefore God exists.

we see once again that the problem lies in the conditional statement – this time statement one. Our world has regularities, without which not. Mathematics is all about the play of regularities, so it isn’t coincidental that some mathematics has applicability. This is not mysterious, and it doesn’t imply anything about supernatural agency. Thus it isn’t reasonable to infer the existence of *any* god, let alone the human-obsessed, son-begetting god adhered to by Dr Craig.