Gravity explained in 761 words

Cosmology, Education, Mathematics, Physics, Relativity, Science, Science communication, The scientific method, Things Explained, , , , , , , ,

People seem to be harbouring the impression that there is no good theory of Gravity yet. I asked a few friends – most thought Newton had explained it, but couldn’t explain it themselves. This is rather sad, 80-odd years after a darn good theory was proposed.

Of course there is still some controvery and the odd contradiction with other beloved theories, but the heart of the General Theory of Relativity really does a great job of explaining gravity and it is really wonderfully beautiful, and can be roughly explained without recourse to jargon and equations.

This is a theory that’s just so darn elegant, it looks, smells and tastes right – once you get it. Of course, the ‘taste’ of a theory doesn’t hold much water; for a theory to survive it needs to make testable predictions (this one does) and needs to survive all manner of logical challenges (so-far-so-good for this one too).

This is not a theory that needs to remain the exclusive domain of physicists, so for my own personal development as a scientist and writer, I thought I might try an exercise in explaining what gravity is – according to the general theory of relativity.

For some reason, my wife thinks this is strange behaviour!

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The story really got started when Einstien realised that someone in an accelerating  spaceship would experience forces indistinguishable from the gravity felt back on Earth. 

He or she could drop things and they would fall to the floor (assuming the spaceship is accellerating upwards)  just as they would fall on earth.

So perhaps that’s all gravity is… some sort of accelleration? Let’s see.

In the spaceship, it’s clear to us that the objects would appear to fall to the floor, but in reality, it is the floor of the spaceship that is rushing up towards the objects – this explains why things fall at the same speed whether heavy or light, matching Galileo’s own test results when he dropped various things, supposedly from the leaning tower of Pisa. It further implies that things will ‘fall’ even if they have no mass at all… such as light beams.

The thought experiment goes thus: Consider if you had a laser-beam shining across the spaceship control room; it would curve slightly downwards, because the light hitting the opposite wall would have been emitted a little time ago, when the spaceship was a little way back, and going a bit slower (remember, its accellerating).

We know the light is not bending, it is just that the source is accellerating, resulting in a curved beam. Imagine a machine-gun spraying bullets across a field – as you swing the gun back and forth the bullets may form curved streams of bullets, but each individual bullet still goes straight.

So Einstein suggested that perhaps light beams will bend in this same way here on earth under a gravitational field. Now Newton’s theory of gravity says light beams may also bend if they have ‘mass’, but the mass of light is a dodgy concept at best (it has inertia but no rest mass, but that’s a whole different blog posting). Anyway, even it it does have mass, it would bend differently from what Einstien predicted. So the race was on to see how much gravity could bend light…

This bending of light prediction was proven by a fellow called Eddington who showed that during a solar eclipse, light from distant stars was indeed bent as it passed near the sun, and by exactly the predicted angle.

Einstein went further though, suggested that light beams on Earth are, just like on the spaceship, really travelling straight, and only appear to bend, and that this can be so if space-time itself is curved. They are going straight, but in curved space.

We know that the shortest distance between two points is a straight line, but if that line is on a curved surface, supposedly straight lines can do strange things – like looping back on themselves. Think of the equator. This model therefore allows things like planets to travel in straight lines around the sun (yes, you read right).

The model has been tested and shown to work, and gives good predictions for planetary motion.

So what can we take home from all this?

Well mainly, if this model is right, we need to let it sink in that gravity may not be a force at all, but an illusion, like the centrifugal ‘force’ you experience when you drive around a corner.

Secondly, it is an open invitation to think about curved space and its marvellous implications!

Extrapolating your way

Education, In the media, Mathematics, Science, Skepticism, The scientific method, , , , , , , ,

There is a very powerful scientific reasoning tool that I use, that, it occurs to me, I wasn’t actually taught… the simple art of extrapolation.

Most people have a pretty good idea of what extrapolating is – its where you look at a trend and predict what will happen if that trend persists. 

For example, if I said it took me 6 months to save £500, I can use extrapolation to predict how long it will take me to save £2000; its something we do all the time – yesterday I was driving down from Bristol, I could count off the the miles, and knowing the distance, I could predict if I would make it for dinner (I didn’t).

Scientists use this too. A good example is the way we can calculate the temperature of “absolute zero” by looking at the volume of a balloon as you heat it up. If you had a balloon at 25C, and you heat it to about 55C its volume would increase by about 10%. What does that tell us? It tells if we cooled it, it would eventually have no volume – and that this would happen at around -275C (-273.15C actually) – absolute zero.

Of course, the method relies upon assumptions – usually the assumption that the trend will continue in the same way (people often use the term “linear” to represent relationships that form straight lines when plotted on a graph).

What if the relationship is non-linear? For example, if little James is 5 feet tall when he is 10, how tall will he be when he is 20? Clearly he won’t be 10ft tall – that is because the relationship between height and age is “non-linear”.

Most of us are smart enough to extrapolate without knowing the jargon, but when the relationships get complicated a bit of maths and jargon can help.

For example, if we want to examine the population of bacteria in a petri dish, or the spread of a virus (or a rumour) through a population, our mental arithmetic is not always up to it. Luckily, some scientists have realised even these complex affairs have some predictability and although “non-linear”, they can still be modelled – graphs can be plotted and extrapolations made.

If this interests you, I refer you to books on epidemiology; I will move onto another sort of extrapolation – one used to check people’s theories by identifying ‘impossible’ extrapolations.

Let’s say, for example, that the want to predict  how the obesity epidemic will progress in the coming decades. If the media says obesity in a certain group increased from 14-24% between 1994 and 2004, and then goes on to predict that obesity will therefore reach 34% by 2014, does this withstand scrutiny?

Never mind that the definition of obesity may be faulty (BMI), never mind that they are extrapoliting from 2 data points – let’s rather ask if the linear trend is justifiable. This can be done by extrapolating the prediction to try to break it. 

If the model is right, obesity will go on increasing and soon enough 100% (or more!) of the population will be obese. This is clearly wrong – obesity is not likely to get everyone – vast swaths of the population are likely to be immunised to some extent against obesity due to active lifestyles and good dietary educations, or perhaps its in their genes, the lucky things. 

The truth will of course be more complex – the first group to become obese will be the most vulnerable, so an increase from 14-24% may incorporate that group, but each successively 10% will be harder fought.  All this is enough to suggest the predictions made for 2014 are doubtful, and those that go further are downright shameless. But it doesn’t stop them…

I am sure you can think of other suspicious trend-based predictions, like those for peak-oil or global warming. They could do with some improvements, so get to it!

 

The trouble with academia…

Science, ,

I have had enough exposure to scientific academia (6 years full time, and now 6 years as an industrial PhD supervisor) to have seriously lost faith. Some of the issues well put by Jonathan Katz, a physicist in his article Don’t become a scientist.

Perhaps ‘lost faith’ is too kind. “Fed up with”, is perhaps more precise. Perhaps I just want to whine. The complaints I would add to Dr Katz are the following opinions I have come across:

  1. The assumption that unless you are a professor at a tedious university, you have no intellect.
  2. The idea that unless your ideas are published in a the ‘preferred’ peer reviewed journal, they are not worth bothering with.
  3. The idea that research with some practical use is somehow inferior.
  4. The idea that anyone deserves to get public money to ponder their theories without a feeling a gratitude and indeed an obligation to pay this kindness back.

Perhaps I am just envious and wish I was a don at Oxford… 😉

How to jump higher

In the media, Physics, Sports, The scientific method, , , , , ,

Have you ever noticed that in order to jump your highest, you need a few preparatory steps and a preparatory hop? Have you ever wondered why?

Well, perhaps sadly, I have. I was wondering if this was mere mental preparation – because surely your ability to jump high is going to be dictated by a) the power in your legs on the one hand, and b) your weight on the other hand?

Turns out it’s not.

But why not? And what do the preparatory leg movements do?

If you think the steps are “warming you muscles up” in preparation for the jump you are, in some sense, right – right that you are preparing, but its not the muscles you are preparing but the tendons…

But what are tendons, and what do they do? Good question.

I used to think that tendons were there to help glue the muscles to the skeleton, like the ropes on sailboats glue the sail to the mast and boom. I later realised that they also help make the body more ergonomic by allowing the muscles to be positioned ‘out-of-the-way’ as is the case for your hands; if all the muscles used for your fingers were actually in your fingers they would be rather fat, and not particularly dexterous.

Likewise, if you had to carry your calf muscles in your feet, it would make your feet somewhat heavier and mean you would have to swing lots of weight back-and-forth, up-and-down when you walk and run. Although the calves and thighs still have to move a bit, tendons have allowed the movement to be reduced substantially by connecting these muscle groups to the feet from safer havens further up the leg. 

Interestingly (to me anyway), this also explains why some people can’t bend their little finger independently of the second (ring) finger – they are sharing tendons that go right up the forearm!

Ah, but none of that explains why we need to hop before we can jump.  True enough – there is another property or function of the tendon that had never occurred to me – until I read “The new science of strong materials” by J.E. Gordon (which I must recommend to scientists of all disciplines, it s a lovely book, I wish I had written it).

So what does a book on material science have to do with tendons? Well, it points out that tendon is unique among materials for its capacity to stretch and in doing so, to store energy. In other words, tendons make remarkably good springs (and explains why animal tendons have been used to make crossbows for thousands of years). 

Springs can be thought of as batteries, you put a bit of effort into stretching them now, and the effort is stored there for later use – to shoot an arrow for example.

So,  it’s simple really, your legs are like a pair of crossbows: if you pre-stretch the tendons before the jump, and then co-ordinate the energy release to coincide with the power stroke of your muscles, you will jump higher.

And finally, it turns out that a rather good way to pre-stretch the tendons is to hop before we jump. So there you are!

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You will probably have noticed that good basket-ballers and high jumpers have taken this further (whether understanding it or not). They run along horizontally at a fair speed (gaining kinetic energy), and then thump down their leg at an angle, transferring all that kinetic energy into their tendons, and then re-cooping it in a vertical jumping burst. I have not done the maths, but I am willing to bet the tendons do much more work than the muscles in the crucial powering phase of the jump.

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Something to try:

Have you ever noticed how much easier it is to do 20 jumps in a row rather than 10 jumps with pauses between each one? This is because the pauses between jumps force you to dissipate the energy in your tendons, and so every jump is pure muscle work.

It is also worth noting that even the task of dissipating the energy stored in tendons is tiring for your body – the muscles actually work against the tendons to turn the energy into heat, which is similar to the work your muscles have to do when you walk down stairs.

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Final word…

The really nerdy readers (those after my own heart) will have further noted that this all goes a long way to explain why PE teachers seem to be so obsessed with stretching. You thought it was just to prevent injury? Think again – while it is essential to warm up and down carefully and stretching once warm does reduce the risk of injury, this is largely because stretching improves the condition of your tendons.

So don’t rush through the stretching next time you go to the gym; stretching and flexing your tendons may well do more to improve your athletic performance than muscle work.  Tendons deserve a part in any workout, please don’t neglect them – they are, after all, the forgotten workhorse of the body.

Information: what exactly is it?

Education, Mathematics, Physics, Science, Science communication, Things Explained, ,

I was walking to the tennis courts in Battersea Park a few years back, when I heard something on my Walkman radio. It stuck with me for years, and until tonight I haven’t followed up on it, read about it or written about it. Though I have told everyone at my work, which has resulted, as usual, in groans about how nerdy I am (and genuine amazement at how I could spend valuable time pondering these things).

What I heard was a very short anecdote about someone who wrote a little regarded paper in the 1940’s (see ref below) in which he made an attempt to define a ‘measure’ for information. Although I never read any more about it (until today), what I heard was enough to set me thinking…

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Now, if you know lots about this subject then bear with me. Those readers who don’t know what he came up with: I challenge you to this question:

  • what contains more information, a phone-number, a ringtone or a photo?

Are they even comparable?

Bits & Bytes…

In this computer age, we already have some clues. We know that text doesn’t use up much disk space, and that photos & video can fill up the memory stick much quicker.

But what about ZIP files? These are a hint that file-size is not a very accurate measure of information content.

So what is a megabyte? Is it just so many transistors on a microchip? Happily, its not, its something much more intuitive and satisfying.

Information: what is it?

If you go to Wikipedia and try to look up Information Theory, within a few seconds you are overrun with jargon and difficult concepts like Entropy; I hope to avoid that.

Let’s rather think about 20 questions. 20 Questions is the game where you have 20 questions to home in on the ‘secret’ word/phrase/person/etc. The key, however, is that the questions need to elicit a yes/no response.

To define information simply: the more questions you need in order to identify a ‘piece of information’, the more information content is embodied in that piece of information (and its context).

This helps us to answer questions like: “How much information is in my telephone number?”

Let’s play 20 questions on this one. How would you design your questions? (Let’s assume we know it has 7 digits)

You could attack it digit by digit: “is the first digit ‘0’? Is the first digit ‘1’? Then changing to the next digit when you get a yes. If the number is 7 digits long, this may take up 70 questions (though in fact if you think a little you will never need more than 9 per digit, and on average you’ll only need about 5 per digit – averaging ~35 in total).

But can you do better? What is the optimum strategy?

Well let’s break down the problem. How many questions do we really need per digit?

We know that there are 10 choices. You could take pot luck, and you could get the right number first time, or you might get it the 9th time (if you get it wrong 9 times, you don’t need a 10th question). However, this strategy will need on average 5 questions.

What about the divide and conquer method? Is it less than 5? If yes, you have halved the options from 10 to 5. Is it less than three? Now you have either 2 or 3 options left. So you will need 3 or 4 questions, depending on your luck, to ID the number.

Aside for nerds: Note now that if your number system only allowed 8 options (the so-called octal system), you would always be able to get to the answer in 3. If you had 16 options (hexadecimal), you would always need 4.

For the decimal system, you could do a few hundred random digits, and find out that you need, on average 3.3219… questions. This is the same as asking “how many times do you need to halve the options until no more than one option remains?’

Aside 2 for nerds : The mathematicians amongst you will have spotted that 23.3219 = 10

Now, we could use 4 questions (I don’t know how to ask 0.32 questions) on each of the 7 digits, and get the phone number, and we will have improved from 35 questions (though variable) to a certain 28 questions.

But we could take the entire number with the divide and conquer method. There are 107  (100 million) options (assuming you can have any number of leading zeroes). How many times would you need to halve that?

1. 50 00o 000
2. 25 000 000
3. ….
…
22. 2.38…
23. 1.19…
24. 0.59…

So we only needed 24 questions. Note that calculators (and MS Excel) have a shortcut to calculate this sort of thing: log2(107) = ~23.25…

OK, so we have played 20 questions. Why? How is the number of questions significant? Because it is actually the accepted measure of information content! This is the famous ‘bit‘ of information. Your 7 digit number contains about 24 bits of information!

Epilogue

As you play with concept, you will quickly see that the amount of information in a number (say the number 42), depends hugely on the number of possible numbers the number could have been. If it could have been literally any number (an infinite set) then, technically speaking, it contains infinite information (see, I’ve proven the number 42 is all-knowing!).

But the numbers we use daily all have context, without context they have no practical use. Any system that may, as part of its working, require ‘any’ number from an infinite set would be unworkable, so this doesn’t crop up often.

Computer programmers are constantly under pressure to ‘dimension’ their variables to the smallest size they can get away with. And once a variable is dimensioned, the number of bits available for its storage is set, and it doesn’t matter what number you store in that variable, it will always require all those bits, because it is the number of possibilities that define the information content of a number, not the size of the number itself.

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I hope that was of interest! Please let me know if I’ve made any errors in my analysis – I do tend to write very late at night 😉

References:

1.  Claude Shannon, “A Mathematical Theory of Communication” 1948

Skepticism: religion’s cancer

Atheism, Evolution, In the media, Skepticism, , , , , ,

Religion has been described as a virus. This is not because it’s ‘bad for you’ necessarily, but rather due to the way it spreads.

It’s not hard to see the parallel: like viruses (and bacteria), religions exist within a population and spread from person to person.

But what about atheism? Is it a viral idea (meme) too?

I will argue that it isn’t. Perhaps it’s more like a cancer, a ‘mutation’ that kills off religious infections.

Cancers are sneaky, because they can occur spontaneously, almost by chance, and are therefore a very statistical phenomenon: your chance of getting cancer is affected by a), your exposure (to carcinogens causing mutation events), and b), your predisposition (genes affecting your ability to cope with the these mutations). 

Your chance of becoming an atheist is likewise affected by a), your exposure (to information about how the world works) and b), your predisposition (intelligence, or ability to apply logic to the information).

I.e. atheism differs from religion in the same way that carcinogens differ from viruses.

Can we develop this idea? I think so.

Let’s look at how you ‘get’ atheism…

Picture it: you’ve been brought up in a good god-fearing, church-going family. You went to Sunday school, you know which of Cain and Abel was the baddy and you can explain to people about how there is good evidence for The Flood. You also have a healthy fear of  sex and the other sins.

But you go to school and you learn about plate tectonics and see how well South America slots into Africa, and then you learn how European bees are not quite the same as African ones, just like Toyota Corollas aren’t, and one day, while looking at the grille of your step-mother’s 1.3GL, and daydreaming about the A-team, a thought strikes you, like a shot of cancer-causing sunshine on that patch of skin on the back of your right shoulder, that cars evolve differently in different counties and maybe that explains all the animals and perhaps God didn’t make a women out of Adam’s rib after all, cos’ that never did make much sense, because a rib is a pretty silly thing to make a women out of anyway.

Catching a dose of Christianity on the other hand, does not come from inside, as the result of reasoning, it comes from outside, from other people.

Most often you will be born into a house absolutely soaked in the infection, you will be infected soon enough, prayers will be said at mealtimes, the church is so big and grand, and the hymns are so catchy, and then they wheel out Christmas and baby Jesus (or baby ‘cheeses’ as my son says)…

But even if you’re not so lucky, there’s hope. You can drop in at a church any time (though Sundays are best I’m told) and the chances are, even if you are down on your luck, short of friends, and even if you aren’t very nice, the sweet people there are quite likely to help you. That feeling of family, of unquestioning acceptance – brings a special warmth to the cockles of the heart.

Once you’re in the door, religion, having evolved pretty niftily, can now play you like a violin. Your emotions, developed to help promote clan solidarity, are hi-jacked and kick in nicely. Did you know, that if you really listen to what these folks say, and really try to feel God’s love, you will indeed feel something! Now that’s a clever infection…

A house price prediction…

Economics, In the media, The scientific method, , , , ,

House prices, like the stock market, are tricky to predict. 

As with the stock market, there are two classes of parameters that affect the prices – the so-called ‘fundamentals’, like supply and demand, the price-to-earnings ratio on the one hand, and the more transient effects like the economic climate and the ever-slippery ‘confidence’.

There has been feverish speculation for years in the UK, and the prices rose for 15 consecutive years, and are at last dropping.

So why did the prices get so high? Many economists would argue it was a classic “bubble”, a self-perpetuating cycle of confidence building more confidence; in other words the fundamentals were being ignored.

Of course, the people found fundamentals they claimed justified the prices; in particular increased demand. Folks living longer, divorce, folks marrying later, immigration, and the breakdown of the family unit; all these things mean we need more houses.

But if these fundamentals were the whole reason, the prices wouldn’t be dropping as they are now. OK, so now most will admit it got out of hand and this is a correction. But how far has it got to correct?

The bubble, it seems to be agreed, was really helped by two factors:

Firstly there was a throttle on the supply – planning permission is notoriously hard to get and the government probably knew it and were happy with prices rising, it made everyone feel prosperous. On a more sinister front, housing developers may have sitting on prime real estate to deliberately keep prices high.

Secondly, there was easy credit – anyone and their dog could get the cash so people who really shouldn’t have been in the game got in and are now out of their league.

But there is a third factor I’ve not seem discussed in the media: the baby-boom generation.

Hasn’t this bubble coincided with the baby-boomer’s ‘rich’ phase – the age from 45-60 when the kids are off and 25 years of mortgage payments have built up the asset list? Surely this is the age-group that is most likely to own big houses, or multiple houses for that matter?

So what will happen now? The bubble has burst, the correction is in full swing, but what will happen in the next 10 years as the baby boomers start retiring, downsizing, and dying? Will this coincide with the next bubble-burst? Will the industry and government look at the population age profile during planning?

I personally hope this is why the market is cock-eyed – why it is that a professional engineer in his mid-thirties with a internationally comparable salary can’t afford more than a mid-terrace house with a 5×5-metre garden…

So I predict (well pray really, if that’s possible for athiests) that we will get into an oversupply situation and that house prices should correct from this ‘second-order’ bubble.

Of course, even if I am right, it may be that the prices are kept up by nasty developers identifying whole towns to ‘let go to ruin’ just to keep the prices high in the next town along…

Celebrity Dynamics

Economics, In the media, Science communication, The scientific method, ,

Celebrity Dynamics. 

The list of people we all ‘know’ isn’t that long, yes, it probably thousands – politicians, actors, singers, historical figures, sports stars – but in a country like the UK, it is still a remarkably small fraction of the populace.

Of course, there are ‘spheres’ – people interested in politics know more politicians, sports fans have more sporting heroes – we here in Cornwall have our local ‘Cornish’ celebrities.

However, if we remembered every celebrity, we would soon run out of space in the public ‘memory’, so we have to be selective.

The media know this – they constantly face choices of which story to follow, and the decisions will often be arbitrary; two minor celebrities did two things today, and we only have 45 seconds of time to fill in our variety news programme – which shall we choose?

This decision process is simple – the editor will pick the celebrity who has more recent ‘hits’ in the news.

Why? Because they know that the audience is more likely to recognise the name – and they know that if the audience hear that name twice it reinforces the memory.

This simple logic creates a very interesting system in which the rise to fame becomes ‘autocatalytic’ – a self-perpetuating, accelerating process. All you need to do is pass some ‘critical point’ of news coverage and you may be in for a ride!

However, we can only hold so many names in the list, so anyone who is out of the news for a time drops off the radar pretty fast, even if they did once enjoy high exposure.

If you are like me, you’ll be thinking of exceptions – folks who just stay famous regardless – do they buck this logic? I don’t think so.

Such people most likely still get exposure, even if its not them in the news – perhaps we see their CD on our shelf, or we talk about their ‘field’ (Thatcherism, Darwinism, Keynesian economics,), and this may be accentuated if their field gets in the news – as has recently been the case for Keynes.

So what value does this theory have?

I think it explains:

  • why so many great deeds don’t lead to fame
  • why often only one person from a high achieving team is ‘selected’ for fame
  • why there’s no such thing as bad publicity
  • local fame does not easily turn to national fame

It also suggests that if you want to be famous, you should:

  • a series of newsworthy events in succession is probably better than a single highly newsworthy achievement
  • if you are in a group/team/band, you need to be the leader or public face of the group
  • you should associate yourself with a newsworthy field, ideally become the posterboy/girl for the field, always dragged out when the field is in the news

And if you want to stay famous once you are you should keep in the public eye:

  • associate yourself with newsworthy events
  • differentiate yourself from other celebrities in your ‘space’ or
  • gang together with other celebrities to create newsworthy events
  • become the posterboy/girl for a newsworthy field/subject, the one dragged out when the field is in the news

Aside:  There seems to be another way to maintain fame:- create mystique, the image of privilege, of some higher plain of existence away from the mundanity of everyday life. People say they like down-to-earth celebrities – that’s because they are very rare – you have to be ‘proper’ famous to stay famous without this tactic! 

Of course, this all assumes you want to be famous! You can equally use the theory to keep a low profile 😉

Good luck either way!

Analogies not equations, please!

Education, Mathematics, Physics, Science, Science communication, The scientific method, , ,

Have you ever noticed how equations look far more complicated and hard to understand than the concept they represent?

I sometimes get myself stuck having to read other people’s work (it’s the ‘peer review process’) and when I first read it, I am often utterly confused, like a person stumbling around a dark room they’ve never been in before. However, because I am expected to make intelligible commentary, I soldier on until I understand what is being said.

Once you understand something, it is hard to remember what you felt like before you understood it. How did that equation look the first time you saw it? I have been thinking about this…

Let’s consider ‘equations’ – a common part of many technical documents. I have found that I always overestimate how clever or useful the equations really are when I first see them. So what does this mean?

It means that using equations to help teach people we risk turning them off by giving them the impression that the work is harder than it is.

Let me give an example:

Maxwell’s wave equations. These are considered (rightly) to be an cornerstone of physics, as they model the behaviour of waves in the inter-related electric and magnetic fields. When I first read them, they were ‘greek’ to me, literally. Here’s a small one:

maxwell-faraday-equation

Obviously, you need to know more to understand what they are about. You need to know what each symbol represents – and you need to know what the operators (the × in this case) actually do. For anyone who has not specifically studied maths at university would then need to backtrack quite far, because in this case the ‘×’ is not the ‘×’ most folks know and love, its the ‘cross product’ which applies to vectors. That even leaves most science graduates cold, draining the joy of discovery for a few hours or days while you go away to learn (or remember) what the heck that means.

But is it all worth it? Is the complexity of partial differential equations and matrix multiplication really required in order to understand what the equation is describing?

Of course not!

So why are equations always wheeled out to ‘explain’ phenomena? This is a failure of teaching. Of science communication. Surely concepts can be explained much better by the use of anecdotes, metaphors & illustrations?

Scientists working at the bleeding edge of science have to be very precise in their logic, and when communicating with one another, equations are undoubtedly very efficient ways to describe hypotheses. And so, while they are good ways for experts to relate, they make it harder for newbies to “break in”, and are dreadful teaching tools.

The Maxwell equations really just describe how waves propagate in a medium – and really its just the full 3-d version of waves in a slinky, or ripples in a pond. The equations, while drawing on complex (and difficult) maths, are describing something the human brain already has an intuitive grip on, because we’ve seen it!

I’m not suggesting we could do away with equations – they are valuable in the predictions they make for those who already understand what they represent – I am just suggesting that equations should be de-emphasised, and only dragged out when the student starts to feel the need to describe the phenomenon mathematically.

So my message to all university lecturers and text-book writers is: describe a phenomenon with the use of analogy, please!

Imaginary numbers challenge

Cosmology, Mathematics, Physics, , , ,

I have a challenge for people who understand imaginary numbers (if that is indeed possible).

Now, I have seen how imaginary numbers can be useful. Just as negative numbers can.

For example, what is 4-6+9?  7. Easy. But your working memory may well have stored ‘-2’ in its mind’s eye during that calculation. But we cannot have -2 oranges. Or travel -2 metres. Oh sure, you can claim 2 metres backwards is -2 metres. I say its +2 metres, the other way (the norm of the vector).

What about a negative bank balance? I say that’s still platonic, a concept. In the real world it means I should hand you some (positive) bank notes.

We use negative numbers as the “left” to the positive’s “right”. Really they are both positive, just in different directions.

Now for imaginary numbers. I have seen how they allow us to solve engineering problems, how the equations for waves seem to rely on them, how the solution of the differential equations in feedback control loops seem to require them.

But I argue that they are just glorified negative numbers. The logarithmic version of the negative number.

So what is my challenge?

Well, the history of mathematics is intertwined with the history of physics. Maths has made predictions that have subsequently helped us to understand things in the real world. Maths models the world well, such as the motion of the planets, or the forces sufferred by current carrying wires in magnetic fields.

But the question is: is there any basis in reality for imaginary numbers? Or the lesser challenge, negative numbers? 

Is there a real world correlation to “i” ? Or is it a mere placeholding convenience?

Or perhaps positive numbers also lack this correlation?