Brian Clegg's Blog, page 7

A Short Infinite Series #5 - my infinity's bigger than yours

An infinite series is a familiar mathematical concept, where '...' effectively indicates 'don't ever stop' - for example 1 + ½ + ¼ + ⅛... an infinite series totalling 2. This, though is the last of a short series of posts about infinity.

Apart from Galileo, as ever standing out from the crowd, pretty well anyone who had dealt with infinity was really handling the potential infinity mentioned by Aristotle – and that’s what the familiar curve of the lemniscate ∞ is the symbol for. Not really infinity, but potential infinity. Towards the end of the 19th century, though, one man took the plunge to think about the real thing. His name was Georg Cantor, and it has been suggested that he went mad as a result of it.
Cantor was born in 1845 and spent all his working life at the university in Halle. This is a German town famous for music, but frankly not for maths. Cantor saw it as a stepping stone to greater things – and it probably would have been, had he not come up with some conclusions that were so mind-bending that at least one other mathematician would set out systematically to ruin his career.SetsCantor’s first great contribution was to formalise the mathematics of sets. Sets had been around really as long as people conceptualized – but Cantor embedded them firmly into mathematics. A set is really just a group of things. They could have something in common – like the set of things that look like an orange, or the set of people with the name Brian – or they could be as disparate as the set of things you thought about today. Cantor built on work of other mathematicians to pull together a picture of how sets operate that lets us do everything from define the numbers to establishing simple mathematical operations. But for our purposes I just want to pick out one aspect of set theory because it is going to be very relevant to the jump from sets to infinity. It’s a property of a set called its cardinality. Let’s think of a two very simple sets – the first is the set of legs on my daughter's dog. The second is the horsemen of the apocalypse. These two sets are said to have the same cardinality if I can pair of the members of the set on a one-to-one basis. Let’s try that. 
Front right, for example could pair up with Death, Front left with Famine, Back right with Pestilence and Back left with War. I’ve exhausted both legs and horsemen, so they have the same cardinality. But – and here’s the clever thing – I needn’t have known how many legs the dog has or how many horsemen there are. Okay, in this example it’s no secret, but it wasn’t necessary to know. (can I just say that no animals were hurt in the making of that graphic).
With set theory in place, Cantor was ready to build on Galileo’s observations about the way that with infinity you could get one-to-one correspondence with a ‘smaller’ infinite set. Remember Galileo’s description of the integers, the counting numbers, and the squares. We can say that the infinite set of counting numbers has the same cardinality as the set of squares, because we can do that one-to-one pairing off, even though we don’t know how many we are dealing with.
And we know that the squares are a subset of the integers. Subset’s a spot of set theory that has escaped into the world, but specifically we mean that all the squares are members of the set integers, but it isn’t the full set – there are members that are squares. Cantor, cunningly, used this strange behaviour to define an infinite set. An infinite set is one that has a one-to-one correspondence with a subset.Aleph nullOnce we’re taking a set theory approach to infinity – and dealing with the true infinity, not Aristotle’s potential infinity – it’s useful to have a different symbol for it. Cantor chose aleph ℵ, the first letter of the Hebrew alphabet, and specifically he called the infinity of the counting numbers aleph zero or aleph null because we can’t assume that all infinities are same, so this is the basic infinity, the infinity of the counting numbers. Aleph null has some pretty strange qualities. We can add one to it and still end up with the same number. You can see why if you imagine doing the one-to-one correspondence starting with an extra value called x, then go on with the rest of the numbers.
What’s more, add aleph null to itself and you get aleph null. Again, we can imagine this happening by setting up a correspondence alternately with two lots of counting numbers. And for that matter you can multiply aleph null by itself – and still get aleph null. Imagine matching each of the numbers with its square. There are enough gaps left to take up all the remaining sets of aleph null.
Mind boggling though this behaviour is, we are actually used to special cases in arithmetic. After all, 0 + 0 = 0 and 1 x 1 = 1. And really we should hardly be surprised if the answer never changed, because surely there can be nothing bigger than infinity. Only that’s exactly what Cantor set out to prove wasn’t true. The cardinality of the rational fractionsWe’ve looked at aleph null squared, but Cantor wanted to check how flexible aleph null really was. He’d checked out the counting numbers, but how about fractions? He started with rational fractions, fractions made out of the ratio of whole numbers. Cantor was to prove that there were also aleph null of these using a delightfully neat proof that requires no maths. Imagine laying out every fraction there is in a huge table. In fact, to make it simple we’re actually repeating many of the fractions. If you look down the diagonal coloured in yellow in the image below, they’re all 1. It doesn’t matter that we’ve got redundancy, the table, continued for ever in both directions, has every single ratio in it.

Now what Cantor did was to set up a repeating path through the table. A simple, repeatable rule that enables us to work through the whole table. Finally, he puts each jump in one to one correspondence with a counting number. We’ve just proved that aleph null applies to the rational fractions as well – they have the same cardinality as the integers because we can match them one-to-one by going through this sequence. Incidentally, though the first one in the video below is the path Cantor used, it’s not the only one that exists. He could just as well have used the second one, for instance… the point is that there is a mechanism to set up a one-to-one correspondence.  What about irrational fractions?Yet these aren’t the only sort of fractions. There are also the irrationals, the numbers that written as decimals go on for ever and ever – presumably to infinity. Do these also squeeze into aleph null? With another blindingly simple proof, Cantor was to show that this wasn’t the case. (I ought to make it clear that what comes next is not the actual proof, but illustrates how it works.)
Let’s imagine putting every single fraction, rational and irrational, between 0 and 1 into a list. If I could really achieve that, then I could use exactly the same one-to-one matching proof, matching each fraction against its position in the list, an so would prove this was another aleph null set. Now to work this proof I need to be able to study sequential numbers in the list. If I have them in order I can’t do that, as two numbers in order only differ right at the end of an infinity long set of digits – so I’m going to scramble the list and pick out the first few randomly selected numbers.
Now let’s look at the diagonal set of digits in those numbers. You could imagine this as a number in its own right: 0.220709 and so on. Now let’s add one to each digit. So instead of reading 0.220709, the diagonal is 0.331810 (9 flips over to zero). 
Finally, let’s compare this new number with our original table. It’s not the first number, because they differ in the first digit. It’s not the second number, because they differ in the second digit. It’s not the third number. And so on. We have generated a number that doesn’t appear in our list. Cantor had proved with beautiful simplicity that you can’t cram all the fractions between 0 and 1 into a list with cardinality aleph null. The count of these fractions was something bigger, something bigger than infinity. Take a moment to think about that. 
Infinity, as people never fail to delight in telling me, is a very big subject. You can read far more in A Brief History of Infinity, which is available from Amazon.co.uk, Amazon.com and Bookshop.orgUsing these links earns us commission at no cost to youImage from Unsplash by Steve Johnson - not this infinity!

These articles will always be free - but if you'd like to support my online work, consider buying a virtual coffee or taking out a membership:Article by Brian Clegg - See all Brian's online articles or subscribe to a weekly email free here

Coffees and memberships

Thank you so much to everyone who has already used the 'Buy me a Coffee' link below to support my online book reviews, general science and writing life articles.

As it says below, my posts on the Popular Science website and here on my blog Now Appearing will always be free, but if you'd really like to help keep me going (and to avoid running intrusive adverts, which I hate) I've introduced a membership scheme that involves a small monthly contribution.

There are three levels:

Bronze - £1 a month (or £10 a year), like the individual coffee purchases, this will help me be able to dedicate the time to writing these posts and reviews, but makes it more secure.Silver - £3 a month (or £30 a year) - by moving up to a coffee a month, I'm adding in additional posts and messages just for silver and gold members, plus discounts on signed books. Membership also includes the option to suggest books for review. There will be still be as many free posts for all readers, but there will be some tasty extras for members.Gold - £5 a month (or £50 a year) - in addition to the Silver benefits you will get a free, signed hardback book (or two paperbacks) at the start of the year if paying the full year, or at the end of the year if paying monthly. You will be given a choice from at least five titles each time, with the book(s) posted to your chosen address.I hope you will consider helping support my online writing - just click the 'Buy me a Coffee' button below.

These articles will always be free - but if you'd like to support my online work, consider buying a virtual coffee:

See all Brian's online articles or subscribe to a weekly email free here

AI and search engines - a dodgy combo?

Search engines are central to our everyday use of the internet - I must use a well-known search engine beginning with G at least a dozen times a day. But the search providers are displaying a worrying trend. Swept along by the enthusiasm for artificial intelligence, most have begun to display or offer an AI summary - in Google's case, this is the first thing you see at the top of the search results. And like all generative AI responses, it doesn't necessarily get it right.

This is quite easy to demonstrate if you make use of a query that pushes the boundary a little. I happened to be writing something about the BICEP2 telescope, located at the South Pole. So, interested to see how the AI would handle it, I asked 'Why was the BICEP2 telescope built at the South Pole?' This is quite a tricky question for an AI to handle - and Google's response demonstrated this powerfully. (The highlighting above was already there, it's not from me.)

It's certainly a good guess that you might locate a telescope at the South Pole because it's dark there, at least for six months of the year, with no light pollution. The problem is, though, that the (now replaced) BICEP2 was a radio/microwave telescope - and a lack of background signals in this part of the electromagnetic spectrum is not described as being dark.

But the real disaster in the confident result produced is that final sentence. The location did not help the telescope to detect primordial gravitational waves - admittedly they did announce that they had... but then they had to withdraw the claim within weeks when it turned out to be a result of polarisation from impact with dust. That final sentence is pure fiction.

Admittedly there is a small print warning that 'Generative AI is experimental' - but that isn't the same as saying that it can't be trusted (and anyway you only see it if you click a drop down to expand the original summary). By making the generative AI result the first thing you see - and let's face it, we're all lazy and often don't dig too far into a search result - there is a real danger that the software's potential for imaginings and hallucinations will be taken as an effective source of information. Surely that's not great?

These articles will always be free - but if you'd like to support my online work, consider buying a virtual coffee:Article by Brian Clegg - See all Brian's online articles or subscribe to a weekly email free here

A Short Infinite Series #4 - Galileo's musing

An infinite series is a familiar mathematical concept, where '...' effectively indicates 'don't ever stop' - for example 1 + ½ + ¼ + ⅛... an infinite series totalling 2. This, though is a short series of posts about infinity.

We're going to take a look at Galileo's surprisingly insightfulcontribution to our understanding of infinity. Galileo Galilei was born in 1564, the son of a musician and scientific dabbler. He tends to be remembered for dropping balls of the Tower of Pisa, something which be probably didn't do (it’s only recorded many years later by his assistant) and for being locked up for daring to suggest that the Earth rotates around the Sun rather than the other way round (not exactly accurate historically). Also inventing the telescope (he didn't) and featuring in that Queen song. But he undertook some remarkable thinking on the subject of infinity.
It took place after his house arrest. Galileo had put together his masterpiece, Discourses and Mathematical Demonstrations Concerning Two New Sciences. He had real trouble getting this published – the Inquisition were not too keen after his previous book – and when it was eventually taken up by the great Dutch publisher Elsevier, Galileo expressed his great surprise that it had been published at all, which he claimed had never been his intention. The book took the form of a conversation between characters, and here, having wondered about what holds matter together, they have a diversion, just for the fun of it, into the nature of infinity. Galileo brings out a number of points, but I'll concentrate on two. The first involves the rotation of a pair of wheels. He starts with wheels with a few sides - in the example below I’ve made them hexagons. These are three dimensional shapes – imagine the hexagons are cut out of sheets of marble. The smaller hexagon is stuck to the larger one, and each of them rests on a rail. I've made a poor quality video below to illustrate this.
We roll the wheel along a little. I'm going to do it twice. First watch what the big wheel does, then the small wheel. The big wheel moves along the track by the length of one of its sides. But not only has the big wheel moved on by that distance, so has the small one. It has to. They’re fixed together. Yet the small one should only have rolled along the track by the distance of its smaller side.  How did it achieve it? It lifts off the rail entirely. There’s a gap of just the right distance.
Now here’s the clever bit. Galileo imagined increasing the number of sides. The more sides, the more sets of small movements along the rail and small gaps you get. Now let’s imagine we take that number of sides to infinity. We end up with circular wheels. Again we roll the two wheels, joined together, along their respective rails. Again they both travel the same distance – in this case a quarter of the circumference of the big wheel. But look what’s happening. The rim of the big wheel has rolled out a quarter of its circumference on its track. The rim of the smaller wheel has rolled out a smaller quarter circumference, but the wheel has travelled the same distance, without ever leaving the track. There were no jumps, or at least so it seems.
Galileo imagined that as the smaller wheel turns there are an infinite number of infinitesimally small gaps, which add up to make the difference between the circumference of the wheel and the distance it moves. After letting this percolate through his brain in the background, Galileo’s thoughtfully challenged character, Simplicio, has a complaint. What Galileo seems to be saying (or technically Salviati, the character that is Galileo’s voice) is that there are an infinite number of points in one circular wheel and an infinite number of points in the other – but somehow, one infinity is bigger than the other.Salviati is rueful. That’s the way it is with infinity: a problem he reckons, of dealing with infinite quantities using our finite minds. And he goes on to show how this is perfectly normal behaviour once you are dealing with infinity.
The simple mathematical tool he uses to demonstrate this is the square (that’s the square of a number, not the shape). Salviati makes sure Simplicio knows what a square is – any number multiplied by itself. So, he imagines going through the integers, multiplying each one by itself. [It’s not rocket science. For every single integer there is a square. We’ve an infinite set of integers, and there’s an infinite set of squares in a one-to-one correspondence. 
But here’s the rub. There are lots of numbers that aren’t squares of anything. So though there’s a square for every single number – an infinite set of them – there are even more individual numbers than there are squares. Arggh. Simplicio’s brain hurts, and it doesn’t surprise us. Galileo has spotted something very special about infinity. The normal rules of arithmetic don’t really apply to it. You can effectively have ‘smaller’ and ‘bigger’ infinities – one a subset of the other that are nonetheless the same size. We’ll come back to this in a big way in the final post of the series.

You can buy my book A Brief History of Infinity  from Amazon.co.uk, Amazon.com and Bookshop.org

Using these links earns us commission at no cost to youImage from Unsplash by Paris Freeman 

These articles will always be free - but if you'd like to support my online work, consider buying a virtual coffee:Article by Brian Clegg - See all Brian's online articles or subscribe to a weekly email free here

Up for a challenge? Take on the Conundrum bonus puzzle

A while ago I wrote a book called Conundrum , which consists of 200 puzzles and ciphers - those who complete the entire book are entered in Conundrum's Hall of Fame, currently featuring just 17 impressive individuals worldwide. I occasionally set a bonus puzzle, open to all, with a free signed book as a prize.

This one comes in the form of a challenge requiring you to put together a number of different elements:

Passing under the seventh Duke, take the date of the crocodile, add the psalm number and divide by the verse to get the answer.

If you are up for the challenge, you can enter your solution on the website here (and see a couple of clues). One winning entrant will be chosen at random - entries to be in by the end of 28 February (GMT). 

Please don't append your entries here - only using the form on the website. I look forward to your suggestions: please do let me know how you got to the answer too in your entry.

You can buy my book Conundrum  from Amazon.co.uk, Amazon.com and Bookshop.org

Using these links earns us commission at no cost to you

These articles will always be free - but if you'd like to support my online work, consider buying a virtual coffee:Article by Brian Clegg - See all Brian's online articles or subscribe to a weekly email free here

A Short Infinite Series #3 - Achilles and the tortoise

An infinite series is a familiar mathematical concept, where '...' effectively indicates 'don't ever stop' - for example 1 + ½ + ¼ + ⅛... an infinite series totalling 2. This, though is a short series of posts about infinity, based on my book A Brief History of Infinity .

At first sight, chains of numbers that go on forever seem harmless and without consequence, but it doesn’t take long to find some that will cause difficulties. Let’s say that rather than just list the numbers we add them all up as we go along to produce a sum. And let’s take a very simple series – just alternating 1 and -1: something like 1-1+1-1+1-1+1-1...
It’s hardly rocket science. We can see how it will total by adding some brackets:(1-1)+(1-1)+(1-1)...
Each 1 is cancelled out by a -1, so the total of the series is 0. Or is it? Just shift the brackets and we still have a series that cancels out, but now we’ve got a 1 left over:1+(-1+1)+(-1+1)...
So the same series has a value of 0 and 1. Scary. This has been rephrased as 'If you turn a light bulb off and on for infinity, does it end up on or off?' It could be either. That is clearly an example told by a mathematician – a physicist would tell you that it will obviously be off, because the bulb will have blown.
Or take the simple series at the top of this post where each item is half the value of the last:1 + ½ + ¼ + ⅛...
It seems, as we add in element after element, that it’s going to eventually reach two, though in practice with any particular number there’s always a little gap left. In fact you could say it added up to two if you had an infinite set of numbers – but what does that mean? And how can an infinite list of things (whatever it is) add up to a finite quantity?
This was the basis of one Zeno’s famous paradoxes. Zeno, who was born as far back as around 539 BC, belonged to the school of Parmenides, which considered reality to be unchanging and movement to be an illusion. Zeno knocked up a number of rather entertaining examples to demonstrate the faulty nature of our attitude to change and motion.
Probably the best known is the arrow, which encourages us to imagine two arrows. One floats stationary in space. The other is flying at full speed. Now catch them at a snapshot in time. How do we tell the difference? For that matter, how does one arrow know to move in the next fraction of time while the other doesn’t?  But the paradox that reflects our sequence concerns Achilles and the Tortoise. This unlikely pair are setting out on a race. Achilles, being after all a hero, gives the slower tortoise a lead. And they’re off.In a very small amount of time, Achilles has reached the Tortoise’s position. But by then, the tortoise has moved on. In an even shorter amount of time, Achilles has reached the Tortoise’s new spot. And again it has moved on. And it doesn’t matter how many times you go through this procedure – an infinite set of times if you like – Achilles will never catch the Tortoise up. 
It’s easy to see the relationship of this paradox and the series if we pretend that Achilles only runs twice as fast as the tortoise (perhaps he’s pulled a hamstring or his Achilles tendon). In the time Achilles covers the first distance, the Tortoise moves half that distance. While Achilles is catching up, the tortoise moves ¼ the original distance. In an infinite number of moves they will only get to twice the original distance (which is where, of course the paradox falls down as Achilles powers through that mark).
There certainly is something unsettling, both about the idea of infinity itself and some infinite series. The Greeks weren’t really sure what to do with it. They called infinity 'apeiron', a word that had the same sort of negative connotations as chaos does today. It meant unbounded, uncontrolled, dangerous.

You can buy my book A Brief History of Infinity  from Amazon.co.uk, Amazon.com and Bookshop.org

Using these links earns us commission at no cost to youImage from Unsplash by Joel Mathey (Achilles is just out of shot)

These articles will always be free - but if you'd like to support my online work, consider buying a virtual coffee:Article by Brian Clegg - See all Brian's online articles or subscribe to a weekly email free here

Quantum mechanics in five minutes

A few years ago I was set the challenge of explaining quantum physics to the then BBC broadcast journalist Robert Peston in 5 minutes.

In practice, of course, this is a pretty much an impossible task, but the idea was to give a quick taster - which I hope it does. 

I've written quite a bit on quantum physics, if it's something you'd like to dig into a bit deeper:

The God Effect - exploring quantum theory's most mind-boggling concept, entanglement with implications from teleportation to unbreakable encryption Cracking Quantum Physics - an illustrated beginner's guide to the quantum world The Quantum Age - focuses on the applications of quantum physics that have transformed our world Quantum Computing - how computers making use of explicit quantum effects have the potential to run algorithms that perform in ways impossible to duplicate with conventional devices.Here's that video:

These articles will always be free - but if you'd like to support my online work, consider buying a virtual coffee:

See all Brian's online articles or subscribe to a weekly email free here

A Short Infinite Series #2 - big numbers

An infinite series is a familiar mathematical concept, where '...' effectively indicates 'don't ever stop' - for example 1 + ½ + ¼ + ⅛... an infinite series totalling 2. This, though is a short series of posts about infinity.

Strictly, this one is just about big numbers - but it's on the way to the real thing. There’s something special about big numbers. It’s almost as if by being able to give a big number a name we demonstrate our power over it – and, of course, the bigger the number is, the more power we have. A classic example of this is in the reported early life of Gautama Buddha. As part of his testing as a young man in an attempt to win the hand of his beloved Gopa, Gautama was required to name numbers up to some huge, totally worthless value – and managed to go even further to show how clever he was. 
But you don’t need to go back in history to examine this fascination. Anyone with children will have heard them counting, running away with sequences of numbers as if they’re trying to find the end. It’s not just for fun, of course. These counting sequences are hammered into us at school to help build familiarity with the numbers. Most of you will also be familiar with rhymes, used for the same purpose. For example, One two three four five, once I caught a fish alive, and so on. But there is a slight danger that arises from the use of these sequences – our tendency to remember in strings of information can become a hindrance to flexibility. Try counting 1 to 10 then 10 to 1 in a language you aren’t totally fluent in – you will find 10 to 1 much harder.
It’s fine giving names to numbers we need to use (or useless ones to show off), but realistically how many of us ever use this number:

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

As it happens it does have a name, one that proved something of a problem for the unfortunate Major Charles Ingram when it was his million pound question on Who Wants to be a Millionaire. He was asked if the number – 1 with 100 noughts after it – was a googol, a megatron, a gigabit or a nanomol. Major Ingram favoured the last of these until a cough from the audience came along to push him in he right direction, and to be honest, who can blame him? A googol frankly sounds childish. And there’s a good reason for that. It is. Literally. 
In 1938, according to maths legend, mathematician Ed Kasner was working on some numbers at home and his nephew, 9-year-old Milton Sirotta, spotted this and said something to the effect of ‘that looks like a google’. I’m not convinced – I just can’t see why Kasner would bother to write that on a blackboard, and think it’s more likely he just asked his nephew, what you call a really, really, REALLY big number (say 1 with 100 noughts after it)’ and up pipes Milton ‘a google.’ 
But this urge to give names to large numbers dates back even further than Buddha. Right back, in fact, to Archimedes, who was born some time around 287 BC. In one rather strange little book, Archimedes made an attempt to count the number of grains of sand that would fill the universe. This clearly wasn’t a practical proposition. But the Greek number system was extremely clumsy, and what Archimedes seems to have been doing in his book, The Sand Reckoner, is to demonstrate that it’s perfectly possible to extend the system as far as you like.
Archimedes starts the book, addressed to King Gelon of Syracuse, by pointing out that some people reckon that there are an infinite number of grains of sand in the world. What he meant by this was not literally infinite, but an uncountable and incalculable number. And then, perversely, Archimedes set out to show this wasn’t true – and just to rub it in, calculated how many grains it would take not only to cover the Earth, but to fill the whole universe. It’s an impressive feat, though we need to be a little careful about what we mean by universe. Archimedes had in mind a picture of everything that had the Earth at the centre of a number of spheres. Around us went the moon, the sun, the planets and finally the stars. So the ‘universe’ he would fill with sand was more like our idea of the solar system – even so, quite a size. Tantalisingly, Archimedes goes a little further. He imagines an even bigger universe, suggested by an off-the-wall theory around at the time that instead of the sun moving around the Earth as it obviously does, the Earth moved around the sun. It’s tantalising because Archimedes’ passing mention of this theory of Aristarchus is the only surviving reference to the earliest known person to spot that we move around the sun rather than the other way around. After making a few assumptions like ‘the diameter of the earth is greater than the moon, and the diameter of the sun is greater than the earth’ (many close to the truth) and some elegant geometry, Archimedes comes to the conclusion that the universe is no more than 10 billion stades across. That’s a measurement based on a stadium, just as we often estimate in football pitches, at around 180 metres, so it probably makes his universe 1,800 million kilometres, which at just outside the orbit of Saturn really isn’t bad. I say probably because stades were a local measure - a stadion was the distance around your local stadium and didn't have a universally accepted value. But it's order of magnitude correct. Archmedes then begins to work up from a grain of sand, through a poppy seed to greater and greater size. He had a bit of a problem, though, as the biggest number in the Greek system was a myriad – 10,000. Archimedes set up a whole supersystem. Everything up to a myriad myriads was the first order. Multiply that by itself and you got the second order. And so on. Do that a myriad myriad times and you’d reached the end of the first period. And so on one again. With a few rules to make this ‘new maths’ work, he was able to put the number of grains of sand in the universe as less than 1,000 units of the seventh order (1051), or in Aristarchus’s bigger universe, less than 10 million of the eighth order (1063). Even by today's standards these are pretty big numbers.

You can buy my book A Brief History of Infinity  from Amazon.co.uk, Amazon.com and Bookshop.org

Using these links earns us commission at no cost to youImage from Unsplash by Keith Hardy - not much sand by Archimedes' reckoning.

These articles will always be free - but if you'd like to support my online work, consider buying a virtual coffee:Review by Brian Clegg - See all Brian's online articles or subscribe to a weekly email free here

Exceptions, proofs, rules and puddings

The English language is a tricksy thing, replete with sayings that can, on the face of it, appear odd, or that get mangled after many repetitions. I recently heard something about one of these on the excellent The Studies Show podcast, hosted by Stuart Ritchie and Tom Chivers, that made me raise an eyebrow, because they claimed my interpretation of a saying was a myth.
The saying in question was 'the exception proves the rule'. I want to come back to that after a brief excursion into another saying that involves puddings.
One of the most cringe-making things for me is when I hear someone on the TV or radio say 'The proof is in the pudding.' This is a totally meaningless statement resulting from mangling the saying 'The proof of the pudding is in the eating'. Anyone using the first version needs to be sent to an English Language re-education camp immediately. But the real version itself can look distinctly confusing. We can prove a mathematical theorem, or that black swans exist... but how do you prove a pudding?
The saying depends on being aware that 'prove' has a less familiar meaning of 'test'. In fact, if you look up 'prove' in the Oxford English Dictionary, the very first definition is 'to make a trial of; to try, to test'. With this in mind, we can see that what's being said here is you can test how good a pudding is by eating it. It's not exactly great wisdom, but it makes sense.
So we move onto that disputed saying 'the exception proves the rule'. This feels weird because you might think that an exception disproves a rule. If a rule is, for example, if we posit a rule that 'all prime numbers are odd' and we point out that 2 is a prime number, this is an exception to that rule, which disproves it. Disproving theories and rules is a central part of doing science or maths.
However, if we use the same meaning of 'test' for 'prove' as we did with the pudding, it makes total sense. We are 'making a trial' of that rule with an exception and breaking it. Yet Chivers and Ritchie (sound like an upmarket jam manufacturer) say that this is a myth.
The basis for this suggestion, I suspect, is that there was a Latin legal phrase 'exceptio probat regulam in casibus non exceptis', which translates as 'the exception proves the rule in cases without exceptions' - which, to be honest, doesn't make much sense except as an indirect way of saying that rules need to exclude situations where the rules don't apply - a total truism. But these aren't the same words, and there is always a danger of deriving English constructs from Latin - this is what led to the bizarre idea that split infinitives should be allowed in English because you can't do them in Latin.
I don't doubt that the Latin phrase may have been the original inspiration of the English phrase. Emphasis on the 'may' - I am not aware of good evidence for the direct link. However, equally there is no doubt that the pudding-style 'proof' here is a valid interpretation of the phrase that frankly is more effective than the Latin use (and the OED supports it as a possible meaning), so it feels wrong to call it a myth.
While we're in the world of puddings, incidentally, I'll finish by briefly mentioning Rutherford's 'plum pudding' model of the atom, where there definitely is a myth involved. I've seen many popular science titles say that in this model the electrons are represented by plums. Unfortunately, plum pudding is an archaic name for a Christmas pudding (see illustration), where 'plum' was as a generic term for dried fruit. Christmas puddings don't contain plums and Rutherford had grape-derived dried fruit such as raisins in mind.
Image from Unsplash by Matt Seymour

These articles will always be free - but if you'd like to support my online work, consider buying a virtual coffee:

See all Brian's online articles or subscribe to a weekly email free here

Coming Soon

After my voluntary break from popular science over the Christmas period, I've also had a dearth of review copies for books out in January. But there's a whole heap of both science and science fiction titles sitting on the review pile - so while reviews themselves are in short supply, here's a few books where the reviews are coming soon:

Already published (awaiting copy) - I nto the Great Wide Ocean - Sönke Johnsen - takes readers inside the peculiar world of the seagoing scientists who are providing tantalizing new insights into how the animals of the open ocean solve the problems of their existence.4 February - Hoodwinked - Mara Einstein - from viral leggings to must-have apps, exposes the hidden parallels between cult manipulation and modern marketing strategies in this eye-opening investigation. Reveals how companies weaponize psychology to transform casual customers into devoted followers.6 February -  Phenomena  - Camille Juzeau and The Shelf Company - from fireflies to the Big Bang, from the magnificent maps of the world’s sands to the anatomy of ice crystals, these 125 illustrated graphic phenomena are designed to take you on a stroll through the vast world of knowledge.13 February - Against the Odds - John and Mary Gribbin - highlights the achievements of women who overcame hurdles and achieved scientific success (although not always as much as they deserved) in spite of male prejudice, as society changed over about 150 years, from the middle of the nineteenth century to the end of the twentieth century. Includes Eunice Newton Foote, who discovered the carbon dioxide greenhouse effect; Chien-Shiung Wu, who discovered the law which allows matter to exist in the Universe today; and Barbara McClintock, who discovered how genes turn on and off.27 February - Pagans - James Alistair Henry - an alternative history crime novel set in a 21st century London where the Norman conquest never happened and Christianity is a minor cult, with modern Britain based on the ancient tribes - but murders still need to be solved.1 March - Amazing Worlds of Science Fiction and Science Fact - Keith Cooper - explores the fictional planets from films such as Star Wars, Dune and Avatar, and discusses how realistic they are based on our current scientific understanding and astronomical observations. 13 March - The Decline and Fall of the Human Empire - Henry Gee - inspired by Edward Gibbon's The History of the Decline and Fall of the Human Empire, Gee follows up his Royal Society Prize winner A (Very) Short History of Life on Earth with a book that plots the future of humanity, suggesting that space habitats may be the best way to keep our species alive.... and many more. Note that the dates are the release date of the books in the UK - I aim to get reviews out as close as possible to this, but it's not a guarantee!

These articles will always be free - but if you'd like to support my online work, consider buying a virtual coffee:

See all Brian's online articles or subscribe to a weekly email free here