I’ve had a few thoughts and discussions since I wrote this article about drawing stars. I thought it was worth sharing.
Stars within stars
The first observation is that stars, drawn in the way I described, contain other stars, nested within one another like a set of Russian dolls. Recall that we use the term ‘n-k star‘ to indicate a star with n points such that, if we draw a circle through all the points and consider the boundary of that circle as split up into n curve segments bounded by the points, then the straight line from one point to another traverses k of those curve segments. Like this, for a 7-3 star:
Yuriy made an interesting observation about the stars in my last article, that the sum of the angles of all the points of an n-k star is 180∘ × (n-2k). In the course of thinking of ways to prove that formula true, I came upon the realisation that a n-k star contains a n-(k-1)star, which contains a n-(k-2) star and so on down to the n-1 star, which is a regular, n-sided polygon.
Here’s a picture that shows this for an 11-5 star.
The red, outer 11-5 star contains a green 11-4 star, which contains a red 11-3 star, which contains a green 11-2 star, which contains a red 11-1 star (polygon). The points of each inner star are the innermost vertices of the star that immediately contains it. Since we will be referring to those vertices again later, let’s make up a name for them. We’ll call such a vertex a ‘tniop‘, since it is in a sense the opposite of a point. The above diagram shows a point and a tniop. We’ll call the stars inside a star ‘sub-stars‘.
We saw in my last essay that, when a star cannot be drawn without taking the pencil off the paper, it is made of a number of ‘component stars’, that are rotated copies of one another. Here is a picture of a 16-6 star, which uses different colours to highlight the two component stars. We have two 8-3 stars, one light blue and one red.
And here is a picture that uses colour variation to show the sub-stars of each of the two components.
The 8-3 light blue star contains an 8-2 star which is made up of two components that are 4-1 stars (also known as ‘squares’) coloured blue and pink. Similarly, the 8-3 red star contains an 8-2 star which is made up of two components that are 4-1 stars (also known as ‘squares’) coloured yellow and green.
The 8-2 stars each contain an 8-1 star (octagon) as the intersection of two squares -pink and dark blue for one octagon and yellow and green for the other.
Finally, those two octagons between them, bound a hexadecagon (16-sided polygon or 16-1 star). So altogether, in the one picture, we have:
- one 16-6 star (red and light blue)
- one 16-5 star (also red and light blue) inside that
- one 16-4 star (pink, green, dark blue and yellow) inside that
- one 16-3 star (also pink, green, dark blue and yellow) inside that
- one 16-2 star (also pink, green, dark blue and yellow) inside that
- one 16-1 star (also pink, green, dark blue and yellow) inside that
- two 8-3 stars (one red, one blue) making up the 16-6 star
- two 8-2 stars (one pink and dark blue, one yellow and green), one inside each of the 8-3 stars
- two 8-1 stars (octagon: one pink and dark blue, one yellow and green), one inside each of the 8-2 stars
- four 4-1 stars (squares: coloured pink, dark blue, yellow and green) which, in pairs, make up the 8-2 stars.
That’s sixteen stars altogether. What a lot of stars in one drawing! Can you see them all?
Here’s a different colouring that makes it easy to see all five 16-point stars:
Although the stars get smaller as k reduces, they do not shrink away to nearly nothing. In fact they get closer together as they go inwards, as if they are asymptotically approaching a circle of some fixed, minimum size.
To investigate this, I drew a 101-50 star:
You’ve probably noticed by now that I’m no longer drawing these by hand. My drawing is much too wobbly to capture the intricacies of stars-within-stars. So I wrote a computer program to draw them for me. I’ll try to remember to attach it at the end of the article, so that those of you who like mucking about with computers can muck about with it.
Anyway, that 101-50 star pretty well killed my hypothesis that the inner stars can’t get very small. In this one they almost disappear out of sight. I like the swirly patterns. I haven’t yet worked out whether they are really features of this very spiky, very complex, star, or whether they are just artefacts of the crudeness introduced by the computer’s need to pixillate.
Here’s a zoomed-in image of the interior of that star. Cool, eh?
This is a low-resolution image. I have saved a moderately high-resolution image of this star here. Zooming in and out is fun. It seems almost fractal as more patterns emerge from the inside when we zoom in. Also, the stars give the illusion that they are rotating as we zoom. To get the best effect you need to download the file (a .png image file) and then open it up, so that zooming is not limited by your internet connection’s speed.
Ratio of Outer to Inner radius
Let me briefly pick up on that idea above about whether there is some minimum inner radius for these stars. For each n, the spikiest n-k star is where k is the largest integer less than n/2, and this contains another k-1 stars, nested one within the other, down to the innermost, which is a n-sided regular polygon. We can work out the ratios of each star to the one immediately inside it, and use that to work out the ratio of the outermost, n-k, star, to the innermost, n-1, star. The attached computer program contains trigonometric formulas to do that. Here are the ratios of the radii of the innermost to the outermost star for each n from 1 to 109:
We observe that the ratios generally go down as n increases, but the decline is not steady. It bumps up and down. I have highlighted the prime numbers with asterisks. Notice how the ratio for those is always lower than for the numbers around them. The two drivers of the ratio seem to be:
- the size of n: the ratio generally declines as n increases; and
- the number of different factors n has. Note how 16 (divisible by 2, 4, 8) has a higher ratio than 15 (divisible by 3, 5) and 56 (divisible by 2, 4, 7, 8, 14) has a higher ratio than 55 (divisible by 5, 11).
This would be interesting to look into further, and to see if there is some neat, sweet, compact formula for the ratio that highlights the relationship to size and number of factors (if there really is one). But I have to stop thinking about that now or I’ll never post this essay.
The most general form of symmetrical stars
We can make an awful lot of stars using the above approach. For an integer n the number of different n-pointed stars is the biggest integer less than n/2.
But in fact there is an infinite number of different n-pointed stars, without having to loosen our standards by allowing asymmetry. After a bit of thought, I realised that the most general form of n-pointed star can be specified by a single number, which is the ratio of its inner radius to its outer radius. The outer radius is the radius of the circle on which the points sit. The inner radius is the radius of the circle on which all the tniops sit. Given n and that ratio – call it θ, we can draw a star as follows:
- Draw two concentric circles with ratio of the inner to the outer radius being θ.
- Mark n equally-spaced dots around the outer circle and draw faint lines connecting each of these to the centre. These will be the points of our star.
- Mark a dot on the inner circle halfway between each of the faint, radial lines drawn in the previous step. These dots will the the tniops of our star.
- Working consistently in one direction around the circle, draw a zig-zag line fro point to tniop to point to tniop and so on, always connecting to the nearest dot.
We will describe such a star as a n/θ star. We use a slash rather than a dash in order not to mix it up with the former type of star. Here is a sequence of six-point stars, with the ratio of the inner to outer radius going from 0.2 up to 0.8:
It is nice that this gives us more options for stars – infinitely many different kinds of star for each n in fact. But they are not as much fun to draw as the n-k stars, and it is harder to make them come out right without geometric instruments – which rules them out as an effective doodling pastime.
Note how, unlike with the n-k stars, the line leading away from a point does not intersect any other point. Nor do we get any inner stars for free. The price of gaining more variety is a loss of structure. The inner structure of a n-k star provides a great richness by enforcing all sorts of relationships between the vertices.
Only very specific values of the radius ratio θ give us n-k stars. I worked out the formula for the ratio by the way, with a bit of trigonometry. The ratio of the tniop radius to the point radius for a n-k star is:
2 sin(π(k-1)/n) sin(πk/n) sin(π(1-(2k-1)/n)) / (sin(2π/2)+sin(π(1-(2k-1)/n)) )
– sin(π((2k-1)/n – ½) )
For a n/θ star, if there are no integers n and k that give values of that formula equal to θ, the star will not have the array of inner stars that a n-k star has.
Computer Program to draw pretty stars
This whole diversion started as an exercise in drawing stars by hand. But there’s a limit to how intricate those stars can get without getting too messy with smeared ink or graphite. For those that like to look at pretty, intricate geometrical pictures, here’s my computer program in R that can draw stars of any of the types discussed.
Sums of angles
For those that like mathematical proofs, there are outlines of proofs here of Yuriy’s observation about the sum of the internal angles at points of a n-k star.
Bondi Junction, April 2017
It has become apparent to me that the world needs another instalment in my series of suggestions for Adult Amusements. There have been complaints. Some are from pedants, who insist that a single monograph about standing on one leg does not constitute a series. Others, more gravely, have expressed concern about the occupational health and safety implications of people trying to balance on one leg while their mind is distracted by other things, like budgets, work-shopping and brain-storming, not to mention trying to be Pro-Active, Customer-focused, Agile, Continuously Improving and Outside the Box all at the same time.
So, belatedly, here it is. I hope that this will be considered less dangerous, being a mostly sedentary activity.
When in business meetings that do not hold us riveted with fascination, we should draw stars!
But not just any old stars. Special stars. Mathematical ones. Stars with prime numbers in them.
It is the dearest wish of every little child, after that of being a firefighter or an astronaut, to draw excellent stars in their pictures. But a wish is one thing, and its fulfilment is another. When as a child I tried to draw stars, the only technique I could think of was to draw a spiky circle. Start anywhere, and draw a perimeter that goes around an imaginary centre, that is a series of spikes. Maybe this works OK for others, but for me it typically produced a result like this (Figure 1):
It invariably goes wonky, because it’s hard to keep track of where the centre is supposed to be, and to make the points point away from that centre. Mine looks like a confused kookaburra.
When one gets a little bit older and more sophisticated, one learns – by instruction or by observation of others – the two standard techniques for drawing stars. These are the six-pointed star, which is made by drawing an upside-down triangle slightly above a right-way up one (Figure 2):
and the five-pointed Pentacle, which requires a little more coordination, but can be done without taking the pencil off the paper (which I call a ‘single pencil stroke’), by following the arrows as shown (Figure 3):
Learning to draw either of these stars is on a par with learning to ride a bike, in terms of the sense of achievement, wonder and progress. All of a sudden, one can construct an image of symmetry and elegance with the stroke of a pencil – or two strokes, in the case of the six-pointer.
I was very happy with this advance in technology for a long time, but then came the day when I hankered after drawing more bristly stars, with seven, ten or twelve points. I tried, but found I was just reverting back to the method of figure 1, and my bent stars just did not satisfy me.
One could of course take out a protractor and compass and, with a bit of preliminary calculation, measure out the exact angles needed for each point, and draw the star using that. But firstly that’s cheating, and against the Spirit of Doodling, and second it might cause others to notice that one is not paying attention to whatever the meeting is discussing.
I thought I was destined to be forever that object of public ridicule – the man with the two-star repertoire. But just as I was starting to come to terms with this being my fate, a discovery came to me in a blinding flash: instead of trying to draw spikes in a circle, I needed to generalise the methods used for the five and six-pointer. Well, to cut a long story short, I tried that, it worked, and now I can draw stars with any number of points up to about fifty.
Here is the method that generalises the way we draw five-pointed stars:
Drawing a star with a single pencil stroke
- Step 1: pick the number of points N, and draw that number of points, as evenly spaced as you can, around the perimeter of an imaginary circle. If there is a large number of points it’s best to first draw points at the 12, 3, 6 and 9 o’clock locations and then put one quarter of the remaining points into each of the four quadrants. To be precise, divide N by 4 to get a quotient Q and a remainder R. Then draw Q points in each of R quadrants of the circle, and Q-1 points in the other quadrants. Ideally, if R=2, adjacent quadrants should not contain the same number of points, but it doesn’t matter very much if that is forgotten.
- Step 2: pick a number K, greater than 1, that has no common factors with N. To make the spikiest possible star (ie with the thinnest spikes), choose K as the largest whole number less than N/2 that has no common factors with N. For instance if N=12 that number is 5. If N=13 it is 6. If N=6, 4 or 3 there is no possible K, and this method cannot be used. I’m pretty sure that, for any N greater than 6, there is at least one K for which this method will work, but I have not proved that yet.
- Step 3, choose your favourite direction in which you want to draw. Unless you are a pan-dimensional creature drawing on paper with three or more dimensions, your only possible choices are clockwise or anti-clockwise.
- Step 4 starting at any point, draw a straight line from that point to the point that is K steps away from it, hopping from point to point around the circumference in the chosen direction. We can call K the ‘side length’, since it is the length of the line that connects one point to another.
- Step 5: repeat step 4 until you get back to the starting point.
If this process is executed carefully, you will have drawn a star that has a point at every one of the points you drew in step 1. And, if you want, you can do all the actual line drawing in steps 4 and 5 without ever taking your pencil off the paper.
Here is a depiction of that process for an eleven-pointed star with side length 5:
And here is a depiction of this process for a sixteen-pointed star with side length 5:
Why do we not allow the side length K to be 1? That’s because if we do that, we just get a N-sided shape which, ignoring any irregularities in our drawing, is a regular polygon, like this, for N=12 (a ‘dodecagon’):
Now the thing about stars is that they are not convex, while regular polygons are. Using the word ‘vertex’ for a place where two edges of a shape meet, an N-pointed star has 2N vertices, of which N are points – the outermost part of a peninsula (if we imagine the shape as an island in an ocean) and the other N are the innermost part of a bay. As we go around the vertices of a star they alternate between inlet and bay. So a regular polygon is not a star because it has no bays, and that’s why K must be more than 1.
Stars with more than one pencil stroke
We observed that the above method does not work for N=6. But we know we can draw a six-pointed star, using two pencil strokes to draw two overlapping triangles. We can use the approach taken there to invent many more stars. In fact, for an N-pointed star there are M different types we can draw, where M is (N+1)/2-2, rounded down to a whole number. Each of these shapes corresponds to using a different value of K, from 2 up to the biggest whole number below N/2.
Here is how we do it:
- Step 1, for picking N and drawing the points around an imaginary circle, is the same as above.
- Step 2. We pick any K as any whole number greater than 1 and less than N/2.
- Do steps 4 and 5 from above. This will draw a shape that is either a star or a polygon. Now comes the tricky bit.
If the shape you drew has not touched all the N points around the circle, repeat the process starting on a point that has not been touched yet. I like doing this with a different colour pencil, as it helps me see the pattern and avoid getting confused.
Repeat that process, using a different colour pencil each time, until all points have been touched.
You will now have a N-pointed star, made up of a number of identical overlapping shapes, which are either all polygons or all stars.
For those that like mathsy stuff, the number of overlapping shapes – the number of pencil strokes required – will be the greatest common factor of N and K. It’s fun to try to work out why that is.
The traditional six-pointed star in figure 2 above is what you get under this method when you use N=6 and K=2. Here are a couple of others:
If we are going to draw a lot of different stars, we need names for them. We could call the star drawn with N points and side length K a ‘N-K star’, so that the pentacle is a 5-2 star and the traditional six-pointer is a 6-2 star.
If we wanted to, where N is even, we could let K be N/2. What we get then is this sort of thing:
The shape we have drawn with each pencil stroke is a single line between a point and the point directly opposite it. Strictly speaking, this too is a star, but I mostly leave it out because it’s not as interesting as the others because (1) everybody knows how to draw a star like that; (2) as any five-year old would tell us, that’s not what stars look like in pictures of things in the night; and (3) it has no inside, so we can’t colour it in all yellow (well, actually the one I drew has a tiny little inside in the middle, because it’s not perfectly symmetrical. But a more accurate drawing would have all the lines going exactly through the middle of the circle, so that there’s no inside at all).
So now you know how to do lots of great stars. You need never be bored in a meeting again. Imagine if you started drawing all the possible stars, starting at the smallest number of points and going up in side-lengths and points until the meeting finished. Leaving out the too-easy ‘thin stars’, you would draw the following stars:
5-2, 6-2, 7-2, 7-3, 8-2, 8-3, 9-2, 9-3, 9-4, 10-2, 10-3, 10-4, 11-2, 11-3, 11-4, 11-5, 12-2, and so on.
Just drawing those, given a due amount of tongue-stuck-into-side-of-mouth-concentration, should be enough to get you through at least a half hour of Death By Powerpoint.
But let’s not forget our roots. With a very few exceptions, we all started off drawing stars like Figure 1. There is a touching ingenuousness about such stars, and I think it’s good to draw them as well. Often really interesting shapes arise when we do, looking like monsters or funny animals. And one good thing about that way is that you don’t have to decide how many points it will have before you begin. You just draw spikes around a circle until you get back to the start. I’ll sign off by doing that for a star with LOTS of points (it ended up being 21), and following it up by a series of the nine different stars with the same number of points drawn by the above method.
I think that each has a certain appeal, in a different way.
‘Partial differentiation’ is an important mathematical technique which, although I have used it for decades, always confused me until a few years ago. When I finally had the blinding insight that de-confused me, I vowed to share that insight so that others could be spared the same trouble (or was it just me that was confused?). It took a while to get around to it, but here it is:
My daughter Eleanor make a drawing for it, of a maths monster (or partial differentiation monster, to be specific) terrorising a hapless student. The picture only displays in a small frame at the linked site, so I’m reproducing it in all its glory here.
Bondi Junction, October 2016
Here’s a piece I wrote explaining the mathematics behind the peculiar phenomenon of acoustic ‘beats’.
It’s a bit maths-y. But for those that don’t love maths quite as much as I do, it also has some interesting graphics and a few rather strange sound clips.
Bondi Junction, August 2016
Mathematics is a bit like plumbing, don’t you think?
Not that I’ve ever done any proper plumbing. Electrical work I can manage, having paid careful attention when doing electricity in our high school physics classes. But plumbing is another matter altogether. Getting those joints water-tight requires just a bit more dexterity than I can manage.
That’s why it’s like maths. It’s those joints that are the problem. Let me explain.
Often in maths, when one is trying to prove some important result, one creates a series of theorems that build on one another to reach the final conclusion. If a proven result along the way is substantial enough – say if it takes more than half a page to prove – we call it a ‘theorem’, otherwise we call it a ‘lemma’. But a lemma is really just another theorem, only a bit smaller than usual.
Now a theorem is something that reads like this:
IF <bunch of premises> is true THEN <conclusion> is true.
Mathematicians try to make their premises as few and weak as possible, and their conclusion as strong as possible. For instance, we could prove a theorem that
IF x=2 THEN x2>0
This theorem is true, and easy to prove. But we can either make the conclusion stronger (ie more specific), turning it into say x2=4, or we can make the premises weaker, turning them into say ‘x is a real number’. We can’t do both however. We could, if we wanted, make the premises weaker and the conclusion stronger, but not by as much as what we just did. We could prove the theorem:
IF x>1 THEN x2>1
Now our premise is weaker (less specific) than before, and our conclusion is stronger (more specific) than before.
How this is like plumbing is as follows. Say we want to prove theorem D and, to do that we need to first prove theorems A, B and C. This might be because we need A to prove B, B to prove C and then C to prove D. So the conclusion of A becomes a premise for B, the conclusion of B becomes a premise for C and so on.
Now unless D is very easy to prove (eg like proving IF x=2 THEN x2>0), we won’t have much room for manouevre. If we make the premises of D too weak then we won’t be able to prove the conclusion, which is our ultimate goal. On the other hand, if we make the premises of D too strong we won’t be able to satisfy them with the conclusion of C.
This is like connecting a series of pipes (got you! And just in time. You thought I’d never get around to the plumbing didn’t you?). Each theorem is a pipe. Each pipe is a straight cylinder except that, at its inflow end, it flares out like a trumpet bell and, at its outflow end, it narrows. This enables pipes of the same diameter to snugly fit together. The end of each pipe has to fit inside the end of the pipe immediately downstream from it if the fit is to be tight enough to avoid a leak.
The inflow end represents the theorem’s premises. The stronger the premises, the narrower the inlet, and the stronger the conclusion the narrower the outlet. Now what happens if we make the premises of pipe D too strong? Then the downstream end (conclusions) of pipe C may not be able to fit inside the premise end of pipe D. If that happens then we need to narrow pipe C, which means making its premises stronger – hence a narrower inlet. That may then create a problem with fitting B into C, and so on. So if we are too ambitious in what we are trying to prove at D, that can create problems all the way back up the stream so that our premises at A are so narrow and restrictive as to make the whole combined theorem of no practical use (because the premises at A will hardly ever be satisfied). The theorem might only apply to left-handed, bearded Scottish taxidermists whose first and last names both begin with ‘X’.
Or, starting at the upstream end, if we start with premises that are too weak – non-specific – at A, our whole series of pipes will have to be so wide that the eventual conclusion at A will be as generalised as an astrologer’s forecast (‘Today will be a good day for working hard and believing in yourself!’). How can I relate this diffuse outcome to plumbing? Hmm, well if the pipe is too wide at the outlet then perhaps there won’t be enough pressure. Yes that’s it, no pressure. It would be like trying to water the garden with a big drainage pipe instead of a hose.
What one finds oneself doing then, as one tries to make the connection from A all the way to D, is going back and forth, loosening a premise here, tightening a conclusion there, then finding that with the premise we just loosened, the conclusion coming out of that pipe is too loose (wide) to fit into the next pipe so we tighten the premise again or alternatively look at the following pipe to see if we can get away with loosening the premises for that.
And now for one last bit that is completely unrelated, except that it also is like plumbing. Recently I’ve been working on something called Homology Theory, which is what you need to be able to tell the difference between a sphere and a donut. This discipline uses techniques of algebra to identify differences between different shapes. It often makes use of chains of things called modules, connected by functions, which are like arrows leading from one module to another. It’s a bit like how you can make a model of a molecule using plasticine for the atoms and straws for the chemical bonds (the modules are the plasticine and the functions are the straws, except it’s important to remember that the functions have a direction, as if water were flowing through the straw). An important part of a function going out of a module is something called the ‘Kernel’, which is a bit like the nucleus of the atom. Each function in a sense ‘projects’ the module from which it starts onto the module at which it ends, and the projection it makes is called the ‘Image’ of the function, which will be part or all of the target module. We call a chain of functions and modules ‘Exact’ if the image of the function coming in is identical to the Kernel of the function going out. Most chains are not Exact, but when a chain is Exact we can prove all sorts of useful things.
So this is more plumbing. If the pipe/straw (function) coming into a joint (module) is too wide (has too big an Image) it won’t fit the Kernel of the pipe/straw (function) going out. Then the liquid will spill out, we’ll have a big mess and we’ll never be able to tell our donuts from our spheres, let alone our circles and Klein Bottles. But if it all fits neatly, we’ll have an Exact Sequence, there’ll be no spillage, and we’ll be able to prove all sorts of useful things without needing a change of clothes.
I’m rather pleased with how many different metaphors I managed to mix together in that last couple of paragraphs. It leaves ‘to take arms against a sea of troubles’ far behind. I think the clarity may have suffered somewhat as a consequence though. Never mind.
Bondi Junction, December 2014
“What is Truth” asked Pilate?
I’ve always been rather fond of that riposte in the New Testament. Good old Pontius sure put those scribes, priests and Pharisees – whatever they were – in their place. Is there more to it than just a great put-down though?
It sounds deep as well, although there’s always the risk that it may be faux deepness, or ‘deepity’ as Daniel Dennett would call it.
Now I’m not going to go all post-modern here, and suggest that everybody has their own version of truth. Nor am I going to suggest that Truth isn’t a useful concept. To ask Freda whether she thinks Bill is telling the truth is a meaningful and useful question. It is actually a speculation about the state of Bill’s mind. Is he telling us what his memory tells him happened, or not? No, what I’m wondering about is what is sometimes called Absolute Truth, the sort of thing exemplified by Bertrand Russell when he says ‘Edinburgh is North of London, whether anybody knows it or not.”
Sometimes Truth is held up as a sort of Holy Grail, the quest for which can fill one’s life with meaning. The Search For Truth is the sort of noble thing to which holy men and scientists alike are said to be dedicated. Indeed, it’s not that long ago that it was a key component of a little motto I made up for myself in an idle moment – “Truth, Beauty and Compassion” – a more inclusive version of St Paul’s triumvirate “Faith, Hope and Love”. I vacillated for a while over whether I should add “Reason” to my motto. I mostly omitted it, because things just seem so much neater in threes, as well as because there seemed to be too much overlap with Truth. Sometimes I included it though, because there seem to be things covered by Reason that are not covered by Truth. For example, one can use Reason to devise a strategy to win the heart of one’s true love, but this has little if anything to do with Truth. Recently though, I have sometimes felt inclined to drop Truth from the motto in favour of Reason, and I blame Kurt Godel for this, as I will explain in the next couple of paragraphs.
Godel and Truth
The logician Kurt Godel is most famous for his First Incompleteness Theorem, which destroyed the most ambitious mathematical project of the early 20th century – the Hilbert Program, which was an attempt to find a way to formally prove everything that was believed to be true about arithmetic and basic mathematics. What Godel’s theorem showed was that, in any logical system that we use to write proofs, there will exist some logical statements that can neither be proved nor disproved, yet are perfectly meaningful. He did this by creating an ingenious variation of the famous ‘Liar Paradox’, which says ‘this statement is false’.
Now, in their attempt to make Godel’s discovery more interesting to a lay audience, some texts and sources claim that the theorem says that there are statements that are ‘true but not provable’. The trouble is that the ‘true’ part of this statement doesn’t have any clear meaning. The closest we can get is to observe that Godel’s theorem shows that there are statements that are unprovable in the chosen logical system, but for which we can prove, using a different logical system, something that is very similar to the original statement. So in that case the word ‘true’ doesn’t relate to some fundamental cosmic reality, but just to what can be proved in an alternative logical system. The word ‘true’ is superfluous as well as ambiguous here, as we can express everything Godel discovered using variants of the word ‘proved’ instead of true.
The realisation that ‘True’ is a useless word in this critical context made me wonder if there is any context in which it expresses something important, that cannot be more accurately described by other words. It is that question that I wish to explore here.
Science and Truth
Let’s start with science. Is that a search for truth? The task of science is to construct theories that relate past experiences (‘observations’) to one another in an ordered way and enable us to make predictions about future experiences, such as the appearance of comets or the effect of mixing hydrochloric acid with sodium hydroxide.
Are scientific theories ever true? Not many scientists or philosophers of science would claim that in any absolute sense. Any scientific theory, however useful, is eligible to be replaced by a more sophisticated one that better explains our observations. The first simple atomic theories were replaced by the Rutherford model in which a nucleus of protons and neutrons was surrounded by electrons whizzing around the outside. That was in turn replaced by Quantum Mechanics, and then we had quarks and other subatomic particles coming into the theory in the 1960s. Now physicists speculate on whether the subatomic particles in turn are just symptoms of some deeper underlying phenomenon such as vibrating strings in eleven dimensional space. It seems natural to suppose that there is some underlying final theory that tells everything, and that the theory is ‘True’ regardless of whether we eventually discover it. But that is by no means the only logical possibility. Here are two others:
- There is an unending descending chain of theories, each more complicated than the one above it and including all its ancestors as special cases that are good approximations under restricted circumstances.
- There are multiple theories that can explain the full range of possible experiences, all as valid as one another. The competing corpuscular and wave theories of light are crude examples of this. In the end, they both proved inadequate to explain all the experimental results obtained, but for a while they were both successful in explaining different aspects of the behaviour of light. There is no apparent reason why there could not be two or more different theories, each of which can by itself explain everything we experience.
Sometimes the Common Sense view of the world is that of course there is an underlying Truth – that quarks and electrons really exist and we just need to find out more about them. While I am passionately in favour of finding out more about quarks, I regard that as a desirable refinement of our theories, not a discovery of Truth. If we are to think in terms of common sense, I cannot help but observe that common sense tells me the table in front of me is a hard, solid, opaque object, not the strange collection of pinpricks and force-fields in empty space that quantum mechanics suggests. That is not to say I don’t ‘believe’ quantum mechanics. Quite the contrary. Rather it is to say that I don’t want to give common sense any privileged role in such a deliberation. And also, that the notion of Truth is compromised. Which is more true, my perception of the table as hard and solid, or the idea of it as an empty space with the odd tiny quark and electromagnetic force field? The practical response is that the question is meaningless. Both representations of the table have their uses, and are complementary. Table as solid object is useful if I want somewhere to put my kettle. Table as quarks is useful if I want to understand why it doesn’t collapse under the weight of the kettle (or how heavy a kettle I can put on it).
Perhaps a belief in Truth equates, in this context at least, to a Realist view of the world – that the world really is made of a specific sort of thing or things, and we just need to find out what it is. It is a belief in Reality, with a capital R.
But hold on, any moderately sane person believes in Reality don’t they? Well, it depends what one means by reality. I am sure I have been guilty of saying ‘so and so lives in a dream world – they have only a passing acquaintance with reality’. My defence is that I do not mean by this that so-and-so doesn’t believe in quarks. Rather, I mean that they do not reason about their experiences in such a way as to give a strong likelihood of their expectations of future experiences being fulfilled. Perhaps so-and-so has gone sun-bathing on the beach in the middle of a summer’s day, under the illusion that a coating of sunscreen will be enough to protect her for two hours in full sun. The ‘reality’ from which she is disconnected is that that evening she will have the experience of red, horribly painful skin, and two days after that she will have the experience of the skin peeling off in sheets. So-and-so’s delusion has nothing to do with her metaphysical opinion of quarks. It is about her inability to use reason to control the nature of her future experiences.
So it seems that when I use the word ‘reality’ in everyday intercourse, I am referring to what experiences we can expect in the future, and what we might be able to do to control those experiences.
Note that Reason, or Rationality, has stepped in here to fill the vacuum left by our jettisoning the notion of Truth.
Truth in other contexts
A belief in Truth is not limited to just quarks. At various times, people have proposed the existence of other forms of truth in the form of objects that exist independently of what people know. Moral rules, aesthetic truths and mathematics are key examples of this. A Moral Realist will claim that moral rules exist, and the purpose of moral education and ethical inquiry is to discover them. A mathematical realist will say the same thing about Pythagoras’s Theorem, or the number six. An aesthetic realist may be a rarer beast, but if I could find one, I would expect them to assert that Mozart’s Requiem is beautiful and Milli Vanilli is not.
Yet there is no more need to believe in such Truths than there is with quarks. I can develop a system of moral rules for my own use, or discover them in my own mind by reflection on my feelings. If I prefer something less subjective, I can define a moral truth to be any feeling about how to treat others that is shared by a large proportion of the human race.
With mathematics I can, if I wish, regard any mathematical concept as an invented idea that helps me arrange things in my mind and reason about what might happen next. I can take the view that everybody invents the mathematics that they use for themselves. So every toddler invents the number six, rather than discovering a pre-existing abstract object. Or perhaps they find a pre-existing pattern for the number six somewhere in their brain, placed there by evolution because of its usefulness to survival.
The more I reflect on this, the less value I see in maintaining a concept of Absolute Truth in the way one thinks about the world.
Consider the binary expansion of Pi. Let F(n) be the statement that, for any m greater than or equal to n, there are more 1s than 0s in the first m bits of that binary expansion. Let S be the statement that there is some number n for which F(n) is true. There is no way of proving or disproving S by trial and error, because we cannot try out what happens with all the infinite number of combinations of values of m and n. Perhaps there’s a clever way of proving it or disproving it that doesn’t require trying an infinite number of combinations, just as we can prove that the sum of two odd numbers is always even, without trying it on all possible pairs of odd numbers. But nobody knows of one, and it seems entirely plausible that there is no such proof.
Yet most people would say that S is either true or false, but we just can’t know which. This seems to be a problem, because there are no words to explain what we mean by true or false here. Let us explore some more.
The Liar Paradox
The Liar Paradox is a statement that generates a contradiction, however you interpret it. One of its more sophisticated versions is ‘This sentence is not true’. If we suppose it’s true then what it says must be true, so it is not true, which contradicts our supposition that it’s true. If we suppose it’s false then what it says cannot be true, so it can’t be true that the sentence is not true, so the sentence must be true, which contradicts our supposition that it’s false. The third possibility is that the sentence is meaningless. But if it’s meaningless then it’s not true, which is exactly what the statement says is the case, so the statement is true, so it’s not meaningless, which contradicts our supposition that it is meaningless.
There are a number of ways to resolve this paradox. Most of them involve questioning what we mean by truth in this context. One possible meaning is provable – we say that something is true if and only if it can be proved. This has shades of Godel’s Theorem. If we take this approach then the paradox disappears because ‘This statement is not provable’ is not paradoxical if we require our proofs to be strictly logical (although one has to have at least a partial understanding of Godel’s theorem to see why that is so).
Another possible meaning is that a sentence is true if it accurately reflects our perception of reality (or just our perception, if we want to avoid the equally troublesome word ‘reality’), false if the opposite of the statement reflects that perception, and meaningless otherwise. This is a popular interpretation of ‘Truth’, which is called the ‘Correspondence Theory of Truth’. With that meaning, it is ‘true’ to say a soccer ball is round because, when we look at it, it looks round. With this meaning, we can resolve the paradox by observing that the sentence is meaningless because it doesn’t say anything about our perception of reality. All it does is talk about itself.
In both these resolutions, we have only been able to make sense out of the sentence by getting rid of the vague, undefined word ‘true’ and replacing it with a clear, pragmatic definition.
This realisation strengthens my feeling that the word True is a troublemaker, that sows confusion wherever it is used. Perhaps we really should dispense with it, except in causal, imprecise, everyday slang. But if so, what should we replace it with?
What can we replace Truth with?
The words ‘true’ and ‘truth’ are so much a part of everyday language that, if we were to discard them, replacement would be required on a large scale. For someone that subscribes to the Correspondence Theory, mentioned above, that theory provides a satisfactory way of defining the terms, and hence justifying their retention. The price that must be paid for that though, is believing that there is such a thing as Ultimate Reality – a unique description of the way things are, to which any other descriptions are only approximations. Furthermore, that description must exist regardless of whether anybody knows it.
Most people seem to hold that belief, and for them the dilemma ends there. But, for the reasons given above, I am skeptical, and certainly see no justification for such a belief in terms of other, more fundamental and intuitive concepts. All we can say with confidence is that we have experiences, and that there are models available – called scientific theories – to connect these experiences together in a patterned way and make predictions about what we may experience in future.
For someone that suffers this same ‘skeptical disability’ as me, the Correspondence Theory is not an option. So what, if anything, can we use instead?
My proposal is to replace it by ‘consistency with experience’. We will call a statement ‘true’ if it is consistent with both actual and potential future experience. By ‘consistent’ we mean there is no possible rational deduction from actual or potential experience that contradicts any part of the statement. So if June tells us that she has not eaten the last biscuit, that is not true if, upon inspecting the biscuit barrel, we would see that there are no biscuits left in there and, if we used an endoscope to inspect her stomach contents, we would find traces of that type of biscuit therein. It would also fail to be true if June’s statement was inconsistent with her own experience – that is, if she has had the experience of taking and eating the last biscuit. It would even fail to be true if June were a sleepwalker who has been known in the past to sleepwalk to the kitchen, take a snack and return to her bed without waking, and she woke this morning with a rash of a kind that she has only ever had before after eating that type of biscuit. In this latter case, June has not experienced eating the biscuit, but she can rationally deduce from her experience of the rash that she has probably eaten it in her sleep.
So that’s my suggested replacement for Truth – consistency with experience. And since the key ingredient for assessing this consistency is Reason, I will adopt that as the third leg of my little motto: Compassion, Beauty and Reason!
Note that we haven’t really got rid of the word Truth. All we’ve done is redefine it. Or rather, defined it because, unless one subscribes to the Correspondence Theory, it was never properly defined in the first place.
This definition has some interesting implications.
Remember the above example of the statement S about the number pi? With this definition we have to conclude that, unless a proof one way or the other exists, S is neither true nor false, because there is no potential experience we could have that would confirm or deny it. In the mathematical context, true means the same thing as ‘provable’.
Further, even if there is a proof or disproof of S that we just haven’t found yet, Godel’s Theorem tells us there will be other statements that can be neither proven nor disproven. So we are now committed to saying that those things are neither true nor false.
On the other hand, we are empowered to say very clearly that provable statements like 2+2=4 are true, because we can easily go through the experience of proving them. Strictly, what we need to say is that ‘In Peano Arithmetic, 2+2=4’, because one could easily invent another system of arithmetic in which 2+2 equalled something else. So the statement is actually about the consistency of what we are saying with the rules of the system – in this case the axioms of Peano Arithmetic.
Here’s another one: ‘The universe will expand forever’. The current state of cosmological knowledge is that we do not know whether that will happen or whether the expansion will eventually slow and then reverse, ending in a Big Crunch. The Big Crunch will happen if the Cosmological Constant is positive, but not if it’s zero or negative. If the constant is positive or negative, we may one day be able to demonstrate that. But if it’s zero, we’ll never be able to demonstrate that, because there’s always the chance thet it’s a positive or negative value too small for us to detect. So if the cosmological constant is zero then the universe will expand forever but we can never know that that’s what it’s going to do. Under our new interpretation of ‘true’, we would have to say that this cosmological statement may be neither true nor false, as there can be no experience that can confirm or deny it. In fact so is the statement ‘The Cosmological Constant is zero’.
If this seems disconcerting, we can comfort ourselves with the idea that, for all the common , everyday uses of the word True, the definition gives us exactly the interpretation that we want:
- “are you telling the Truth” means ‘is what you are telling me now consistent with your own experience of what you are talking about?’
- “I feel cold” is true if the person saying it is currently having the experience of feeling cold.
So we can preserve a role for the word ‘True’ in our everyday language. What I think we can’t do without having to make unfounded assumptions about the existence and nature of Ultimate Reality, is elevate the concept to some universal principle that guides our understanding of the universe. “Reason” does a much better, and more practical, job.
Andrew Kirk, Bondi Junction, February 2012
It occurred to me a few days after finishing the above essay that, if somebody nevertheless wishes to retain a notion of Absolute Truth in their worldview, perhaps for aesthetic reasons, or just because it fits with long-accustomed habit, one way to do so is to incorporate an omniscient conscious entity in the worldview. That way, Absolute Truth can simply be defined as what it is that this entity knows. This is entirely consistent with the above suggestion of defining truth in terms of experience because, as the entity is omniscient, it experiences every object or event.
This entity may sound a bit like what some people call God. But I should note that other attributes ascribed to God by the most popular Middle-Eastern religions of Christianity and Islam, such as omnipotence and being a creator, law maker and law enforcer, are not necessary in order to take this route to believing in Absolute Truth. Nor does this approach necessitate that there be only one omniscient entity.
If the entity is infinite then it could even know the truth status of the mathematical proposition S above. Knowing the truth status of S requires knowing an infinite amount of information but perhaps that would not be a problem for an infinite entity.
This approach has echoes of the way that George Berkeley completed his Idealist theory of existence. Berkeley’s theory says that only ideas exist, not matter, and he addressed the problem of whether objects exist when nobody is looking at or thinking of them by saying that they still exist as ideas in God’s mind.
Personally, I like the idea of a universe without Absolute Truth and have no need or wish to hypothesise such an entity. But it may be of comfort to those who do like the notion of Absolute Truth to know that there is a rational basis on which they can do so.
In discussions about free will, consciousness or interpretations of quantum mechanics people talk a great deal about whether the world is deterministic, random or something else. Well I don’t know what the ‘something else’ bit could be but I’m also starting to wonder whether the idea of randomness makes any sense.
The general idea of the distinction between a deterministic and a random universe seems to be that, in the former, events are somehow ‘fixed’ before they occur, whereas in the latter they are not – there are multiple different possible events. That sounds clear enough if you don’t think too hard about it, but if we do then we come up against the question of what do we mean by fixed? Fixed how, and by whom?
The apparent answer is that they are fixed by the ‘laws of nature’. But what are the laws of nature? Isn’t this some fairly heavy duty reifying to posit that there exist some actual laws, perhaps dwelling in some Platonic kingdom of Forms or written on magical stone tablets? Sure we have useful laws like Schrodinger’s and Einstein’s equations, but all we know for sure is that these are ideas we use to make sense of what we see and make predictions about the future. Whether they have some mysterious metaphysical existence independent of human minds is an entirely separate matter. It seems quite unlikely to me, given that every century or so we have to tweak the laws we use when further experiments and theorising show they are not completely accurate.
A Platonist might argue that there really are laws of nature out there, which determine how the universe behaves. OK if that’s the case then what about the law that is simply a description of where every single particle in the universe will be at every instant in the history of time? This is essentially a very, very long shopping list, but it also happens to perfectly describe the behaviour of the universe. Let’s call it the M-law, as it is the ‘mother of all laws’. Is that a law of nature? If it is then the universe cannot be random because everything that ever happens is described by the M-law.
So what are our choices about universe types? If we deny that laws of nature have any existence independent of human brains then everything in the universe is random because there is nothing that fixes it. If we assert the separate existence of laws of nature then nothing is random because it is all predicted by the M-law. Or, we could try and be picky and assert the existence of laws but only count them if they meet certain criteria. But what would such criteria be?
One option is to say that a law only counts if it can be known prior to the event we want to use it to predict. That would certainly disqualify the M-law. The trouble is, it would also disqualify everything else, because we cannot prove any of the laws. All we can do is build up supportive evidence for them, and there is never enough to be certain.
Sure, says the law-enthusiast, you can’t ever be completely certain, but in practice, if we are 99% certain of something, we would consider that good enough. Alright then. That does seem a bit arbitrary, but let’s go with it and see where it leads. What can we say about the motions of the planets prior to the discoveries of Kepler, Galileo and Newton? Back then we knew nothing of the ‘laws’ we now use to describe those motions, so under this criterion those laws didn’t apply, which apparently makes the motions of planets in 1347 CE random. Is that what we want?
‘Ah yes’ replies the enthusiast, ‘but you are limiting yourself to what we managed to work out based on our imperfect interpretation of the available information and our limited ability to make observations. The validity of a law should be based on all the information that was available up to that time, to an omniscient – but not future-seeing – observer, who was able to develop the best possible theories with the available information.’
Well I have to say this is getting even weirder and more implausible! We now have an ideal scientist-observer that is our yardstick for what constitutes a law of nature. If we go with that, and accept some arbitrary threshold of confidence – say 99% – on the validity of a law (leaving aside the very difficult question of how we would try to implement that threshold and whether it would be possible to validly calculate probabilities against it) then maybe we have arrived at a definition that could be pressed into use for laws of nature while excluding the M-law.
But – haven’t we ended up with a definition of randomness that is entirely epistemological? We have effectively defined a random event as one for which our ideal observer could not have 99% confidence beforehand of what the outcome would be. And the trouble with that is that we can no longer make the distinction that metaphysicists like to make between epistemological uncertainty over deterministic but chaotic events such as a coin toss, and ‘genuine’ randomness such as the decay of a radioactive isotope. With our new definition both of these types of uncertainty are ‘merely’ epistemological, and there is no such thing as metaphysical or ontological randomness. If we take this path we have to conclude that all randomness is epistemological, and there is not distinction from ‘metaphysical randomness’.
One last point to wrap up with. I went searching for a mathematical definition of randomness and came up with a complete blank. There are definitions of random variable, random (stochastic) process, probability space and various other related objects. But none of them have anything in them that captures the idea of ‘metaphysical undeterminedness’ that lurks under the popular conception of randomness. In fact, rather oddly, of the various interpretations of quantum physics, the only one that has close parallels to any of those mathematical objects in the field of probability theory is the ‘many worlds interpretation’, which looks very like the peculiar object that is a ‘stochastic process in continuous time’. That is ironic, as the many-worlds interpretation is regarded as a ‘deterministic’ interpretation, standing in contrast to the most popular ‘Copenhagen interpretation’ which is regarded as indeterministic, ie random.
Andrew Kirk Bondi Junction July 2012