What Are Numbers?

The Nature of Numbers

Along with Thoughts Without Words and The Ape Which Asked Why

by Gwydion M. Williams

An interesting discussion on numbers emerged on the Aubane Mailing List, which is an invitation-only mailing list.  It began on ‘Pi Day’[1], an unofficial holiday the celebrate the number ‘Pi’.

For those unclear, or think it is just about circles, here is how the Wiki defines it:

“The number π … is a mathematical constant. Originally defined as the ratio of a circle’s circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics. It is approximately equal to 3.14159. It has been represented by the Greek letter “π” since the mid-18th century, though it is also sometimes spelled out as “pi”. It is also called Archimedes’ constant.

“Being an irrational number, π cannot be expressed as a common fraction (equivalently, its decimal representation never ends and never settles into a permanently repeating pattern).”[2]

With the agreement of the participants, I am writing it up in a more accessible form.

Pat Muldowney, March 14th

Pi day today, March 14,


Buffon’s Needles :


Jack Lane

What else did Buffon do for a living?

Is this not a way to try to come to terms with an infinite problem which Pi is and the mind can’t cope with it?

Pat Muldowney

There are infinitely many whole numbers

1, 2, 3, …..

going on forever.

I don’t know whether the mind can cope with that or not.

But if it can, then maybe it can also cope with pi.

Jack Lane

I think the difference with Pi is that is supposed to be able to measure (exactly) something that exists, a circle, but the answer can be infinite. I think that blows the mind – it does mine at any rate.

Pat Muldowney

That’s reasonable I suppose. And there is a strong historical precedent.

If you have a square with sides 1 inch long, then you can measure exactly the area of the square — it’s one square inch.

But several thousand years ago the Pythagoras school discovered that even though they could see exactly the diagonal line of the square, they could not measure it exactly.

Its length is 1.41421356237…. (the square root of 2) going on forever without end, in an unpredictable way.

Just like pi = 3.141592…. And completely different from whole numbers 1, 2, 3,… or fractions 1/2, 3/5, 2/7 etc etc.

This blew the mind of the Pythagoreans, it seems.

There are actually far more of these kind of numbers than any other kind, such as fractions.


As to “countable”, the whole numbers and the fractions are countable. The numbers like pi and square root of 2 etc are not countable, and there are a great many more of them.

Peter Brooke

Would I be right in thinking the problem of measuring the diagonal is quite different from the problem of measuring the circle? In the case of the diagonal it’s a straight line so it surely can be measured. The difficulty comes in reconciling the spans you use to do it with the spans (inches, centimetres or whatever) you used for the sides of the square. Using the same units, it’s the square root of 2 but you could measure it using slightly different units. The problem with the circle though is that it is of a radically different nature from the straight line. It can only be measured by pretending that it is a multisided polygon. Hence infinitesimal calculus, which is a convenient fiction for measuring what can’t be measured. The same problem comes in trying to measure time. You have to pretend that it is space i.e. a fourth dimension or a series of very small static units as in quantum physics. As for infinity in something that exists, even a straight line is, in theory at least, infinitely divisible. See Pascal on the subject of the infinitely great and the infinitely small.

Pat Muldowney

“The value of the number pi has been calculated to a new world record length of 31 trillion digits, far past the previous record of 22 trillion. Emma Haruka Iwao, a Google employee from Japan, found the new digits with the help of the company’s cloud computing service.  “[3]

Re diagonal of square. If you change the units (i.e. use a ruler which measures the diagonal to an exact number of units), then that ruler will not measure the sides exactly. You end up with the same problem. For the circumference of a circle — if you lay a piece of string along the circumference you can then lay that length of string out as a straight line and measure it with a straight ruler; and you still have the same problem with pi. – The straightened out string does not give a length you can read from the ruler.

Peter Brooke

In this case then the problem seems to be just to do with the nature of numbers as a way of interpreting reality, not a problem in the reality itself. Basically numbers imply a jump, or, if you prefer, a quantum leap, from 1 to 2, whereas reality is continuous. So by numbers you can only get an approximation, albeit a very close approximation, to the realty they affect to describe. Just like the picture on a digital screen, made up of millions of little dots or the frames in a celluloid film which run by quickly give an illusion of continuous movement. They can only give an approximation to, not a perfect representation of, the thing they describe. Which is not surprising really when you think of the even more dubious relation of words to whatever it is the words are trying to describe.


There was incidentally a programme on the radio recently about a meeting between the Australian poet Les Murray, who wrote a poem I like called ‘Portrait of the Autist as a New World Driver’ and a fellow-autist who had been invited to Oxford to recite pi by memory over a period of I think two days.

Jack Lane

Peter, you are quite right about the inadequacy of numbers and words to describe reality but can we think without words?

Peter Brooke

I’m not suggesting we do away with words or numbers just that we don’t confuse a problem that arises from our means of interpreting reality (even when these are the best means possible) with a problem in the reality itself. I think one of the arguments of Ernst Mach, the mathematician who inspired the Bolshevik empirio-critics, was that the reality that could be constructed by numbers (mechanics) was superior, more amenable to human manipulation, to the messy reality constructed by … nature? God? Whether or not he is right about its superiority he was recognising that it is different.

Pat Muldowney

In light of the contributions, and in fairness to the “empirio-criticist” side, I’d like to post something about Godel’s incompleteness theorem

(Just a brief mention of Godel, nothing boring, (or at least no worse than previous posts))

It’s as if some hardcore theologian, using the Proofs by Causation, by Order, by Design etc etc., set out to establish that there is no God; and succeeded.

Thus vindicating the “empirio-critical” method by subverting  “empirio-criticism” itself. Which is probably beyond the reach of the theologian.

Peter Brooke

This is almost exactly what was done by William of Ockham in Christianity and al-Ghazali in Islam. Except that they weren’t trying to prove there is no God, only that the existence of God is not accessible to logical proof.

Pat Muldowney

Not taking sides here — but isn’t “logical proof” the territory owned by empirio-criticists and their ilk?

I studied Sheehan’s Apologetics for two years solid at school. It didn’t make me any more or any less religious. Seems the force of Sheehan’s logic had nothing to do with it.

Theologians using logic to prove that logic doesn’t work is no more convincing than empirio-criticists using theology to prove that religion doesn’t work.

‘Message Server’[4]

Peter, you are quite right about the inadequacy of numbers and words to describe reality but can we think without words?

Gwydion M. Williams

I am agreeing with Peter’s definition.

We can also manage without words.  Sometimes have ideas which are hard to put into words.  And a child who has not learned to speak gives evidence of some quite complex thinking.

Animals can definitely think about problems and solve them, though such language as they have would not be adequate to speak about it.

“Use a stick to get at the out-of-reach food item”, for instance.

And those taught a human sign language have proved unable to say anything so complex.

Gwydion M. Williams

Would it be OK with everyone if I organised this interesting discussion into chronological order and published it in the next Problems?  I’d send everyone a draft to confirm I had got their contribution right.

And correct spellings, of course.

But that’s not a circle, Mr Buffon

The discussion ended there.  But if you were wondering about Pat Muldowney’s mention of ‘Buffon’s Needles’, this is one of the many places where Pi crops up without any obvious relationship to circles:

“Asked to get an estimate for the famed mathematical constant pi (π), you might do what the ancient Greeks did: Divide the circumference of a circle by its diameter. Here you can estimate pi by a less conventional method: the random tossing of toothpicks…

“Count the total number of toothpicks you tossed. Also count the number of toothpicks that touch or cross one of your lines. Do not count any toothpicks that missed the paper or poked out beyond the paper’s edge.

“Divide the total number of toothpicks you threw by the number that touched a line.

“This is your approximation of pi, or 3.14…

“This surprising method of calculating pi, known as Buffon’s Needle, was first discovered in the late 18th century by French naturalist Georges-Louis Leclerc, Comte de Buffon. Count Buffon was inspired by a then-popular game of chance that involved tossing a coin onto a tiled floor and betting on whether it would land entirely within one of the tiles.”[5]

So Pi is an unexplained feature of our universe, first encountered in the context of circles but more basic than that.  And essential to know approximately if you need to measure circles, of course.  Which does not resolve the questions asked.

Later Comment by Peter Brooke

I could have added that I wasn’t suggesting the theologians were trying to prove that logic doesn’t work, only that it doesn’t work if the intention is to prove the existence of God, which can only be known empirically, i.e. by experience. In the case of William of Ockham his ‘nominalism’ facilitated the opening up of the whole field of scientific endeavour. Al Ghazali’s arguments contributed to closing it down. I also didn’t understand Pat’s remark that ‘logical proof’ is ‘the territory owned by empirio-criticists and their ilk’. They were, as the name suggests, empiricists, arguing that the whole of reality can only be known empirically, i.e. by experience, i.e. by subjectively experienced sensation. We never got Pat’s mailing on Godel’s incompleteness theorem.

I disagree with Pat’s understanding of Machism/empirio-criticism. Something of my rather different understanding of Machism can be found in my essay Materialist Theories of Consciousness.[6]  This attempts some understanding of what empiriocriticism, or empiriomonism, was all about

Later Comment by Pat Muldowney

I wonder if that’s the whole story. I vaguely remember reading Materialism and Empirio-criticism by Lenin. Anything which was not dialectical materialism was reducible to Berkeleyan solipsism. Which sounds a bit extreme. Apparently Empirio-critics were Machists. Einstein was Machist. That was the trend of the time, including Bertrand Russell, G.H. Hardy and David Hilbert. They wanted to axiomatise knowledge, so everything was to be deducible (“logically provable”) from certain given basic principles or axioms. As in Russell’s effort (Principia Mathematica) which famously managed to define the number 1 at around page 300 or something. (That’s not quite as bad as it sounds. Defining/deducing “1” is probably the hardest part.) Godel’s incompleteness theorem more or less sunk the project, using its own tools and philosophy. Interesting enough, but nobody is much bothered by it one way or the other.

Later Comments by Gwydion M. Williams

Regarding language, I got hold of a book called Nim Chimpsky: The Chimp Who Would be Human.[7]  This confirmed that a chimp who learns sign language will mostly use it to get things.  That they have thoughts that they never sign about, whereas a child may well talk about their own plans and thoughts.  Will mostly ask a trusted adult for help if they hit a problem.

Children also repeatedly ask ‘why’, and should always be answered.  Chimps etc. have never been reported as doing so.  This may indeed be the key difference – humans are the animals that ask WHY?

Quite separately from the e-mail debate, I had expressed a view that maths was just a language that allowed some unfamiliar things to be talked about.  This was in an article called Fear of Socialism: The Real Issue for the ‘Vainglorious Seven’.  I condemned their passive acceptance of the New Right values that had risen in the 1980s:

“Despite a highly successful Mixed Economy following Roosevelt’s semi-socialist New Deal, a wave of protestors wrote hymns to Imaginary Capitalism, claiming it the source of all things good and nothing bad.  They devised fancy maths, but it is unconnected to how real economies work.  Maths is a set of languages that allow exact and testable descriptions of weird events, particularly for subatomic particles and for the universe as a whole.  But you can write gibberish in those same languages, like any other.

“To speak of ‘the snowy vistas of the mountains of East Anglia’ is valid English, but unreal.  Or you could imagine men in bowler hats, black leather jackets and tartan kilts doing a clog-dance at the centre of Lords Cricket Ground, while the London Symphony Orchestra plays Pop Goes the Weasel.  But this won’t actually occur.”[8]

I later tracked down a relevant remark by noted biologist J B S Haldane:

“In scientific thought we adopt the simplest theory which will explain all the facts under consideration and enable us to predict new facts of the same kind. The catch in this criterion lies in the world ‘simplest.’ It is really an aesthetic canon such as we find implicit in our criticisms of poetry or painting. The layman finds such a law as dx/dt = K(d^2x/dy^2) much less simple than ‘it oozes,’ of which it is the mathematical statement. The physicist reverses this judgment, and his statement is certainly the more fruitful of the two, so far as prediction is concerned. It is, however, a statement about something very unfamiliar to the plain man, namely the rate of change of a rate of change.”[9]

To understand an ooze and predict what it will do, you need the language of maths.  And yet the maths for gases is only an approximation for real gases: the behaviour of an ideal gas is close enough for most purposed.[10]  I assume that real oozes likewise are different from an ‘ideal ooze’, but we mostly do not need to worry about this.

If I’ve understood it correctly, similar issues apply regarding the ‘Speed of Light’.  This was found as an observable fact by a Danish astronomer called Ole Romer.   He noticed mysterious variations in the movements of the moons of Jupiter, and realised that they made sense if light had a large but finite speed.  Using what was then the best estimate of Jupiter’s distance, he got a value that in modern terms would be about 220,000 kilometres per second.  This was 26% lower than the true value of 299,792 km/s: a remarkable success given the general state of knowledge at the time.

When Maxwell worked out the basic maths for electricity and magnetism, he found they predicted electro-magnetic waves with a speed suspiciously similar to the known speed of light – now measured much more accurately by a variety of methods.  There was a reasonable belief that light, electricity and magnetism were somehow connected.   And independently of Maxwell’s work, scientists had already discovered what we now call infrared and ultraviolet, beyond the spectrum of visible light.

It was left to a German physicist called Heinrich Hertz to conclusively prove the existence of the electromagnetic waves that Maxwell theorized.  He produced some new and unsuspected waves with a frequency of about four meters,  and showed that they behaved like light-waves.  But if he noticed the interesting fact that these waves could go through walls, he didn’t see this as important.  It was left to Marconi to realise that radio waves might be useful for communication, and develop them as such.  That, incidentally, is one of many examples of why it is wise to let scientists investigate things with no obvious usefulness: they may have unexpected value.

It was also reasonable to think that the speed of electromagnetic waves to vary depending on the speed of the source.  But attempts to measure this failed: and meantime Einstein had noticed that Maxwell’s maths implied that the speed would be constant.  That led on to Relativity.  But to make definite predictions based on seeing the universe as a single space-time in which time’s velocity might vary, Einstein needed the specialised mathematical language of Tensor Calculus, which he had been taught as a student but since forgotten about.  Using this, he was able to explain a baffling oddity in the orbit of Mercury.  And to make specific and unexpected predictions about how far stars would be seen to be displaced when their light passed near the sun, seen only in a total eclipse.

It was the eclipse data that made Einstein famous.  Finding a fancy explanation for the known puzzle of Mercury’s orbit was less impressive: it is fairly easy to fit your theory to known facts.  But the eclipse results were the first of many that common-sense Newtonian gravity could not explain.

Various TV dramatizations get this wrong: Newtonian gravity could live with light being subject to gravity, and the idea of what we now call a Black Hole had been predicted by amateur astronomer John Michell in 1783.[11]  But the size of the shift was different according to Einstein, and the hard data backed him.

Was that the Last Word?  Until the 1960s, it was widely believed that the mathematics of relativity were just a good approximation, to be replaced by better theories just as Einstein replaced Newton.  To the surprise of many, every measurement we have so far done has exactly matched Einstein’s predictions.   Yet we still use Newton’s much simpler maths for working out how to send space probes round the solar system.  And it remains possible that relativity too will in time be replaced.

Physicist now see the ‘Speed of Light’ as a fundamental of our space-time.  Any massless particle should travel at exactly that speed, even if it is unconnected to electro-magnetism.  For a long time it was believed that this was true for neutrinos: but the fact that they can transform into a different variety of neutrinos is believed to mean that they do have a mass, though a very tiny one.

General Relativity says that gravitational waves will travel at the speed of light.  As of now, there is a single measurement of an event that was also seen in various sorts of electromagnetic radiation,  and the speed appears identical.

[1] https://en.wikipedia.org/wiki/Pi_Day

[2] https://en.wikipedia.org/wiki/Pi

[3] https://www.bbc.com/news/technology-47524760

[4] This contributor chose to remain anonymous

[5] https://www.exploratorium.edu/snacks/pi-toss

[6] http://www.peterbrooke.org/politics-and-theology/consciousness/

[7] Hess, Elizabeth.  Nim Chimpsky: The Chimp Who Would be Human by. Bantam Books 2008.

[8] https://gwydionwilliams.com/40-britain/fear-of-socialism-by-the-labour-defectors/

[9] Haldane, John Burdon Sanderson.  Possible Worlds, 1927.

[10] https://en.wikipedia.org/wiki/Ideal_gas

[11] See https://en.wikipedia.org/wiki/John_Michell for details