### The problem of mathematics

*Throughout this post I shall use the British shortening of the word "mathematics".*

*I struggle with maths. Well, ok, as the proud possessor of a maths 'A'-level and a finance MBA I suppose I don't struggle with it that much, really, compared to many people. But in my head I struggle. I look at a page of equations and my eyes glaze over. I can't look at a graph and imagine the equations that explain it. Nor can I look at an equation and work out what shape of graph it would create. I can visualise the forces in a mechanical system, but converting that into algebra is a bit of a problem as I can never remember what the letters stand for. I have the same problem with econometrics too - even when the terms are clearly documented at the top of the page, by the time I'm halfway down it I've forgotten what they mean and have no idea what the equations are supposed to show.*

Many people have the same problem when faced with a musical score. But I don't. I can look at a page of complex musical notation and hear in my head how the music would sound. And as far as I remember I have always been able to do this. Because I can hear the music on the page, I can sight-read anything. It took me a long time to realise that the reason I find it difficult to teach sight-reading is that the majority of people can't look at the score and hear the music as I do, so they have to do it in a completely different way that is foreign to me.

So, you see, I have a problem with mathematics that I don't have with music. But I don't think this problem has anything to do with my ability to "do" maths. I think it is a language problem.

Let me explain. Both written music and maths are systems of notation. They are both ways of describing the workings of real phenomena in an organised and comprehensible manner. But they are not the phenomena themselves. The full score of a Beethoven symphony is no more "that" symphony than a DSGE model "is" the economic phenomenon it portrays. In each case the phenomenon is only partially explained by the notation that is used to explain it. And the further removed the system of notation is from the reality of the phenomenon it is attempting to describe, the less accurate it is. Much contemporary music, for example, is not easily rendered in traditional Western musical notation.

Similarly, languages are systems of notation. They are a way of "making present" thoughts and ideas in a manner that is comprehensible by others. But they are only a partial representation of those thoughts and ideas, and there may be other, better ways of representing them. The use of symbols in maths is itself a derivative of language: but I would argue that maths is actually a language in its own right. Mathematical symbols are, if you like, "words", and the equations that are constructed from them are phrases and sentences: the manipulations and transformations that the equations go through are the mathematical equivalent of grammar.

In languages there seems to be a definite "transition" between having a working knowledge of the language and being fluent. That transition occurs when people stop mentally translating from one language to another in order to understand, and simply understand IN the language. People describe this as "thinking" in the language. A similar transition seems to occur in reading, when people move from reading the words and mentally visualising what each word represents, to reading (or perhaps "living") the story. My ability to read a page of musical notation and hear the music that it represents also appears to be similar: I don't need the notation translated into something else in order to understand it.

Does the same sort of transition happen in maths? I think it does. Unlike me, real mathematicians do not need their symbols translated into English (or whatever their native language is) in order to understand what they mean: they understand directly what those symbols represent. And they understand directly what equations represent, too, and how they can be manipulated and transformed to create different effects. I wish I did. I need the transformation explained in English in order to understand it in maths. I don't "think" in maths as I do in both English and music. I have a "working knowledge" of maths: but in music, I am fluent.

Which causes me a problem when someone dismisses music as "just maths", and insists that maths is the fundamental language that underpins everything and I really should understand it in order to be a proper musician. I find myself rebelling against the notion that a language in which I don't feel comfortable is "better" than a language in which I am fluent. That's like telling me that I should write in Latin, not English, because English is (partly) derived from Latin. If the purpose of language is to communicate thoughts and ideas and describe real phenomena, surely the "best" language to use is the one in which I am most at home? Any other would be less effective, simply because I do not know it as well.

But I have another, and perhaps more fundamental, problem with maths. The person who criticised me for not being fluent in maths as I am in music was making a value judgement about maths as a language. He assumes that mathematical language is ALWAYS the best way of describing and explaining real phenomena - including musical sounds and patterns. I think this is completely wrong and very dangerous.

Earlier in this post, I commented that both maths and music are at best partial explanations of the phenomena they represent. They are approximations. The trouble is, we tend to confuse the paper approximation of the phenomenon with the phenomenon itself - and in so doing we ignore or eliminate those aspects of the phenomenon that don't fit with the system of notation that we have used. Generally in Western music we understand this now, though for a long time we did not: generations of music teachers drummed into their students that they must "play what is on the page", leaving little opportunity for personal expression. But the advent of forms of music such as jazz, that could not be adequately portrayed by Western musical notation, has forced musicians to develop other forms of notation that better represent the newer musical forms - and perhaps more importantly, has forced people to learn music by listening and copying rather than reading. But you will still find jazz standards forced into Western notation in sheet music books produced for use by the general public. They are twisted, tormented mockeries of the real thing.

Historically scientists have shown themselves very willing to invent new systems of mathematics if the old ones don't fit with new discoveries. But at the moment we appear to have some idolisation of mathematics going on. I don't believe that all real phenomena are adequately explainable by our mathematics: I think that other languages - including ones we haven't invented yet - may be equally useful. But if we insist, as my critic did, that everything is "just maths", then we exclude the possibility of expressing real phenomena in other ways, and in that way potentially limit the means by which we can better understand the universe we inhabit. This, to me, would be a serious loss.

And there is another loss, too. In Western music we have experimented with an entirely mathematical approach to composition, and rejected it - because although it resulted in melody and harmony, it lost something fundamental. It lost its humanity. Music is made by people for people: when we reduce it to a series of graphs and a set of mathematical formulae programmed into a computer, we have eliminated its purpose. I suppose musicians of the future could simply be handed a graphical trace of sound patterns and told to re-create them, but this would simply be reheating someone else's creation, rather than creating music themselves. It is the approximate nature of musical notation that makes it possible for performers to express their own personalities through their playing or singing and share with the composer in the creation process. It is this that makes every performance of a Beethoven symphony different, that means that Beethoven symphonies are to a degree still being written today. Music that does not admit the possibility of variation from performance to performance is inhuman.

The dominance of mathematics in other disciplines such as economics tends to exclude people like me, who are not fluent in that language, and in so doing it makes those disciplines elitist. And the problem with elites is that they become divorced from reality: they regard their creations as set in stone and invariable, but the real world varies and changes, and humanity with it. Just as mathematical dominance in music resulted in inhuman musical creations, while advanced mathematics dominates economic thinking, economics is in danger of becoming inhuman.

It is all about fractality.

ReplyDeleteJust study mother nature instead of your maths instructor ;-)

I never said you cannot be proper musician or even a good one without understand maths, that like saying Ted Hughes is a rubbish poet cause he speaks and writes in a very northern dialect.

ReplyDeletePrices are a good way of understanding why math is indeed a language and not a science.

The word fairness is only found in the English language, there are 21 different meanings thus to be frank, the idea of fairness is so abstract as to be what you want it to be.

A price is a communication, sometimes that communication is misunderstood or its not very clear, or even someone made it up like fairness.

The importance of math is that it is a precise language, everything aspires to be it. Its rules are underpinning reality. Fairness unless you are talking about a 50/50 situation in nature which is rare, is therefore not a part of reality.

Either mathematics is too big for the human mind or the human mind is more than a machine - Kurt Godel (1906-1978)

And no I only have the faintest idea of Gödel's incompleteness theorems

Musical notation is an extremely precise language, but it is inadequate as a descriptor of music in reality. So with maths. It is that very precision that makes both musical notation and mathematics inadequate, because reality is not precise and not fully understood.

DeleteI do not agree that the rules of maths "underpin reality". Reality is bigger than any language we can invent: mathematical rules are simply our interpretation of reality as we understand it. And other languages do not "aspire to be maths". The very fluidity and imprecision of languages is what gives them their ability to express abstract concepts and permit debate about meaning.

Excellently put!

DeleteI view maths as the "language of logic", but logic is not all there is (just as Spock)

That said, I think views like Sean's have been furthered by the fact that physical scientists have been able to make predictions about real-world phenomena based on extrapolations from mathematical models created to describe previous discoveries: for example, the fact that "black holes" were described before we have any empirical evidence for them. That has given people the idea that there must be some great abstract truth in the universe that Maths somehow describe. I am not trully sold on this notion either.

As an engineer I know that maths cannot, and does not adequately describe the real world; it is a rather limited approximation. The real world is the orchestra and musicians with their instruments or voices. In engineering terms it is an aeroplane or ship's hull moving through a fluid and the interaction between them. Mathematics cannot describe these adequately which is why we have to use wind tunnels, computer models and trial and error. Just like an orchestra rehearsing. (Maths is not all it's cracked up to be!) Economists who believe maths has all the answers simply demonstrates how misguided they are.

ReplyDeleteComplexity of the modern world is made up from many sectors. History, psychology, systematics, fractality, math, order and chaos. The bottom line is that use of debt, to pay of debt, will lead to lack of trust and because of that people will walk away from people in politics and central banking (planning) Near zero interest rate will feed the bankers with blood from the working class who will be left with alone in debt and slavery. .. " Debt is slavery of the free." ~ Publilius Syrus

ReplyDeleteErnest Rutherford famously said "Physics is the only real science. The rest are just stamp collecting." Despite not being a physicist, I can see some truth in this statement. There is something very abstract about the world of physics and there is an argument that in many way the other sciences are simply applications of physics with specific domain knowledge/observations of top.

ReplyDeleteIn many ways much the same could be said about mathematics and much of human knowledge. Mathematics provides the logical underpinning to describe all that much of what goes on the world around us. Differential equations can be used to describe a lot of the physical interactions around us. Game theory can describe quite a lot of human and animal interactions (as demonstrated by the beautiful pose of Richard Dawkins).

However the problem with this statement is that extrapolating from simple system to complex systems is very difficult/almost impossible. This is because (as Henri Poincaire discovered with his analysis of the solar system and Lorenz rediscovered) when you have more than two interacting bodies you have most likely have a chaotic system. This means that to make meaningful predictions you need to know the initial conditions infinitely precisely (which is impossible).

Likewise game theory quickly becomes very complex (and optimum solutions very difficult to determine) when you factor in social factors such as punishment and stigma. So much so that in such situations its generally better explanation of observations than predictions. [Although tragically it seems to be a lots of people on both the left and the right who willfully misunderstand such complexities try to slander this area of research by associating it with extreme selfish capitalism]

The conclusion that I draw from all of this is that after experimentation with a good analogous real world model systems (which isn't always possible or easy) mathematics is the best predictive tool we have. Unfortunately once you have any level of ambiguity and complexity, its pretty lousy....

From what you say, I think you are (like me) a Nominalist when it comes to mathematics, the problem you have is that the majority of people, particularly physical scientists and mathematicians, are Realists. I have a few posts (rants?) on this:

ReplyDeletehttp://magic-maths-money.blogspot.co.uk/2012/01/why-dont-more-mathematicians-see.html

http://magic-maths-money.blogspot.co.uk/2012/09/parades-end.html

http://magic-maths-money.blogspot.co.uk/2011/12/does-finance-need-scientific-revolution.html

Francis, That’s a great article.

ReplyDeleteMaths is fine where relevant variables, quantities etc can be measured VERY accurately. E.g. we can measure the tinsile strength of steel VERY accurately, which means that maths is fine for designing bridges.

In economics, it’s just laughable to claim that the variables etc can be measured accurately. Plus there is normally a large amount of disgreement as to what variables etc are relevant to a particular problem. So the mathematical models are usually just fantasy.

I loved maths, it was always like solving a fun puzzle. I might have been described as fluent in maths. I miss it...

ReplyDeleteI took up another science, and like maths, the theoretical models are more of a guide than a concrete representation of the world. However its still a very useful guide that can help you understand what is happening in a system, and how to achieve the desired outcome.

Frances, I have the problem you describe in reverse, but it seems to me that it relates very much to my inability to 'do' music. I can (or once could) read musical notation tolerably well, and translate it to piano keys or guitar fingerings or whatever, but I don't know what it's going to sound like until I actually play it (or, better, someone else does). On the other hand, if you show me an equation "y=ax^2 + bx + c" it speaks to me of a parabola, as a mathematical concept, not just an English word. This doesn't change if you represent it instead as "parabola(a,b,c)" ; the choice of notation is just a question of convenience.

ReplyDeleteI have no objection to the use of whatever mathematics works best in economic models. For me, the problem arises when the modeller loses sight of the money and thinks only about the maths. Because money, not mathematics, is what economics is about.

Paul,

DeleteI'm not convinced this is a "doing" problem. I can read and hear music I can't play - including instruments I have never played in my life, though obviously I've heard them (or I couldn't imagine the sound). In some ways being able to hear the music on the page is annoying, because I know how I think it SHOULD sound and I can't necessarily reproduce that in practice.

I don't agree that economics is about money. At its fundamental level it's about exchange of resources between people for mutual benefit (some people regard it as being about allocation of scarce resources). Money is a secondary feature - it makes exchange easier because it provides a generally accepted way of valuing resources and solves intertemporal difficulties.

Interesting post. I am a good mathematician and a relatively poor musician: I can make all the right sounds but can't hear the music off the page.

ReplyDeleteI'm also relatively good at French and mediocre at Italian. Ok. I'm rubbish at Italian except when drunk. However I believe that if I were to practise more (Italian, not drinking), then I would get better. As I did with French.

I don't think that Economics being maths-heavy excludes you any more than an orchestra excludes me and my Grade 6 Viola. It just asks you to go away and do a little more practice if you want to join in. Maths is very learnable as a language. The concepts it describes and produces are hard and might be limited by ones intelligence, but the language itself? Just put in the hard yards.

You say that you wish you understood maths like you understood music. I wish I understood Italian like I understood French. I recognise that work, not wishing, is the answer.

Hmm. In addition to the studies I mentioned, I spent 17 years doing fairly geeky financial things on computers in banks. Over my life I would say I've actually done more maths than music. Yet I still can't think in maths as I do in music. Maybe we are dealing with something more fundamental than simply "work"?

DeleteMy (perhaps jaundiced) view is that economists are more intent on building mathematical models than they are with accurately representing the underlying economic reality. They seem to believe that if the Greek letters they use in their models are not sub or super scripted then they are somehow failing.

ReplyDeleteMy experience is that they spend very little time explaining the economic concepts or ideas they purport to communicate before they launch into opaque mathematical representations of that which they are purporting to describe and analyse. Consequently, many able people are simply turned off because of the pseudo science in which economic concepts and ideas are presented.

Just recently John Kay wrote in his blog that wages are determined according to marginal productivity. Really? Marginalism requires familiarity with calculus and the ascertainment of the firm's cost and revenue curves. I suggest that at best some employers will meet the first criterion and no employers will meet the second. Employers are very practical people and do not sit around estimating cost and revenue curves. Nor do accountants. If they did, they would not attempt to fit curves to the data and would settle for a straight line instead. And yet, economics teaching pushes the classical paradigm of marginalism as if it were true.

There is a role for Maths and Stats in Economics but not to the extent or to the complexity with these two disciplines are currently being misapplied.

Rant over.

Frances,

ReplyDeleteI enjoyed reading your post and felt moved as you described how you hear the music as you read the notation. I do also think, however, that you're mischaracterizing mathematics (which being from left of the Atlantic, I'll shorten as "math").

You wrote, "[b]oth written music and maths are systems of notation." You make the valid distinction between music and written music, but not for math. Math is no more a language or system of notation than is music. The Ancient Greeks used very different notation than we use today, but the Pythagorean Theorem is not altered in the slightest. It's independent of any notation used to describe it. The notation used to describe or exposit mathematics is most certainly not the mathematics itself.

I think this confusion arises from math--I should say a very small part of math--being used in practical ways. As such, math is used merely as a tool, and its ability to describe, to notate, is seen as its primary usefulness. But what is being described mathematically is not math. People often say that such and such real world phenomenon is "mathematically proven" to occur. If the phenomenon doesn't occur is the math wrong? Of course not. It never was, nor ever could be, proven mathematically because the objects of the real world are not mathematical objects.

Said another way, scientists, economists, engineers, etc. use math in the same way as the military may use music to rouse soldiers in battle. Whatever the reasons for music having this effect on soldiers, it is not music itself.

So, I think the problem is not mathematics, but rather how other disciplines use (a very small part of) mathematics.

William

William, I don't think this is equivalent.

DeleteI read a page of musical notation and I "hear" a real phenomenon - musical sounds. You are in effect saying that maths doesn't need to have any relationship to real phenomena - it exists "in itself" irrespective of the way the real world behaves. That's like saying that musical notation has a meaning of its own independent of its use as descriptor of real music.

What is the purpose of maths if not to describe real world phenomena?

Frances,

DeleteMathematics--to mathematicians--is an art in and of itself, just as music is to musicians. Musicians don't make music to achieve some other end, though it may be used as such. They make music to make music. It's the same with math.

I can't do better than to quote England's greatest mathematician since Newton, G.H. Hardy, from his wonderful little book "A Mathematician's Apology":

"A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas."

And:

"I am interested in mathematics only as a creative art."

I imagine that when you "hear" the music, it's because the music is part of you--you feel it--and the musical notation is invoking that music in your mind's ear. But those sounds you "hear" are "real" in a very different way than are physical vibrations occurring in the "real world" when someone plays music on a physical, real, instrument.

When mathematicians (professional or otherwise) read a page of mathematical notation, if they understand it, are feeling--"seeing" if you will--the mathematical objets being described. Perhaps an example of what I mean by "mathematical object" will make this clear:

Take a circle. Can you draw one? You can try, but it will only be some rough drawing representing a genuine mathematical object that we call a circle. Circles don't have rough edges. In fact the set of points that make them up have no width. But any drawing you make will have rough edges that have a width. Moreover, if you try to measure its radius you'll only be close, never exact. The unit circle has a radius of exactly one. You'll never be able to draw or measure a length of exactly one no matter how hard you try.

The point I'm making is that a circle--a mathematical circle--exists in a different world--though no less "real"--than our real world of pencils and paper and rulers and compasses. We can represent a unit circle by drawing one--perhaps to stimulate our sluggish imagination--with a compass. Or we can express it analytically as x^2 + y^2 = 1. But regardless of notation, the mathematical object is the same.

Now if someone wanted to make, say, a coin, he or she may benefit by using the properties of a real circle--a mathematical circle--but the coin will never be a circle. It will never have all the properties of a circle. In the mathematical world, circles exist smaller than an atom and larger than the universe. A coin maker, no matter how talented, can not make such coins. But a mathematician can make such circles.

Sorry for such a long response.

William

This comment has been removed by the author.

DeleteFrances,

ReplyDeleteAt the risk of being even more verbose, I'd like to comment more on your statement: "That's like saying that musical notation has a meaning of its own independent of its use as descriptor of real music."

I'm not saying that (though it might, but that's another discussion). I'm saying that I believe you are confusing math with mathematical notation. It seems that you believe they are one and the same. They are not.

To put it as an analogy (with the caveat that all analogies break down when pushed too far):

Math is to music as mathematical notation is to musical notation, and also as mathematical objects (such as circles) are to musical objects (such as sounds and timing).

William

William,

DeleteNo, I don't accept your analogy.

I can only "hear" the music on the page because at some time I have actually heard those sounds in reality - just not organised in that particular way. I can't "hear" the sound of an instrument I have never heard in reality. Whereas you are arguing that maths describes something that not only doesn't exist in reality, it can NEVER exist in reality. It exists only in your imagination. That is fundamentally different from the way in which I "hear" music.

Musical notation depicts a real-world phenomenon. You are effectively saying that mathematical notation doesn't depict anything in the real world.

I don't have so much difficulty with maths when I'm using it to solve real-world problems - in physics, for example. My main problem there is remembering what the symbols represent. But I struggle with the abstract use of maths that you describe, precisely BECAUSE it has no real-world application. I can't visualise it.

Frances,

DeleteThe sounds that you have heard in reality, if the physicists are right, are from air molecules bouncing off each other. If there is no air there can be no sound. This is the real world phenomenon. Musical notation cannot depict the real world phenomenon since it has nothing to say about air molecules and sound waves.

But this is not what you mean when you say that you hear the music. The music you hear is not air moving, but a reaction in your brain--whether in the presence of music being played (air moving) or from memory. Now your mind can alter this music. But this altered music in your mind may not be able to be reproduced in the "real world" of cellos and violins and drums and air. But just because music in your mind may not be exactly reproducible does not mean it's not real. Furthermore, musical notation is describing the music you hear in your brain, not the air molecules.

Likewise, the genesis of mathematical objects are often phenomena in the real world. But just as the music in your mind is not air molecules bouncing around, the mathematical objects in your mind are not the physical objects of the real world. Let's take the coin example again:

If someone sees a coin on a table, and you ask them what shape they see, undoubtedly they will say that they see a circle. But do they actually see a circle? Unless they're standing directly above the coin, they will not see a circle but rather a more oval like shape. Why then would they say that they see a circle? It's because the brain makes an idealization. Those idealizations become the starting points of mathematical objects. But once they become mathematical objects, they are no longer part of the everyday world of coins, but rather part of the mathematical world of circles. And once they become part of the mathematical world, they take on a different reality.

William

The music I "hear" has no real existence. It exists only in my mind. I know that. But it is a reorganised memory of real sounds, not an abstraction of something that can never be real.

DeleteYou seem to think that what exists only in your imagination is "real". It isn't - it is fantasy. Let me put it to you that even when you have mathematical notation that describes a perfect circle, what you visualise from that exists only in YOUR mind, not in anyone else's. You have no way of sharing that visualisation with others - no way of knowing that they see what you see. And because you cannot reproduce that perfect circle, because it does not exist in the natural world, you can never share your visualisation with anyone else. You are effectively saying that real mathematical objects exist only in your mind.

Whereas although I know others don't hear exactly what I hear when they look at that page, I don't regard what I "hear" as definitive. Real music is really played, in the real world, by real people on real instruments, and others can hear it too.

Certainly maths doesn't have to describe anything in the physical world. But nevertheless it exists as something more than a fantasy.

DeleteTo take a famous example: there is no physical application I can think of for what is called "Fermat's Last Theorem". And I can't see how the conjecture could even be expressed without using mathematical language (certainly the associated Taniyama–Shimura–Weil conjecture could not). Nevertheless, Wiles' proof is real, in the sense that it makes us confident that there is no integer solution to the equation: if we encountered a sufficiently advanced alien civilization, and if that civilization had considered the problem, we would expect it to have proved the same thing.

What's the purpose of it? It satisfies human curiosity. I suppose that there are a lot of people, most of whom can't (like me) follow the proof themselves, who nevertheless feel some pleasure at knowing it exists.

Really this debate is touching on the nature of reality. Is reality limited to things we can observe with our senses, or is there a reality "beyond" that of things we can imagine with our mind but not observe with our senses?

DeleteDSGE, dynamic stochastic general equilibrium, is a triple oxymoron.

ReplyDeleteStochastic means randomly determined.* When some thing is mathematically treated as random any dynamic math goes out the window.

If some thing is randomly determined it is not in equilibrium. Try throwing dice and say it or the result is in equilibrium.

Equilibrium and Dynamic contradict each other.

Time look at it side ways to toss it all together. If you buy DSGE what else would you buy?

*Stochastic: randomly determined; having a random probability distribution or pattern that may be analyzed statistically but may not be predicted precisely.

Frances Coppola's comment "I can't look at a graph and imagine the equations that explain it."

ReplyDeleteBecause it is a difficult problem! It is not your fault. I do not know of a general solution or method do do it!

That is a known hard problem related to "curve fitting"! Which is not good enough unless you know the equation or it's approximation ahead of time! It is a guessing game. Maybe there are better ways.

Computer graphing software and graphing calculators help.

Benefits:

1. Graphing software helps you to improve your ability to imagine the graph that comes from equations. Especially, if you also "break the equation into pieces" and graph the pieces individually. This is also helpful when working with pencil and paper. You can also play with the parameters. With experience, one learns what the graphs of basic equations and functions look like. Then it is easier to attempt going from graph to possible equations or approximations equations.

a. But, it is hardest to go from graph to equation (like you mentioned) rather than the other way. It is the same problem as attempting to get the equation from data. And, that is very difficult when the equations are not known. With things that can be exponential it helps try logarithmic graphs. Easy with software. You see curve fitting is trying to match data or curves to a guess of an equation. The guess makes it hard. It is a big problem. Lots of people would like the general solution to that problem!

b. Examples of curve fitting might be least squares fit to a line using points from the graph, And, least squares fit to any specified equation.

2. Graphing software and calculators are more efficient and you can play more in less time.

Is music written in base ten? maths is.. perhaps therein lies your problem.

ReplyDeleteWestern music is in base 8 or 12, sometimes 5. Music of other cultures is more diverse, though base 5 is surprisingly common - the pentatonic scale is probably the nearest we have to a universal musical base equivalent to the decimal in mathematics.

DeleteMaths doesn't have to be base 10. It can in theory be any base. Base 2 (binary) is important in computer science, as is base 16 (hexadecimal).

In the novel Watership Down, rabbits worked in base 4.