Let's talk first for a while about history.
So, where did all the mathematical notation that we use today come from?
Well, that's all bound up with the history of mathematics itself, so we have to talk a bit about that. People often have this view that mathematics is somehow the way it is because that's the only conceivable way it could be. That somehow it's capturing what arbitrary abstract systems are like.
One of the things that's become very clear to me from the big science project that I've been doing for the past nine years is that such a view of mathematics is really not correct. Mathematics, as it's practiced, isn't about arbitrary abstract systems. It's about the particular abstract system that happens to have been historically studied in mathematics. And if one traces things back, there seem to be three basic traditions from which essentially all of mathematics as we know it emerged: arithmetic, geometry, and logic.
All these traditions are quite old. Arithmetic comes from Babylonian times, geometry perhaps from then but certainly from Egyptian times, and logic from Greek times.
And what we'll see is that the development of mathematical notation--the language of mathematics--had a lot to do with the interplay of these traditions, particularly arithmetic and logic.
One thing to realize is that these three traditions probably came from rather different things, and that's greatly affected the kinds of notation they use.
Arithmetic presumably came from commerce and from doing things like counting money, though it later got pulled into astrology and astronomy. Geometry presumably came from land surveying and things like that. And logic we pretty much know came from trying to codify arguments made in natural language.
It's notable, by the way, that another very old kind of (formal) tradition that I'll talk about a bit more later--grammar--never really got integrated with mathematics, at least not until extremely recently.
So let's talk about notation that was used in the different early traditions for mathematics.
First, there's arithmetic. And the most basic thing for arithmetic is numbers. So what notations have been used for numbers?
Well, the first representations for numbers that we recognize are probably notches in bones made perhaps 25,000 years ago. They typically worked in unary: to represent 7, you made 7 notches, and so on.
Well, of course we don't absolutely know these were the first representations of numbers. I mean, we might not have found any artifacts from earlier representations. But also, if someone had invented a really funky representation for numbers, and put it in, for example, their cave painting, we might not know that it was a representation for numbers; it might just look like a piece of decoration to us.
So numbers can be represented in unary. And that idea seems to have gotten reinvented quite a few times, in many different parts of the world.
But when one looks at what happened beyond that, there's quite a bit of diversity. It reminds one a little of the different kinds of constructions for sentences or verbs or whatever that got made up in different natural languages.
And actually one of the biggest issues with numbers, I suspect, had to do with a theme that'll show up many more times: just how much correspondence should there really be between ordinary natural language and mathematical language?
Here's the issue: it has to do with reusing digits, and with the idea of positional notation.
You see, in natural language one tends to have a word for "ten", another word for "a hundred", another for "a thousand", "a million" and so on. But we know that mathematically we can represent ten as "one zero" (10), a hundred as "one zero zero" (100), a thousand as "one zero zero zero" (1000), and so on. We can reuse that 1 digit, and make it mean different things depending on what position it appears in the number.
Well, this is a tricky idea, and it took thousands of years for people generally to really grock it. And their failure to grock it had big effects on the notation that they used, both for numbers and other things.
As happens so often in human history, the right idea was actually had very early, but somehow didn't win for a long time. You see, more than 5000 years ago, the Babylonians--and probably the Sumerians before them--had the idea of positional notation for numbers. They mostly used base 60--not base 10--which is actually presumably where our hours, minutes, seconds scheme comes from. But they had the idea of using the same digits to represent multiples of different powers of 60.
Here's an example of their notation.
You see from this picture why archeology is hard. This is a very small baked piece of clay. There are about half a million of these little Babylonian tablets that have been found. And about one in a thousand--about 400 of them altogether--have math on them. Which, by the way, is a somehat higher fraction of texts having math in them than I think you'd find on the web today, although until MathML gets more common, it's a little hard to figure that out.
But anyway, the little markings on this tablet look somewhat like little tiny bird foot impressions. But about 50 years ago it finally got figured out that this cuneiform tablet from the time of Hammurabi--around 1750 BC--is actually a table of what we now call Pythagorean triples.
Well, this fine abstract Babylonian scheme for doing things was almost forgotten for nearly 3000 years. And instead, what mostly was used, I suspect, were more natural-language-based schemes, where there were different symbols for tens, hundreds, etc.
So, for example, in Egyptian the symbol for a thousand was a lotus flower icon, and a hundred thousand a bird, and so on. Each different power of ten had a different symbol for it.
Then there was another big idea, which by the way the Babylonians didn't have, nor did the Egyptians. And that was to have actual characters for digits: not to make up a 7 digit with 7 of something, and so on.
The Greeks--perhaps following the Phoenicians--did have this idea, though. Actually, they had a slightly different idea. Their idea was to label the sequence of numbers by the sequence of letters in their alphabet. So alpha was 1, beta was 2, and so on.
So here's how a table of numbers look in Greek notation.
(I guess this is how the sysadmins at Plato's Academy would have customized their version of Mathematica; their virtual Mathematica version -600 or whatever.)
There are all kinds of problems with this scheme for numbers. For example, there's a serious versioning problem: even if you decide to drop letters from your alphabet, you have to leave them in your numbers, or else all your previously-written numbers get messed up.
So that means that there are various obsolete Greek letters left in their number system: like koppa for 90 and sampi for 900. But since--as a consequence of my fine English classical education--I included these in the character set for Mathematica, our Greek number form works just fine here.
A while after this Greek scheme for numbers, the Romans came up with their number form that we're still familiar with today.
And while it's not clear that their number-letters actually started as letters, they certainly ended up that way.
So let's try Roman number form.
It's also a rather inconvenient scheme, particularly for big numbers.
It has all sorts of other fun features though. For example, the length of the representation of a number increases fractally with the size of the number.
And in general, for big numbers, schemes like this get in a lot of trouble. Like when Archimedes wrote his very nice sand reckoner paper about estimating the number of grains of sand that it would take to fill the universe--which he figured was about . (I think the right answer is about .) Well, he ended up pretty much just using words, not notation, to describe numbers that big.
But actually, there was a more serious conceptual problem with the letters-as-numbers idea: it made it very difficult to invent the concept of symbolic variables--of having some symbolic thing that stands for a number. Because any letter one might use for that symbolic thing could be confused with a piece of the number.
The general idea of having letters stand symbolically for things actually came quite early. In fact, Euclid used it in his geometry.
We don't have any contemporary versions of Euclid. But a few hundred years later, there are at least versions of Euclid. Here's one written in Greek.
And what you can see on these geometrical figures are points symbolically labeled with Greek letters. And in the description of theorems, there are lots of things where points and lines and angles are represented symbolically by letters. So the idea of having letters symbolically stand for things came as early as Euclid.
Actually, this may have started happening before Euclid. If I could read Babylonian I could probably tell you for sure. But here is a Babylonian tablet that relates to the square root of two that uses Babylonian letters to label things.
I guess baked clay is a more lasting medium that papyrus, so one actually knows more about what the original Babylonians wrote than one knows what people like Euclid wrote.
Generally, this failure to see that one could name numerical variables is sort of an interesting case of the language or notation one uses preventing a certain kind of thinking. That's something that's certainly discussed in ordinary linguistics. In its popular versions, it's often called the Sapir-Whorf hypothesis.
And of course those of us who've spent some part of our lives designing computer languages definitely care about this phenomenon. I mean, I certainly know that if I think in Mathematica, there are concepts that are easy for me to understand in that language, and I'm quite sure they wouldn't be if I wasn't operating in that language structure.
But anyway, without variables things were definitely a little difficult. For example, how do you represent a polynomial?
Well, Diophantus--the guy who did Diophantine equations--had the problem of representing things like polynomials around 150 AD. He ended up with a scheme that used explicit letter-based names for squares, cubes, and things. Here's how that worked.
At least to us now, Diophantus' notation for polynomials looks extremely hard to understand. It's an example of a notation that's not a good notation. I guess the main reason--apart from the fact that it's not very extensible--is that it somehow doesn't make clear the mathematical correspondences between different polynomials and it doesn't highlight the things that we think are important.
There are various other schemes people came up with for doing polynomials without variables, like a Chinese scheme that involved making a two dimensional array of coefficients.
The problem here, again, is extensibility. And one sees this over and over again with graphically-based notations: they are limited by the two dimensions that are available on a piece of paper or papyrus, or whatever.
OK, so what about letters for variables?
For those to be invented, I think something like our modern notation for numbers had to be invented. And that didn't happen for a while. There are a few hints of Hindu-Arabic notation in the mid-first-millennium AD. But it didn't get really set up until about 1000 AD. And it didn't really come to the West until Fibonacci wrote his book about calculating around 1200 AD.
Fibonacci was, of course, also the guy who talked about Fibonacci numbers in connection with rabbits, though they'd actually come up more than a thousand years earlier in connection with studying forms of Indian poetry. And I always find it one of those curious and sobering episodes in the history of mathematics that Fibonacci numbers--which arose incredibly early in the history of western mathematics and which are somehow very obvious and basic--didn't really start getting popular in mainstream math until maybe less than 20 years ago.
Anyway, it's also interesting to notice that the idea of breaking digits up into groups of three to make big numbers more readable is already in Fibonacci's book from 1202, I think, though he talked about using overparens on top of the numbers, not commas in the middle.
After Fibonacci, our modern representation for numbers gradually got more popular, and by the time books were being printed in the 1400s it was pretty universal, though there were still a few curious pieces of backtracking.
But still there weren't really algebraic variables. Those didn't get started pretty much until Vieta at the very end of the 1500s, and they weren't common until way into the 1600s. So that means people like Copernicus didn't have them. Nor for the most part did Kepler. So these guys pretty much used plain text, or sometimes things structured like Euclid.
By the way, even though math notation hadn't gotten going very well by their time, the kind of symbolic notation used in alchemy, astrology, and music pretty much had been developed. So, for example, Kepler ended up using what looks like modern musical notation to explain his "music of the spheres" for ratios of planetary orbits in the early 1600s.
Starting with Vieta and friends, letters routinely got used for algebraic variables. Usually, by the way, he used vowels for unknowns, and consonants for knowns.
Here's how Vieta wrote out a polynomial in the scheme he called "zetetics" and we would now call symbolic algebra:
You see, he uses words for the operations, partly so the operations won't be confused with the variables.
So how did people represent operations?
Well, the idea that operations are even something that has to represented probably took a long time to arrive. The Babylonians didn't usually use operation symbols: for addition they mostly just juxtaposed things. And generally they tended to put things into tables so they didn't have to write out operations.
The Egyptians did have some notation for operations--they used a pair of legs walking forwards for plus, and walking backwards for minus--in a fine hieroglyphic tradition that perhaps we'll even come back to a bit in future math notation.
But the modern + sign--which was probably a shorthand for the Latin "et" for "and"--doesn't seem to have arisen until the end of the 1400s.
But here, from 1579, is something that looks almost modern, particular being written in English, until you realize that those funny squiggles aren't x's--they're special non-letter characters that represent different powers of the variable.
In the early to mid-1600s there was kind of revolution in math notation, and things very quickly started looking quite modern. Square root signs got invented: previously Rx--the symbol we use now for medical prescriptions--was what was usually used. And generally algebraic notation as we know it today got established.
One of the people who was most serious about this was a fellow called William Oughtred. One of the things he was noted for was inventing a version of the slide rule. Actually he's almost like an unknown character. He wasn't a major research mathematician, but he did some nice pedagogical stuff, with people like Christopher Wren as his students. It's curious that I'd certainly never heard about him in school--especially since it so happens that he went to the same high school as me, though 400 years earlier. But the achievement of inventing a slide rule was not sufficiently great to have landed him a place in most mathematical history.
But anyway, he was serious about notation. He invented the cross for multiplication, and he argued that algebra should be done with notation, not with words, like Vieta had done. And actually he invented quite a bit of extra notation, like these kinds of squiggles, for predicates like IntegerQ.
Well, after Oughtred and friends, algebraic notation pretty quickly settled in. There were weird sidetracks--like the proposal to use waxing and waning moon symbols for the four operations of arithmetic: a fine example of poor, nonextensible design. But basically modern stuff was being used.
Here is an example.
This is a fragment of Newton's manuscript for the Principia that shows Newton using basically modern looking algebraic notation. I think Newton was the guy, for example, who invented the idea that you can write negative powers of things instead of one over things and so on. The Principia has rather little notation in it, except for this algebraic stuff and Euclid-style presentation of other things. And actually Newton was not a great notation enthusiast. He barely even wanted to use the dot notation for his fluxions.
But Leibniz was a different story. Leibniz was an extremely serious notation buff. Actually, he thought that having the right notation was somehow the secret to a lot of issues of human affairs in general. He made his living as a kind of diplomat-analyst, shuttling between various countries, with all their different languages, and so on. And he had the idea that if one could only create a universal logical language, then everyone would be able to understand each other, and figure out anything.
There were other people who had thought about similar kinds of things, mostly from the point of view of ordinary human language and of logic. A typical one was a rather peculiar character called Ramon Lull, who lived around 1300, and who claimed to have developed various kinds of logic wheels--that were like circular slide rules--that would figure out answers to arbitrary problems about the world.
But anyway, what Leibniz really brought to things was an interest also in mathematics. What he wanted to do was somehow to merge the kind of notation that was emerging in mathematics into a precise version of human language to create a mathematics-like way of describing and working out any problem--and a way that would be independent, and above, all the particular natural languages that people happened to be using.
Well, like many other of his projects, Leibniz never brought this to fruition. But along the way he did all sorts of math, and was very serious about developing notation for it. His most famous piece of notation was invented in 1675. For integrals, he had been using "omn.", presumably standing for omnium. But on Friday October 29, 1675 he wrote the following on a piece of paper.
What we see on this fragment of paper for the first time is an integral sign. He thought of it as an elongated S. But it's obviously exactly our modern integral sign. So, from the very first integral sign to the integral sign we use today, there's been very little change.
Then on Thursday November 11 of the same year, he wrote down the "d" for derivative. Actually, he said he didn't think it was a terribly good notation, and he hoped he could think of a better one soon. But as we all know, that didn't happen.
Well, Leibniz corresponded with all kinds of people about notation. He thought of himself as chairing what would now be called a standards committee for mathematical notation. He had the point of view that notation should somehow be minimal. So he said things like, "Why use a double set of two dots for proportion, when you could use just one?"
He tried a few ideas that haven't worked out. For example, whereas he used letters to stand for variables, he used astronomical signs to stand for complete expressions: kind of an interesting idea, actually.
And he had a notation like this for functions.
In:= f[x] + g[x] + f[x, y] + h[u, v] //LeibnizForm
In:= f[x] + g[x] + f[x, y] + h[u, v] + h[x] + f[a, b, c] //LeibnizForm
Well, apart from these things, and with a few exceptions like the "square intersection" sign he used for equal, Leibniz pretty much settled on the notation that still gets used today.
Euler, in the 1700s, was then a big systematic user of notation. But he pretty much followed what Leibniz had set up. I think, though, that he was the first serious user of Greek as well as Roman letters for variables.
There are some other pieces of notation that came shortly after Leibniz. This next example is actually from a book under the Newton brand name published a few years after Newton died. It's a text book of algebra and it shows very traditional algebraic notation already being printed.
Here is a book by L'Hospital, printed around the same time, showing pretty much standard algebraic notation.
And finally, here is an example from Euler, showing very much modern notation for integrals and things.
One of the things that Euler did that is quite famous is to popularize the letter for pi--a notation that had originally been suggested by a character called William Jones, who thought of it as a shorthand for the word perimeter.
So the notation of Leibniz and friends pretty much chugged along unchanged for quite a while. A few pieces of streamlining happened, like x x becoming always written as . But not much got added.
But then one gets to the end of the 1800s, and there's another burst of notational activity, mostly associated with the development of mathematical logic. There was some stuff done by physicists like Maxwell and Gibbs, particularly on vectors and vector analysis, as an outgrowth of the development of abstract algebra. But the most dramatic stuff got done by people starting with Frege around 1879 who were interested in mathematical logic.
What these people wanted to do was a little like part of what Leibniz had wanted to do. They wanted to develop notation that would represent not only formulas in mathematics, but also deductions and proofs in mathematics. Boole had shown around 1850 that one could represent basic propositional logic in mathematical terms. But what Frege and people wanted to do was to take that further and represent predicate logic and, they hoped, arbitrary mathematical arguments in mathematical terms and mathematical notation.
Frege decided that to represent what he wanted to represent, he should use a kind of graphical notation. So here's a piece of his so-called "concept notation."
Unfortunately, it's very hard to understand. And actually if one looks at all of notational history, essentially every piece of graphical notation that anyone's ever tried to invent for anything seems to have had the same problem of being hard to understand. But in any case, Frege's notation definitely didn't catch on.
Then along came Peano. He was a major notation enthusiast. He believed in using a more linear notation. Here's a sample:
Actually, in the 1880s Peano ended up inventing things that are pretty close to the standard notations we use for most of the set-theoretical concepts.
But,a little like Leibniz, he wasn't satisfied with just inventing a universal notation for math. He wanted to have a universal language for everything. So he came up with what he called Interlingua, which was a language based on simplified Latin. And he ended up writing a kind of summary of mathematics--called Formulario Mathematico--which was based on his notation for formulas, and written in this derivative of Latin that he called Interlingua.
Interlingua, like Esperanto--which came along about the same time--didn't catch on as an actual human language. But Peano's notation eventually did. At first nobody much knew about it. But then Whitehead and Russell wrote their Principia Mathematica, and in it they used Peano's notation.
I think Whitehead and Russell probably win the prize for the most notation-intensive non-machine-generated piece of work that's ever been done. Here's an example of a typical page from Principia Mathematica.
They had all sorts of funky notations in there. In fact, I'm told--in a typical tale often heard of authors being ahead of their publishers--that Russell ended up having to get fonts made specially for some of the notation they used.
And, of course, in those days we're not talking about TrueType or Type 1 fonts; we're talking about pieces of lead. And I'm told that Russell could actually be seen sometimes wheeling wheelbarrows full of lead type over to the Cambridge University Press so his books could be appropriately typeset.
Well, for all that effort, the results were fairly grotesque and incomprehensible. I think it's fairly clear that Russell and Whitehead went too far with their notation.
And even though the field of mathematical logic scaled back a bit from Russell and Whitehead, it's still the field that has the most complicated notation of any, and the least standardization.
But what about what's normally thought of as more mainstream mathematics?
For a while, at the beginning of the 1900s, there was almost no effect from what had been done in mathematical logic. But then, when the Bourbaki movement in France started taking root in the 1940s or so, there was suddenly a change.
You see, Bourbaki emphasized a much more abstract, logic-oriented approach to mathematics. In particular, it emphasized using notation whenever one could, and somehow minimizing the amount of potentially imprecise text that had to be written.
Starting around the 1940s, there was a fairly sudden transition in papers in pure mathematics that one can see by looking at journals or ICM proceedings or something of that kind. The transition was from papers that were dominated by text, with only basic algebra and calculus notation, to ones that were full of extra notation.
Of course, not all places where math gets used followed this trend. It's kind of like what's often done in linguistics of ordinary natural languages. One can see when different fields that use mathematics peeled off from the main trunk of mathematical development by looking at what vintage of mathematical notation they use. So, for example, we can tell that physics pretty much peeled off around the end of the 1800s, because that's the vintage of math notation that it uses.
There's one thing that comes through in a lot of this history, by the way: notation, like ordinary language, is a dramatic divider of people. I mean, there's somehow a big division between those who read a particular notation and those who don't. It ends up seeming rather mystical. It's like the alchemists and the occultists: mathematical notation is full of signs and symbols that people don't normally use, and that most people don't understand.
It's kind of curious, actually, that just recently there've been quite a few consumer product and service ads that have started appearing that are sort of centered around math notation. I think for some reason in the last couple of years, mathematical notation is becoming chic. Here's an example of an ad that's running right now.
But the way one tends to see math notation used, for example in math education, reminds me awfully of things like symbols of secret societies and so on.
Well, so that's a rough summary of some of the history of mathematical notation.
And after all that history, there's a certain notation that's ended up being used. Apart from a few areas like mathematical logic, it's become fairly standardized. There are not a lot of differences in how it's used. Whatever ordinary language a book or paper is written in, the math pretty much always looks the same.