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MathML International Conference Abstract Most mathematical notation now in use is between one and five hundred years old. I will review how it developed, with precursors in antiquity and the Middle Ages, through its definition at the hands of Leibniz, Euler, Peano and others, to its widespread use in the nineteenth and twentieth centuries.

I will discuss the extent to which mathematical notation is like ordinary human language—albeit international in scope.

I will show that some general principles that have been discovered for ordinary human language and its history apply to mathematical notation, while others do not. Given its historical basis, it might have been that mathematical notation—like natural language—would be extremely difficult for computers to understand.

But over the past five years we have developed in Mathematica capabilities for understanding something very close to standard mathematical notation. I will discuss some of the key ideas that made this possible, as well as some features of mathematical notation that we discovered in doing it.

Large mathematical expressions—unlike pieces of ordinary text—are often generated automatically as results of computations. I will discuss issues involved in handling such expressions and making them easier for humans to understand.

Traditional mathematical notation represents mathematical objects but not mathematical processes. I will discuss attempts to develop notation for algorithms, and experiences with these in APL, Mathematica, theorem-proving programs and other systems.

Ordinary language involves strings of text; mathematical notation often also involves two-dimensional structures. I will discuss how mathematical notation might make use of more general structures, and whether human cognitive abilities would be up to such things.

The scope of a particular human language is often claimed to limit the scope of thinking done by those who use it.

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I will discuss the extent to which traditional mathematical notation may have limited the scope of mathematics, and some of what I have discovered about what generalizations of mathematics might be like.

Introduction When this conference was first being put together, people thought it would be good to have someone talk about general issues of mathematical notation.

And there was an obvious candidate speaker—a certain Florian Cajori—author of a classic book entitled A History of Mathematical Notation. But upon investigation, it turned out that there was a logistical problem in inviting the esteemed Dr. Cajori—he has been dead for no less than seventy years.

So I guess I'm the substitute. And I think there weren't too many other possible choices. Because it turns out that so far as we could all tell, there's almost nobody who's alive today who's really thought that much about basic issues regarding mathematical notation.

In the past, the times these things have ever been thought about, even a bit, have mostly coincided with various efforts to systematize mathematics. So Leibniz and other people were interested in these things in the mids. Babbage wrote one of his rather ponderous polemics on the subject in And at the end of the s and the beginning of the s, when abstract algebra and mathematical logic were really getting going, there was another burst of interest and activity.

But after that, pretty much nothing. But in a sense it's not surprising that I've been interested in this stuff. Because with Mathematica, one of my big goals has been to take another big step in what one can think of as the systematization of mathematics.

And I guess in general my goal with Mathematica was somehow to take the general power of computation and harness it for all kinds of technical and mathematical work.

There are really two parts to that: One of the big achievements of Mathematica, as probably most of you know, was to figure out a very general way to have the computations inside work and also be practical—based on just doing transformations on symbolic expressions, with the symbolic expressions representing data or programs or graphics or documents or formulas or whatever.

But just being able to do computations isn't enough; one also has to have a way for people to tell Mathematica what computations they want done. And basically the way that people seem to communicate anything sophisticated that they want to communicate is by using some kind of language.

Normally, languages arise through some sort of gradual historical process of consensus. But computer languages have historically been different. The good ones tend to get invented pretty much all at once, normally by just one person.

So what's really involved in doing that? Well, at least the way I thought about it for Mathematica was this: I tried to think of all the possible computations that people might want to do, and then I tried to see what chunks of computational work came up over and over again.

And then essentially I gave names to those chunks, and we implemented the chunks as the built-in functions of Mathematica.Algebra I For Dummies, 2nd Edition () was previously published as Algebra I For Dummies, 2nd Edition ().

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