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The three most important things to remember are: Explain each step in your working clearly. Lay out your explanation clearly. Use correct math —make sure that what you write is mathematically correct.
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Featured Video. Cite this Article Format. Russell, Deb. Frege decided that to represent what he wanted to represent, he should use a kind of graphical notation.
Then along came Peano. He was a major notation enthusiast. He believed in using a more linear notation. Actually, in the s Peano ended up inventing things that are pretty close to the standard notations we use for most of the set-theoretical concepts.
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. At first nobody much knew about it. They had all sorts of funky notations in there. Well, for all that effort, the results were fairly grotesque and incomprehensible. For a while, at the beginning of the s, 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 s 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 s, 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. 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. It ends up seeming rather mystical. I think for some reason in the last couple of years, mathematical notation is becoming chic.
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. Whatever ordinary language a book or paper is written in, the math pretty much always looks the same.
But now the question is: can computers be set up to understand that notation? For ordinary human language, people have been making grammars for ages. Certainly lots of Greek and Roman philosophers and orators talked about them a lot. And not only have there been grammars for language; in the last centuries or so, there have been endless scholarly works on proper language usage and so on.
But despite all this activity about ordinary language, essentially absolutely nothing has been done for mathematical language and mathematical notation. In the early s mathematical logicians talked quite a bit about different layers in well-formed mathematical expressions: variables inside functions inside predicates inside functions inside connectives inside quantifiers. But not really about what this meant for the notation for the expressions. Things got a little more definite in the s, when Chomsky and Backus, essentially independently, invented the idea of context-free languages.
The idea came out of work on production systems in mathematical logic, particularly by Emil Post in the s. But, curiously, both Chomsky and Backus came up with the same basic idea in the s. And he certainly noticed that algebraic expressions could be represented by context-free grammars.
Chomsky applied the idea to ordinary human language. And he pointed out that to some approximation ordinary human languages can be represented by context-free grammars too. But the thing that I always find remarkable, and scientifically the most important, is that to a first approximation it is true that ordinary human languages are context-free. But neither seems to have looked at more advanced kinds of math than simple algebraic language.
And, so far as I can tell, nor has almost anyone else since then. But if you want to see if you can interpret mathematical notation, you have to know what kind of grammar it uses. Now I have to tell you that I had always assumed that mathematical notation was too haphazard to be used as any kind of thing that a computer could reasonably interpret in a rigorous way.
But at the beginning of the s we got interested in making Mathematica be able to interact with mathematical notation. And so we realized that we really had to figure out what was going on with mathematical notation. Neil Soiffer had spent quite a number of years working on editing and interpreting mathematical notation, and when he joined our company in , he started trying to convince me that one really could work with mathematical notation in a reasonable way, for both output and input.
Well, actually, one already learned something from output. One learned that at least at some level, a lot of mathematical notation could be represented in some kind of context-free form. Because one knew that in TeX, for instance, one could set things up in a tree of nested boxes. But how about input? Well, one of the biggest things was something that always comes up in parsing: if you have a string of text, with operands and operators, how do you tell what groups with what?
What does it mean? Well, to know that you have to know the precedence of the operators—which ones bind tighter to their operands and so on. But I decided to actually take a look at it. So I went through all sorts of math books, and started asking all kinds of people how they would interpret random lumps of mathematical notation. We can say with pretty much confidence that this is the precedence table that people imagine when they look at pieces of mathematical notation.
Having found this fact, I got a lot more optimistic about us really being able to interpret mathematical notation input. One way one could always do this is by having templates.
Like one has a template for an integral sign, and one just fills stuff into the integrand, the variable, and so on. And when the template pastes into a document it looks right, but it still maintains its information about what template it is, so a program knows how to interpret it. And indeed various programs work like this. Because as soon as you try to type fast—or do editing—you just keep on finding that your computer is beeping at you, and refusing to let you do things that seem like you should obviously be able to do.
Letting people do free-form input is much harder. Well, basically one needs a completely rigorous and unambiguous syntax for math. Obviously, one can have such a syntax if one just uses regular computer language like string-based syntax. So what can one do about it? Well, of course that would be absurdly confusing. What should the special one look like?
Well, the idea we had—actually I think I was in the end responsible for it—was to use double-struck characters. We tried all sorts of other graphical forms. But the double struck idea was the best. Partly because it sort of follows a convention in math of having notation for specific objects be double struck. So, for example, a capital R in mathematical text might be a variable. But double struck R represents a specific object: the set of all real numbers.
And it works like this:. Or a variable? Things get horribly confused. And you end up with a completely well defined syntax. It works like this:. It turns out that there are actually very few tweaks that one has to make to the core of mathematical notation to make it unambiguous.
Of course, to make it really nice, there are lots of details that have to be right. One has to actually be able to type things in an efficient and easy-to-remember way. We thought very hard about that.
And we came up with some rather nice general schemes for it. One of them has to do with entering things like powers as superscripts. Well, having a clean set of principles like that is crucial to making this whole kind of thing work in practice.
But it does. And the point is that this expression is completely understandable to Mathematica , so you can evaluate it. And the thing that comes out is the same kind of object as the input, and you can edit it, pick it apart, use its pieces as input, and so on.
So instead of just having things like prefix operators, we also have things like overfix operators, and so on. And it certainly has all the various compactifying and structuring features of ordinary math notation. And the important thing is that nobody who knows ordinary math notation would be at all confused about what the expression means. Like the way trig functions are written, and so on. Well, I would argue rather strongly that the Mathematica StandardForm, as we call it, is a better and clearer version of this expression.
But if one wants to be fully compatible with traditional textbooks one needs something different. And the actual TraditionalForm I get always contains enough internal information that it can unambiguously be turned back into StandardForm. But the TraditionalForm looks just like traditional math notation. With all the slightly crazy things that are in traditional math notation, like writing sin squared x, instead of sin x squared, and so on. You may notice those jaws on the right-hand side of the cell.
We can edit just fine. Actually, we have a few hundred rules that are heuristics for understanding traditional form expressions.
And they work fairly well. Sufficiently well, in fact, that one can really go through large volumes of legacy math notation—say specified in TeX—and expect to convert it automatically to unambiguously meaningful Mathematica input. But with math there is. Of course, there are some things with math, particularly on the output side, that are a lot trickier than text. Part of the issue is that with math one can expect to generate things automatically.
But with math, you do a computation, and out comes a huge expression. So then you have to do things like figure out how to break the expression into lines elegantly, which is something we did a lot of work on in Mathematica. And that means there are nasty problems like that you can be typing more characters, but suddenly your cursor jumps backwards. Well, that particular problem I think we solved in a particularly neat way.
Did you see that? There was a funny blob that appeared just for a moment when the cursor had to move backwards. Perhaps you noticed the blob.
Physiologically, I think it works by using nerve impulses that end up not in the ordinary visual cortex, but directly in the brain stem where eye motion is controlled.
So it works by making you subconsciously move your eyes to the right place. Does that mean we should turn everything Mathematica can do into math-like notation? Should we have special characters for all the various operations in Mathematica? We could certainly make very compact notation that way. But would it be sensible?
Would it be readable? One could have no special notation. Then one has Mathematica FullForm. But that gets pretty tiresome to read. The other possibility is that everything could have a special notation.
Well, then one has something like APL—or parts of mathematical logic. Think about Unix. In early versions of Unix it seemed really nice that there were just a few quick-to-type commands. But then the system started getting bigger. And after a while there were zillions of few-letter commands. And the whole thing started looking completely incomprehensible.
People can handle a modest number of special forms and special characters. Maybe a few tens of them. But not more. And if you try to give them more, particularly all at once, they just become confused and put off. Well, one has to qualify that a bit. There are, for example, lots of relational operators. And, of course, it is in principle possible for people to learn lots of lots of different characters. Because languages like Chinese and Japanese have thousands of ideograms.
But it takes people many extra years of school to learn to read those languages, compared to ones that just use alphabets. Well, Cantor introduced a Hebrew aleph for his infinite cardinal numbers. But there are no other characters that have really gotten imported from other languages. Well, I was curious what that distribution was like for letters in math. So I had a look in MathWorld , which is a large website of mathematical information that has about 10, entries and looked at what the distribution of different letters was.
We can see that lowercase is the most common followed by , , , , etc. But what notation is good to use? Most people who actually use math notation have some feeling for that. Because anything you type will be unambiguously understandable. But for TraditionalForm, it would be good to have some principles. Perhaps to finish off, let me talk a little about the future of mathematical notation.
And with the right drawing of characters, quite a few of these could be made perfectly to fit in with other mathematical characters. Well, the most obvious possibility is notation for representing programs as well as mathematical operations.
In Mathematica , for instance, there are quite a few textual operators that are used in programs. Because we picked the ASCII characters well, one can often get special characters that are visually very similar but more elegant.
And what makes all this work is that the parser for Mathematica can accept both the special character and non-special character forms of these kinds of operators.
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