`abstract\`

style anyway. My
hacked rtftohtml recognizes that.
[Chris... I'll probably move that table to a config file and send you diffs
when I do.]
You may make and distribute verbatim copies of this document, provided the copyright notice and permission are intact on all such copies. \

I mentioned in the meeting** that I thought we needed a "new math" formalism that was E-text friendly. Since such a radical view is likely to have some opposition, I thought I would set down here what I meant and why I think we have to go this route. \

** See note below for a clue as to the original audience of this paper.\

The issues raised here are crucial for UU-GNA,** because a large part of what it hopes to accomplish is teaching Physical, Mathematical, and Computer Science, and related fields like Economics and Management Science or Accounting, all of which use Mathematical formalism.\

** Usenet University, now called Global Network Academy. See news://alt.uu.future. This paper should also be of interest to the TEI project, which, for a scholarly project, is notably deficient in the means of representing formulas and statistical text--even when such things are becoming commonplace in, say, Linguistics or even English Literature. (See, I said I would publish my thoughts). \

My discussion has three parts:\

Part I: Answering Obvious Objections\

Part II: Why Patches won't work\

Part III: What I Mean by a New Notation\

The annoying paragraph marks are a pathetic attempt on my part to add HTML markup by hand, so this document can be read on the World Wide Web. They are easy to remove.\

\

1. Math is a universal language, understood by all scientists and mathematicians; a new notation is "going it alone".\

2. An education based on a variant notation would isolate the individual from this world community.\

3. E-text will never be used for *hand* calculation the way we solve algebra problems on paper, and without teaching the paper formalism the student will not have an efficient formalism for their own calculations--an essential goal of any Math education.\

4. Learning two formalisms will confuse the student.\

5. It is impossible to communicate mathematical ideas in E-text. You need TeX at a minimum and possibly postscript or bitmaps\

6. Any Science textbook will have to have graphics anyway; formulas are just another kind of graphic.\

Let me take these one at a time:\

1. Math is a universal language, understood by all scientists and mathematicians; a new notation is "going it alone".\

Math is also a very heterogeneous formalism. There is no reason that multiple formalisms for the same thing can coexist, like (a) vectors as boldface letters and (b) vectors with subscripted coordinates. The first view is good for abstract reasoning about vectors as entities and the second is necessary for computer computations.\

Often, we can pick the notation that is most friendly to E-text: e.g. "continuing fraction" notation for continued fractions or "einstein summation" notation for sums over indices.\

2. An education based on a variant notation would isolate the individual from this world community.\

True. There is no way presently to be exclusively educated by E-text and still be part of the "world community". We shouldn't make all our materials conform to conventionality--they won't. \

Instead we should aim at supplementing traditional education in useful ways. Books work the same way. You can't be educated exclusively by reading on your own and seem "normal". Abraham Lincoln is an example of brilliantly self-educated person who was not "normal". \

3. E-text will never be used for *hand* calculation the way we solve algebra problems on paper, and without teaching the paper formalism the student will not have an efficient formalism for their own calculations--an essential goal of any Math education.\

This argument reminds me of arguments that the Chinese notation for words is better than an alphabetic one because it represents the ideas in an (abstract) pictorial fashion. However, I am willing to grant the premiss that hand calculation in traditional notation is essential for the working scientist, since it is at least as hard to convince Mathematicians to give up their notation as to tell the Japanese that they should just use Romaji to write their language and then they won't have any problems in the modern world (this has been seriously advocated by Westerners, believe it or not).\

I would also point out that calculating with a spreadsheet and programming are equally essential to the working scientist and *are* done in the native E-text medium (in fact it is easier to run a program if you have E-text--you don't have to type).\

So this argument cuts both ways. Obviously, it behoves an E-text on Physics to emphasize programming, just as it behoves a paper text to emphasize hand calculation. We should play our strengths and innovate to cover our weaknesses.\

4. Learning two formalisms will confuse the student.\

Bull. I can learn French and not impair my understanding of English. In fact, learning a foreign language helps. Programming probably helps my understanding of Math. Learning two notations will drive home to the student the underlying, abstract idea very forcefully. I think someone who knows how to write an algebra formula *and* write a program to evaluate it will not be confused but be a superior student.\

The prevalent notation has ambiguities that positively *confuse* all but the brightest students, such as the difference between a function and its value, implicitly confusing in [y=f(x)]. Another notation, such as [f: x->y] actually helps the student. An E-text specific notation may help pedagogy, not hinder it. 5. It is impossible to communicate mathematical ideas in E-text. You need TeX at a minimum and possibly postscript or bitmaps.\

It may take us a while to develop formalisms that rival paper (paper formalism has a 500+ year head start), but we can do it.\

6. Any Science textbook will have to have graphics anyway; formulas are just another kind of graphic.\

First of all, a textbook with graphics will not work on a text-based monitor. If we accept this view we loose a *lot* of people, many of whom, now that computers capable of reading text can cost $100 used, stand to benefit from access to science texts.\

But we don't *have* to loose those people. It may be true that "a lot of science is spatial intuition; you just can't teach it without diagrams", but a lot of science is written without many pictures, especially the theoretical parts. Of course there are lots of formulas. I think if we could get E-text to express Math notation we would go a long way towards providing usable science texts, even without graphics. Also, we would reduce the burden on authors, who then only have to make graphics for the graphics, not for the formulas too.\

Even if you use graphics and the student has a bitmapped screen, it is often more efficient to give the student a program or formula for plotting or drawing the graphics, and not a picture, so that whatever tools the student's platform has can interpret the instructions and draw a picture. That way you get more potential students. It is even better if the instructions can be mailed. In this sense, TIFF is better than Macintosh PICT or an X bitmap, and Embedded Postscript is better than TIFF.\

\

First of all (minor point), if it's not easy, people won't use it. Patches may be possible, but I advocate the surer path of making things Easy On The Humans. Who's going to digitize a picture and put in a cross-reference if they can do the same thing in a few keystrokes? Let's invent cool notations so they can do it. 9 out of 10 scientists use clumsy E-text formulas anyway when they talk to their friends by E-mail. If they see a good notation, they take it and use it. Period. Let's invent better ones so people can do what they're going to do anyway, only better. Even if the Textbooks are in TeX, the tutorial sessions will be in off-the-cuff E-mail, using the best text-based notation we can devise. In fact, a lot of what we need has been invented already, since people have been trying to figure out how to type formulas in E-text ever since FORTRAN.\

TeX is close to optimal for a notation that is easy to write, universal, and looks good on paper. If we have software that seamlessly converts TeX to a bitmapped image on MOSAIC we're done, right? \

Nope. I really don't want to loose all the potential students (*millions* of them) who don't have access to bitmapped terminals. Even when bitmapped terminals become the only ones around, you will always miss people who have the "wrong" platform unless you use plain text.\

In short,\

1. E-text is or can be a poor person's medium.\

It is quite possible to create textbooks that cost 50 cents. On CD ROM that can be reduced to pennies. I want to see textbooks that can be read on *any* platform and cheap enough to be purchased by any library, even ones without internet connections. This doesn't mean that every UU-GNA book has to be a candidate for this E-text library, but some should; and GNA is in a good position to be the standard setter for books that are distributed this way.\

The book that gets GNA the most bang for its buck will be a textbook that can be viewed on the WWW running MOSAIC *and* made available (after conversion from HTML) in text form. If any of these books are going to be about Science or Math, then the issues I'm raising will have to be solved.\

2. E-text is not and never will be able to show Math formulas.\

Not unless we do something to enhance the ability of the medium to represent Math--like invent new notations. TeX can be printed or converted to a bitmap, but it cannot be read on the screen as easily as other notations we might possibly invent. \

3. We're never going to outgrow E-text. \

Somewhere, 20 years from now, there will be text-based terminals,** and those people will want to learn Science and Math. If we can figure out notations we'll be closer to the goal of creating the texts they'll need.\

** I don't know whether these will be the 1,000,000 terminals in the very poorest third world country, which can only afford Western castoffs, or the 20-year old terminals that we are refurbishing because resource depletion has driven the cost of all manufacturing through the roof; but I'm sure they will be there and that UU-GNA's texts will be on them.\

To put it another way, when text-based communication has been around another 20 years, we will have invented display formats that are capable of representing Math (and every other subject of human communication); the question is whether GNA wants to be a leader or a follower in this field. If it chooses to be a leader, it can establish the standards, design the software, and not have to retrofit all its books.\

\

** available by anon. FTP from \

gopher://world.std.com:70/obi/Networking/John.Goodwin.\

\

FORTRAN and similar languages have tackled this problem admirably. Probably a C-like notation would be preferred today (except for exponents, which would likely use the FORTRAN). One problem with C as a standard is its use of E-mail unsafe characters, whence the infamous ANSI trigraphs. We need workable alternates for those.\

The basic and successful pattern of programming languages should be followed. Here is a list of 10 innovations that programmers invented to break out of the paradigm of Mathematical Prose, yet remain close enough to it to enable humans to use computers effectively:\

1. Expand variable names to many letters to compensate for the paucity of symbols.\

2. Represent subscipts and superscripts in the usual way, with parenthesis notation.\

3. Break up complex formulas with "temporary" variables.\

4. In fact, instead of forcing every statement into the paradigm of an Equality, force it into the paradigm of an Algorithm.\

5. Break up algorithms, just like formulas, by functional decomposition.\

6. Use indentation to represent "nesting" within algorithms.\

7. Use loops instead of summations.\

8. Integrate logical relations into numeric ones, introducing a hierarchy of precedence that makes this idea natural, given mathematical speech habits.\

9. Provide easy means of mapping text on to numbers, and numbers on to text (numerals).\

10. Introduce "hypertext" links, first via line numbering, then via structured programming and its conditional branching.\

I contend that each of these is essentially an adaptation to the medium in which the mathematical problem was being formulated--E-text. We often forget, now that we are trying to break out of the "procedural" paradigm, just how radical it was in its day! We can still learn from these innovations, none of which were "obvious" to Mathematicians trained to think of Math as the science of solving equalities for unknows.\

We can also relax and allow a number of informal notations that would choke mechanical parsers, but are easy for humans. For example, if we are willing to restrict ourselves to 26 possible variables (in short communications we typically are), we can abbreviate the formal [C(i,j)] to [Cij], as in\

Cij = Aij + Bij.\

We might also find use for the function notation becoming common place in Object-Oriented languages, and use something like [x.f] for f(x). (I know; I hate postfix notation too, but it's easy to type).\

I mentioned above that fractions are a special problem. We can try several notations:\

(a + b)/(c + d) num = a + b den = c + d frac a + b c + d Each of these has its strengths and weaknesses. The last explicitly shows the syntax tree and may ultimately be the most workable solution for representing Math that *wants* to be non-linear and refuses to look good streched out in a straight line. In any event, trees are easier to read than (((a+b)/(c+d))*c)+a. I mean\

a + factor c (a + b)/(c + d), where "factor" is a placeholder for the product of the following lines. This notation is quite adaptable. and does *not* depend on synchronizing lines the way mathematical text is sometimes typed on a typewriter, with powers in one line and variables in another:\

2 2 x + y = 1 (it does depend, however, on linebreaks; so reformatting has to be suppressed).\

"Makefiles" provide another interesting paradigm, that of rules and actions, which might provide inspiration for math notation.\

\

1. A good way to write numbers in "statistical text". (Can we do better than ugly things like 1.0E2 or 6+/-0.2?)\

2. How to distinguish things like the variable a from ordinary text-- I mean how to distinguish things like [a] from ordinary text. (TeX uses math italics and a dollar-sign delimiter)\

3. A variety of display formats for equations, program samples, algorithms and pseudo-code, and the means of numbering equations.**\

4. Various fonts or delimiters to represent entities like variable names (math italics), program names (small caps), and so on.\

** a bad way to do this is by tagging fonts explictly. TeX is not so brilliant here.\

\

Tab formatted text works, but tabs are not E-mail safe. Commas, as in Comma Separated Value (CSV) format are safe, however. CSV text does not make for an interesting read, but if you add spaces it does nicely:\

\ 9, 8, 18 7, -2, 4 3, 0, 2 \

There are some problems:\

Many spreadsheets won't accept formulas, just values. It would be nice to be able to provide instant spreadsheets for students with the *formulas*. \

DIF format isn't fun to read, but about as close as you can get to a universal, text-based interchange format. We may need to invent another one. Ideally, one could click on the object and get a table you can manipulate in a separate window; or better, by using something like AppleEvents on a Macintosh, bring up the local spreadsheet program, whatever it may be. The beauty of this would be if all this functionality could be invoked by the tag \

Possible enhancements to the MOSAIC/UU-GNA learning environment:\

Drag and drop matrix multiplication.\

What-if plug-in for formula values (click on a variable and type the number for each independent variable. Option of single numeric value, a spreadsheet, or graphic output).\

Click on common constants like [pi] or [e] and other numeric entities to see their value.\

Automatic graphing for formulas. (Select [x] and [y]--your choice of plots. You control plotting range, axis labels, etc. through a popup menu)\

Or maybe we just write an (enhanced) HTML to Mathematica converter and bring up Mathematica (or your favorite high-end Math tool) whenever you click on a formula.\

These sort of enhancements could make MOSAIC (or whatever) an interactive learning environment on the high end, without compromising the visual display of the low end. They would be activated by tags, just like hypertext links. Now here's a *real* use for SGML.\

What I'm trying to get across is the idea that Mathematics isn't an endless sea of notation. If you restrict you horizons to simple, algebra-like formulas, and go for the gaudier stuff later, but expand your idea of the functionality you want, then you really can do a lot with the right, standard, notation.\

A uniform notation that *can be written in E-text* is absolutely crucial to distributed applications that provide the sort of high end functionality discussed here. After all, that is what a programming language is; and we all know that without a programming language you don't get functionality out of a computer. The learning environment is not different.\

If we are serious about creating Math texts, then both the low tech and high tech ends demand that we devise a new language for representing Math formulas. We get the most mileage if this language is easy to read at the low end *and* functional at the high end.\

TeX is a good language for writing formulas that can be printed; \

FORTRAN is a good language for writing formulas that can be evaluated algorithmically;\

Now we need a symbolic language for representing formulas that humans can see and interact with on a computer screen. \

=John Goodwin= jgoodwin@adcalc.fnal.gov (Fermilab)| | You may make and distribute verbatim copies of this document, provided | the copyright notice and permission are intact on all such copies. \

One of my favorite books of the last few years is a simple book for
Macintosh users titled "A Mac is not a typewriter." The existence of the
keyboard is one of the major barriers to unleashing the power of the
computer. I think John's comments are just a particular version of this -
e-text is more than ASCII, EBCDIC, ISO-LATIN II, or something like that.
To wit:
- Science, 17 September 1993, has a two page article on the Xerox PARC
(Palo Alto Research Center), and its computers here, there, and everywhere,
approach. Not a keyboard in site. Digital ink, ubiquituous computing.
(Scientific American had a full issue either last year or the year before
with a similar article.)
- For the last week I've been using an Apple Newton. The leap between hand
writing recognition to mathematics recognition is just a thin PCMIA card
away.
- "Exploring Mathematics with Mathematica" is a book I bought a couple of
years ago that included not just the electronic "text" on a CD-ROM, but the
EXECUTABLE program. It was not just mathematics to read, it was
mathematics to DO. E-math should not be imagined as purely static
representation, but as dynamic, actionable, thinking system.
- With the right software, e-text-based e-math might even be a
mathematician! Puppet? Robot? Sega Virtual Reality Mathmatical Theme
Park?
Yes, we've got ASCII today, but that's just an historical anomaly.
Steve

| | FORTRAN and similar languages have tackled this problem admirably. | Probably a C-like notation would be preferred today (except for | exponents, which would likely use the FORTRAN). One problem with C as | a standard is its use of E-mail unsafe characters, whence the infamous | ANSI trigraphs. We need workable alternates for those.\

I disagree strongly with these statements. Literate Programming is an
attemp to FIX the serious problems with C and FORTRAN style notation. C
and FORTRAN can be very difficult to understand. Particularly if you are
doing vector or tensor calculations. The notation used is almost
unintelligable. Mathmatical notation is much cleaner. That is why, using
literate programming, it is desirable to include as part of the program the
full mathmatical description.
Literate programming uses additional fonts and symbols to make the code
much more intelligable. Instead of using .NE., <>, or != to mean "not
equals", literate programs print and display the code using a standard
mathmatical not-equals sign. Assignments are shown using a left pointing
arrow. There are literate programming tools for C, C++, Fortran, Pascal,
Modula 2, and several others, as well as some "generic" literate
programming tools that let you define you own syntax. Using literate
programming, array indicies are printed as subscripts.
The proposal John Goodwin makes, is in complete opposition to one of the
basic ideas behind one of Usenet's newest newsgroups:
comp.programming.literate.
Instead of flaming the proposal in detail, I am making an open invitation
to read comp.programming.literate (also available via a mailserver at
LitProg@SHSU.edu).
--
Jeffrey M\\kern-.05em\\raise.5ex\\hbox{\\b c}\\kern-.05emArthur
a.k.a. Jeffrey McArthur ATLIS Publishing
phone: (301) 210-6655 12001 Indian Creek Court
fax: (301) 210-4999 Beltsville, MD 20705
EMAIL: j_mcarthur@bix.com

I'm not a mathematician, so I can't enter the argument pro or con the
overall proposal, but in relation to #5, it's worth pointing out that TeX
remains available free (unless you buy one of the commercial versions) so
there really is no reason why every mathematician shouldn't have a copy, as
it runs on most every machine in existence. The developments currently
under way in MIME, WWW and such like tend to indicate that a email header
identifying TeX content is a useable solution, but we need to wait for
mailing systems to implement this.
An alternative is full SGML math markup: again, this would have the
advantage of universality and international standardization, but what I
hear from the people involved is that there is still a lot of discussion to
go through before it becomes a useable reality.
///Peter

[Peter Flynn] | I'm not a mathematician, so I can't enter the argument pro or con the | overall proposal, but in relation to #5, it's worth pointing out that | TeX remains available free (unless you buy one of the commercial | versions) so there really is no reason why every mathematician | shouldn't have a copy, as it runs on most every machine in existence. | The developments currently under way in MIME, WWW and such like tend to | indicate that a email header identifying TeX content is a useable | solution, but we need to wait for mailing systems to implement this. TeX is a usable solution but splicing in into HTML purely for the purpose of displaying equations is not. Practical problems such as avaliability of header files are raised. TeX is a large system and not particularly fast, it would not be practical to use it as part of an editing browser for WWW. | An alternative is full SGML math markup: again, this would have the | advantage of universality and international standardization, but what I | hear from the people involved is that there is still a lot of | discussion to go through before it becomes a useable reality. A third alternative is presentation based SGML markup. This is what HTML and HTML+ are. They are presentation based but the presentation model is not fully defined. Thus it is still possible to use the same source for very different output formats - eg postscript and linemode terminal. But one thing SGML certainly will not bring is "universality and international standardisation". If it did it would be the first standard ever to manage that! Whatever the future there will be much more than one document format. That is merely a fact of life. TeX will exist for many years as the de facto standard for academic computing. Proprietary formats will abound. All SGML can hope to be that is "universal" is an interchange format. I currently have a presentation based markup that deals with almost all standard mathematical notations and uses only 5 tags. A discussion document should be out soon - once the formatting routines are complete. Hopefully there will be some gifs/jpegs of the examples to demonstrate it can be done. Practically it is little harder than the existing \