Category Archives: language

The Linguistic Sherlock Holmes

Spysweeper is a program on our laptop that scans the disk, finds Bad Files and presents them for your inspection.   The list of Bad Files appears and under it is a button that says Quarantine Selected.  You are supposed to click that button to quarantine them.  When this happens I have been annoyed because until I clicked the button I hadn’t made the choice to quarantine them.

The other day I realized that the phrase on the button was meant to say “Quarantine the selected files”, not “You have selected quarantine”.  So I decided that the author of that program was a native speaker of a Slavic language.  Slavic languages use participles that way.  So do some Western European languages but a native speaker of one of them would probably write “Quarantine the selected”, still not idiomatic but not confusing.

Any time you want a linguistic puzzle detected, just ask me.

Of course, I have no evidence that I was correct…

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An early revolution in notation

This is a reprint of an article by Jeffrey A. Tucker. Musical notation was invented by Guido six hundred years before Viète and Descartes invented modern algebraic notation. –cw

Guido the Great, by Jeffrey A. Tucker

This essay is adapted from Jeffrey’s new book, Sing Like a Catholic.

The people who make modern inventions are often celebrated for improving our lives. But what about those one thousand years ago who laid the very technological foundation of civilization as we know it? They too served the world, but with the primary purpose of contributing to the faith. I’m thinking here of those who solved the architectural problem to build the great cathedrals of the middle ages, and the scientists of the period who took the first steps toward modern medical knowledge.

Also we don’t often consider the innovations in art that make all music possible. There is one person who stands out here: the late 10th and early 11th century Benedictine monk named Guido d’Arezzo (991/992–after 1033). He is credited with fantastic musical innovations that led to the creation of the modern system of notes and staffs, and also the organization of scales that allowed for teaching and writing music.

His contributions have usually been seen as technical innovations and evaluated as such, though known only inside a small circle of music historians. Without his contribution, the music you hear on your iPod and on the radio would not likely exist.

A new book by Angelo Rusconi, synopsized by Patrick Reynolds from the Italian, and appearing in Goldberg #46 (2007), offers a more complete picture of what drove Guido, and the results will be very exciting for anyone who seeks to understand how any serious innovation upsets the status quo, makes enemies, causes a bit of social upheaval, and ultimately makes the world a better place.

Consider the technical feat that Guido undertook. Imagine a world without printed music. How would you go about conveying a tune in printed form? It’s one thing to render words on paper in a way that others can read them. But what about sound? It floats in the air and resists having a physical presence at all.

How can you share the melody without singing it for them, by just writing things down? People had tried since the ancient world without success. Some attempts in the 8th and 9th centuries came a bit closer (but the results look like chicken scratch to us). It was Guido who made the breakthrough with lines and scales that illustrate for the eye what the voice is to sing, and precisely so. His innovation was a beautiful integration of art and science.

And what a remarkable innovation, if you think about it. From the beginning of time until his time, the teaching of music was done by a tiny and ever-arrogant cartel of masters. You had to sing exactly as they instructed you. If they weren’t around, you were stuck. They held the monopoly. To become a master of music, you had to study under one of the greats, and then receive the blessing to become a teacher yourself, and you know that they wanted to limit their numbers. One can imagine that you had to be sycophantic to even get your foot in the door.

Guido’s innovation busted up the cartel. Rusconi shows that Guido’s primary interest was in notating not just music in general but Gregorian chant in particular. He was frustrated that the chant was passed on by oral tradition only. He worried that melodies would be lost, especially given the then new fashion for multi-part improvisation.

So while writers have usually treated Guido as an innovator, what’s been forgotten is that his innovations were driven by the desire to conserve and preserve for future generations. The desire to maintain the chant and pass it on was the key issue for him; the technical aspects of the music and writing were merely tools and not ends in themselves.

And there was an interesting sociological element here. He had become seriously annoyed at the cartel of chant masters and the power they exercised over the monastic community. He wanted the chant to be freed and put into the hands of everyone both inside and outside the monastery walls.

For this reason, his first great project was a notated Antiphoner, a book of melodies: “For, in such a ways, with the help of God I have determined to notate this antiphoner, so that hereafter through it, any intelligent and diligent person can learn a chant, and after he has learned well part of it through a teacher, he recognizes the rest unhesitatingly by himself without a teacher.”

He goes further. Without a written form of music “wretched singers and pupils of singers, even if they should sing every day for a hundred years, will never sing by themselves without a teacher one antiphon, not even a short one, wasting so much time in singing that they could have spent better learning thoroughly sacred and secularly writing.”

The elite musicians resisted his attempt to democratize the knowledge and conserve time. Guido did whatever great innovator does: he freed up resources for other uses even while improving lives.

But as a result of his innovation, his monastery in Pomposa, Italy, tossed him out into the snow. He then went to the Pope, who was very impressed, and gave him a letter of support. With the letter in hand, he went to the Bishop of Arezzo, who took him in so that he could continue his preaching and his work.

Now, one can’t but think of mistakes that have been made over the years with the Gregorian chant: the attempt to keep it the private preserve of musicologists; the dominance of singers by a single master who believes that he knows the one true way; the perception that chant is only for monasteries but not regular people; and on and on.

Here we see Guido embodying the same principle that drove the Solesmes monastery at the early part of the restoration efforts in the late 19th century: innovation in order to preserve, teach, and distribute this glorious music as widely as possible, in the service of the faith. They had the right ordering of priorities: technical innovation in the service of preserving universal truth.

This story illustrates a general principle in the history of technology. There does seem to be a real pattern here. There are those who believe that innovation is for everyone and ought to be accessible to all – that everyone should be permitted to have access to the forms and structures that make for progress. This side loves technical innovation not for its own sake but in the service of great goals.

Then there is the other side, which is reactionary, hates technical innovation, wants to reserve technical forms to a tiny elite, fears freedom, detests the idea of human choice, and advances a kind of Gnosticism over technical forms – always wants it to remain the private preserve of the elect who appoint each other and operate as a kind of guild. This Gnostic guild wants to guard and exclude and privatize, and the people are ultimately their enemy.

This perspective hearkens back to the ancient world where priests served the philosopher kings, and sparingly hand out religious truth to the masses based on what they believe they should know in the service of their agenda.

One can detect these two tendencies from all ages.

Guido didn’t patent his innovation. He didn’t copyright his music. The legal means weren’t available to him, and he wouldn’t have used them if they were. His whole point was to uplift the whole of the culture. Keeping his innovation to himself would have been contrary to that goal. As a result, his use of the staff spread widely. His innovation was infinitely reproducible, and it changed the world.

In his pedagogy, he adapted an existing song to illustrate the scale: Ut Queant Laxis, a hymn to St. John the Baptist, who was then considered the patron saint of singers. On the first syllable of each ascending note, the words were Ut, Re, Mi, Fa, Sol – the very foundation of music pedagogy to this day: do, re, mi, etc.

A millennium later, Guido’s innovation is still with us!

Here is a model for our Arial and all Arial.

March 4, 2009

Jeffrey Tucker [send him mail] is editorial vice president of www.Mises.org.

Copyright © 2009 by LewRockwell.com. Permission to reprint in whole or in part is gladly granted, provided full credit is given.

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Space is real

In the late eighties, I was at a church service on the Sunday when they talk about the budget (usually once a year in mainline Protestant churches). After the talk, ten members of the congregation marched up front each carrying a sign with one letter on it. They arranged themselves to spell

I realized then that the use of computers had changed the way we think about spaces. Now a space is something (a particular character) instead of nothing.

This meant Mathematica could use the space as a symbol for multiplication, and everyone under 50 understands it.

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Typical examples

There is a sense in which a robin is a typical bird and a penguin is not a typical bird. (See Reference 1.) Mathematical objects are like this and not like this.

A mathematical definition is simply a list of properties. If an object has all the properties, it is an example of the definition. If it doesn’t, it is not an example. So in some basic sense no example is any more typical than any other. This is in the sense of rigorous thinking described in the abstractmath chapter on images and metaphors.

In fact mathematicians are often strongly opinionated about examples: Some are typical, some are trivial, some are monstrous, some are surprising. A dihedral group is more nearly a typical finite group than a cyclic group. The real numbers on addition is not typical; it has very special properties. The monster group is the largest finite simple group; hardly typical. Its smallest faithful representation as complex matrices involves matrices that are of size 196,883. The monster group deserves its name. Nevertheless, the monster group is no more or less a group than the trivial group.

People communicating abstract math need to keep these two ideas in mind. In the rigorous sense, any group is just as much a group as any other. But when you think of an example of a group, you need to find one that is likely to provide some information. This depends on the problem. For some problems you might want to think of dihedral groups. For others, large abelian p-groups. The trivial group and the monster group are not usually good examples. This way of thinking is not merely a matter of psychology, but it probably can’t be made rigorous either. It is part of the way a mathematician thinks, and this aspect of doing math needs to be taught explicitly.

Reference 1. Lakoff, G. (1986), Women, Fire, and Dangerous Things. The University of Chicago Press. He calls typical examples “prototypical”.

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Mathematical definitions

The definition of a concept in math has properties that are different from definitions in other subjects:

• Every correct statement about the concept follows logically from its definition.
• An example of the concept fits all the requirements of the definition (not just most of them).
• Every math object that fits all the requirements of the definition is an example of the concept.
• Mathematical definitions are crisp, not fuzzy.
• The definition gives a small amount of structural information and properties that are enough to determine the concept.
• Usually, much else is known about the concept besides what is in the definition.
• The info in the definition may not be the most important things to know about the concept.
• The same concept can have very different-looking definitions. It may be difficult to prove they give the same concept.
• Math texts use special wording to give definitions. Newcomers may not understand that the intent of an assertion is that it is a definition.

How many college math teachers ever explain these things?

I will expand on some of these concepts in future posts.

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Twin Stories I: Political Correctness

1. Cannon to the left of us. A few years ago, you could be rebuked in Glasgow if you asked for black coffee. You had to ask for “coffee without cream”. This reportedly happened in Glasgow according to the Global Language Monitor.

Apparently the Enforcers were sometimes a bit confused. The report said that when one person in Glasgow asked for coffee without milk, the server said: “We don’t have any milk, you’ll have to have coffee without cream.”

I remember reading about an edict from the London County Council saying the same thing, but haven’t found a reference to it on the web. As I remember it, Londoners made a big joke out of it. When they would go outside on a moonless night, they would say, “It’s a night without cream.”

By the way, there is a problem with tea. “Black tea” refers to the color of the leaves, not to any milk withheld. (*Sigh*) The English language has as many traps for us as, well, math.

2. Cannon to the right of us. The American Life League has objected to the following press release from Krispy Kreme Doughnuts, Inc:

Krispy Kreme Doughnuts, Inc. (NYSE: KKD) is honoring American’s sense of pride and freedom of choice on Inauguration Day, by offering a free doughnut of choice to every customer on this historic day, Jan. 20. By doing so, participating Krispy Kreme stores nationwide are making an oath to tasty goodies — just another reminder of how oh-so-sweet “free” can be.

The League says: “Celebrating his inauguration with “Freedom of Choice” doughnuts – only two days before the anniversary of the Supreme Court decision to decriminalize abortion – is not only extremely tacky, it’s disrespectful and insensitive and makes a mockery of a national tragedy.”

Mark Liberman of The Language Log has something to say about this. Wait till he hears that Krispy Kreme backed down and stopped using the word “choice”.

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Parameters

I have been putting off writing the section on parameters in abstractmath.org because I don’t very well understand the connections between the different ways the word is used.

A parametrized family of functions, say f_a(x) = x^2+a, is easily handled by using the adjunction that makes the category of sets cartesian closed. That is, the three ways of looking at it, function of x parametrized by a, a function of a parametrized by x, and a function of two variables, are all naturally equivalent in a completely satisfactory way.

When you talk about taking a curve defined by an equation, e.g. x^2 + y^2 =1, and parametrizing it, that is a different situation; I have never thought about it from a categorical point of view but it is not hard to explain at the level of abstractmath.org.

What I don’t understand is if there is any real connection between these two meanings. I would welcome comments!

There are other meanings of “parameter”. One is the usage, “An important parameter of a finite group is its order.” I am not sure how often this occurs, and I look at the usage with some disapproval. The order of a finite group is more like an invariant (e.g. under isomorphism) than a parameter. Now do I have to write about invariants, too? Ye gods, do I have to do everything myself?

Another usage is in programming languages, where it can be analyzed simply as a variable. But they have different kinds of parameters that are implemented differently.

What I really need to do is spend a day searching for the word in JStor. I swore when I started doing abstractmath.org I wasn’t going to do any more lexicographical research (after the Handbook.). But maybe looking around for a few hours won’t hurt.

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Automatic spelling using different systems

In a previous post, I discussed how we should read in the near future, with an electronic Reader that allowed searching and access to all texts. One particular service by the Reader that it is reasonable to expect is to provide the text in the spelling system you want to use. Of course, each spelling system would have to be implemented by someone or some organization.

This is already known to be possible, since there is software that will read printed text to you. Software that vocalizes has to identify how each word is pronounced. If that can be done it should be reasonable to hope to have software that implements some alternative spelling system.

As I mentioned in my post on automatic spelling reform, this vocalizing software is pretty good: For example, the English reader in Excel reads “I have read [pronounced “red”] the book” and “I will read [“reed”] the book” correctly. The software can never be perfect (people can’t read every text completely correctly), but it is already good enough.

Imagine you could push a button and get:

a. The text, in any language, printed in the International Phonetic Alphabet.
b. Russian text printed with accents added.
c. Latin text with long marks added.
d. French text printed with dots under silent letters.
e. English text printed in American spelling, or in British spelling.
f. English text printed using your favorite spelling reform.
g. Mandarin in either standard characters or simplified characters.
h. Mandarin in some phonetic transcription that shows tones. (This is likely to be much harder to carry out than the previous tasks. Maybe it can’t be done satisfactorily.)

One type of reform for English would be text printed with the standard spelling but with diacritic marks added to indicate the pronunciation. This would keep the standard spelling in front of someone learning English but would clue her in on how to pronounce it as well. Here is an example I cooked up:

This is just a demonstration of what ought to be possible. Some features of this particular system:

a. Dot under a letter indicates it is silent.
b. Dot over a vowel indicates it is schwa.
c. Long and short vowels indicated by usual American dictionary symbols.
d. Other common vowel pronunciations indicated by diacritics.
e. Rare pronunciations, as in “to” and “of”, indicated by a small letter written above.
f. Each consonant has zero or one pronunciation that is unmarked.
g. All pronunciations are marked for vowels.

None of these choices may be ideal. I made choices that I hope require as little memorization as possible. For example, the reader need not memorize the rules for the two pronunciations of the past ending –ed or the plural ending –s. The example above does not indicate stress, but that should be done too.

Note that this system is not meant to be written or typed by anyone. It will be created by a computer program running on text in ordinary English spelling.

To provide all this would provide lots of coding and each specific ability might have to be purchased separately, although it seems to me that it would be in the interest of a country trying to promote its language to provide appropriate software free.

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Reading in the near future

I find reading, especially reading technical exposition, is frustrating because there are so many ways it could be made better in this age of computers. This post will suggest some of the improvements that could be made. I hope dozens of people are already working hard on these ideas. If not, then get busy!

These ideas are based on the following assumptions, which I think could reasonably be realized in the next few years. Indeed, most of them already exist in some form.

  1. We will read on a Reader that is some sort of lightweight computer not much bigger than a book. It may even be a piece of paper that displays the text electronically, but it must have a method for input.
  2. The Reader will display text and pictures in color.
  3. The Reader will allow searching the book and the internet.
  4. The Reader will not have to be connected to a power source all the time.
  5. The Reader will have the computing and storage capacity of a present-day laptop.
  6. The Reader will have approximately the resolution that a paper book has.

Amazon’s Kindle has some of these properties. The idea is to be able to read text in the same comfortable way we read a book, but with the ability to do other things, some of them problematical, that I will talk about here.

Links and searching

Searching and links are widely used now and I think they are the best thing since sliced bread. One of the sites that makes the best use of links is Wikipedia, and some of the news blogs do a pretty good job too.

In these Modern Times it is annoying to have to read a dead old pack of paper (dopop, pronounce “dope-op”) instead of an interactive book that better fits the habits of us grasshoppers. Technical books are the most annoying, but even novels sometime make a reference to a character not mentioned for awhile or to some obscure rock group, and it would be great to stop right there and find out more about them. (By the way, in Ancient Times – 1980 – mystery novelists had a Rule that no two main characters in a novel would have names starting with the same letter. This is no longer true. Why?)

It would be straightforward to provide the ability to search the book you are reading, as well as to mark some phrase and search the internet for it. On the other hand, links in general have to be supplied by the author or at least by someone familiar with the ideas in the text. That is labor intensive. A Wikipedia-like solution to this problem would be to allow other people to add links either freely or under editorial control. (“Read Hamlet, with links provided by the Helsingborg University History Department” – or “with links provided by Linkopedia”.)

Wikipedia itself seems to have people who spend some time adding links to its own articles, and I think in some cases they even have robots to do it. For example, most years in recorded history have an article devoted to it, and a mention of a year in some other article seems to quickly acquire a link to the article about that year.

Every piece of text should be obtainable and most of them should be free

One of my practices in writing abstractmath.org is to provide links that are mostly to electronic texts. (There are a couple of lists of dopops here and there in it.) That is because most younger people and many older ones have the following characteristics:

  • They will not go to the library to look something up in a book. Indeed, I won’t do that except in desperate circumstances. Recently, I visited a college math department and asked them if they wanted some of my books. They had a departmental library of books suitable for students but their reply was they wanted to get rid of what they had – the students did not use them.
  • They do not have the money to buy most books. Peter Johnstone’s two-volume book on toposes costs $576! Even Timothy Gower’s new Princeton Companion to Mathematics costs $100, which most math students can’t afford if it is not required by a course.

I have been frustrated many times by clicking on a link to a math or linguistics article only to discover that I have to Pay Money to read it. This happens even though Case Western Reserve University (where I am professor emeritus) makes many journals free for downloading. I have notices that this situation is worse in math education and in philosophy than it is in math; indeed many researchers in those fields don’t post their papers on their own websites (many others do). This is not the way Modern Scientific Research should be run!

This problem does not have an easy solution and I am not clever enough to propose one. I am aware that some universities as well as Google are working on this, but it will be a long time coming. People who do research or expository writing should be paid for what they do, but I don’t know how to arrange it so that some poor young person named Ramanujan can look up the latest research on Eisenstein series and pay only an anna for it.

I have more to say, but it must wait until a later post.

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Renaming technical concepts

Here are some thoughts about the names of mathematical objects. I don’t make recommendations about how to name things; I am just analyzing some aspects of how names are given and used. I have written about some of these ideas in abstractmath.org, under Names and Semantic Contamination.

Some objects have names from Latin or Greek, such as “matrix” or “homomorphism”, that don’t give the reader a clue as to what they mean, unless the reader has a substantial vocabulary of Latin and Greek roots.

Some are named after people, such as “Riemann sum” and “Hausdorff space”. They don’t have suggestive names either. Well, they suggest that the person they are named after discovered it, but that is not always true; for example L’Hôpital’s rule was discovered by some Bernoulli or other.

You could call both types of names learnèd names.

Others concepts have names that are English words, such as “slope” or “group”. I will call them commonword names. Some of these suggest some aspect of their meaning; “slope” certainly does and so do “truth set” and “variable”. But “group” only suggests that it is a bunch of things; it does not suggest the primary group datum, namely the binary operation. Not only that, but too many commonword names suggest the wrong ideas, for example “real” and “imaginary”.

In contrast, learnèd names don’t usually suggest the wrong things, but they can and do intimidate people.

One upon a time, Roger Godement and Peter J. Huber came up with an important construction for adjunctions in category theory. They called it the standard construction. That commonword name communicates very little. They named it that because it kept coming up in their work. Well, derivatives and integrals are each more deserving of the name. Eilenberg and Moore renamed them triples, which suggests nothing useful except that the concept is given by three data. Well, so are rings. Saunders Mac Lane renamed them again, calling them monads, a learnèd name that suggests nothing except possibly an illusory connection with a certain philosophical concept.

Perhaps learnèd names are better, since they don’t suggest the wrong things. In that case “monad” is better than the other names, but I have a personal prejudice since I have co-authored two books that called them “triples”.

Some writers of popularizations of math and science avoid using the names of certain concepts that suggest the wrong things. In Symmetry and the Monster, by Mark Ronan, the author talks about “atoms of symmetry” instead of “simple groups”, on the grounds that “simple group” is misleading (the Monster Group is simple!) and doesn’t suggest the important property they have. He called involutions “mirror symmetries”, which is appopriately suggestive. Centralizers of involutions became “cross-sections”, which I don’t understand; it must be based on a way of thinking about them that I am not aware of. He doesn’t change the name of the Monster Group, though; that is a terrific name.

Frank Wilczek, in The Lightness of Being, used “core theory” for the theory in particle physics that is commonly called the “standard model”. I suppose that really is more suggestive of its current place in physics, since as far as I know all modern theories build on it.

Marcus du Sautoy, in The Music of the Primes (HarperCollins, 2003), also introduces new names for concepts. His description of the meanings of the many concepts he discusses uses some great metaphors that clearly communicate the ideas. He talks about the “landscape” of the zeta function, how Riemann “extended the landscape to the west”, and refers to its zeroes as its places “at sea level”. But he also calls them by their normal mathematical name “zeroes”. (I could have done without his reference to the “ley line of zeroes”.) He refers to modular arithmetic as “clock calculators” and in one parenthetical remark explains that modular arithmetic is what he means.

Summary

The problem with learnèd names is that they don’t give you a clue about the meaning, and for some students (co-intimidators) they induce anxiety.

The problems with commonword naming are that what a commonword name suggests can give you only one connotation and it is hard to find the best one, and almost any choice produces a metaphor that suggests some incorrect ideas. Furthermore, beginning abstract math students are way too likely to be stuck on one metaphor per mathematical object and commonword names only encourage this behavior. I have written about that here and here.

One problem with popular renaming is that the interested reader has a hard time searching the internet for more information about it, unless she noticed that one place in the book where the fact that it was not the standard name was mentioned.

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