Category Archives: language of math

language of math, math

Studying math using linguistics

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The Handbook and part of the abstractmath website involve studying math from the point of view of linguistics.  I thought I was all by myself until K. L. O’Halloran’s book came out:

O’Halloran, K. L. (2005), Mathematical Discourse: Language, Symbolism And Visual Images. Continuum International Publishing Group.

She is a semioticist.

Now another new book also looks at math from a linguistic point of view, this time in connection with how mathematical discourse works in other languages:

Barton, Bill (2009), The Language of Mathematics: Telling Mathematical Tales. Springer.

It is described in Reidar Mosvold’s math ed blog.

By the way, lots of books and articles that use phrases such as “the language of math” are not written from a linguistics point of view at all.

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abstractmath.org, language, language of math, math, representations, understanding math

Mathematical definitions

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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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abstractmath.org, exposition, Handbook, language of math, math, proof, understanding math

Abstractmath.org after four years

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I have been working on the abstractmath website for about four years now (with time off for three major operations). Much has been written, but there are still lots of stubs that need to be filled in. Also much of it needs editing for stylistic uniformity, and for filling in details and providing more examples in some hastily written sections that read like outlines. Not to mention correcting errors, which seem to multiply when I am not looking. The website consists of four main parts and some ancillary chapters. I will go into more detail about some of the parts in later articles.

The languages of math. This is a description of mathematical English and the symbolic language of math (which are two different languages!) with an emphasis on the problems they cause people new to abstract math (roughly, math after calculus). At this point, I have completed a fairly thorough edit of the whole chapter that makes it almost presentable. Start with the Introduction.

Proofs. Mathematical proofs are a central problem for abstract math newbies. People interested in abstract math must learn to read and understand proofs. A proof is narrated in mathematical English. A proof has a logical structure. The reader must extract the logical structure from the narrative form. The chapter on proofs gives examples of proofs and discusses the logical structure and its relationship with the narration. The introduction to the chapter on proofs tells more about it.

Understanding math. There are certain barriers to understanding math that are difficult to get over. Mathematicians, math educators and philosophers work on various aspects of these problems and this chapter draws on their work and my own observations as a mathematician and a teacher.

All true statements about a math object must follow from the definition. That sounds clear enough. But in fact there are subtleties about definitions teachers may not tell students about because they are not aware of them themselves. For example, a definition can really mislead you about how to think about a math object.

The section on math objects breaks new ground (in my opinion) about how to think about them. I also discuss representations and models and images and metaphors (which I think is especially important), and in shorter articles about other topics such as abstraction and pattern recognition.

Doing math. This chapter points out useful behaviors and dysfunctional behaviors in doing math, with concrete examples. Beginners need to be told that when proving an elementary theorem they need to rewrite what is to be proved according to the definitions. Were you ever told that? (If you went to a Jesuit high school, you probably were.) Beginners need to be told that they should not try the same computational trick over and over even though it doesn’t work. That they need to look at examples. That they need to zoom in and out, looking at a detail and then the big picture. We need someone to make movies illustrating these things.

These other articles are outside the main organization:
Topic articles. Sets, real numbers, functions, and so on. In each case I talk just a bit about the topic to get the newbie over the initial hump.
Diagnostic examples. Examples chosen to evoke a misunderstanding, with a link to where it is explained. This needs to be greatly expanded.
Attitudes. This explains my point of view in doing abstractmath.org. I expect to rewrite it.

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exposition, language, language of math, math, representations

Renaming technical concepts

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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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exposition, language, language of math, math

Two science books in the modern expository style

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Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, By Mark Ronan. Oxford University Press, 2006.

This is an excellent way for the non-mathematician to learn about what is going on in the attempt to classify symmetries by discovering all the finite “simple groups”. The last one found was the Monster Group and the classification was completed in 1982. This book is full of fascinating information about how this came about and the tantalizing connections between physics and the Monster that have been discovered since.

The Lightness of Being, by Frank Wilczek. Basic Books, 2008

This book is an exposition for the layman of the modern theory of particle physics – the Standard Model, Supersymmetry and other possible extensions. I recommend it for anyone interested in the subject.

These two books are examples of the modern trend in science expository writing, using metaphors, anecdotes, graphs and speculation to try to communicate an understanding of how the scientists involved think about the subject and what their motivations are. Ronan and Wilczek use much the same approach that I have been using in abstractmath.org and it has made me think about what works and what doesn’t.

I will be writing about my reactions to the writing in such books in future posts.

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language, language of math, math, representations

Default meanings

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Concerning Michael Barr’s comments on my post on terminology, I remember going to a meeting of topologists in 1965 or 1966 in which people kept spouting nonsense about free groups. The reason it was nonsense was that they were talking about free abelian groups without saying so. That may have been the first time I became aware of default meaning in different groups of mathematicians.

I became aware of default meanings in ethnic and regional groups long before that, when I joined the Air Force after never having been outside the deep south and discovered that other people thought “sweet milk” and “ink pen” were weird things to say.

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category theory, language of math, math

Cribles

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Mark Meckes’ comments on technical words in English reminds me of an incident from the Ancient Days, namely 1966. Well, 1966-ish. A visiting Belgian mathematician in my department talked about category theory. One concept that came up was that of “crible”, which was its name in French. It is a family of arrows with a common target (the sources can vary). He couldn’t think of the English translation of “crible” so he said something like this: “The best way I can describe this is to think of a soldier in the trenches in World War I who suddenly stands up and is shot full of holes by many machine gun bullets.”

We were completely baffled by this explanation. Is there an English word that describes a person with lots of bullet holes? You can see where the picture comes from by thinking of the target as a person with lots of arrows stuck in him, like Saint Sebastian, or Hagar the Horrible on a bad day.

The English word he wanted is “sieve” and that is the usual name of the concept today. In the sixties, many English speaking mathematicians called it “crible” but that usage died out as far as I know. A few tried to pronounce it the French way, but no one understood them, so they spelled it, and then most people said “cribble”.

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language, language of math, math, representations

Set notation

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Students commonly think that the notation “{Ø}” denotes the empty set. Many secondary school teachers think this, too.

Mistakes in reading math notation occur because the reader’s understanding of the notation system is different from the author’s. The most common bits of the symbolic language of math have fairly standard interpretations that most mathematicians agree on most of the time. Students develop their own non-standard interpretation for many reasons, including especially cognitive dissonance from ordinary usage and ambiguous statements by teachers.

I believe (from teaching experience) that when a student sees “{1, 2, 3, 5}” they think, “That is the set 1, 2, 3 and 5”. The (incorrect) rule they follow is that the curly braces mean that what is inside them is a set. So clearly “{Ø}” is the empty set because the symbol for the empty set is inside the braces.

However, “1, 2, 3 and 5” is not a set, it is the names of four integers. A set is not its elements. It is a single mathematical object that is different from its elements but determined exactly by what its elements are. The correct understanding of set notation is that what is inside the braces is an expression that tells you what the elements of the set are. This expression may be a list, as in “{1, 2, 3, 5}”, or it may be a statement in setbuilder format, as in “{x x > 1}”. According to this rule, “{Ø}” denotes the singleton set whose only element is the empty set.

This posting is based on the belief that that mathematical notation has a standard, (mostly) agreed-on interpretation. I made this attitude explicit in the second paragraph. Teachers rarely make it explicit; they merely assume it if they think about it at all.

The student’s interpretation is a natural one. (Proof: So many of them make that interpretation!) Did the teacher tell the student that math notation has a standard interpretation and that this is not always what an otherwise literate person would expect? Did the teacher explain the specific and rather subtle rule about set notation that I described two paragraphs above? If not, the student does not deserve to be ridiculed for making this mistake.

Many people who get advanced degrees in math understood the correct rule for set notation when they first learned it, without having to be told. Being good at abstract math requires that kind of talent, which is linguistic as well as mathematical. Most students in abstract math classes are not going to get an advanced degree in math and don’t have that talent. They need to be taught things explicitly that the hotshots knew without being told. If all math teachers had this attitude there would be fewer people who hate math.

PS: My claim about how students think that leads them to believe that “{Ø}” denotes the empty set is a testable claim. There are many reports in the math ed literature from investigators who have been able to get students to talk about what they understand, for example, while working a word problem, but I don’t know of any reports about my assertion about “{Ø}” .

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language of math

Special cases

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Adam Barr’s blog “Proudly Serving My Corporate Masters” has a post on the controversy over whether a Square class should be defined in object oriented programming as inheriting from a Rectangle class. You can find lots of argument about that by typing “square rectangle Liskov” (without the quotes!) into Google. This controversy is based on technical aspects of OOP, but the controversy about whether a square is a rectangle has been around for years.

According to modern mathematical usage, a square is a special case of a rectangle. (Euclid did not include squares in rectangles, although I can’t give a reference to this.) This post concerns some of the issues in math and logic connected with special cases.

A rectangle is given by two parameters, the length and the height. Call them L and H. A square is the special case of a rectangle satisfying L = H. (Note: I am talking about a rectangle independently of how it drawn in the plane. You need six parameters to define a rectangle embedded in the plane.)

Rectangles and squares are an example of the following phenomenon: A certain family of math objects is defined by parameters. A subfamily is defined by setting two (or more) of the parameters equal, or by fixing the value of one of the parameters. Such a subfamily may be called an equationally defined family or a variety. Sometimes such a subfamily is said to be degenerate.

Mathworld refers to a square as a degenerate rectangle. This says something interesting about the images and metaphors we use in math. I would have said a square is a specially nice kind of rectangle. Saying it is degenerate disses it. Of course, the technical definition of “degenerate” says nothing about a square’s moral failings, but the word’s connotations are negative.

Not all special cases fit this equational pattern. For many mathematical structures, the definition allows the structure with empty underlying set to be an example. For example, a topological space is allowed to be empty. If you define semigroup to be a set with an associative binary operation, that definition allows the underlying set to be empty, but in fact some texts explicitly add a requirement that excludes the empty set. This has been the source of considerable irritation between researchers in universal algebra and researchers in categorical model theory.

If your definition of the structure includes the requirement that some element with a certain property exists, then the structure must be nonempty. That’s why groups are nonempty – they have to have an identity. So the behavior of special cases depends on the logical form of the definition. In particular you could define rectangles as having two parameters L and H together with the requirement that L not equal H. Then squares would not be rectangles. But that is not the way we do it nowaways, not for rectangles.

But in some other math definitiions, we rule out two parameters being equal. The definition of field always requires 0 not equal 1, although Mathworld appears to add the requirement only as an afterthought. The definition of Boolean algebra sometimes requires that T not equal F and sometimes not. MathWorld gives several definitions, some allowing T = F and some not. Wikipedia’s definition of Boolean algebra said until recently that a Boolean algebra has “…two elements called 0 and 1…” It did not say they must be distinct in the axiom but in the examples it said that the two-element Boolean algebra is the simplest example. Since 18 October 2006 Wikipedia has said they must be “distinct.” This change was added by SixWingedSeraph, who is either a close relative of mine or not depending on your attitude toward special cases.

There must be some fairly deep seated psychology underlying the decision to allow some special cases and exclude others. Certainly the naïve abstract math newbie tends to assume (subconsciously) that special cases should be excluded, with the consequence that the uppity math majors (and sometimes the teacher) disses the student as stupid, etc.

Note: When I wrote this, Microsoft Word put a diaeresis over the “I” in “naïve” without asking me first. Who said it could?

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language, language of math, math

The languages of mathematics

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Conjecture: Mathematical English (ME) and the symbolic language of math (SL) are two distinct languages, not dialects of the same language.

I have asserted this in several places (Handbook, abstractmath.org) but I am not a linguist and it could be that linguists would disagree with this conjecture, or that the study of a mathematical corpus would reveal that another theoretical take on the situation would be more appropriate.

Some relevant points are listed below. I intend to expand on them in later posts.

1) Is ME a dialect of English or a register of English? Or does it have some other relationship to English?

2) ME appears to have several dialects or registers. One register is that used for what mathematicians call “formal proofs”. These are not formal in the sense of first order predicate logic, but their language is constrained, with the intent of making it easier to see the logical structure of the argument. Another register is that of “intuitive [or informal] explanations”. This is more like standard English.

3) The SL is clearly not a spoken language. It is a two-dimensional written language using symbols from English and other languages and some symbols native only to math. People do try to speak formulas aloud occasionally but this is well known to be difficult and can be done successfully only for fairly simple expressions.

4) There are other non-spoken languages such as ASL for example. I don’t know whether there are other non-spoken languages that are written. I don’t think dead languages count.

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