Category Archives: Handbook

category theory, exposition, Handbook, language, language of math, math, understanding math

Mathematical and linguistic ability

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This post uses MathJax.  If you see mathematical formulas with dollar signs around them, or badly formatted formulas, try refreshing the screen.  Sometimes you have to do it two or three times.

Some personal history

When I was young, I was your typical nerdy geek.  (Never mind what I am now that I am old.)

In high school, I was fascinated by languages, primarily by their structure.  I would have wanted to become a linguist if I had known there was such a thing.  I was good at grasping the structure of a language and read grammars for fun. I was only pretty good at picking up vocabulary. I studied four different languages in high school and college and Turkish when I was in the military.  I know a lot about their structure but am not fluent in any of them (possibly including English).

After college, I decided to go to math grad school.  This was soon after Sputnik and jobs for PhD's were temporarily easy to get.

I always found algebra easy.  When I had to learn other symbolic languages, for example set theory, first order logic, and early programming languages, I found them easy too.  I had enough geometric insight that I did well in all my math courses, but my real strength was in learning languages. 

When I got a job at (what is now) Case Western Reserve University, I began learning category theory and a bit of cohomology of groups. I wrote a paper about group automorphisms that got into Transactions of the AMS.  (Full disclosure: I am bragging). 

The way Saunders Mac Lane did cohomology, he used "$+$" as a noncommutative operation.  No problem with that, I did lots of calculations in his notation.  In reading category theory I learned how to reason using commutative diagrams.  That is radically different from other math — it isn't strings of symbols — but I caught on. I read Beck's thesis in detail.  Beck wrote functions on the right (unlike Mac Lane) which I adapted to with no problem.  In fact my automorphisms paper and many others in those days was written with functions on the right. 

Later on in my career, I learned to program in Forth reasonably well. It is a reverse Polish language. Then (by virtue of summer grants in the 1990's) to use Mathematica, which I now use a lot:  I am an "experienced" user but not an "expert".

Learning foreign languages in studying math

I taught mostly engineering students during my 35 years at CWRU (especially computer engineering). When I used a text (including my own discrete math class notes) some students pleaded with me not to use $P\wedge Q$ and $P \vee Q$ but let them use $PQ$ and $P+Q$ like they did in their CS courses.  Likewise $1$ and $0$ instead of T and F.  Many of them simply could not switch easily between different codes.  Similar problems occurred in classes in first order logic. 

In the early days of calculators when most of them were reverse Polish, some students never mastered their use. 

These days, a common complaint about Mathematica is that it is a difficult language to learn; at the MAA meeting in Madison (where I am as I write this) they didn't even staff a booth.  Apparently too many of the professors can't handle Mathematica.

I gave up writing papers with functions on the right because several professional mathematicians complained that they found them too hard to read. I guess not all professional mathematicians can switch code easily, either. 

There are many great mathematicians whose main strength is geometric understanding, not linguistic understanding.  Nevertheless, to become a mathematician you have to have enough linguistic ability to learn…

Algebra

The big elephant in the room is ordinary symbolic algebra as is used in high school algebra and precalculus.  This of course causes difficulty among first year calculus students, too, but college profs are spared the problem that high school teachers have with a large percentage of the students never really grasping how algebra works.  We don't see those students in STEM courses.

It is surely the case that algebra is a difficult and unintuitive foreign language.  I have carried on about this in my stuff about the languages of math in my abstractmath site. 

Some students already in college don't really understand expressions such as $x^2$.  You still get some who sporadically think it means $2x$.  (They don't always think that, but it happens when they are off guard.)  Lots of them don't understand the difference between $x^2$ and $2^x$.

In complicated situations, students don't grasp the difference between an expression such as $x^2+2x+1$ and a statement like $x^2+2x+1=0$.  Not to mention the difference between the way $x^2+2x+1=0$ and $x^2+2x+1=(x+1)^2$ are different kinds of statements even though the difference is not indicated in the syntax.

There are many irregularities and ambiguities (just like any natural language — the symbolic language of math is a natural language!): consider $\sin xy$, $\sin x + y$, $\sin x/y$.  (Don't squawk to me about order of operators.  That's as bad as aus, außer, bei, mit, zu.  German can't help it, but mathematical notation could.)

One monstrous ambiguity is $(x,y)$, which could be an ordered pair, the GCD, or an open interval.  I found an example of two of those in the same sentence in the Handbook of Mathematical Discourse, and today in a lecture I saw someone use it with two meanings about three inches apart on a transparency.

Anyway, the symbolic language of math is difficult and we don't teach it well.

Structuring calculations

There are other ways to structure calculations that are much more transparent.  Most of them use two or three dimensions.

  • Spreadsheets: It is easy to approximate the zeros of a function using a spreadsheet and changing the input till you get the value near zero. Why can't middle school students be taught that?
  • Bret Victor has made suggestions for easy ways to calculate things.
  • My post Visible Algebra I suggest a two-dimensional approach to putting together calculations.  (There are several more posts coming about that idea.)
  • Mathematica interactive demos could maybe be provided in a way that would allow them to be joined together to make a complicated calculation. (Modules such as an inverse image constructor.)  I have not tried to do this.

A lot of these alternatives work better because they make full use of two dimensions.  Toolkits could be made for elementary school students (there are some already but I am not familiar with them).  

It is impractical to expect that every high school student master basic algebraic notation.  It is difficult and we don't know how to teach it to everyone. With the right toolkits, we could provide everyone, not just students, to put together usable calculations on their computer and experiment with them.  This includes working out the effect of different payment periods on loans, how much paint you need for a room, and many other things.

STEM students will still have to learn algebraic notation as we use it now.  It should be taught as a foreign language with explicit instruction in its syntax (sentences and terms, scope of an operator, and so on), ambiguities and peculiarities.

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Conceptual blending

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This post uses MathJax.  If you see formulas in unrendered TeX, try refreshing the screen.

A conceptual blend is a structure in your brain that connects two concepts by associating part of one with part of another.  Conceptual blending is a major tool used by our brain to understand the world.

The concept of conceptual blend includes special cases, such as representations, images and conceptual metaphors, that math educators have used for years to understand how mathematics is communicated and how it is learned.  The Wikipedia article is a good starting place for understanding conceptual blending. 

In this post I will illustrate some of the ways conceptual blending is used to understand a function of the sort you meet with in freshman calculus.  I omit the connections with programs, which I will discuss in a separate post.

A particular function

Consider the function $h(t)=4-(t-2)^2$. You may think of this function in many ways.

FORMULA:

$h(t)$ is defined by the formula $4-(t-2)^2$.

  • The formula encapsulates a particular computation of the value of $h$ at a given value $t$.
  • The formula defines the function, which is a stronger statement than saying it represents the function.
  • The formula is in standard algebraic notation. (See Note 1)
  • To use the formula requires one of these:
    • Understand and use the rules of algebra
    • Use a calculator
    • Use an algebraic programming language. 
  • Other formulas could be used, for example $4t-t^2$.
    • That formula encapsulates a different computation of the value of $h$.

TREE: 

$h(t)$ is also defined by this tree (right).
  • The tree makes explicit the computation needed to evaluate the function.
  • The form of the tree is based on a convention, almost universal in computing science, that the last operation performed (the root) is placed at the top and that evaluation is done from bottom to top.
  • Both formula and tree require knowledge of conventions.
  • The blending of formula and tree matches some of the symbols in the formula with nodes in the tree, but the parentheses do not appear in the tree because they are not necessary by the bottom-up convention.
  • Other formulas correspond to other trees.  In other words, conceptually, each tree captures not only everything about the function, but everything about a particular computation of the function.
  • More about trees in these posts:

GRAPH:

$h(t)$ is represented by its graph (right). (See note 2.)

  • This is the graph as visual image, not the graph as a set of ordered pairs.
  • The blending of graph and formula associates each point on the (blue) graph with the value of the formula at the number on the x-axis directly underneath the point.
  • In contrast to the formula, the graph does not define the function because it is a physical picture that is only approximate.
  • But the formula does represent the function.  (This is "represents" in the sense of cognitive psychology, but not in the mathematical sense.)
  • The blending requires familiarity with the conventions concerning graphs of functions. 
  • It sets into operation the vision machinery of your brain, which is remarkably elaborate and powerful.
    • Your visual machinery allows you to see instantly that the maximum of the curve occurs at about $t=2$. 
  • The blending leaves out many things.
    • For one, the graph does not show the whole function.  (That's another reason why the graph does not define the function.)
    • Nor does it make it obvious that the rest of the graph goes off to negative infinity in both directions, whereas that formula does make that obvious (if you understand algebraic notation).  

GEOMETRIC

The graph of $h(t)$ is the parabola with vertex $(2,4)$, directrix $x=2$, and focus $(2,\frac{3}{4})$. 

  • The blending with the graph makes the parabola identical with the graph.
  • This tells you immediately (if you know enough about parabolas!) that the maximum is at $(2,4)$ (because the directrix is vertical).
  • Knowing where the focus and directrix are enables you to mechanically construct a drawing of the parabola using a pins, string, T-square and pencil.  (In the age of computers, do you care?)

HEIGHT:

$h(t)$ gives the height of a certain projectile going straight up and down over time.

  • The blending of height and graph lets you see instantly (using your visual machinery) how high the projectile goes. 
  • The blending of formula and height allows you to determing the projectile's velocity at any point by taking the derivative of the function.
  • A student may easily be confused into thinking that the path of the projectile is a parabola like the graph shown.  Such a student has misunderstood the blending.

KINETIC:

You may understand $h(t)$ kinetically in various ways.

  • You can visualize moving along the graph from left to right, going, reaching the maximum, then starting down.
    • This calls on your experience of going over a hill. 
    • You are feeling this with the help of mirror neurons.
  • As you imagine traversing the graph, you feel it getting less and less steep until it is briefly level at the maximum, then it gets steeper and steeper going down.
    • This gives you a physical understanding of how the derivative represents the slope.
    • You may have seen teachers swooping with their hand up one side and down the other to illustrate this.
  • You can kinetically blend the movement of the projectile (see height above) with the graph of the function.
    • As it goes up (with $t$ increasing) the projectile starts fast but begins to slow down.
    • Then it is briefly stationery at $t=2$ and then starts to go down.
    • You can associate these feelings with riding in an elevator.
      • Yes, the elevator is not a projectile, so this blending is inaccurate in detail.
    • This gives you a kinetic understanding of how the derivative gives the velocity and the second derivative gives the acceleration.

OBJECT:

The function $h(t)$ is a mathematical object.

  • Usually the mental picture of function-as-object consists of thinking of the function as a set of ordered pairs $\Gamma(h):=\{(t,4-(t-2)^2)|t\in\mathbb{R}\}$. 
  • Sometimes you have to specify domain and codomain, but not usually in calculus problems, where conventions tell you they are both the set of real numbers.
  • The blend object and graph identifies each point on the graph with an element of $\Gamma(h)$.
  • When you give a formal proof, you usually revert to a dry-bones mode and think of math objects as inert and timeless, so that the proof does not mention change or causation.
    • The mathematical object $h(t)$ is a particular set of ordered pairs. 
    • It just sits there.
    • When reasoning about something like this, implication statements work like they are supposed to in math: no causation, just picking apart a bunch of dead things. (See Note 3).
    • I did not say that math objects are inert and timeless, I said you think of them that way.  This post is not about Platonism or formalism. What math objects "really are" is irrelevant to understanding understanding math [sic].

DEFINITION

A definition of the concept of function provides a way of thinking about the function.

  • One definition is simply to specify a mathematical object corresponding to a function: A set of ordered pairs satisfying the property that no two distinct ordered pairs have the same second coordinate, along with a specification of the codomain if that is necessary.
  • A concept can have many different definitions.
    • A group is usually defined as a set with a binary operation, an inverse operation, and an identity with specific properties.  But it can be defined as a set with a ternary operation, as well.
    • A partition of a set is a set of subsets of a set with certain properties. An equivalence relation is a relation on a set with certain properties.  But a partition is an equivalence relation and an equivalence relation is a partition.  You have just picked different primitives to spell out the definition. 
    • If you are a beginner at doing proofs, you may focus on the particular primitive objects in the definition to the exclusion of other objects and properties that may be more important for your current purposes.
      • For example, the definition of $h(t)$ does not mention continuity, differentiability, parabola, and other such things.
      • The definition of group doesn't mention that it has linear representations.

SPECIFICATION

A function can be given as a specification, such as this:

If $t$ is a real number, then $h(t)$ is a real number, whose value is obtained by subtracting $2$ from $t$, squaring the result, and then subtracting that result from $4$.

  • This tells you everything you need to know to use the function $h$.
  • It does not tell you what it is as a mathematical object: It is only a description of how to use the notation $h(t)$.

Notes

1. Formulas can be give in other notations, in particular Polish and Reverse Polish notation. Some forms of these notations don't need parentheses.

2. There are various ways to give a pictorial image of the function.  The usual way to do this is presenting the graph as shown above.  But you can also show its cograph and its endograph, which are other ways of representing a function pictorially.  They  are particularly useful for finite and discrete functions. You can find lots of detail in these posts and Mathematica notebooks:

3. See How to understand conditionals in the abstractmath article on conditionals.

References

  1. Conceptual blending (Wikipedia)
  2. Conceptual metaphors (Wikipedia)
  3. Definitions (abstractmath)
  4. Embodied cognition (Wikipedia)
  5. Handbook of mathematical discourse (see articles on conceptual blend, mental representation, representation, and metaphor)
  6. Images and Metaphors (article in abstractmath)
  7. Links to G&G posts on representations
  8. Metaphors in Computing Science (previous post)
  9. Mirror neurons (Wikipedia)
  10. Representations and models (article in abstractmath)
  11. Representations II: dry bones (article in abstractmath)
  12. The transition to formal thinking in mathematics, David Tall, 2010
  13. What is the object of the encapsulation of a process? Tall et al., 2000.

 

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The meaning of the word “superposition”

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This is from the Wikipedia article on Hilbert's 13th Problem as it was on 31 March 2012:

[Hilbert’s 13th Problem suggests this] question: can every continuous function of three variables be expressed as a composition  of finitely many continuous functions of two variables? The affirmative answer to this general question was given in 1957 by Vladimir Arnold, then only nineteen years old and a student of Andrey Kolmogorov. Kolmogorov had shown in the previous year that any function of several variables can be constructed with a finite number of three-variable functions. Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering Hilbert's question.  

In their paper A relation between multidimensional data compression and Hilbert’s 13th  problem,  Masahiro Yamada and Shigeo Akashi describe an example of Arnold's theorem this way: 

Let $f ( \cdot , \cdot, \cdot )$ be the function of three variable defined as \(f(x, y, z)=xy+yz+zx\), $x ,y , z\in \mathbb{C}$ . Then, we can easily prove that there do not exist functions of two variables $g(\cdot , \cdot )$ , $u(\cdot, \cdot)$ and $v(\cdot , \cdot )$ satisfying the following equality: $f(x, y, z)=g(u(x, y),v(x, z)) , x , y , z\in \mathbb{C}$ . This result shows us that $f$ cannot be represented any 1-time nested superposition constructed from three complex-valued functions of two variables. But it is clear that the following equality holds: $f(x, y, z)=x(y+z)+(yz)$ , $x,y,z\in \mathbb{C}$ . This result shows us that $f$ can be represented as a 2-time nested superposition.

The article about superposition in All about circuits says:

The strategy used in the Superposition Theorem is to eliminate all but one source of power within a network at a time, using series/parallel analysis to determine voltage drops (and/or currents) within the modified network for each power source separately. Then, once voltage drops and/or currents have been determined for each power source working separately, the values are all “superimposed” on top of each other (added algebraically) to find the actual voltage drops/currents with all sources active. 

Superposition Theorem in Wikipedia:

The superposition theorem for electrical circuits states that for a linear system the response (Voltage or Current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, while all other independent sources are replaced by their internal impedances.

Quantum superposition in Wikipedia:  

Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system — such as an electron — exists partly in all its particular, theoretically possible states (or, configuration of its properties) simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations (as described in interpretation of quantum mechanics).

Mathematically, it refers to a property of solutions to the Schrödinger equation; since theSchrödinger equation is linear, any linear combination of solutions to a particular equation will also be a solution of it. Such solutions are often made to be orthogonal (i.e. the vectors are at right-angles to each other), such as the energy levels of an electron. By doing so the overlap energy of the states is nullified, and the expectation value of an operator (any superposition state) is the expectation value of the operator in the individual states, multiplied by the fraction of the superposition state that is "in" that state

The CIO midmarket site says much the same thing as the first paragraph of the Wikipedia Quantum Superposition entry but does not mention the stuff in the second paragraph.

In particular, the  Yamada & Akashi article describes the way the functions of two variables are put together as "superposition", whereas the Wikipedia article on Hilbert's 13th calls it composition.  Of course, superposition in the sense of the Superposition Principle is a composition of multivalued functions with the top function being addition.  Both of Yamada & Akashi's examples have addition at the top.  But the Arnold theorem allows any continuous function at the top (and anywhere else in the composite).  

So one question is: is the word "superposition" ever used for general composition of multivariable functions? This requires the kind of research I proposed in the introduction of The Handbook of Mathematical Discourse, which I am not about to do myself.

The first Wikipedia article above uses "composition" where I would use "composite".  This is part of a general phenomenon of using the operation name for the result of the operation; for examples, students, even college students, sometimes refer to the "plus of 2 and 3" instead of the "sum of 2 and 3". (See "name and value" in abstractmath.org.)  Using "composite" for "composition" is analogous to this, although the analogy is not perfect.  This may be a change in progress in the language which simplifies things without doing much harm.  Even so, I am irritated when "composition" is used for "composite".

Quantum superposition seems to be a separate idea.  The second paragraph of the Wikipedia article on quantum superposition probably explains the use of the word in quantum mechanics.

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Mathematical usage

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Comments about mathematical usage, extending those in my post on abuse of notation.

Geoffrey Pullum, in his post Dogma vs. Evidence: Singular They, makes some good points about usage that I want to write about in connection with mathematical usage.  There are two different attitudes toward language usage abroad in the English-speaking world. (See Note [1])

  • What matters is what people actually write and say.   Usage in this sense may often be reported with reference to particular dialects or registers, but in any case it is based on evidence, for example citations of quotations or a linguistic corpus.  (Note [2].)  This approach is scientific.
  • What matters is what a particular writer (of usage or style books) believes about  standards for speaking or writing English.  Pullum calls this "faith-based grammar".  (People who think in this way often use the word "grammar" for usage.)  This approach is unscientific.

People who write about mathematical usage fluctuate between these two camps.

My writings in the Handbook of Mathematical Discourse and abstractmath.org are mostly evidence based, with some comments here and there deprecating certain usages because they are confusing to students.  I think that is about the right approach.  Students need to know what is actual mathematical usage, even usage that many mathematicians deprecate.

Most math usage that is deprecated (by me and others) is deprecated for a reason.  This reason should be explained, and that is enough to stop it being faith-based.  To make it really scientific you ought to cite evidence that students have been confused by the usage.  Math education people have done some work of this sort.  Most of it is at the K-12 level, but some have worked with college students observing the way the solve problems or how they understand some concepts, and this work often cites examples.

Examples of usage to be deprecated

 

Powers of functions

f^n(x) can mean either iterated composition or multiplication of the values.  For example, f^2(x) can mean f(x)f(x) or f(f(x)).  This is exacerbated by the fact that in undergrad calculus texts,  \sin^{-1}x refers to the arcsine, and \sin^2 x refers to \sin x\sin x.  This causes innumerable students trouble.  It is a Big Deal.

In

Set "in" another set.  This is discussed in the Handbook.  My impression is that for students the problem is that they confuse "element of" with "subset of", and the fact that "in" is used for both meanings is not the primary culprit.  That's because most sets in practice don't have both sets and non-sets as elements.  So the problem is a Big Deal when students first meet with the concept of set, but the notational confusion with "in" is only a Small Deal.

Two

This is not a Big Deal.  But I have personally witnessed students (in upper level undergrad courses) that were confused by this.

Parentheses

The many uses of parentheses, discussed in abstractmath.  (The Handbook article on parentheses gives citations, including one in which the notation "(a,b)" means open interval once and GCD once in the same sentence!)  I think the only part that is a Big Deal, or maybe Medium Deal, is the fact that the value of a function f at an input x can be written either  "f\,x" or as "f(x)".  In fact, we do without the parentheses when the name of the function is a convention, as in \sin x or \log x, and with the parentheses when it is a variable symbol, as in "f(x)".  (But a substantial minority of mathematicians use f\,x in the latter case.  Not to mention xf.)  This causes some beginning calculus students to think "\sin x" means "sin" times x.

More

The examples given above are only a sampling of troubles caused by mathematical notation.   Many others are mentioned in the Handbook and in Abstractmath, but they are scattered.  I welcome suggestions for other examples, particularly at the college and graduate level. Abstractmath will probably have a separate article listing the examples someday…

Notes

[1] The situation Pullum describes for English is probably different in languages such as Spanish, German and French, which have Academies that dictate usage for the language.  On the other hand, from what I know about them most speakers of those languages ignore their dictates.

[2] Actually, they may use more than one corpus, but I didn't want to write "corpuses" or "corpora" because in either way I would get sharp comments from faith-based usage people.

References on mathematical usage

Bagchi, A. and C. Wells (1997), Communicating Logical Reasoning.

Bagchi, A. and C. Wells (1998)  Varieties of Mathematical Prose.

Bullock, J. O. (1994), ‘Literacy in the language of mathematics’. American Mathematical Monthly, volume 101, pages 735743.

de Bruijn, N. G. (1994), ‘The mathematical vernacular, a language for mathematics with typed sets’. In Selected Papers on Automath, Nederpelt, R. P., J. H. Geuvers, and R. C. de Vrijer, editors, volume 133 of Studies in Logic and the Foundations of Mathematics, pages 865  935.  

Epp, S. S. (1999), ‘The language of quantification in mathematics instruction’. In Developing Mathematical Reasoning in Grades K-12. Stiff, L. V., editor (1999),  NCTM Publications.  Pages 188197.

Gillman, L. (1987), Writing Mathematics Well. Mathematical Association of America

Higham, N. J. (1993), Handbook of Writing for the Mathematical Sciences. Society for Industrial and Applied Mathematics.

Knuth, D. E., T. Larrabee, and P. M. Roberts (1989), Mathematical Writing, volume 14 of MAA Notes. Mathematical Association of America.

Krantz, S. G. (1997), A Primer of Mathematical Writing. American Mathematical Society.

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

Pimm, D. (1987), Speaking Mathematically: Communications in Mathematics Classrooms.  Routledge & Kegan Paul.

Schweiger, F. (1994b), ‘Mathematics is a language’. In Selected Lectures from the 7th International Congress on Mathematical Education, Robitaille, D. F., D. H. Wheeler, and C. Kieran, editors. Sainte-Foy: Presses de l’Université Laval.

Steenrod, N. E., P. R. Halmos, M. M. Schiffer, and J. A. Dieudonné (1975), How to Write Mathematics. American Mathematical Society.

Wells, C. (1995), Communicating Mathematics: Useful Ideas from Computer Science.

Wells, C. (2003), Handbook of Mathematical Discourse

Wells, C. (ongoing), Abstractmath.org.

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Two

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The post Are these questions unambiguous? in the blog Explaining Mathematics concerns the funny way mathematicians use the number “two” (Note [3]).  This is discussed in Abstractmath.org, based on usage quotations (see Note [1]) in the Handbook of Mathematical Discourse. They are citations  54, 119, 220, 229, 260, 322, 323 and 338.  The list is in the online version of the Handbook (see Note [2]) which takes forever to load.  (There is a separate file for users of the paperback book but it is currently trashed.)

The usage quirk concerning “two” is exemplified by statements such as these:

  1. The sum of any two even integers is even.
  2. Courant gives Leibniz’ rule for finding the Nth derivative of the product of two functions.  (This is from Citation 323.)
  3. Are there two positive integers m and n, both greater than 1, satisfying mn=9? (This is from Explaining Mathematics.)

Statements 1 and 2 are of course true.  They are still true if the “two” things are the same.  Mathematicians generally assume that such a statement includes the case where the two things are the same.  If the case that they are the same is excluded, the statement becomes an unnecessarily weak assertion.

Statement 3, in my opinion, is badly written.  If the two positive integers have to be distinct, the answer is “no”.   I think any competent mathematical writer would write something like, “There are not two distinct integers m and n both greater than 1 for which mn = 9″.

It is fair to say that when mathematicians refer to “two integers” in statements like these, they are allowed to be the same.  If they can’t be the same for the sentence to remain true, they will (or at least should) insert a word such as “distinct”.

Of course, in some sentences the two integers can’t be the same because of some condition imposed in the context.  That doesn’t happen in the citations I have listed.  Maybe someone can contribute an example.

Notes

[1] In the Handbook, usage quotations are called “citations”.  It appears to me that the commonest name for citations among lexicographers is “usage quotations”, so I will start calling them that.

[2] I created the online version of the Handbook hastily in 2006.  It needs work, since it has TeX mistakes (which may irritate you but should not interfere with readability) and omits the quotations, illustrations, and some backlinks, including backlinks for the citations.  Some Day When I Get A Round Tuit…

[3] This funny property of “two” was discussed many years ago by Steenrod or Knuth or someone, and is mentioned in a paper by Susanna Epp, but I don’t currently have access to any of the references.

 

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Abuse of notation

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I have recently read the Wikipedia article on Abuse of Notation (this link is to the version of 29 December 2011, since I will eventually edit it).  The Handbook of Mathematical Discourse and abstractmath.org mention this idea briefly.  It is time to expand the abstractmath article and to redo parts of the Wikipedia article, which  contains some confusions.

This is a preliminary draft, part of which I’ll incorporate into abstractmath after you readers make insightful comments :).

The phrase “Abuse of Notation” is used in articles and books written by research mathematicians.  It is part of Mathematical English.  This post is about

  • What “abuse of notation” means in mathematical writing and conversation.
  • What it could be used to mean.
  • Mathematical usage in general.  I will discuss this point in the context of the particular phrase “abuse of notation”, not a bad way to talk about a subject.

Mathematical Usage

Sources

If I’m going to write about the usage of Mathematical English, I should ideally verify what I claim about the usage by finding citations for a claim: documented quotations that illustrate the usage.  This is the standard way to produce any dictionary.

There is no complete authoritative source for usage of words and phrases in Mathematical English (ME), or for that matter for usage in the Symbolic Language (SL).

  • The Oxford Concise Dictionary of Mathematics [2] covers technical terms and symbols used in school math and in much of undergraduate math, but not so much of research math.  It does not mention being based on citations and it hardly talks about usage at all, even for notorious student-confusing notations such as “\sin^k x“. But it appears quite accurate with good explanations of the math it covers.
  • I wrote Handbook of Mathematical Discourse to stimulate investigations into mathematical usage.  It describes a good many usages in Mathematical English and the Symbolic Language, documented with citations of quotations, but is quite incomplete (as I said in its Introduction).  The Handbook has 428 citations for various usages.  (They are at the end of the on-line PDF version. They are not in the printed book, but are on the web with links to pages in the printed book.)
  • MathWorld has an extensive list of mathematical words, phrases and symbols, and accurate definitions or descriptions of them, even for a great many advanced research topics. It also frequently mentions usage (see formula and inverse sine), but does not give citations.
  • Wikipedia has the most complete set of definitions of mathematical objects that I know of.  The entries sometimes mention usage. I have not detected any entry that gives citations for usage.  Not that that should stop anyone from adding them.

Teaching mathematical usage

In explaining mathematical usage to students, particularly college-level or higher math students, you have choices:

  1. Tell them what you think the usage of a word, phrase, or symbol is, without researching citations.
  2. Tell them what you think the usage ought to be.
  3. Tell them what you think the usage is, supported by citations.

(1) has the problem that you can be wrong.  In fact when I worked on the Handbook I was amazed  at how wrong I could be in what the usage was, in spite of the fact that I had been thinking about usage in ME and SL since I first started teaching (and kept a folder of what I had noticed about various usages).  However,  professional mathematicians generally have a reasonably accurate idea about usage for most things, particularly in their field and in undergraduate courses.

(2) is dangerous.  Far too many mathematicians (but nevertheless a minority), introduce usage in articles and lecturing that is not common or that they invented themselves. As a result their students will be confused in trying to read other sources and may argue with other teachers about what is “correct”.  It is a gross violation of teaching ethics to tell the students that (for example) “x > 0″ allows x = 0 and not mention to them that nearly all written mathematics does not allow that.  (Did you know that a small percentage of mathematicians and educators do use that meaning, including in some secondary institutions in some countries?  It is partly Bourbaki’s fault.)

(3) You often can’t tell them what the usage is, supported by citations, because, as mentioned above, documented mathematical usage is sparse.

I think people should usually choose (1) instead of (2).  If they do want to introduce a new usage or notation because it is “more logical” or because “my thesis advisor used it” or something, they should reconsider.  Most such attempts have failed, and thousands of students have been confused by the attempts.

Abuse of notation

“Abuse of notation” is a phrase used in mathematical writing to describe terminology and notation that does not have transparent meaning. (Transparent meaning is described in some detail under “compositional” in the Handbook.)

Abuse of notation was originally defined in French, where the word “abus” does not carry the same strongly negative connotation that it does in English.

Suppression of parameters

One widely noticed practice called “abuse of notation”  is the use of the name of the underlying set of a mathematical structure to refer to a structure. For example, a group is a structure (G,\text{*}) where G is a set and * is a binary operation with certain properties. The most common way to refer to this structure is simply to call it G. Since any set of cardinality greater than 1 has more than one group structure on it, this does not include all the information needed to determine the group. This type of usage is cited in 82 below.  It is an example of suppression of parameters.

Writing “\log x” without mentioning the base of the logarithm is also an example of suppression of parameters.  I think most mathematicians would regard this as a convention rather than as an abuse of notation.  But I have no citations for this (although they would probably be easy to find).  I doubt that it is possible to find a rational distinction between “abuse of notation” and “convention”; it is all a matter of what people are used to saying.

Synecdoche

The naming of a structure by using the name of its underlying set is also an example of synecdoche, the naming of a whole by a part (for example, “wheels” to mean a car).

Another type of synecdoche that has been called abuse of notation is referring to an equivalence class by naming one of its elements.  I do not have a good quotation-citation that shows this use.  Sometimes people write 2 + 4 = 1 when they are working in the Galois field with 5 elements.  But that can be interpreted in more than one way.  If GF[5] consists of equivalence classes of integers (mod 5) then they are indeed using 2 (for example) to stand for the equivalence class of 2.  But they could instead define GF[5] in the obvious way with underlying set {0,1,2,3,4}.  In any case, making distinctions of that sort is pedantic, since the two structures are related by a natural isomorphism (next paragraph!)

Identifying objects via isomorphism

This is quite commonly called “abuse of notation” and is exemplified in citations 209, 395 and AB3.

Overloaded notation

John Harrison, in [1], uses “abuse of notation” to describe the use of a function symbol to apply to both an element of its domain and a subset of the domain.  This is an example of overloaded notation.  I have not found another citation for this usage other than Harrison and I don’t remember anyone using it.  Another example of overloaded notation is the use of the same symbol “\times” for multiplication of numbers, matrices and 3-vectors.  I have never heard that called abuse of notation.  But I have no authority to say anything about this usage because I haven’t made the requisite thorough search of the literature.

Powers of functions

The Wikipedia Article on abuse of notation (29 Dec 2011 version) mentions the fact that f^2(x) can mean either f(x)f(x) or f(f(x)).   I have never heard this called abuse of notation and I don’t think it should be called that.  The notation “f^2(x)” can in ordinary usage mean one of two things and the author or teacher should say which one they mean.  Many math phrases or symbolic expressions  can mean more than one thing and the author generally should say which.  I don’t see the point of calling this phenomenon abuse of notation.

Radial concept

The Wikipedia article mentions phrases such as “partial function”.  This article does provide a citation for Bourbaki for calling a sentence such as “Let f:A\to B be a partial function” abuse of notation.  Bourbaki is wrong in a deep sense (as the article implies).  There are several points to make about this:

  • Some authors, particularly in logic, define a function to be what most of us call a partial function.  Some authors  require a ring to have a unit and others don’t.  So what?
  • The phrase “partial function” has a standard meaning in math:  Roughly “it is a function except it is defined on only part of its domain”.  Precisely, f:A\to B is a partial function if it is a function f:A'\to B for some subset A' of A.
  • A partial function is not in general a function.  A stepmother is not a mother.  A left identity may not be an identity, but the phrase “left identity” is defined precisely.   An incomplete proof is not a proof, but you know what the phrase means! (Compare “expectant mother”).   This is the way we normally talk and think.  See the article “radial concept” in the Handbook.

Other uses

AB4 involves a redefinition of  “\in” in a special case.  Authors redefine symbols all the time.  This kind of redefinition on the fly probably should be avoided, but since they did it I am glad they mentioned it.

I have not talked about some of the uses mentioned in the Wikipedia article because I don’t yet understand them well enough.  AB1 and AB2 refer to a common use with pullback that I am not sure I understand (in terms of how they author is thinking of it).  I also don’t understand AB5.  Suggestions from readers would be appreciated.

Kill it!

Well, it’s more polite to say, we don’t need the phrase “abuse of notation” and it should be deprecated.

  • The use of the word “abuse” makes it sound like a bad thing, and most instances of abuse of notation are nothing of the sort.  They make mathematical writing much more readable.
  • Nearly everywhere it is used it could just as well be called a convention.  (This requires verification by studying math texts.)

Citations

The first three citations at in the Handbook list; the numbers refer to that list’s numbering. The others I searched out for the purpose of this post.

82. Busenberg, S., D. C. Fisher, and M. Martelli (1989), Minimal periods of discrete and smooth orbits. American Mathematical Monthly, volume 96, pages 5–17. [p. 8. Lines 2–4.]

Therefore, a normed linear space is really a pair (\mathbf{E},\|\cdot\|) where \mathbf{E} is a linear vector space and \|\cdot\|:\mathbf{E}\to(0,\infty) is a norm. In speaking of normed spaces, we will frequently abuse this notation and write \mathbf{E} instead of the pair (\mathbf{E},\|\cdot\|).

209. Hunter, T. J. (1996), On the homology spectral sequence for topological Hochschild homology. Transactions of the American Mathematical Society, volume 348, pages 3941–3953. [p. 3934. Lines 8–6 from bottom.]

We will often abuse notation by omitting mention of the natural isomorphisms making \wedge associative and unital.

395. Teitelbaum, J. T. (1991), ‘The Poisson kernel for Drinfeld modular curves’. Journal of the American Mathematical Society, volume 4, pages 491–511. [p. 494. Lines 1–4.]

\ldots may find a homeomorphism x:E\to \mathbb{P}^1_k such that \displaystyle x(\gamma u) = \frac{ax(u)+b}{cx(u)+d}. We will tend to abuse notation and identify E with \mathbb{P}^1_k by means of the function x.

AB1. Fujita, T. On the structure of polarized manifolds with total deficiency one.  I. J. Math. Soc. Japan, Vol. 32, No. 4, 1980.

Here we show examples of symbols used in this paper \ldots

L_{T}: The pull back of L to a space T by a given morphism T\rightarrow S . However, when there is no danger of confusion, we OFTEN write L instead of L_T by abuse of notation.

AB2. Sternberg, S. Minimal coupling and the symplectic mechanics of a classical
particle in the presence of a Yang-Mills field. Physics, Vol. 74, No. 12, pp. 5253-5254, December 1977.

On the other hand, let us, by abuse of notation, continue to write \Omega for the pullback of \Omega from F to P \times F by projection onto the second factor. Thus, we can write \xi_Q\rfloor\Omega = \xi_F\rfloor\Omega and \ldots

AB3. Dobson, D, and Vogel, C. Convergence of an iterative method for total variation denoising. SIAM J. Numer. Anal., Vol. 34, pp. 1779, October, 1997.

Consider the approximation

(3.7) u\approx U\stackrel{\text{def}}{=}\sum_{j=1}^N U_j\phi_j \ldots

In an abuse of notation, U will represent both the coefficient vector \{U_j\}_{j=1}^N and the corresponding linear combination (3.7).

AB4. Lewis, R, and Torczon, V. Pattern search algorithms for bound constrained minimization.  NASA Contractor Report 198306; ICASE Report No. 96-20.

By abuse of notation, if A is a matrix, y\in A means that the vector y is a column of A.

AB5. Allemandi, G, Borowiecz, A. and Francaviglia, M. Accelerated Cosmological Models in Ricci squared Gravity. ArXiv:hep-th/0407090v2, 2008.

This allows to reinterpret both f(S) and f'(S) as functions of \tau in the expressions:
\begin{equation*}\begin{cases} f(S) = f(F(\tau)) = f(\tau )\\ f'(S) = f'(F(\tau )) = f'(\tau )\end{cases}\end{equation*}
following the abuse of notation f(F(t )) = f(t ) and f'(F(t )) = f'(t ).

References

[1] Harrison, J. Criticism and reconstruction, in Formalized Mathematics (1996).

[2] Clapham, C. and J. Nicholson.  Oxford Concise Dictionary of Mathematics, Fourth Edition (2009).  Oxford University Press.

 

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Etymology

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Retire

I was recently asked about the etymology of the English word “retire”(in connection with quitting work).  It comes from Old French “retirer”, compounded from “re” (meaning “back”, a prefix used in Latin) and the Old French verb “tirer” meaning something like “pull” (which comes from a Germanic language, not Latin, and is related to “tier”, but not apparently to “tire”).

Its earliest citations in the Oxford English Dictionary show meanings such as

  • Pull back or retreat from the enemy.
  • To move back for safety or storage (“they retired to their houses”).
  • Leave office or work permanently.

All these meanings appear in print in the 16th century.

What good does it do to know this?  Not much.  You can’t explain the modern meaning of a word knowing the meaning of its ancient roots.

In the case of “retire”, I can make up a story of meanings changing using a chain of metaphors.

  1. “Retirer” in French meant literally “pull back” in the physical sense, for example pulling on a dog’s leash to drag it back so it won’t get into a fight with another dog. This literal meaning has not survived in the English word “retire” (nor, I think, in the French word “retirer”).
  2. In the 12th century (sez the OED without citation) the French word was used to refer to an army pulling back from a battle.  This is clearly a metaphor based on the literal meaning.  In a phrase such as “The Army retired from battle” it has become intransitive, but perhaps people once said things like “The General retired the Army from battle”.  Note that in modern English we could use the exact same metaphor with “pull back”: “The General pulled the Army back from battle”, although “withdrew” would be more common.
  3. Now someone comes along and uses the metaphor “going to work is like being in a battle”, and says things like “He retired from his job”.   This happened in English before 1533 and the usage has survived to this day.  It is probably the commonest meaning of the word “retire” now.

Now all that is a story I made up.  It is plausible, but it might have happened in a different way.  It is not at all likely we will discover the workings of metaphors in the minds of people who lived 600 years ago.  (Conceivably someone could have written down their thoughts about the word “retire” and it will be discovered in an odd subcrypt of Durham Cathedral and some linguist would get very excited, but I could win the lottery, too).

That’s why knowing the original literal meaning of the roots of a modern English word really means nothing about the modern meaning.  There could have been many steps along the way where a metaphorical usage became the standard meaning, then someone took the standard meaning and used it in another metaphor, maybe many times.  And metaphors aren’t the only method.  Words can change meaning because of misunderstanding, specialization, generalization, use in secret languages that become public, and so on.

I didn’t include etymology in the Handbook, mainly for this reason.  But there are certain mathematical words where knowing the metaphor or even the literal meaning can be of help.  I’ll write about that in a separate article.

 

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Multivalued Functions

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Multivalued functions

I am reconstructing the abstractmath website and am currently working on the part on functions. This has generated some bloggable blustering.

The phrase multivalued function refers to an object that is like a function {f:S\rightarrow T} except that for {s\in S}, {f(s)} may denote more than one value. Multivalued functions arose in considering complex functions such as {\sqrt{z}}. Another example: the indefinite integral is a multivalued operator.

It is useful to think of a multivalued function as a function although it violates one of the requirements of being a function (being single-valued).

A multivalued function {f:S\rightarrow T} can be modeled as a function with domain {S} and codomain the set of all subsets of {T}. The two meanings are equivalent in a strong sense (naturally equivalent). Even so, it seems to me that they represent two different ways of thinking about multivalued functions.: “The value may be any of these things…” as opposed to “The value is this whole set of things.”) The “value may be any of these…” idea has a perfectly good mathematical model: a relation (set of ordered pairs) from {S} to {T} which is the inverse of a surjective function.

Phrases such as “multivalued function” and “partial function” upset some uptight types who say things like, “But a multivalued function is not a function!”. A stepmother is not a mother, either.

I fulminated at length about this in the Handbook article on radial category. (This is conceptual category in the sense of Lakoff, Women, fire and dangerous things, University of Chicago, 1986.). The Handbook is on line, but it downloads very slowly, so I have extracted the particular page on radial categories here.

Functions generate a radial category of concepts in mathematics. There are lots of other concepts in math that have generated radial categories. Think of “incomplete proof” or “left identity”. Radial categories are a basic mechanism of the way we think and function in the world. They should not be banished from mathematics.

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Presenting math on the web

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This is a long post about ways to present math on the web, in the context of what I have done with The Handbook of Mathematical Discourse and abstractmath.org (Abmath).  “Ways to present math” include both organization and production technology.

The post is motivated by and focused on my plans to reconstruct Abmath this fall, when I will not be teaching.    During the last couple of years I have experimented with several possibilities for the reconstruction (while doing precious little on the actual website) and have come to a tentative conclusion about how I will do it.  I am laying all this out here, past history and future plans, in the hope that readers will have suggestions that will help the process (or change my mind).

I set out to write both the Handbook and Abmath using ideas about how math should be presented on the web.  They came out differently.  Now I think I went wrong with some of the ways in which I organized Abmath and that I need to reconstruct it so that it is more like the Handbook.  On the other hand, I have decided to stick with the production method I used for Abmath. I will explain.

Organization

My concept for both these works was that they  would have these properties:

1) Each work would be a cloud of articles. They would have little or no hierarchy.  They would consist of lots of short articles, not organized into chapters, sections and subsections.

2) The articles would be densely hyperlinked with each other and with the rest of the web. The reader would use the links to move from article to article. The articles might occur in alphabetical order in the production file but to the reader the order would be irrelevant.

I wanted the works to be organized that way because that is what I wanted from an information-presenting website.  I want it that way because I am a grasshopper. Wikipedia and n-lab are each organized as a cloud of articles. I started writing the Handbook in the late nineties before Wikipedia began.

The Handbook exists in two forms. The web version is a hypertext PDF file that consists of short articles with extensive interlinking. The printed book has the same short articles arranged in alphabetical order. In the book form, the links are replaced by page indices (“paper hyperlinks”). In both forms some links are arranged as lists  of related topics.

Abstractmath.org is a large, interlinked collection of html pages.  They are organized in four large sections with many subsections.

Many entrances

For this cloud of articles arrangement to work, there must be many entrances into the website, so that a reader can find what they want. The Handbook has a list of entries in alphabetical order. Certain entries (for example the entries on attitudes, on behaviors, and on multiple meanings) have internal lists of links to examples of what that entry discusses.  In addition, the paper version has an index that (in theory) provides links to all important occurrences of each concept in the book.  This index is not included in the current hypertext version, although the LaTeX package hyperref would make it possible to include it.  On the other hand, the hypertext version has the PDF search capability.

Abmath has a table of contents, listing articles in hierarchical form, as well as an index, which is different from the Handbook index in that it gives only one link from each word or phrase. In addition, it has header sections that briefly describe the contents of each main section and (in some cases) subsection, and also a Diagnostic Examples section (currently fragmentary)in which each entry provides a description of a particular problem that someone may have in understanding abstract math, with links to where it is discussed. The website currently has no search capability.

The Handbook is really a cloud of articles, and Abmath is not. I made a serious mistake imposing a hierarchy on Abmath, and that is the main thing I want to correct when I reconstruct it.  Basically, I want to dissolve the hierarchy into a cloud of articles.

Production methods

The Handbook was composed using LaTeX.  It originally existed in hypertext form (in a PDF file) and lived on the web for several years, generating many useful suggestions. I wrote a LaTeX header that could be set to produce PDF output with hyperlinks or PDF output formatted as a book with paper hyperlinks; that form was eventually published as a book.

I used a number of Awk programs to gather the various kinds of links.  For example, every entry referring to a math word that has multiple meanings was marked and an Awk program gathered them into a list of links.

I generated the html pages for Abmath using Microsoft Word and MathType.  MathType is very easy to use and has the capability (recently acquired) of converting all math entries that it generated  into TeX. The method used for Abmath has several defects.  You can’t apply Awk (or nowadays Python) programs to a Word document since it is in a proprietary format.  Another problem is that the appearance of the result varies with browser.

But the Abmath method also has advantages.  It produces html documents which can be read in windows that you can make narrower or wider and the text will adjust.  PDF files are fixed width and rigid, and I find clicking on links requires you to be annoyingly precise with your fingers.

So my original thought was to go back to LaTeX for the new version of Abmath. There are several ways to produce html files from LaTeX, and converting the MathType entries to TeX provides a big headstart on converting the Word files into text files.  Then I could use Awk to do a lot of bookkeeping and cut the hyperlink errors, the way I did with the Handbook.

So at first I was quite nostalgic about the wonderful time I had doing the Handbook in LaTeX — until I remembered all the fussing I did to include illustrations and marginal remarks. (I couldn’t just put the illo there and leave it.) Until I remembered how slowly the resulting PDF file loads because there seems to be no way to break it into individual article files without breaking the links.

And then I found that (as far as I could determine) there is no HTMLTeX that produces a reasonable HTML file from any TeX file the way PDFTeX produces a PDF file from any TeX file, using Knuth’s  TeX program. In fact all the TeX to HTML systems I investigated don’t use Knuth’s program at all — they just have code in some programming language that reads a TeX file and interprets what the programmer felt like interpreting.  I would love to be contradicted concerning this.

So now my thought is to stick with Word and MathType.  And to do textual manipulation I will have to learn Word Basic.  I just ordered two books on Word Basic. I would rather learn Python, but I have to work with what I have already done.  Stay tuned.

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Expository writing in the future

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I have written a lot about math exposition in the past. [Note 1.] Lately I have been thinking about the effect of technological change on exposition.

Texting

A lot of commentators have complained that their students’ writing style has “deteriorated” because of texting, specifically their use of abbreviations and acronyms.

Last January I resumed teaching mathematics after an exactly ten year lapse. My students and I email a lot, post on message boards, hand in homework, write up tests. I have seen very few “lol”s and “cu”s and the like, mostly in emails and almost entirely from students whose native language is not English. (See Note 1.)

As far as I can see the students’ written language has not deteriorated. In fact I think native English speakers write better English than they did ten years ago. (But Minnesota has a considerably better educational system than Ohio.)

Besides, if lol and cu become part of the written language, so what? Many Old Fogies may find it jarring, but Old Fogies die and their descendants talk however they want to.

Bulleted lists

I have been using Powerpoint part of the time in teaching (I had already given some talks using it). People complain about that affecting our style, too. But I think that in particular bulleted and numbered lists are great. I wish people would use them more often. Consider this passage from a recent version of Thomas’ Calculus [1]:

\displaystyle \int_a^bx\,dx=\dfrac{b^2}{2}-\dfrac{a^2}{2}\quad (a< b)\quad\quad\quad(1)

This computation gives the area of a trapezoid. Equation (1) remains valid when {a} and {b} are negative. When {a<b<0}, the definite integral value … is a negative number, the negative of the area of the trapezoid dropping down to the line {y=x} below the {x}-axis. When {a<0} and {b>0}, Equation (1) is still valid and the definite integral gives the difference between two areas …

It would be much better to write something like this:

Equation (1) is valid for any {a} and {b}.

  • When {a} and {b} are positive, Equation (1) gives the area of a trapezoid.
  • When {a} and {b} are both negative, the result is negative and is the negative of the area…
  • When {a<0} and {b>0}, the result is the difference between two areas…

That is much easier to read than the first version, in which you have to parse through the paragraph detecting that it states parallel facts. That is not terribly difficult but it slows you down. Especially in this case where the sentences are not written in parallel and contain remarks about validity in scattered places when in fact the equation is valid for all cases.

This book does use numbered or lettered lists in many other places.

The future is upon us

Lots of lists and illustrations require more paper. This will go away soon. Some future edition of the book on an e-reader could contain this list of facts as a nicely spaced list, much easier to grasp, and could contain three graphs, with {a} and {b} respectively left of the {x}-axis, straddling it, and to the right of it. This will cost some preparation time but no paper and computer memory at the scale of a book is practically free.

I use bulleted lists a lot in abstractmath, as here. Abstractmath is intended to be read on the computer. It is not organized linearly and a paper copy would not be particularly useful.

By the way, since the last time I looked at this page all the bullets have been replaced with copyright signs. (In three different browsers!) Somebody’s been Messing With Me. AArgH.

The Irish mystery writer Ken Bruen regularly uses lists, without bullets or numbers. Look at page 3 of The Killing of the Tinkers.

Some people find bulleted lists jarring simply because they are new. I think some are academic snobs who diss anything that sounds like something a business person would do. See my remarks at the end of the section on texting.

Notes

1. You can see much of what I have said on this blog about exposition by reading the posts labeled “exposition” (scroll down to the list of categories in the left column.) See also Varieties of Mathematical Prose by Atish Bagchi and me.

2. Foreign language speakers also write things like “Hi Charles” instead of “Dear Professor Wells” or using no greeting at all (which is probably the best thing to do). Dealing with a foreign language requires familiarity with the local social structure and customs of address, of being aware of levels of the various formal and informal registers, and so on. When we lived in Switzerland, how was I to know that “Ciao” went with “du” and “wiederluege” went with “Sie”? (If I remember correctly. Ye Gods, that was 35 years ago.)

References

1. Thomas’ Calculus, Early Transcendentals, Eleventh Edition, Media Upgrade. Pearson Education, 2008.

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