Category Archives: language

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

Abuse of notation

10 Comments

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.

 

Send to Kindle
exposition, language, math, Uncategorized, understanding math

Tangents

2 Comments

The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Tangent Line.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

This is an experiment in exposition of the mathematical concepts of tangent.  It follows the same pattern as my previous post on secant, although that post has explanations of my motivation for this kind of presentation that are not repeated here.

Tangent line

A line is tangent to a curve (in the plane) at a given point if all the following conditions hold (Wikipedia has more detail.):

  1. The line is a straight line through the point.
  2. The curve goes through that point.
  3. The curve is differentiable in a neighborhood of the point.
  4. The slope of the straight line is the same as the derivative of the curve at that point.

In this picture the curve is $ y=x^3-x$ and the tangent is shown in red. You can click on the + signs for additional controls and information.

Etymology and metaphor

The word “tangent” comes from the Latin word for “touching”. (See Note below.) The early scholars who talked about “tangent” all read Latin and knew that the word meant touching, so the metaphor was alive to them.

The mathematical meaning of “tangent” requires that the tangent line have slope equal to the derivative of the curve at the point of contact. All of the red lines in the picture below touch the curve at the point (0, 1.5). None of them are tangent to the curve there because the curve has no derivative at the point:

The curve in this picture is defined by

The mathematical meaning restricts the metaphor. The red lines you can generate in the graph all touch the curve at one point, in fact at exactly at one point (because I made the limits on the slider -1 and 1), but there are not tangent to the curve.

Tangents can hug!

On the other hand, “touching” in English usage includes maintaining contact on an interval (hugging!) as well as just one point, like this:

The blue curve in this graph is given by

The green curve is the derivative dy/dx. Notice that it has corners at the endpoints of the unit interval, so the blue curve has no second derivative there. (See my post Curvature).

Tangent lines in calculus usually touch at the point of tangency and not nearby (although it can cross the curve somewhere else). But the red line above is nevertheless tangent to the curve at every point on the curve defined on the unit interval, according to the definition of tangent. It hugs the curve at the straight part.

The calculus-book behavior of tangent line touching at only one point comes about because functions in calculus books are always analytic, and two analytic curves cannot agree on an open set without being the same curve.

The blue curve above is not analytic; it is not even smooth, because its second derivative is broken at $x=0$ and $x=1$. With bump functions you can get pictures like that with a smooth function, but I am too lazy to do it.

Tangent on the unit circle

In trigonometry, the value of the tangent function at an angle $ \theta$ erected on the x-axis is the length of the segment of the tangent at (1,0) to the unit circle (in the sense defined above) measured from the x-axis to the tangent’s intersection with the secant line given by the angle. The tangent line segment is the red line in this picture:


This defines the tangent function for $ -\frac{\pi}{2} < x < \frac{\pi}{2}$.

The tangent function in calculus

That is not the way the tangent function is usually defined in calculus. It is given by \tan\theta=\frac{\sin\theta}{\cos\theta}, which is easily seen by similar triangles to be the same on -\frac{\pi}{2} < x < \frac{\pi}{2}.

We can now see the relationship between the geometric and the $ \frac{\sin\theta}{\cos\theta}$ definition of the tangent function using this graph:


The red segment and the green segment are always the same length.
It might make sense to extend the geometric definition to $ \frac{\pi}{2} < x < \frac{3\pi}{2}$ by constructing the tangent line to the unit circle at (-1,0), but then the definition would not agree with the $ \frac{\sin\theta}{\cos\theta}$ definition.

References

Send to Kindle
exposition, language, language of math, math, Mathematica, understanding math

Case Study in Exposition: Secant

3 Comments

The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code comes from several Mathematica notebooks lists in the References. The notebooks are available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Pictures, metaphors and etymology

Math texts and too many math teachers do not provide enough pictures and metaphors to help students understand a concept.  I suspect that the etymology of the technical terms might also be useful. This post is an experimental exposition of the math concept of “secant” that use pictures, metaphors and etymology to describe the concept.

The exposition is interlarded with comments about what I am doing and why.  An exposition directly aimed at students would be slimmer — but some explanations of why you are doing such and such in an exposition are not necessarily out of place every time!

Secant Line

The word “secant” is used in various related ways in math.  To start with, a secant line on a curve is the unique line determined by two distinct points on the curve, like this:


The word “secant” comes from the Latin word for “cut”, which came from the Indo-European root “sek”, meaning “cut”.  The IE root also came directly into English via various Germanic sound changes to give us “saw” and “sedge”.

The picture

Showing pictures of mathematical objects that the reader can fiddle with may make it much easier to understand a new concept.  The static picture you get above by keeping your mitts off the sliders requires imagining similar lines going through other pairs of points. When you wiggle the picture you see similar lines going through other pairs of points.  You also get a very strong understanding of how the secant line is a function of the two given points.  I don’t think that is obvious to someone without some experience with such things.

This belief contains the hidden claim that individuals vary a lot on how they can see the possibilities in a still picture that stands as an example of a lot of similar mathematical objects.  (Math books are full of such pictures.)  So people who have not had much practice learning about possible variation in abstract structures by looking at one motionless one will benefit from using movable parametrized pictures of various kinds.  This is the sort of claim that is amenable to field testing.

The metaphor

Most metaphors are based on a physical phenomenon.  The mathematical meanings of “secant” use the metaphor of cutting.  When the word “secant” was first introduced by a European writer (see its etymology) in the 16th century, the word really was a metaphor.   In those days essentially every European scholar read Latin. To them “secant” would transparently mean “cutting”.  This is not transparent to many of us these days, so the metaphor may be hidden.

If you examine the metaphor you realize that (like all metaphors) it involves making some remarkably subtle connections in your brain.

  • The straight line does not really cut the curve.  Indeed, the curve itself is both an abstract object that is not physical, so can’t be cut, and also the picture you see on the screen, which is physical, but what would it mean to cut it?  Cut the screen?  The line can’t do that.
  • You can make up a story that (for example) the use was suggested by the mental image of a mark made by a knife edge crossing the plane at points a and b that looks like it is severing the curve.
  • The metaphor is restricted further by saying that it is determined by two points on the curve.   This restriction turns the general idea of secant line into a (not necessarily faithful!) two-parameter family of straight lines.  You could define such a family by using one point on the curve and a slope, for example.  This particular way of doing it with two points on the curve leads directly to the concept of tangent line as limit.

Secant on circle

Another use of the word “secant” is the red line in this picture:


This is the secant line on the unit circle determined by the origin and one point on the circle, with one difference: The secant of the angle is the line segment between the origin and the point on the curve.  This means it corresponds to a number, and that number is what we mean by “secant” in trigonometry.

To the ancient Greeks, a (positive) number was the length of a line segment.

The Definition

The secant of an angle $\theta$ is usually defined as $\frac{1}{\cos\theta}$, which you can see by similar triangles is the length of the red line in the picture above.

In many parts of the world, trig students don’t learn the word “secant”. They simply use $\frac{1}{\cos\theta}$.

This illustrates important facts about definitions:

  • Different equivalent definitions all make the same theorems true.
  • Different equivalent definitions can give you a very different understanding of the concept.

The red-line-segment-in-picture definition gives you a majorly important visual understanding of the concept of “secant”.  You can tell a lot from its behavior right off (it goes to infinity near $\pi/2$, for example).

The definition $\sec\theta=\frac{1}{\cos\theta}$ gives you a way of computing $\sec\theta$.  It also reduces the definition of $ \sec\theta$ to a previously known concept.

It used to be common to give only the $ \frac{1}{\cos\theta}$ definition of secant, with no mention of the geometric idea behind it.  That is a crime.  Yes, I know many students don’t want to “understand” stuff, they only want to know how to do the problems.  Teachers need to talk them out of that attitude.  One way to do that in this case is to test them on the geometric definition.

Etymology

This idea was known to the Arabs, and brought into European view in the 16th century by Danish mathematician Thomas Fincke in “Geometria Rotundi” (1583), where the first known use of the word “secant” occurs.  I have not checked, but I suspect from the title of the book that the geometric definition was the one he used in the book.

It wold be interesting to know the original Arabic name for secant, and what physical metaphor it is based on.  A cursory search of the internet gave me the current name in Arabic for secant but nothing else.

Graph of the secant function

The familiar graph of the secant function can be seen as generated by the angle sweeping around the curve, as in the picture below. The two red line segments always have the same length.


References

Mathematica notebooks used in this post:

 

Send to Kindle
exposition, Handbook, language, language of math, Uncategorized, understanding math

Etymology

Leave a comment

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.

 

Send to Kindle
exposition, Handbook, language, language of math, math

Expository writing in the future

3 Comments

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.

Send to Kindle
exposition, Handbook, language, language of math, math, representations

Technical meanings clash with everyday meanings

7 Comments

Recently (see note [a]) on MathOverflow, Colin Tan asked [1] “What does ‘kernel’ mean in ‘integral kernel’?”  He had noticed the different use of the word in referring to the kernels of morphisms.

I have long thought [2] that the clash between technical meanings and everyday meaning of technical terms (not just in math) causes trouble for learners.  I have recently returned to teaching (discrete math) and my feeling is reinforced — some students early in studying abstract math cannot rid themselves of thinking of a concept in terms of familiar meanings of the word.

One of the worst areas is logic, where “implies” causes well-known bafflement.   “How can ‘If P then Q’ be true if P is false??”  For a large minority of beginning college math students, it is useless to say, “Because the truth table says so!”.  I may write in large purple letters (see [3] for example) on the board and in class notes that The Definition of a Technical Math Concept Determines Everything That Is True About the Concept but it does not take.  Not nearly.

The problem seems to be worse in logic, which changes the meaning of words used in communicating math reasoning as well as those naming math concepts. But it is bad enough elsewhere in math.

Colin’s question about “kernel” is motivated by these feelings, although in this case it is the clash of two different technical meanings given to the same English word — he wondered what the original idea was that resulted in the two meanings.  (This is discussed by those who answered his question.)

Well, when I was a grad student I made a more fundamental mistake when I was faced with two meanings of the word “domain” (in fact there are at least four meanings in math).  I tried to prove that the domain of a continuous function had to be a connected open set.  It didn’t take me all that long to realize that calculus books talked about functions defined on closed intervals, so then I thought maybe it was the interior of the domain that was a, uh, domain, but I pretty soon decided the two meanings had no relation to each other.   If I am not mistaken Colin never thought the two meanings of “kernel” had a common mathematical definition.

It is not wrong to ask about the metaphor behind the use of a particular common word for a technical concept.  It is quite illuminating to get an expert in a subject to tell about metaphors and images they have about something.  Younger mathematicians know this.  Many of the questions on MathOverflow are asking just for that.  My recollection of the Bad Old Days of Abstraction and Only Abstraction (1940-1990?) is that such questions were then strongly discouraged.

Notes

[a] The recent stock market crash has been blamed [4] on the fact that computers make buy and sell decisions so rapidly that their actions cannot be communicated around the world fast enough because of the finiteness of the speed of light.  This has affected academic exposition, too.  At the time of writing, “recently” means yesterday.

References

[1] Colin Tan, “What does ‘kernel’ mean in ‘integral kernel’?

[2] Commonword names for technical concepts (previous blog).

[3] Definitions. (Abstractmath).

[4] John Baez, This weeks finds in mathematical physics, Week 297.

Send to Kindle
abstractmath.org, Handbook, language, language of math, math, Proposals for research, representations, understanding math

Syntax Trees in Mathematicians’ Brains

11 Comments

Understanding the quadratic formula

In my last post I wrote about how a student’s pattern recognition mechanism can go awry in applying the quadratic formula.

The template for the quadratic formula says that the solution of a quadratic equation of the form ${ax^2+bx+c=0}$ is given by the formula

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

When you ask students to solve ${a+bx+cx^2=0}$ some may write

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

instead of

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2c}$

That’s because they have memorized the template in terms of the letters ${a}$, ${b}$ and ${c}$ instead of in terms of their structural meaning — $ {a}$ is the coefficient of the quadratic term, ${c}$ is the constant term, etc.

The problem occurs because there is a clash between the occurrences of the letters “a”, “b”, and “c” in the template and in the equation to solve. But maybe the confusion would occur anyway, just because of the ordering of the coefficients. As I asked in the previous post, what happens if students are asked to solve $ {3+5x+2x^2=0}$ after having learned the quadratic formula in terms of ${ax^2+bx+c=0}$? Some may make the same kind of mistake, getting ${x=-1}$ and ${x=-\frac{2}{3}}$ instead of $ {x=-1}$ and $ {x=-\frac{3}{2}}$. Has anyone ever investigated this sort of thing?

People do pattern recognition remarkably well, but how they do it is mysterious. Just as mistakes in speech may give the linguist a clue as to how the brain processes language, students’ mistakes may tell us something about how pattern recognition works in parsing symbolic statements as well as perhaps suggesting ways to teach them the correct understanding of the quadratic formula.

Syntactic Structure

“Structural meaning” refers to the syntactic structure of a mathematical expression such as ${3+5x+2x^2}$. It can be represented as a tree:

(1)

This is more or less the way a program compiler or interpreter for some language would represent the polynomial. I believe it corresponds pretty well to the organization of the quadratic-polynomial parser in a mathematician’s brain. This is not surprising: The compiler writer would have to have in mind the correct understanding of how polynomials are evaluated in order to write a correct compiler.

Linguists represent English sentences with syntax trees, too. This is a deep and complicated subject, but the kind of tree they would use to represent a sentence such as “My cousin saw a large ship” would look like this:

Parsing by mathematicians

Presumably a mathematician has constructed a parser that builds a structure in their brain corresponding to a quadratic polynomial using the same mechanisms that as a child they learned to parse sentences in their native language. The mathematician learned this mostly unconsciously, just as a child learns a language. In any case it shouldn’t be surprising that the mathematicians’s syntax tree for the polynomial is similar to the compiler’s.

Students who are not yet skilled in algebra have presumably constructed incorrect syntax trees, just as young children do for their native language.

Lots of theoretical work has been done on human parsing of natural language. Parsing mathematical symbolism to be compiled into a computer program is well understood. You can get a start on both of these by reading the Wikipedia articles on parsing and on syntax trees.

There are papers on students’ misunderstandings of mathematical notation. Two articles I recently turned up in a Google search are:

Both of these papers talk specifically about the syntax of mathematical expressions. I know I have read other such papers in the past, as well.

What I have not found is any study of how the trained mathematician parses mathematical expression.

For one thing, for my parsing of the expression $ {3+5x+2x^2}$, the branching is wrong in (1). I think of ${3+5x+2x^2}$ as “Take 3 and add $ {5x}$ to it and then add ${2x^2}$ to that”, which would require the shape of the tree to be like this:

I am saying this from introspection, which is dangerous!

Of course, a compiler may group it that way, too, although my dim recollection of the little bit I understand about compilers is that they tend to group it as in (1) because they read the expression from left to right.

This difference in compiling is well-understood.  Another difference is that the expression could be compiled using addition as an operator on a list, in this case a list of length 3.  I don’t visualize quadratics that way but I certainly understand that it is equivalent to the tree in Diagram (1).  Maybe some mathematicians do think that way.

But these observations indicate what might be learned about mathematicians’ understanding of mathematical expressions if linguists and mathematicians got together to study human parsing of expressions by trained mathematicians.

Some educational constructivists argue against the idea that there is only one correct way to understand a mathematical expression.  To have many metaphors for thinking about math is great, but I believe we want uniformity of understanding of the symbolism, at least in the narrow sense of parsing, so that we can communicate dependably.  It would be really neat if we discovered deep differences in parsing among mathematicians.  It would also be neat if we discovered that mathematicians parsed in generally the same way!

Send to Kindle
language

Isaac buys him a prism

3 Comments

I say things like “I need to buy me some new shoes” fairly often.  This marks me as being a native of  the American South.   This construction  was recently discussed extensively in the article On beyond personal datives? in the Language Log.  Some of the commenters quoted examples that were far more elaborate than anything I would say, but I am a rather diluted Southerner.  (I’ll cry me a river because I have lost me most of my heritage.)

I was charmed recently to discover that a certain Physicist was in spirit a fellow Southerner.  In a paper by Isaac Newton in the Philosophical Transactions in 1671, he wrote:

Sir: To perform my late promise to you, I shall without further ceremony acquaint you, that in the beginning of the Year 1666 (at which time I applyed my self to the grinding of Optick glasses of other figures than Spherical,) I procured me a Triangular glass-Prisme, to try therewith the celebrated Phoenomena of Colours.

Later on he wrote

…that the difference ‘twixt the length of the Image, and diameter of the hole, through which the light was transmitted, was proportionable to their distance.

I think he has been channeling Pogo, who is also a Southerner.

I recommend looking at his paper.  I didn’t realize what an excellent science writer he was.

By the way, I thought I remembered that Newton’s trip to buy a prism was described in one of the volumes of Neal Stephenson’s The Baroque Cycle, but a search of Google Books doesn’t reveal it.

Send to Kindle
language

BC and AD

Leave a comment

I forget half the time that to be politically correct I should use BCE and CE instead of BC and AD.  I have a better idea.  I decree that from now on, BC means Backward Counting and AD means Advancing Denumeration.  At least in what I write.  Unless I forget.

Send to Kindle
language

More about pronouncing "a" and "the"

2 Comments

I eavesdropped on my grandson dictating a story for his granny to type.  He always pronounced “a” as a schwa.  I mentioned in an earlier post that Barak Obama and Hilary Clinton both normally pronounce it “ay” in speeches and wondered if this is a generational change.  The grandsonian evidence suggests not.  But:

  1. Do Obama and Clinton pronounce it that way in ordinary conversation?  I bet not.
  2. Is there a Speech Making School for Politicians that has them do this, or is this the result of their unconsciously adopting a Speech Making Register?  I’ll bet the latter.

The grandson also regularly said “thee” for “the” before vowel sounds and used the schwa before consonants.  This makes me want to go back to You Tube and eavesdrop on politicians some more.

I might add that if he stopped after “the” to let poor old granny catch up with her typing, he used the schwa even thought what came next was a vowel.  However, this may not prove anything since he may not have had the next word in mind.

Years ago, I was fascinated listening to John Kennedy, who pronounced r at the end of a word only if the next word began with a vowel, as in “The far east” but “The fah boundaries of…”.  I thought that it was remarkable that he did this even if he paused before second word.

Send to Kindle