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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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Definition of “function”

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I have made a major revision of the abstractmath.org article Functions: Specification and Definition.   The links from the revised article lead into the main abstractmath website, but links from other articles on the website still go back to the old version. So if you click on a link in the revised version, make it come up in a new window.

I expect to link the revision in after I make a few small changes, and I will take into account any comments from you all.

Remarks

1.  You will notice that the new version is in PDF instead of HTML.  A couple of other articles on the website are already in PDF, but I don’t expect to continue replacing HTML by PDF.   It is too much work.  Besides, you can’t shrink it to fit tablets.

2. It would also have been a lot of work to adapt the revision so that I could display it directly on Word Press.  In some cases I have written revisions first in WP and then posted them on the abmath website.  That is not so difficult and I expect to do it again.

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Experiment with abstractmath.org

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This is a rant about technical problems with creating abstractmath.org.  You will not get great new insights into mathematical language.  You will not get any purty pictures, either.  But if you read the following anyway and have suggestions, I would appreciate them no end.

I have long been frustrated with the process I used to create articles for abstractmath.  The process has been this:  I write the article in Word using MathType, then use their facility for generating an html file that uses pictures (stored in separate files) for the more complicated math expressions, then load them into the abstractmath website.

There are many good and bad things about this. Two of the most aggravating:

  • It is difficult to change links if I reorganize something.  With a TeX file I could write WinEdt macros to do it, but the Word macro language makes manipulating links (and doing many other things) a %#!!*.
  • The documents look different in different browsers.  IE Explorer 8 does the best job, Chrome looks uglier, and Firefox is the ugliest.  After a document has been posted for a while, sometimes I open it to discover weird things, such as the recent discovery that some of my bullets had turned into copyright signs (Firefox turns bullets into double hyphens that are close to invisible).

For the past few weeks I have experimented with generating PDF documents using PDFLaTeX.   I did this with the section called Functions: Notation and Terminology, the only one that is posted so far.  Posting it required 45 minutes  of fixing links in other articles by hand.

Creating that section in TeX was a pain.  I used GrindEQ Math to convert the original document to TeX, which required a great deal of preprocessing (mostly to recover the links, but for some formatting things too) and postprocessing to fix many many things.  I also had to recreate the sidebars by hand; I used wrapfig. Wrapfig does not work well.

In the process of converting this and some other files that I may post later, I created a bunch of macros in the Word macro language (horrible, although in principle I believe in OOP) and the WindEdt macro language, which is pretty good.  Even with the macros, it is a lot of labor to do the conversion.

I have decided to abandon the effort.  I may post a few more articles that I have already transferred to  PDF, but I doubt I will revise any more from scratch.

One thing that has changed since I started doing the conversions was that a recent revision of MathType allows you to type the equations directly in TeX and to toggle back and forth between MathType form and TeX form.  Before that you had to select symbols from a palette.  That was the single most frustrating thing about using Word with MathType.  I type fluently in TeX but I couldn’t use it.  Now that I can type the TeX in directly the prospect of editing articles and writing new ones using Word is much less painful.

So one direction I will go in is to revise the articles already on the web using Word and MathType.  I also expect to kill a good many of the incomplete or less well thought out ones in favor of links to Wikipedia.

But there is another direction, opened up by Mathematica’s Computable Document files.  I have published some experiments with this in the last four posts here.  I expect to be able to turn some of the articles in abstractmath.org into computable documents.  The reader will have to have to have the free CDF Player on their computer, but if you download it once you have it forever.

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Endograph and cograph of real functions

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This post is covered by the Creative Commons Attribution – ShareAlike 3.0 License, which means you may use, adapt and distribute the work provided you follow the requirements of the license.

Introduction

In the article Functions: Images and Metaphors in abstractmath I list a bunch of different images or metaphors for thinking about functions. Some of these metaphors have realizations in pictures, such as a graph or a surface shown by level curves. Others have typographical representations, as formulas, algorithms or flowcharts (which are also pictorial). There are kinetic metaphors — the graph of {y=x^2} swoops up to the right.

Many of these same metaphors have realizations in actual mathematical representations.

Two images (mentioned only briefly in the abstractmath article) are the cograph and the endograph of a real function of one variable. Both of these are visualizations that correspond to mathematical representations. These representations have been used occasionally in texts, but are not used as much as the usual graph of a continuous function. I think they would be useful in teaching and perhaps even sometimes in research.

A rough and unfinished Mathematica notebook is available that contains code that generate graphs and cographs of real-valued functions. I used it to generate most of the examples in this post, and it contains many other examples. (Note [1].)

The endograph of a function

In principle, the endograph (Note [2]) of a function {f} has a dot for each element of the domain and of the codomain, and an arrow from {x} to {f(x)} for each {x} in the domain. For example, this is the endograph of the function {n\mapsto n^2+1 \pmod 11} from the set {\{0,1,\ldots,10\}} to itself:


“In principle” means that the entire endograph can be shown only for small finite functions. This is analogous to the way calculus books refer to a graph as “the graph of the squaring function” when in fact the infinite tails are cut off.

Real endographs

I expect to discuss finite endographs in another post. Here I will concentrate on endographs of continuous functions with domain and codomain that are connected subsets of the real numbers. I believe that they could be used to good effect in teaching math at the college level.

Here is the endograph of the function {y=x^2} on the reals:

I have displayed this endograph with the real line drawn in the usual way, with tick marks showing the location of the points on the part shown.

The distance function on the reals gives us a way of interpreting the spacing and location of the arrowheads. This means that information can be gleaned from the graph even though only a finite number of arrows are shown. For example you see immediately that the function has only nonnegative values and that its increase grows with {x}.(See note [3]).

I think it would be useful to show students endographs such as this and ask them specific questions about why the arrows do what they do.

For the one shown, you could ask these questions, probably for class discussion rather that on homework.

  • Explain why most of the arrows go to the right. (They go left only between 0 and 1 — and this graph has such a coarse point selection that it shows only two arrows doing that!)
  • Why do the arrows cross over each other? (Tricky question — they wouldn’t cross over if you drew the arrows with negative input below the line instead of above.)
  • What does it say about the function that every arrowhead except two has two curves going into it?

Real Cographs

The cograph (Note [4] of a real function has an arrow from input to output just as the endograph does, but the graph represents the domain and codomain as their disjoint union. In this post the domain is a horizontal representation of the real line and the codomain is another such representation below the domain. You may also represent them in other configurations (Note [5]).

Here is the cograph representation of the function {y=x^2}. Compare it with the endograph representation above.

Besides the question of most arrows going to the right, you could also ask what is the envelope curve on the left.

More examples

Absolute value function

Arctangent function

Notes

[1] This website contains other notebooks you might find useful. They are in Mathematica .nb, .nbp, or .cdf formats, and can be read, evaluated and modified if you have Mathematica 8.0. They can also be made to appear in your browser with Wolfram CDF Player, downloadable free from Wolfram site. The CDF player allows you to operate any interactive demos contained in the file, but you can’t evaluate or modify the file without Mathematica.

The notebooks are mostly raw code with few comments. They are covered by the Creative Commons Attribution – ShareAlike 3.0 License, which means you may use, adapt and distribute the code following the requirements of the license. I am making the files available because I doubt that I will refine them into respectable CDF files any time soon.

[2] I call them “endographs” to avoid confusion with the usual graphs of functions — — drawings of (some of) the set of ordered pairs {x,f(x)} of the function.

[3] This is in contrast to a function defined on a discrete set, where the elements of the domain and codomain can be arranged in any old way. Then the significance of the resulting arrangement of the arrows lies entirely in which two dots they connect. Even then, some things can be seen immediately: Whether the function is a cycle, permutation, an involution, idempotent, and so on.

Of course, the placement of the arrows may tell you more if the finite sets are ordered in a natural way, as for example a function on the integers modulo some integer.

[4] The text [1] uses the cograph representation extensively. The word “cograph” is being used with its standard meaning in category theory. It is used by graph theorists with an entirely different meaning.

[5] It would also be possible to show the domain codomain in the usual {x-y} plane arrangement, with the domain the {x} axis and the codomain the {y} axis. I have not written the code for this yet.

References

[1] Sets for Mathematics, by F. William Lawvere and Robert Rosebrugh. Cambridge University Press, 2003.

[2] Martin Flashman’s website contains many exampls of cographs of functions, which he calls mapping diagrams.

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Skills needed for learning languages and math

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Learning a language involves a variety of skills, and so does learning math. Some skills are apparently needed for both, but others are distinct.

Learning languages and learning math

Some years ago I sat in on a second year college Spanish class. Most of the other students were ages 18-24. The students showed a wide spectrum of ability.

  • Some were quite fluent and conversed easily. Others struggled to put a sentence together.
  • Some had trouble with basic grammar, for example adjective-noun agreement (number and gender). I would not have thought second year students would do that. Some also had trouble with verbs. Spanish verbs are generally difficult, but second year students shouldn’t have trouble with using “canta” with singular subjects and “cantan” with plural ones.
  • Some had trouble reading aloud, stumbling over pronunciation, such putting the accent in the right place in real time and pronouncing some letters correctly (“ll”, “e”, intervocalic “s”). The rules for accent and pronouncing letters are very easy in Spanish, and I was surprised that second year students would have difficulty with them. But the speech of most of them sounded good to me.

I can read Spanish pretty well, but have had very little practice speaking or writing it. I comprehend some of what they say on Univision (soap operas are particularly easy, but I still miss more than half of it), but then I am hard of hearing. I used to have a reasonable ability to speak and understand street German; judging from experience I think it would come back rapidly if we went to live in a German-speaking city again. I can easily read math papers written in Spanish or in German, but I couldn’t come close to giving a math lecture in either language.

Some find learning rules of pronunciation that are different from English very hard, like the Spanish students I mentioned above. I know that some people can’t keep “ei” and “ie” straight in German, and some Russian students find it hard to get used to the Cyrillic alphabet. I find that part of language learning easy. I also find learning grammar and using it in real time fairly easy. I have more difficulty remembering vocabulary.

Learning the new sounds of a language is an entirely different problem from learning the rules of pronunciation.

Mathematical ability

Some difficulties that students have with the symbolic language of math [1] are probably the same kind of difficulties that language students have with learning another language.

When I have taught elementary logic, I usually have a scattering of students who can’t keep the symbols {\land} and {\lor} separate. (See Note [a].) Some even have the same trouble with intersection and union of sets. This is sort of like differentiating “ie” and “ei” in German, except that the latter distinction runs into cognitive dissonance [2] caused by the usual English pronunciation.

Of course, both language students and math students have immense problems with cognitive dissonance in areas other than symbol-learning. For example, many technical words in math have meanings different from ordinary English usage, such as “if”, “group”, and “category”. Language students have difficulties with “false friends” such as “Gift”, which is the German word for “poison”, and very common words such as prepositions, which can have several different translations into English depending on context — and many prepositions in other European languages look like English prepositions. (Note [b]).

On the other hand, some types of mathematical learning seem to involve problems language students don’t run into.

Substitution, for example, appears to me to cause conceptual difficulties that are not like anything in learning language. But I would like to hear examples to the contrary.

If {f(x) = x^2+3x+1}, then {f(x+1)= (x+1)^2+3(x+1)+1}. Is there anything like this in natural languages? And simplifying this to {x^4+5 x^2+5} is not like anything in natural language either — is it?

Is there anything in learning natural languages that is like thinking of an element of a set? Or like the two-level quantification involved in understanding the definition of continuity?

Is there anything in learning math that involves the same kind of difficulty as learning to pronounce a new sound in another language? (Well, making a speech sound involves moving parts of your mouth in three dimensions, and some people find visualizing 3D shapes difficult. But that seems like a stretch to me).

A proposal for investigation

Students show a wide variety of conceptual skills. Some skills seem to be required both in learning mathematics and in learning a foreign language. Others are different. Also, there is a difference between learning school math and learning abstract math at the college level (Note [c]).

TOPIC FOR RESEARCH

  • Identify the types of concept formation that learning a foreign language and learning math have in common.
  • Determine if “being good at languages” and “being good at mathematics” are correlated at the high school level.
  • Ditto for college-level abstract math.

Undoubtedly math teachers and language teachers have written about certain specific issues of the sort I have discussed, but I think we need a systematic comparative investigation of skills involved in the tasks of learning languages and learning math.

I have made proposals for research concerning various other questions with math ed, particularly in connection with linguistics. I will install a new topic “Proposals for research” in my “List of categories” (on the left side of the screen under “Recent posts”) and mark this and other articles that contain such proposals.

Notes

[a]. That is why, in the mathematical reasoning sections of abmath, for example [3], I use the usual English wordings of mathematical assertions instead of systematically using logical symbolism. For many students, introducing symbols and then immediately using them to talk about the subtleties of meaning and usage puts a difficult burden on some of the students. (I do define the symbols in asides).

This may not be the right thing to do. If a student finds it hard to learn to use symbols easily and fluently, should they be studying math?

[b]. I once knew a teenage German who spoke pretty good English, but he could not bear to use the English possessive case. That’s because German young people (assuming I understand this correctly) hate to say things like “Das Auto meines Vaters” and instead say “Das Auto von meinem Vater”. Unfortunately this resulted in his saying in English “The car from my father”, “The girlfriend from my brother” and so on…

[c]. I have been concerned primarily with understanding the difficulties students have when starting to study abstract math after they have had calculus. I have seen many students ace calculus and flunk abstract algebra or logic. There is a wall to fall off of there. The only organization I know of concerned with this is RUME, although it is involved with college calculus as well as what comes after.

References

[1] The symbolic language of math.

[2] Cognitive dissonance.

[3] Conditional assertions.

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Representations 2

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Introduction

In a recent post I began a discussion of the mental, physical and mathematical representations of a mathematical object. The discussion continues here. Mathematicians, linguists, cognitive scientists and math educators have investigate some aspects of this topic, but there are many subtle connections between the different ideas which need to be studied.

I don’t have any overall theoretical grasp of these relationships. What I will do here is grope for an overall theory by mentioning a whole bunch of fine points. Some of these have been discussed in the literature and some (as far as I know) have not been discussed.  Many of them (I hope)  can be described as “an obvious fact about representations but no one has pointed it out before”.  Such fine points could be valuable; I think some scholars who have written about mathematical discourse and math in the classroom are not aware of many of these facts.

I am hoping that by thrashing around like this here (for graphs of functions) and for other concepts (set, function, triangle, number …) some theoretical understanding may emerge of what it means to understand math, do math, and talk about math.

The graph of a function

Let’s look at the graph of the function {y=x^3-x}.

What you are looking at is a physical representation of the graph of the function. The graph creates in your brain a mental representation of the graph of the function. These are subtly related to each other and to the mathematical definition of the graph.

Fine points

  1. The mathematical definition [2] of the graph of this function is: The set of ordered pairs of numbers {(x,x^3-x)} for all real numbers {x}.
  2. In the physical representation, each point {(x,x^3-x)} is shown in a location determined by the conventional {x-y} coordinate system, which uses a straight-line representation of the real numbers with labels and ticks.
    • The physical representation makes use of the fact that the function is continuous. It shows the graph as a curving line rather than a bunch of points.
    • The physical representation you are looking at is not the physical representation I am looking at. They are on different computer screens or pieces of paper. We both expect that the representations are very similar, in some sense physically isomorphic.
    • “Location” on the physical representation is a physical idea. The mathematical location on the mathematical graph is essentially the concept of the physical location refined as the accuracy goes to infinity. (This last statement is a metaphor attached to a genuine mathematical construction, for example Cauchy sequences.)
  3. The mathematical definition of “graph” and the physical representation are related by a metaphor. (See Note 1.)
    • The physical curve in blue in the picture corresponds via the metaphor to the graph in the mathematical sense: in this way, each location on the physical curve corresponds to an ordered pair of the form {(x,x^3-x)}.
    • The correspondence between the locations and the pairs is imperfect. You can’t measure with infinite accuracy.
    • The set of ordered pairs {(x,x^3-x)} form a parametrized curve in the mathematical sense. This curve has zero thickness. The curve in the physical representation has positive thickness.
    • Not all the points in the mathematical graph actually occur on the physical curve: The physical curve doesn’t show the left and right infinite tails.
    • The physical curve is drawn to show some salient characteristics of the curve, such as its extrema and inflection points. This is expected by convention in mathematical writing. If the graph had left out a maximum, for example, the author would be constrained (by convention!) to say so.
    • An experienced mathematician or advanced student understands the fine points just listed. A newbie may not, and may draw false conclusions about the function from the graph. (Note 2.)
  4. If you are a mathematician or at least a math student, seeing the physical graph shown above produces a mental image(see Note 3.) of the graph in your mind.
  5. The mathematical definition and the mental image are connected by a metaphor. This is not the same metaphor as the one that connects the physical representation and the mathematical definition.
    • The curve I visualize in my mental representation has an S shape and so does the physical representation. Or does it? Isn’t the S-ness of the shape a fact I construct mentally (without consciously intending to do so!)?
    • Does the curve in the mental rep have thickness? I am not sure this is a meaningful question. However, if you are a sufficiently sophisticated mathematician, your mental image is annotated with the fact that the curve has zero thickness. (See Note 4.)
    • The curve in your mental image of the curve may very well be blue (just because you just looked at my picture) but you must have an annotation to the effect that that is irrelevant! That is the essence of metaphor: Some things are identified with each other and others are emphatically not identified.
    • The coordinate axes do exist in the physical representation and they don’t exist in the mathematical definition of the graph. Of course they are implied by the definition by the properties of the projection functions from a product. But what about your mental image of the graph? My own image does not show the axes, but I do “know” what the coordinates of some of the points are (for example, {(-1,0)}) and I “see” some points (the local maximum and the local minimum) whose coordinates I can figure out.

Notes

1. This is metaphor in the sense lately used by cognitive scientists, for example in [6]. A metaphor can be described roughly as two mental images in which certain parts of one are identified with certain parts of another, in other words a pushout. The rhetorical use of the word “metaphor” requires it to be a figure of speech expressed in a certain way (the identification is direct rather than expressed by “is like” or some such thing.)  In my use in this article a metaphor is something that occurs in your brain.  The form it takes in speech or writing is not relevant.

2. I have noticed, for example, that some students don’t really understand that the left and right tails go off to infinity horizontally as well as vertically.   In fact, the picture above could mislead someone into thinking the curve has vertical asymptotes: The right tail looks like it goes straight up.  How could it get to x equals a billion if it goes straight up?

3. The “mental image” is of course a physical structure in your brain.  So mental representations are physical representations.

4. I presume this “annotation” is some kind of physical connection between neurons or something.  It is clear that a “mental image” is some sort of physical construction or event in the brain, but from what little I know about cognitive science, the scientists themselves are still arguing about the form of the construction.  I would appreciate more information on this. (If the physical representation of mental images is indeed still controversial, this says nothing bad about cognitive science, which is very new.)

References

[1] Mental Representations in Math (previous post).

[2] Definitions (in abstractmath).

[3] Lakoff, G. and R. E. Núñez (2000), Where Mathematics Comes From. Basic Books.

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The Mathematical Definition of Function

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Introduction

This post is a completely rewritten version of the abstractmath article on the definition of function. Like every part of abstractmath, the chapter on functions is designed to get you started thinking about functions. It is no way complete. Wikipedia has much more complete coverage of mathematical functions, but be aware that the coverage is scattered over many articles.

The concept of function in mathematics is as important as any mathematical idea. The mathematician’s concept of function includes the kinds of functions you studied in calculus but is much more abstract and general. If you are new to abstract math you need to know:

  • The precise meaning of the word “function” and other concepts associated with functions. That’s what this section is about.
  • Notation and terminology for functions. (That will be a separate section of abstractmath.org which I will post soon.)
  • The many different kinds of functions there are. (See Examples of Functions in abmath).
  • The many ways mathematicians think about functions. The abmath article Images and Metaphors for Functions is a stub for this.

I will use two running examples throughout this discussion:

  • {F} is the function defined on the set {\left\{1,\,2,3,6 \right\}} as follows: {F(1)=3,\,\,\,F(2)=3,\,\,\,F(3)=2,\,\,\,F(6)=1}. This is a function defined on a finite set by explicitly naming each value.
  • {G} is the real-valued function defined by the formula {G(x)={{x}^{2}}+2x+5}.

Specification of function

We start by giving a specification of “function”. (See the abstractmath article on specification.) After that, we get into the technicalities of the definitions of the general concept of function.

Specification: A function {f} is a mathematical object which determines and is completely determined bythe following data:

  • {f} has a domain, which is a set. The domain may be denoted by {\text{dom }f}.
  • {f} has a codomain, which is also a set and may be denoted by {\text{cod }f}.
  • For each element {a} of the domain of {f}, {f} has a value at {a}, denoted by {f(a)}.
  • The value of {f} at {a} is completely determined by {a} and {f} .
  • The value of {f} at {a} must be an element of the codomain of {f}.

The operation of finding {f(a)} given {f} and {a} is called evaluation.

Examples

  • The definition above of the finite function {F} specifies that the domain is the set {\left\{1,\,2,\,3,\,6 \right\}}. The value of {F} at each element of the domain is given explicitly. The value at 3, for example, is 2, because the definition says that {F(2) = 3}. The codomain of {F} is not specified, but must include the set {\{1,2,3\}}.
  • The definition of {G} above gives the value at each element of the domain by a formula. The value at 3, for example, is {G(3)=3^2+2\cdot3+5=20}. The definition does not specify the domain or the codomain. The convention in the case of functions defined on the real numbers by a formula is to take the domain to be all real numbers at which the formula is defined. In this case, that is every real number, so the domain is {{\mathbb R}}. The codomain must include all real numbers greater than or equal to 4. (Why?)

Comment: The formula above that defines the function G in fact defines a function of complex numbers (even quaternions).

Definition of function

In the nineteenth century, mathematicians realized that it was necessary for some purposes (particularly harmonic analysis) to give a mathematical definition of the concept of function. A stricter version of this definition turned out to be necessary in algebraic topology and other fields, and that is the one I give here.

To state this definition we need a preliminary idea.

The functional property

A set R of ordered pairs has the functional property if two pairs in R with the same first coordinate have to have the same second coordinate (which means they are the same pair).

Examples

  • The set {\{(1,2), (2,4), (3,2), (5,8)\}} has the functional property, since no two different pairs have the same first coordinate. It is true that two of them have the same second coordinate, but that is irrelevant.
  • The set {\{(1,2), (2,4), (3,2), (2,8)\}} does not have the functional property. There are two different pairs with first coordinate 2.
  • The graphs of functions in beginning calculus have the functional property.
  • The empty set {\emptyset} has the functional property .

Example: Graph of a function defined by a formula

The graph of the function {G} given above has the functional property. The graph is the set

\displaystyle \left\{ (x,{{x}^{2}}+2x+5)\,\mathsf{|}\,x\in {\mathbb R} \right\}.

If you repeatedly plug in one real number over and over, you get out the same real number every time. Example:

  • if {x = 0}, then {{{x}^{2}}+2x+5=5}.  You get 5 every time you plug in 0.
  • if {x = 1}, then {{{x}^{2}}+2x+5=8}.
  • if {x =-2}, then {{{x}^{2}}+2x+5=5}.

This set has the functional property because if {x} is any real number, the formula {{{x}^{2}}+2x+5} defines a specific real number. (This description of the graph implicitly assumes that {\text{dom } G={\mathbb R}}.)  No other pair whose first coordinate is {-2} is in the graph of {G}, only {(-2, 5)}. That is because when you plug {-2} into the formula {{{x}^{2}}+2x+5}, you get {5} every time. Of course, {(0, 5)} is in the graph, but that does not contradict the functional property. {(0, 5)} and {(-2, 5)} have the same second coordinate, but that is OK.

How to think about the functional property

The point of the functional property is that for any pair in the set of ordered pairs, the first coordinate determines what the second one is. That’s why you can write “{G(x)}” for any {x } in the domain of {G} and not be ambiguous.

Mathematical definition of function

A function{f} is a mathematical structure consisting of the following objects:

  • A set called the domain of {f}, denoted by {\text{dom } f}.
  • A set called the codomain of {f}, denoted by {\text{cod } f}.
  • A set of ordered pairs called the graph of { f}, with the following properties:
    • {\text{dom } f} is the set of all first coordinates of pairs in the graph of {f}.
    • Every second coordinate of a pair in the graph of {f} is in {\text{cod } f} (but {\text{cod } f} may contain other elements).
    • The graph of {f} has the functional property. Using arrow notation, this implies that {f:A\rightarrow B}.

Examples

  • Let {F} have graph {\{(1,2), (2,4), (3,2), (5,8)\}} and define {A = \{1, 2, 3, 5\}} and {B = \{2, 4, 8\}}. Then {F:A\rightarrow B} is a function.
  • Let {G} have graph {\{(1,2), (2,4), (3,2), (5,8)\}} (same as above), and define {A = \{1, 2, 3, 5\}} and {C = \{2, 4, 8, 9, 11, \pi, 3/2\}}. Then {G:A\rightarrow C} is a (admittedly ridiculous) function. Note that all the second coordinates of the graph are in {C}, along with a bunch of miscellaneous suspicious characters that are not second coordinates of pairs in the graph.
  • Let {H} have graph {\{(1,2), (2,4), (3,2), (5,8)\}}. Then {H:A\rightarrow {\mathbb R}} is a function.

According to the definition of function, {F}, {G} and {H} are three different functions.

Identity and inclusion

Suppose we have two sets A and B with {A\subseteq B}.

  • The identity function on A is the function {{{\text{id}}_{A}}:A\rightarrow A} defined by {{{\text{id}}_{A}}(x)=x} for all{x\in A}. (Many authors call it {{{1}_{A}}}).
  • The inclusion function from A to B is the function {i:A\rightarrow B} defined by {i(x)=x} for all {x\in A}. Note that there is a different function for each pair of sets A and B for which {A\subseteq B}. Some authors call it {{{i}_{A,\,B}}} or {\text{in}{{\text{c}}_{A,\,B}}}.

Remark The identity function and an inclusion function for the same set A have exactly the same graph, namely {\left\{ (a,a)|a\in A \right\}}.

Graphs and functions

  • If {f} is a function, the domain of {f} is the set of first coordinates of all the pairs in {f}.
  • If {x\in \text{dom } f}, then {f(x)} is the second coordinate of the only ordered pair in {f} whose first coordinate is {x}.

Examples

The set {\{(1,2), (2,4), (3,2), (5,8)\}} has the functional property, so it is the graph of a function. Call the function {H}. Then its domain is {\{1,2,3,5\}} and {H(1) = 2} and {H(2) = 4}. {H(4)} is not defined because there is no ordered pair in H beginning with {4} (hence {4} is not in {\text{dom } H}.)

I showed above that the graph of the function {G}, ordinarily described as “the function {G(x)={{x}^{2}}+2x+5}”, has the functional property. The specification of function requires that we say what the domain is and what the value is at each point. These two facts are determined by the graph.

Other definitions of function

Because of the examples above, many authors define a function as a graph with the functional property. Now, the graph of a function {G} may be denoted by {\Gamma(G)}.  This is an older, less strict definition of function that doesn’t work correctly with the concepts of algebraic topology, category theory, and some other branches of mathematics.

For this less strict definition of function, {G=\Gamma(G)}, which causes a clash of our mental images of “graph” and “function”. In every important way except the less-strict definition, they ARE different!

A definition is a device for making the meaning of math technical terms precise. When a mathematician think of “function” they think of many aspects of functions, such as a map of one shape into another, a graph in the real plane, a computational process, a renaming, and so on. One of the ways of thinking of a function is to think about its graph. That happens to be the best way to define the concept of function.  (It is the less strict definition and it is a necessary concept in the modern definition given here.)

The occurrence of the graph in either definition doesn’t make thinking of a function in terms of its graph the most important way of visualizing  it. I don’t think it is even in the top three.

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Operation as metaphor in math

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Operation: Is it just a name or is there a metaphor behind it?

A function of the form {f:S\times S\rightarrow S} may be called a binary operation on {S}. The main point to notice is that it takes pairs of elements of {S} to the same set {S}.

A binary operation is a special case of n-ary operation for any natural number {n}, which is a function of the form {f:S^n\rightarrow S}. A {1}-ary (unary) operation on {S} is a function from a set to itself (such as the map that takes an element of a group to its inverse), and a {0}-ary (nullary) operation on {S} is a constant.

It is useful at times to consider multisorted algebra, where a binary operation can be a function {f:S_1\times S_2\rightarrow S_3} where the {S_i} are possibly different sets. Then a unary operation is simply a function.

Calling a function a multisorted unary operation suggest a different way of thinking about it, but as far as I can tell the difference is only that the author is thinking of algebraic operations as examples. This does not seem to be a different metaphor the way “function as map” and “function as transformation” are different metaphors. Am I missing something?

In the 1960’s some mathematicians (not algebraists) were taken aback by the idea that addition of real numbers (for example) is a function. I observed this personally. I don’t think any mathematician would react this way today.

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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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