All posts by SixWingedSeraph

Charles Wells. Professor Emeritus at Case Western Reserve University. Now living in Minneapolis, Minnesota, USA. CWRU provides support for this blog with software and library privileges.
abstractmath.org, exposition, Handbook, language, language of math, math

The meaning of the word “superposition”

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

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

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

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

The article about superposition in All about circuits says:

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

Superposition Theorem in Wikipedia:

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

Quantum superposition in Wikipedia:  

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

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

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

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

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

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

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

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abstractmath.org, exposition, math, Mathematica, Proposals for research, representations, Uncategorized, understanding math

Offloading chunking

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In my previous post I wrote about the idea of offloading abstraction, the sort of things we do with geometric figures, diagrams (that post emphasized manipulable diagrams), drawing the tree of an algebraic expression, and so on.  This post describes a way to offload chunking.  

Chunking

I am talking about chunking in the sense of encapsulation, as some math ed. people use it.  I wrote about it briefly in [1], and [2] describes the general idea.  I don't have a good math ed reference for it, but I will include references if readers supply them.  

Chunking for some educators means breaking a complicated problem down into pieces and concentrating on them one by one.  That is not really the same thing as what I am writing about.  Chunking as I mean it enables you to think more coherently and efficiently about a complicated mathematical structure by objectifying some of the data in the structure.  

Project 

This project an example of how chunking could be made visible in interactive diagrams, so that the reader grasps the idea of chunking.  I guess I am chunking chunking.  

Here is a short version of an example of chunking worked out in ridiculous detail in reference [1]. 

Let \[f(x)=.0002{{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}\]  How do I know it is never negative?  Well, because it has the form (a positive number)(times)(something)$^6$.    Now (something)$^6$ is ((something)$^3)^2$ and a square is always nonnegative, so the function is (positive)(times)(nonnegative), so it has to be nonnegative.  

I recognized a salient fact about .0002, namely that it was positive: I grayed out (in my mind) its exact value, which is irrelevant.  I also noticed a salient fact about \[{{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}\] namely that it was (a big mess that I grayed out)(to the 6th power).  And proceeded from there.  (And my chunking was inefficient; for example, it is more to the point that .0002 is nonnegative).

I believe you could make a movie of chunking like this using Mathematica CDF.  You would start with the formula, and then as the voiceover said "what's really important is that .0002 is nonnegative" the number would turn into a gray cloud with a thought balloon aimed at it saying "nonnegative".  The other part would turn into a gray cloud to the sixth, then the six would break into 3 times 2 as the voice comments on what is happening.  

It would take a considerable amount of work to carry this out.  Lots of decisions would need to be made.  

One problem is that Mathematica doesn't provide a way to do voiceovers directly (as far as I know).  Perhaps you could make a screen movie using screenshot software in real time while you talked and (offscreen) pushed buttons that made the various changes happen.

You could also do it with print instead of voiceover, as I did in the example in this post. In this case you need to arrange to have the printed part and the diagram simultaneously visible.  

I may someday try my hand at this.  But I would encourage others to attack this project if it interests them.  This whole blog is covered by the Creative Commons Attribution – ShareAlike 3.0 License", which means you may use, adapt and distribute the work freely provided you follow the requirements of the license.

I have other projects in mind that I will post separately.

References

  1. Abstractmath article on chunking.
  2. Wikipedia on chunking
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exposition, math, Mathematica, representations, understanding math

Offloading abstraction

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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.


The diagram above shows you the tangent line to the curve $y=x^3-x$ at a specific point.  The slider allows you to move the point around, and the tangent line moves with it. You can click on one of the plus signs for options about things you can do with the slider.  (Note: This is not new.  Many other people have produced diagrams like this one.)

I have some comments to make about this very simple diagram. I hope they raise your consciousness about what is going on when you use a manipulable demonstration.

Farming out your abstraction load

A diagram showing a tangent line drawn on the board or in a paper book requires you visualize how the tangent line would look at other points.  This imposes a burden of visualization on you.  Even if you are a new student you won't find that terribly hard (am I wrong?) but you might miss some things at first:

  • There are places where the tangent line is horizontal.
  • There are places where some of the tangent lines cross the curve at another point. Many calculus students believe in the myth that the tangent line crosses the curve at only one point.  (It is not really a myth, it is a lie.  Any decent myth contains illuminating stories and metaphors.)
  • You may not envision (until you have some experience anyway) how when you move the tangent line around it sort of rocks like a seesaw.

You see these things immediately when you manipulate the slider.

Manipulating the slider reduces the load of abstract thinking in your learning process.     You have less to keep in your memory; some of the abstract thinking is offloaded onto the diagram.  This could be described as contracting out (from your head to the picture) part of the visualization process.  (Visualizing something in your head is a form of abstraction.)

Of course, reading and writing does that, too.  And even a static graph of a function lowers your visualization load.  What interactive diagrams give the student is a new tool for offloading abstraction.

You can also think of it as providing external chunking.  (I'll have to think about that more…)

Simple manipulative diagrams vs. complicated ones

The diagram above is very simple with no bells and whistles.  People have come up with much more complicated diagrams to illustrate a mathematical point.  Such diagrams:

  • May give you buttons that give you a choice of several curves that show the tangent line.
  • May give a numerical table that shows things like the slope or intercept of the current tangent line.
  • May also show the graph of the derivative, enabling you to see that it is in fact giving the value of the slope.

Such complicated diagrams are better suited for the student to play with at home, or to play with in class with a partner (much better than doing it by yourself).  When the teacher first explains a concept, the diagrams ought to be simple.

Examples

  • The Definition of derivative demo (from the Wolfram Demonstration Project) is an example that provides a table that shows the current values of some parameters that depend on the position of the slider.
  • The Wolfram demo Graphs of Taylor Polynomials is a good example of a demo to take home and experiment extensively with.  It gives buttons to choose different functions, a slider to choose the expansion point, another one to choose the number of Taylor polynomials, and other things.
  • On the other hand, the Wolfram demo Tangent to a Curve is very simple and differs from the one above in one respect: It shows only a finite piece of the tangent line.  That actually has a very different philosophical basis: it is representing for you the stalk of the tangent space at that point (the infinitesimal vector that contains the essence of the tangent line).
  • Brian Hayes wrote an article in American Scientist containing a moving graph (it moves only  on the website, not in the paper version!) that shows the changes of the population of the world by bars representing age groups.  This makes it much easier to visualize what happens over time.  Each age group moves up the graph — and shrinks until it disappears around age 100 — step by step.  If you have only the printed version, you have to imagine that happening.  The printed version requires more abstract visualization than the moving version.
  • Evaluating an algebraic expression requires seeing the abstract structure of the expression, which can be shown as a tree.  I would expect that if the students could automatically generate the tree (as you can in Mathematica)  they would retain the picture when working with an expression.  In my post computable algebraic expressions in tree form I show how you could turn the tree into an evaluation aid.  See also my post Syntax trees.

This blog has a category "Mathematica" which contains all the graphs (many of the interactive) that are designed as an aid to offloading abstraction.

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exposition, math, Sketches and forms, Uncategorized

Idempotents by sketches and forms

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This post provides a detailed description of an example of a mathematical structure presented as a sketch and as a form.  It is a supplement to my article An Introduction to forms.  Most of the constructions I mention here are given in more detail in that article.
 
It helps in reading this post to be familiar with the basic ideas of category, including commutative diagram and limit cone, and of the concepts of logical theory and model in logic.
 

Sketches and forms

A sketch of a mathematical structure is a collection of objects and arrows that make up a digraph (directed graph), together with some specified cones, cocones and diagrams in the digraph.  A model of the sketch is a digraph morphism from the digraph to some category that takes the cones to limit cones, the cocones to colimit cocones, and the diagrams to commutative diagrams.  A morphism of models of a sketch from one model to another in the same category is a natural transformation.  Sketches can be used to define all kinds of algebraic structures in the sense of universal algebra, and many other types of structures (including many types of categories).  

There are many structures that sketches cannot sketch.  Forms were first defined in [4].  They can define anything a sketch can define and lots of other things.  [5] gives a leisurely description of forms suitable for people who have a little bit of knowledge of categories and [1] gives a more thorough description.  

An idempotent is a very simple kind of algebraic structure.  Here I will describe both a sketch and a form for idempotents. In another post I will do the same for binops (magmas).

Idempotent

An idempotent is a unary operation $u$ for which $u^2=u$.

  • If $u$ is a morphism in a category whose morphisms are set functions, a function $u:S\to S$ is an idempotent if $u(u(x))=u(x)$ for all $x$ in the domain.  
  • Any identity element in any category is an idempotent.
  • A nontrivial example is the function $u(x,y):=(-x,0)$ on the real plane.  

Any idempotent $u$ makes the following diagram commute

and that diagram can be taken as the definition of idempotent in any category.

The diagram is in green.  In this post (and in [5]) diagrams in the category of models of a sketch or a form are shown in green.

A sketch for idempotents

The sketch for idempotents contains a digraph with one object and one arrow from that object to itself (above left) and one diagram (above right).  It has no cones or cocones.  So this is an almost trivial example.  When being expository (well, I can hardly say "when you are exposing") your first example should not be trivial, but it should be easy.  Let's call the sketch $\mathcal{S}$.

  • The diagram looks the same as the green diagram above.  It is in black, because I am showing things in syntax (things in sketches and forms) in black and semantics (things in categories of models) in green.
  • The green diagram is a commutative diagram in some category (unspecified).  
  • The black diagram is a diagram in a digraph. It doesn't make sense to say it is commutative because digraphs don't have composition of arrows.
  • Each sketch has a specific digraph and lists of specific diagrams, cones and cocones.  The left digraph above is not in the list of diagrams of $\mathcal{S}$ (see below).

The definition of sketch says that every diagram in the official list of diagrams of a given sketch must become a commutative diagram in a model.  This use of the word "become" means in this case that a model must be a digraph morphism $M:\mathcal{S}\to\mathcal{C}$ for some category $\mathcal{C}$ for which the diagram below commutes.

This sketch generates a category called the Theory ("Cattheory" in [5]) of the sketch $\mathcal{S}$, denoted by $\text{Th}(\mathcal{S})$.  It is roughly the "smallest" category containing $f$ and $C$ for which the diagrams in $\mathcal{S}$ are commutative.  
 
This theory contains the generic model $G:\mathcal{S}\to \text{Th}(\mathcal{S})$ that takes $f$ and $C$ to themselves.
  • $G$ is "generic" because anything you prove about $G$ is true of every model of $\mathcal{S}$ in any category.
  • In particular, in the category $\text{Th}(\mathcal{S})$, $G(f)\circ G(f)=G(f)$.  
  • $G$ is a universal morphism in the sense of category theory: It lifts any model $M:\mathcal{S}\to\mathcal{C}$ to a unique functor $\bar{M}=M\circ G:\text{Th}(\mathcal{S})\to\mathcal{C}$ which can therefore be regarded as the same model.  See Note [2].
SInce models are functors, morphisms between models are natural transformations.  This gives what you would normally call homomorphisms for models of almost any sketchable structure.  In [2] you can find a sketch for groups, and indeed the natural transformations between models are group homomorphisms.

Sketching categories

You can sketch categories with a sketch CatSk containing diagrams and cones, but no cocones.  This is done in detail in [3]. The resulting theory $\text{Th}(\mathbf{CatSk})$ is required to be the least category-with-finite-limits generated by $\mathcal{S}$ with the diagrams becoming commutative diagrams and the cones becoming limit cones.  This theory is the FL-Theory for categories, which I will call ThCat (suppressing mention of FL).  

Doctrines

In general the theory of a particular kind of structure contains a parameter that denotes its doctrine. The sketch $\mathcal{S}$ for idempotents didn't require cones, but you can construct theories $\text{Th}(\mathcal{S})$, $\text{Th} (\text{FP},\mathcal{S})$ and $\text{Th}(\text{FL},\mathcal{S})$ for idempotents (FP means it is a category with finite products).  

In a strong sense, all these theories have the same models, namely idempotents, but the doctrine of the theory allows you to use more mechanisms for proving properties of idempotents.  (The doctrine for $\text{Th}(\mathcal{S})$ provides for equational proofs for unary operations only, a doctrine which has no common name such as FP or FS.)  The paper [1] is devoted to explicating proof in the context of forms, using graphs and diagrams instead of formulas that are strings of symbols.

Describing composable pairs of arrows

The form for any type of structure is constructed using the FL theory for some type of category, for example category with all limits, cartesian closed category, topos, and so on.  The form for idempotents can be constructed in ThCat (no extra structure needed).  The form for reflexive function spaces (for example) needs the FL theory for cartesian closed categories (see [5]).

Such an FL theory must contain objects $\text{ob}$ and $\text{ar}$ that become the set of objects and the set of arrows of the category that a model produces.  (Since FL theories have models in any category with finite limits, I could have said "object of objects" and "object of arrows".  But in this post I will talk about only models in Set.)

ThCat contains an object  $\text{ar}_2$ that represents composable pairs of arrows.  That requires a cone to define it:

This must become a limit cone in a model.

  • I usually show cones in blue. 
  • $\text{dom}$ and $\text{cod}$ give (in a model) the domain and codomain of an arrow.
  • $\text{lfac}$ gives the left factor and $\text{rfac}$ gives the right factor. It is usually useful to give suggestive names to some of the projections in situations like this, since they will be used elsewhere (where they will be black!).
  • The objects and arrows in the diagram (including $\text{ar}_2$) are already members of the FL theory for categories.
  • This diagram is annotated in green with sample names of objects and arrows that might exist in a model.  Atish and I introduced that annotation system in [1] to help you chase the diagram and think about what it means.

This cone is a graph-based description of the object of composable arrows in a category (as opposed to a linguistic or string-based description).

Describing endomorphisms

Now an idempotent must be an endomorphism, so we provide a cone describing the object of endomorphisms in a category. This cone already exists in the FL theory for categories.

  • $\text{loop}$ is a monomorphism (in fact a regular mono because it is the mono produced by an equalizer) so it is not unreasonable to give the element annotation for $\text{endo}$ and $\text{ar}$ the same name.
  • "$\text{dc}$" takes $f$ to its domain and codomain. 
  • $\text{loop}$ and "$\text{dc}$" were not created when I produced the cone above.  They were already in the FL theory for categories.
 
Since the cone defining $\text{ar}_2$ is a limit cone (in the Theory, not in a model), if you have any other commutative cone (purple) to that cone, a unique arrow (red) $\text{diag}$ automatically is present as shown below:

This particular purple cone is the limit cone defining $\text{endo}$ just defined.  Now $\text{diag}$ is a specific arrow in the FL theory for categories. In a model of the theory (which is a category in Set or in some other category) takes an endomorphism to the corresponding pair of composable arrows.

The object of idempotents

Now using these arrows we can define the object $\text{idm}$ of idempotents using the diagram below. See Note [3].

 

 

 

 

 

Idm is an object in ThCat.  In any category, in other words in any model of ThCat, idm becomes the set of idempotent arrows in that category.

In the terminology of [5], the object idm is the form for idempotents, and the cone it is the limit of is the description of idempotent.  

Now take ThCat and adjoin an arrow $g:1\to\text{idm}$.  You get a new FL category I will call the FL-theory of the form for idempotents.  A model of the theory of the form in Set  is a category with a specified idempotent. A particular example of a model of the form idm in the category of real linear vector spaces is the map $u(x,y):=(-x,0)$ of the (set of points of) the real plane to itself (it is an idempotent endomorphism of $\textbf{R}^2$).  

This example is typical of forms and their models, except in one way:  Idempotents are also sketchable, as I described above.  Many mathematical structures can be perceived as models of forms, but not models of sketches, such as reflexive function spaces as in [5].

Notes

[1] The diagrams shown in this post were drawn in Mathematica.  The code for them is shown in the notebook SketchFormExamples.nb .  I am in the early stages of developing a package for drawing categorical diagrams in Mathematica, so this notebook shows the diagrams defined in very primitive machine-code-like Mathematica.  The package will not rival xypic for TeX any time soon.  I am doing it so I can produce diagrams (including 3D diagrams) you can manipulate.

[2] In practice I would refer to the names of the objects and arrows in the sketch rather than using the M notation:  I might write $f\circ f=f$ instead of $M(f)\circ M(f)=M(f)$ for example.  Of course this confuses syntax with semantics, which sounds like a Grievous Sin, but it is similar to what we do all the time in writing math:  "In a semigroup, $x$ is an idempotent if $xx=x$."  We use same notation for the binary operation for any semigroup and we use $x$ as an arbitrary element of most anything.  Actually, if I write $f\circ f=f$ I can claim I am talking in the generic model, since any statement true in the generic model is true in any model.  So there.

[3] In the Mathematica notebook SketchFormExamples.nb in which I drew these diagrams, this diagram is plotted in Euclidean 3-space and can be viewed from different viewpoints by running your cursor over it.

References

[1] Atish Bagchi and Charles Wells, Graph-Base Logic and Sketches, draft, September 2008, on ArXiv.

[2] Michael Barr and Charles Wells, Category Theory for Computing Science (1999). Les Publications CRM, Montreal (publication PM023).

[3] Michael Barr and Charles Wells, Toposes, Triples and Theories (2005). Reprints in Theory and Applications of Categories 1.

[4] Charles Wells, A generalization of the concept of sketch, Theoretical Computer Science 70, 1990

[5] Charles Wells, An Introduction to forms.

 

 

 

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computer science, language, language of math, math, Mathematica

Bugs in English and in math

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Everyone knows that computer programs have bugs.  In fact, languages have bugs, too, although we don't usually call them that.  

Bugs in English 

  

Right

Q: "Should I turn left at the next corner?" A: "Right".  Probably most Americans who drive now know this bug.  The answer could mean "yes" or "turn right".  So we have to stop and think how to answer this question.  That makes it a bug.  

Too, two

Comment: " We will take Route 30".  Answer: "We will take Route 30 too".  This bug is probably responsible for the survival of the word "also".  

Note that unlike the case of "right", this is a bug only of spoken English.

Subject and predicate

In Comma rule found dysfunctional, I wrote about the problem that in formal English writing there is no way to indicate where the subject ends and the predicate begins.  This causes a problem reading complicated sentences with many clauses such as academic writing often uses.  Of course, one way around this is to write short, simple sentences!  (That sounds like the subject of a future blog…) 

Bugs in the symbolic language of math

  

Fractions

In both Excel and Mathematica, "1/2*3" means 3/2. Now, I would think "1/2a" means "1/(2a)", but younger mathematicians are taught PEMDAS (see Purplemath), which says that division and multiplication have the same precedence and operations are evaluated from left to right.  

 If in Mathematica you define a function f[a_] := 1/2a, f[3] evaluates to 3/2, so Mathematica (and most other computer languages) agree with PEMDAS. (Note: When you write 1/2a in a Mathematica notebook, it automatically puts a space between the 2 and the a, and space in Mathematica means times, so it does warn you.)

Nevertheless, my ancient education would lead me to write (1/2)a for that meaning.  This means I must learn to write 1/(2a) for the other meaning instead of 1/2a.  

Questions:

  • Did the language really change or was I always "doing it wrong"?  I would like to hear from other ancient mathematicians.  (But I don't know very many who would read blogs or Purplemath.)
  • Should such a phenomenon be called a bug? 

Repeated exponentiation

In Excel, "2^2^3" means $(2^2)^3$, in other words, 64.  In Mathematica, it means $2^{(2^3)}=2^8=256$.  My impression is that most mathematicians expect it to mean $2^{(2^3)}$.  

References: This post in Walking Randomly, my post Mathematical Usage, Wikipedia's article.  

Exponentiation on functions is ambiguous

If $f:\mathbb{R}\to\mathbb{R}$ is a function, $f^2(x)$ can mean either $f(f(x))$ or $f(x)f(x)$, and both usages are common.  You should tell your students about this because no one is ever going to make one of the usages go away.

A far worse catastrophe is the fact that in calculus books, $\sin^2x=(\sin\,x)(\sin\,x)$ but $\sin^{-1}x=\text{arcsin}\,x$.  I betcha (lived in Minnesota four years now) we could succeed with a campaign to convince calc book publishers to always write $(\sin\,x)^2$ and $\arcsin\,x$.  

Bugs in the Mathematical Dialect of English

The mathematical dialect of English is what I call Mathematical English in the abstractmath website.  It is a different language from the symbolic language, which is not a dialect of English.

I have written about the problems with Mathematical English in a ridiculous number of places.  (See references in The Handbook of Mathematical Discourse).  It is normal for a dialect of a language to use words and grammatical structures that in the original language mean different things.  (See Dialects below).

Words with different meanings

  • A set is a group in standard English, but not in math English.  
  • The number 2+3i is a real number in standard English, but not in math English.  
  • And so on.

Use of adjectives and prefixes

  • A "noncommutative ring" has commutative addition.
  • A "semigroup" has a fully defined binary operation.

If, then

The bug that grabs math newbies by the throat and won't let go is the meaning of "If P, then Q".  

  • "If a number is divisible by 4, then it is even" in math dialect means a number not divisible by 4 might be even anyway.
  • "If you eat your broccoli you will get your dessert" in standard American Parental English does not mean you might get your dessert if you don't eat your broccoli.

And then there is the phenomenon of Vacuous Implication, which leaves students gasping and writhing.

About "dialects"

Most Americans are not familiar with dialects in the sense I am using the word here, since the only really different dialects we have are Gullah and Hawaiian Pidgin, both of which are very hard to understand; although for example Appalachian English and African-American urban vernacular [1] are dialects of a milder sort.  I grew up in Savannah and heard diluted Gullah sometimes on the street (didn't understand much).  I am also rather familiar with Züritüütsch since we lived in Zürich for a year.   

What the rest of the world call dialects have many distinctive properties:

  • They have nonstandard pronunciation to the point where they are difficult to understand. 
  • They have differences in grammar.  (Both Gullah and especially Hawaiian Creole have differences in grammar from Standard English.) 
  • They have differences in vocabulary, enough sometimes to cause misunderstanding.

I grew up speaking an Atlanta dialect, which really did have differences in all those parameters.  But what people today call a Southern accent is really just an accent (minor variations in pronunciation), not a dialect.  

Hawaiian Creole, and possibly Gullah, but not the other dialects I mentioned, are singled out by linguists as creoles because they been modified heavy influence from another language.  Züritüütsch is not a creole, but it is quite difficult for native German-speakers to understand.  The Swiss situation particularly emphasizes the distinction between "dialect" and "accent".  The typical native of Zürich speaks Züritüütsch and also speaks standard German with a Swiss accent.  

Reference

[1] What Language Is (And What It Isn't and What It Could Be) by John H. McWhorter. Gotham, 2011.

 

 

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category theory, computer science, math, Sketches and forms

An Introduction to Forms

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In 2009, I wrote a sequence of posts on this blog explaining the concept of form that I introduced in [1].  I have now updated and combined them into an article [2].  The posts no longer exist on the blog. The article contains links to other papers on forms.

[1] A generalization of the concept of sketch, Theoretical Computer Science 70, 1990.

[2] An Introduction to forms.

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Abstract objects

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Some thoughts toward revising my article on mathematical objects.  

Mathematical objects are a kind of abstract object.  There are lots of abstract objects that are not mathematical objects,  For example, if you keep a calendar or schedule for appointments, that calendar is an abstract object.  (This example comes from [2]). 

It may be represented as a physical object or you may keep it entirely in your head.  I am not going to talk about the latter possibility, because I don't know what to say.

  1. If it is a paper calendar, that physical object represents the information that is contained in your calendar.  
  2. Same for a calendar on a computer, but that is stored as magnetic bits on a disk or in flash memory. A computer program (part of the operating system) is required to present it on the screen in such a way that you can read it.  Each time you open it, you get a new physical representation of the calendar.

Your brain contains a module (see [5], [7]) that interprets the representation in (1) or (2) and which has connections with other modules in your brain for dates, times, locations and whether the appointment is for a committee, a medical exam, or whatever.  

The calendar-interpreter module in your brain is necessary for the physical object to be a calendar.  The physical object is not in itself your calendar.  The calendar in this sense does not exist in the physical world.  It is abstract.  Since we think of it as a thing, it is an abstract object.

The abstract object "my calendar" affects the physical world (it causes you to go to the dentist next Tuesday).  The relation of the abstract object to the physical world is mediated by whatever physical object you call your calendar along with the modules in the brain that relate to it.  The modules in the brain are actions by physical objects, so this point of view does not involve Cartesian style dualism.

Note:  A module is a meme.  Are all memes modules?  This needs to be investigated.  Whatever they are, they exist as physical objects in people's brains.

Mathematical objects

A rigorous proof of a theorem about a mathematical object tends to refer to the object as if it were absolutely static and did not affect anything in the physical world.  I talked about this in [10], where I called it the dry bones representation of a mathematical object.  Mathematical objects don't have to be thought of this way, but (I suggest) what makes them mathematical objects is that they can be thought of in dry bones mode.  

If you use calculus to figure out how much fuel to use in a rocket to make it go a mile high, then actually use that amount in the rocket and send it off, your calculations have affected your physical actions, so you were thinking of the calculations as an abstract object.  But if you sit down to check your calculations, you concentrate on the steps one by one with the rules of algebra and calculus in mind.  You are looking at them as inert objects, like you would look at a bone of a dinosaur to see what species it belongs to. From that point of view your calculations form a mathematical object, because you are using the dry-bones approach.

Caveat

All this blather is about how you should think about mathematical objects.  It can be read as philosophy, but I have no intention of defending it as philosophy.  People learning abstract math at college level have a lot of trouble thinking about mathematical objects as objects, and my intention is to start clarifying some aspects of how you think about them in different circumstances.  (The operative word is "start" — there is a lot more to be said.)

About the exposition of this post (a commercial)

You will notice that I gave examples of abstract objects but did not define the word "abstract object".  I did the same with mathematical objects.  In both cases, I put the word "abstract object" or "mathematical object" in boldface at a suitable place in the exposition.

That is not the way it is done in math, where you usually make the definition of a word in a formal way, marking it as Definition, putting the word in bold or italics, and listing the attributes it must have.  I want to point out two things:

  • For the most part, that behavior is peculiar to mathematics.
  • This post is not a presentation of mathematical ideas.  

This gives me an opportunity for a commercial:  Read what we have written about definitions in References [1], [3] and [4].

References

  1. Atish Bagchi and Charles Wells, Varieties of Mathematical Prose, 1998.
  2. Reuben Hersh, What is mathematics, really? Oxford University Press, 1997. 
  3. Charles Wells, Handbook of Mathematical Discourse.
  4. Charles Wells, Mathematical objects in abstractmath.org
  5. Math and modules of the mind (previous post)
  6. Mathematical Concepts (previous post)
  7. Thinking about abstract math (previous post)
  8. Terrence W. Deacon, Incomplete Nature.  W. W. Norton, 2012. [I have read only a little of this book so far, but I think he is talking about abstract objects in the sense I have described above.]
  9. Gideon Rosen, Abstract Objects.  Stanford Encyclopedia of Philosophy.
  10. Representations II: Dry Bones (previous post)

 

http://plato.stanford.edu/entries/abstract-objects/

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Whole numbers

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Sue Van Hattum wrote in response to a recent post:

I’d like to know what you think of my ‘abuse of terminology’. I teach at a community college, and I sometimes use incorrect terms (and tell the students I’m doing so), because they feel more aligned with common sense.

To me, and to most students, the phrase “whole numbers” sounds like it refers to anything that doesn’t need fractions to represent it, and should include negative numbers. (It then, of course, would mean the same thing that the word integers does.) So I try to avoid the phrase, mostly. But I sometimes say we’ll use it with the common sense meaning, not the official math meaning.

Her comments brought up a couple of things I want to blather about.

Official meaning

There is no such thing as an "official math meaning".  Mathematical notation has no governing authority and research mathematicians are too ornery to go along with one anyway.  There is a good reason for that attitude:  Mathematical research constantly causes us to rethink the relationship among different mathematical ideas, which can make us want to use names that show our new view of the ideas.  An excellent example of that is the evolution of the concept of "function" over the past 150 years, traced in the Wikipedia article.

What some "authorities" say about "whole number":

  • MathWorld  says that "whole number" is used to mean any of these:  Any positive integer, any nonnegative integer or any integer.
  • Wikipedia also allows all three meanings.
  • Webster's New World dictionary (of which I have been a consultant, but they didn't ask me about whole numbers!) gives "any integer" as a second meaning.
  • American Heritage Dictionary give "any integer" as the only meaning.
  • Someone stole my copy of Merriam Webster.

Common Sense Meaning

Mathematicians think about and talk any particular kind of math object using images and metaphors.  Sometimes (not very often) the name they give to a math object embodies a metaphor.  Examples:

  • A complex number is usually notated using two real parameters, so it looks more complicated than a real number.
  • "Rings" were originally called that because the first examples were integers (mod n) for some positive integer, and you can think of them as going around a clock showing n hours.

Unfortunately, much of the time the name of a kind of object contains a suggestive metaphor that is bad,  meaning that it suggests an erroneous picture or idea of what the object is like.

  • A "group" ought to be a bunch of things.  In other words, the word ought to mean "set".
  • The word "line" suggests that it ought to be a row of points.  That suggests that each point on a line ought to have one next to it.  But that's not true on the "real line"!

Sue's idea that the "common sense" meaning of "whole number" is "integer" refers, I think, to the built-in metaphor of the phrase "whole number" (unbroken number).

I urge math teachers to do these things:

  • Explain to your students that the same math word or phrase can mean different things in different books.
  • Convince your  students to avoid being fooled by the common-sense (metaphorical meaning) of a mathematical phrase.

 

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Mathematical usage

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

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

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

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

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

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

Examples of usage to be deprecated

 

Powers of functions

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

In

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

Two

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

Parentheses

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

More

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

Notes

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

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

References on mathematical usage

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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