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Demonstrating the inverse image of an interval

2011/09/07 SixWingedSeraph 2 Comments

This post has been superseded by the post Inverse Image Demo Revisited

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Representations 2

2011/05/25 Charles Wells Leave a comment

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

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 ${x}$ .
In the physical representation, each point is shown in a location determined by the conventional 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.)
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.)
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.
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

2011/05/20 Charles Wells Leave a comment

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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Mental, Physical and Mathematical Representations

2011/05/17 Charles Wells 2 Comments

For a given mathematical object, a mathematician may have:

A mental representation of the object. This can be a metaphor, a mental image, or a kinesthetic understanding of the object.
A physical representation of the object. This may be a (physical) picture or drawing or three-dimensional model of the object.
A mathematical definition and one or more mathematical representations for the object. Such a representation is itself a mathematical object.

The boldface things in this list are related to each other in lots of ways, and they are fuzzy and overlap and don’t include every phenomenon connected with a math object.

I have written about these things ([1], [2], [3], [4]). So have lots of other people. In this post I summarize these ideas. I expect to write about particular examples later on and will use this as a reference.

Two Examples

The following examples point out a few of the relationships between the ideas in boldface above. There is much more to understand.

Function as black box

The idea that a function is a black box or machine with input and output is a metaphor for a function.

A is a metaphor for B means that A and B are cognitively pasted together in such a way that the behavior of A is in many ways like the behavior of B. Such a thing is both useful and dangerous, dangerous because there will be ways in which A behaves that suggest inappropriate ideas about B.

The function as machine is a good metaphor: for example functional composition involves connecting the output of one machine to the input of another, and the inverse function is like running the machine backward.

The function as machine is a bad metaphor: For example, it is wrong to think you could build a machine to calculate any given function exactly. But you can still imagine such a machine, given by a specification (it outputs the value of the function at a given input) and then, in your imagination, connecting the input of one to the output of another must perforce calculate the composite of the corresponding functions.

Like any metaphor, this is a mental representation. That means the metaphor has a physical instantiation in your brain. So a metaphor has a physical representation.

Different people won’t have quite the same concept of a particular metaphor. So a metaphor will have lots of slightly different physical representations, but mathematicians form a community, and communication between mathematicians fine-tunes the different physical instantiations so that they correspond more closely to each other. This is the sense in which mathematical objects have a shared existence in a community as Reuben Hersh has suggested.

A function is a mathematical object, which can be rigorously specified as a set of ordered pairs together with a domain and a codomain. There is a cognitive relationship between the concepts of function as math object and function as black box with input and output.

Triangle

A triangle can be drawn, or created on a computer and a physical image printed out. You may also have a mental image of the triangle.

The physical and the mental images are not the same thing, but they are definitely related. The relationship is mediated by the neuronal circuitry behind your retinas, which performs a highly sophisticated transformation of the pixels on your retina into an organized physical structure in your brain, connected to various other neurons.

This circuitry exists because it helps us get a useful understanding of the world through our eyes. So a picture of a triangle takes advantage of pre-existing neuron structure to generate a useful mental representation that helps us understand and prove things about triangles.

This mental representation also lives in a community of mathematician. Like any community, it has subgroups with “dialects” — varying understanding of representation.

For example, a mathematician who looks at the triangle below sees a triangle that looks like a right triangle. A student sees a triangle that is a right triangle.

This is “sees” in the sense of what their brain reports after all that processing. The mathematician’s brain connects the “triangle I am seeing” module (in their brain) to the “looks like a right triangle” module, but does not connect it to the “is a right triangle” module because they don’t see any statement in the surrounding text that it is a right triangle. The student, on the other hand, fallaciously makes the connection to “is a right triangle” directly.

In some sense, a student who does not make that connection directly is already a mathematician.

A triangle also exists as a mathematical object in your and my brain. It is described by a formal mathematical definition. The pictures of triangles you see above do not fit this definition. For one thing, the line segments in the pictures have thickness. But the pictures trigger a reaction in your neurons that causes your brain to cognitively paste together the line segments in the drawing to the segments required by the formal definition. This is a kind of metaphor of concrete-to-abstract that connects drawings to math objects that mathematicians use all the time.

Note that this “concrete-to-abstract metaphor” itself has a physical existence in your brain. It drops, for example, the property of thickness that the line segments in the drawing have when matching them (in the metaphor) with the line segments in the corresponding abstract triangle. On the other hand, it preserves the sense the all three angles in the triangle are acute. The abstract mathematical concept of triangle (the generic triangle) has no requirement on the angles except that they add up to pi.

Summary

The discussions above describe a few of the complex and subtle relationships that exist between

Mental representations of math objects
Physical representations of math objects
Formally defined math objects and their formally defined representations.

I have purported to discuss how mathematics is understood (especially in connection with language) in several articles and a book but only a few of the relationships I just described are mentioned in any of those articles. Perhaps one or two things I said caused you to react: “Actually, that’s obviously true but I never thought of it before”. (Much the way I had mathematicians in the ’60’s tell me, “I see what you mean that addition is a function of two variables, but I never thought of it that way before”.) (I was a brash category theorist wannabe then.)

A lot of research has been done on understanding math, and some research has been done on mathematical discourse. But what has been done has merely exposed the fin of the shark.

References

[1] Images and metaphors (in abstractmath).

[2] Representations and Models (in abstractmath).

[3] Mathematical Concepts (previous blog).

[4] Mental Representations in Math (previous blog).

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Operation as metaphor in math

2011/01/17 Charles Wells 3 Comments

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

2011/01/03 Charles Wells Leave a comment

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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Function as map

2010/05/24 Charles Wells Leave a comment

This is a first draft of an article to eventually appear in abstractmath.

Images and metaphors

To explain a math concept, you need to explain how mathematicians think about the concept. This is what in abstractmath I call the images and metaphors carried by the concept. Of course you have to give the precise definition of the concept and basic theorems about it. But without the images and metaphors most students, not to mention mathematicians from a different field, will find it hard to prove much more than some immediate consequences of the definition. Nor will they have much sense of the place of the concept in math and applications.

Teachers will often explain the images and metaphors with handwaving and pictures in a fairly vague way. That is good to start with, but it’s important to get more precise about the images and metaphors. That’s because images and metaphors are often not quite a good fit for the concept — they may suggest things that are false and not suggest things that are true. For example, if a set is a container, why isn’t the element-of relation transitive? (A coin in a coinpurse in your pocket is a coin in your pocket.)

“A metaphor is a useful way to think about something, but it is not the same thing as the same thing.” (I think I stole that from the Economist.) Here, I am going to get precise with the notion that a function is a map. I am acting like a mathematician in “getting precise”, but I am getting precise about a metaphor, not about a mathematical object.

A function is a map

A map (ordinary paper map) of Minnesota has the property that each point on the paper represents a point in the state of Minnesota. This map can be represented as a mathematical function from a subset of a 2-sphere to ${{\mathbb R}^2}$ . The function is a mathematical idealization of the relation between the state and the piece of paper, analogous to the mathematical description of the flight of a rocket ship as a function from ${{\mathbb R}}$ to ${{\mathbb R}^3}$ .

The Minnesota map-as-function is probably continuous and differentiable, and as is well known it can be angle preserving or area preserving but not both.

So you can say there is a point on the paper that represents the location of the statue of Paul Bunyan in Bemidji. There is a set of points that represents the part of the Mississippi River that lies in Minnesota. And so on.

A function has an image. If you think about it you will realize that the image is just a certain portion of the piece of paper. Knowing that a particular point on the paper is in the image of the function is not the information contained in what we call “this map of Minnesota”.

This yields what I consider a basic insight about function-as-map: The map contains the information about the preimage of each point on the paper map. So:

The map in the sense of a “map of Minnesota” is represented by the whole function, not merely by the image.

I think that is the essence of the metaphor that a function is a map. And I don’t think newbies in abstractmath always understand that relationship.

A morphism is a map

The preceding discussion doesn’t really represent how we think of a paper map of Minnesota. We don’t think in terms of points at all. What we see are marks on the map showing where some particular things are. If it is a road map it has marks showing a lot of roads, a lot of towns, and maybe county boundaries. If it is a topographical map it will show level curves showing elevation. So a paper map of a state should be represented by a structure preserving map, a morphism. Road maps preserve some structure, topographical maps preserve other structure.

The things we call “maps” in math are usually morphisms. For example, you could say that every simple closed curve in the plane is an equivalence class of maps from the unit circle to the plane. Here equivalence class meaning forget the parametrization.

The very fact that I have to mention forgetting the parametrization is that the commonest mathematical way to talk about morphisms is as point-to-point maps with certain properties. But we think about a simple closed curve in the plane as just a distorted circle. The point-to-point correspondence doesn’t matter. So this example is really talking about a morphism as a shape-preserving map. Mathematicians introduced points into talking about preserving shapes in the nineteenth century and we are so used to doing that that we think we have to have points for all maps.

Not that points aren’t useful. But I am analyzing the metaphor here, not the technical side of the math.

Groups are functors

People who don’t do category theory think the idea of a mathematical structure as a functor is weird. From the point of view of the preceding discussion, a particular group is a functor from the generic group to some category. (The target category is Set if the group is discrete, Top if it is a topological group, and so on.)

The generic group is a group in a category called its theory or sketch that is just big enough to let it be a group. If the theory is the category with finite products that is just big enough then it is the Lawvere theory of the group. If it is a topos that is just big enough then it is the classifying topos of groups. The theory in this sense is equivalent to some theory in the sense of string-based logic, for example the signature-with-axioms (equational theory) or the first order theory of groups. Johnstone’s Elephant book is the best place to find the translation between these ideas.

A particular group is represented by a finite-limit-preserving functor on the algebraic theory, or by a logical functor on the classifying topos, and so on; constructions which bring with them the right concept of group homomorphisms as well (they will be any natural transformations).

The way we talk about groups mimics the way we talk about maps. We look at the symmetric group on five letters and say its multiplication is noncommutative. “Its multiplication” tells us that when we talk about this group we are talking about the functor, not just the values of the functor on objects. We use the same symbols of juxtaposition for multiplication in any group, “ ${1}$ ” or “ ${e}$ ” for the identity, “ ${a^{-1}}$ ” for the inverse of ${a}$ , and so on. That is because we are really talking about the multiplication, identity and inverse function in the generic group — they really are the same for all groups. That is because a group is not its underlying set, it is a functor. Just like the map of Minnesota “is” the whole function from the state to the paper, not just the image of the function.

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category theory, math, understanding math

Just-in-time foundations

2009/11/13 Charles Wells 1 Comment

Introduction

In MathOverflow, statements similar to the following two occurred in comments:

Sets and functions do not form a category
Categories and functors do not form a category.

I cannot find either one of them now, but I want to talk about them anyway.

If you look at the definition of categories in various works (for example references [1] through [3] below) you find that the objects and arrows of a category must each form a “collection” or “class” together with certain operations. The authors all describe the connection with Grothendieck’s concept of “universe” and define “large categories” and “small categories” in the usual way. So Statement 1 above is simply wrong.

Statement 2 is more problematic. The trouble is that if the word “categories” includes large categories then the objects do not form a set even in the second universe. You have to go to the third universe.

Now there is a way to define categories where this issue does not come up. It allows us to think about categories without having a particular system such as ZF and universes in mind.

A syntactic definition of category

A category consists of objects and arrows, together with four methods of construction M1 – M4 satisfying laws L1 -L7. I treat “object” and “arrow” as predicates: object[f] means f is an object and arrow[a] means a is an arrow. “=” means equals in the mathematical sense.

M1 Source If arrow[f], object[f.source].

M2 Target If arrow [f], object[f.target].

M3 Identity If object[a], arrow[a.identity].

M4 Comp If arrow[g] and arrow[f] and f.target = g.source, then arrow[(g,f).comp].

L1. If object[a], a.identity.source = a.

L2. If object[a], a.identity.target = a.

L3. If arrow[g] and arrow[f] and f.target = g.source, then (g,f).comp.source = f.source.

L4. If arrow[g] and arrow[f] and f.target = g.source, then (g,f).comp.target = g.target.

L5. If object[a] and arrow[f] and f.source = a, then (f, a.identity) = f.

L6. If object[a] and arrow[g] and g.target = a, then (a.identity, g) = g.

L7. If arrow[h] and arrow[g] and arrow[f] and h.source= g.target and g.source = f.target, then (h,(g,f).comp = ((h,g).comp, f.comp).

Remarks on this definition

1. I have deliberately made this definition look like a specification in an object oriented program (see [6]), although the syntax is not the same as any particular oo language. It is as rigorous a mathematical definition as you could want, and it could presumably be compiled in some oo language, except that I don’t know if oo languages allow the conditional definition of a method as given in M4.

2. I could have given the definition in mathematical English, for example “If f is an arrow then the source of f is an object”. My point in providing the impenetrable definition above is to make a connection (admittedly incompletely established) with a part of math (the theory of oo languages) that is definitely rigorous but is not logic. An informal definition in math English of course could also be transformed rigorously into first order logic.

3. This definition is exactly equivalent to the FL sketch for categories given in my post [5]. That sketch has models in many categories, not just Set, as well as its generic model living in the corresponding FL-cattheory (or in the classifying topos it generates).

4. Saunders Mac Lane defined metacategory in precisely this way in [1]. That was of course before anyone every heard of oo languages. I think he should have made that the definition of category.

Just-in-time foundations

Mathematicians work inside the categories Set (sets and functions) and Cat (categories and functors) all the time, including functors to or from Cat or Set. When they consider a category, the use theorems that follow from the definition above. They do not have to have foundations in mind.

Once in awhile, they are frustrated because they cannot talk about the set of objects of some category. For example, Freyd’s solution set condition is required to prove the existence of a left adjoint because of that problem. The ss condition is a work-around for a familiar obstruction to an easy way to prove something. I can imagine coming up with such a work-around without ever giving a passing thought to foundations, in particular without thinking of universes.

When you work with a mathematical object, the syntax of the definitions and theorems give you all you need to justify the claim that something is a theorem. You absolutely need models of the theory to think up and understand proofs, but the models could be sets or classes with structure, or functors (as in sketch theory), or you may work with generic models which may require you to use intuitionistic reasoning. You don’t have to have any particular kind of model in mind when you work in Set or Cat.

When you do run into something like the impossibility of forming the set of objects of some category (which happens in any model theory environment that uses classical rather than intuitionistic reasonins) then you may want to consider an approach through some theory of foundations. That is what most mathematicians do: they use just-in-time foundations. For example, in a particular application you may be happy to work in a topos with a set-of-all-objects, particularly if you are a certain type of computer scientists who lives in Pittsburgh. You may be happy to explicitly consider universes, although I am not aware of any category-theoretical results that do explicitly mention universes.

But my point is that most mathematicians think about foundations only when they need to, and most mathematicians never need to think about foundations in their work. Moral: Don’t think in terms of foundations unless you have to.

This point of view is related to the recent discussions of pragmatic foundations [7] [8].

Side remark

The situation that you can’t always construct a set of somethings is analogous to the problem that you have in working with real numbers: You can’t name most real numbers. This may get in the way of some analyst wanting to do something, I don’t know. But in any branch of math, there are obstructions to things you want to do that really do get in your way. For example, in beginning linear algebra, it may have occurred to you, to your annoyance, that if you have the basis of a subspace you can extend it to the basis for the whole space, but if you have a basis of the whole space, and a subspace, the basis may not contain a basis of the subspace.

References and links

Saunders Mac Lane, Categories for the working mathematician. Springer-Verlag, 1971.
Wikipedia article on category theory
Michael Barr and Charles Wells, Category Theory for Computing Science, Third Edition (1999). Les Publications CRM, Montreal (publication PM023).
Discussion of functions in abstractmath.org.
Definitions into Mathematical Objects 7.
Object oriented programming in Wikipedia.
M. Gelfand, We Do Not Choose Mathematics as Our Profession, It Chooses Us: Interview with Yuri Manin.
Discussion in n-category cafe.

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category theory, exposition, language, language of math, math, representations

Composites of functions

2009/09/11 Charles Wells 3 Comments

In my post on automatic spelling reform, I mentioned the various attempts at spelling reform that have resulted in both the old and new systems being used, which only makes things worse. This happens in Christian denominations, too. Someone (Martin Luther, John Wesley) tries to reform things; result: two denominations. But a lot of the time the reform effort simply disappears. The Chicago Tribune tried for years to get us to write “thru” and “tho” — and failed. Nynorsk (really a language reform rather than a spelling reform) is down to 18% of the population and the result of allowing Nynorsk forms to be used in the standard language have mostly been nil. (See Note 1.)

In my early years as a mathematician I wrote a bunch of papers writing functions on the right (including the one mentioned in the last post). I was inspired by some algebraists and particularly by Beck’s Thesis (available online via TAC), which I thought was exceptionally well-written. This makes function composition read left to right and makes the pronunciation of commutative diagrams get along with notation, so when you see the diagram below you naturally write h = fg instead of h = gf. Composite

Sadly, I gave all that up before 1980 (I just looked at some of my old papers to check). People kept complaining. I even completely rewrote one long paper (Reference [3]) changing from right hand to left hand (just like Samoa). I did this in Zürich when I had the gout, and I was happy to do it because it was very complicated and I had a chance to check for errors.

Well, I adapted. I have learned to read the arrows backward (g then f in the diagram above). Some French category theorists write the diagram backward, thus:

CompositeOp

But I was co-authoring books on category theory in those days and didn’t think people would accept it. Not to mention Mike Barr (not that he is not a people, oh, never mind).

Nevertheless, we should have gone the other way. We should have adopted the Dvorak keyboard and Betamax, too.

Notes

[1] A lifelong Norwegian friend of ours said that when her children say “boka” instead of “boken” it sound like hillbilly talk does to Americans. I kind of regretted this, since I grew up in north Georgia and have been a kind of hillbilly-wannabe (mostly because of the music); I don’t share that negative reaction to hillbillies. On the other hand, you can fageddabout “ho” for “hun”.

References

[1] Charles Wells, Automorphisms of group extensions, Trans. Amer. Math. Soc, 155 (1970), 189-194.

[2] John Martino and Stewart Priddy, Group extensions and automorphism group rings. Homology, Homotopy and Applications 5 (2003), 53-70.

[3] Charles Wells, Wreath product decomposition of categories 1, Acta Sci. Math. Szeged 52 (1988), 307 – 319.

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Tag Archives: function

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Just-in-time foundations

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