## Modules for mathematical objects

A recent article in Scientific American mentions discusses the idea that concepts are represented in the brain by clumps of neurons.  Other neuroscientists have proposed that each concept is distributed among millions of neurons, or that each concept corresponds to one neuron.

I have written many posts about the idea that:

• Each mathematical concept is embodied in some kind of module in the brain.
• This idea is a useful metaphor for understanding how we think about mathematical objects.
• You don't have to know the details of the method of storage for this metaphor to be useful.
• The metaphor clears up a number of paradoxes and conundrums that have agitated philosophers of math.

The SA article inspired me to write about just how such a module may work in some specific cases.

## Integers

Mathematicians normally thinks of a particular integer, say $42$, as some kind of abstract object, and the decimal representation "42" as a representation of the integer, along with XLII and 2A$_{16}$.  You can visualize the physical process like this:

• The mathematician has a module Int (clump of neurons or whatever) that represents integers, and a module FT that represents the particular integer $42$.
• There is some kind of asymmetric three-way connection from FT to Int and a module EO (for "element of" or "IS_A").
• That the connection is "asymmetric" means that the three modules play different roles in the connection, meaning something like "$42$ IS_A Integer"
• The connection is a physical connection, not a sentence, and when  FT is alerted ("fired"?), Int and EO are both alerted as well.
• That means that if someone asks the mathematician, "Is $42$ an integer?", they answer immediately without having to think about it — it is a random access concept like (for many people) knowing that September has 30 days.
• The module for $42$ has many other connections to other modules in the brain, and these connections vary among mathematicians.

The preceding description gives no details about how the modules and interconnections are physically processed.  Neuroscientists probably would have lots of ideas about this (with no doubt considerable variation) and would criticize what I wrote as misrepresenting the physical details in some ways.  But the physical details are their job, not mine.  What I claim is that this way of thinking makes it plausible to view abstract objects and their properties and relationships as physical objects in the brain.  You don't have to know the details any more than you have to know the details of how a rainbow works to see it (but you know that a rainbow is a physical phenomenon).

This way of thinking provides a metaphor for thinking about math objects, a metaphor that is plausibly related to what happens in the real world.

### Students

A student may have a rather different representation of $42$ in the brain.  For one thing, their module for $42$ may not distinguish the symbol "42" from the number $42$, which is an abstract object.   As a result they ask questions such as, "Is $42$ composite in hexadecimal?"  This phenomenon reveals a complicated situation.

• People think they are talking about the same thing when in fact their internal modules for that thing may be very differently connected to other concepts in their brain.
• Mathematicians generally share many more similarities in their modules for $42$ than people in general do.  When they differ, the differences may be of the sort that one of them is a number theorist, so knows more about $42$ (for example, that it is a Catalan number) than another mathematician does.  Or has read The Hitchhiker's Guide to the Galaxy.
• Mathematicians also share a stance that there are right and wrong beliefs about mathematical objects, and that there is a received method for distinguishing correct from erroneous statements about a particular kind of object. (I am not saying the method always gives an answer!).
• Of course, this stance constitutes a module in the brain.
• Some philosophers of education believe that this stance is erroneous, that the truth or falsity of statements are merely a matter of social acceptance.
• In fact, the statements in purple are true of nearly all mathematicians.
• The fact that the truth or falsity of statements is merely a matter of social acceptance is also true, but the word "merely" is misleading.
• The fact is that overwhelming evidence provided by experience shows that the "received method" (proof) for determining the truth of math statements works well and can be depended on. Teachers need to convince their students of this by examples rather that imposing the received method as an authority figure.

### Real numbers

A mathematician thinks of a real number as having a decimal representation.

• The representation is an infinitely long list of decimal digits, together with a location for the decimal point. (Ignoring conventions about infinite strings of zeroes.)
• There is a metaphor that you can go along the list from left to right and when you do you get a better approximation of the "value" of the real number. (The "value" is typically thought of in terms of the metaphor of a point on the real line.)
• Mathematicians nevertheless think of the entries in the decimal expansion of a real number as already in existence, even though you may not be able to say what they all are.
• There is no contradiction between the points of view expressed in the last two bullets.
• Students frequently do not believe that the decimal entries are "already there".  As a result they may argue fiercely that $.999\ldots$ cannot possibly be the same number as $1$.  (The Wikipedia article on this topic has to be one of the most thoroughly reworked math articles in the encyclopedia.)

All these facts correspond to modules in mathematicians' and students' brains.  There are modules for real number, metaphor, infinite list, decimal digit, decimal expansion, and so on.  This does not mean that the module has a separate link to each one of the digits in the decimal expansion.  The idea that there is an entry at every one of the infinite number of locations is itself a module, and no one has ever discovered a contradiction resulting from holding that belief.

## References

• Brain cells for Grandmother, by Rodrigo Quian Quiroga, Itzhak Fried and Christof Koch.  Scientific American, February 2013, pages 31ff.

### Notes on Viewing

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

## Explaining math

I am in the process of writing an explanation of monads for people with not much math background.  In that article, I began to explain my ideas about exposition for readers at that level and after I had written several paragraphs decided I needed a separate article about exposition.  This is that article. It is mostly about language.

## Who is it written for?

### Interested laypeople

There are many recent books explaining some aspect of math for people who are not happy with high school algebra; some of them are listed in the references.  They must be smart readers who know how to concentrate, but for whom algebra and logic and definition-theorem-proof do not communicate.  They could be called interested laypeople, but that is a lousy name and I would appreciate suggestions for a better name.

### Math newbies

My post on monads is aimed at people who have some math, and who are interested in "understanding" some aspect of "higher math"; not understanding in the sense of being able to prove things about monads, but merely how to think about them.   I will call them math newbies.  Of course, I am including math majors, but I want to make it available to other people who are willing to tackle mathematical explanations and who are interested in knowing more about advanced stuff.

These "other people" may include people (students and practitioners) in other science & technology areas as well as liberal-artsy people.  There are such people, I have met them.  I recall one theologian who asked me about what was the big deal about ruler-and-compass construction and who seemed to feel enlightened when I told him that those constructions preserve exactly the ideal nature of geometric objects.  (I later found out he was a famous theologian I had never heard of, just like Ngô Bảo Châu is a famous mathematician nonmathematicians have never heard of.)

## Algebra and other foreign languages

If you are aiming at interested laypeople you absolutely must avoid algebra.  It is a foreign language that simply does not communicate to most of the educated people in the world.  Learning a foreign language is difficult.

So how do you avoid algebra?  Well, you have to be clever and insightful.  The book by Matthew Watkins (below) has absolutely wonderful tricks for doing that, and I think anyone interested in math exposition ought to read it.  He uses metaphors, pictures and saying the same thing in different words. When you finish reading his book, you won't know how to prove statements related to the prime number theorem (unless you already knew how) but you have a good chance of understanding the statement of some theorem in that subject. See my review of the book for more details.

If your article is for math newbies, you don't have to avoid algebra completely.  But remember they are newbies and not as fluent as you are — they do things analogous to "Throw Mama from the train a kiss" and "I can haz cheeseburger?".  But if you are trying to give them some way of thinking about a concept, you need many other things (metaphors, illustrative applications, diagrams…)  You don't need the definition-theorem-proof style too common in "exposition".  (You do need that for math majors who want to become professional mathematicians.)

### Unfamiliar notation

In writing expositions for interested laypeople or math newbies, you should not introduce an unfamiliar notation system (which is like a minilanguage).  I expect to write the monad article without commutative diagrams.  Now, commutative diagrams are a wonderful invention, the best way of writing about categories, and they are widely used by other than category theorists.  But to explain monads to a newbie by introducing and then using commutative diagrams is like incorporating a short grammar of Spanish which you will then use in an explanation of Sancho Panza's relationship with Don Quixote.

The abstractmath article on and, or and not does not use any of the several symbolic notations for logic that are in use.  The explanations simply use "and", "or" and "not".  I did introduce the notation, but didn't use it in the explanations.  When I rewrite the article I expect to put the notation at the end of the article instead of in the middle.  I expect to rewrite the other articles on mathematical reasoning to follow that practice, too.

## Technical terminology

This is about the technical terminology used in math.  Technical terminology belongs to the math dialect (or register) of English, which is not a foreign language in the same sense as algebra.  One big problem is changing the meaning of ordinary English words to a technical meaning.  This requires a definition, and definitions are not something most people take seriously until they have been thoroughly brainwashed into using mathematical methodology.  Math majors have to be brainwashed in this way, but if you are writing for laypeople or newbies you cannot use the technology of formal definition.

### Groups, simple groups

"You say the Monster Group is SIMPLE???  You must be a GENIUS!"  So Mark Ronan in his book (below) referred to simple groups as atoms.  Marcus du Sautoy calls them building blocks.  The mathematical meaning of "simple group" is not a transparent consequence of the meanings of "simple" and "group". Du Sautoy usually writes "group of symmetries" instead of just "group", which gives you an image of what he is talking about without having to go into the abstract definition of group. So in that usage, "group" just means "collection", which is what some students continue to think well after you give the definition.

A better, but ugly, name for "group" might be "symmetroid". It sounds technical, but that might be an advantage, not a disadvantage. "Group" obviously means any collection, as I've known since childhood. "Symmetroid" I've never heard of so maybe I'd better find out what it means.

In beginning abstract math courses my students fervently (but subconsciously) believe that they can figure out what a word means by what it means already, never mind the "definition" which causes their eyes to glaze over. You have to be really persuasive to change their minds.

### Prime factorization

Matthew Watkins referred to the prime factorization of an integer as a cluster. I am not sure why Watkins doesn't like "prime factorization", which usually refers to an expression such as  $p^{n_1}_1p^{n_2}_2\ldots p^{n_k}_k$.  This (as he says) has a spurious ordering that makes you have to worry about what the uniqueness of factorization means. The prime factorization is really a multiset of primes, where the order does not matter.

Watkins illustrates a cluster of primes as a bunch of pingpong balls stuck together with glue, so the prime factorization of 90 would be four smushed together balls marked 2, 3, 3 and 5. Below is another way of illustrating the prime factorization of 90. Yes, the random movement programming could be improved, but Mathematica seduces you into infinite playing around and I want to finish this post. (Actually, I am beginning to think I like smushed pingpong balls better. Even better would be a smushed pingpong picture that I could click on and look at it from different angles.)

### Metaphors, pictures, graphs, animation

Any exposition of math should use metaphors, pictures and graphs, especially manipulable pictures (like the one above) and graphs.  This applies to expositions for math majors as well as laypeople and newbies.  Calculus and other texts nowadays have begun doing this, more with pictures than with metaphors.

I was turned on to these ideas as far back as 1967 (date not certain) when I found an early version of David Mumford's "Red Book", which I think evolved into the book The Red Book of Varieties and Schemes.  The early version was a revelation to me both about schemes and about exposition. I have lost the early book and only looked at the published version briefly when it appeared (1999).  I remember (not necessarily correctly) that he illustrated the spectrum as a graph whose coordinates were primes, and generic points were smudges.  Writing this post has motivated me to go to the University of Minnesota math library and look at the published version again.

## References

### Notes on Viewing

• This post uses MathJax. If you see mathematical formulas with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.
• To manipulate the demos in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. The code for the demos is in the Mathematica notebook algebra2.nb.

## Metaphors in computing science 2

In Metaphors in Computer Science 1, I discussed some metaphors used when thinking about various aspects of computing.  This is a continuation of that post.

### Metaphor: A program is a list of instructions.

• I discussed this metaphor in detail in the earlier post.
• Note particularly that the instructions can be in a natural or a programming language. (Is that a zeugma?)  Many writers would call instructions in a natural language an algorithm.
• I will continue to use “program” in the broader sense.

### Metaphor: A programming language is a language.

• This metaphor is a specific conceptual blend that associates the strings of symbols that constitute a program in a computer language with text in a natural language.
• The metaphor is based on some similarities between expressions in a programming language and expressions in a natural language.
• In both, the expressions have a meaning.
• Both natural and programming languages have specific rules for constructing well-formed expressions.
• This way of thinking ignores many deep differences between programming languages and natural languages. In particular, they don’t talk about the same things!
• The metaphor has been powerful in suggesting ways of thinking about computer programs, for example semantics (below) and ambiguity.

### Metaphor: A computer program is a list of statements

• A consequence of this metaphor is that a computer program is a list of symbols that can be stored in a computer’s memory.
• This metaphor comes with the assumption that if the program is written in accordance with the language’s rules, a computer can execute the program and perhaps produce an output.
• This is the profound discovery, probably by Alan Turing, that made the computer revolution possible. (You don’t have to have different physical machines to do different things.)
• You may want me to say more in the heading above: “A computer program is a list of statements in a programming language that satisfies the well-formedness requirements of the language.”  But the point of the metaphor is only that a program is a list of statements.  The metaphor is not intended to define the concept of “program”.

### Metaphor: A program in a computer language has meanings.

A program is intended to mean something to a human reader.

• Some languages are designed to be easily read by a human reader: Cobol, Basic, SQL.
• Their instructions look like English.
• The algorithm can nevertheless be difficult to understand.
• Some languages are written in a dense symbolic style.
• In many cases the style is an extension of the style of algebraic formulas: C, Fortran.
• Other languages are written in a notation not based on algebra:  Lisp, APL, Forth.
• The boundary between “easily read” and “dense symbolic” is a matter of opinion!

A program is intended to be executed by a computer.

• The execution always involves translation into intermediate languages.
• Most often the execution requires repeated translation into a succession of intermediate languages.
• Each translation requires the preservation of the intended meaning of the program.
• The preservation of intended meaning is what is usually called the semanticsof a programming language.
• In fact, the meaning of the program to a person could be called semantics, too.
• And the human semantics had better correspond in “meaning” to the machine semantics!
• The actual execution of the program requires successive changes in the state of the computer.
• By “state” I mean a list of the form of the electrical charges of each unit of memory in the computer.
• Or you can restrict it to the relevant units of memory, but spelling that out is horrifying to contemplate.
• The resulting state of the machine after the program is run is required to preserve the intended meaning as well as all the intermediate translations.
• Notice that the actual execution is a series of physical events.  You can describe the execution in English or in some notation, but that notation is not the actual execution.

#### References

Conceptual blend (Wikipedia)

Conceptual metaphors (Wikipedia)

Images and Metaphors (article in abstractmath)

Semantics in computer science (Wikipedia)

## Conceptual blending

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

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

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

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

### A particular function

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

#### FORMULA:

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

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

#### TREE:

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

#### GRAPH:

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

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

#### GEOMETRIC

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

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

#### HEIGHT:

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

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

#### KINETIC:

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

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

#### OBJECT:

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

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

#### DEFINITION

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

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

#### SPECIFICATION

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

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

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

## Notes

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

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

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

## References

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

## Metaphors in computing science I

Michael Barr recently told me of a transcription of a talk by Edsger Dijkstra dissing the use of metaphors in teaching programming and advocating that every program be written together with a proof that it works.  This led me to think about the metaphors used in computing science, and that is what this post is about.  It is not a direct answer to what Dijkstra said.

We understand almost anything by using metaphors.  This is a broader sense of metaphor than that thing in English class where you had to say "my love is a red red rose" instead of "my love is like a red red rose".  Here I am talking about conceptual metaphors (see references at the end of the post).

### Metaphor: A program is a set of instructions

You can think of a program as a list of instructions that you can read and, if it is not very complicated, understand how to carry them out.  This metaphor comes from your experience with directions on how to do something (like directions from Google Maps or for assembling a toy).   In the case of a program, you can visualize doing what the program says to do and coming out with the expected output. This is one of the fundamental metaphors for programs.

Such a program may be informal text or it may be written in a computer language.

#### Example

A description of how to calculate $n!$ in English could be:  "Multiply the integers $1$ through $n$".  In Mathematica, you could define the factorial function this way:

fac[n_] := Apply[Times, Table[i, {i, 1, n}]]

This more or less directly copies the English definition, which could have been reworded as "Apply the Times function to the integers from $1$ to $n$ inclusive."  Mathematica programmers customarily use the abbreviation "@@" for Apply because it is more convenient:

Fac[n_]:=Times @@ Table[i, {i, 1, 6}]

As far as I know, C does not have list operations built in.  This simple program gives you the factorial function evaluated at $n$:

j=1;  for (i=2; i<=n; i++)   j=j*i; return j;

This does the calculation in a different way: it goes through the numbers $1, 2,\ldots,n$ and multiplies the result-so-far by the new number.  If you are old enough to remember Pascal or Basic, you will see that there you could use a DO loop to accomplish the same thing.

#### What this metaphor makes you think of

Every metaphor suggests both correct and incorrect ideas about the concept.

• If you think of a list of instructions, you typically think that you should carry out the instructions in order.  (If they are Ikea instructions, your experience may have taught you that you must carry out the instructions in order.)
• In fact, you don't have to "multiply the numbers from $1$ to $n$" in order at all: You could break the list of numbers into several lists and give each one to a different person to do, and they would give their answers to you and you would multiply them together.
• The instructions for calculating the factorial can be translated directly into Mathematica instructions, which does not specify an order.   When $n$ is large enough, Mathematica would in fact do something like the process of giving it to several different people (well, processors) to speed things up.
• I had hoped that Wolfram alpha would answer "720" if I wrote "multiply the numbers from $1$ to $6$" in its box, but it didn't work.  If it had worked, the instruction in English would not be translated at all. (Note added 7 July 2012:  Wolfram has repaired this.)
• The example program for C that I gave above explicitly multiplies the numbers together in order from little to big.  That is the way it is usually taught in class.  In fact, you could program a package for lists using pointers (a process taught in class!) and then use your package to write a C program that looks like the  "multiply the numbers from $1$ to $n$" approach.  I don't know much about C; a reader could probably tell me other better ways to do it.

So notice what happened:

• You can translate the "multiply the numbers from $1$ to $n$" directly into Mathematica.
•  For C, you have to write a program that implements multiplying the numbers from $1$ to $n$. Implementation in this sense doesn't seem to come up when we think about instruction sets for putting furniture together.  It is sort of like: Build a robot to insert & tighten all the screws.

Thus the concept of program in computing science comes with the idea of translating the program instruction set into another instruction set.

• The translation provided above for Mathematica resembles translating the instruction set into another language.
• The two translations I suggested for C (the program and the definition of a list package to be used in the translation) are not like translating from English to another language.  They involve a conceptual reconstruction of the set of instructions.

Similarly, a compiler translates a program in a computer language into machine code, which involves automated conceptual reconstruction on a vast scale.

#### Other metaphors

• C or Mathematica as like a natural language in some ways
• Compiling (or interpreting) as translation

Computing science has used other VIM's (Very Important Metaphors) that I need to write about later:

• Semantics (metaphor: meaning)
• Program as text – this allows you to treat the program as a mathematical object
• Program as machine, with states and actions like automata and Turing machines.
• Specification of a program.  You can regard  "the product of the numbers from $1$ to $n$" as a specification.  Notice that saying "the product" instead of "multiply" changes the metaphor from "instruction" to "specification".

#### References

Conceptual metaphors (Wikipedia)

Images and Metaphors (article in abstractmath)

Images and Metaphors for Sets (article in abstractmath)

Images and Metaphors for Functions (incomplete article in abstractmath)

## Whole numbers

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.

## Freezing a family of functions

To manipulate the diagrams in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. The Mathematica notebooks used here are listed in the references below.

### Some background

• Generally, I have advocated using all sorts of images and metaphors to enable people to think about particular mathematical objects more easily.
• In previous posts I have illustrated many ways (some old, some new, many recently using Mathematica CDF files) that you can provide such images and metaphors, to help university math majors get over the abstraction cliff.
• When you have to prove something you find yourself throwing out the images and metaphors (usually a bit at a time rather than all at once) to get down to the rigorous view of math [1], [2], [3], to the point where you think of all the mathematical objects you are dealing with as unchanging and inert (not reacting to anything else).  In other words, dead.
• The simple example of a family of functions in this post is intended to give people a way of thinking about getting into the rigorous view of the family.  So this post uses image-and-metaphor technology to illustrate a way of thinking about one of the basic proof techniques in math (representing the object in rigor mortis so you can dissect it).  I suppose this is meta-math-ed.  But I don’t want to think about that too much…
• This example also illustrates the difference between parameters and variables. The bottom line is that the difference is entirely in how we think about them. I will write more about that later.

### A family of functions

This graph shows individual members of the family of functions $$y=a\sin\,x$$ for various values of $latex a$. Let’s look at some of the ways you can think about this.

• Each choice of  “shows the function for that value of the parameter $latex a$”.  But really, it shows the graph of the function, in fact only the part between $latex x=-4$ and $latex x= 4$.
• You can also think of it as showing the function changing shape as $latex a$ changes over time (as you slide the controller back and forth).

Well, you can graph something changing over time by introducing another axis for time.  When you graph vertical motion of a particle over time you use a two-dimensional picture, one axis representing time and the other the height of the particle. Our representation of the function $latex y=a\sin\,x$ is a two-dimensional object (using its graph) so we represent the function in 3-space, as in this picture, where the slider not only shows the current (graph of the) function for parameter value $latex a$ but also locates it over $latex a$ on the $latex z$ axis.

The picture below shows the surface given by $latex y=a\sin\,x$ as a function of both variables $latex a$ and $latex x$. Note that this graph is static: it does not change over time (no slide bar!). This is the family of functions represented as a rigorous (dead!) mathematical object.

If you click the “Show Curves” button, you will see a selection of the curves in middle diagram above drawn as functions of $latex x$ for certain values of $latex a$. Each blue curve is thus a sine wave of amplitude $latex a$. Pushing that button illustrates the process going on in your mind when you concentrate on one aspect of the surface, namely its cross-sections in the $latex x$ direction.

Reference [4] gives the code for the diagrams in this post, as well as a couple of others that may add more insight to the idea. Reference [5] gives similar constructions for a different family of functions.

### References

1. Rigorous view in abstractmath.org
2. Representations II: Dry Bones (post)
3. Representations III: Rigor and Rigor Mortis (post)
4. FamiliesFrozen.nb,  FamiliesFrozen.cdf (Mathematica file used to make this post)
5. AnotherFamiliesFrozen.nbAnotherFamiliesFrozen.cdf (Mathematica file showing another family of functions)

### The abstraction cliff

In universities in the USA, a math major typically starts with calculus, followed by courses such as linear algebra, discrete math, or a special intro course for math majors (which may be taken simultaneously with calculus), then go on to abstract algebra, analysis, and other courses involving abstraction and proofs.

At this point, too many of them hit a wall; their grades drop and they change majors.  They had been getting good grades in high school and in calculus because they were strong in algebra and geometry, but the sudden increase in abstraction in the newer courses completely baffles them. I believe that one big difficulty is that they can't grasp how to think about abstract mathematical objects.  (See Reference [9] and note [a].)   They have fallen off the abstraction cliff.  We lose too many math majors this way. (Abstractmath.org is my major effort to address the problems math majors have during or after calculus.)

This post is a summary of the way I see how mathematicians and students think about math.  I will use it as a reference in later posts where I will write about how we can communicate these ways of thinking.

### Concept Image

In 1981, Tall and Vinner  [5] introduced the notion of the concept image that a person has about a mathematical concept or object.   Their paper's abstract says

The concept image consists of all the cognitive structure in the individual's mind that is associated with a given concept. This may not be globally coherent and may have aspects which are quite different from the formal concept definition.

The concept image you may have of an abstract object generally contains many kinds of constituents:

• visual images of the object
• metaphors connecting the object to other concepts
• descriptions of the object in mathematical English
• descriptions and symbols of the object in the symbolic language of math
• kinetic feelings concerning certain aspects of the object
• how you calculate parameters of the object
• how you prove particular statements about the object

This list is incomplete and the items overlap.  I will write in detail about these ideas later.

The name "concept image" is misleading [b]), so when I have written about them, I have called them metaphors or mental representations as well as concept images, for example in [3] and [4].

### Abstract mathematical concepts

This is my take on the notion of concept image, which may be different from that of most researchers in math ed. It owes a lot to the ideas of Reuben Hersh [7], [8].

• An abstract mathematical concept is represented physically in your brain by what I have called "modules" [1] (physical constituents or activities of the brain [c]).
• The representation generally consists of many modules.  They correspond to the list of constituents of a concept image given above.  There is no assumption that all the modules are "correct".
• This representation exists in a semi-public network of mathematicians' and students' brains. This network exercises (incomplete) control over your personal representation of the abstract structure by means of conversation with other mathematicians and reading books and papers.  In this sense, an abstract concept is a social object.  (This is the only point of view in the philosophy of math that I know of that contains any scientific content.)

### Notes

[a]  Before you object that abstraction isn't the only thing they have trouble with, note that a proof is an abstract mathematical object. The written proof is a representation of the abstract structure of the proof.  Of course, proofs are a special kind of abstract structure that causes special problems for students.

[b] Cognitive science people use "image" to include nonvisual representations, but not everyone does.  Indeed, cognitive scientists use "metaphor" as well with a broader meaning than your high school English teacher.  A metaphor involves the cognitive merging of parts of two concepts (specifically with other parts not merged). See [6].

[c] Note that I am carefully not saying what the modules actually are — neurons, networks of neurons, events in the brain, etc.   From the point of view of teaching and understanding math, it doesn't matter what they are, only that they exist and live in a society where they get modified by memes  (ideas, attitudes, styles physically transmitted from brain to brain by speech, writing, nonverbal communication, appearance, and in other ways).

### References

1. Math and modules of the mind (previous post)
2. Mathematical Concepts (previous post)
3. Mental, physical and mathematical representations (previous post)
4. Images and Metaphors (abstractmath.org)
5. David Tall and Schlomo Vinner, Concept Image and Concept Definition in Mathematics with particular reference to limits and continuity, Journal Educational Studies in Mathematics, 12 (May, 1981), no. 2, 151–169.
6. Conceptual metaphor (Wikipedia article).
7. What is mathematics, really? by Reuben Hersh, Oxford University Press, 1999.  Read online at Questia.
8. 18 Unconventional Essays on the Nature of Mathematics, by Reuben Hersh. Springer, 2005.
9. Mathematical objects (abstractmath.org).

## Liberal-artsy people

I graduated from Oberlin College with a B.A. as a math major and minors in philosophy and English literature, with only three semesters of science courses.  I was and am "liberal-artsy".   As professor of math at Case Western Reserve University,  I had lots of colleagues in both pure and applied math who started out with B.Sc. degrees. We did not always understand each other very well!

Caveat: "Liberal-artsy" and "Narrowly Focused B.Sc. type" (I need a better name) are characteristics that people may have in varying amounts, and many professors in science and math have both characteristics.   I do, myself, although I am more L.A. that B.Sc.  Furthermore, I know nothing about any sociological or cognitive-science research on these characteristics.  I am making it all up as I write.  (This is a blog post, not a tome.)

I recently posted on secants and  tangents.  These articles were deliberately aimed to tickle the interests of L.A.  students.

Liberal-artsy types want to know about connections between concepts.  In each post, I wrote on both common meanings of the words (secant line and function, tangent line and function) and the close connections between them.  Some trig teachers / trig texts tell students about these connections but too many don't.   On the other hand, many B.Sc. types are left cold by such discussions.  B.Sc. types are goal-oriented and want to know a) how do I use it? b) how do I calculate it?  They get impatient when you talk about anything else.  I say point out these connections anyway.

L.A. types want to know about the reason for the name of a concept.  The post on secants refers to the metaphor that "secant" means "cutting". This is based on the etymology of "secant", which is hidden to many students  because it is based on Latin.  The post makes the connection that the "original" definition of "secant" was the length of a certain line segment generated by an angle in the unit circle. The post on tangents makes an analogous connection, and also points out that most tangent lines that students see touch the curve at only a single point, which is not a connotation of the English word "touch".

Many people think they have learned something when they know the etymology of a word.  In fact, the etymology of a word may have little or nothing to do with its current meaning, which may have developed over many centuries of metaphors that become dead, generate new metaphors that become dead, umpteen times, so that the original meaning is lost.  (The word "testimony" cam from a Latin phrase meaning hold your testicles, which is really not related to its meaning in present-day English.)

So I am not convinced that etymologies of names can help much in most cases.  In particular, different mathematical definitions of the same concept can be practically disjoint in terms of the data they use, and there is no one "correct" definition, although there may be only one that motivates the name.  (There often isn't a definition that motivates the name.  Think "group".)  But I do know that when I mention the history of a name of a concept in class, some students are fascinated and ask me questions about it.

L.A. types are often fascinated by ETBell-like stories about the mathematician who came up with a concept, and sometimes the stories illuminate the mathematical idea.  But L. A. types often are interested anyway.  It's funny when you talk about such a thing in class, because some students visibly tune out while others noticeably perk up and start paying attention.

So who should you cater to?  Answer:  Both kinds of students.  (Tell interesting stories, but quickly and in an offhand way.)

The posts on secants and tangents also experimented with using manipulable diagrams to illustrate the ideas.  I expect to write about that more in another post.

For more about the role of definitions, check out the abmath article and also Timothy Gowers' post on definitions (one of a series of excellent posts on working with abstract math).

Posted in exposition, math, understanding math. Tags:. 2 Comments »

## Tangents

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

To manipulate the diagrams in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. The Mathematica notebooks used here are listed in the references below.

### Tangent line

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

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

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

Etymology and metaphor

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

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

The curve in this picture is defined by

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

Tangents can hug!

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

The blue curve in this graph is given by

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

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

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

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

### Tangent on the unit circle

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

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

The tangent function in calculus

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

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

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

The root is “tangent-” but the nominative is “tangens”. This is normal behavior for Latin participles, but it causes angst among people who write about etymology of trig functions. Ignore “tangens”, it is the root that matters.

The Latin root comes from the Indo-European root “tag”. Don’t worry about how the n got into “tangent”; this is typical Indo-European behavior. The English word “tag” does not come from the IE root, but “thwack” may.