Introducing abstract topics

I have been busy for the past several years revising abstractmath.org (abmath). Now I believe, perhaps foolishly, that most of the articles in abmath have reached beta, so now it is time for something new.

For some time I have been considering writing introductions to topics in abstract math, some typically studied by undergraduates and some taken by scientists and engineers. The topics I have in mind to do first include group theory and category theory.

The point of these introductions is to get the student started at the very beginning of the topic, when some students give up in total confusion. They meet and fall off of what I have called the abstraction cliff, which is discussed here and also in my blog posts Very early difficulties and Very early difficulties II.

I may have stolen the phrase “abstraction cliff” from someone else.

Group theory

Group theory sets several traps for beginning students.

Multiplication table

• A student may balk when a small finite group is defined using a set of letters in a multiplication table.
“But you didn’t say what the letters are or what the multiplication is?”
• Such a definition is an abstract definition, in contrast to the definition of “prime”, for example, which is stated in terms of already known entities, namely the integers.
• The multiplication table of a group tells you exactly what the binary operation is and any set with an operation that makes such a table correct is an example of the group being defined.
• A student who has no understanding of abstraction is going to be totally lost in this situation. It is quite possible that the professor has never even mentioned the concept of abstract definition. The professor is probably like most successful mathematicians: when they were students, they understood abstraction without having to have it explained, and possibly without even noticing they did so.

Cosets

• Cosets are a real killer. Some students at this stage are nowhere near thinking of a set as an object or a thing. The concept of applying a binary operation on a pair of sets (or any other mathematical objects with internal structure) is completely foreign to them. Did anyone ever talk to them about mathematical objects?
• The consequence of this early difficulty is that such a student will find it hard to understand what a quotient group is, and that is one of the major concepts you get early in a group theory course.
• The conceptual problems with multiplication of cosets is similar to those with pointwise addition of functions. Given two functions $f,g:\mathbb{R}\to\mathbb{R}$, you define $f+g$ to be the function $(f+g)(x):=f(x)+g(x)$ Along with pointwise multiplication, this makes the space of functions $\mathbb{R}\to\mathbb{R}$ a ring with nice properties.
• But you have to understand that each element of the ring is a function thought of as a single math object. The values of the function are properties of the function, but they are not elements of the ring. (You can include the real numbers in the ring as constant functions, but don’t confuse me with facts.)
• Similarly the elements of the quotient group are math objects called cosets. They are not elements of the original group. (To add to the confusion, they are also blocks of a congruence.)

Isomorphic groups

• Many books, and many professors (including me) regard two isomorphic groups as the same. I remember getting anguished questions: “But the elements of $\mathbb{Z}_2$ are equivalence classes and the elements of the group of permutations of $\{1,2\}$ are functions.”
• I admit that regarding two isomorphic groups as the same needs to be treated carefully when, unlike $\mathbb{Z}_2$, the group has a nontrivial automorphism group. ($\mathbb{Z}_3$ is “the same as itself” in two different ways.) But you don’t have to bring that up the first time you attack that subject, any more than you have to bring up the fact that the category of sets does not have a set of objects on the first day you define categories.

Category theory

Category theory causes similar troubles. Beginning college math majors don’t usually meet it early. But category theory has begun to be used in other fields, so plenty of computer science students, people dealing with databases, and so on are suddenly trying to understand categories and failing to do so at the very start.

The G&G post A new kind of introduction to category theory constitutes an alpha draft of the first part of an article introducing category theory following the ideas of this post.

Objects and arrows are abstract

• Every once in a while someone asks a question on Math StackExchange that shows they have no idea that an object of a category need not have elements and that morphisms need not be functions that take elements to elements.
• One questioner understood that the claim that a morphism need not be a function meant that it might be a multivalued function.

Duality

• That misunderstanding comes up with duality. The definition of dual category requires turning the arrows around. Even if the original morphism takes elements to elements, the opposite morphism does not have to take elements to elements. In the case of the category of sets, an arrow in $\text{Set}^{op}$ cannot take elements to elements — for example, the opposite of the function $\emptyset\to\{1,2\}$.
• The fact that there is a concrete category equivalent to $\text{Set}^{op}$ is a red herring. It involves different sets: the function corresponding to the function just mentioned goes from a four-element set to a singleton. But in the category $\text{Set}^{op}$ as defined it is simply an arrow, not a function.

Not understanding how to use definitions

• Some of the questioners on Math Stack Exchange ask how to prove a statement that is quite simple to prove directly from the definitions of the terms involved, but what they ask and what they are obviously trying to do is to gain an intuition in order to understand why the statement is true. This is backward — the first thing you should do is use the definition (at least in the first few days of a math class — after that you have to use theorems as well!
• I have discussed this in the blog post Insights into mathematical definitions (which gives references to other longer discussions by math ed people). See also the abmath section Rewrite according to the definitions.

How an introduction to a math topic needs to be written

The following list shows some of the tactics I am thinking of using in the math topic introductions. It is quite likely that I will conclude that some tactics won’t work, and I am sure that tactics I haven’t mentioned here will be used.

• The introductions should not go very far into the subject. Instead, they should bring an exhaustive and explicit discussion of how to get into the very earliest part of the topic, perhaps the definition, some examples, and a few simple theorems. I doubt that a group theory student who hasn’t mastered abstraction and what proofs are about will ever be ready to learn the Sylow theorems.
• You can’t do examples and definitions simultaneously, but you can come close by going through an example step by step, checking each part of the definition.
• There is a real split between students who want the definitions first
(most of whom don’t have the abstraction problems I am trying to overcome)
and those who really really think they need examples first (the majority)
because they don’t understand abstraction.

• When you introduce an axiom, give an example of how you would prove that some binary operation satisfies the axiom. For example, if the axiom is that every element of a group must have an inverse, right then and there prove that addition on the integers satisfies the axiom and disprove that multiplication on integers satisies it.
• When the definition uses some undefined math objects, point out immediately with examples that you can’t have any intuition about them except what the axioms give you. (In contrast to definition of division of integers, where you and the student already have intuitions about the objects.)
• Make explicit the possible problems with abstractmath.org and Gyre&Gimble) will indeed find it difficult to become mathematical researchers — but not impossible!
• But that is not the point. All college math professors will get people who will go into theoretical computing science, and therefore need to understand category theory, or into particle physics, and need to understand groups, and so on.
• By being clear at the earliest stages of how mathematicians actually do math, they will produce more people in other fields who actually have some grasp of what is going on with the topics they have studied in math classes, and hopefully will be willing to go back and learn some more math if some type of math rears its head in the theories of their field.
• Besides, why do you want to alienate huge numbers of people from math, as our way of teaching in the past has done?
• “Our” means grammar school teachers, high school teachers and college professors.

Acknowledgment

Thanks to Kevin Clift for corrections.



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Insights into mathematical definitions

My general practice with abstractmath.org has been to write about the problems students have at the point where they first start studying abstract math, with some emphasis on the languages of math. I have used my own observations of students, lexicographical work I did in the early 2000’s, and papers written by workers in math ed at the college level.

A few months ago, I finished revising and updating abstractmath.org. This took rather more than a year because among other things I had to reconstitute the files so that the html could be edited directly. During that time I just about quit reading the math ed literature. In the last few weeks I have found several articles that have changed my thinking about some things I wrote in abmath, so now I need to go back and revise some more!

In this post I will make some points about definitions that I learned from the paper by Edwards and Ward and the paper by Selden and Selden

I hope math ed people will read the final remarks.

Peculiarities of math definitions

When I use a word, it means just what I choose it to mean–neither more nor less.” — Humpty Dumpty

A mathematical definition is fundamentally different from other sorts of definitions in two different ways. These differences are not widely appreciated by students or even by mathematicians. The differences cause students a lot of trouble.

List of properties

One of the ways in which a math definition is different from other kinds is that the definition of a math object is given by accumulation of attributes, that is, by listing properties that the object is required to have. Any object defined by the definition must have all those properties, and conversely any object with all the properties must be an example of the type of object being defined. Furthermore, there is no other criterion than the list of attributes.

Definitions in many fields, including some sciences, don’t follow this rule. Those definitions may list some properties the objects defined may have, but exceptions may be allowed. They also sometimes give prototypical examples. Dictionary definitions are generally based on observation of usage in writing and speech.

Imposed by decree

One thing that Edwards and Ward pointed out is that, unlike definitions in most other areas of knowledge, a math definition is stipulated. That means that meaning of (the name of) a math object is imposed on the reader by decree, rather than being determined by studying the way the word is used, as a lexicographer would do. Mathematicians have the liberty of defining (or redefining) a math object in any way they want, provided it is expressed as a compulsory list of attributes. (When I read the paper by Edwards and Ward, I realized that the abstractmath.org article on math definitions did not spell that out, although it was implicit. I have recently revised it to say something about this, but it needs further work.)

An example is the fact that in the nineteenth century some mathe­maticians allowed $1$ to be a prime. Eventually they restricted the definition to exclude $1$ because including it made the statement of the Fundamental Theorem of Arithmetic complicated to state.

Another example is that it has become common to stipulate codomains as well as domains for functions.

Student difficulties

Giving the math definition low priority

Some beginning abstract math students don’t give the math definition the absolute dictatorial power that it has. They may depend on their understanding of some examples they have studied and actively avoid referring to the definition. Examples of this are given by Edwards and Ward.

Arbitrary bothers them

Students are bothered by definitions that seem arbitrary. This includes the fact that the definition of “prime” excludes $1$. There is of course no rule that says definitions must not seem arbitrary, but the students still need an explanation (when we can give it) about why definitions are specified in the way they are.

What do you DO with a definition?

Some students don’t realize that a definition gives a magic formula — all you have to do is say it out loud.
More generally, the definition of a kind of math object, and also each theorem about it, gives you one or more methods to deal with the type of object.

For example, $n$ is a prime by definition if $n\gt 1$ and the only positive integers that divide $n$ are $1$ and $n$. Now if you know that $p$ is a prime bigger than $10$ then you can say that $p$ is not divisible by $3$ because the definition of prime says so. (In Hogwarts you have to say it in Latin, but that is no longer true in math!) Likewise, if $n\gt10$ and $3$ divides $n$ then you can say that $n$ is not a prime by definition of prime.

The paper by Bills and Tall calls this sort of thing an operable definition.

The paper by Selden and Selden gives a more substantial example using the definition of inverse image. If $f:S\to T$ and $T’\subseteq T$, then by definition, the inverse image $f^{-1}T’$ is the set $\{s\in S\,|\,f(s)\in T’\}$. You now have a magic spell — just say it and it makes something true:

• If you know $x\in f^{-1}T’$ then can state that $f(x)\in T’$, and all you need to justify that statement is to say “by definition of inverse image”.
• If you know $f(x)\in T’$ then you can state that $x\in f^{-1}T’$, using the same magic spell.

Theorems can be operable, too. Wiles’ Theorem wipes out the possibility that there is an integer $n$ for which $n^{42}=365^{42}+666^{42}$. You just quote Wiles’ Theorem — you don’t have to calculate anything. It’s a spell that reveals impossibilities.

What the operability of definitions and theorems means is:

A definition or theorem is not just a static statement,it is a weapon for deducing truth.

Some students do not realize this. The students need to be told what is going on. They do not have to be discarded to become history majors just because they may not have the capability of becoming another Andrew Wiles.

Final remarks

I have a wish that more math ed people would write blog posts or informal articles (like the one by Edwards and Ward) about what that have learned about students learning math at the college level. Math ed people do write scholarly articles, but most of the articles are behind paywalls. We need accessible articles and blog posts aimed at students and others aimed at math teachers.

And feel free to steal other math ed people’s ideas (and credit them in a footnote). That’s what I have been doing in abstractmath.org and in this blog for the last ten years.

References

• Bills, L., & Tall, D. (1998). Operable definitions in advanced mathematics: The case of the least upper bound. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 104-111). Stellenbosch, South Africa: University of Stellenbosch.
• B. S. Edwards, and M. B. Ward, Surprises from mathematics education research: Student (mis) use of mathematical definitions (2004). American Mathematical Monthly, 111, 411-424.
• G. Lakoff, Women, Fire and Dangerous
Things
. University of Chicago Press, 1990. See his discussion of concepts and prototypes.
• J. Selden and A. Selden, Proof Construction Perspectives: Structure, Sequences of Actions, and Local Memory, Extended Abstract for KHDM Conference, Hanover, Germany, December 1-4, 2015. This paper may be downloaded from Academia.edu.
• A Handbook of mathematical discourse, by Charles Wells. See concept, definition, and prototype.
• Definitions, article in abstractmath.org. (Some of the ideas in this post have now been included in this article, but it is due for another revision.)
• Definitions in logic and mathematics in Wikipedia.

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Very early difficulties II

This is the second part of a series of posts about certain difficulties math students have in the very early stages of studying abstract math. The first post, Very early difficulties in studying abstract math, gives some background to the subject and discusses one particular difficulty: Some students do not know that it is worthwhile to try starting a proof by rewriting what is to be proved using the definitions of the terms involved.

Math StackExchange

The website Math StackExchange is open to any questions about math, even very easy ones. It is in contrast with Math OverFlow, which is aimed at professional mathematicians asking questions in their own field.

Math SE contains many examples of the early difficulties discussed in this series of posts, and I recommend to math ed people (not just RUME people, since some abstract math occurs in advanced high school courses) that they might consider reading through questions on Math SE for examples of misunderstanding students have.

There are two caveats:

• Most questions on Math SE are at a high enough level that they don’t really concern these early difficulties.
• Many of the questions are so confused that it is hard to pinpoint what is causing the difficulty that the questioner has.

Connotations of English words

The terms(s) defined in a definition are often given ordinary English words as names, and the beginner automatically associates the connotations of the meaning of the English word with the objects defined in the definition.

Infinite cardinals

If $A$ if a finite set, the cardinality of $A$ is simply a natural number (including $0$). If $A$ is a proper subset of another set $B$, then the cardinality of $A$ is strictly less than the cardinality of $B$.

In the nineteenth century, mathematicians extended the definition of cardinality for infinite sets, and for the most part cardinality has the same behavior as for finite sets. For example, the cardinal numbers are well-ordered. However, for infinite sets it is possible for a set and a proper subset of the set to have the same cardinality. For example, the cardinality of the set of natural numbers is the same as the cardinality of the set of rational numbers. This phenomenon causes major cognitive dissonance.

Question 1331680 on Math Stack Exchange shows an example of this confusion. I have also discussed the problem with cardinality in the abstractmath.org section Cardinality.

Morphism in category theory

The concept of category is defined by saying there is a bunch of objects called objects (sorry bout that) and a bunch of objects called morphisms, subject to certain axioms. One requirement is that there are functions from morphisms to objects choosing a “domain” and a “codomain” of each morphism. This is spelled out in Category Theory in Wikibooks, and in any other book on category theory.

The concepts of morphism, domain and codomain in a category are therefore defined by abstract definitions, which means that any property of morphisms and their domains and codomains that is true in every category must follow from the axioms. However, the word “morphism” and the talk about domains and codomains naturally suggests to many students that a morphism must be a function, so they immediately and incorrectly expect to evaluate it at an element of its domain, or to treat it as a function in other ways.

Example

If $\mathcal{C}$ is a category, its opposite category $\mathcal{C}^{op}$ is defined this way:

• The objects of $\mathcal{C}^{op}$ are the objects of $\mathcal{C}$.
• A morphism $f:X\to Y$ of $\mathcal{C}^{op}$ is a morphism from $Y$ to $X$ of $\mathcal{C}$ (swap the domain and codomain).

In Question 980933 on Math SE, the questioner is saying (among other things) that in $\text{Set}^{op}$, this would imply that there has to be a morphism from a nonempty set to the empty set. This of course is true, but the questioner is worried that you can’t have a function from a nonempty set to the empty set. That is also true, but what it implies is that in $\text{Set}^{op}$, the morphism from $\{1,2,3\}$ to the empty set is not a function from $\{1,2,3\}$ to the empty set. The morphism exists, but it is not a function. This does not any any sense make the definition of $\text{Set}^{op}$ incorrect.

Student confusion like this tends to make the teacher want to have a one foot by six foot billboard in his classroom saying

A MORPHISM DOESN’T HAVE TO BE A FUNCTION!

However, even that statement causes confusion. The questioner who asked Question 1594658 essentially responded to the statement in purple prose above by assuming a morphism that is “not a function” must have two distinct values at some input!

That questioner is still allowing the connotations of the word “morphism” to lead them to assume something that the definition of category does not give: that the morphism can evaluate elements of the domain to give elements of the codomain.

So we need a more elaborate poster in the classroom:

The definition of “category” makes no requirement
that an object has elements
or that morphisms evaluate elements.

As was remarked long long ago, category theory is pointless.

English words implementing logic

There are lots of questions about logic that show that students really do not think that the definition of some particular logical construction can possibly be correct. That is why in the abstractmath.org chapter on definitions I inserted this purple prose:

A definition is a totalitarian dictator.

It is often the case that you can explain why the definition is worded the way it is, and of course when you can you should. But it is also true that the student has to grovel and obey the definition no matter how weird they think it is.

Formula and term

In logic you learn that a formula is a statement with variables in it, for example “$\exists x((x+5)^3\gt2)$”. The expression “$(x+5)^3$” is not a formula because it is not a statement; it is a “term”. But in English, $H_2O$ is a formula, the formula for water. As a result, some students have a remarkably difficult time understanding the difference between “term” and “formula”. I think that is because those students don’t really believe that the definition must be taken seriously.

Exclusive or

Question 804250 in MathSE says:

“Consider $P$ and $Q$. Let $P+Q$ denote exclusive or. Then if $P$ and $Q$ are both true or are both false then $P+Q$ is false. If one of them is true and one of them is false then $P+Q$ is true. By exclusive or I mean $P$ or $Q$ but not both. I have been trying to figure out why the truth table is the way it is. For example if $P$ is true and $Q$ is true then no matter what would it be true?”

I believe that the questioner is really confused by the plus sign: $P+Q$ ought to be true if $P$ and $Q$ are both true because that’s what the plus sign ought to mean.

Yes, I know this is about a symbol instead of an English word, but I think the difficulty has the same dynamics as the English-word examples I have given.

If I have understood this difficulty correctly, it is similar to the students who want to know why $1$ is not a prime number. In that case, there is a good explanation.

Only if

The phrase “only if” simply does not mean the same thing in math as it does in English. In Question 17562 in MathSE, a reader asks the question, why does “$P$ only if $Q$” mean the same as “if $P$ then $Q$” instead of “if $Q$ then $P$”?

Many answerers wasted a lot of time trying to convince us that “$P$ only if $Q$” mean the same as “if $P$ then $Q$” in ordinary English, when in fact it does not. That’s because in English, clauses involving “if” usually connote causation, which does not happen in math English.

Consider these two pairs of examples.

1. “I take my umbrella only if it is raining.”
2. “If I take my umbrella, then it is raining.”
3. “I flip that switch only if a light comes on.”
4. “If I flip that switch, a light comes on.”

The average non-mathematical English speaker will easily believe that (1) and (4) are true, but will balk and (2) and (3). To me, (3) means that the light coming on makes me flip the switch. (2) is more problematical, but it does (to me) have a feeling of causation going the wrong way. It is this difference that causes students to balk at the equivalence in math of “$P$ only if $Q$” and “If $P$, then $Q$”. In math, there is no such thing as causation, and the truth tables for implication force us to live with the fact that these two sentences mean the same thing.

Henning Makholm’ answer to Question 17562 begins this way: “I don’t think there’s really anything to understand here. One simply has to learn as a fact that in mathematics jargon the words ‘only if’ invariably encode that particular meaning. It is not really forced by the everyday meanings of ‘only’ and’ if’ in isolation; it’s just how it is.” That is the best way to answer the question. (Other answerers besides Makholm said something similar.)

I have also discussed this difficulty (and other difficulties with logic) in the abmath section on “only if“.

References

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Very early difficulties in studying abstract math

Introduction

There are a some difficulties that students have at the very beginning of studying abstract math that are overwhelmingly important, not because they are difficult to explain but because too many teachers don’t even know the difficulties exist, or if they do, they think they are trivial and the students should know better without being told. These difficulties cause too many students to give up on abstract math and drop out of STEM courses altogether.

I spent my entire career in math at Case Western Reserve University. I taught many calculus sections, some courses taken by math majors, and discrete math courses taken mostly by computing science majors. I became aware that some students who may have been A students in calculus essentially fell off a cliff when they had to do the more abstract reasoning involved in discrete math, and in the initial courses in abstract algebra, linear algebra, advanced calculus and logic.

That experience led me to write the Handbook of Mathematical Discourse and to create the website abstractmath.org. Abstractmath.org in particular grew quite large. It does describe some of the major difficulties that caused good students to fall of the abstraction cliff, but also describes many many minor difficulties. The latter are mostly about the peculiarities of the languages of math.

I have observed people’s use of language since I was like four or five years old. Not because I consciously wanted to — I just did. When I was a teenager I would have wanted to be a linguist if I had known what linguistics is.

I will describe one of the major difficulties here (failure to rewrite according to the definition) with an example. I am planning future posts concerning other difficulties that occur specifically at the very beginning of studying abstract math.

Rewrite according to the definition

To prove that a statement
involving some concepts is true,
start by rewriting the statement
using the definitions of the concepts.

Example

Definition

A function $f:S\to T$ is surjective if for any $t\in T$ there is an $s\in S$ for which $f(s)=t$.

Definition

For a function $f:S\to T$, the image of $f$ is the set $\{t\in T\,|\,\text{there is an }s\in S\text{ for which }f(s)=t\}$

Theorem

Let $f:S\to T$ be a function between sets. Then $f$ is surjective if and only if the image of $f$ is $T$.

Proof

If $f$ is surjective, then the statement “there is an $s\in S$ for which $f(s)=t$” is true for any $t\in T$ by definition of surjectivity. Therefore, by definition of image, the image of $f$ is $T$.

If the image of $f$ is $T$, then the definition of image means that there is an $s\in S$ for which $f(s)=t$ for any $t\in T$. So by definition of surjective, $f$ is surjective.

“This proof is trivial”

The response of many mathematicians I know is that this proof is trivial and a student who can’t come up with it doesn’t belong in a university math course. I agree that the proof is trivial. I even agree that such a student is not a likely candidate for getting a Ph.D. in math. But:

• Most math students in an American university are not going to get a Ph.D. in math. They may be going on in some STEM field or to teach high school math.
• Some courses taken by students who are not math majors take courses in which simple proofs are required (particularly discrete math and linear algebra). Some of these students may simply be interested in math for its own sake!

A sizeable minority of students who are taking a math course requiring proofs need to be told the most elementary facts about how to do proofs. To refuse to explain these facts is a disfavor to the mathematics community and adds to the fear and dislike of math that too many people already have.

These remarks may not apply to students in many countries other than the USA. See When these problems occur.

“This proof does not describe how mathematicians think”

The proof I wrote out above does not describe how I would come up with a proof of the statement, which would go something like this: I do math largely in pictures. I envision the image of $f$ as a kind of highlighted area of the codomain of $f$. If $f$ is surjective, the highlighting covers the whole codomain. That’s what the theorem says. I wouldn’t dream of writing out the proof I gave about just to verify that it is true.

More examples

Abstractmath.org and Gyre&Gimble contain several spelled-out theorems that start by rewriting according to the definition. In these examples one then goes on to use algebraic manipulation or to quote known theorems to put the proof together.

This post contains testable claims

Herein, I claim that some things are true of students just beginning abstract math. The claims are based largely on my teaching experience and some statements in the math ed literature. These claims are testable.

When these problems occur

In the United States, the problems I describe here occur in the student’s first or second year, in university courses aimed at math majors and other STEM majors. Students typically start university at age 18, and when they start university they may not choose their major until the second year.

In much of the rest of the world, students are more likely to have one more year in a secondary school (sixth form in England lasts two years) or go to a “college” for a year or two before entering a university, and then they get their bachelor’s degree in three years instead of four as in the USA. Not only that, when they do go to university they enter a particular program immediately — math, computing science, etc.

These differences may mean that the abstract math cliff occurs early in a student’s university career in the USA and before the student enters university elsewhere.

In my experience at CWRU, some math majors fall of the cliff, but the percentage of computing science students having trouble was considerably greater. On the other hand, more of them survived the discrete math course when I taught it because the discrete math course contain less abstraction and more computation than the math major courses (except linear algebra, which had a balance similar to the discrete math course — and was taken by a sizeable number of non-math majors).

References

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Context

This is a revised draft of the abstractmath.org article on context in math texts. Note: WordPress changed double primes into quotes. Tsk.

Context

Written and especially spoken language depends heavily on the context – the physical surroundings, the preceding conversation, and social and cultural assumptions.  Mathematical statements are produced in such contexts, too, but here I will discuss a special thing that happens in math conversation and writing that does not seem to happen much in other sorts of discourse:

The meanings of expressions
in both the symbolic language and math English
change from phrase to phrase
as the speaker or writer changes the constraints on them.

Example

In a math text, before the occurrence of a phrase such as “Let $n=3$”, $n$ may be known only as an integer variable.  After the phrase, it means specifically $3$.  So this phrase changes the meaning of $n$ by constraining $n$
to be $3$.  We say the context of occurrences of “$n$” before the phrase requires only that $n$ be an integer, but after the occurrence the context requires $n=3$.

Definition

In this article, the context at a particular location in mathematical discourse is the sum total of what the reader or listener can know about the symbols and names used in the discourse when they have read everything up to that location.

Remarks

• Each clause can change the meaning of or constraints on one or more symbols or names. The conventions in effect during the discourse can also put constraints on the symbols and names.
• Chierchia and McConnell-Ginet give a mathematical definition of context in the sense described here.
• The references to “before” and “after” the phrase “Let $3$” refer to the physical location in text and to actual time in spoken math. There is more about this phenomenon in the Handbook of Mathematical Discourse, page 252, items (f) and (g).
• Contextual changes of this sort take place using the pretense that you are reading the text in order, which many students and professionals do not do (they are “grasshoppers”).
• I am not aware of much context-changing in everyday speech. One place it does occur is in playing games. For example, during some card games the word “trumps” changes meaning from time to time.
• In symbolic logic, the context at a given place may be denoted by “$\Gamma$”.

Detailed example of a math text

Here is a typical example of a theorem and its proof.  It is printed twice, the second time with comments about the changes of context.  This is the same proof that is already analyzed practically to death in the chapter on presentation of proofs.

First time through

Definition: Divides

Let $m$ and $n$ be integers with $m\ne 0$. The statement “$m$ divides $n$” means that there is an integer $q$ for which $n=qm$

Theorem

Let $m$, $n$ and $p$ be integers, with $m$ and $n$ nonzero, and suppose $m$ divides $n$ and $n$ divides $p$.  Then $m$ divides $p$.

Proof

By definition of divides, there are integers $q$ and $q’$ for which $n=qm$ and $p=q’n$. We must prove that there is an integer $q”$ for which $p=q”m$. But $p=q’n=q’qm$, so let $q”=q’q$.  Then $p=q”m$.

Second time, with analysis

 Definition: Divides Begins a definition. The word “divides” is the word being defined. The scope of the definition is the following paragraph. Let $m$ and $n$ be integers $m$ and $n$ are new symbols in this discourse, constrained to be integers. with $m\ne 0$ Another constraint on $m$. The statement “$m$ divides $n$ means that” This phrase means that what follows is the definition of “$m$ divides $n$” there is an integer $q$ “There is” signals that we are beginning an existence statement and that $q$ is the bound variable within the existence statement. for which $n=qm$ Now we know that “$m$ divides $n$” and “there is an integer $q$ for which $m=qn$” are equivalent statements.  Notes: (1) The first statement would only have implied the second statement if this had not been in the context of a definition. (2) After the conclusion of the definition, $m$, $n$ and $q$ are undefined variables. Theorem This announces that the next paragraph is a statement has been proved. In fact, in real time the statement was proved long before this discourse was written, but in terms of reading the text in order, it has not yet been proved. Let $m$, $n$ and $p$ be integers, “Let” tells us that the following statement is the hypothesis of an implication, so we can assume that $m$, $n$ and $p$ are all integers.  This changes the status of $m$ and $n$, which were variables used in the preceding paragraph, but whose constraints disappeared at the end of the paragraph.  We are starting over with $m$ and $n$. with $m$ and $n$ nonzero. This clause is also part of the hypothesis. We can assume $m$ and $n$ are constrained to be nonzero. and suppose $m$ divides $n$ and $n$ divides $p$. This is the last clause in the hypothesis. We can assume that $m$ divides $n$ and $n$ divides $p$. Then $m$ divides $p$. This is a claim that $m$ divides $p$. It has a different status from the assumptions that $m$ divides $n$ and $n$ divides $p$. If we are going to follow the proof we have to treat $m$ and $n$ as if they divide $n$ and $p$ respectively. However, we can’t treat $m$ as if it divides $p$. All we know is that the author is claiming that $m$ divides $p$, given the facts in the hypothesis. Proof An announcement that a proof is about to begin, meaning a chain of math reasoning. The fact that it is a proof of the Theorem just stated is not explicitly stated. By definition of divides, there are integers $q$ and $q’$ for which $n=qm$ and $p=q’n$. The proof uses the direct method (rather than contradiction or induction or some other method) and begins by rewriting the hypothesis using the definition of “divides”. The proof does not announce the use of these techniques, it just starts in doing it. So $q$ and $q$’ are new symbols that satisfy the equations $n=qm$ and $p=q’n$. The phrase “by definition of divides” justifies the introduction of $q$ and $q’$. $m$, $n$ and $p$ have already been introduced in the statement of the Theorem. We must prove that there is an integer $q”$ for which $p=q”m$. Introduces a new variable $q”$ which has not been given a value. We must define it so that $p=q”m$; this requirement is justified (without saying so) by the definition of “divides”. But $p=q’n=q’qm$, This is a claim about $p$, $q$, $q’$, $m$ and $n$.  It is justified by certain preceding sentences but this justification is not made explicit. Note that “$p=q’n=q’qm$” pivots on $q’n$, in other words makes two claims about it. so let $q”=q’q$. We have already introduced $q”$; now we give it the value $q”=q’q$. Then $p=q”m$ This is an assertion about $p$, $q”$ and $n$, justified (but not explicitly — note the hidden use of associativity) by the previous claim that $p=q’n=q’qm$. The proof is now complete, although no statement asserts that it is.

Remark

If you have some skill in reading proofs, all the stuff in the right hand column happens in your brain without, for the most part, your being conscious of it.

Acknowledgment

Thanks to Chris Smith for correcting errors.

References for “context”

Chierchia, G. and S. McConnell-Ginet
(1990), Meaning and Grammar. The MIT Press.

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

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

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The Mathematics Depository: A Proposal

Introduction

This post is about taking texts written in mathematical English and the symbolic language and encoding it in a formal language that could be tested by an automated proof verifier. This is a very difficult undertaking, but we could get closer and closer to a working system by a worldwide effort continuing over, probably, decades. The system would have to contain many components working together to create incremental improvements in the process.

This post, which is a first draft, outlines some suggestions as to how this could work. I do not discuss the encoding required, which is not my area of expertise. Yes, I understand that coding is the hard part!

Much work has been done by computing scientists in developing proof checking and proof-finding programs. Work has also been done, primarily by math education workers but also by some philosophers and computing scientists, in uncovering the many areas where ordinary math language is ambiguous and deviates from ordinary English usage. These characteristics confuse students and also make it hard to design a program that can interpret the language. I have been working in that area mostly from the math ed point of view for the last twenty years.

The Reference section lists many references to the problem of parsing mathematical English, some from the point of view of automatic translation of math language into code, but most from the point of view of helping students understand how to understand it.

The Mathematics Depository

I imagine a system for converting documents written in math language into machine-readable language and testing their claims. An organization, call it the Mathematics Depository, would be developed that is supported by many countries, organizations and individual supporters. It should consist of several components listed below, no doubt with other components as we become aware of needing them. The organization would be tasked with supporting and improving these components over time.

The main parts of the system

Each component is linked to a more detailed description that is given later in this post.

• A Proof Verifier (PV), that inputs a proof and determines if it is correct.
• A specification of a supported subset of Mathematical English and the symbolic language, that I will call Strict Math English (SME).
• A Text-SME Converter, a program that would input a text written in ordinary math English that has been annotated by a knowledgeable person and convert it into SME.
• An SME-PV Converter that will convert text written in SME into code that can be directly read by the Proof Verifier.
• One or more Automatic Theorem Provers, that to begin with can take fairly simple conjectures written in SME and sometimes succeed in proving them.
• An Annotation System containing an Annotation Editor that would allow a person to use SME to annotate an article written in ordinary math English so that it could be read by the Text-SME Converter.
• A Data Base that would include the texts that have been collected in this endeavor, along with the annotations and the results of the proof checking.
• A Data Base Miner that would watch for patterns in the annotations as new papers were submitted. The operators might also program it to watch for patterns in other aspects of the operation.

These facilities would be organized so that the systems work together, with the result that the individual components I named improve over time, both automatically and via human intervention.

Flow of Work

1. A math text is submitted.
2. If it is already in Strict Math English (SME), it is input to the Proof Verifier (PV).
3. Otherwise, the math text is input into the Annotation System.
4. The resulting SME text is input into the Text-SME Converter.
5. The output of the Text-SME Converter is input into the Proof Verifier.
6. The PV incorporates each definition in the text into the context of the math text. This is a specific meaning of the word “context”, including a list of the status of variables (bound, unbound, type, and so on), meanings of technical words, and other facts created in the text. “Context” is described informally in my article Context in abstractmath.org. That article gives references to the formal literature.
7. In my experience mathematicians spend only a little time reading arguments step by step as described in the Context article. They usually look at a theorem and try to figure it out themselves, “cheating” occasionally by glancing at parts of the proof.

8. Each mathematical assertion in the text is marked as a claim.
9. The checking process records those claims occurring in the proof that are not proved in the text, along with any references given to other texts.
10. If a reference to a result in another text is made, the PV looks for the result in the Database. If it does not find it, the PV incorporates the result and its location in the Database as an externally proven but untested claim.
11. If no reference or proof for a claim is given, the PV checks the Database to see if it has already been proved.
12. Any claim in the current text not shown as proven in the Database is submitted to the Automatic Theorem Prover (ATP). The output of the ATP is put in the database (proved, counterexample found, or unable to determine truth).
13. If a segment of text is presented as a proof, it is input into the PV to be verified.
14. The PV reports the result for each claimed proof, which can consist of several possibilities:
• A counterexample for a proof is found, so the claim that the proof was supposed to report is false.
• The proof contains gaps, so the claim is unsettled.
• The proof is reported as correct.
15. At the end of the process, all the information gathered is put into the Database:
• The original text showing all the annotations.
• The text in SME.
• All claims, with their status (proven true, proven false, truth unknown, reference if one was given).
• Every proof, with its status and the entire context at each step of the proof.

Details

The proof verifier

• Proof checking programs have been developed over the last thirty or so years. The MD should write or adapt one or more Proof Verifiers and improve it incrementally as a result of experience in running the system. In this post I have assumed the use of just one Proof Verifier.
• The Proof Verifier should be designed to read the output of the SME-PV converter.
• The PV must read a whole math text in SME, identify and record each claim and check each proof (among other things). This is different from current proof verifiers, which take exactly one proof as input.
• The PV must create the context of each proof and change it step by step as it reads each syntactic fragment of the math text.
• Typically the context for a claimed proof is built up in the whole math text, not just in the part called “Proof”.
• The PV should automatically query the Data Base for unproved steps in a proof in the input text to see if they have already been verified somewhere else. These results should be quoted in a proof verifier output.
• The PV should also automatically submit steps in the proof that haven’t been verified to the Automatic Theorem Provers and wait for the step to be verified or not.
• The Proof Verifier should output details of the result of the checking whether it succeeded in verifying the whole input text or not. In particular, it should list steps in proofs it failed to verify, including steps in proofs for which the input text cited the proof in some other paper, in the MD system or not.
• The Proof Verifier should be available online for anyone to submit, in SME, a mathematical text claiming to prove a theorem. Submission might require a small charge.

Strict Math English

• One of the most important aspects of the system would be the simultaneous incremental updating of the SME and the SME-PV Converter.
• The idea is that SME would get more and more inclusive of the phrases and clauses it allows.

Example: Universal Assertions

At the start SME might allow these statements to be recognized as the same universal assertion:

• “$\forall x(x^2+1\gt0)$”
• “For all [every, any] $x$, $x^2+1\gt0$.” (universality asserted using an English word.)
• “For all [every, any] $x$, $x^2+1$ is positive.”

As time goes on, a person or the Data Base Miner might detect that many annotators also recognized these statements as saying the same thing:

• “$x^2+1\gt0\,\,\,\,\,(\text{all } x)$” (as a displayed statement)
• “$x^2+1$ is positive for every $x$.” Universality asserted using an adjective in a postposited phrase.
• “$x^2+1$ is always positive.” Universality hidden in a postposited adverb that seems to be referring to time!
• There are more examples in my article Universally True Assertions. See also Susanna Epp’s article on quantification for other problems in this area.

These other variations would then be added to the Strict Math Language. (This is only an example of how the system would evolve. I have no doubt that in fact all the terminology mentioned above would be included at the outset, since they are all documented in the math ed literature.)

Even at the start, SME will include phrases and clauses in the English language as well as symbolic expressions. It is notorious that automatically parsing general English sentences is difficult and that the ubiquity of metaphors makes it essentially impossible to reliably construct the meaning of a sentence. That is why SME must start with a very narrow subset of math English. But even in early days, it should include some stereotyped metaphors, such as using “always” in universal assertions.

The SME-PV Converter

• The SME-PV Converter would read documents written in SME and convert them into code readable by the proof checking program, as well as by the automatic theorem provers.
• Such a program is essentially the subject of Ganesingalam’s book.
• Converting SME so that the Proof Verifier can handle it involves lots of subtleties. For example, if the text says, “For any $x$, $x^2+1\gt0$”, the translation has to recognize not only that this is a universally quantified statement with $x$ as the bound variable, but that $x$ must be a real number, since complex numbers don’t do greater-than.
• Frequent revisions of the SME-PV Converter will be necessary since its input language, the SME, will be constantly expanded.
• It may be that the output language of the SME-PV Converter (which the Proof Verifier and Automatic Theorem Provers read) will require only infrequent revisions.

The Automatic Theorem Provers

• The system could support several ATP’s, each one adapted to read the output of the SME-PV Converter.
• The Automatic Theorem Provers should provide output in such a way that the Proof Verifier can include in its report the positive or negative results of the Theorem Prover in detail.

The Annotation System

• The Annotation system would facilitate construction of a data structure that connects each annotation to the specific piece of text it rewrites. The linking should be facilitated by the Annotation Editor.
• For example, an annotation that is meant to explain that the statement (in the input text) “$x^2+1$ is always greater than $0$” is to be translated as “$\forall x(x^2+1\gt0)$” (which is presumably allowed by SME) should cause the first statement to be be linked to the second statement. The first statement, the one in the input text, should not be changed. This will enable the Data Base Miner to find patterns of similar text being annotated in similar ways.
• The annotations should clarify words, symbolic expressions and sentences in the input text to allow the Proof Verifier to input them correctly.
• In particular, every claim that a statement is true should be marked as a proposed theorem, and similarly every proof should be marked as a proof and every definition should be marked as a definition. Such labeling is often omitted in the math literature. Annotators would have to recognize segments of the text as claims, proofs and definitions and annotate them as such.
• The annotations would be written in the current version of Strict Math English. Since SME is frequently updated, the instructions for the annotator would also have to be frequently updated.

Examples

• If a paper used the word “domain” without defining it, the annotator would clarify whether it meant an open connected set, a type of ring, a type of poset, or the domain of a function. See Example 1
• Annotators will note instances in which the same text will use a symbol with two different meanings. See Example 2.
• In a phrase, a single occurrence of a symbol can require an annotation that assigns more than one attribute to the symbol. See Example 3.

The Annotation Editor

• The annotators should be provided with an Annotation Editor designed specifically for annotation.
• The editor should include a system of linking an annotation to the exact phrase it annotates that is easy for a person reading the annotated document to understand it as well as providing the information to the Text-SME Converter.

The Annotators

• Great demands will be made of an annotator.
• They must understand the detailed meaning of the text they annotate. This means they must be quite familiar with the field of math the text is concerned with.
• They must learn SME. I know for a fact that many mathematicians are not good at learning foreign languages. It will help that SME will be a subset of the full language of math.
• All this means that annotators must be chosen carefully and paid well. This means that not very many papers will get annotated by paid annotators, so that there will have to be some committee that chooses the papers to be annotated. This will be a genuine bottleneck.
• One thing that will help in the long run is that the SME should evolve to include more features of the general language of math, so many mathematicians will actually write their papers in SME and submit it directly to the Depository. (“Long run” may mean more than ten years).

The Text-to-SME Converter

• This converter takes a math text in ordinary Math English that has been annotated and convert it into SME.
• The format for feeding it to the Automatic Theorem Prover may very well have to be different from the format to be read by a human. Both formats should be saved.

The Data Base

• The Data Base would contain all math papers that have been run through the Proof Verifier, along with the results found by the Proof Verifier. A paper should be included whether or not every claim in the paper was verified.
• Funding agencies (and private individuals) might choose particularly important papers and pay more money for annotation for those than for other papers.
• Mathematicians in a particular field could be hired to annotate particular articles in their field, using a standard annotation language that would develop through time.
• The annotated papers would be made freely available to the public.
• It will no doubt prove useful for the Data Base to contain many other items. Possibilities:
• A searchable list of all theorems that have been verified.
• A glossary: a list of math words that have been defined in the papers in the Depository. This will include synonyms and words with multiple meanings.

The Data Base Miner

Watch for patterns

The DBM would watch for patterns in annotation as new annotated papers were submitted. It should probably look only at annotated papers whose proofs had been verified. The patterns might include:

• Correlation between annotations that associate particular meanings to particular words or symbols with the branch of math the paper belongs to. See Example 1.
• Noting that a particular format of combining symbols usually results in the same kind of annotation. See Example 4.
• Providing data in such a way that lexicographers studying math English could make use of them. My Handbook began with my doing lexicographical research on math English, but I found it so slow that when I started abstractmath.org I resolved not to such research any more. Nevertheless, it needs to be done and the Database should make the process much easier.

Statistical translation

Since the annotated papers will be stored in the Data Base, the Data Base Miner could use the annotations in somewhat the same way some language translators work (in part): to translate a phrase, it will find occurrences of the phrase in the source language that have been translated into the target language and use the most common translation. In this case the source language is the paper (in English) and the target language is in annotated math English readable by the Proof Verifier. Once the Database includes most of the papers ever published (twenty years from now?), statistical translation might actually become useful.

Examples

Example 1: Meaning varies with branch of math

• Field” means one thing in an algebra paper and another in a mathematical physics paper.
• Domain” means
• An open connected set in topology.
• A type of ring in algebra.
• A type of poset in theoretical computing science.
• The domain of a function –everywhere in math, which makes it seem that this is going to be very hard to distinguish without human help!
• Log” usually implies base $2$ in the computing world, base $10$ in engineering (but I am not sure how prevalant this meaning is there), and base $e$ in pure math. With exceptions!
• Example 2: Meaning varies even in the same article

• The notation “$(a,b)$” can mean an ordered pair, an open interval, or the GCD. What’s worse, there are many instances where the symbol is used without definition. Citation 139 in the Handbook provides a single sentence in which the first two meanings both occur:

$\dots$ Richard Darst and Gerald Taylor investigated the differentiability of functions $f^p$ (which for our purposes we will restrict to $(0,1)$) defined for each $p\geq1$ by$F(x):= \begin{cases} 0 & \text{if }x\text{ is irrational}\\ \displaystyle{\frac{1}{n^p}} & \text{if }x = \displaystyle{\frac{m}{n}}\text{ with }(m,n)=1\\ \end{cases}$

The sad thing is that any mathematician will know immediately what each occurrence means. This may be a case where the correct annotation will never be automatically detectable.

Example 3: One mention of a symbol may require several meanings

In the sentence, “This infinite series converges to $\zeta(2)=\frac{\pi^2}{6}\approx 1.65$,” the annotator would provide two pieces of information about “$\frac{\pi^2}{6}$”, namely that it is both the right constituent of the equation “$\zeta(2)=\frac{\pi^2}{6}$” and the left constituent of the approximation statement “$\frac{\pi^2}{6}\approx 1.65$” — and that these two statements were the constituents of an asserted conjunction. (See my post Pivoted symbols.)

Example 4: Function to a power

Some expressions not in the SME will almost always be annotated in the same way. This makes it discoverable by the Data Base Miner.

• “$\sin^{-1}x$” always means $\arcsin x$.
• For positive $n$, “$\sin^n x$” always means $(\sin x)^n$. It never means the $n$-fold application of $\sin$ to $x$.
• In contrast, for an arbitrary function symbol, $f^n(x)$ will often be annotated as $n$-fold application of $f$ and also often as $f(x)^n$. (And maybe those last two possibilities are correlated by branch of math.)

References

I believe that work in formal verification has tended to overlook the work on math language difficulties in math ed, so I have included some articles from that specialty.

The following are posts from my blog Gyre&Gimble. They are in reverse chronological order.

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Abstraction and the axiomatic method

This post will become an article in abstractmath.org.

Abstraction

An abstraction of a concept $C$ is a concept $C’$ with these properties:

• $C’$ includes all instances of $C$ and
• $C’$ is constructed by taking as axioms certain assertions that are true of all instances of $C$.

There are two major situations where abstraction is used in math.

• $C$ may be a familiar concept or property that has not yet been given a math definition.
• $C$ may already have a mathe­matical definition using axioms. In that case the abstraction will be a generalization of $C$.

In both cases, the math definition may allow instances of $C’$ that were not originally thought of as being part of $C$.

Example: Relations

Mathematicians have made use of relations between math objects since antiquity.

• For real numbers $r$ and $s$. “$r\lt x$” means that $r$ is less than $s$. So the statement “$5\lt 7$” is true, but the statement “$7\lt 5$” is false. We say that “$\lt$” is a relation on the real numbers. Other relations on real numbers denoted by symbols are “$=$” and “$\leq$”.
• Suppose $m$ and $n$ are positive integers. $m$ and $n$ are said to be relatively prime if the greatest common divisor of $m$ and $n$ is $1$. So $5$ and $7$ are relatively prime, but $15$ and $21$ are not relatively prime. So being relatively prime is a relation on positive integers. This is a relation that does not have a commonly used symbol.
• The concept of congruence of triangles has been used for a couple of millenia. In recent centuries it has been denoted by the symbol “$\cong$”. Congruence is a relation on triangles.

One could say that a relation is a true-or-false statement that can be made about a pair of math objects of a certain type. Logicians have in fact made that a formal definition. But when set theory came to be used around 100 years ago as a basis for all definitions in math, we started using this definition:

A relation on a set $S$ is a set $\alpha$ of ordered pairs of elements of $S$.

“$\alpha$” is the Greek letter alpha.

The idea is that if $(s,t)\in\alpha$, then $s$ is related by $\alpha$ to $t$, then $(s,t)$ is an element of $\alpha$, and if $s$ is not related by $\alpha$ to $t$, then $(s,t)$ is not an element of $\alpha$. That abstracts the everyday concept of relationship by focusing on the property that a relation either holds or doesn’t hold between two given objects.

For example, the less-than relation on the set of all real numbers $\mathbb{R}$ is the set $\alpha:=\{(r,s)|r\in\mathbb{R}\text{ and }s\in\mathbb{R}\text{ and }r\lt s\}$ In other words, $r\lt s$ if and only if $(r,s)\in \alpha$.

Example

A consequence of this definition is that any set of ordered pairs is a relation. Example: Let $\xi:=\{(2,3),(2,9),(9,1),(9,2)\}$. Then $\xi$ is a relation on the set $\{1,2,3,9\}$. Your reaction may be: What relation IS it? Answer: just that set of ordered pairs. You know that $2\xi3$ and $2\xi9$, for example, but $9\xi1$ is false. There is no other definition of $\xi$.

Yes, the relation $\xi$ is weird. It is an arbitrary definition. It does not have any verbal description other than listing the element of $\xi$. It is probably useless. Live with it.

The symbol “$\xi$” is a Greek letter. It looks weird, so I used it to name a weird relation. Its upper case version is “$\Xi$”, which is even weirder. I pronounce “$\xi$” as “ksee” but most mathematicians call it “si” or “zi” (rhyming with “pie”).

Defining a relation as any old set of ordered pairs is an example of a reconstructive generalization.

$n$-ary relations

Years ago, mathematicians started coming up with things that were like relations but which involved more than two elements of a set.

Example

Let $r$, $s$ and $t$ be real numbers. We say that “$s$ is between $r$ and $t$” if $r\lt s$ and $s\lt t$. Then betweenness is a relation that is true or false about three real numbers.

Mathematicians now call this a ternary relation. The abstract definition of a ternary relation is this: A ternary relation on a set $S$ is a set of ordered triple of elements of $S$. This is an reconstructive generalization of the concept of relation that allows ordered triples of elements as well as ordered pairs of elements.

In the case of betweenness, we have to decide on the ordering. Let us say that the betweenness relation holds for the triple $(r,s,t)$ if $r\lt s$ and $s\lt t$. So $(4,5,7)$ is in the betweenness relation and $(4,7,5)$ is not.

You could argue that in the sentence, “$s$ is between $r$ and $t$”, the $s$ comes first, so that we should say that the betweenness relation (meaning $r$ is between $s$ and $t$) holds for $(r,s,t)$ if $s\lt r$ and $r\lt t$. Well, when you write an article you can write it that way. But I am writing this article.

Nowadays we talk about $n$-ary relations for any positive integer $n$. One consequence of this is that if we want to talk just about sets of ordered pairs we must call them binary relations.

When I was a child there was only one kind of guitar and it was called “a guitar”. (My older cousin Junior has a guitar, but I had only a plastic ukelele.) Some time in the fifties, electrically amplified guitars came into being, so we had to refer to the original kind as “acoustic guitars”. I was a teenager when this happened, and being a typical teenager, I was completely contemptuous of the adults who reacted with extreme irritation at the phrase “acoustic guitar”.

The axiomatic method

The axiomatic method is a technique for studying math objects of some kind by formulating them as a type of math structure. You take some basic properties of the kind of structure you are interested in and set them down as axioms, then deduce other properties (that you may or may not have already known) as theorems. The point of doing this is to make your reasoning and all your assumptions completely explicit.

Nowadays research papers typically state and prove their theorems in terms of math structures defined by axioms, although a particular paper may not mention the axioms but merely refer to other papers or texts where the axioms are given.  For some common structures such as the real numbers and sets, the axioms are not only not referenced, but the authors clearly don’t even think about them in terms of axioms: they use commonly-known properties (or real numbers or sets, for example) without reference.

The axiomatic method in practice

Typically when using the axiomatic method some of these things may happen:

• You discover that there are other examples of this system that you hadn’t previously known about.  This makes the axioms more broadly applicable.
• You discover that some properties that your original examples had don’t hold for some of the new examples.  Depending on your research goals, you may then add some of those properties to the axioms, so that the new examples are not examples any more.
• You may discover that some of your axioms follow from others, so that you can omit them from the system.

Example: Continuity

A continuous function (from the set of real numbers to the set of real numbers) is sometimes described as a function whose graph you can draw without lifting your chalk from the board.  This is a physical description, not a mathe­matical definition.

In the nineteenth century, mathe­ma­ticians talked about continuous functions but became aware that they needed a rigorous definition.  One possibility was functions given by formulas, but that didn’t work: some formulas give discontinuous functions and they couldn’t think of formulas for some continuous functions.

This description of nineteenth century math is an oversimpli­fication.

Cauchy produced the definition we now use (the epsilon-delta definition) which is a rigorous mathe­matical version of the no-lifting-chalk idea and which included the functions they thought of as continuous.

To their surprise, some clever mathe­maticians produced examples of some weird continuous functions that you can’t draw, for example the sine blur function.  In the terminology in the discussion of abstraction above, the abstraction $C’$ (epsilon-delta continuous functions) had functions in it that were not in $C$ (no-chalk-lifting functions.) On the other hand, their definition now applied to functions between some spaces besides the real numbers, for example the complex numbers, for which drawing the graph without lifting the chalk doesn’t even make sense.

Example: Rings

Suppose you are studying the algebraic properties of numbers.  You know that addition and multiplication are both associative operations and that they are related by the distributive law:  $x(y+z)=xy+xz$. Both addition and multiplication have identity elements ($0$ and $1$) and satisfy some other properties as well: addition forms a commutative group for example, and if $x$ is any number, then $0\cdot x=0$.

One way to approach this problem is to write down some of these laws as axioms on a set with two binary operations without assuming that the elements are numbers. In doing this, you are abstracting some of the properties of numbers.

Certain properties such as those in the first paragraph of this example were chosen to define a type of math structure called a ring. (The precise set of axioms for rings is given in the Wikipedia article.)

You may then prove theorems about rings strictly by logical deduction from the axioms without calling on your familiarity with numbers.

When mathematicians did this, the following events occurred:

• They discovered systems such as matrices whose elements are not numbers but which obey most of the axioms for rings.
• Although multiplication of numbers is commutative, multiplication of matrices is not commutative.
• Now they had to decide whether to require commutative of multiplication as an axioms for rings or not.  In this example, historically, mathe­maticians decided not to require multi­plication to be commutative, so (for example) the set of all $2\times 2$ matrices with real entries is a ring.
• They then defined a commutative ring to be a ring in which multi­plication is commutative.
• So the name “commutative ring” means the multiplication is commutative, because addition in rings is always commutative. Mathematical names are not always transparent.

• You can prove from the axioms that in any ring, $0 x=0$ for all $x$, so you don’t need to include it as an axiom.

Nowadays, all math structures are defined by axioms.

Other examples

• Historically, the first example of something like the axiomatic method is Euclid’s axiomatization of geometry.  The axiomatic method began to take off in the late nineteenth century and now is a standard tool in math.  For more about the axiomatic method see the Wikipedia article.
• Partitions. and equivalence
relations
are two other concepts that have been axiomatized. Remarkably, although the axioms for the two types of structures are quite different, every partition is in fact an equivalence relation in exactly one way, and any equivalence relation is a partition in exactly one way.

Remark

Many articles on the web about the axiomatic method emphasize the representation of the axiom system as a formal logical theory (formal system).
In practice, mathematicians create and use a particular axiom system as a tool for research and understanding, and state and prove theorems of the system in semi-formal narrative form rather than in formal logic.

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Rabbits out of a hat

This is a revision and expansion of the entry on rabbits in the abstractmath article Dysfunctional attitudes and behaviors.

Rabbits

Sometimes when you are reading or listening to a proof you will find yourself following each step but with no idea why these steps are going to give a proof. This can happen with the whole structure of the proof or with the sudden appearance of a step that seems like the prover pulled a rabbit out of a hat . You feel as if you are walking blindfolded.

Example (mysterious proof structure)

The lecturer says he will prove that for an integer $n$, if $n^2$ is even then $n$ is even. He begins the proof: Let $n^2$ be odd” and then continues to the conclusion, “Therefore $n$ is odd.”

Why did he begin a proof about being even with the assumption that $n$ is odd?

The answer is that in this case he is doing a proof by contrapositive . If you don’t recognize the pattern of the proof you may be totally lost. This can happen if you don’t recognize other forms, for example contradiction and induction.

Example (rabbit)

You are reading a proof that $\underset{x\to2}{\mathop{\lim }}{{x}^{2}}=4$. It is an $\varepsilon \text{-}\delta$ proof, so what must be proved is:

• For any positive real number $\varepsilon$,
• there is a positive real number $\delta$ for which:
• if $\left| x-2 \right|\lt\delta$ then
• $\left| x^2-4 \right|\lt\varepsilon$.

Proof

Here is the proof, with what I imagine might be your agitated reaction to certain steps. Below is a proof with detailed explanations .

1) Suppose $\varepsilon \gt0$ is given.

2) Let $\delta =\text{min}\,(1,\,\frac{\varepsilon }{5})$ (the minimum of the two numbers 1 and $\frac{\varepsilon}{5}$ ).

Where the *!#@! did that come from? They pulled it out of thin air! I can’t see where we are going with this proof!

3) Suppose that $\left| x-2 \right|\lt\delta$.

4) Then $\left| x-2 \right|\lt1$ by (2) and (3).

5) By (4) and algebra, $\left|x+2 \right|\lt5$.

Well, so what? We know that $\left| x+39\right|\lt42$ and lots of other things, too. Why did they do this?

6) Also $\left| x-2 \right|\lt\frac{\varepsilon }{5}$ by (2) and(3).

7) Then $\left| {{x}^{2}}-4\right|=\left| (x-2)(x+2) \right|\lt\frac{\varepsilon }{5}\cdot 5=\varepsilon$ by (5) and (6). End of Proof.

Remarks

This proof is typical of proofs in texts.

• Steps 2) and 5) look like they were rabbits pulled out of a hat.
• The author gives no explanation of where they came from.
• Even so, each step of the proof follows from previous steps, so the proof is correct.
• Whether you are surprised or not has nothing to do with whether it is correct.
• In order to understand a proof, you do not have to know where the rabbits came from.
• In general, the author did not think up the proof steps in the order they occur in the proof. (See this remark in the section on Forms of Proofs.) See also look ahead.

Proof with detailed explanations

1. Suppose $\varepsilon >0$ is given. (We are starting a proof by universal generalization.)
2. Let $\delta$ be the minimum of the two numbers $1$ and $\frac{\varepsilon}{5}$). (Rabbit out of the hat. You can “let” any symbol mean anything you want, so this is a legitimate thing to do even if you don’t see where this is all going.{
3. Suppose $\left|x-2\right|\lt\delta$. (We are about to prove the conditional statement “If $\left| x-2 \right|\lt\delta$ then $\left| {{x}^{2}}-4 \right|\lt\varepsilon$” and we are proceeding by the direct method.)
4. Then $\left| x-2 \right|\lt 1$ by (2) and (3). (The fact that $\delta =\text{min}\,(1,\,\frac{\varepsilon }{5})$ means that $\delta \le 1$ and that $\delta \le \frac{\varepsilon }{5}$. Since $\left| x-2 \right|\lt \delta$, the statement $\left| x-2 \right|\lt 1$ follows by transitivity of “$\lt$”. This is another rabbit. WHY do we want $\left| x-2 \right|\lt 1$? Be Patient.)
5. By (4) and algebra, $\left| x+2 \right|\lt 5$. ($\left| x-2 \right|\lt 1$ means that $-1\lt x-2\lt 1$. Add $4$ to each term in this equation to get $3\lt x+2\lt 5$. This is another rabbit, but it is a correct statement!)
6. Also $\left| x-2 \right|\lt \frac{\varepsilon }{5}$ by (2) and (3). ((2) says that $\delta\le\frac{\varepsilon }{5}$ and (3) says that $\left| x-2 \right|\lt\delta$, so $\left| x-2 \right|\lt \frac{\varepsilon }{5}$ follows by transitivity.)
7. Then $\left| {{x}^{2}}-4\right|=\left| (x-2)(x+2) \right|\lt\frac{\varepsilon }{5}\cdot 5=\varepsilon$ by (5) and (6). End of Proof. (This last statement actually shows the algebra.)

Coming up with that proof

The author did not think up the proof steps in the order they occur in the proof. She looked ahead at the goal of proving that $\left| {{x}^{2}}-4\right|\lt\varepsilon$ and thought of factoring the left side. Now she must prove that $\left| (x-2)(x+2) \right|\lt\varepsilon$

But if $\left|x-2\right|$ is small then $x$ has to be close to $2$, so that $x + 2$ can’t be too big. Since the only restriction on $\delta$ is that it has to be positive, let’s restrict it to being smaller than $1$. (The choice of $1$ is purely arbitrary. Any positive real number would do.)

In that case step (5) shows that $\left|x+2\right|\lt5$.. So how small do you have to make to make $\varepsilon$? In other words, how small do you have to make $\delta$ to make $\left| 5(x-2) \right|\lt\varepsilon$ (remembering that $\left| x-2 \right|\lt\delta$). Well, clearly $\frac{\varepsilon }{5}$ will do!

That explains her choice of $\delta$ be the minimum of the two numbers $1$ and $\frac{\varepsilon}{5}$. Notice that that choice is made very early in the proof but it was made only after experimenting with the sizes of $\left|x-2\right|$ and $\left|x+2\right|$.

You can check that if she had chosen to restrict $\delta$ to being less than 42, then she would need $\delta =\text{min}\,(42,\,\frac{\varepsilon }{47})$.

Acknowledgments

Thanks to Robert Burns for corrections and suggestions

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Forms of proofs

Abstractmath.org is a website I have been maintaining since 2005. It is intended for people beginning the study of abstract math, often a course that requires proofs and thinking about mathematical structures. The Introduction to the website and the article Attitude explain the website in more detail.

One of the chapters in abstractmath.org covers Proofs. As everywhere in abstractmath.org, there is no attempt at complete coverage: the emphasis is on aspects that cause difficulty for abstraction-newbies. In the case of proofs, this includes sections on how proofs are written (math language is a big emphasis all over abstractmath.org). One of those sections is Forms of Proof. This post is a fairly extensive revision of that section.

More than half of the section on Proofs has already been revised (the ones entitled “abstractmath.org 2.0)”, and my current task is to finish that revision.

Normally, I post the actual article here on Gyre&Gimble, but something has changed in the operation of WordPress which causes the html processor to obey linebreaks in the input, which would make the article look chaotic.

So this time, I have to ask you to click a button to read the revised section on Forms of Proof. I apologize for the excessive effort by your finger.