Please read this post at abstractmath.org. I originally posted the document here but some of the diagrams would not render, and I haven’t been able to figure out why. Sorry for having to redirect.
Send to KindleMathematics is communicated using two languages: Mathematical English and the symbolic language of math (more about them in two languages).
This post is a collection of examples of the sorts of trouble that the two languages cause beginning abstract math students. I have gathered many of them here since they are scattered throughout the literature. I would welcome suggestions for other references to problems caused by the languages of math.
In many of the examples, I give links to the literature and leave you to fish out the details there. Almost all of the links are to documents on the internet.
There is an extensive list of references.
Scattered through this post are conjectures. Like most of my writing about difficulties students have with math language, these conjectures are based on personal observation over 37 years of teaching mostly computer engineering and math majors. The only hard research of any sort I have done in math ed consists of the 426 citations of written mathematical writing included in the Handbook of Mathematical Discourse.
This post is an attempt to gather together the ways in which math language causes trouble for students. It is even more preliminary and rough than most of my other posts.
I expect to revise this article periodically as I find more references and examples and understand some of the topics better. Suggestions would be very welcome.
I have occasionally had students tell me that have great difficulty understanding a complicated symbolic expression. They can’t just look at it and learn something about what it means.
Consider the symbolic expression \[\displaystyle\left(\frac{x^3-10}{3 e^{-x}+1}\right)^6\]
Now, I could read this expression aloud as if it were text, or more precisely describe it so that someone else could write it down. But if I am in math mode and see this expression I don’t “read” it, even to myself.
I am one of those people who much of the time think in pictures or abstractions without words. (See references here.)
In this case I would look at the expression as a structured picture. I could determine a number of things about it, and when I was explaining it I would point at the board, not try to pronounce it or part of it:
There is much more about this example in Zooming and Chunking.
There are many high school students stymied by algebra, never do well at it, and hate math as a result. I have known many such people over the years. A revealing remark that I have heard many times is that “algebra is totally meaningless to me”. This is sometimes accompanied by a remark that geometry is “obvious” or something similar. This may be because they think they have to “read” an algebraic expression instead of studying it as they would a graph or a diagram.
Many beginning abstractmath students have difficulty understanding a symbolic expression like the one above. Could this be cause by resistance to treating the expression as a structure to be studied?
A symbolic assertion (“formula” to logicians) can be embedded in a math English sentence in different ways, requiring the symbolic assertion to be pronounced in different ways. The assertion itself is not modified in any way in these different situations.
I used the phrase “symbolic assertion” in abstractmath.org because students are confused by the logicians’ use of “formula“.
In everyday English, “$\text{H}_2\text{O}$” is the “formula” for water, but it is a term, not an assertion.
“For every real number $x\gt0$ there is a real number $y$ such that $x\gt y\gt0$.”
The consequence is that the symbolic assertion, in this case “$x\gt0$”, does not reveal that role it plays in the math English sentence that it is embedded in.
Many of the examples occurring later in the post are also examples of context-sensitive pronunciation.
Many students are subconsciously bothered by the way the same symbolic expression is pronounced differently in different math English sentences.
This probably impedes some students’ progress. Teachers should point this phenomenon out with examples.
Students should be discouraged from pronouncing mathematical expressions.
For one thing, this could get you into trouble. Consider pronouncing “$\sqrt{3+5}+6$”. In any case, when you are reading any text you don’t pronounce the words, you just take in their meaning. Why not take in the meaning of algebraic expressions in the same way?
A parenthetic assertion is a symbolic assertion embedded in a sentence in math English in such a way that is a subordinate clause.
In the math English sentence
“For every real number $x\gt0$ there is a real number $y$ such that $x\gt y\gt0$”
mentioned above, the symbolic assertion “$x\gt0$” plays the role of a subordinate clause.
It is not merely that the pronunciation is different compared to that of the independent statement “$x\gt0$”. The math English sentence is hard to parse. The obvious (to an experienced mathematician) meaning is that the beginning of the sentence can be read this way: “For every real number $x$, which is bigger than $0$…”.
But new student might try to read it is “For every real number $x$ is greater than $0$ …” by literally substituting the standalone meaning of “$x\gt0$” where it occurs in the sentence. This makes the text what linguists call a garden path sentence. The student has to stop and start over to try to make sense of it, and the symbolic expression lacks the natural language hints that help understand how it should be read.
Note that the other two symbolic expressions in the sentence are not parenthetic assertions. The phrase “real number” needs to be followed by a term, and it is, and the phrase “such that” must be followed by a clause, and it is.
David Butler’s post Contrapositive grammar has other good examples.
Math texts are in general badly written. Students need to be taught how to read badly written math as well as how to write math clearly. Those that succeed (in my observation) in being able to read math texts often solve the problem by glancing at what is written and then reconstructing what the author is supposedly saying.
Some students are baffled, or at least bothered consciously or unconsciously, by parenthetic assertions, because the clues that would exist in a purely English statement are missing.
Nevertheless, many if not most math students read parenthetic assertions correctly the first time and never even notice how peculiar they are.
What makes the difference between them and the students who are stymied by parenthetic assertions?
There is another conjecture concerning parenthetic assertions below.
The word “if” in definitions does not mean the same thing that it means in other math statements.
Context-sensitive meaning occurs in ordinary English as well. Think of a strike in baseball.
The nearly universal custom of using “if” to mean “if and only if” in definitions makes it a harder for students to understand implication.
This custom is not the major problem in understanding the role of definitions. See my article Definitions.
In a course in group theory, a lecturer may say at one point, “Let $F:G\to H$ be a homomorphism”, and at another point, “Let $g\in G$”.
In the first sentence, $G$ refers to the group, and in the second sentence it refers to the underlying set of the group.
This usage is almost universal. I think the difficulty it causes is subtle. When you refer to $\mathbb{R}$, for example, you (usually) are referring to the set of real numbers together with all its canonical structure. The way students think of it, a real number comes with its many relations and connections with the other real numbers, ordering, field properties, topology, and so on.
But in a group theory class, you may define the Klein $4$-group to be $\mathbb{Z}_2\times\mathbb{Z}_2$. Later you may say “the symmetry group of a rectangle that is not a square is the Klein $4$-group.” Almost invariably some student will balk at this.
Referring to a group by naming its underlying set is also an example of synecdoche.
Students expect every important set in math to have a canonical structure. When they get into a course that is a bit more abstract, suddenly the same set can have different structures, and math objects with different underlying sets can have the same structure. This catastrophic shift in a way of thinking should be described explicitly with examples.
Way back when, it got mighty upsetting when the earth started going around the sun instead of vice versa. Remind your students that these upheavals happen in the math world too.
A particular text may refer to the identity element of any group as $e$.
This is as far as I know not a problem for students. I think I know why: There is a generic identity element. The identity element in any group is an instantiation of that generic identity element. The generic identity element exists in the sketch for groups; every group is a functor defined on that sketch. (Or if you insist, the generic identity element exists in the first order theory for groups.) I suspect mathematicians subconsciously think of identity elements in this way.
Matrix multiplication is not commutative. A student may forget this and write $(A^2B^2=(AB)^2$. This also happens in group theory courses.
This problem occurs because the symbolic language uses the same symbol for many different operations, in this case the juxtaposition notation for multiplication. This phenomenon is called overloaded notation and is discussed in abstractmath.org here.
Noncommutative binary operations written using juxtaposition cause students trouble because going to noncommutative operations requires abandoning some overlearned reflexes in doing algebra.
Identity elements seem to behave the same in any binary operation, so there are no reflexes to unlearn. There are generic binary operations of various types as well. That’s why mathematicians are comfortable overloading juxtaposition. But to get to be a mathematician you have to unlearn some reflexes.
Sometimes you need to reword a math statement that contains symbolic expressions. This particularly causes trouble in connection with negation.
The English language is notorious among language learners for making it complicated to negate a sentence. The negation of “I saw that movie” is “I did not see that movie”. (You have to put “d** not” (using the appropriate form of “do”) before the verb and then modify the verb appropriately.) You can’t just say “I not saw that movie” (as in Spanish) or “I saw not that movie” (as in German).
The method in English used to negate a sentence may cause problems with math students whose native language is not English. (But does it cause math problems with those students?)
These examples are difficulties caused by not understanding the math. They are not directly caused by difficulties with the languages of math.
Suppose you want to prove:
“If $f:\mathbb{R}\to\mathbb{R}$ is differentiable, then $f$ is continuous”.
A good way to do this is by using the contrapositive. A mechanical way of writing the contrapositive is:
“If $f$ is not continuous, then $f:\mathbb{R}\to\mathbb{R}$ is not differentiable.”
That is not good. The sentence needs to be massaged:
“If $f:\mathbb{R}\to\mathbb{R}$ is not continuous, then $f$ is not differentiable.”
Even better would be to write the original sentence as:
“Suppose $f:\mathbb{R}\to\mathbb{R}$. Then if $f$ is differentiable, then $f$ is continuous.”
This is discussed in detail in David Butler’s post Contrapositive grammar.
Students need to be taught to understand parenthetic assertions that occur in the symbolic language and to learn to extract a parenthetic assertion and write it as a standalone assertion ahead of the statement it occurs in.
The scope of a word or variable consists of the part of the text for which its current definition is in effect.
There are many more examples in the abstractmath.org article Scope.
If you are a grasshopper (you like to dive into the middle of a book or paper to find out what it says), knowing the scope of a variable can be hard to determine. It is particularly difficult for commonly used words or symbols that have been defined differently from the usual usage. You may not suspect that this has happened since it might be define once early in the text. Some books on writing mathematics have urged writers to keep global definitions to a minimum. This is good advice.
Finding the scope is considerably easier when the text is online and you can search for the definition.
Knowing the scope of a word or variable can be difficult. It is particular hard when the word or variable has a large scope (chapter or whole book.)
Variables are often introduced in math writing and then used in the subsequent discussion. In a complicated discussion, several variables may be referred to that have different statuses, some of them introduced several pages before. There are many particular ways discussed below that can cause trouble for students. This post is restricted to trouble in connection with the languages of math. The concept of variable is difficult in itself, not just because of the way the math languages represent them, but that is not covered here.
Much of this part of the post is based on work of Susanna Epp, including three papers listed in the references. Her papers also include many references to other work in the math ed literature that have to do with understanding variables.
See also Variables in abstractmath.org and Variables in Wikipedia.
Students blunder by forgetting the type of the variable they are dealing with. The example given previously of problems with matrix multiplication is occasioned by forgetting the type of a variable.
Students sometimes have problems because they forget the data type of the variables they are dealing with. This is primarily causes by overloaded notation.
If you define $y=x^2+1$, then $x$ is an independent variable and $y$ is a dependent variable. But dependence and independence of variablesare more general than that example suggests.
In an epsilon-delta proof of the limit of a function (example below,) $\varepsilon$ is independent and $\delta$ is dependent on $\varepsilon$, although not functionally dependent.
Distinguishing dependent and independent variables causes problems, particularly when the dependence is not clearly functional.
I recently ran across a discussion of this on the internet but failed to record where I saw it. Help!
This causes trouble with integration, among other things. It is discussed in abstractmath.org in Variables and Substitution. I expect to add some references to the math ed literature soon.
Some of these variables may be given by existential instantiation, in which case they are dependent on variables that define them. Others may be given by universal instantiation, in which case the variable is generic; it is independent of other variables, and you can’t impose arbitrary restrictions on it.
A theorem that an object exists under certain conditions allows you to name it and use it by that name in further arguments.
Suppose $m$ and $n$ are integers. Then by definition, $m$ divides $n$ if there is an integer $q$ such that $n=qm$. Then you can use “$q$” in further discussion, but $q$ depends on $m$ and $n$. You must not use it with any other meaning unless you start a new paragraph and redefine it.
So the following (start of a) “proof” blunders by ignoring this restriction:
Theorem: Prove that if an integer $m$ divides both integers $n$ and $p$, then $m$ divides $n+p$.
“Proof”: Let $n = qm$ and $p = qm$…”
It is a theorem that for any integer $n$, there is no integer strictly between $n$ and $n+1$. So if you are given an arbitrary integer $k$, there is no integer strictly between $k$ and $k+1$. There is no integer between $42$ and $43$.
By itself, universal instantiation does not seem to cause problems, provided you pay attention to the types of your variables. (“There is no integer between $\pi$ and $\pi+1$” is false.)
However, when you introduce variables using both universal and existential quantification, students can get confused.
Consider the definition of limit:
Definition: $\lim_{x\to a} f(x)=L$ if and only if for every $\epsilon\gt0$ there is a $\delta\gt0$ for which if $|x-a|\lt\delta$ then $|f(x)-L|\lt\epsilon$.
A proof for a particular instance of this definition is given in detail in Rabbits out of a Hat. In this proof, you may not put constraints on $\epsilon$ except the given one that it is positive. On the other hand, you have to come up with a definition of $\delta$ and prove that it works. The $\delta$ depends on what $f$, $a$ and $L$ are, but there are always infinitely many values of $\delta$ which fit the constraints, and you have to come up with only one. So in general, two people doing this proof will not get the same answer.
Susanna Epp’s paper Proof issues with existential quantification discusses the problems that students have with both existential and universal quantification with excellent examples. In particular, that paper gives examples of problems students have that are not hinted at here.
A nearly final version of The Handbook of Mathematical Discourse is available on the web with links, including all the citations. This version contains some broken links. I am unable to recompile it because TeX has evolved enough since 2003 that the source no longer compiles. The paperback version (without the citations) can be bought as a book here. (There are usually cheaper used versions on Amazon.)
Abstractmath.org is a website for beginning students in abstract mathematics. It includes most of the material in the Handbook, but not the citations. The Introduction gives you a clue as to what it is about.
My take on the two languages of math are discussed in these articles:
The Language of Mathematics, by Mohan Ganesalingam, covers these two languages in more detail than any other book I know of. He says right away on page 18 that mathematical language consists of “textual sentences with symbolic material embedded like ‘islands’ in the text.” So for him, math language is one language.
I have envisioned two separate languages for math in abstractmath.org and in the Handbook, because in fact you can in principle translate any mathematical text into either English or logical notation (first order logic or type theory), although the result in either case would be impossible to understand for any sizeable text.
Context-sensitive interpretation.
These topics have a strong overlap with the topics with the same name in abstractmath.org. They are included here because the Handbook contains links to citations of the usage.
Syntactic and semantic thinkers
Technical meanings clash with everyday meanings
Three kinds of mathematical thinkers
Variations in meaning in math.
Contrapositive grammar, blog post by David Butler.
Proof issues with existential quantification, by Susanna Epp.
The role of logic in teaching proof, by Susanna Epp (2003).
The language of quantification in mathematics instruction, by Susanna Epp (1999).
The Language of Mathematics: A Linguistic and Philosophical Investigation
by Mohan Ganesalingam, 2013. (Not available from the internet.)
On the communication of mathematical reasoning, by Atish Bagchi, and Charles Wells (1998a), PRIMUS, volume 8, pages 15–27.
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Send to KindleThis is a revision of the section of abstractmath.org on notation for sets.
The following notation for sets of numbers is fairly standard.
Until the 1930’s, Germany was the world center for scientific and mathematical study, and at least until the 1960’s, being able to read scientific German was was required of anyone who wanted a degree in science. A few years ago I was asked to transcribe some hymns from a German hymnbook — not into English, but merely from fraktur (the old German alphabet) into the Roman alphabet. I sometimes feel that I am the last living American to be able to read fraktur easily.
The expression “$x\in A$” means that $x$ is an element of the set $A$. The expression “$x\notin A$” means that $x$ is not an element of $A$.
“$x\in A$” is pronounced in any of the following ways:
A set with a small number of elements may be denoted by listing the elements inside braces (curly brackets). The list must include exactly all of the elements of the set and nothing else.
The set $\{1,\,3,\,\pi \}$ contains the numbers $1$, $3$ and $\pi $ as elements, and no others. So $3\in \{1,3,\pi \}$ but $-3\notin \{1,\,3,\,\pi \}$.
If $a$ occurs in a list notation, then $a$ is in the set the notation defines. If it does not occur, then it is not in the set.
When I say “$a$ occurs” I don’t mean it necessarily occurs using that name. For example, $3\in\{3+5,2+3,1+2\}$.
For example, $\{2,5,6\}$ and $\{5,2,6\}$ are the same set.
$\{2,5,6\}$, $\{5,2,6\}$, $\{2,2,5,6 \}$ and $\{2,5,5,5,6,6\}$ are all different representations of the same set. That set has exactly three elements, no matter how many numbers you see in the list notation.
Multisets may be written with braces and repeated entries, but then the repetitions mean something.
When (some of) the elements in list notation are themselves sets (more about that here), care is required. For example, the numbers $1$ and $2$ are not elements of the set \[S:=\left\{ \left\{ 1,\,2,\,3 \right\},\,\,\left\{ 3,\,4 \right\},\,3,\,4 \right\}\]The elements listed include the set $\{1, 2, 3\}$ among others, but not the number $2$. The set $S$ contains four elements, two sets and two numbers.
Another way of saying this is that the element relation is not transitive: The facts that $A\in B$ and $B\in C$ do not imply that $A\in C$.
Suppose $P$ is an assertion. Then the expression “$\left\{x|P(x) \right\}$” denotes the set of all objects $x$ for which $P(x)$ is true. It contains no other elements.
In these examples, $n$ is an integer variable and $x$ is a real variable..
A set can be expressed in many different ways in setbuilder notation. For example, $\left\{ x|x\gt6 \right\}=\left\{ x|x\ge 6\text{ and }x\ne 6 \right\}$. Those two expressions denote exactly the same set. (But $\left\{x|x^2\gt36 \right\}$ is a different set.)
In certain areas of math research, setbuilder notation can go seriously wrong. See Russell’s Paradox if you are curious.
An expression may be used left of the vertical line in setbuilder notation, instead of a single variable.
You can use an expression on the left side of setbuilder notation to indicate the type of the variable.
The unit interval $I$ could be defined as \[\mathbb{I}=\left\{x\in \mathrm{R}\,|\,0\le x\le 1 \right\}\]making it clear that it is a set of real numbers rather than, say rational numbers. You can always get rid of the type expression to the left of the vertical line by complicating the defining condition, like this:\[\mathbb{I}=\left\{ x|x\in \mathrm{R}\text{ and }0\le x\le 1 \right\}\]
Other kinds of expressions occur before the vertical line in setbuilder notation as well.
The set\[\left\{ {{n}^{2}}\,|\,n\in \mathbb{Z} \right\}\]consists of all the squares of integers; in other words its elements are 0,1,4,9,16,…. This definition could be rewritten as $\left\{m|\text{ there is an }n\in \mathrm{}\text{ such that }m={{n}^{2}} \right\}$.
Let $A=\left\{1,3,6 \right\}$. Then $\left\{ n-2\,|\,n\in A\right\}=\left\{ -1,1,4 \right\}$.
Be careful when you read such expressions.
The integer $9$ is an element of the set \[\left\{{{n}^{2}}\,|\,n\in \text{ Z and }n\ne 3 \right\}\]It is true that $9={{3}^{2}}$ and that $3$ is excluded by the defining condition, but it is also true that $9={{(-3)}^{2}}$ and $-3$ is not an integer ruled out by the defining condition.
Sets. Previous post.
Toby Bartels for corrections.
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Send to KindleAbstractmath.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.
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CONTENTS The truth table for conditionals |
This section is concerned with logical constructions made with the connective called the conditional operator. In mathematical English, applying the conditional operator to $P$ and $Q$ produces a sentence that may bewritten, “If $P$, then $Q$”, or “$P$ implies$Q$”. Sentences of this form are conditional assertions.
Conditional assertions are at the very heart of mathematical reasoning. Mathematical proofs typically consist of chains of conditional assertions.
Some of the narrative formats used for proving conditional assertions are discussed in Forms of Proof.
A conditional assertion “If $P$ then $Q$” has the precise truth table shown here.
| $P$ | $Q$ | If $P$ then $Q$ | |
| T | T | T | |
| T | F | F | |
| F | T | T | |
| F | F | T |
The meaning of “If $P$ then $Q$” is determined entirely by the truth values of $P$ and $Q$ and this truth table. The meaning is not determined by the usual English meanings of the words “if” and “then”.
The truth table is summed up by this purple pronouncement:
The Prime Directive of conditional assertions:
A conditional assertion is true unless
the hypothesis is true and the conclusion is false.
That means that to prove “If $P$ then $Q$” is FALSE
you must show that $P$ is TRUE(!) and $Q$ is FALSE.
The Prime Directive is harder to believe in than leprechauns. Some who are new to abstract math get into an enormous amount of difficulty because they don’t take it seriously.
The statement “if $n\gt 5$, then $n\gt 3$” is true for all integers
$n$.
For more about this, see Understanding conditionals.
Most of the time in mathematical writing the conditional assertions which are actually stated involve assertions containing variables, and the claim is typically that the assertion is true for all instances of the variables. Assertions involving statements without variables occur only implicitly in the process of checking instances of the assertions. That is why a statement such as, “If $2\gt 5$ then $2\gt 3$” seems awkward and unfamiliar.
It is unfamiliar and occurs rarely. I mention it here because of the occurrence of vacuous truths, which do occur in mathematical writing.
The set $\{x|P(x)\}$ is the set of exactly all $x$ for which $P(x)$ is true. It is called the truth set of $P(x)$.
“If $P(x)$ then $Q(x)$” means that $\{x|P(x)\}\subseteq
\{x|Q(x)\}$. We say $P(x)$ is stronger than $Q(x)$, meaning that $P$ puts more requirements on $x$ than $Q$ does. The objects $x$ that make $P$ true necessarily make $Q$ true, so there might be objects making $Q$ true that don’t make $P$ true.
The statement “$x\gt4$” is stronger than the statement “$x\gt\pi$”. That means that $\{x|x\gt4\}$ is a proper subset of $\{x|x\gt\pi\}$. In other words, $\{x|x\gt4\}$ is “smaller” than $\{x|x\gt\pi\}$ in the sense of subsets. For example, $3.5\in\{x|x\gt\pi\}$ but $3.5\notin\{x|x\gt4\}$. This is a kind of reversal (a Galois correspondence) that confused many of my students.
“Smaller” means the truth set of the stronger statement omits elements that are in the truth set of the weaker statement. In the case of finite truth sets, “smaller” also means it has fewer elements, but that does not necessarily work for infinite sets, such as in the example above, because the two truth sets $\{x|x\gt4\}$ and $\{x|x\gt\pi\}$ have the same cardinality.
Making a statement stronger
makes its truth set “smaller”.
In the assertion “If $P$, then $Q$”:
Conditionals such as “If $P$ then $Q$” are also called implications , but be wary: “implication” is a technical term and does not fit the meaning of the word in conversational English.
The last two lines of the truth table for conditional assertions mean that if the hypothesis of the assertion is false, then the assertion is automatically true.
In the case that “If $P$ then $Q$” is true because $P$ is false, the assertion is said to be vacuously true.
The word “vacuous” refers to the fact that in the vacuous case the conditional assertion says nothing interesting about either $P$ or $Q$. In particular, the conditional assertion may be true even if the conclusion is false (because of the last line of the truth table).
Both these statements are vacuously true!
If $A$ is any set then $\emptyset\subseteq A.$ Proof (rewrite by definition): You have to prove that if $x\in\emptyset$, then $x\in A$. But the statement “$x\in\emptyset$” is false no matter what $x$ is, so the statement “$\emptyset\subseteq A$” is vacuously true.
Vacuous truth can cause surprises in connection with certain concepts which are defined using a conditional assertion.
Although vacuous truth may be disturbing when you first see it, making either statement in the example false would result in even more peculiar situations. For example, if you decided that “If $P$ then $Q$” must be false when $P$ and $Q$ are both false, you would then have to say that this statement
is not always true (substitute $3$ for $m$ and $4$ for $n$ and you get both $P$ and $Q$ false). This would surely be an unsatisfactory state of affairs.
A conditional assertion may be worded in various ways. It takes some practice to get used to understanding all of them as conditional.
Our habit of swiping English words and phrases and changing their meaning in an unintuitive way causes many problems for new students, but I am sure that the worst problem of that kind is caused by the way conditional assertions are worded.
The most common ways of wording a conditional assertion with hypothesis $P$ and conclusion $Q$ are:
Math logic is notorious for the many different symbols used by different authors with the same meaning. This is in part because it developed separately in three different academic areas: Math, Philosophy and Computing Science.
All the statements below mean the same thing. In these statements $n$ is an integer variable.
Since “$P(x)\supset Q(x)$” means that $\{x|P(x)\}\subseteq
\{x|Q(x)\}$, there is a notational clash between implication written “$\supset $” and inclusion written “$\subseteq $”. This is exacerbated by the two meanings of the inclusion symbol “$\subset$”.
These ways of wording conditionals cause problems for students, some of them severe. They are discussed in the section Understanding conditionals.
The logical symbols “$\to$”, “$\Rightarrow$”,
“$\supset$” are frequently used when writing on the blackboard, but are not common in texts, except for texts in mathematical logic.
If you know some logic, you may know that there is a subtle difference between the statements
Here is a concrete example:
Note that the subject of sentence (1) is the (variable) number $x$, but the subject of sentence (2) is the assertion
“$x\lt2$”. Behind this is a distinction made in formal logic between the material conditional “if $P$ then $Q$” (which means that $P$ and $Q$ obey the truth table for “If..then”) and logical consequence ($Q$ can be proved given $P$). I will ignore the distinction here, as most mathematicians do except when they are proving things about logic.
In some texts, $P\Rightarrow Q$ denotes the material conditional and $P\to Q$ denotes logical consequence.
A conditional assertion containing a variable that is true for any value of the correct type of that variable is a universally true conditional assertion. It is a special case of the general notion of universally true assertion.
These are all assertions of the form “If $P(x)$, then $Q(x)$”. In (1), the hypothesis is the assertion “$x\lt5$”; in (2), it is the assertion “$n^2$ is even”, using an adjective to describe property that $n^2$ is even; in (3), it is the assertion “$x$ is an integer”, using a noun to assert that $x$ has the property of being an integer. (See integral.)
The sentences listed in the example above provide ways of expressing universally true conditionals in English. They use “for all” or “for any”, You may also use these forms (compare in this discussion of universal assertions in general.)
Sometimes mathematicians write, “If the function $f$ is differentiable, then it is continuous.” At least sometimes, they mean that every function that is differentiable is continuous. I suspect that this usage occurs in texts written by non-native-English speakers.
There are other ways of expressing universal conditionals that are disguised, because they are not conditional assertions in English.
Let $C(f)$ mean that $f$ is continuous and and $D(f)$ mean that $f$ is differentiable. The (true) assertion
can be said in the following ways:
The converse of a conditional assertion “If $P$ then $Q$” is “If $Q$ then $P$”.
Whether a conditional assertion is true
has no bearing on whether its converse it true.
The contrapositive of a conditional assertion “If $P$ then $Q$” is “If not $Q$ then not $P$.”
A conditional assertion and its contrapositive
are both true or both false.
The contrapositive of
“If $x > 3$, then $x > 2$”
is (after a little translation)
“If $x\leq2$ then $x\leq3$.”
For any number $x$, these two statements are both true or both false.
This means that if you prove “If not $P$ then not $Q$”, then you have also proved “If $P$ then $Q$.”
You can prove an assertion by proving its contrapositive.
This is called the contrapositive method and is discussed in detail in this section.
So a conditional assertion and its contrapositive have the same truth value. Two assertions that have the same truth value are said to be equivalent. Equivalence is discussed with examples in the Wikipedia article on necessary and sufficient.
As you can see from the preceding discussions, statements of the form “If $P$ then Q” don’t mean the same thing in math that they do in ordinary English. This causes semantic contamination.
In ordinary English, “If $P$ then $Q$” can suggest order of occurrence. For example, “If we go outside, then the neighbors will see us” implies that the neighbors will see us after we go outside.
Consider “If $n\gt7$, then $n\gt5$.” If $n\gt7$, that doesn’t mean $n$ suddenly gets greater than $7$ earlier than $n$ gets greater than $5$. On the other hand, “$n\gt5$ is necessary for $n\gt7$” (which remember means the same thing as “If $n\gt7$, then $n\gt5$) doesn’t mean that $n\gt5$ happens earlier than $n\gt7$. Since we are used to “if…then” having a timing implication, I suspect we get subconscious dissonance between “If $P$ then $Q$” and “$Q$ is necessary for $P$” in mathematical statements, and this dissonance makes it difficult to believe that that can mean the same thing.
“If $P$ then $Q$” can also suggest causation. The the sentence, “If we go outside, the neighbors will see us” has the connotation that the neighbors will see us because we went outside.
The contrapositive is “If the neighbors won’t see us, then we don’t go outside.” This English sentence seems to me to mean that if the neighbors are not around to see us, then that causes us to stay inside. In contrast to contrapositive in math, this means something quite different from the original sentence.
For some instances of the use of “if…then” in English, the truth table is different.
Consider: “If you eat your vegetables, you can have dessert.” Every child knows that this means they will get dessert if they eat their vegetables and not otherwise. So the truth table is:
| $P$ | $Q$ | If $P$ then $Q$ | |
| T | T | T | |
| T | F | F | |
| F | T | F | |
| F | F | T |
In other words, $P$ is equivalent to $Q$. It appears to me that this truth table corresponds to English “if…then” when a rule is being asserted.
These examples show:
The different ways of expressing conditional assertions
may mean different things in English.
How can you get to the stage where you automatically understand the meaning of conditional assertions in math English?
You need to understand the equivalence of these formulations so well that it is part of your unconscious reaction to conditionals.
How can you gain that intuitive understanding? One way is by doing abstract math regularly for several years! (Of course, this is how you gain expertise in anything.) In other words, Practice, Practice!
But it may help to remember that when doing proofs, we must take the rigorous view of mathematical objects:
The integers (like all math objects) just sit there, not doing anything and not affecting anything. $10$ is not greater than $4$ “because” it is greater than $7$. There is no “because” in rigorous math. Both facts, $10\gt4$ and $10\gt7$, are eternally true.
Eternal is how we think of them – I am not making a claim about “reality”.
These three observations describe the same set of facts about a bunch of things (integers) that just sit there in their various relationships without changing, moving or doing anything. If you keep these remarks in mind, you will eventually have a natural, unforced understanding of conditionals in math.
None of this means you have to think of mathematical objects as dead and fossilized all the time. Feel free to think of them using all the metaphors and imagery you know, except when you are reading or formulating a proof written in mathematical English. Then you have to be rigorous!
The truth table for conditional assertions may be summed up by saying: The conditional assertion “If $P$, then $Q$” is true unless $P$ is true and $Q$ is false.
This fits with the major use of conditional assertions in reasoning:
In symbols:
(1) When “If $P$ then $Q$” and $P$ are both true,
(2) then $Q$ must be true as well.
Modus Ponens is the most used method of deduction of all.
Modus ponens is not a method of proving conditional assertions. It is a method of using a conditional assertion in the proof of another assertion. Methods for proving conditional assertions are found in the chapter Forms of proof.
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Send to KindleThis post outlines the way that proofs are used in mathematical writing. I have been revising the chapter on Proofs in abstractmath.org, and I felt that giving an overview would keep my mind organized when I was enmeshed in writing up complicated details.
Added 2014-12-19
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Send to KindleIn some situations you may have conflicting information from different sources about a subject. The resulting confusion in your thinking is called cognitive dissonance.
It may happen that a person suffering cognitive dissonance suppresses one of the ways of understanding in order to resolve the conflict. For example, at a certain stage in learning English, you (small child or non-native-English speaker) may learn a rule that the past tense is made from the present form by adding “-ed”. So you say “bringed” instead of “brought” even though you may have heard people use “brought” many times. You have suppressed the evidence in favor of the rule.
Some of the ways cognitive dissonance can affect learning math are discussed here
We think about math objects using metaphors, as we do with most concepts that are not totally concrete. The metaphors are imperfect, suggesting facts about the objects that may not follow from the definition. This is discussed at length in the section on images and metaphors here.
Mathematicians think of the real numbers as constituting a line infinitely long in both directions, with each number as a point on the line. But this does not mean that you can think of the line as a row of points. Between any two points there are uncountably many other points. See density of the reals.
One of the most intransigent examples of metaphorical contamination occurs when students think about countably infinite sets. Their metaphor is that a sequence such as the set of natural numbers $\{0,1,2,3,4,\ldots\}$ “goes on forever but never ends”. The metaphor mathematicians have in mind is quite different: The natural numbers constitute the set that contains every natural number right now.
An excruciating example of this is the true statement
“$.999\ldots=1.0$.” The notion that it can’t be true comes from thinking of “$0.999\ldots$” as consisting of the list of numbers \[0.9,0.99,0.999,0.9999,0.99999,\ldots\] which the student may say “gets closer and closer to $1.0$ but never gets there”.
Now consider the way a mathematician thinks: The numbers are all already there, and they make a set.
The proof that $.999\ldots=1.0$ has several steps. In the list below, I have inserted some remarks in red that indicate areas of abstract math that beginning students have trouble with.
The problem that occurs with the word “definition” in this case is that a definition appears to be a dictatorial act. The student needs to know why you made this definition. This is not a stupid request. The act can be justified by the way the definition gets along with the algebraic and topological characteristic of the real numbers.
Each one of these steps should be made explicit. Even the Wikipedia article, which is regarded as a well written document, doesn’t make all of the points explicit.
Many math objects have names that are ordinary English words.
(See names.) So the person learning about them is faced with two inputs:
It is easy and natural to suppress the information given by the definition (or part of it) and rely only on the English meaning. But math does not work that way:
If another source of understanding contradicts the definition
THE DEFINITION WINS.
The connotations of a name may fit the concept in some ways and not others. Infinite cardinal numbers are a notorious example of this: there are some ways in which they are like numbers and other in which they are not.
For a finite set, the cardinality of the set is the number of elements in the set. Long ago, mathematicians started talking about the cardinality of an infinite set. They worked out a lot of facts about that, for example:
The teacher may even say that there are just as many points on the real line as in the real points. And know-it-all math majors will say that to their friends.
Many students will find that totally bizarre. Essentially, what has happened is that the math dictators have taken the phrase “cardinality” to mean what it usually means for finite sets and extend it to infinite sets by using a perfectly consistent (and useful) definition of “cardinality” which has very different properties from the finite case.
That causes a perfect storm of cognitive dissonance.
Math majors must learn to get used to situations like this; they occur in all branches of math. But it is bad behavior to use the phrase “the same number of elements” to non-mathematicians. Indeed, I don’t think you should use the word cardinality in that setting either: you should refer to a “one-to-one correspondence” instead and admit up front that the existence of such a correspondence is quite amazing.
Let’s look at the word “series”in more detail. In ordinary English, a series is a bunch of things, one after the other.
In mathematics an infinite series is an object expressed like this:
\[\sum\limits_{k=1}^{\infty
}{{{a}_{k}}}\]
where the ${{a}_{k}}$ are numbers. It has partial sums
\[\sum\limits_{k=1}^{n}{{{a}_{k}}}\]
For example, if ${{a}_{k}}$ is defined to be $1/{{k}^{2}}$ for positive integers $k$, then
\[\sum\limits_{k=1}^{6}{{{a}_{k}}}=1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}=\frac{\text{5369}}{\text{3600}}=\text{
about }1.49\]
This infinite series converges to $\zeta (2)$, which is about $1.65$. (This is not obvious. See the Zeta-function article in Wikipedia.) So this “infinite series” is really an infinite sum. It does not fit the image given by the English word “series”. The English meaning contaminates the mathematical meaning. But the definition wins.
The mathematical word that corresponds to the usual meaning of “series” is “sequence”. For example, $a_k:=1/{{k}^{2}}$ is the infinite sequence $1,\frac{1}{4},\frac{1}{9},\frac{1}{16}\ldots$ It is not an infinite series.
“Only if” is also discussed from a more technical point of view in the article on conditional assertions.
In math English, sentences of the form $P$ only if $Q$” mean exactly the same thing as “If $P$ then $Q$”. The phrase “only if” is rarely used this way in ordinary English discourse.
Sentences of the form “$P$ only if $Q$” about ordinary everyday things generally do not mean the same thing as “If $P$ then $Q$”. That is because in such situations there are considerations of time and causation that do not come up with mathematical objects. Consider “If it is raining, I will carry an umbrella” (seeing the rain will cause me to carry the umbrella) and “It is raining only if I carry an umbrella” (which sounds like my carrying an umbrella will cause it to rain). When “$P$ only if $Q$” is about math objects,
there is no question of time and causation because math objects are inert and unchanging.
Students sometimes flatly refuse to believe me when I tell them about the mathematical meaning of “only if”. This is a classic example of semantic contamination. Two sources of information appear to contradict each other, in this case (1) the professor and (2) a lifetime of intimate experience with the English language. The information from one of these sources must be rejected or suppressed. It is hardly surprising that many students prefer to suppress the professor’s apparently unnatural and usually unmotivated claims.
The following cause more minor cognitive dissonance.
Besides the examples given above, you can find many others in these two works:
Send to KindleI have been working my way through abstractmath.org, revising the articles and turning them into pure HTML so they will be easier to update. In some cases I am making substantial revisions. In particular, many of the articles need a more modern point of view.
The math community’s understanding of sets and structures has changed because of category theory and will change
because of homotopy type theory.
This post considers some issues and possibilities concerning the chapter on sets.
The references listed at the end of the article include several about homotopy type theory. They provide different viewpoints and require different levels of sophistication.
The abmath article Specification of sets specifies what a set is in this way:
A set is a single math object distinct from but completely determined by what its elements are.
I have used this specification for sets since the eighties, first in my Discrete Math lecture notes and then in abstractmath.org. It has proved useful because it is quite simple and the statement implies lots of immediate consequences. Each of the first four consequences in this list below exposes a confusion that some students have.
Those consequences make the spec a useful teaching tool. But if a beginning abstract math student gets very far in their studies, some complications come up.
In the late nineteenth century, math people started formally defining particular math structures such as groups and various
kinds of spaces. This was normally done by starting with a set and adding structure.
You may think that “starting with a set and adding structure” brushes a lot of complications under the rug. Well, don’t look under the rug, at least not right now.
The way they thought about sets was a informal version of what is now called naive set theory. In particular, they freely defined particular sets using what is essentially setbuilder notation, producing sets in a way which (I claim) satisfies my specification.
Then along came Russell’s paradox. In the context of this discussion, the paradox implied that the spec for sets is not a definition.The spec provides a set of necessary conditions for being a set. But it is not sufficient. You can say “Let $S$ be the set of all sets that…[satisfy some condition]” until you are blue in the face, but there are conditions (including the empty condition) that don’t define a set.
The Zermelo-Fraenkel axioms were designed to provide a definition that didn’t create contradictions. The axioms accomplish this by creating a sort of hierarchy that requires that each set must be defined in terms of sets defined previously. They provide a good way (but not the only one) of providing a way of legitimizing our use of sets in math.
Observe that the “set of all sets” is certainly not “defined” in terms of previously defined sets!
During those days there was a movement to provide a solid foundation for mathematics. After Zermelo-Fraenkel came along, the progress of thinking seemed to be:
That list is oversimplified. In particular, the development of predicate logic was essential to this approach, but I can’t write about everything at once.
This leads to monsters such as the notorious definition of ordered pair:
The ordered pair $(a,b)$ is the set $\{a,\{b\}\}$.
This leads to the ludicrous statement that $a$ is an element of $(a,b)$ but that $b$ is not.
By saying every math object may be modeled as a set with structure, ZF set theory becomes a model of all of math. This approach gives a useful proof that all of math is as consistent as ZF set theory is.
But many mathematicians jumped to the conclusion that every math object must be a set with structure. This approach does not match the way mathematicians think about math objects. In particular, it makes computerized proof assistance hard to use because you have to translate your thinking into sets and first order logic.
“A mathematical object is determined by the role it plays in a category.” — A. Grothendieck
In category theory, you define math structures in terms of how they relate to other math structures. This shifts the emphasis from
What is it?
to
What are its properties?
For example, an ordered pair is a mathematical object $p$ determined by these properties:
“Categorical” here means “as understood in category theory”. It unfortunately has a very different meaning in model theory (set of axioms with only one model up to isomorphism) and in general usage, as in “My answer is categorically NO” said by someone who is red in the face. The word “categorial” has an entirely different meaning in linguistics. *Sigh*.
William Lawvere has produced an axiomatization of the category of sets.
The most accessible introduction to it that I know of is the article Rethinking set theory, by Tom Leinster. This axiomatization defines sets by their relationship with each other and other math objects in much the same way as the categorical definition of (for example) groups gives a definition of groups that works in any category.
The word set as used informally has two different meanings.
One of the great improvements in mathematics that homotopy type theory supplies is a systematic way of keeping track of the isomorphisms, the isomorphisms between the isomorphisms, and so on ad infinitum (literally). But note: I am just beginning to understand htt, so regard this remark as something to be suspicious of.
I suggest that we keep the word “set” for the traditional concept and call the strict categorical concept an urset.
The traditional set $\{3,5\}$ consists of the unique two-element urset coindexed on the integers.
A (ur)set $S$ coindexed by a math structure $A$ is a monic map from $S$ to the underlying set of $A$. In this example, the map has codomain the integers and takes one element of the two-element urset to $3$ and the other to $5$.
Note added 2014-10-05 in response to Toby Bartels’ comment: I am inclined to use the names “abstract set” for “urset” and “concrete set” for coindexed sets when I revise the articles on sets. But most of the time we can get away with just “set”.
There is clearly no isomorphism of coindexed sets from $\{3,4\}$ to $\{3,5\}$, so those two traditional sets are not equal in the category of coindexed sets.
I made up the phrase “coindexed set” to use in this sense, since it is a kind of opposite of indexed set. If terminology for this already exists, lemme know. Linguists will tell you they use the word “coindexed” in a different sense.
The concept of “element” in categorical thinking is very different from the traditional idea, where an element of a set can be any mathematical object. In categorical thinking, an element of an object $A$ of a category $\mathbf{C}$ is an arrow $1\to A$ where $1$ is the terminal object. Thus $4$ as an integer is the arrow $1\to \mathbb{Z}$ whose unique value is the number $4$.
In the usage of category theory, the arrow $1\to\mathbb{R}$ whose value is the real number $4$ is a different math object from the arrow $1\to\mathbb{Z}$ whose value is the integer $4$.
A category theorist will probably agree that we can identify the integer $4$ with the real number $4$ via the well known canonical embedding of the ring of integers into the field of real numbers. But in categorical thinking you have to keep all such embeddings in mind; you don’t say the integer $4$ is the same thing as the real number $4$. (Most computer languages keep them distinct, too.)
This difference is actually not hard to get used to and is in fact an improvement over traditional set theory. When you do category theory you use lots of commutative diagrams. The embeddings show up as monic arrows and are essential in keeping the different objects ($\mathbb{Z}$ and $\mathbb{R}$ in the example) separate.
The paper Relating first-order set theory and elementary toposes, by Awodey, Butz, Simpson and Streicher, introduces a concept of “structural system of inclusions” that appears to me to restore the idea of object being an element of more than one set for many purposes.
Homotopy type theory allows an object to have only one type, with much the same effect as in the categorical approach.
The arrow $1\to \mathbb{Z}$ that picks out the integer $4$ is a constant function. It is useful to think of any arrow $A\to B$ of any category as a variable element (or generalized element) of the object $B$. For example, the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=x^2$ allows you to
think of $x^2$ as a variable number with real parameter. This is another way of thinking about the “$y$” in the equation $y=x^2$, which is commonly called a dependent variable.
One way to think about $y$ is that some statements about it are true, some are false, and many statements are neither true nor false.
This way of thinking about variable objects clears up a lot of confusion about variables and deserves to be more widely used in teaching.
The book Category theory for computing science provides some examples of the use of variable elements as a way of thinking about categorical ideas.
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Send to KindleThis is a draft of an addition to the entry Alphabets in abstractmath.org.
Mathematicians use the word script to refer to two rather different styles. Both of them apply only to uppercase letters.
| $A$: $\scr{A}$ | $H$: $\scr{H}$ | $O$: $\scr{O}$ | $V$: $\scr{V}$ |
| $B$: $\scr{B}$ | $I$: $\scr{I}$ | $P$: $\scr{P}$ | $W$: $\scr{W}$ |
| $C$: $\scr{C}$ | $J$: $\scr{J}$ | $Q$: $\scr{Q}$ | $X$: $\scr{X}$ |
| $D$: $\scr{D}$ | $K$: $\scr{K}$ | $R$: $\scr{R}$ | $Y$: $\scr{Y}$ |
| $E$: $\scr{E}$ | $L$: $\scr{L}$ | $S$: $\scr{S}$ | $Z$: $\scr{Z}$ |
| $F$: $\scr{F}$ | $M$: $\scr{M}$ | $T$: $\scr{T}$ | |
| $G$: $\scr{G}$ | $N$: $\scr{N}$ | $U$: $\scr{U}$ |
| $A$: $\cal{A}$ | $H$: $\cal{H}$ | $O$: $\cal{O}$ | $V$: $\cal{V}$ |
| $B$: $\cal{B}$ | $I$: $\cal{I}$ | $P$: $\cal{P}$ | $W$: $\cal{W}$ |
| $C$: $\cal{C}$ | $J$: $\cal{J}$ | $Q$: $\cal{Q}$ | $X$: $\cal{X}$ |
| $D$: $\cal{D}$ | $K$: $\cal{K}$ | $R$: $\cal{R}$ | $Y$: $\cal{Y}$ |
| $E$: $\cal{E}$ | $L$: $\cal{L}$ | $S$: $\cal{S}$ | $Z$: $\cal{Z}$ |
| $F$: $\cal{F}$ | $M$: $\cal{M}$ | $T$: $\cal{T}$ | |
| $G$: $\cal{G}$ | $N$: $\cal{N}$ | $U$: $\cal{U}$ |
Thanks to JM Wilson for suggesting this topic and to the various people on Math Stack Exchange and Math Educators Stack Exchange who discussed script and cal.
Send to KindleThis post discusses some ideas I have for improving abstractmath.org.
The Handbook was kind of a false start on abmath, and is the source of much of the material in abmath. It still contains some material not in abmath, particularly the citations.
By citations I mean lexicographical citations: examples of the usage from texts and scholarly articles.
I published the Handbook of mathematical discourse in 2003. The first link below takes you to an article that describes what the Handbook does in some detail. Briefly, the Handbook surveys the use of language in math (and some other things) with an emphasis on the problems it causes students. Its collection of citations of usage could some day could be the start of an academic survey of mathematical language. But don’t expect me to do it.
The Handbook exists as a book and as two different web versions. I lost the TeX source of the Handbook a few years after I published the book, so none of the different web versions are perfect. Version 2 below is probably the most useful.
Soon after the Handbook was published, I started work on abstractmath.org, which I abbreviate as abmath. It is intended specifically for people beginning to study abstract math, which means roughly post-calculus. I hope their teachers will read it, too. I had noticed when I was teaching that many students hit a big bump when faced with abstraction, and many of them never recovered. They would typically move into another field, often far away from STEM stuff.
These abmath articles give more detail about the purpose of this website and the thinking behind the way it is presented:
Abmath is written for students of abstract math and other beginners to tell them about the obstacles they may meet up with in learning abstract math. It is not a scholarly work and is not written in the style of a scholarly work. There is more detail about its style in my rant in Attitude.
Scholarly works should not be written in the style of a scholarly work, either.
Every time I revise an article I find myself rewriting overly formal parts. Fifty years of writing research papers has taken its toll. I must say that I am not giving this informalization stuff very high priority, but I will continue doing it.
One major difference concerns the citations in the Handbook. I collected these in the late nineties by spending many hours at Jstor and looking through physical books. When I started abmath I decided that the website would be informal and aimed at students, and would contain few or no citations, simply because of the time it took to find them.
The Handbook had both sidebars on every page of the paper version containing a reference index to words on that page, and also on many pages boxouts with comments. It was written in TeX. I had great difficulty using TeX to control the placement of both the sidebars and especially the boxouts. Also, you couldn’t use TeX to let the text expand or contract as needed by the width of the user’s screen.
Abmath uses boxouts but not sidebars. I wrote Abmath using HTML, which allows it to be presented on large or small screens and to have extensive hyperlinks.
HTML also makes boxouts easy.
The arrival of tablets and i-pods has made it desirable to allow an abmath page to be made quite narrow while still readable. This makes boxouts hard to deal with. Also I have gotten into the habit of posting revisions to articles on Gyre&Gimble, whose editor converts boxouts into inline boxes. That can probably be avoided.
I have to decide whether to turn all boxouts into inline small-print paragraphs the was you see them in this article. That would make the situation easier for people reading small screens. But in-line small-print paragraphs are harder to associate to the location you want them to refer, in contrast to boxouts.
For the first few years, I used Microsoft Word with MathType, but was plagued with problems described in the link below. Then I switched to writing directly in HTML. The articles of abmath labeled “abstractmath.org 2.0” are written in this new way. This makes the articles much, much easier to update. Unfortunately, Word produces HTML that is extraordinarily complicated, so transforming them into abmath 2.0 form takes a lot of effort.
Abmath does not have enough illustrations and diagrams. Gyre&Gimble has many posts with static illustrations, some of them innovative. It also has some posts with interactive demos created with Mathematica. These demos require the reader to download the CDF Player, which is free. Unfortunately, it is available only for Windows, Mac and Linux, which precludes using them on many small devices.
Abmath includes most of the ideas about language in the Handbook (rewritten and expanded) and adds a lot of new material.
The language articles would greatly benefit from more illustrations. In particular:
Abmath discusses how we understand math and strategies for doing math in some detail. This part is based on my own observations during 35 years of teaching, as well as extensive reading of the math ed literature. The math ed literature is usually credited in footnotes.
Understanding how to think about mathematical objects is, I believe, one of the most difficult hurdles newbies have to overcome in learning abstract math. This is one area that the math ed community has focused on in depth.
The first two links below are take you to the two places in abmath that discuss this problem. The first article has links to some of the math ed literature.
An important point about math objects that needs to be brought out more is that everything in math is a math object. Obviously math structures are math objects. But the symbol “$\leq$” in the statement “$35\leq45$” denotes a math object, too. And a proof is a math object: A proof written on a blackboard during a lecture does not look like it is an instance of a rigorously defined math object, but most mathematicians, including me, believe that in principle such proofs can be transformed into a proof in a formal logical system. Formal logics, such as first order logic, are certainly math objects with precise mathematical definitions. Definitions, math expressions and theorems are math objects, too. This will be spelled out in a later post.
Classically, math structures have been presented as sets with structure, with the structure being described in terms of subsets and functions. My chapter on math structures only waves a hand at this. This is a decidedly out-of-date way of doing it, now that we have category theory and type theory. I expect to post about this in order to clarify my thinking about how to introduce categorical and type-theoretical ideas without writing a whole book about it.
Abmath includes discussions
of the problems students have with certain particular types of structures. These sections talk mostly about how to think about these structure and some particular misunderstandings students have at the most basic levels.
These articles are certainly not proper introductions to the structures. Abmath in general is like that: It tells students about some aspects of math that are known to cause them trouble when they begin studying abstract math. And that is all it does.
This chapter covers several aspects of proofs that cause trouble for students, the logical aspects and also the way proofs are written.
It specifically does not make use of any particular symbolic language for logic and proofs. Some math students are not good at learning languages, and I didn’t see any point in introducing a specific language just to do rudimentary discussions about proofs and logic. The second link below discusses linguistic ability in connection with algebra.
I taught logic notation as part of various courses to computer engineering students and was surprised to discover how difficult some students found using (for example) $p+q$ in one course and $p\vee q$ in another. Other students breezed through different notations with total insouciance.
Much of the chapter on proofs is badly written. When I get around to upgrading it to abmath 2.0 I intend to do a thorough rewrite, which I hope will inspire ideas about how to conceptually improve it.
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