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Context

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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

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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.

  7. Each mathematical assertion in the text is marked as a claim.
  8. 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.
  9. 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.
  10. If no reference or proof for a claim is given, the PV checks the Database to see if it has already been proved.
  11. 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).
  12. If a segment of text is presented as a proof, it is input into the PV to be verified.
  13. 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.
  14. 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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    Variations in meaning in math

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    Words in a natural language may have different meanings in different social groups or different places.  Words and symbols in both mathematical English and the symbolic language vary according to specialty and, occasionally, country (see convention, default).  And words and symbols can change their meanings from place to place within the same mathematical discourse (see scope).

    This article mostly provides pointers to other articles in abstractmath.org that give more details about the ideas.

    Conventions

    A convention in mathematical discourse is notation or terminology used with a special meaning in certain contexts or in certain fields. Articles and books in a specialty do not always clue you in on these conventions.

    Some conventions are nearly universal in math.

    Example 1

    The use of “if” to mean “if and only if” in a definition is a convention. More about this here. This is a hidden definition by cases. “Hidden” means that no one tells the students, except for Susanna Epp and me.

    Example 2

    Constants or parameters are conventionally denoted by a, b, … , functions by f, g, … and variables by x, y,…. More.

    Example 3

    Referring to a group (or other mathematical structure) and its underlying set by the same name is a convention.  This is an example of both synecdoche and context-sensitive.

    Example 4

    The meaning of ${{\sin }^{n}}x$ in many calculus books is:

    • The inverse sine (arcsin) if $n=-1$.
    • The mult­iplica­tive power for positive $n$; in other words, ${{\sin }^{n}}x={{(\sin x)}^{n}}$ if $n\ne -1$.

    This, like Example 1, is a definition by cases. Unlike Example 1, calculus books often make it explicit. Explicit or not, this usage is an abomination.

    Some conventions are pervasive among math­ematicians but different conventions hold in other subjects that use mathematics.

    • Scientists and engineers may regard a truncated decimal such as 0.252 as an approximation, but a mathematician is likely to read it as an exact rational number, namely $\frac{252}{1000}$.
    • In most computer languages a distinction is made between real numbers and integers;
      42 would be an integer but 42.0 would be a real number.  Older mathematicians may not know this.
    • Mathematicians use i to denote the imaginary unit. In electrical engineering it is commonly denoted j instead, a fact that many mathematicians are un­aware of. I first learned about it when a student asked me if i was the same as j.

    Conventions may vary by country.

    • In France and possibly other countries schools may use “positive” to mean “nonnegative”, so that zero is positive. 
    • In the secondary schools in some places, the value of sin x may be computed clockwise starting at (0,1)  instead of counterclockwise starting at (1,0).  I have heard this from students. 

    Conventions may vary by specialty within math.

    “Field” and “log” are examples. 

    Defaults

    An interface to a computer program may have many possible choices for the user to make. In most cases, the interface will use certain choices automatically when the user doesn’t specify them.  One says the program defaults to those choices.  

    Examples

    • A word processing program may default to justified paragraphs and insert mode, but allow you to pick ragged right or typeover mode.

      • I have spent a lot of time in both Minne­sota and Georgia and the remarks about skiing are based on my own observation. But these usages are not absolute. Some affluent Geor­gians may refer to snow skiing as “skiing”, for example, and this usage can result in a put-down if the hearer thinks they are talking about water skiing. One wonders where the boundary line is. Perhaps people in Kentucky are confused on the issue.

    • There is a sense in which the word “ski” defaults to snow skiing in Minnesota and to water skiing in Georgia.
    • “CSU” defaults to Cleveland State University in northern Ohio and to Colorado State University in parts of the west.

    Math language behaves in this way, too.

    Default usage in mathematical discourse

    Symbols

    • In high school, $\pi$ refers by default to the ratio of the circumference of a circle to its diameter.  Students are often quite surprised when they get to abstract math courses and discover the many other meanings of $\pi $ (see here).
    • Recently authors in the popular literature seem to think that $\phi$ (phi) defaults to the golden ratio.  In fact, a search through the research literature shows very few hits for $\phi$ meaning the golden ratio: in other words, it usually means something else. 
    • The set $\mathbb{R}$ of real numbers has many different group structures defined on it but “The group $\mathbb{R}$” essentially always means that the group operation is ordinary addition.  In other words, “$\mathbb{R}$” as a group defaults to +.  Analogous remarks apply to “the field $\mathbb{R}$”. 
    • In informal conversation among many analysts, functions are continuous by default.
    • It used to be the case that in informal conversations among topologists, “group” defaulted to Abelian group. I don’t know whether that is still true or not.

    Remark

    This meaning of “default” has made it into dictionaries only since around 1960 (see the Wikipedia entry). This usage does not carry a derogatory connotation.   In abstractmath.org I am using the word to mean a special type of convention that imposes a choice of parameter, so that it is a special case of both “convention” and “suppression of parameters”.

    Scope

    Both mathematical English and the symbolic language have a feature that is uncommon in ordinary spoken or written English:  The meaning of a phrase or a symbolic expression can be different in different parts of the discourse.   The portion of the text in which a particular meaning is in effect is called the scope of the meaning.  This is accomplished in several ways.

    Explicit statement

    Examples

    • “In this paper, all groups are abelian”.  This means that every instance of the word “group” or any symbol denoting a group the group is constrained to be abelian.   The scope in this case is the whole paper.   See assumption.
    • “Suppose (or “let” or “assume”) $n$ is divisible by $4$”. Before this statement, you could not assume $n$ is divisible by $4$. Now you can, until the end of the current paragraph or section.

    Definition

    The definition of a word, phrase or symbol sets its meaning.  If the word definition is used and the scope is not given explicitly, it is probably the whole discourse.

    Example

    “Definition.  An integer is even if it is divisible by 2.”  This is marked as a definition, so it establishes the meaning of the word “even” (when applied to an integer) for the rest of the text. 

    If

    Used in modus ponens (see here) and (along with let, usually “now let…”) in proof by cases.

    Example(modus ponens)

    Suppose you want to prove that if an integer $n$ is divisible by $4$ then it is even. To show that it is even you must show that it is divisible by $2$. So you write:

    • “Let $n$ be divisible by $4$. That means $n=4k$ for some integer $k$. But then $n=2(2k)$, so $n$ is even by definition.”

    Now if you start a new paragraph with something like “For any integer $n\ldots$” you can no longer assume $n$ is divisible by $4$.

    Example (proof by cases)

    Theorem: For all integers $n$, $n^2+n+1$ is odd.

    Definitions:

    • “$n$ is even” means that $n=2s$ for some integer $s$.
    • “$n$ is odd” means that $n=2t+1$ for some integer $t$.

    Proof:

    • Suppose $n$ is even. Then

      \[\begin{align*}
      n^2+n+1&=4s^2+2s+1\\
      &=2(2s^2+s)+1\\
      &=2(\text{something})+1
      \end{align*}\]

      so $n^2+n+1$ is odd. (See Zooming and Chunking.)

    • Now suppose $n$ is odd. Then

      \[\begin{align*}
      n^2+n+1&=(2t+1)^2+2t+1+1\\
      &=4t^2+4t+1+2t+1+1\\
      &=2(2t^2+3t)+3\\
      &=2(2t^2+3t+1)+1\\
      &=2(\text{something})+1
      \end{align*}\]

      So $n^2+n+1$ is odd.

    Remark

    The proof I just gave uses only the definition of even and odd and some high school algebra. Some simple grade-school facts about even and odd numbers are:

    • Even plus even is even.
    • Odd plus odd is even.
    • Even times even is even.
    • Odd times odd is odd.

    Put these facts together and you get a nicer proof (I think anyway): $n^2+n$ is even, so when you add $1$ to it you must get an odd number.

    Bound variables

    A variable is bound if it is in the scope of an integral, quantifier, summation, or other binding operators.  More here.

    Example

    Consider this text:

    Exercise: Show that for all real numbers $x$, it is true that $x^2\geq0$. Proof: Let $x=-2$. Then $x^2=(-2)^2=4$ which is greater than $0$. End of proof.”

    The problem with that text is that in the statement, “For all real numbers $x$, it is true that $x^2\geq0$”, $x$ is a bound variable. It is bound by the universal quantifier “for all” which means that $x$ can be any real number whatever. But in the next sentence, the meaning of $x$ is changed by the assumption that $x=-2$. So the statement that $x\geq0$ only applies to $-2$. As a result the proof does not cover all cases.

    Many students just beginning to learn to do proofs make this mistake. Fellow students who are a little further along may be astonished that someone would write something like that paragraph and might sneer at them. But this common mistake does not deserve a sneer, it deserves an explanation. This is an example of the ratchet effect.

    Variable meaning in natural language

    Meanings commonly vary in natural language because of conventions and defaults. But varying in scope during a conversation seems to me uncommon.

    It does occur in games. In Skat and Bridge, the meaning of “trump” changes from hand to hand. The meaning of “strike” in a baseball game changes according to context: If the current batter has already had fewer than two strikes, a foul is a strike, but not otherwise.

    I have not come up with non-game examples, and anyway games are played by rules that are suspiciously like mathematical axioms. Perhaps you can think of some non-game occasions in which meaning is determined by scoping that I have overlooked. 

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    Rigorous proofs

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    Rich and rigorous

    When we try to understand a math statement, we visualize what the statement says using metaphors, images and kinetic feelings to feel how it is true, or to suggest that the statement is not true.

    If we are convinced that it is true, we may then want to prove it. Doing that involves pitching out all the lovely pictures and metaphors and gestures and treating the mathematical objects involved in the proof as static and inert. “Static” means the object does not change. “Inert” means that it does not affect anything else. I am saying how we think about math objects for the purpose of rigorous proof. I am not saying anything about “what math objects are”.

    In this post I give a detailed example of a proof of the rigorous sort.

    Example

    Informal statement

    First, I’ll describe this example in typical spoken mathematical English. Suppose you suspect that the following statement is true:

    Claim: Let $f(x)$ be a differentiable function with $f'(a)=0$.
    Going from left to right, suppose the graph of $f(x)$ goes UP before $x$ reaches $a$ and then DOWN for $x$ to the right of $a$
    Then $a$ has to be a local maximum of the function.

    This claim is written in informal math English. Mathematicians talk like that a lot. In this example they will probably wave their hands around in swoops.

    The language used is an attempt to get a feeling for the graph going up to $(a,f(a))$ and then falling away from it. It uses two different metaphors for $x\lt a$ and $x\gt a$. I suspect that most of us would want to clean that up a bit even in informal writing.

    A more formal statement

    Theorem: Let $f$ be a real valued differentiable function defined on an open interval $R$. Let $a$ be a number in $R$ for which $f'(a)=0$. Suppose that for all $x\in R$, $f$ increases for $x\lt a$ and decreases for $x\gt a$. Then $f(a)$ is a maximum of $f$ in $R$.

    Proof

    1. By definition of derivative, \[\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=0.\]
    2. By definition of limit, then for any positive $\epsilon$ there is a positive $\delta$ for which if $0\lt|x-a|\lt\delta$ then \[\left|\frac{f(x)-f(a)}{x-a}\right|\lt\epsilon.\]
    3. By requiring that $\delta\lt 1$, it follows from (2) that for any positive $\epsilon$, there is a positive $\delta$ for which if $0\lt|x-a|\lt\delta$, then $|f(x)-f(a)|\lt\epsilon$.
    4. “$f$ increases for $x\lt a$” means that if $x$ and $y$ are numbers in $R$ and $x\lt y\lt a$, then $f(x)\lt f(y)$.
    5. “$f$ decreases for $x\gt a$” means that if $x$ and $y$ are numbers in $R$ and $a\lt x\lt y$, then $f(x)\gt f(y)$.
    6. “$f(a)$ is a maximum of $f$ in $R$” means that for $x\in R$, if $x\neq a$, then $f(x)\lt f(a)$.
    7. Suppose that $x\in R$ and $x\lt a$. (The case that $x\gt a$ has a symmetric proof.)
    8. Given $\epsilon\gt0$ with $\delta$ as given by (3), choose $y\in R$ such that $x\lt y\lt a$ and $|f(y)-f(a)|\lt\epsilon$.
    9. By (4), $f(x)\lt f(y)$. So by (8), \[\begin{align*}
      f(x)-f(a)&=
      f(x)-f(y)+f(y)-f(a)\\ &\lt f(y)-f(a)\\ &\leq|f(y)-f(a)|\lt\epsilon\end{align*}\]
      so that $f(x)\lt f(a)+\epsilon$. By inserting “$-f(y)+f(y)$” into the second formula, I am “adding zero cleverly”, an example of pulling a rabbit out of a hat. Students hate that. But you have to live with it; as long as the statements following are correct, it makes a valid proof. Rabbit-out-of-a-hat doesn’t make a proof wrong, but it does make you wonder how the author thought of it. Live with it.
    10. Since (9) is true for all positive $\epsilon$, it follows that $f(x)\leq f(a)$.
    11. By the same argument as that leading up to (10), $f(\frac{x-a}{2})\leq f(a)$.
    12. Since $f(x)\lt f(\frac{x-a}{2})$, it follows that $f(x)\lt f(a)$ as required.

    About the proof

    This proof is intended to be a typical “rigorous” proof. I suspect it tends to be more rigorous than most mathematicians would find necessary,

    Extensionality

    The point about “rigor”, about insisting that the objects be static and inert, is that this causes symbols and expression to retain the same meaning throughout the text. This is one aspect of extensionality.

    Of course, some of the symbols denote variables, or variable objects. This does not mean they are “varying”. I am taking this point of view: A variable refers to a math object but you don’t know what it is. Constraints such as $x\lt a$ rule out some possible values but don’t generally tell you exactly what $x$ is. There is more about this in Variable Objects

    The idea in (6), for example, is that $y$ denotes a real number. You don’t know which number it is, but you do know some facts about it: $x\lt y\lt a$, $|f(y)\lt f(a)|\lt\epsilon$ and so on. Similarly you don’t know what function $f$ is, but you do know some facts about it: It is differentiable, for example, and $f'(a)=0$.

    My statement that the variables aren’t “varying” means specifically that each unbound occurrence of the variable refers to the same value as any other occurrence, unless some intervening remark changes its meaning. For example, the references to $x$ in (7) through (10) refer to the same value it has in (6), and (10), in particular, constitutes a statement that the claim about $x$ is correct.

    Checkability

    The elimination of metaphors that lets the proof achieve rigor is part of a plan in the back of the mind of at least some mathematicians who write proofs. The idea is that the proof be totally checkable:

    • Every statement in the proof has a semantics, a meaning, that is invariant (given the remark about variables above).
    • Each statement is justified by some of the previous statements. This justification is given by two systems that the reader is supposed to understand.
    • One system is the rules of symbol manipulation that are applied to the symbolic expressions, ordinary algebra, and higher-level manipulations used in particular branches of math.
    • The other system consists of the rules of logical reasoning that justify the claims that each statement follows logically from preceding ones.
    • These two systems are really branches of one system, the entire system of math computation and reasoning. It can be obscure which system is being used in a particular step.

    Suppression of reasons

    The logical and symbolic-manipulation reasons justifying the deductions may not be made completely explicit. In fact, for many steps they may not be mentioned at all, and for others, one or two phrases may be used to give a hint. This is standard practice in writing “rigorous” proofs. That is a descriptive statement, made without criticism. Giving all the reasons is essentially impossible without a computer.

    I am aware that some work has been done to write proof checkers that can read a theorem like the one we are considering, stated in natural language, and correctly implement the semantics I have described in this list. I don’t know of any references to such work and would appreciate information about it.

    Suppression of reasons makes it difficult to mechanically check a proof written in this standard “rigorous” writing style. Basically, you must be at at least the graduate student level to be able to make sense of what is said, and even experienced math research people find it difficult to read a paper in a very different field. Writing the proof so that it can be checked by a proof checker requires understanding of the same sort, and it typically makes the proof much longer.

    One hopeful new approach is to write the proofs using homotopy type theory. The pioneers in that field report that the proofs don’t expand nearly as much as is required by first order logic.

    Examples of suppression

    Here are many examples of suppression in the $\epsilon$-$\delta$ proof above. This is intended to raise your consciousness concerning how nearly opaque writing in math research is to anyone but the cognoscenti.

    • The first sentence of the theorem names $R$ and $f$ and puts constraints on them that can be used to justify statements in the proof. The naming of $R$ and $f$ requires that every occurrence of $R$ in the proof refers to the same mathematical object, and similarly for $f$.

    Remark: The savvy reader “knows” the facts stated in (a), possibly entirely subconsciously. For many of us there is no conscious thought of constraints and permanence of naming. My goal is to convince those who teach beginning abstract math course to become conscious of these phenomena. This remark applies to all the following items as well.

    • The second sentence gives $a$ a specific meaning that will be maintained throughout the proof. It also puts constraints on $a$ and an additional constraint on $f$.
    • The third sentence gives a constraint on $R$, $f$ and $a$. It does not give a constraint on $x$, which is a bound variable. Nor does it name $x$ as a specific number with the same meaning in the rest of the proof. (That happens later).
    • The fact that the first three sentences impose constraints on various objects is signaled by the fact that the sentences are introduced by “let” and “suppose”. The savvy reader knows this.
    • The fourth sentence announces that “$f(a)$ is a maximum of $f$ in $R$” is a consequence of the constraints imposed by the preceding three sentences. (In other words, it follows from the context.) This is signaled by the word “then”.
    • The fact that the paragraph is labeled “Theorem” informs us that the fourth sentence is therefore a statement of what is to be proved, and that every constraint imposed by the first three sentences of the Theorem may be used in the proof.
    • In the proof, statements (1), (4), (5) and (6) rewrite the statements in the theorem according to the definitions of the words involved, namely “derivative: “increases”, “decreases” and “maximum”. Rewriting statements according to the definitions of the words involved is a fundamental method for starting a proof.
    • (2) follows from (1) by rewriting using the definition of “limit”. Note that pattern-matching against the definition of limit requires understanding that there is a zero inside the absolute value signs that is not written down. Could a computer proof-checker handle that?
    • (3) follows from (2). The reader or proof-checker must:
      • Know that it is acceptable to put an upper bound on $\delta$ in the definition of limit.
      • Notice that you can move $|x-a|$ out of the denominator because $x\neq a$ by (2).
    • The conclusion in (6) that we much show that $f(x)\lt f(a)$ is now the statement we must prove.

    Remark: In the following items, I mention the context of the proof. I am using the word informally here. It is used in some forms of formal logic with a related but more precise meaning. The context consists of the variables you must hold in your head as you read each part of the proof, along with their current constraints. “Current” means the “now” that you are in when considering the step of the proof you are reading right now. I give some references at the end of the post.

    • At the point between (6) and (7), our context consists of $a$, $R$ and $f$ all subject to some constraints. $x$ is not yet in the context of our proof because its previous occurrences in the theorems and in (1) through (6) have been bound, mostly by an unexpressed universal quantifier. Now we are to think of $x$ as a specific number bound by some constraints.
    • The statement in (7) that the case $x\gt a$ as a symmetric proof is a much higher-level claim than the other steps in this proof, even though in fact it is not very high level compared to statements such as “An application of Serre’s spectral sequence shows$\ldots$”. Most mathematicians with even a little experience will read this statement and accept it in the confidence that they will know how to swap “$\lt$” and “$\gt$” in the proof in the correct way (which is a bit picky) to provide a dual proof. Some students might write out the dual proof to make sure they understand it (more likely because writing it out was a class assignment). I await the day that an automated proof checker can handle a statement like this.
    • (8) introduces three new math objects $\epsilon$, $\delta$ and $y$ subject to several constraints. The symbols occur earlier but they are all bound. $\epsilon$ will be fixed in our context from now until (10). The others don’t appear later.
    • (9) consists of several steps of algebraic computation. A cognoscent (I am tired of writing “savvy”) reader first looks at the computation as a whole and notices that it deduces that $|f(x)-f(a)|\lt\epsilon$, which is almost what is to be proved. This helps the reader understand the reason for the calculation. No mention whatever is made in this step of all this stuff that should go through your mind (or the proof-checker’s “mind”).
    • The computations in (9) are are basic algebra not explained step by step, except that the remark that $f(x)\lt f(y)$ explains how you get $f(x)-f(y)+f(y)-f(a) \lt f(y)-f(a)$.
    • (10) banishes $\epsilon$ from the context by universally quantifying over it. That $f(x)\leq f(a)$ follows by the garbage-dump-in-Star-Wars trick that often baffles first year analysis students: Since for all positive $\epsilon$, $f(x)\lt f(a)+\epsilon$, then $f(x)\leq f(a)$. (See also Terry Tao’s article in Tricks Wiki.)
    • (11) “By the same argument as leading up to (10)” puts some demands on the reader, who has to discover that you have to go back to (7) and do the following steps with a new context using a value of $x$ that is halfway closer to $a$ than the “old” $x$ was. This means in particular that the choice of $\frac{x-2}{2}$ is unnecessarily specific. But it works.
    • (12) suppresses the reference to (11).

      • References

      I have written extensively on these topics. Here are some links.

      Rich-rigorous bifurcation in math thinking

    The symbolic language

    Math English and the language of proofs

    Proofs and context

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    Semantics of algebra I

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

    In the post Algebra is a difficult foreign language  I listed some of the difficulties of the syntax of the symbolic language of math (which includes high school algebra and precalculus).  The semantics causes difficulties as well.  Again I will list some examples without any attempt at completeness.

    The status of the symbolic language as a language

    There is a sharp distinction between the symbolic language of math and mathematical English, which I have written about in The languages of math and in the Handbook of mathematical discourse. Other authors do not make this sharp distinction (see the list of references at the end of this post). The symbolic language occurs embedded in mathematical English and the embedding has its own semantics which may cause great difficulty for students.

    The symbolic language of math can be described as a natural formal language. Pieces of it were invented by mathematicians and others over the course of the last several hundred years. Individual pieces (notation such as "$3x+1=2y$") can be given a strictly formal syntax, but the whole system is ambiguous, inconsistent, and context-sensitive.  When you get to the research level, it has many dialects: Research mathematicians in one field may not be able to read research articles in a very different field.

    Examples

    I think the examples below will make these claims plausible.  This should be the subject of deep research.

    Superscripts and functions

    • A superscript, as in $5^2$ or $x^3$, has a pretty standard meaning denoting a power, at least until you get to higher level stuff such as tensors.  
    • A function can be denoted by a letter, symbol, or string, and the notation $f(x)$ refers to the value of the function at input $x$.  

    For functions defined on numbers, it is common in precalculus and higher to write $f^2(x)$ to denoted $(f(x))^2=f(x)\,f(x)$.  Since the value of certain multiletter functions are commonly written without the parentheses (for example, $\sin\,x$), one writes $\sin^2x$ to mean $(\sin\,x)^2$.

    The notation $f^n$ is also widely used to mean the $n$th iterate of $f$ (if it exists), so $f^3(x)=f(f(f(x)))$ and so on.  This leads naturally to writing $f^{-1}(x)$ for the inverse function of $f$; this is common notation whether the function $f$ is bijective or not (in which case $f^{-1}$ is set-valued).  Thus $\sin^{-1}x$ means $\arcsin\,x$.

    It is notorious that words in mathematical English have different meanings in different texts.  This is an example in the symbolic language (and not just at the research level) of a systematic construction that can give expressions that have ambiguous meanings.

    This phenomenon is an example of why I say the symbolic language of math is a natural formal language: I have described a natural extension of notation used with multiplication of values that has been extended to being used for the binary operation of composition.  And that leads to students thinking that $\sin^{-1}x$ means $\frac{1}{\sin\,x}$. 

    History can overtake notation, too: Mathematicians probably took to writing $\sin\,x$ instead of $\sin(x)$ because it saves writing.  That was not very misleading in the old days when mathematical variables were always single symbols.  But students see multiletter variable names all the time these days (in programming languages, Excel and elsewhere), so of course some of them think $\sin\,x$ means $\sin$ times $x$. People who do this are not idiots.

    Juxtaposition

    Juxtaposition of two symbols means many different things.

    • If $m$ and $n$ are numbers, $mn$ denotes the product of the two numbers.
      • Multiplication is commutative, so $mn$ and $nm$ denote the same number, but they correspond to different calculations.  
    • If $M$ and $N$ are matrices, $MN$ denotes the matrix product of the two matrices.
      • This is a binary operation but it is not the same operation denoted by juxtaposition of numbers. (In fact it involves both addition and multiplication of numbers.)
      • Now $MN$ may not be the same matrix as $NM$.
    • If $A$ and $B$ are points in a geometric drawing, $AB$ denotes the line segment from $A$ to $B$.
      • This is a function of two variables denoting points whose value is a line segment.  
      • It is not what is usually called a binary operation, although as an opinionated category theorist I would call it a multisorted binary operation.
      • It is commutative, but it doesn't make sense to ask if it is associative.

    This phenomenon is called overloaded notation.  

    • In order to understand the meaning of the juxtaposition of symbols, you have to know the type of the variables.
    • ​The surrounding text may tell you specifically the variables denote matrices or whatever. So this is an instance of context-sensitive semantics. 
      • Students tend to expect that they know what any formula means in isolation from the text.  It may make them very sad to discover that this doesn't work — once they believe it, which can take quite a while.
    • In many cases the problem is alleviated by the use of convention.
      • Matrices are usually denoted by capital letters, numbers by lower case letters.
      • But points in geometry are usually denoted by capital letters too.  So you have to know that referring to a geometric diagram is significant to understanding the notation. This is an indirect form of context-sensitivity.  Did any teacher every point this out to students?  Does it appear anywhere in print?

    The earlier example of $\sin^{-1}x$ is a case which is not context-sensitive.  Knowing the types of the variables won't help.  Of course, if the author explains which meaning is meant, that explanation is within the context of the book!  That is not a lot of help for grasshoppers like me that look back and forth at different parts of a math book instead of reading it straight through..  

    Equations

    Consider the expressions

    1. $x^2-5x+4=0$
    2. $x^2+y^2=1$
    3. $x^2+2x+1=(x+1)^2$

    They are assertions that two expressions have the same value. A strictly logical view of an equation containing variables is that it puts a constraint on the variables.  It is true of some numbers (or pairs of numbers) and false of others.  That is the defining property of an equation. Equation 1 requires that $x=1$ or $x=4$.  Equation 2 imposes a constraint which is satisfied by uncountably many pairs of real numbers, and is also not true of uncountably many pairs. But equation 3 puts no constraint on the variable.  It is true of every number $x$.

    A strictly logical view of symbolic notation does math a disservice.  Here, the notion that an equation is by definition a symbolic statement that has a truth set and a falsity set may be correct but it is not the important thing about any particular equation. When we read and do math we have many different metaphors and images about a concept.  The definition of a kind of object is often in terms of things that may not be the most important things to know about it.  (One of the most important fact about groups is that it is an abstraction of symmetries, which the axioms don't mention at all.)

    Equation 1. is something that would make most people set out to discover the truth set.  Equation 2. calls out for drawing its graph.  Equation 3. being an identity means that is useful in algebraic reasoning.  The images they call up are different and what you do with them is different.  The images and metaphors that cluster around a concept are an important part of the semantics of the symbolic language.

    I expect to post separately about the semantics of variables and about the semantics of symbolic language embedded in mathematical English.

    References

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