Tag Archives: language

Pattern recognition and me

Recently, I revised the abstractmath.org article on pattern recognition. Doing that that prompted me to write about my own experiences with patterns. Recognizing patterns is something that has always delighted me: it is more of a big deal for me than it does for many other people. That, I believe, is what led me into doing research in math.

I have had several experiences with déjà vu, which is the result of pattern recognition with one pattern hidden. That will be a separate post. I expect to post about my experiences in recognizing patterns in math as well.

Patterns in language

As a teenager I was a page in the Savannah Public Library. There I discovered grammars for many languages. The grammars of other languages are astonishingly different from each other and are full of obscurities that I love to detect. Until I went to college, I was the only person I knew who read grammars for fun.

I am using the word “grammar” in the sense that linguists use it: patterns in our speech and writing, mostly unnoticed, that help express what we want to say)

The word “grammar” is also used to mean rules laid down by the ruling classes about phrases like “between you and I” and the uses of “whom”. Such rules primarily divide the underprivileged from the privileged, and many will disappear when the older members of the privileged class die (but they will think of new ones).

Grammar-induced glee


I got pretty good at reading and speaking Russian when I was a student (1959-62), but most of it has disappeared. In 1990, we hosted a Russian cello student with the Soviet-American Youth Orchestra for a couple of days. I could hardly say anything to him. One time he noticed one of our cats and said “кошка”, to which I replied “два кошки” (“two cats”). He responded by correcting me: “две кошки”. Then I remembered that the word for “two” in Russian is the only word in the language that distinguishes gender in the plural. I excitedly went around telling people about this until I realized that no one cared.


Recently I visited a display about the Maya at the Minnesota Science Museum that had all its posters in English and Spanish. I discovered a past subjunctive in one of the Spanish texts. That was exciting, but I had no one to be excited with.

The preceding paragraph is an example of a Pity Play.

Just the other day our choir learned a piece for Christmas with Spanish words. It had three lines in a row ending in a past subjunctive. (It is in rhyming triples and if you use all first conjugation verbs they rhyme.) Such excitement.


During the Cold War, I spent 18 months at İncirlik Air Base in Turkey. Turkish is a wonderful language for us geeks, very complicated yet most everything is regular. Like a computer language.

I didn’t know about computer languages during the Cold War, although they were just beginning to be used. I did work on a “computer” that you programmed by plugging cables into holes in various ways.

In Turkish, to modify a noun by a noun, you add an ending to the second noun. “İş Bankası” (no dot over the i) means “business bank”. (We would say “commercial bank”.) “İş” means “business” and “bank” by itself is “banka”. Do you think this is a lovably odd pattern? Well I do. But that’s the way I am.

A spate of spit

We live a couple blocks from Minnehaha Falls in Minneapolis. Last June the river flooded quite furiously and I went down to photograph it. I thought to my self, the river is in full spate. I wondered if the word “spate” came from the same IE root as the word “spit”. I got all excited and went home and looked it up. (No conclusion –it looks like it might be but there is no citation that proves it). Do you know anyone who gets excited about etymology?

Secret patterns in nature

All around us there are natural patterns that people don’t know about.

Cedars in Kentucky

For many years, we occasionally drove back and forth between Cleveland (where we lived) and Atlanta (where I had many relatives). We often stopped in Kentucky, where Jane grew up. It delighted me to drive by abandoned fields in Kentucky where cedars were colonizing. (They are “red cedars,” which are really junipers, but the name “cedar” is universal in the American midwest.)

What delighted me was that I knew a secret pattern: The presence of cedars means that the soil is over limestone. There is a large region including much of Kentucky and southern Indiana that lies over limestone underneath.

That gives me another secret: When you look closely at limestone blocks in a building in Bloomington, Indiana, you can see fossils. (It is better if the block is not polished, which unfortunately the University of Indiana buildings mostly are.) Not many people care about things like this.

The bump on Georgia

The first piece of pattern recognition that I remember was noticing that some states had “bumps”. This resulted in a confusing conversation with my mother. See Why Georgia has a bump.

Maybe soon I will write about why some states have panhandles, including the New England state that has a tiny panhandle that almost no one knows about.

Minnesota river

We live in Minneapolis now and occasionally drive over the Mendota Bridge, which crosses the Minnesota River. That river is medium sized, although it is a river, unlike Minnehaha Creek. But the Minnesota River Valley is a huge wide valley completely out of proportion with its river. This peculiarity hides a Secret Story that even many Minnesotans don’t know about.

The Minnesota River starts in western Minnesota and flows south and east until it runs into the Mississippi River. The source of the Red River is a few miles north of the source of the Minnesota. It flows north, becoming the boundary with North Dakota and going by Fargo and through Winnipeg and then flows into Lake Winnipeg. Thousands of years ago, all of the Red River was part of the Minnesota River and flowed south, bringing huge amounts of meltwater from the glaciers. That is what made the big valley. Eventually the glaciers receded far enough that the northern part of the river changed direction and started flowing north, leaving the Minnesota River a respectable medium sized river in a giant valley.

The Mendota Bridge is also one of the few places in the area where you can see the skyscrapers of Minneapolis and of St Paul simultaneously.


Baroque music

I love baroque music because of patterns such as fugues, which I understood, and the harmony it uses, which I still don’t understand. When I was 10 years old I had already detected its different harmony and asked my music teacher about it. She waved her hands and declaimed, “I don’t understand Bach.” (She was given to proclamations. Once she said, “I am never going out of the State of Georgia again because in Virginia they put mayonnaise on their hamburgers!”)

Some baroque music uses a ground bass, which floored me when I first heard it. I went on a rampage looking for records of chaconnes and passacaglias. Then I discovered early rock music (Beatles, Doors) and figured out that they sometimes used a ground bass too. That is one of the major attractions of rock music for me, along with its patterns of harmony.

Shape note music

Some shape note tunes (for example, Villulia), as well as some early rock music, has a funny hollow sound that sounds Asian to me. I delight in secretly knowing why: They use parallel fifths.

The Beatles have one song (I have forgotten which) that had a tune which in one place had three or four beats in a row that were sung on the same pitch — except once, when the (third I think) beat was raised a fourth. I fell in love with that and excitedly pointed it out to people. They looked at me funny. Later on, I found several shape note tunes that have that same pattern.


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The Greek alphabet in math

This is a revision of the portion of the article Alphabets in abstractmath.org that describes the use of the Greek alphabet by mathematicians.

Every letter of the Greek alphabet except omicron is used in math. All the other lowercase forms and all those uppercase forms that are not identical with the Latin alphabet are used.

  • Many Greek letters are used as proper names of mathe­ma­tical objects, for example $\pi$. Here, I provide some usages that might be known to undergraduate math majors.  Many other usages are given in MathWorld and in Wikipedia. In both those sources, each letter has an individual entry.
  • But any mathematician will feel free to use any Greek letter with a meaning different from common usage. This includes $\pi$, which for example is often used to denote a projection.
  • Greek letters are widely used in other sciences, but I have not attempted to cover those uses here.

The letters

  • English-speaking mathematicians pronounce these letters in various ways.  There is a substantial difference between the way American mathe­maticians pronounce them and the way they are pronounced by English-speaking mathe­maticians whose background is from British Commonwealth countries. (This is indicated below by (Br).)
  • Mathematicians speaking languages other than English may pronounce these letters differently. In particular, in modern Greek, most Greek letters are pro­nounced differ­ently from the way we pronounce them; β for example is pro­nounced vēta (last vowel as in "father").
  • Newcomers to abstract math often don’t know the names of some of the letters, or mispronounce them if they do.  I have heard young mathe­maticians pronounce $\phi $ and $\psi $ in exactly the same way, and since they were writing it on the board I doubt that anyone except language geeks like me noticed that they were doing it.  Another one pronounced $\phi $ as  “fee” and $\psi $ as “fie”.

Pronunciation key

  • ăt, āte, ɘgo (ago), bĕt, ēve, pĭt, rīde, cŏt, gō, ŭp, mūte.
  • Stress is indicated by an apostrophe after the stressed syllable, for example ū'nit, ɘgō'.
  • The pronunciations given below are what mathematicians usually use. In some cases this includes pronunciations not found in dictionaries.


Alpha: $\text{A},\, \alpha$: ă'lfɘ. Used occasionally as a variable, for example for angles or ordinals. Should be kept distinct from the proportionality sign "∝".


Beta: $\text{B},\, \beta $: bā'tɘ or (Br) bē'tɘ. The Euler Beta function is a function of two variables denoted by $B$. (The capital beta looks just like a "B" but they call it “beta” anyway.)  The Dirichlet beta function is a function of one variable denoted by $\beta$.


Gamma: $\Gamma, \,\gamma$: gă'mɘ. Used for the names of variables and functions. One familiar one is the $\Gamma$ function. Don’t refer to lower case "$\gamma$" as “r”, or snooty cognoscenti may ridicule you.

Delta: $\Delta \text{,}\,\,\delta$: dĕltɘ. The Dirac delta function and the Kronecker delta are denoted by $\delta $.  $\Delta x$ denotes the change or increment in x and $\Delta f$ denotes the Laplacian of a multivariable function. Lowercase $\delta$, along with $\epsilon$, is used as standard notation in the $\epsilon\text{-}\delta$ definition of limit.

Epsilon: $\text{E},\,\epsilon$ or $\varepsilon$: ĕp'sĭlɘn, ĕp'sĭlŏn, sometimes ĕpsī'lɘn. I am not aware of anyone using both lowercase forms $\epsilon$ and $\varepsilon$ to mean different things. The letter $\epsilon $ is frequently used informally to denoted a positive real number that is thought of as being small. The symbol ∈ for elementhood is not an epsilon, but many mathematicians use an epsilon for it anyway.

Zeta: $\text{Z},\zeta$: zā'tɘ or (Br) zē'tɘ. There are many functions called “zeta functions” and they are mostly related to each other. The Riemann hypothesis concerns the Riemann $\zeta $-function.

Eta: $\text{H},\,\eta$: ā'tɘ or (Br) ē'tɘ. Don't pronounce $\eta$ as "N" or you will reveal your newbieness.

Theta: $\Theta ,\,\theta$ or $\vartheta$: thā'tɘ or (Br) thē'tɘ.  The letter $\theta $ is commonly used to denote an angle. There is also a Jacobi $\theta $-function related to the Riemann $\zeta $-function. See also Wikipedia.

Iota: $\text{I},\,\iota$: īō'tɘ. Occurs occasionally in math and in some computer languages, but it is not common.

Kappa: $\text{K},\, \kappa $: kă'pɘ. Commonly used for curvature.

Lambda: $\Lambda,\,\lambda$: lăm'dɘ. An eigenvalue of a matrix is typically denoted $\lambda $.  The $\lambda $-calculus is a language for expressing abstract programs, and that has stimulated the use of $\lambda$ to define anonymous functions. (But mathematicians usually use barred arrow notation for anonymous functions.)

Mu: $\text{M},\,\mu$: mū.  Common uses: to denote the mean of a distribution or a set of numbers, a measure, and the Möbius function. Don’t call it “u”. 

Nu: $\text{N},\,\nu$: nū.    Used occasionally in pure math,more commonly in physics (frequency or a type of neutrino).   The lowercase $\nu$ looks confusingly like the lowercase upsilon, $\upsilon$. Don't call it "v".

Xi: $\Xi,\,\xi$: zī, sī or ksē. Both the upper and the lower case are used occasionally in mathe­matics. I recommend the ksee pronunciation since it is unambiguous.

Omicron: $\text{O, o}$: ŏ'mĭcrŏn.  Not used since it looks just like the Roman letter.

Pi: $\Pi \text{,}\,\pi$: pī.  The upper case $\Pi $ is used for an indexed product.  The lower case $\pi $ is used for the ratio of the circumference of a circle to its diameter, and also commonly to denote a projection function or the function that counts primes.  See default.

Rho: $\text{P},\,\rho$: rō. The lower case $\rho$ is used in spherical coordinate systems.  Do not call it pee.

Sigma: $\Sigma,\,\sigma$: sĭg'mɘ. The upper case $\Sigma $ is used for indexed sums.  The lower case $\sigma$ (don't call it "oh") is used for the standard deviation and also for the sum-of-divisors function.

Tau: $\text{T},\,\tau$ or τ: tăoo (rhymes with "cow"). The lowercase $\tau$ is used to indicate torsion, although the torsion tensor seems usually to be denoted by $T$. There are several other functions named $\tau$ as well.

Upsilon: $\Upsilon ,\,\upsilon$  ŭp'sĭlŏn. (Note: I have never heard anyone pronounce this letter, and various dictionaries suggest a ridiculous number of different pronunciations.) Rarely used in math; there are references in the Handbook.

Phi: $\Phi ,\,\phi$ or $\varphi$: fē or fī. Used for the totient function, for the “golden ratio” $\frac{1+\sqrt{5}}{2}$ (see default) and also commonly used to denote an angle.  Historically, $\phi$ is not the same as the notation $\varnothing$ for the empty set, but many mathematicians use it that way anyway, sometimes even calling the empty set “fee” or “fie”. 

Chi: $\text{X},\,\chi$: kī.  (Note that capital chi looks like $\text{X}$ and capital xi looks like $\Xi$.) Used for the ${{\chi }^{2}}$distribution in statistics, and for various math objects whose name start with “ch” (the usual transliteration of $\chi$) such as “characteristic” and “chromatic”.

Psi: $\Psi, \,\psi$; sē or sī. A few of us pronounce it as psē or psī to distinguish it from $\xi$.  $\psi$, like $\phi$, is often used to denote an angle.

Omega: $\Omega ,\,\omega$: ōmā'gɘ. $\Omega$ is often used as the name of a domain in $\mathbb{R}^n$. The set of natural numbers with the usual ordering is commonly denoted by $\omega$. Both forms have many other uses in advanced math.  

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Thinking about thought

Modules of the brain

Cognitive neuroscientists have taken the point of view that concepts, memories, words, and so on are represented in the brain by physical systems: perhaps they are individual neurons, or systems of structures, or even waves of discharges. In my previous writing I have referred to these as modules, and I will do that here. Each module is connected to many other modules that encode various properties of the concept, thoughts and memories that occur when you think of that concept (in other words stimulate the module), and so on.

How these modules implement the way we think and perceive the world is not well understood and forms a major research task of cognitive neuroscience. The fact that they are implemented in physical systems in the brain gives us a new way of thinking about thought and perception.


The grandmother module

There is a module in your brain representing the concept of grandmother. It is likely to be connected to other modules representing your actual grandmothers if you have any memory of them. These modules are connected to many others — memories (if you knew them), other relatives related to them, incidents in their lives that you were told about, and so on. Even if you don’t have any memory of them, you have a module representing the fact that you don’t have any memory of them, and maybe modules explaining why you don’t.

Each different aspect related to “grandmother” belongs to a separate module somehow connected to the grandmother module. That may be hard to believe, but the human brain has over eighty billion neurons.

A particular module connected with math

There is a module in your brain connected with the number $42$. That module has many connections to things you know about it, such as its factorization, the fact that it is an integer, and so on. The module may also have connections to a module concerning the attitude that $42$ is the Answer. If it does, that module may have a connection with the module representing Douglas Adams. He was physically outside your body, but is the number $42$ outside your body?

That has a decidedly complicated answer. The number $42$ exists in a network of brains which communicate with each other and share some ideas about properties of $42$. So it exists socially. This social existence occasionally changes your knowledge of the properties of $42$ and in particular may make you realize that you were wrong about some of its aspects. (Perhaps you once thought it was $7\times 8$.)

This example suggests how I have been using the module idea to explain how we think about math.

A new metaphor for understanding thinking

I am proposing to use the idea of module as a metaphor for thinking about thinking. I believe that it clarifies a lot of the confusion people have about the relation between thinking and the real world. In particular it clarifies why we think of mathematical objects as if they were real-world objects (see Modules and math below.)

I am explicitly proposing this metaphor as a successor to previous metaphors drawn from science to explain things. For example when machines became useful in the 18th century many naturalists used metaphors such as the Universe is a Machine or the Body is a Machine as a way of understanding the world. In the 20th century we fell heavily for the metaphor that the Mind Is A Computer (or Program). Both the 18th century and the 20th century metaphors (in my opinion) improved our understanding of things, even though they both fell short in many ways.

In no way am I claiming that the ways of thinking I am pushing have anything but a rough resemblance to current neuroscientists’ thinking. Even so, further discoveries in neuroscience may give us even more insight into thinking that they do now. Unless at some point something goes awry and we have to, ahem, think differently again.

For thousands of years, new scientific theories have been giving us new metaphors for thinking about life, the universe and everything. I am saying here is a new apple on the tree of knowledge; let’s eat it.

The rest of this post elaborates my proposed metaphor. Like any metaphor, it gets some things right and some wrong, and my explanations of how it works are no doubt full of errors and dubious ideas. Nevertheless, I think it is worth thinking about thought using these ideas with the usual correction process that happens in society with new metaphors.

Our theory of the world

We don’t have any direct perception of the “real world”; we have only the sensations we get from those parts of our body which sense things in the world. These sensations are organized by our brain into a theory of the world.

  • The theory of the world says that the world is “out there” and that our sensory units give us information about it. We are directly aware of our experiences because they are a function of our brain. That the experiences (many of them) originate from outside our body is a very plausible theory generated by our brain on the bases of these experience.
  • The theory is generated by our brain in a way that we cannot observe and is out of our control (mostly). We see a table and we know we can see in in daytime but not when it is dark and we can bump into it, which causes experiences to occur via our touch and sound facilities. But the concept of “table” and the fact that we decide something is or is not a table takes place in our brain, not “out there”.
  • We do make some conscious amendments to the theory. For example, we “know” the sky is not a blue shell around our world, although it looks like it. That we think of the apparent blue surface as an artifact of our vision processing comes about through conscious reasoning. But most of how we understand the world comes about subconsciously.
  • Our brain (and the rest of our body) does an enormous amount of processing to create the view of the world that we have. Visual perception requires a huge amount of processing in our brain and the other sensory methods we use also undergo a lot of processing, but not as much as vision.
  • The theory of the world organizes a lot of what we experience as interaction with physical objects. We perceive physical objects as having properties such as persistence, changing with time, and so on. Our brains create the concept of physical object and the properties of persistence, changing, and particular properties an individual object might have.
  • We think of the Mississippi River as an object that is many years old even though none of its current molecules are the same as were in the river a decade ago. How is it one thing when its substance is constantly changing? This is a famous and ancient conundrum which becomes a non-problem if you realize that the “object” is created inside your brain and imposed by your thinking on your understanding of the world.
  • The notion that semantics is a connection between our brain and the outside world has also become a philosophical conundrum that vanishes if we understand that the connection with the outside world exists entirely inside our theory, which is entirely within our brain.


Our brain also has a theory of society We are immersed in a world of people, that we have close connections with some of them and more distant connections with many other via speech, stories, reading and various kinds of long-distance communications.

  • We associate with individual people, in our family and with our friend. The communication is not just through speech: it involves vision heavily (seeing what The Other is thinking) and probably through pheromones, among other channels. For one perspective on vision, see The vision revolution, by Mark Changizi. (Review)
  • We consciously and unconsciously absorb ideas and attitudes (cultural and otherwise) from the people around us, especially including the adults and children we grow up with. In this way we are heavily embedded in the social world, which creates our point of view and attitudes by our observation and experience and presumably via memes. An example is the widespread recent changes in attitudes in the USA concerning gay marriage.
  • The theory of society seems to me to be a mechanism in our brain that is separate from our theory of the physical world, but which interacts with it. But it may be that it is better to regard the two theories as modules in one big theory.

Modules and math

The module associated with a math object is connected to many other modules, some of which have nothing to do with math.

  • For example, they may have have connections to our sensory organs. We may get a physical feeling that the parabola $y=x^2$ is going “up” as $x$ “moves to the right”. The mirror neurons in our brain that “feel” this are connected to our “parabola $y=x^2$” module. (See Constructivism and Platonism and the posts it links to.)
  • I tend to think of math objects as “things”. Every time I investigate the number $111$, it turns out to be $3\times37$. Every time I investigate the alternating group on $6$ letters it is simple. If I prove a new theorem it feels as if I have discovered the theorem. So math objects are out there and persistent.
  • If some math calculation does not give the same answer the second time I frequently find that I made a mistake. So math facts are consistent.
  • There is presumably a module that recognizes that something is “out there” when I have repeatable and consistent experiences with it. The feeling originates in a brain arranged to detect consistent behavior. The feeling is not evidence that math objects exist in some ideal space. In this way, my proposed new way of thinking about thought abolishes all the problems with Platonism.
  • If I think of two groups that are isomorphic (for example the cyclic group of order $3$ and the alternating group of rank $3$), I picture them as in two different places with a connection between the two isomorphic ones. This phenomenon is presumably connected with modules that respond to seeing physical objects and carrying with them a sense of where they are (two different places). This is a strategy my brain uses to think about objects without having to name them, using the mechanism already built in to think about two things in different places.


Many of the ideas in this post come from my previous writing, listed in the references. This post was also inspired by ideas from Chomsky, Jackendoff (particularly Chapter 9), the Scientific American article Brain cells for Grandmother by Quian Quiroga, Fried and Koch, and the papers by Ernest and Hersh.


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Algebra is a difficult foreign language

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 a previous post, I said that the symbolic language of mathematics is difficult to learn and that we don't teach it well. (The symbolic language includes as a subset the notation used in high school algebra, precalculus, and calculus.) I gave some examples in that post but now I want to go into more detail.  This discussion is an incomplete sketch of some aspects of the syntax of the symbolic language.  I will write one or more posts about the semantics later.

The languages of math

First, let's distinguish between mathematical English and the symbolic language of math. 

  • Mathematical English is a special register or jargon of English. It has not only its special vocabulary, like any jargon, but also used ordinary English words such as "If…then", "definition" and "let" in special ways. 
  • The symbolic language of math is a distinct, special-purpose written language which is not a dialect of the English language and can in fact be read by mathematicians with little knowledge of English.
    • It has its own symbols and rules that are quite different from spoken languages. 
    • Simple expressions can be pronounced, but complicated expressions may only be pointed to or referred to.
  • A mathematical article or book is typically written using mathematical English interspersed with expressions in the symbolic language of math.

Symbolic expressions

A symbolic noun (logicians call it a term) is an expression in the symbolic language that names a number or other mathematical object, and may carry other information as well.

  • "3" is a noun denoting the number 3.
  • "$\text{Sym}_3$" is a noun denoting the symmetric group of order 3.
  • "$2+1$" is a noun denoting the number 3.  But it contains more information than that: it describes a way of calculating 3 as a sum.
  • "$\sin^2\frac{\pi}{4}$" is a noun denoting the number $\frac{1}{2}$, and it also describes a computation that yields the number $\frac{1}{2}$.  If you understand the symbolic language and know that $\sin$ is a numerical function, you can recognize "$\sin^2\frac{\pi}{4}$" as a symbolic noun representing a number even if you don't know how to calculate it.
  • "$2+1$" and "$\sin^2\frac{\pi}{4}$" are said to be encapsulated computations.
    • The word "encapsulated" refers to the fact that to understand what the expressions mean, you must think of the computation not as a process but as an object.
    • Note that a computer program is also an object, not a process.
  • "$a+1$" and "$\sin^2\frac{\pi x}{4}$" are encapsulated computations containing variables that represent numbers. In these cases you can calculate the value of these computations if you give values to the variables.  

symbolic statement is a symbolic expression that represents a statement that is either true or false or free, meaning that it contains variables and is true or false depending on the values assigned to the variables.

  • $\pi\gt0$ is a symbolic assertion that is true.
  • $\pi\lt0$ is a symbolic assertion that it is false.  The fact that it is false does not stop it from being a symbolic assertion.
  • $x^2-5x+4\gt0$ is an assertion that is true for $x=5$ and false for $x=1$.
  • $x^2-5x+4=0$ is an assertion that is true for $x=1$ and $x=4$ and false for all other numbers $x$.
  • $x^2+2x+1=(x+1)^2$ is an assertion that is true for all numbers $x$. 

Properties of the symbolic language

The constituents of a symbolic expression are symbols for numbers, variables and other mathematical objects. In a particular expression, the symbols are arranged according to conventions that must be understood by the reader. These conventions form the syntax or grammar of symbolic expressions. 

The symbolic language has been invented piecemeal by mathematicians over the past several centuries. It is thus a natural language and like all natural languages it has irregularities and often results in ambiguous expressions. It is therefore difficult to learn and requires much practice to learn to use it well. Students learn the grammar in school and are often expected to understand it by osmosis instead of by being taught specifically.  However, it is not as difficult to learn well as a foreign language is.

In the basic symbolic language, expressions are written as strings of symbols.

  • The symbolic language gives (sometimes ambiguous) meaning to symbols placed above or below the line of symbols, so the strings are in some sense more than one dimensional but less than two-dimensional.
  • Integral notation, limit notation, and others, are two-dimensional enough to have two or three levels of symbols. 
  • Matrices are fully two-dimensional symbols, and so are commutative diagrams.
  • I will not consider graphs (in both senses) and geometric drawings in this post because I am not sure what I want to write about them.

Syntax of the language

One of the basic methods of the symbolic language is the use of constructors.  These can usually be analyzed as functions or operators, but I am thinking of "constructor" as a linguistic device for producing an expression denoting a mathematical object or assertion. Ordinary languages have constructors, too; for example "-ness" makes a noun out of a verb ("good" to "goodness") and "and" forms a grouping ("men and women").

Special symbols

The language uses special symbols both as names of specific objects and as constructors.

  • The digits "0", "1", "2" are named by special symbols.  So are some other objects: "$\emptyset$", "$\infty$".
  • Certain verbs are represented by special symbols: "$=$", "$\lt$", "$\in$", "$\subseteq$".
  • Some constructors are infixes: "$2+3$" denotes the sum of 2 and 3 and "$2-3$" denotes the difference between them.
  • Others are placed before, after, above or even below the name of an object.  Examples: $a'$, which can mean the derivative of $a$ or the name of another variable; $n!$ denotes $n$ factorial; $a^\star$ is the dual of $a$ in some contexts; $\vec{v}$ constructs a vector whose name is "$v$".
  • Letters from other alphabets may be used as names of objects, either defined in the context of a particular article, or with more nearly global meaning such as "$\pi$" (but "$\pi$" can denote a projection, too).

This is a lot of stuff for students to learn. Each symbol has its own rules of use (where you put it, which sort of expression you may it with, etc.)  And the meaning is often determined by context. For example $\pi x$ usually means $\pi$ multiplied by $x$, but in some books it can mean the function $\pi$ evaluated at $x$. (But this is a remark about semantics — more in another post.)

"Systematic" notation

  • The form "$f(x)$" is systematically used to denote the value of a function $f$ at the input $x$.  But this usage has variations that confuse beginning students:
    • "$\sin\,x$" is more common than "$\sin(x)$".
    • When the function has just been named as a letter, "$f(x)$" is more common that "$fx$" but many authors do use the latter.
  • Raising a symbol after another symbol commonly denotes exponentiation: "$x^2$" denotes $x$ times $x$.  But it is used in a different meaning in the case of tensors (and elsewhere).
  • Lowering a symbol after another symbol, as in "$x_i$"  may denote an item in a sequence.  But "$f_x$" is more likely to denote a partial derivative.
  • The integral notation is quite complicated.  The expression \[\int_a^b f(x)\,dx\] has three parameters, $a$, $b$ and $f$, and a bound variable $x$ that specifies the variable used in the formula for $f$.  Students gradually learn the significance of these facts as they work with integrals. 


Variables have deep problems concerned with their meaning (semantics). But substitution for variables causes syntactic problems that students have difficulty with as well.

  • Substituting $4$ for $x$ in the expression $3+x$ results in $3+4$. 
  • Substituting $4$ for $x$ in the expression $3x$ results in $12$, not $34$. 
  • Substituting "$y+z$" in the expression $3x$ results in $3(y+z)$, not $3y+z$.  Some of my calculus students in preforming this substitution would write $3\,\,y+z$, using a space to separate.  The rules don't allow that, but I think it is a perfectly natural mistake. 

Using expressions and writing about them

  • If I write "If $x$ is an odd integer, then $3+x$ is odd", then I am using $3+x$ in a sentence. It is a noun denoting an unspecified number which can be constructed in a specified way.
  • When I mention substituting $4$ for $x$ in "$3+x$", I am talking about the expression $3+x$.  I am not writing about a number, I am writing about a string of symbols.  This distinction causes students major difficulties and teacher hardly ever talk about it.
  • In the section on variables, I wrote "the expression $3+x$", which shows more explicitly that I am talking about it as an expression.
    • Note that quotes in novels don't mean you are talking about the expression inside the quotes, it means you are describing the act of a person saying something.
  • It is very common to write something like, "If I substitute $4$ for $x$ in $3x$ I get $3 \times 4=12$".  This is called a parenthetic assertion, and it is literally nonsense (it says I get an equation).
  • If I pronounce the sentence "We know that $x\gt0$" we pronounce "$x\gt0$" as "$x$ is greater than zero",  If I pronounce the sentence "For any $x\gt0$ there is $y\gt0$ for which $x\gt y$", then I pronounce the expression "$x\gt0$" as "$x$ greater than zero$",  This is an example of context-sensitive pronunciation
  • There is a lot more about parenthetic assertions and context-sensitive pronunciation in More about the languages of math.


I have described some aspects of the syntax of the symbolic language of math. Learning that syntax is difficult and requires a lot of practice. Students who manage to learn the syntax and semantics can go on to learn further math, but students who don't are forever blocked from many rewarding careers. I heard someone say at the MathFest in Madison that about 25% of all high school students never really understand algebra.  I have only taught college students, but some students (maybe 5%) who get into freshman calculus in college are weak enough in algebra that they cannot continue. 

I am not proposing that all aspects of the syntax (or semantics) be taught explicitly.  A lot must be learned by doing algebra, where they pick up the syntax subconsciously just as they pick up lots of other behavior-information in and out of school. But teachers should explicitly understand the structure of algebra at least in some basic way so that they can be aware of the source of many of the students' problems. 

It is likely that the widespread use of computers will allow some parts of the symbolic language of math to be replaced by other methods such as using Excel or some visual manipulation of operations as suggested in my post Mathematical and linguistic ability.  It is also likely that the symbolic language will gradually be improved to get rid of ambiguities and irregularities.  But a deliberate top-down effort to simplify notation will not succeed. Such things rarely succeed.




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Metaphors in computing science 2

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

Metaphor: A program is a list of instructions.

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

Metaphor: A programming language is a language.

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

Metaphor: A computer program is a list of statements

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

Metaphor: A program in a computer language has meanings.

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

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

A program is intended to be executed by a computer.

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


Conceptual blend (Wikipedia)

Conceptual metaphors (Wikipedia)

Images and Metaphors (article in abstractmath)

Semantics in computer science (Wikipedia)

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I was recently asked about the etymology of the English word “retire”(in connection with quitting work).  It comes from Old French “retirer”, compounded from “re” (meaning “back”, a prefix used in Latin) and the Old French verb “tirer” meaning something like “pull” (which comes from a Germanic language, not Latin, and is related to “tier”, but not apparently to “tire”).

Its earliest citations in the Oxford English Dictionary show meanings such as

  • Pull back or retreat from the enemy.
  • To move back for safety or storage (“they retired to their houses”).
  • Leave office or work permanently.

All these meanings appear in print in the 16th century.

What good does it do to know this?  Not much.  You can’t explain the modern meaning of a word knowing the meaning of its ancient roots.

In the case of “retire”, I can make up a story of meanings changing using a chain of metaphors.

  1. “Retirer” in French meant literally “pull back” in the physical sense, for example pulling on a dog’s leash to drag it back so it won’t get into a fight with another dog. This literal meaning has not survived in the English word “retire” (nor, I think, in the French word “retirer”).
  2. In the 12th century (sez the OED without citation) the French word was used to refer to an army pulling back from a battle.  This is clearly a metaphor based on the literal meaning.  In a phrase such as “The Army retired from battle” it has become intransitive, but perhaps people once said things like “The General retired the Army from battle”.  Note that in modern English we could use the exact same metaphor with “pull back”: “The General pulled the Army back from battle”, although “withdrew” would be more common.
  3. Now someone comes along and uses the metaphor “going to work is like being in a battle”, and says things like “He retired from his job”.   This happened in English before 1533 and the usage has survived to this day.  It is probably the commonest meaning of the word “retire” now.

Now all that is a story I made up.  It is plausible, but it might have happened in a different way.  It is not at all likely we will discover the workings of metaphors in the minds of people who lived 600 years ago.  (Conceivably someone could have written down their thoughts about the word “retire” and it will be discovered in an odd subcrypt of Durham Cathedral and some linguist would get very excited, but I could win the lottery, too).

That’s why knowing the original literal meaning of the roots of a modern English word really means nothing about the modern meaning.  There could have been many steps along the way where a metaphorical usage became the standard meaning, then someone took the standard meaning and used it in another metaphor, maybe many times.  And metaphors aren’t the only method.  Words can change meaning because of misunderstanding, specialization, generalization, use in secret languages that become public, and so on.

I didn’t include etymology in the Handbook, mainly for this reason.  But there are certain mathematical words where knowing the metaphor or even the literal meaning can be of help.  I’ll write about that in a separate article.


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