Category Archives: representations

Every post that talks about representation of mathematical objects in the most general sense.

astounding math, math, representations, understanding math

Naming real numbers

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I am considering an Astounding Math story about how you can’t name an arbitrary real number. In this blog I will describe some of the math technicalities, teaching problems and writing problems that arise in writing the story. It would be great if other math popularizers would blog about the problems they faced in their writing and what decisions they made about them.

Consider first that we can name every rational number. You can describe rational numbers in terms of equivalence classes of expressions of the form “m/n” where m and n are names of integers, n not zero. Then any expression of the form “m/n” names a specific rational number and every rational number can be named in this way. Yes, the naming can be made unique by using expressions in lowest terms, but that is not the point here, which is that in theory you really can name every rational number.

Every real number has a decimal expansion. The expansion is nearly unique: some rational numbers have two different decimal expansions, but that is all (see (2) below). We may define a decimal expansion precisely using the “regular” expression [-]?[0..9]*[.][0..9]^(infinity). This means:

One minus sign or nothing
followed by
A string of any finite length of decimal digits
followed by
A decimal point
followed by
An infinite string of decimal digits.

This is not really a regular expression since regex’s don’t allow you to specify an infinite string. But it is a precise definition of decimal expansion, and every decimal expansion refers to a specific real number. You can then define the real numbers as equivalence classes of regular expressions of decimal digits, with each equivalence class containing one or two members.

The Astounding thing is that as a result of this construction

a) You can describe precisely the set of real numbers.

b) Each real number has a description as an infinitely long decimal expansion.
c) You cannot give a name to every real number. That’s because the description is an infinite sequence and you cannot give every infinite sequence even in theory (see (3).)

d) So when mathematicians deal with real numbers, they are dealing with things that in most cases they cannot refer to.

Complications

My purpose in writing Astounding Math Stories is to get people who are already somewhat familiar with math to have their consciousness raised about all the fascinating things that go on in math. This requires a delicate balancing act when I write them.

1) I get comments from readers like this one: “That is not astounding. I already knew about it.” This is probably inevitable. In the case of the names of real numbers, however, I’ll bet there are practicing mathematicians who understand item (d) implicitly but have never heard it said out loud.

2) You have to say precisely which two infinite sequences are in the same class. When I was teaching discrete math in the eighties and nineties, I realized that I had never seen this written out explicitly. Every description depended heavily on pattern recognition, as in the description “I am referring to the phenomenon that for example 0.9999… denotes the same number as 1.0000…” (See remark (6).) I included a nearly explicit description in my discrete math class notes (page 12).

Perhaps this problem should be slurred over. Really every real number has one decimal expansion. That thing with the 9’s is just a technicality. (This makes me a heretic. Mathematicians don’t usually say things like that.)

3) You cannot give the name of every real number because the set of linguistic expressions is countable and the set of infinitely long decimal expansions is uncountable. Do I just quote this fact? Do I write another Astounding Math Story about infinite cardinality? Probably.

Some would object that you can’t give the name of every rational number either. But there is a name (a finitely long linguistic expression) for every rational number. You can’t in physical fact “give” the humongous ones but that is a practical problem. In contrast, most real numbers have no linguistic expression naming them.

4) I need to keep the demands on the reader as low as is reasonable, but not lower. A minor example in this case is that I express everything in terms of decimal expansion instead of binary expansion or Cauchy sequences (more abstract) or Dedekind cuts (even more abstract). In theory, binary expansions are not any more abstract than decimal expansions and require less data, but in fact most of the people I am trying to attract are less familiar with binary than decimal, and that drags on their understanding.

5) When I say “You can describe a real number as an infinitely long decimal expansion” you run into the ubiquitous difficulties math-newbies have with infinite sequences. Namely, they think of then as progressing through time, so you never get to “the end”. In fact, experienced mathematicians think of an infinite sequence as existing all at once: every entry is there now.

Students complain that they can’t “visualize” the entries all at once, but that is not the point: You are not suppose to visualize the whole sequence at once, you are suppose to think and talk about the entries as if they are all there. That is, assent to the concept that the whole thing is “there”, which is not the same thing as visualizing it. (I also wrote about this phenomenon in abstractmath and in a previous blog.)

So when I write about the infinitely long decimal expansions I know many readers’ understanding will falter right there; they will not be able to take in the rest of what I write. What do I do about this? Well, I suppose I could include the last paragraph!

Note: This discussion is not about what infinite sequences “really are”, but about how you think about them. This way of thinking about them has been around for a couple of centuries and have produced many useful theorems and no known contradictions. Philosophers may have a problem with this point of view, but mathematicians don’t.

6) Studies show that most math students do not believe that 0.999… is the same number as 1.000… Some mathematicians I know say things like: “Why are you writing for people like that? They are too stupid to understand anything about abstract math.” But it is time mathematicians stopped insisting that there is no point in getting people who are not especially talented in math interested in math, or trying to explain anything to them. In fact, it appears to me that this elitist attitude is in the process of dying out. It had better be dying out.

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exposition, language, language of math, math, representations

Renaming technical concepts

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Here are some thoughts about the names of mathematical objects. I don’t make recommendations about how to name things; I am just analyzing some aspects of how names are given and used. I have written about some of these ideas in abstractmath.org, under Names and Semantic Contamination.

Some objects have names from Latin or Greek, such as “matrix” or “homomorphism”, that don’t give the reader a clue as to what they mean, unless the reader has a substantial vocabulary of Latin and Greek roots.

Some are named after people, such as “Riemann sum” and “Hausdorff space”. They don’t have suggestive names either. Well, they suggest that the person they are named after discovered it, but that is not always true; for example L’Hôpital’s rule was discovered by some Bernoulli or other.

You could call both types of names learnèd names.

Others concepts have names that are English words, such as “slope” or “group”. I will call them commonword names. Some of these suggest some aspect of their meaning; “slope” certainly does and so do “truth set” and “variable”. But “group” only suggests that it is a bunch of things; it does not suggest the primary group datum, namely the binary operation. Not only that, but too many commonword names suggest the wrong ideas, for example “real” and “imaginary”.

In contrast, learnèd names don’t usually suggest the wrong things, but they can and do intimidate people.

One upon a time, Roger Godement and Peter J. Huber came up with an important construction for adjunctions in category theory. They called it the standard construction. That commonword name communicates very little. They named it that because it kept coming up in their work. Well, derivatives and integrals are each more deserving of the name. Eilenberg and Moore renamed them triples, which suggests nothing useful except that the concept is given by three data. Well, so are rings. Saunders Mac Lane renamed them again, calling them monads, a learnèd name that suggests nothing except possibly an illusory connection with a certain philosophical concept.

Perhaps learnèd names are better, since they don’t suggest the wrong things. In that case “monad” is better than the other names, but I have a personal prejudice since I have co-authored two books that called them “triples”.

Some writers of popularizations of math and science avoid using the names of certain concepts that suggest the wrong things. In Symmetry and the Monster, by Mark Ronan, the author talks about “atoms of symmetry” instead of “simple groups”, on the grounds that “simple group” is misleading (the Monster Group is simple!) and doesn’t suggest the important property they have. He called involutions “mirror symmetries”, which is appopriately suggestive. Centralizers of involutions became “cross-sections”, which I don’t understand; it must be based on a way of thinking about them that I am not aware of. He doesn’t change the name of the Monster Group, though; that is a terrific name.

Frank Wilczek, in The Lightness of Being, used “core theory” for the theory in particle physics that is commonly called the “standard model”. I suppose that really is more suggestive of its current place in physics, since as far as I know all modern theories build on it.

Marcus du Sautoy, in The Music of the Primes (HarperCollins, 2003), also introduces new names for concepts. His description of the meanings of the many concepts he discusses uses some great metaphors that clearly communicate the ideas. He talks about the “landscape” of the zeta function, how Riemann “extended the landscape to the west”, and refers to its zeroes as its places “at sea level”. But he also calls them by their normal mathematical name “zeroes”. (I could have done without his reference to the “ley line of zeroes”.) He refers to modular arithmetic as “clock calculators” and in one parenthetical remark explains that modular arithmetic is what he means.

Summary

The problem with learnèd names is that they don’t give you a clue about the meaning, and for some students (co-intimidators) they induce anxiety.

The problems with commonword naming are that what a commonword name suggests can give you only one connotation and it is hard to find the best one, and almost any choice produces a metaphor that suggests some incorrect ideas. Furthermore, beginning abstract math students are way too likely to be stuck on one metaphor per mathematical object and commonword names only encourage this behavior. I have written about that here and here.

One problem with popular renaming is that the interested reader has a hard time searching the internet for more information about it, unless she noticed that one place in the book where the fact that it was not the standard name was mentioned.

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language, language of math, math, representations

Default meanings

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Concerning Michael Barr’s comments on my post on terminology, I remember going to a meeting of topologists in 1965 or 1966 in which people kept spouting nonsense about free groups. The reason it was nonsense was that they were talking about free abelian groups without saying so. That may have been the first time I became aware of default meaning in different groups of mathematicians.

I became aware of default meanings in ethnic and regional groups long before that, when I joined the Air Force after never having been outside the deep south and discovered that other people thought “sweet milk” and “ink pen” were weird things to say.

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math, representations, understanding math

Representations IV

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Mark Meckes recently commented on my post on writing Astounding Math Stories that students “usually think of decimal expansions as formal expressions”. His point is well taken, but I would go further and say that they think of real numbers as decimal expansions, hence as formal expressions.

For example, 1/3 is approximately 0.333… However, 1/3 is exactly 1/3. The expression “1/3” is more exact than the decimal expansion. Similarly \sqrt 2 is defined exactly as the positive real number whose square is 2. And of course there are still other representations of some or all real numbers, for example binary notation, representations as limits, as solutions of equations, and so on.

The main thing to understand is that every interesting mathematical object has several representations, each representation coming from a different system of metaphors. And if you are going to understand math you have to be aware of various representations of the same object and hold (some of) the details of several of them in your head at once. Even “2 + 3 = 5″ is talking about two different representations of a number simultaneously. William Thurston once said that it was a revelation to him as a child that when you divide 127 by 23 you get 127/23. That notation”127/23” tells you two things: exactly what the number is and one way it is related to other numbers. That kind of phenomenon is what makes math work.

I went on and on about this stuff in abstractmath.org here and here.

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math, proof, representations, understanding math

Representations III: Rigor and Rigor Mortis

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In a recent blog post , I talked about the particular mental representation (“dry bones”) of math that we use when we are being “rigorous” – we think of mathematical objects as inert, not changing and affecting nothing. There is a reason why we use this representation, and I didn’t say anything about that.

Rigor requires that we use classical logical reasoning: The logical connectives, implication in particular, are defined by truth tables. They have no temporal or causal connotations. That is not like everyday reasoning about things that affect each other and change over time. (See Note 1).

Example: “A smooth function that is increasing at $x = a$ and decreasing at $x = b$ has to turn around at some point $m$ between $a$ and $b$. Being smooth, its derivative must be $0$ at $m$ and its second derivative must be negative near m since the slope changes from positive to negative, so m must occur at a maximum”. This is a convincing intuitive argument that depends on our understanding of smooth functions, but it would not be called “rigorous” by many of us. If someone demands a complete rigorous proof we probably start arguing with epsilons and deltas, and our arguments will be about the function and its values and derivatives as static objects, each thought of as an unchanging whole mathematical object just sitting there for our inspection. That is the dry-bones representation.

In other words, we use the dry bones representation to make classical first order logic correct, in the sense that classical reasoning about the statements we make become sound, as they are obviously not in everyday reasoning.

This point may have implications for mathematical education at the level where we teach proofs. Perhaps we should be open with students about images and metaphors, about how they suggest applications and suggest what may be true, but they have to “go dead” when we set out to prove something rigorously. We have been doing exactly that at the blackboard in front of our students, but we rarely point it out explicitly. It is not automatically the case that this explicit approach will turn out to help very many students, but it is worth investigating. (See Note 2).

It may also have implications for the philosophy of math.

Note 1: The statement “If you eat all your dinner you can have dessert” does not fit the truth table for classical (material) implication in ordinary discourse, where it means: “You can’t have dessert until you eat your dinner”. Not only is there a temporal element here, but there is a causal element which makes the statement false if the hypothesis and conclusion are both false. Some philosophers say that implication in English has classical implication as its primary meaning, but idiomatic usage modifies it according to context. I find that hard to believe. I don’t believe any translation is going on in your head when you hear that sentence: you get its nonclassical meaning immediately and directly with no thought of the classical vacuous-implication idea.

Note 2: I used to think that being explicit about the semiotic aspects of various situations that take place in the classroom could only help students, but in fact it appears to scare some of them. “I can’t listen to what you say AND keep in mind the subject matter AND keep in mind rules about the differences in syntax and semantics in mathematical discourse AND keep in mind that the impersonality of the discourse may trigger alienation in my soul AND…” This needs investigation.

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math, proof, representations, understanding math

Representations II: Dry Bones

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In my abstract math website here I wrote about “two levels of images and metaphors” in math, the rich and the rigorous. There are several things wrong with that presentation and I intend to rewrite it. This post is a first attempt to get things straight.

When we are trying to understand or explain math, we may use various kinds of images and metaphors about the subject matter to construct a colorful and rich representation of the mathematical objects and processes involved. I described some of these briefly in the previous post on representations.

When we set out to prove some math statement, we go into what I called “rigorous mode”. We feel that we have to forget about all the color and excitement of the rich view. We must think of math objects as totally inert and static. The don’t move or change over time and they don’t interact with other objects or the real world. In other words, pretend that all math objects are dead.

We don’t always go all the way into this rigorous mode, but if we use an image or metaphor in a proof and someone challenges us about it, we may rewrite that part to get rid of the colorful representation and replace it by a calculation or line of reasoning that refers to the math objects as if they were inert and static – dead.

I now think that “rigorous mode” is a misleading description. The description of math objects as inert and static is just another representation. We need a name for this representation; I thought about using “the dead representation” and “the leached out representation” (the name comes from a remark by Steven Pinker), but my working name in this post is the dry bones representation (from the book of Ezekiel).

Well, there is a sense in which the dry bones representation is not just another representation. It is unusual because it is a representation of every mathematical object. Most representations, images, metaphors, models of math objects apply only to some objects. You can say that the function $y = 25 – t^2$ “rises and then falls” but you can’t say the monster group rises and falls. The dry bones representation applies to all objects. Its representation of that function, or of the monster group, is that it is one object, all there all at once, not changing, not affecting anything, a kind of

dead totality.

When we do math, we hold several representations of what we are working with in our heads all at once. When writing about them we use metaphors in passing, perhaps implicitly. We use symbolic representations embedded in the prose as well as graphs and other visual representations, fluently and usually without much explicit notice. One of those representations is the dry bones representation. It is specially associated with rigorous reasoning, but other representations occur in mathematical reasoning as well. To call it a “mode” is to suggest that it is the only thing happening, and that is not always true. In fact I suspect that it the dry bones representation is rarely the only representation around, but that would require lexicographical work on a mathematical corpus (another kind of dead body!).

I expect to rewrite the chapter on images and metaphors to capture these ideas, as well as to give it more prominence instead of being buried in the middle of a discussion of the general idea of images and metaphors.

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Technical words in English

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English is unusual among major languages in the number of technical words borrowed from other languages instead of being made up from native roots. We have some, listed under suggestive names. But how can you tell from looking at them what “parabola” or “homomorphism” mean?

The English word “carnivore” (from Latin roots) can be translated as “Fleischfresser” in German; to a German speaker, that word means literally “meat eater”. So a question such as “What does a carnivore eat” translates into something like, “What does a meat-eater eat?” (And do they do it in Grant’s tomb?) Similarly the word for “plane” (ebene) looks like “flat”.

Chinese is another language that forms words in that way: see the discussion of “diagonal” in Julia Lan Dai’s blog. (I stole the carnivore example from her blog, too.)

The result is that many technical words in English do not suggest their meaning at all to a reader not familiar with the subject. Of course, in the case of “carnivore” if you know Latin, French or Spanish you are likely to guess the meaning, but it is nevertheless true that English has a kind of elitist stratum of technical words that provide little or no clue to their meaning. German has a much smaller elitist stratum of words. I don’t know about Chinese.

This is a problem in all technical fields, not just in math.

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language, language of math, math, representations

Set notation

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Students commonly think that the notation “{Ø}” denotes the empty set. Many secondary school teachers think this, too.

Mistakes in reading math notation occur because the reader’s understanding of the notation system is different from the author’s. The most common bits of the symbolic language of math have fairly standard interpretations that most mathematicians agree on most of the time. Students develop their own non-standard interpretation for many reasons, including especially cognitive dissonance from ordinary usage and ambiguous statements by teachers.

I believe (from teaching experience) that when a student sees “{1, 2, 3, 5}” they think, “That is the set 1, 2, 3 and 5”. The (incorrect) rule they follow is that the curly braces mean that what is inside them is a set. So clearly “{Ø}” is the empty set because the symbol for the empty set is inside the braces.

However, “1, 2, 3 and 5” is not a set, it is the names of four integers. A set is not its elements. It is a single mathematical object that is different from its elements but determined exactly by what its elements are. The correct understanding of set notation is that what is inside the braces is an expression that tells you what the elements of the set are. This expression may be a list, as in “{1, 2, 3, 5}”, or it may be a statement in setbuilder format, as in “{x x > 1}”. According to this rule, “{Ø}” denotes the singleton set whose only element is the empty set.

This posting is based on the belief that that mathematical notation has a standard, (mostly) agreed-on interpretation. I made this attitude explicit in the second paragraph. Teachers rarely make it explicit; they merely assume it if they think about it at all.

The student’s interpretation is a natural one. (Proof: So many of them make that interpretation!) Did the teacher tell the student that math notation has a standard interpretation and that this is not always what an otherwise literate person would expect? Did the teacher explain the specific and rather subtle rule about set notation that I described two paragraphs above? If not, the student does not deserve to be ridiculed for making this mistake.

Many people who get advanced degrees in math understood the correct rule for set notation when they first learned it, without having to be told. Being good at abstract math requires that kind of talent, which is linguistic as well as mathematical. Most students in abstract math classes are not going to get an advanced degree in math and don’t have that talent. They need to be taught things explicitly that the hotshots knew without being told. If all math teachers had this attitude there would be fewer people who hate math.

PS: My claim about how students think that leads them to believe that “{Ø}” denotes the empty set is a testable claim. There are many reports in the math ed literature from investigators who have been able to get students to talk about what they understand, for example, while working a word problem, but I don’t know of any reports about my assertion about “{Ø}” .

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More about neurons and math

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In the last post I talked about a neuron assembly in the brain that when it fires makes you feel you have been in the current situation before, and another neuron assembly that makes you feel that you are dealing with a persistent object with consistent behavior. I want to make it clear that I don’t know precisely how these brain functions are implemented, and I know of no research literature on these topics.

Brain research has shown that many different kinds of behavior, including thinking about different real and unreal things, causes activity in specific parts of the brain. I claim that the idea that there is a déjà-vu site and a persistent-thing-recognizing site is plausible and consistent with what we know about the brain. And they are far more plausible than any explanation of déjà-vu as coming from past lives or any explanation that mathematical objects are real and live in some ideal non-physical realm that we have no evidence for at all.

Another point: If our perception that when we think about and calculate with math objects we are dealing with things that are “out there” comes from the way our brains are organized, then we mathematicians should feel free to think about them and talk about them that way. We are making use of a brain mechanism that presumably evolved to cope with physical reality, as well as a general metaphor-mechanism that everyone makes use of to think about both physical and non-physical situations in a productive and creative way.

This point of view about metaphors has a lot of literature: see the section “about metaphors” here.

Again, it is a reasonable hypothesis that the metaphor-mechanism is implemented in some physical way in the brain that involves neurons and their connections.

To sum up, when we mathematicians think and act like Platonists we are using some of the main mechanisms of our brain for learning and creativity, and we should go ahead and be Platonists in action, without feeling embarrassed about it and without subscribing to any idealistic airy-fairyness.

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Mathematical Objects are "out there"?

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(This article is continued in More about math and neurons).

Sometimes we have a feeling of déjà vu in a situation where we know we have never been before. I have had two very strong occurrences of that in my life. One was when I saw St Cuthbert’s Church in Wells in England, and the other was the first time I saw St Martin’s in the Fields in London. Now my ancestors are mostly from England, some even from the south of England (the English ancestors of most white southern Americans are from the north of England, as are some of mine). Was this ancestral memory? Was it memories from a Previous Life? Well, I didn’t believe that, but the feeling was remarkably strong.

Many years later I discovered the reasons for the feelings in both cases. Adelbert Stone Chapel on the Case Western Reserve University campus in Cleveland (where I taught for 35 years) is an exact copy (on the outside) of St Cuthbert’s Church. Independent Presbyterian Church in Savannah is a three quarters size copy of St Martin’s in the Fields, and when I lived in Savannah as a teenager I frequently rode past that church on the bus.

There is presumably a neuron assembly (or something like that) in the brain devoted to recognizing things “I have seen before”. No doubt this can be triggered in the brain by mistake. Being triggered does not have to mean you had a previous life, it may mean a mistake in the recognition devices in your brain. The fact that I eventually understood my two experiences is in fact irrelevant. If you have the feeling of déjà vu and know you haven’t been there before and you never are able to explain it it still doesn’t prove you had a previous life or anything else supernatural. The feeling means only that a certain part of your brain was triggered and you don't know why.

When I deal with mathematical objects such as numbers, spaces, or groups I tend to think of them as “things” that are “out there”. Every time I investigate the number 42, it is even. 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.

There is also presumably another neuron assembly that recognizes that something is “out there” when I have repeatable and consistent experiences with it. Every time I push the button on my car door the door will open, except sometimes and then I consistently discover that it is locked and can be unlocked with my key. Every time I experiment with the number 111 it turns out to be 3 times 37. If some math calculation does not give the same answer the second time I frequently find that I made a mistake. I know this feeling of consistent “out there” behavior does not prove that numbers and other math objects are physical objects. The feeling originates in a brain arranged to detect consistent behavior. The feeling is not evidence that math objects exist in some ideal space.

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