Category Archives: representations

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

exposition, language, language of math, math, representations

Commonword names for technical concepts

2009/06/30 Charles Wells 4 Comments

In a previous post I talked about the use of commonword names for technical concepts, for example, “simple group” for a group with no proper normal subgroups. This makes the monster group a simple group! Lay readers on the subject might very well feel terminally put-down by such usage. (If he calls that “simple” he must be a genius. How could I ever understand that? See note 1.) Mark Ronan used of “atom of symmetry” instead of “simple group” in his book Symmetry and the Monster, probably for some such reason.

Recently I had what used to be called a CAT scan and (perhaps) what used to be called a PET scan on the same day. The medically community now refers to CT scan or nuclear imaging. This may be because too many clients were thinking of doing sadistic testing on cats or other pets. But I have not been able to confirm that.

The nurse called the CT scan an x-ray. Well, of course, it is an x-ray, but it is an x-ray with tomography. She explicitly said that calling CT scans x-rays was common usage in their lab. In the past, other medical people have said to me, “It used to be called CAT scan but now it is CT scan.” But no one said why.

The situation about PET scan is more complicated. I didn’t raise the question with the nurse, and Wikipedia has separate articles about PET scans and nuclear imaging, even though they both use positrons and tomography. The chemicals mentioned for PET are isotopes of low-atomic-number elements, whereas the nuclear medicine article mentions technetium99 as the most commonly used isotope. Nowhere does it explain the difference. I wrote a querulous note in the comments section of the NM article requesting clarification.

Note 1. “If he calls that ‘simple’ he must be a genius. How could I ever understand that?” Do not dismiss this as the reaction of a stupid person. This kind of ready-to-be-intimidated attitude is very common among intelligent, educated, but non-technically-oriented people. If mathematicians dismiss people like that we will continue to find mathematics anathema among educated people. We need people to feel that they understand something about what mathematicians do (I use that wording advisedly). Even if you are an elitist who doesn’t give a damn about ordinary people, remember who funds the NSF. See co-intimidator.

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language of math, math, representations, understanding math

Mathematical concepts

2009/06/10 Charles Wells 5 Comments

This post was triggered by John Armstrong’s comment on my last post.

We need to distinguish two ideas: representations of a mathematical concept and the total concept. (I will say more about terminology later.)

Example: We can construct the quotient of the kernel of a group homomorphism by taking its cosets and defining a multiplication on them. We can construct the image of the homomorphism by take the set of values of the homomorphism and using the multiplication induced by the codomain group. The quotient group and the image are the same mathematical structure in the sense that anything useful you can say about one is true of the other. For example, it may be useful to know the cardinality of the quotient (image) but it is not useful to know what its elements are.

But hold on, as the Australians say, if we knew that the codomain was an Abelian group then we would know that the quotient group was abelian because the elements of the image form a subgroup of the codomain. (But the Australians I know wouldn’t say that.)

Now that kind of thinking is based on the idea that the elements of the image are “really” elements of the codomain whereas elements of the quotients are “really” subsets of the domain. That is outmoded thinking. The image and the quotient are the same in all important aspects because they are naturally isomorphic. We should think of the quotient as just as much as subgroup of the codomain as the image is. John Baez (I think) would say that to ask whether the subgroup embedding is the identity on elements or not is an evil question.

Let’s step back and look at what is going on here. The definition of the quotient group is a construction using cosets. The definition of the image is a construction using values of the homomorphism. Those are two different specific representations of the same concept.

But what is the concept, as distinct from its representations? Intuitively, it is

All the constructions made possible by the definition of the concept.
All the statements that are true about the concept.

(That is not precise.)

The total concept is like the clone plus the equational theory of a specific type of algebra in the sense of universal algebra. The clone is all the operations you can construct knowing the given signature and equations and the equational theory is the set of all equations that follow from them. That is one way of describing it. Another is the monad in Set that gives the type of algebra — the operations are the arrows and the equations are the commutative diagrams.

Note: The preceding description of the monad is not quite right. Also the whole discussion omits mention of the fact that we are in the world (doctrine) of universal algebra. In the world of first order logic, for example, we need to refer to the classifying topos of the category of algebras of that type (or to its first order theory).

Terminology

We need better terminology for all this. I am not going to propose better terminology, so this is a shaggy dog story.

Math ed people talk about a particular concept image of a concept as well as the total schema of the concept.

In categorical logic, we talk about the sketch or presentation of the concept vs. the theory. The theory is a category (of the kind appropriate to the doctrine) that contains all the possible constructions and commutative diagrams that follow from the presentation.

In this post I have used “total concept” to refer to the schema or theory. I have referred the particular things as “representations” (for example construct the image of a homomorphism by cosets or by values of the homomorphism).

“Representation” does not have the same connotations as “presentation”. Indeed a presentation of a group and a representation of a group are mathematically two different things. But I suspect they are two different aspects of the same idea.

All this needs to be untangled. Maybe we should come up with two completely arbitrary words, like “dostak” and “dosh”.

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language of math, math, representations, understanding math

Different names for the same thing

2009/06/10 Charles Wells 6 Comments

I recommend reading the discussion (to which I contributed) of the post “Why aren’t all functions well-defined?” on Gower’s Weblog. It resulted in an insight I should have had a long time ago.

I have been preaching the importance of different ways of thinking about a math object (different images, metaphors, mental representations — there are too many names for this in the math ed literature). Well, mathematicians at least occasionally use different names for a type of math object to indicate how they are thinking about it.

Examples

We talk about a relation and we talk about multivalued functions. Those are two different ways of talking about the same thing (they are the same by an adjunction). A relation is a predicate. A multivalued function is a function except that it can have more than one output for a given input. But they are the same thing.

We talk about an equivalence relation and we talk about a partition of a set (or a quotient set). The category of equivalence relations and the category of partitions of sets are naturally isomorphic, not merely equivalent. But one is a special kind of relation and the other is a grouping.

Let’s be open about what we do

We should be explicit about the way we think about and do math. We have several different ways to think about any interesting type of math object and we should push this practice to students as being absolutely vital. In particular we (some of us) use different names sometimes for the same object and we refuse to give them up, muttering about “reductionism” and “nothing buttery”.

Some students arrive in class already as (pedantic?)(geeky?) as many mathematicians (I am a recovering pedant myself). We need to be up front about this phenomenon and explain the value of thinking and talking about the same thing in different ways, even using different words.

It used to be different but now it’s the same

A kind of opposite phenomenon occurs with some students and mathematicians of a certain personality type. Consider the name “multivalued function”. Of course a multivalued function is not (necessarily) a function. Your mother-in-law is not your mother, either. I go on about this (using ideas from Lakoff) in the Handbook under “radial concept”. Pedantic types can’t stand this kind of usage. “A multivalued function can’t be a function”. “Equivalence relations and partitions are not the same thing because one is a relation and the other is a set of sets.” “The image of a homomorphism and the quotient by its kernel are not the same thing because…”

This attitude makes me tired. Put your hands on the tv screen and think like a category theorist.

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abstractmath.org, language of math, math, representations, understanding math

Variables

2009/05/15 Charles Wells 4 Comments

One of the themes of abstractmath.org is that we should pay attention to how we think about mathematical objects. This is not the same questions as “What are mathematical objects?”. This post addresses the question: How do we think about variables? What follows are extracts from newly rewrittens sections from Variables and Substitution and Mathematical Objects.

Role playing

If the author says “x is a real variable” then x plays the role of a real number in whatever expression it occurs in. It is like an actor in a play. If the producer says Dwayne will play Polonius you know that Dwayne will hide behind a curtain at a certain point in the play. When x occurs in the expression $x^3-1$ you know that if a number is substituted for x in the expression, the expression will then denote the result of cubing the number and subtracting 1 from it.

Slot or cell

The variable x is a slot into which you can put any real number. If you plug 3 into x in the expression $x^3-1$ you will get 26.

This is like a blank cell in a spreadsheet. If you define another cell with the formula “ $=x^3-1$ ” and put 3 in the cell representing x, the other cell will contain 26.

What’s wrong with this metaphor: In Excel, a blank cell is automatically set to 0. To be a better metaphor the cell shouldn’t have a value until it is given one, and the cell with the formula “ $= x^3-1$ ” should say “undefined!”. (I am not saying this would make Excel a better spreadsheet. Excel was not invented so that I could make a point about variables.)

Variable mathematical object

The two metaphors above refer to the name x. You can instead think of x as a variable mathematical object, meaning x is a genuine mathematical object, but with limitations about what you can say or think about it. This sort of thinking works for both the symbolic language and mathematical English, and it works for any kind of mathematical structure (“Let G be an Abelian group…”), not just numbers in a symbolic expression. There are two related points of view:

1. Some statements about the object are neither true nor false.

This means x is a genuine mathematical object and you can make assertions about it, but some of the assertions might have no truth value. From “Let x be a real number” you know these things:

The assertion “Either x > 0 or $x \leq 0$ ” is true.
The assertion “ $x^2 = -1$ ” is false.
The assertion “x > 0” is neither true nor false.

The assertion “x is a real number” is in a certain sense the most general true statement you can make about x. In other words, x is a mathematical object given by an incomplete specification, so you are limited in what you can say about it or in what conclusions you can draw about it.

If you say, “Let n be an integer divisible by 4, you cannot assume it is 8 or 12, for example. In other words, the statement “n is divisible by 4” is true, and “n = 3” is false, but the statement “n = 8” is neither true nor false, and you can’t derive any conclusions from n being 8.

2. The object is fixed but some things are not known about it.

If you say x is a real number, you know x is a real number (duh) and:

You know x is either positive or nonnegative.

You know $x^2$ is not equal to any negative number.

You don’t know whether x is positive or not.

This way of looking at it involves thinking of x as a particular real number. During the process of solving the equation $x^2-5x=-6$ you are thinking of x as a specific real number, but you don’t know which one.

These points of view (1) and (2) provide genuinely different metaphors for variables. In (1) I say certain statements are neither true nor false, but (2) suggests that all statements about the object are either true or false but you don’t know which. However, note when solving the equation
$x^2-5x=-6$ that, when you are finished, you still don’t know whether x = 2 or x = 3. This factcauses me cognitive dissonance, but the point of view that some statements are neither true nor false upsets other people. I prefer (1) over (2) but I have to admit that (1) is much less familiar to most mathematicians.

View (1) is advocated by category theorists because it allows you to think of a quantity holistically as a single thing rather than as a table of values. The height of a cannonball is different at different times but the “height” is nevertheless one continuous mathematical quantity. People who know more about history than I do believe that that is the simple and uncomplicated way nineteenth-century mathematicians thought about variable quantities.

We need good tools to do math. This means good images and metaphors as well as good tools for reasoning. Having simple and uncomplicated ways to think about math objects (along with guidelines for the way you think about them, such as dropping the law of the excluded middle in some cases!) is every bit as important as making sure our reasoning follows carefully thought out rules that lead from truth only to truth.

Note: Heyting valued logic actually provides sound but non-classical reasoning for thinking about variable objects, but most mathematicians with sound intuitions nevertheless use classical reasoning and come up with correct conclusions. Some of us are now in the practice of using non-classical logic to study differentials and other things, and that is a Good Thing, but it would be a complete misunderstanding if you read this post as advocating that mathematicians change over to that way of doing things. This post is about how we think about variability.

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Mental Representations in Math

2009/04/03 Charles Wells 3 Comments

This post is part of the abstractmath article on images and metaphors. I have had some new insights into the subject of mental representations and have incorporated them in this rewritten version (which omits some examples). I would welcome comments.

Mathematicians who work with a particular kind of mathematical object have mental representations of that type of object that help them understand it. These mental representations come in various forms:

Visual images, for example of what a right triangle looks like.
Notation, for example visualizing the square root of 2 by the symbol “ $\sqrt{2}$ “. Of course, in a sense notation is also a physical representation of the number. An important fact: A mathematical object may be referred to by many different notations. There are examples here and here. (If you think deeply about the role notation plays in your head and on paper you can easily get a headache.)
Kinetic understanding, for example thinking of the value of a function as approaching a limit by imagining yourself moving along the graph of the function.
Metaphorical understanding, for example thinking of a function such as as a machine that turns one number into another: for example, when you put in 3 out comes 9. See also literalism and this post on Gyre&Gimble.

Example

Consider the function $h(t)=25-(t-5)^2$ . The chapter on images and metaphors for functions describes many ways to think about functions. A few of them are considered here.

Visual images You can picture this function in terms of its graph, which is a parabola. You can think of it more physically, as like the Gateway Arch. The graph visualization suggests that the function has a single maximum point that appears to occur at t = 5.

I personally use visual placement to remember relationships between abstract objects, as well. For example, if I think of three groups, two of which are isomorphic (for example $C_2$ and $\text{Alt}_3$ .), I picture them as in different places with a connection between the two isomorphic ones. I know of no research on this.

Notation You can think of the function as its formula . The formula tells you that its graph will be a parabola (if you know that quadratics give parabolas) and it tells you instantly without calculus that its maximum will be at (see ratchet effect).

Another formula for the same function is $-t^2+10t$ . The formula is only a representation of the function. It is not the same thing as the function. The functions h(t) and k(t) defined on the real numbers by $h(t)=25-(t-5)^2$ and $k(t)=-t^2+10t$ are the same function; in other words, h = k.

Kinetic The function h(t) could model the height over time of a physical object, perhaps a ball shot vertically upwards on a planet with no atmosphere. You could think of the ball starting at time t = 0 at elevation 0, reaching an elevation of (for example) 16 units at time t = 2, and landing at t = 10. You are imagining a physical event continuing over time, not just as a picture but as a feeling of going up and down (see mirror neuron). This feeling of the ball going up and down is attached in your brain to your understanding of the function h(t).

Although h(t) models the height of the ball, it is not the same thing as the height of the ball. A mathematical object may have a relationship in our mind to physical processes or situations but is distinct from them.

According to this report, kinetic understanding can also help with learning math that does not involve pictures. I know that when I think of evaluating the function at 3, I visualize 3 moving into the x slot and then the formula transforming itself into 10. I remember doing this even before I had ever heard of the Transformers.

Metaphor One metaphor for functions is that it is a machine that turns one number into another. For example, the function h(t) turns 0 into 0 (which is therefore a fixed point) and 5 into 25. It also turns 10 into 0, so once you have the output you can’t necessarily tell what the input was (the function is not injective).

More examples

¨ “Continuous functions don’t have gaps in the graph“. This is a visual image.
¨ You may think of the real numbers as lying along a straight line (the real line) that extends infinitely far in both directions. This is both visual and a metaphor (a real number “is” a place on the real line).
¨ You can think of the set containing 1, 3 and 5 and nothing else in terms of its list notation {1, 3, 5}. But remember that {5, 1,3} is the same set. In other words the list notation has irrelevant features – the order in which the elements are listed in this case.
The interior of a closed curve or a sphere is called that because it is like the interior in the everyday sense of a bucket or a house. Note The boundary of a real-life container such as a bucket has thickness, in contrast to the boundary of a closed curve or a sphere. This observation illustrates my description of a metaphor as identifying part of one situation with part of another. One aspect is emphasized; another aspect, where they may differ, is ignored.

Uses of mental representations

Integers and metaphors make up what is arguably the most important part of the mathematician’s understanding of the concept.

Mental representations of mathematical objects using metaphors and images are necessary for understanding and communicating about them (especially with types of objects that are new to us).
They are necessary for seeing how the theory can be applied.
They are useful for coming up with proofs.

Many representations

Different mental representations of the same kind of object help you understand different aspects of the object.

Every important mathematical object has many representations and skilled mathematicians generally have several of them in mind at once.

New concepts and old ones

We especially depend on metaphors and images to understand a math concept that is new to us. But if we work with it for awhile, finding lots of examples, and eventually proving theorems and providing counterexamples to conjectures, we begin to understand the concept in its own terms and the images and metaphors tend to fade away from our awareness…

Then, when someone asks us about this concept that we are now experts with, we trundle out our old images and metaphors – and are often surprised at how difficult and misleading our listener finds them!

Some mathematicians retreat from images and metaphors because of this and refuse to do more than state the definition and some theorems about the concept. They are wrong to do this. That behavior encourages the attitude of many people that

mathematicians can’t explain things
math concepts are incomprehensible or bizarre
you have to have a mathematical mind to understand math

All three of these statements are half-truths. There is no doubt that a lot of abstract math is hard to understand, but understanding is certainly made easier with the use of images and metaphors

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Proofs without dry bones

2009/03/24 Charles Wells 3 Comments

I have discussed images, metaphors and proofs in math in two ways:

(A) A mathematical proof

A monk starts at dawn at the bottom of a mountain and goes up a path to the top, arriving there at dusk. The next morning at dawn he begins to go down the path, arriving at dusk at the place he started from on the previous day. Prove that there is a time of day at which he is at the same place on the path on both days.

Proof: Envision both events occurring on the same day, with a monk starting at the top and another starting at the bottom at the same time and doing the same thing the monk did on different days. They are on the same path, so they must meet each other. The time at which they meet is the time required.

This example comes from Fauconnier, Mappings in Thought and Language, Cambridge Univ. Press, 1997. I discuss it in the Handbook, pages 46 and 153. See the Wikipedia article on conceptual blending.

(B) Rigor and rigor mortis

The following is quoted from a previous post here. See also the discussion in abstractmath.

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 here. They can involve thinking of abstract things moving and changing and affecting each other.

When we set out to prove some math statement, we go into what I have called “rigorous mode”. We feel that we have to forget some of the color and excitement of the rich view. We must think of math objects as inert and static. They 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 didn’t contradict myself.
I want to clear up some tension between these two ideas.

The argument in (A) is a genuine mathematical proof, just as it is written. It contains hidden assumptions (enthymemes), but all math proofs contain hidden assumptions. My remarks in (B) do not mean that a proof is not a proof until everything goes dead, but that when challenged you have to abandon some of the colorful and kinetic reasoning to make sure you have it right. (This is a standard mathematical technique (note 1).)

One of the hidden assumptions in (A) is that two monks walking the opposite way on the path over the same interval of time will meet each other. This is based on our physical experience. If someone questions this we have several ways to get more rigorous. One many mathematicians might think of is to model the path as a curve in space and consider two different parametrizations by the unit interval that go in opposite directions. This model can then appeal to the intermediate value theorem to assert that there is a point where the two parametrizations give the same value.

I suppose that argument goes all the way to the dead. In the original argument the monk is moving. But the parametrized curve just sits there. The parametrizations are sets of ordered pairs in R x (R x R x R). Nothing is moving. All is dry bones. Ezekiel has not done his thing yet.

This technique works, I think, because it allows classical logic to be correct. It is not correct in everyday life when things are moving and changing and time is passing.

Avoid models; axiomatize directly
But it certainly is not necessary to rigorize this argument by using parametrizations involving the real numbers. You could instead look at the situation of the monk and make some axioms the events being described. For example, you could presumably make axioms on locations on the path that treat the locations as intervals rather than as points.

The idea is to make axioms that state properties that intervals have but doesn’t say they are intervals. For example that there is a relation “higher than” between locations that must be reflexive and transitive but not antisymmetric. I have not done this, but I would propose that you could do this without recreating the classical real numbers by the axioms. (You would presumably be creating the intuitionistic real numbers.)

Of course, we commonly fall into using the real numbers because methods of modeling using real numbers have been worked out in great detail. Why start from scratch?

About the heading on this section: There is a sense in which “axiomatizing directly” is a way of creating a model. Nevertheless there is a distinction between these two approaches, but I am to confused to say anything about this right now.

First order logic.
It is commonly held that if you rigorize a proof enough you could get it all the way down to a proof in first order logic. You could do this in the case of the proof in (A) but there is a genuine problem in doing this that people don’t pay enough attention to.

The point is you replace the path and the monks by mathematical models (a curve in space) and their actions by parametrizations. The resulting argument calls on well known theorems in real analysis and I have no doubt can be turned into a strict first order logic argument. But the resulting argument is no longer about the monk on the path.

The argument in (A) involves our understanding of a possibly real physical situation along with a metaphorical transference in time of the two walks (a transference that takes place in our brain using techniques (conceptual blending) the brain uses every minute of every day). Changing over to using a mathematical model might get something wrong. Even if the argument using parametrized curves doesn’t have any important flaws (and I don’t believe it does) it is still transferring the argument from one situation to another.

Conclusion:
Mathematical arguments are still mathematical arguments whether they refer to mathematical objects or not. A mathematical argument can be challenged and tested by uncovering hidden assumptions and making them explicit as well as by transferring the argument to a classical mathematical situation.

Note 1. Did you ever hear anyone talking about rigor requiring making images and metaphors dead? This is indeed a standard mathematical technique but it is almost always suppressed, or more likely unnoticed. But I am not claiming to be the first one to reveal it to the world. Some of the members of Bourbaki talked this way. (I have lost the reference to this.)

They certainly killed more metaphors than most mathematicians.

Note 2. This discussion about rigor and dead things is itself a metaphor, so it involves a metametaphor. Metaphors always have something misleading about them. Metametaphorical statements have the potential of being far worse. For example, the notion that mathematics contains some kind of absolute truth is the result of bad metametaphorical thinking.

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An early revolution in notation

2009/03/05 Charles Wells Leave a comment

This is a reprint of an article by Jeffrey A. Tucker. Musical notation was invented by Guido six hundred years before Viète and Descartes invented modern algebraic notation. –cw

Guido the Great, by Jeffrey A. Tucker

This essay is adapted from Jeffrey’s new book, Sing Like a Catholic.

The people who make modern inventions are often celebrated for improving our lives. But what about those one thousand years ago who laid the very technological foundation of civilization as we know it? They too served the world, but with the primary purpose of contributing to the faith. I’m thinking here of those who solved the architectural problem to build the great cathedrals of the middle ages, and the scientists of the period who took the first steps toward modern medical knowledge.

Also we don’t often consider the innovations in art that make all music possible. There is one person who stands out here: the late 10^th and early 11^th century Benedictine monk named Guido d’Arezzo (991/992–after 1033). He is credited with fantastic musical innovations that led to the creation of the modern system of notes and staffs, and also the organization of scales that allowed for teaching and writing music.

His contributions have usually been seen as technical innovations and evaluated as such, though known only inside a small circle of music historians. Without his contribution, the music you hear on your iPod and on the radio would not likely exist.

A new book by Angelo Rusconi, synopsized by Patrick Reynolds from the Italian, and appearing in Goldberg #46 (2007), offers a more complete picture of what drove Guido, and the results will be very exciting for anyone who seeks to understand how any serious innovation upsets the status quo, makes enemies, causes a bit of social upheaval, and ultimately makes the world a better place.

Consider the technical feat that Guido undertook. Imagine a world without printed music. How would you go about conveying a tune in printed form? It’s one thing to render words on paper in a way that others can read them. But what about sound? It floats in the air and resists having a physical presence at all.

How can you share the melody without singing it for them, by just writing things down? People had tried since the ancient world without success. Some attempts in the 8^th and 9^th centuries came a bit closer (but the results look like chicken scratch to us). It was Guido who made the breakthrough with lines and scales that illustrate for the eye what the voice is to sing, and precisely so. His innovation was a beautiful integration of art and science.

And what a remarkable innovation, if you think about it. From the beginning of time until his time, the teaching of music was done by a tiny and ever-arrogant cartel of masters. You had to sing exactly as they instructed you. If they weren’t around, you were stuck. They held the monopoly. To become a master of music, you had to study under one of the greats, and then receive the blessing to become a teacher yourself, and you know that they wanted to limit their numbers. One can imagine that you had to be sycophantic to even get your foot in the door.

Guido’s innovation busted up the cartel. Rusconi shows that Guido’s primary interest was in notating not just music in general but Gregorian chant in particular. He was frustrated that the chant was passed on by oral tradition only. He worried that melodies would be lost, especially given the then new fashion for multi-part improvisation.

So while writers have usually treated Guido as an innovator, what’s been forgotten is that his innovations were driven by the desire to conserve and preserve for future generations. The desire to maintain the chant and pass it on was the key issue for him; the technical aspects of the music and writing were merely tools and not ends in themselves.

And there was an interesting sociological element here. He had become seriously annoyed at the cartel of chant masters and the power they exercised over the monastic community. He wanted the chant to be freed and put into the hands of everyone both inside and outside the monastery walls.

For this reason, his first great project was a notated Antiphoner, a book of melodies: “For, in such a ways, with the help of God I have determined to notate this antiphoner, so that hereafter through it, any intelligent and diligent person can learn a chant, and after he has learned well part of it through a teacher, he recognizes the rest unhesitatingly by himself without a teacher.”

He goes further. Without a written form of music “wretched singers and pupils of singers, even if they should sing every day for a hundred years, will never sing by themselves without a teacher one antiphon, not even a short one, wasting so much time in singing that they could have spent better learning thoroughly sacred and secularly writing.”

The elite musicians resisted his attempt to democratize the knowledge and conserve time. Guido did whatever great innovator does: he freed up resources for other uses even while improving lives.

But as a result of his innovation, his monastery in Pomposa, Italy, tossed him out into the snow. He then went to the Pope, who was very impressed, and gave him a letter of support. With the letter in hand, he went to the Bishop of Arezzo, who took him in so that he could continue his preaching and his work.

Now, one can’t but think of mistakes that have been made over the years with the Gregorian chant: the attempt to keep it the private preserve of musicologists; the dominance of singers by a single master who believes that he knows the one true way; the perception that chant is only for monasteries but not regular people; and on and on.

Here we see Guido embodying the same principle that drove the Solesmes monastery at the early part of the restoration efforts in the late 19^th century: innovation in order to preserve, teach, and distribute this glorious music as widely as possible, in the service of the faith. They had the right ordering of priorities: technical innovation in the service of preserving universal truth.

This story illustrates a general principle in the history of technology. There does seem to be a real pattern here. There are those who believe that innovation is for everyone and ought to be accessible to all – that everyone should be permitted to have access to the forms and structures that make for progress. This side loves technical innovation not for its own sake but in the service of great goals.

Then there is the other side, which is reactionary, hates technical innovation, wants to reserve technical forms to a tiny elite, fears freedom, detests the idea of human choice, and advances a kind of Gnosticism over technical forms – always wants it to remain the private preserve of the elect who appoint each other and operate as a kind of guild. This Gnostic guild wants to guard and exclude and privatize, and the people are ultimately their enemy.

This perspective hearkens back to the ancient world where priests served the philosopher kings, and sparingly hand out religious truth to the masses based on what they believe they should know in the service of their agenda.

One can detect these two tendencies from all ages.

Guido didn’t patent his innovation. He didn’t copyright his music. The legal means weren’t available to him, and he wouldn’t have used them if they were. His whole point was to uplift the whole of the culture. Keeping his innovation to himself would have been contrary to that goal. As a result, his use of the staff spread widely. His innovation was infinitely reproducible, and it changed the world.

In his pedagogy, he adapted an existing song to illustrate the scale: Ut Queant Laxis, a hymn to St. John the Baptist, who was then considered the patron saint of singers. On the first syllable of each ascending note, the words were Ut, Re, Mi, Fa, Sol – the very foundation of music pedagogy to this day: do, re, mi, etc.

A millennium later, Guido’s innovation is still with us!

Here is a model for our Arial and all Arial.

March 4, 2009

Jeffrey Tucker [send him mail] is editorial vice president of www.Mises.org.

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language, math, representations, understanding math

Typical examples

2009/02/11 Charles Wells 2 Comments

There is a sense in which a robin is a typical bird and a penguin is not a typical bird. (See Reference 1.) Mathematical objects are like this and not like this.

A mathematical definition is simply a list of properties. If an object has all the properties, it is an example of the definition. If it doesn’t, it is not an example. So in some basic sense no example is any more typical than any other. This is in the sense of rigorous thinking described in the abstractmath chapter on images and metaphors.

In fact mathematicians are often strongly opinionated about examples: Some are typical, some are trivial, some are monstrous, some are surprising. A dihedral group is more nearly a typical finite group than a cyclic group. The real numbers on addition is not typical; it has very special properties. The monster group is the largest finite simple group; hardly typical. Its smallest faithful representation as complex matrices involves matrices that are of size 196,883. The monster group deserves its name. Nevertheless, the monster group is no more or less a group than the trivial group.

People communicating abstract math need to keep these two ideas in mind. In the rigorous sense, any group is just as much a group as any other. But when you think of an example of a group, you need to find one that is likely to provide some information. This depends on the problem. For some problems you might want to think of dihedral groups. For others, large abelian p-groups. The trivial group and the monster group are not usually good examples. This way of thinking is not merely a matter of psychology, but it probably can’t be made rigorous either. It is part of the way a mathematician thinks, and this aspect of doing math needs to be taught explicitly.

Reference 1. Lakoff, G. (1986), Women, Fire, and Dangerous Things. The University of Chicago Press. He calls typical examples “prototypical”.

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Mathematical definitions

2009/02/11 Charles Wells 1 Comment

The definition of a concept in math has properties that are different from definitions in other subjects:

• Every correct statement about the concept follows logically from its definition.
• An example of the concept fits all the requirements of the definition (not just most of them).
• Every math object that fits all the requirements of the definition is an example of the concept.
• Mathematical definitions are crisp, not fuzzy.
• The definition gives a small amount of structural information and properties that are enough to determine the concept.
• Usually, much else is known about the concept besides what is in the definition.
• The info in the definition may not be the most important things to know about the concept.
• The same concept can have very different-looking definitions. It may be difficult to prove they give the same concept.
• Math texts use special wording to give definitions. Newcomers may not understand that the intent of an assertion is that it is a definition.

How many college math teachers ever explain these things?

I will expand on some of these concepts in future posts.

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Math and the Modules of the Mind

2009/01/30 Charles Wells 6 Comments

I have written (references below) about the way we seem to think about math objects using our mind’s mechanisms for thinking about physical objects. What I want to do in this post is to establish a vocabulary for talking about these ideas that is carefully enough defined that what I say presupposes as little as possible about how our mind behaves. (But it does presuppose some things.) This is roughly like Gregor Mendel’s formulation of the laws of inheritance, which gave precise descriptions of how characteristics were inherited while saying nothing at all about the mechanism.

I will use module as a name for the systems in the mind that perform various tasks.

Examples of modules

a) We have an “I’ve seen this before module” that I talked about here.

b) When we see a table, our mind has a module that recognizes it as a table, a module that notes that it is nearby, and in particular a module that notes that it is a physical object. The physical-object module is connected to many other modules, including for example expectations of what we would feel if we touched it, and in particular connections to our language-producing module that has us talk about it in a certain way (a table, the table, my table, and so on.)

c) We also have a module for abstract objects. Abstract objects are discussed in detail in the math objects chapter of abstractmath.org. A schedule is an abstract object, and so is the month of November. They are not mathematical objects because they affect people and change over time. (More about this here.) For example, the statement “it is now November” is true sometimes and false sometimes. Abstract objects are also not abstractions, like “beauty” and “love” which are not thought of as objects.

d) We talk about numbers in some ways like we talk about physical objects. We say “3 is a number”. We say “I am thinking of the only even prime”. But if we point and say, “Look, there is a 3”, we know that we have shifted ground and are talking about, not the number 3, but about a physical representation of the number 3. That’s because numbers trigger our abstract object module and our math object module, but not our physical object module. (Back and fill time: if you are not a mathematician, your mind may not have a math object module. People are not all the same.)

More about modules

My first choice for a name for these systems would have been object, as in object-oriented programming, but this discussion has too many things called objects already. Now let’s clear up some possible misconceptions:

e) I am talking about a module of the mind. My best guess would be that the mind is a function of the brain and its relationship with the world, but I am not presuppposing that. Whatever the mind is, it obviously has a system for recognizing that something is a physical object or a color or a thought or whatever. (Not all the modules are recognizers; some of them initiate actions or feelings.)

f) It seems likely that each module is a neuron together with its connections to other neurons, with some connections stronger than others (our concepts are fuzzy, not Boolean). But maybe a module is many neurons working together. Or maybe it is like a module in a computer program, that is instantiated anew each time it is called, so that a module does not have a fixed place in the brain. But it doesn’t matter. A module is whatever it is that carries out a particular function. Something has to carry out such functions.

Math objects

The modules in a mathematician’s mind that deal with math objects use some of the same machinery that the mind uses for physical objects.

g) You can do things to them. You can add two numbers. You can evaluate a function at an input. You can take the derivative of some functions.

h) You can discover properties of some kinds of math objects. (Every differentiable function is continuous.)

i) Names of some math objects are treated as proper nouns (such as “42”) and others as common nouns (such as “a prime”.)

I maintain that these phenomena are evidence that the systems in your mind for thinking about physical objects are sometimes useful for thinking about math objects.

Different ways of thinking about math objects.

j) You can construct a mathematical object that is new to you. You may feel that you invented it, that it didn’t exist before you created it. That’s your I just created this module acting. If you feel this way, you may think math is constantly evolving.

k) Many mathematicians feel that math objects are all already there. That’s a module that recognizes that math objects don't come into or go out of existence.

l) When you are trying to understand math objects you use all sorts of physical representations (graphs, diagrams) and mental representations (metaphors, images). You say things like, “This cubic curve goes up to positive infinity in the negative direction” and “This function vanishes at 2” and “Think of a Möbius strip as the unit square with two parallel sides identified in the reverse direction.”

m) When you are trying to prove something about math objects mathematicians generally think of math objects as eternal and inert (not affecting anything else). For example, you replace “the slope of the secant gets closer and closer to the slope of the tangent” by an epsilon-delta argument in which everything you talk about is treated as if it is unchanging and permanent. (See my discussion of the rigorous view.)

Consequences

When you have a feeling of déjà vu, it is because something has triggered your “I have seen this before” module (see (a)). It does not mean you have seen it before.

When you say “the number 3” is odd, that is a convenient way of talking about it (see (d) above), but it doesn’t mean that there is really only one number three.

If you say the function x^2 takes 3 to 9 it doesn’t have physical consequences like “Take me to the bank” might have. You are using your transport module but in a pretend way (you are using the pretend module!).

When you think you have constructed a new math object (see (j)), your mental modules leave you feeling that the object didn’t exist before. When you think you have discovered a new math object (see (k)), your modules leave you feeling that it did exist before. Neither of those feelings say anything about reality, and you can even have both feelings at the same time.

When you think about math objects as eternal and inert (see (m)) you are using your eternal and inert modules in a pretend way. This does not constitute an assertion that they are eternal and inert.

Is this philosophy?

My descriptions of how we think about math are testable claims about the behavior of our mind, expressed in terms of modules whose behavior I (partially) specify but whose nature I don’t specify. Just as Mendel’s Laws turned out to be explained by the real behavior of chromosomes under meiosis, the phenomena I describe may someday turn out to be explained by whatever instantiation the modules actually have – except for those phenomena that I have described wrongly, of course – that is what “testable” means!

So what I am doing is science, not philosophy, right?

Now my metaphor-producing module presents the familiar picture of philosophy and science as being adjacent countries, with science intermittently taking over pieces of philosophy’s territory…

Links to my other articles in this thread

Math objects in abstractmath.org
Mathematical objects are “out there”?
Neurons and math
A scientific view of mathematics (has many references to what other people have said about math objects)
Constructivism and Platonism

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Examples

Role playing

Slot or cell

Variable mathematical object

1. Some statements about the object are neither true nor false.

2. The object is fixed but some things are not known about it.

Example

More examples

Uses of mental representations

Many representations

New concepts and old ones

math, language and other things that may show up in the wabe