Category Archives: math

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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abstractmath.org, astounding math, math, proof

Writing Astounding Math Stories

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I have written a second Astounding Math Story, this one about factoring integers and primality testing, published here. (I announced ASM here.)

These stories are aimed at people interested in math but not very far along in studying abstract math, the same audience that my abstractmath website is concerned with. That makes the stories hard to write.

For one thing, they need to be streamlined as much as possible. Each story should explain one astounding fact clearly, making as few mathematical demands on the reader as possible. I have put in links, mostly to Wikipedia, to explain concepts used in the story. This story has lots of footnotes telling about further developments and fine points.

Another point is that it is sometimes hard to convince students that they need to be astounded! I mentioned the phenomenon that primality testing is faster than factorization many times in my teaching (mostly to computing science students). Often I had to work hard to get them to realize that there was something shocking about this: Being a composite means having proper factors, but the fast ways of discovering compositeness tell you it is composite without giving any clue as to what the proper factors are. Research mathematicians are familiar with the idea of proving something exists without being able to say what it is, but students often have to be led by the nose to grasp this idea.

With this story I have experimented with making it a dialog. That makes it easier to write about a conflict between new ideas and old presuppositions. I would love to get comments about how these stories are written as well as the math involved.

Experienced research mathematicians will probably not be Astounded by these stories. But the ones I am writing about have often given mathematicians in previous centuries a lot of trouble — they seemed unbelievable or contradictory. And modern students have trouble with these ideas, too. In the current ASM I mention Kronecker’s problem with nonconstructive existence proofs.

One of the worst problems comes with infinite decimal expansions of real numbers (which I intend to write about). You can prove that 1.000… – .999… but the students don’t really believe it. That may be because they don’t really believe that all the decimal digits are really there. (A lot of philosophers don’t believe this either, but almost all research mathematicians talk and act as if they do.)

By the way, it is amazing how often there is an article in Wikipedia that says just what you want the reader to know (and of course usually a lot more) and does it pretty well. Lately I have run across just one exception, the articles on context. I have been writing about context in mathematical writing for abmath (don’t look, it isn’t there yet) and have wound up going into more detail than I wanted because I could not refer to Wikipedia.

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What I learned in school is the Only Truth

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Michael Barr recently commented on another post about a Dutch student who insisted that the words “long” and “short” in English referred to qualitative differences such as that between “ride” and “rid”, whereas linguists use the words to refer to temporal length, such as the different in the vowels between “hid” and “hit”.

I assume the student acquired the qualitative meaning from English courses in school; that meaning is very still widely used in English classes in the USA and Britain, so that (I’ll bet) nearly any person on the street in the USA would expect the meaning of “long” and “short” to be the difference between “ride” and “rid”.

This is an example of a phenomenon mathematicians have to put up with too. We know that the same word or symbol can have many different meanings in math, but people who know a little math assume that all meanings that they learned in whatever math courses they took are universal and set in granite. They are startled that “pi” can mean anything other that what they think it means. Someone recently started talking to me about “phi” as if I should know what it means, but I recovered fairly quickly, since I had become vaguely aware that it means the so-called golden ratio to laymen. In my experience mathematicians mostly use phi to denote some function.

When I taught, I was constantly in trouble with students who told me that 0 was not a natural number if the textbook said it was, or was a natural number if the textbook said it wasn’t, because that was the definition in some previous course they had taken.

I have written about this phenomenon here and here.

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category theory, language of math, math

Cribles

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Mark Meckes’ comments on technical words in English reminds me of an incident from the Ancient Days, namely 1966. Well, 1966-ish. A visiting Belgian mathematician in my department talked about category theory. One concept that came up was that of “crible”, which was its name in French. It is a family of arrows with a common target (the sources can vary). He couldn’t think of the English translation of “crible” so he said something like this: “The best way I can describe this is to think of a soldier in the trenches in World War I who suddenly stands up and is shot full of holes by many machine gun bullets.”

We were completely baffled by this explanation. Is there an English word that describes a person with lots of bullet holes? You can see where the picture comes from by thinking of the target as a person with lots of arrows stuck in him, like Saint Sebastian, or Hagar the Horrible on a bad day.

The English word he wanted is “sieve” and that is the usual name of the concept today. In the sixties, many English speaking mathematicians called it “crible” but that usage died out as far as I know. A few tried to pronounce it the French way, but no one understood them, so they spelled it, and then most people said “cribble”.

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A Scientific View of Mathematics

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Over most of the history of human thinking, both philosophers and theologians have come up with explanations of some natural phenomenon, only to be faced with scientific investigations that give a successful evidence-based explanation of the phenomenon they have written about.

The theologians and philosophers eventually lose the argument. The word “eventually” means that (1) the scientific investigation has to produce a pretty solid theory that explains a lot of the evidence and (2) the older theologians and philosophers have to die, since they rarely change their minds about stuff after the age of 50. (Some theologians and philosophers still continue to argue for things such as flat earth, intelligent design and dualism, but they are not really engaging in the intellectual world’s ongoing conversation.)

It is now possible to investigate the theory and practice of mathematics using evidence-based scientific reasoning. In particular, recent findings in neuroscience and cognitive theory make it plausible to provide a description of mathematics that is based on the interaction between brain, body and culture. By continuing to study mathematics and its practice scientifically we can hope to come up with a theory of mathematics that will be a part of cognitive theory.

I have been writing about bits and pieces of this idea for a long time, and so have many others. What I am going to give here is a lacunary sketch of my current thoughts with references. Most of these ideas originated with other people!

Math is an activity of our brains.
Our brains contain ideas. These ideas are real physical structures, organizations of neurons or something. [MO], [TaBa2002] I will call them PSB’s (physical structures in the brain). This is early days in neuroscience and exactly how the ideas exist physically in the brain is still controversial. I assert only that ideas are physical, nature in part yet to be determined.
Among our ideas are representations of objects and lists of rules.

Objects are represented in our brains.
The objects represented in our brains may be physical objects, fictional objects or abstract objects, including mathematical objects [AbMO], [Her97].

There is presumably a PSB that is triggered when you think about any kind of object. It is triggered if you think about the Parthenon, Sherlock Holmes, or the function f that takes a real number x to x^2. (If you are not an experienced mathematician, you might in fact not think about f as an object, but rather as rule or procedure. This can cause serious difficulties for students in calculus classes who are faced with such concepts as “the derivative of f”.)

There is no doubt another PSB’s that recognize that the Parthenon is a physical object in contrast to the squaring function, which is an abstract object. But our brain clearly recognizes both as an object because we talk about physical, fictional and abstract objects using the same grammatical structures and we think about them using similar mental operations. [AbLM]

The fact that we think and talk about the set of all real numbers (for example) as an object is explained by this PSB. It does not imply that the set of all real numbers exists anywhere, physically or ideally.

Our brains are organized to follow rules.
We apparently have a PSB that implements a rule-following mode. (This has been studied but I don’t have a good reference for it.) We learn and follow rules very easily when we play any game, baseball, chess or whatever. We also learn rules for algebraic manipulation and for mathematical reasoning using the rule-following mode. People who are good at math seem to engage the rule-following mode easily in these situations.

Math is communicated among people using the languages of math.
Math has several languages. Mathematical English is a special dialect of English with some disconcertingly different rules. Other major languages have a similar special dialect. The symbolic language of math is a special purpose system that is largely independent of any particular natural language. Graphs, geometric drawings and diagrams form a system for communication as well. The various systems are intertwined with each other in conversation, in lectures and in written math. [AbLM], [O’H], [Wel2003].

When we do math we think about math objects as if they were things.
Conceiving of math as talking about (abstract) objects enables us to think about it using the machinery in our brain we use to think about physical objects. This machinery is highly developed and uses metaphors and physical reasoning (maybe using mirror neurons). We could not do math without it. [LakNun], [WMN], [AbImMet].

Useful mathematical ideas tend to come from our physical experiences with our body and the world.
This is a thesis of [LakNun]. This understanding of the origins of mathematical objects might be developable into an explanation of the “unreasonable effectiveness of mathematics”. Note that you have to explain the effectiveness of mathematical reasoning as well as the usefulness of the objects we talk about.

Mathematical computation and reasoning lead to consistent results.
When we find inconsistent results using math we expect to find a mistake somewhere, and we usually do. This claim is about both numerical and algebraic computation and also formal mathematical reasoning. This phenomenon gives us confidence that mathematical processing is dependable.

This is what was behind my point about actual infinity in [AI]. When we envision the real numbers (for example) as an infinite set that exists all at once, and follow the correct rules of mathematical reasoning, it all works.

It’s also true that we now know of genuine limitations to what we can know, because of incompleteness results as well as cardinality results that say, for example, that there is an uncountably infinite number of real numbers that we cannot refer to individually.

Since in particular we can’t prove the consistency of a system within the system, our experience of consistent results is the only evidence we have that mathematics really
works and can be applied to the world.

References

[AbImMet] Images and metaphors
[AbLM] The languages of mathematics
[AbMO] Mathematical objects
[AI] Actual infinity
[MO] Gyre&Gimble (2007), Mathematical objects are “out there” ?
[LM] Gyre&Gimble (2007), The languages of mathematics
[NM] Gyre&Gimble (2007), More about neurons and math
[RIII] Gyre&Gimble (2008), Representations III: Rigor and rigor mortis.
[Her97] Hersh, R. What is Mathematics, Really? Oxford University Press, 1997. ISBN 978-0195113686
[LakNun] Lakoff, G. and R. E. Nüñez (2000). Where Mathematics Comes From. Basic Books. ISBN 978-0465037711.
[O’H] O’Halloran, K. L. (2005), Mathematical Discourse: Language, Symbolism And Visual Images. Continuum International Publishing Group. ISBN 978-0826468574
[TaBa2002] Tall, David and Barnard, Tony (2002). Cognitive units, connections and compression in mathematical thinking
[Ta2001] Tall, David (2001). Natural and formal infinities.
[Wel2003] Wells, C. (2003). The Handbook of Mathematical Discourse.
[WMN] Wikipedia on mirror neurons.

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Actual Infinity

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The infinite sequence 1, 1/2, 1/3, 1/4, … is commonly described as “getting closer and closer to 0”. This fits with the mental representation (metaphor) students have of the sequence as something whose successive entries are revealed or created over time.

People who are used to abstract math have another handle on that sequence: It is the function whose value at each positive integer n is 1/n. Stated that way, the sequence is being pictured as existing all at once (another metaphor). It is a completed infinity or actual infinity. Completed infinities occur all over modern math and mathematicians rarely remark on them. But some people outside math, in particular philosophers, get all hot and bothered about it.

I have not found a good reference on the internet to the idea of completed infinity. The article on actual infinity in Wikipedia gives lots of reasons against the idea but few reasons in favor of it. Many other articles on the internet also mostly discuss the opposition to the idea.

The phrase “the function whose value at each positive integer n is 1/n” is a definite and clear description of the sequence. Mathematicians have proved all sorts of statement about the sequence using classical logic; it is a Cauchy sequence converging to 0, for example. These statements all seem to be consistent with each other and with other parts of math. To put this in another way, we have a clear syntax for talking about this sequence and others, and we have built up a fund of statements about it that all hang together.

As always when talking about anything, we use metaphors and images when talking about this sequence. The image of the sequence as getting closer and closer to zero is one, which we could call the Xeno image. The image of the sequence as existing all at once is another, let’s say the actual-infinity image.

Neither image says anything about physical reality. The sequence is of course represented physically in our brains by arrangements of neurons or some such thing, but in no way does that imply that every entry is represented in our brain. What is represented in the brain are properties of the sequence, the metaphors and images we have mentioned (and others), relationships with similar sequences, and so on, what the math ed people call our schema of the concept. (See [1] and [2].)

I propose that the trouble philosophers and students have with the actual-infinity image occurs because there is a physical arrangement in a brain that serves to recognize big bunches of individual things. (Compare this post.) It may be triggered, for example, if you look out of a high window and see a big crowd of people standing in the street. You have a person-recognizer in your brain and you also have a big-bunch recognizer in your brain, which works with the person-recognizer to identify a crowd of people. (I talked about a similar idea, the I-saw-this-before recognizer, here).

A smart person soon realizes that there is a difference between a crowd of people and an infinite sequence, namely that the crowd is large but finite whereas the sequence is infinite. Well, that is true. So what? The crowd may be finite but you still can’t hold an image of each individual in the crowd in your head. The finite-infinite difference is indeed a difference, just as the idea that one is composed of physical objects (people) and the other is composed of mathematical objects (numbers). You can reason about both the crowd and the sequence, make discoveries about them, and so on.

Why is there a difficulty? Perhaps it is because a device in your brain that is usually used for bunches of physical objects is now being triggered by a bunch of abstract objects. That can be disconcerting.

But: You don’t have to get all upset about something just because it disconcerts you.

Now, then, no one ever needs to worry abou
t actual infinity again. (Fat chance).

REFERENCES

[1] Ed Dubinsky and Michael A. McDonald, APOS: A Constructivist Theory of Learning

in Undergraduate Mathematics Education Research

[2] David Tall, Reflections on APOS theory in Elementary and Advanced Mathematical Thinking.

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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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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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math

The Big Number Conjecture Conjecture

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Mathematicians have long noticed that in many fields, theorems have exceptions for small integers. Some theorems for compact differentiable manifolds can be proved for n bigger than 4, but things go haywire for 1, 2, 3, 4, especially 4. The finite simple groups have all been classified as being in one of several infinite families, with a finite list of exceptions, the largest being of order less than 10^54. (Well, that is small relative to most numbers!) The largest exceptional Lie group is a manifold of dimension 248.

Perhaps math gets better behaved for very large integers. This suggests a conjecture:

THE BIG NUMBER CONJECTURE CONJECTURE (sic)
If P(n) is a mathematical statement with one free variable n that ranges over the positive integers, then there is a number B_P depending only on the form of P with the property that, in order to prove that P(n) is true for all positive integers, it is sufficient to prove P(n) for all positive integers less than B_P.

Remarks:

a) This is precisely a conjecture that a meaningful conjecture exists.
b) The BNC is not a proper conjecture until I define “mathematical statement” precisely. Anyway, it may be true for some forms of statements and not others.
c) B_P has to depend on P because you could replace P(n) by P(f(n)) where f(n) is some slow growing function, such as the greatest integer in log log n.
d) But the dependence of B_P on P must be on the FORM of P in some sense (number of quantifiers or some such thing). Otherwise the conjecture is trivially true.

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Representations I

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This is the first of a series of blogs about representations in mathematics in a very broad sense.

Every kind of representation associates one kind of object with another kind of object, with the association limited to certain aspects of the objects. The way this association is limited is not always or even usually made explicit. There are many examples of different sorts of representations on the abstractmath website in the understanding math chapter, particularly in the articles on models and representations, and on images and metaphors. I intend to reorganize this material because my understanding of the situation has changed over the past year, so I will say some things here in g&g and hope for an informative reaction.

This posting is a summary of the various kinds of representation I want to talk about. The links above have more detail about many of them.

A representation can be physical, mental or mathematical, and what it represents can be a physical process or a mathematical object or other concepts.

Examples

  • The printed graph of a function or an icosahedron made out of plastic are physical representations of math objects.

  • What you picture in your mind when you think about the graph of a particular function is a mental representation of a math object.

  • Your visualization of a particle going faster or slower on a path may be a mental representation of both a physical process and a function of time that models the movement of the particle.
  • A matrix representation of a group, or a string of digits in base 10 notation, are mathematical representations of a mathematical objects.
  • The function describing the movement of the physical particle just mentioned is a mathematical model of a physical process.

Terminology
Words used for special types of representations are models, images, and metaphors.

  • A model may be a mathematical representation of a physical process.
  • A model in logic is a mathematical representation of a logical theory (which is a mathematical object).
  • A model may also be a physical representation (usually 3D) of a geometric object, such as that plastic icosahedron.
  • An image is a physical representation, a picture, of a mathematical object.
  • In Mathematics Education, the word “image” (concept image) is used to refer to a mental representation of a math object which may or may not be pictorial.

    • Metaphors

    Metaphors are one of the Big New Things in cognitive science and the word has had its meaning extended so much from the grammatical meaning that it may be referred to as a conceptual metaphor.

    • When you say the function f(x) = x^2 “goes to infinity when x gets large” you are using a metaphor.
    • When you think of the set of real numbers as an infinitely long line you are using a conceptual metaphor.

    Stay tuned…

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