language of math, math, representations, understanding math

Different names for the same thing

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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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Distributive plurals

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A statement in English such as “all squared nonzero real numbers are positive” is called a distributive plural.  This means that the statement “the square of x is positive” is true for every nonzero real number.  It can be translated directly into symbolic notation:  \forall x\,(\text{if }x\ne 0\text{ then }{{x}^{2}}>0)

Not all statements involving plurals in English are distributive plurals.  The statement “The agents are surrounding the building” does not imply that Agent James is surrounding the building.  This type of statement is called a collective plural. Such a statement cannot be translated directly into a statement involving a universal quantifier.  More about this here.  This discussion on Wordwizard suggests that there may be a difference between British and American usage.

The word “distributive” as used here is analogous to the distributive law of arithmetic.  If the set of things referred to is finite, for example the set {-2, -1, 1, 3} then one can say  that “\forall x\,({{x}^{2}}>0)” is equivalent to “{{(-2)}^{2}}>0\text{ and }{{(-1)}^{2}}>0\text{ and }{{\text{1}}^{2}}>0\text{ and }{{\text{3}}^{2}}>0”.

I once found a report on the internet that a Quaker Oats box contained this exhortation: “Eating a good-sized bowl of Quaker Oatmeal for 30 days will actually help remove cholesterol from your body.”  This undoubtedly exhibits a confusion between distributive plurals and the other kind of plural, but I don’t understand the connection well enough to explain it.

I can no longer find the report on the internet.  This may mean the Quaker Oats box with that label never existed.

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Handbook now online

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I have placed an interactive version of the Handbook of Mathematical Discourse on line here. Its formatting is still a little rough, and it omits the quotations and illustrations from the printed book. It also needs the backlinks from the citations and bibliography reactivated. I will do that when I Get Around To It.

Now I can refer to the Handbook via a direct link from a blog post or from abstractmath, and you can click on a lexicographical citation and go directly to the text of the citation.

Comments and error reports are welcome.

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Variables

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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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    Reticulating Splines?

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    My backup program (Mozy) went awry.  I followed their instructions for repairing it (delete certain files) and it started up.  It took a long time to get going (but it finally did), and during that time it emitted several messages.  One of them was “Reticulating splines”.

    This causes me considerable Cognitive Dissonance (sometimes known as CD).  In fact, this causes me Dismaying Virtual Disturbance (otherwise known as DVD).  Reticulating splines is all about continuous  (indeed differentiable) things.  Backing up digital files is all about discrete things.  How can you apply splines to a discrete process?

    No doubt some reader will tell me.

    POSTSCRIPT:  I just found out, from here. Now I have Egg On My Face (EOMF) because I wrote a blog about something without looking it up on Google. Sigh.

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    Sets don't have to be homogeneous?

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    Colm Bhandal commented on my article on sets in abstractmath.org.

    Let me first of all say that I am impressed with your website. It gave
    me a few very good insights into set notation. Now, I’ll get straight
    to the point. While reading your page, I came across a section
    claiming that:

    “Sets do not have to be homogeneous in any sense”

    This confused me for a while, as I was of the opinion that all objects in a set were of the same type. After thinking about it for a while, I came to a conclusion:

    A set defines a level of abstraction at which all objects are homogeneous, though they may not be so at other levels of abstraction.

    Taking the example on your page, the set {PI^2, M, f, 42, -1/e^2} contains two irrational numbers, a matrix, a function, and a whole number. Thus, the elements are not homogeneous from one perspective (level of abstraction as I call it) in that they are spread across four known sets. However, in another sense they are homogeneous, in that they are all mathematical objects. Sure, this is a very high level of abstraction: A mathematical object could be a lot of things,
    but it still allows every object in the set to be treated homogeneously i.e. as mathematical objects.

    You are right.  I think I had better say “The elements of a set do not have to be ‘all of the same kind’ in the sense of that phrase in everyday speech.”  Of course, a mathematician would say the elements of a set S are “all of the same kind”, the “kind” being elements of S.
     
    Apparently, according to the way our brains work, there are natural kinds and artificial kinds.  There is something going on in my students’ minds that cause them to be bothered by sets like that given about or even sets such as {1,3,5,6,7,9,11} (see the Handbook, page 279).   Philosophers talk about “natural kinds” but they seem to be referring to whether they exist in the world.  What I am talking about is a construct in our brain that makes “cat” a natural kind and “blue-eyed OR calico cat” an artificial kind.  Any teacher of abstract math knows that this construct exists and has to be overcome by talking about how sets can be arbitrary, functions can be arbitrary, and so on, and that’s OK.

     This distinction seems to be built into our brains.  A large part of abstractmath.org is devoted to pointing out the clashes between mathematical thinking and everyday thinking. 

    Disclaimer:  When I say the distinction is “built into our brains” I am not claiming that it is or is not inborn; it may be a result of cultural conditioning. What seems most likely to me is that our brains are wired to think in terms of natural kinds, but culture may affect which kinds they learn.  Congnitive theorists have studied this; they call them “natural categories” and the study is part of prototype theory.  I seem to remember reading that they have some evidence that babies are born with the tendency to learn natural categories, but I don’t have a reference.

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    Studying math using linguistics

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    The Handbook and part of the abstractmath website involve studying math from the point of view of linguistics.  I thought I was all by myself until K. L. O’Halloran’s book came out:

    O’Halloran, K. L. (2005), Mathematical Discourse: Language, Symbolism And Visual Images. Continuum International Publishing Group.

    She is a semioticist.

    Now another new book also looks at math from a linguistic point of view, this time in connection with how mathematical discourse works in other languages:

    Barton, Bill (2009), The Language of Mathematics: Telling Mathematical Tales. Springer.

    It is described in Reidar Mosvold’s math ed blog.

    By the way, lots of books and articles that use phrases such as “the language of math” are not written from a linguistics point of view at all.

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    How "math is logic" ruined math for a generation

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    Mark Meckes responded to my statement

    But it seems to me that this sort of thinking has mostly resulted in people thinking philosophy of math is merely a matter of logic and set theory.  That point of view has been ruinous to the practice of math.

    with this comment:

    I may be misreading your analysis of the second straw man, but you seem to imply that “people thinking philosophy of math is merely a matter of logic and set theory” has done great damage to mathematics. I think that’s quite an overstatement. It means that in practice, mathematicians find philosophy of mathematics to be irrelevant and useless. Perhaps philosophers of mathematics could in principle have something to say that mathematicians would find helpful but in practice they don’t; however, we’re getting along quite well without their help.

    On the other hand, maybe you only meant that people who think “philosophy of math is merely a matter of logic and set theory” are handicapped in their own ability to do mathematics. Again, I think most mathematicians get along fine just not thinking about philosophy.

    Mark is right that at least this aspect of philosophy of math is irrelevant and useless to mathematicians.  But my remark that the attitude that “philosophy of math is merely a matter of logic and set theory” is ruinous to math was sloppy, it was not what I should have said.    I was thinking of a related phenomenon which was ruinous to math communication and teaching.

    By the 1950’s many mathematicians adopted the attitude that all math is is theorem and proof.  Images, metaphors and the like were regarded as misleading and resulting in incorrect proofs.  (I am not going to get into how this attitude came about).     Teachers and colloquium lecturers suppressed intuitive insights and motivations in their talks and just stated the theorem and went through the proof.

    I believe both expository and research papers were affected by this as well, but I would not be able to defend that with citations.

    I was a math student 1959 through 1965.  My undergraduate calculus (and advanced calculus) teacher was a very good teacher but he was affected by this tendency.  He knew he had to give us intuitive insights but he would say things like “close the door” and “don’t tell anyone I said this” before he did.  His attitude seemed to be that that was not real math and was slightly shameful to talk about.  Most of my other undergrad teachers simply did not give us insights.

    In graduate school I had courses in Lie Algebra and Mathematical Logic from the same teacher.   He was excellent at giving us theorem-proof lectures, much better than most teachers, but he never gave us any geometric insights into Lie Algebra (I never heard him say anything about differential equations!) or any idea of the significance of mathematical logic.  We went through Killing’s classification theorem and Gödel’s incompleteness theorem in a very thorough way and I came out of his courses pleased with my understanding of the subject matter.  But I had no idea what either one of them had to do with any other part of math.

    I had another teacher for several courses in algebra and various levels of number theory.   He was not much for insights, metaphors, etc, but he did do well in explaining how you come up with a proof.  My teacher in point set topology was absolutely awful and turned me off the Moore Method forever.   The Moore method seems to be based on: don’t give the student any insights whatever. I have to say that one of my fellow students thought the Moore method was the best thing since sliced bread and went on to get a degree from this teacher.

    These dismal years in math teaching lasted through the seventies and perhaps into the eighties.  Apparently now younger professors are much more into insights, images and metaphors and to some extent into pointing out connections with the rest of math and science.  Since I have been retired since 1999 I don’t have much exposure to the newer generation and I am not sure how thoroughly things have changed.

    One noticeable phenomenon was that category theorists (I got into category theory in the mid seventies) were very assiduous in lectures and to some extent in papers in giving motivation and insight.  It may be that attitudes varied a lot between different disciplines.

    This Dark Ages of math teaching was one of the motivations for abstractmath.org.  My belief is that not only should we give the students insights, images and metaphors to think about objects, and so on, but that we should be upfront about it:   Tell them what we are doing (don’t just mutter the word “intuitive”) and point out that these insights are necessary for understanding but are dangerous when used in proofs.  Tell them these things with examples. In every class.

    My other main motivation for abstractmath.org was the way math language causes difficulties.  But that is another story.

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    Mental Representations in Math

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    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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