All posts by Charles Wells

math, proof, representations, understanding math

Representations II: Dry Bones

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

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

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

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

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

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

dead totality.

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

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

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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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math, understanding math

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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    astounding math, math

    Astounding Math Stories

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    Astounding Math Stories

    I am in the process of selling my old collection of Astounding Science Fiction on Ebay. It occurred to me that math needs an Astounding Math magazine. It could contain short descriptions of some mathematical facts that are really weird that could awaken people’s interest in math. Well, geeky people anyway.

    These astounding facts also illustrate some important ideas in math. For one thing, some of them are not astounding when you get the right representation or proof (for example e^{iπ} = –1). And some are simply frauds (infinite cardinals).

    Each of the examples below will be fleshed out and given references. The first one, the Perrin function, has already been posted on the Astounding Math Stories website. Comments and suggestions are earnestly desired.

    Perrin pseudoprimes

    The Perrin function P(n) is a certain easily-defined function on the natural numbers with this property:

    For all integers n e^{iπ} = –1

    This is Astounding. Who would have thought that the numbers e, i and π would be related in this way? Well, actually, it is not hard to understand why it is true if you use Euler’s formula in the context of the Argand representation. And the fact that Euler’s formula works is not very difficult to understand. (Perhaps Euler’s formula is nevertheless Astounding. Feynman thought it was.) This is an excellent example of the ratchet effect: An amazing or incomprehensible statement about math suddenly becomes totally obvious and you can’t understand why you didn’t understand it before!

    Infinite cardinals

    There are “as many” integers as there are rational numbers. This is Astounding!

    Really? In fact, this statement is a fraud. It depends on defining the cardinality of a set in terms of bijections (not a fraud) and then referring to the cardinality of the integers or rationals in terms of words like “many”.

    What is happening is that, for finite sets, the cardinality function on sets Card(S) means the same as #(S). the number of elements of S. On infinite sets, the cardinality function does not have some of the familiar properties of the number of elements of a finite set; in particular, it can happen that S is a proper subset of T but Card(S) = Card(T). Unless you specifically state, “When I say S has as many elements as T, I mean Card(S) = Card(T)”, then you are deliberately using the word “many” with a nonstandard meaning that the listener may not know. This is like the politician who told his audience that his opponent was a “sexagenerian”. Playing on someone’s ignorance is FRAUD.

    Composite numbers
    It is possible to prove that an integer n is composite without knowing any nontrivial factors of n. This is Astounding! How could you show it is composite without finding a nontrivial factor? This is a consciousness-raising example that shows that just because you can’t think of how to do something doesn’t mean it is impossible.

    To show that n is not a prime, all you have to do is find an integer a for which a^n – a is not divisible by n (Fermat’s Little Theorem). That is not even very hard to prove.

    Humongous numbers

    As far as we know, π(x) − li(x) changes sign for the first time at some humongous number. 1.397×10^316 is a LARGE number. Other even huger numbers are Moser’s number and Graham’s number. One could also refer to Ackerman’s function. All these are consciousness raising examples that show how big numbers can get.

    Many years ago the Little Girl Next Door asked me what the next number after a trillion is. I said, a trillion and one. She was Disgusted.

    Function Spaces

    The Astounding thing about function spaces is that each point in a function space is a function. A whole function, like sin x or the Riemann Zeta function. Not its values, not its formula, but the whole thing. This story is going to be difficult to write, but if I can carry it off it may help the student along the way to understanding the concept of an encapsulated mathematical object.

    Other topics:

    The monster group

    1^2+2^2+3^2+…+24^2 = 702 and the Leech Lattice

    41 and 163

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    exposition, language

    Automatic spelling reform

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    The English language badly needs spelling reform. It is becoming a widely used language and the spelling is a real hindrance in learning it. But spelling reform has two big problems:

    Resistance
    Both the French and the Germans have tried rather minor spelling reforms in recent years that have utterly failed. The Chinese Communists made substantial changes in the characters used in Chinese and succeeded where they had dictatorial control, but failed in the diaspora. As a result, people educated in Taiwan and Hong Kong can’t read stuff printed on the mainland and vice versa. On the other hand, Greek spelling reform, mostly a matter of simplifying the accents put on vowels, seems to have succeeded.

    Any English spelling reform would succeed at most partially, resulting in texts being written in two spellings, the old one and the new one. People who grew up on the old one probably could learn to read the new one, but never as easily as they read the old one. And conversely.

    Dialect differences
    Americans outside the south pronounce “Mary”, “merry” and “marry” the same. Southerners and Britishers distinguish between two or three of them.

    Britishers pronounce “Wanda” and “wander” the same. Americans pronounce them differently.

    Most Americans pronounce “bother” and “father” so they rhyme. Some Americans pronounce “cot” and “caught” the same. Canadians and Britishers distinguish these pairs.

    The people who don’t distinguish between two phonemes have to learn different spellings for words that sound the same, or else people who DO distinguish them have to whether the writer meant merry or marry, for example.
     
     
    Technology comes to the rescue
    The text-to-speech system in Excel 2003 pronounces both of the following sentences correctly:
    “We will record the song and I will make a record of it.”
    “I will read the book and when I have read it I will tell you.”
    It no doubt makes mistakes in some situations, too.
    This system presumably operates by at least partially parsing the sentences, looking at context and perhaps using other methods as well. So it would be possible to devise a system that would convert text on the fly from a traditional spelling to a reformed spelling. This would probably work well most of the time and could allow several different spelling systems to flourish. When books involving fixed print on paper become obsolete, as they surely will, this will solve the problem.
    One obvious way to do this is to add diacritics and accent marks to the existing spelling.
     
    Note
    On this blog I once proposed that subject and predicate phrases in English be color-coded. Writers would not want to do this by hand, but when sentence parsing gets good enough (maybe it already is) this could be done automatically in the same way as different spellings.
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    understanding math

    Mighty Mathematician Shows Idiosyncrasy

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    I started attending a dance aerobics class about eight years ago. For the first few weeks I had considerable trouble because I copied what the instructor did, and everyone else copied the mirror image of what the instructor did. When she raised her right arm I raised my right arm and everyone else raised their left arms.

    Pretty soon I learned to do what everyone else did, except for occasional lapses. This morning I thought about mentioning this idiosyncrasy on this blog and got confused enough that I had to stop dead and start over — and then it took me starting over twice to get back into the exercise.

    I thought smugly that this was all because I am a Mathematician and understand Coordinate Systems intuitively. Or at least because I am a Boy and can Read Maps Intuitively. However, none of the other mathematicians and scientists in the class were doing this.

    Two people occasionally reverse what they do on purpose and pretend to bump into other people. They do this very skillfully. One is an English Professor and the other is a botanist. They are both much better at this dancing thing than I am.

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    language, representations

    Technical words in English

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

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

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

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

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

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    language, language of math, math, representations

    Set notation

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

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

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

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

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

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

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

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

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    language

    Most annoying bug in English?

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    I said in a recent posting, “Perhaps the most annoying bug in English is the fact that “two”, “to” and “too” are pronounced identically.” That may be the most annoying bug in spoken English but the lack of a gender neutral third person pronoun is annoying in both spoken and written English.

    This bug may be on its way to being eliminated by using “they” for “he or she”. This has got me to thinking about the future of the language. The plural “you” has already replaced the singular “thou/thee” in English. Perhaps someday “they” will oust “he” and “she” completely and become the only third person pronoun, or perhaps more likely the animate third person pronoun, contrasting with “it”.

    English has a compulsory plural for nouns, like other European languages (except Turkish) and unlike most East Asian languages. But if the second and third person pronouns stop marking the plural, will nouns be far behind? A few dialects already eliminate it — notably in the West Indies. This development will only be encouraged as English becomes more and more everyone’s second language. I hope someone reads this in 2107 and puts me down in history as a linguistic Notradamus.

    There is lots of interesting stuff about singular “they” on the web: See the comments on World Wide Words and by Geoff Pullum . (Was the previous sentence a zeugma?)

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