All posts by SixWingedSeraph

Charles Wells. Professor Emeritus at Case Western Reserve University. Now living in Minneapolis, Minnesota, USA. CWRU provides support for this blog with software and library privileges.
exposition, math, representations, Uncategorized, understanding math

Visible Algebra I

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This is the first in a series of articles about how algebra could be implemented without using the standard language of algebra that so many people find difficult. The code for the graphs are in the Mathematica notebook Algebra1.nb.

An algebra problem

Suppose you are designing a window that is in the shape of a rectangle surmounted by a semicircle, shown above for the window with width 2 and rectangle height 3. 

This example occurs in a tiresomely familiar calculus problem where you put a constraint on the perimeter of the window, thus turning it into a one-variable problem, then finding the values of the width and height that give the maximum area.  In this post, I am not going to get that far.  All I will do is come up with a calculation for the area.  I will describe a way you might do it on a laptop five or ten years from now. 

You have an algebra application that shows a screen with some operations that you may select to paste into your calculation.  The ones we use are called plus, times, power, value and input. You choose a function called value, and label it "Area of window". You recognize that the answer is the sum of the areas of the rectangle and the area of the semicircle, so you choose plus and attach to it two inputs which you label "area of rectangle" and "area of semicircle", like this:

 

The notational metaphor is that the computation starts at the bottom and goes upward, performing the operations indicated.

You know (or are told by the system) that the area of a rectangle is the product of its width and height, so you replace the value called "area of rectangle" with a times button and attach two values called $w$ and $h$:

 

You also determine that the area under the semicircle is half the area of a circle of radius $r$ (where $r$ must be calculated).

 

You have a function for the area of a circle of radius $r$, so you attach that:

Finally, you use the fact that you know that the semicircle has a radius which is half the width of the rectangle.

Now, to make the calculation operational, you attach two inputs named "width" and "height" and feed them into the values $w$ and $h$.  When you type numbers into these buttons, the calculation will proceed upward and finally show the area of the window at the top.

In a later post I will produce a live version of this diagram.  (Added 2012-09-08: the live version is here.) Right now I want to get this post out before I leave for MathFest.  (I might even produce the live version at MathFest, depending on how boring the talks are.) 

You can see an example of a live calculation resembling this in my post A visualization of a computation in tree form.

Remarks

Who

  • This calculation might be a typical exercise for a student part way along learning basic algebra. 
  • College students and scientists and engineers would have a system with a lot more built-in functions, including some they built themselves.

Syntax

  • Once you have grasped the idea that the calculation proceed upward from the inputs, carrying out the operations shown, this picture is completely self-explanatory.
    • Well, you have to know what the operations do.
    • The syntax for standard algebra is much more difficult to learn (more later about this).
  • The syntax actually used in later years may not look like mine.
    • For one thing, the flow might run top down or left to right instead of bottom up. 
    • Or something very different might be used. What works best will be discovered by using different approaches.
  • The syntax is fully two-dimensional, which makes it simple to understand (because it uses the most powerful tool our brain has: the visual system).
    • The usual algebraic code was developed because people used pencil and paper. 
    • I would guess that the usual code has fractional dimension about 1.2. 
    • The tree syntax would require too much writing with pencil and paper.  That is alleviated on a computer by using menus.
    • Once you construct the computation and input some data it evaluates automatically.
  • It may be worthwhile to use 3D syntax.  I have an experiment with this in my post Showing categorical diagrams in 3D.

Later posts will cover related topics:

  • The difficulties with standard algebraic notation.  They are not trivial.
  • Solving equations in tree form.
  • Using properties such as associativity and commutativity in tree form.
  • Using this syntax with calculus.
  • The deep connection with Lawvere theories and sketches.

References

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exposition, language of math, math, representations, Uncategorized, understanding math

The mystery of the prime numbers: a review

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The Mystery of the Prime Numbers: Secrets of Creation v. 1, Matthew Watkins (Author), Matt Tweed (Illustrator). Inamorata Press, 2010.

The author and illustrator describe some of the mysteries of the distribution of primes, ending with Riemann's harmonic decomposition of the distribution. (If you don't understand what all that means, you will if you read the book and concentrate a lot). It is the first part of a trilogy which the author promises will culminate in a connection with quantum theory.  I can't wait.

The most remarkable thing about this book is the presentation.  You do not have to understand symbolic math.  The ideas are communicated using many, many pictures and metaphors. For example, you have a number of apples and you can't arrange them into a rectangle (a row doesn't count as a rectangle) then the number is a prime. Technical terminology is avoided, especially when the terminology consists of ordinary English words with new meanings.  So a prime factorization of a number is a "cluster" — and of course it really is since the order the primes occur in is irrelevant. On other occasions he will use the technical term (for example "distribution of primes" but warn you against your understanding being contaminated by the everyday meaning such as "distribute two pencils to each student").

The author is not afraid of saying the same thing several times, using different metaphors and rewording.  He will notice that some ideas will make you uncomfortable, such as the prime number theorem which "ought" to tell you the exact number of primes less than a number instead of merely estimate it.  (How many of your teachers ever admitted that an idea may make you uncomfortable and this is why it does…)  His explanation of functions and graphs is pictorial (using the endograph — arrows from one place to another on the real line) and kinetic (a ball rising from the x axis, hitting the graph, and being knocked horizontally to the y axis).

This makes me believe a reader who finds algebraic equations hard to understand and the nomenclature baffling can still get a reasonable mental picture of what primes are and what the Prime Number Theorem and Riemann's theorem on the distribution are actually saying.  It won't be easy: such a reader will have to concentrate and stop and think a lot (and I hope doodle pictures) and at times it will be slow going.  But this book makes it possible for someone who has no self-confidence in their ability to understand math to understand some deep stuff that many mathematicians find as astonishing as anything in math.

If we had a hundred authors writing books like this about different parts of math we would (in the long run) have fewer people who hate math or claim it all sounds like gibberish.

The long time reader of Gyre & Gimble will note that I have been saying for years that math should be explained like this.  So naturally I am pleased to recommend this book.  (Even so, all the times I taught primes I never thought of defining a prime number by saying you can't arrange your trinkets into a rectangle.  Oh, the chagrin…)

 

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computer science, exposition, language, language of math, math, representations, understanding math

Metaphors in computing science 2

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In Metaphors in Computer Science 1, I discussed some metaphors used when thinking about various aspects of computing.  This is a continuation of that post.

Metaphor: A program is a list of instructions.

  • I discussed this metaphor in detail in the earlier post.
  • Note particularly that the instructions can be in a natural or a programming language. (Is that a zeugma?)  Many writers would call instructions in a natural language an algorithm.
  • I will continue to use “program” in the broader sense.

Metaphor: A programming language is a language.

  • This metaphor is a specific conceptual blend that associates the strings of symbols that constitute a program in a computer language with text in a natural language.
  • The metaphor is based on some similarities between expressions in a programming language and expressions in a natural language.
    • In both, the expressions have a meaning.
    • Both natural and programming languages have specific rules for constructing well-formed expressions.
  • This way of thinking ignores many deep differences between programming languages and natural languages. In particular, they don’t talk about the same things!
  • The metaphor has been powerful in suggesting ways of thinking about computer programs, for example semantics (below) and ambiguity.

Metaphor: A computer program is a list of statements

  • A consequence of this metaphor is that a computer program is a list of symbols that can be stored in a computer’s memory.
  • This metaphor comes with the assumption that if the program is written in accordance with the language’s rules, a computer can execute the program and perhaps produce an output.
  • This is the profound discovery, probably by Alan Turing, that made the computer revolution possible. (You don’t have to have different physical machines to do different things.)
  • You may want me to say more in the heading above: “A computer program is a list of statements in a programming language that satisfies the well-formedness requirements of the language.”  But the point of the metaphor is only that a program is a list of statements.  The metaphor is not intended to define the concept of “program”.

Metaphor: A program in a computer language has meanings.

A program is intended to mean something to a human reader.

  • Some languages are designed to be easily read by a human reader: Cobol, Basic, SQL.
    • Their instructions look like English.
    • The algorithm can nevertheless be difficult to understand.
  • Some languages are written in a dense symbolic style.
    • In many cases the style is an extension of the style of algebraic formulas: C, Fortran.
    • Other languages are written in a notation not based on algebra:  Lisp, APL, Forth.
  • The boundary between “easily read” and “dense symbolic” is a matter of opinion!

A program is intended to be executed by a computer.

  • The execution always involves translation into intermediate languages. 
    • Most often the execution requires repeated translation into a succession of intermediate languages.
    • Each translation requires the preservation of the intended meaning of the program.
  • The preservation of intended meaning is what is usually called the semanticsof a programming language.
    • In fact, the meaning of the program to a person could be called semantics, too.
    • And the human semantics had better correspond in “meaning” to the machine semantics!
  • The actual execution of the program requires successive changes in the state of the computer.
    • By “state” I mean a list of the form of the electrical charges of each unit of memory in the computer.
    • Or you can restrict it to the relevant units of memory, but spelling that out is horrifying to contemplate.
    • The resulting state of the machine after the program is run is required to preserve the intended meaning as well as all the intermediate translations.
    • Notice that the actual execution is a series of physical events.  You can describe the execution in English or in some notation, but that notation is not the actual execution.

References

Conceptual blend (Wikipedia)

Conceptual metaphors (Wikipedia)

Images and Metaphors (article in abstractmath)

Semantics in computer science (Wikipedia)

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computer science, exposition, language of math, math, Mathematica, representations, Uncategorized, understanding math

A visualization of a computation in tree form

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Live evaluation of expressions in TreeForm 3, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The code is ad-hoc.  It might be worthwhile for someone to design a package that produces this sort of tree for any expression. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

This demonstration shows the step by step computation of the value of the expression $3x^2+2(1+y)$ shown as a tree.  By moving the first slider from right to left, you go through the six steps of the computation. You may select the values of $x$ and $y$ with the second and third sliders.  If you click on the plus sign next to a slide, a menu opens up that allows you to make the slider move automatically, shows the values, and other things.

Note that subtrees on the same level are evaluate left to right.  Parallel processing would save two steps.

A previous post related to this post is Making visible the abstraction in algebraic notation.

  

 
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exposition, Handbook, language of math, math, representations, understanding math

Conceptual blending

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This post uses MathJax.  If you see formulas in unrendered TeX, try refreshing the screen.

A conceptual blend is a structure in your brain that connects two concepts by associating part of one with part of another.  Conceptual blending is a major tool used by our brain to understand the world.

The concept of conceptual blend includes special cases, such as representations, images and conceptual metaphors, that math educators have used for years to understand how mathematics is communicated and how it is learned.  The Wikipedia article is a good starting place for understanding conceptual blending. 

In this post I will illustrate some of the ways conceptual blending is used to understand a function of the sort you meet with in freshman calculus.  I omit the connections with programs, which I will discuss in a separate post.

A particular function

Consider the function $h(t)=4-(t-2)^2$. You may think of this function in many ways.

FORMULA:

$h(t)$ is defined by the formula $4-(t-2)^2$.

  • The formula encapsulates a particular computation of the value of $h$ at a given value $t$.
  • The formula defines the function, which is a stronger statement than saying it represents the function.
  • The formula is in standard algebraic notation. (See Note 1)
  • To use the formula requires one of these:
    • Understand and use the rules of algebra
    • Use a calculator
    • Use an algebraic programming language. 
  • Other formulas could be used, for example $4t-t^2$.
    • That formula encapsulates a different computation of the value of $h$.

TREE: 

$h(t)$ is also defined by this tree (right).
  • The tree makes explicit the computation needed to evaluate the function.
  • The form of the tree is based on a convention, almost universal in computing science, that the last operation performed (the root) is placed at the top and that evaluation is done from bottom to top.
  • Both formula and tree require knowledge of conventions.
  • The blending of formula and tree matches some of the symbols in the formula with nodes in the tree, but the parentheses do not appear in the tree because they are not necessary by the bottom-up convention.
  • Other formulas correspond to other trees.  In other words, conceptually, each tree captures not only everything about the function, but everything about a particular computation of the function.
  • More about trees in these posts:

GRAPH:

$h(t)$ is represented by its graph (right). (See note 2.)

  • This is the graph as visual image, not the graph as a set of ordered pairs.
  • The blending of graph and formula associates each point on the (blue) graph with the value of the formula at the number on the x-axis directly underneath the point.
  • In contrast to the formula, the graph does not define the function because it is a physical picture that is only approximate.
  • But the formula does represent the function.  (This is "represents" in the sense of cognitive psychology, but not in the mathematical sense.)
  • The blending requires familiarity with the conventions concerning graphs of functions. 
  • It sets into operation the vision machinery of your brain, which is remarkably elaborate and powerful.
    • Your visual machinery allows you to see instantly that the maximum of the curve occurs at about $t=2$. 
  • The blending leaves out many things.
    • For one, the graph does not show the whole function.  (That's another reason why the graph does not define the function.)
    • Nor does it make it obvious that the rest of the graph goes off to negative infinity in both directions, whereas that formula does make that obvious (if you understand algebraic notation).  

GEOMETRIC

The graph of $h(t)$ is the parabola with vertex $(2,4)$, directrix $x=2$, and focus $(2,\frac{3}{4})$. 

  • The blending with the graph makes the parabola identical with the graph.
  • This tells you immediately (if you know enough about parabolas!) that the maximum is at $(2,4)$ (because the directrix is vertical).
  • Knowing where the focus and directrix are enables you to mechanically construct a drawing of the parabola using a pins, string, T-square and pencil.  (In the age of computers, do you care?)

HEIGHT:

$h(t)$ gives the height of a certain projectile going straight up and down over time.

  • The blending of height and graph lets you see instantly (using your visual machinery) how high the projectile goes. 
  • The blending of formula and height allows you to determing the projectile's velocity at any point by taking the derivative of the function.
  • A student may easily be confused into thinking that the path of the projectile is a parabola like the graph shown.  Such a student has misunderstood the blending.

KINETIC:

You may understand $h(t)$ kinetically in various ways.

  • You can visualize moving along the graph from left to right, going, reaching the maximum, then starting down.
    • This calls on your experience of going over a hill. 
    • You are feeling this with the help of mirror neurons.
  • As you imagine traversing the graph, you feel it getting less and less steep until it is briefly level at the maximum, then it gets steeper and steeper going down.
    • This gives you a physical understanding of how the derivative represents the slope.
    • You may have seen teachers swooping with their hand up one side and down the other to illustrate this.
  • You can kinetically blend the movement of the projectile (see height above) with the graph of the function.
    • As it goes up (with $t$ increasing) the projectile starts fast but begins to slow down.
    • Then it is briefly stationery at $t=2$ and then starts to go down.
    • You can associate these feelings with riding in an elevator.
      • Yes, the elevator is not a projectile, so this blending is inaccurate in detail.
    • This gives you a kinetic understanding of how the derivative gives the velocity and the second derivative gives the acceleration.

OBJECT:

The function $h(t)$ is a mathematical object.

  • Usually the mental picture of function-as-object consists of thinking of the function as a set of ordered pairs $\Gamma(h):=\{(t,4-(t-2)^2)|t\in\mathbb{R}\}$. 
  • Sometimes you have to specify domain and codomain, but not usually in calculus problems, where conventions tell you they are both the set of real numbers.
  • The blend object and graph identifies each point on the graph with an element of $\Gamma(h)$.
  • When you give a formal proof, you usually revert to a dry-bones mode and think of math objects as inert and timeless, so that the proof does not mention change or causation.
    • The mathematical object $h(t)$ is a particular set of ordered pairs. 
    • It just sits there.
    • When reasoning about something like this, implication statements work like they are supposed to in math: no causation, just picking apart a bunch of dead things. (See Note 3).
    • I did not say that math objects are inert and timeless, I said you think of them that way.  This post is not about Platonism or formalism. What math objects "really are" is irrelevant to understanding understanding math [sic].

DEFINITION

A definition of the concept of function provides a way of thinking about the function.

  • One definition is simply to specify a mathematical object corresponding to a function: A set of ordered pairs satisfying the property that no two distinct ordered pairs have the same second coordinate, along with a specification of the codomain if that is necessary.
  • A concept can have many different definitions.
    • A group is usually defined as a set with a binary operation, an inverse operation, and an identity with specific properties.  But it can be defined as a set with a ternary operation, as well.
    • A partition of a set is a set of subsets of a set with certain properties. An equivalence relation is a relation on a set with certain properties.  But a partition is an equivalence relation and an equivalence relation is a partition.  You have just picked different primitives to spell out the definition. 
    • If you are a beginner at doing proofs, you may focus on the particular primitive objects in the definition to the exclusion of other objects and properties that may be more important for your current purposes.
      • For example, the definition of $h(t)$ does not mention continuity, differentiability, parabola, and other such things.
      • The definition of group doesn't mention that it has linear representations.

SPECIFICATION

A function can be given as a specification, such as this:

If $t$ is a real number, then $h(t)$ is a real number, whose value is obtained by subtracting $2$ from $t$, squaring the result, and then subtracting that result from $4$.

  • This tells you everything you need to know to use the function $h$.
  • It does not tell you what it is as a mathematical object: It is only a description of how to use the notation $h(t)$.

Notes

1. Formulas can be give in other notations, in particular Polish and Reverse Polish notation. Some forms of these notations don't need parentheses.

2. There are various ways to give a pictorial image of the function.  The usual way to do this is presenting the graph as shown above.  But you can also show its cograph and its endograph, which are other ways of representing a function pictorially.  They  are particularly useful for finite and discrete functions. You can find lots of detail in these posts and Mathematica notebooks:

3. See How to understand conditionals in the abstractmath article on conditionals.

References

  1. Conceptual blending (Wikipedia)
  2. Conceptual metaphors (Wikipedia)
  3. Definitions (abstractmath)
  4. Embodied cognition (Wikipedia)
  5. Handbook of mathematical discourse (see articles on conceptual blend, mental representation, representation, and metaphor)
  6. Images and Metaphors (article in abstractmath)
  7. Links to G&G posts on representations
  8. Metaphors in Computing Science (previous post)
  9. Mirror neurons (Wikipedia)
  10. Representations and models (article in abstractmath)
  11. Representations II: dry bones (article in abstractmath)
  12. The transition to formal thinking in mathematics, David Tall, 2010
  13. What is the object of the encapsulation of a process? Tall et al., 2000.

 

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Making visible the abstraction in algebraic notation

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Handmade Exp Tree.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Algebraic notation

Algebraic notation contains a hidden abstract structure coded by apparently arbitrary conventions that many college calculus students don't understand completely. This very simple example shows one of the ways in which calc students may be confused:

  1. $x+2y$
  2. $(x+2)y$

Students often mean to express formula 2 when they write something like \[x\!\!+\!\!2\,\,\,\,y\] (with a space).  This is a perfectly natural way to write it. But it is against the rules, I presume because in handwriting it is not clear when you mean a space and when you don't. 

Formula 1 can also be written as $x+(2y)$, and if it were usually written that way students (I predict) would be less confused.   Always writing it this way would exacerbate the clutter of parentheses but would allow a simple rule:

Evaluate every expression inside parentheses first, starting with the innermost.

Using trees for algebra

Writing algebraic expressions as a tree (as in computing science)

  • makes it obvious what gets evaluated first
  • uses no parentheses at all.

An example of using the tree of an expression to do calculations is available in Expressions.nb (requires Mathematica) and Expressions.cdf (requires CDF player only) on my Mathematica website.  I could imagine using tree expressions instead of standard notation as the normal way of doing things. That would require working on Ipads or some such and would take a big amount of investment in software making it intuitive and easy to use.  No, I am not going to embark on such an adventure, but I think it ought to be attempted.  (Brett Victor has many ideas like this.)

Transforming algebraic notation into trees

The two manipulable diagrams below show the algebraic notation being transformed into tree form.  I expect that this will make the abstract structure more concrete for many students and I encourage others to show it to their students.  Note that the tree form makes everything explicit.>

After I return from a ten-day trip I will explore the possibility of making the expression-to-tree transformer turn the expression into an evaluable tree as in Expressions.nb and Expressions.cdf.  In the I hope not to distant future students should have access to many transformers that morph expressions from one form into another.  Such transformers are much more politically correct than Optimus Prime.

Offloading chunking and Computable algebraic expressions in tree form are earlier posts related to this post.

 

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Metaphors in computing science I

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(This article is continued in Metaphors in computing science II)

Michael Barr recently told me of a transcription of a talk by Edsger Dijkstra dissing the use of metaphors in teaching programming and advocating that every program be written together with a proof that it works.  This led me to think about the metaphors used in computing science, and that is what this post is about.  It is not a direct answer to what Dijkstra said. 

We understand almost anything by using metaphors.  This is a broader sense of metaphor than that thing in English class where you had to say "my love is a red red rose" instead of "my love is like a red red rose".  Here I am talking about conceptual metaphors (see references at the end of the post).  

Metaphor: A program is a set of instructions

You can think of a program as a list of instructions that you can read and, if it is not very complicated, understand how to carry them out.  This metaphor comes from your experience with directions on how to do something (like directions from Google Maps or for assembling a toy).   In the case of a program, you can visualize doing what the program says to do and coming out with the expected output. This is one of the fundamental metaphors for programs. 

Such a program may be informal text or it may be written in a computer language.

Example

A description of how to calculate $n!$ in English could be:  "Multiply the integers $1$ through $n$".  In Mathematica, you could define the factorial function this way:

fac[n_] := Apply[Times, Table[i, {i, 1, n}]]

This more or less directly copies the English definition, which could have been reworded as "Apply the Times function to the integers from $1$ to $n$ inclusive."  Mathematica programmers customarily use the abbreviation "@@" for Apply because it is more convenient:

Fac[n_]:=Times @@ Table[i, {i, 1, 6}]

As far as I know, C does not have list operations built in.  This simple program gives you the factorial function evaluated at $n$:

 j=1;  for (i=2; i<=n; i++)   j=j*i; return j;  

This does the calculation in a different way: it goes through the numbers $1, 2,\ldots,n$ and multiplies the result-so-far by the new number.  If you are old enough to remember Pascal or Basic, you will see that there you could use a DO loop to accomplish the same thing.

What this metaphor makes you think of

Every metaphor suggests both correct and incorrect ideas about the concept.  

  • If you think of a list of instructions, you typically think that you should carry out the instructions in order.  (If they are Ikea instructions, your experience may have taught you that you must carry out the instructions in order.)  
  • In fact, you don't have to "multiply the numbers from $1$ to $n$" in order at all: You could break the list of numbers into several lists and give each one to a different person to do, and they would give their answers to you and you would multiply them together.
  • The instructions for calculating the factorial can be translated directly into Mathematica instructions, which does not specify an order.   When $n$ is large enough, Mathematica would in fact do something like the process of giving it to several different people (well, processors) to speed things up.
  • I had hoped that Wolfram alpha would answer "720" if I wrote "multiply the numbers from $1$ to $6$" in its box, but it didn't work.  If it had worked, the instruction in English would not be translated at all. (Note added 7 July 2012:  Wolfram has repaired this.)
  • The example program for C that I gave above explicitly multiplies the numbers together in order from little to big.  That is the way it is usually taught in class.  In fact, you could program a package for lists using pointers (a process taught in class!) and then use your package to write a C program that looks like the  "multiply the numbers from $1$ to $n$" approach.  I don't know much about C; a reader could probably tell me other better ways to do it.

So notice what happened:

  • You can translate the "multiply the numbers from $1$ to $n$" directly into Mathematica.
  •  For C, you have to write a program that implements multiplying the numbers from $1$ to $n$. Implementation in this sense doesn't seem to come up when we think about instruction sets for putting furniture together.  It is sort of like: Build a robot to insert & tighten all the screws.

Thus the concept of program in computing science comes with the idea of translating the program instruction set into another instruction set.

  • The translation provided above for Mathematica resembles translating the instruction set into another language. 
  • The two translations I suggested for C (the program and the definition of a list package to be used in the translation) are not like translating from English to another language.  They involve a conceptual reconstruction of the set of instructions.

Similarly, a compiler translates a program in a computer language into machine code, which involves automated conceptual reconstruction on a vast scale.

Other metaphors

In writing about this, I have brought in other metaphors, for example:

  • C or Mathematica as like a natural language in some ways 
  • Compiling (or interpreting) as translation

Computing science has used other VIM's (Very Important Metaphors) that I need to write about later:

  • Semantics (metaphor: meaning)
  • Program as text — this allows you to treat the program as a mathematical object
  • Program as machine, with states and actions like automata and Turing machines.
  • Specification of a program.  You can regard  "the product of the numbers from $1$ to $n$" as a specification.  Notice that saying "the product" instead of "multiply" changes the metaphor from "instruction" to "specification".

References

Conceptual metaphors (Wikipedia)

Images and Metaphors (article in abstractmath)

Images and Metaphors for Sets (article in abstractmath)

Images and Metaphors for Functions (incomplete article in abstractmath)

 

 
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Improved clouds

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome.

The source code for these demos is Animated Riemann.nb at my Mathematica Site. The notebook is is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

The animated clouds show two hundred precalculated clouds for each picture, so you get the same clouds each time you run the animation.  It would have taken too long to generate the random clouds on the fly.  Each list of two hundred took about seven minutes to create on my computer.

In my post Riemann Clouds Improved I showed examples of clouds of values of Riemann sums in such a way that you could see the convergence to the value, the efficiency of the midpoint rule, and other things.  Here I include two Riemann sums that are shown

  • as manipulable graphs,
  • in clouds in an animated form.

Each manipulable graph (see Elaborate Riemann Sums Demo) has a slider to choose the mesh (1/n) of the partitions.  The small plus sign besides the slider gives you additional options. The buttons allow you to choose the type of partition and the type of evaluation points.

Each cloud shows a collection of values of random Riemann sums of the function, plotted by size of mesh (an upper bound on the width of the largest subdivision) and the value of the sum.  The cloud shows how the sums converge to the value of the integral. 

Every dot represents a random partition.  The sums with blue dots have random valuation points, the green dots use the left side of the subdivision, the brown dots the right side, and the red dots the midpoint.  The clouds may be suitable for students to study.  Some possible questions they could be asked to do are listed at the end.

Pressing the starter shows many clouds in rapid succession.  I don't know how much educational value that has but I think it is fun, and fun is worthwhile in itself.

Quarter Circle

Manipulable graph:

Animated cloud

 

Sine wave

Manipulable graph:

Animated cloud

Questions

I am not sure of the answers to some of these myself.

  • Why is the accuracy generally better for the sine wave than for the quarter circle?  
  • Why are the green dots above all the others and the brown dots below all the others in the quarter circle?
  • Why are they mixed in with the others for the sine curve?  In fact why do they tend upward? (Going from right to left, in other words in the direction of more accuracy).
  • Why are the midpoint sums so much more accurate?
  • Why do they tend downward for the sine wave?
  • Is it an optical illusion or do they also tend downward for the quarter circle? 

Notes:

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More about the definition of function

1 Comment

Maya Incaand commented on my post Definition of "function":

Why did you decide against "two inequivalent descriptions in common use"?  Is it no longer true?

This question concerns [1], which is a draft article.  I have not promoted it to the standard article in abstractmath because I am not satisfied with some things in it. 

More specifically, there really are two inequivalent descriptions in common use.  This is stated by the article, buried in the text, but if you read the beginning, you get the impression that there is only one specification.  I waffled, in other words, and I expect to rewrite the beginning to make things clearer.

Below are the two main definitions you see in university courses taken by math majors and grad students.  A functional relation has the property that no two distinct ordered pairs have the same first element.

Strict definition: A function consists of a functional relation with specified codomain (the domain is then defined to be the set of first elements of pairs in the relation).  Thus if $A$ and $B$ are sets and $A\subseteq B$, then the identity function $1_A:A\to A$ and the inclusion function $i:A\to B$  are two different functions.

Relational definition: A function is a functional relation.  Then the identity and inclusion functions are the same function.  This means that a function and its graph are the same thing (discussed in the draft article).

These definitions are subject to variations:

Variations in the strict definition: Some authors use "range" for "codomain" in the definition, and some don't make it clear that two functions with the same functional relation but different codomains are different functions.

Variations in the relational definition: Most such definitions state explicitly that the domain and range are determined by the relation (the set of first coordinates and the set of second coordinates). 

Formalism

There are many other variations in the formalism used in the definition.  For example, the strict definition can be formalized (as in Wikipedia) as an ordered triple $(A, B, f)$ where $A$ and $B$ are sets and $f$ is a functional relation with the property thar every element of $A$ is the first element of an ordered pair in the relation.  

You could of course talk about an ordered triple $(A,f,B)$ blah blah.  Such definitions introduce arbitrary constructions that have properties irrelevant to the concept of function.  Would you ever say that the second element of the function $f(x)=x+1$ on the reals is the set of real numbers?  (Of course, if you used the formalism $(A,f,B)$ you would have to say the second element of the function is its graph! )

It is that kind of thing that led me to use a specification instead of a definition.  If you pay attention to such irrelevant formalism there seems to be many definitions of function.  In fact, at the university level there are only two, the strict definition and the relational definition.  The usage varies by discipline and age.  Younger mathematicians are more likely to use the strict definition.  Topologists use the strict definition more often than analysts (I think).

Usage

There is also variation in usage.

  • Most authors don't tell you which definition they use, and it often doesn't matter anyway. 
  • If an author defines a function using a formula, there is commonly an implicit assumption that the domain includes everything for which the formula is well-defined.  (The "everything" may be modified by referring to it as an integer, real, or complex function.)

Definitions of function on the web

Below are some definitions of function that appear on the web.  I have excluded most definitions aimed at calculus students or below; they often assume you are talking about numbers and formulas.  I have not surveyed textbooks and research papers.  That would have to be done for a proper scholarly article about mathematical usage of "function". But most younger people get their knowledge from the web anyway.

  1. Abstractmath draft article: Functions: Specification and Definition.  (Note:  Right now you can't get to this from the Table of Contents; you have to click the preceding link.) 
  2. Gyre&Gimble post: Definition of "function"
  3. Intmath discussion of function  Function as functional relation between numbers, with induced domain and range.
  4. Mathworld definition of function Functional-relation definition.  Defines $F:A\to B$ in a way that requires $B$ to be the image.
  5. Planet Math definition of function Strict definition.
  6. Prime Encyclopedia of Mathematics Functional-relation definition.
  7. Springer Encyclopedia of Math definition of function  Strict definition, except not clear if different codomains mean different functions.
  8. Wikipedia definition of function Discusses both definitions.
  9. Wisconsin Department of Public Instruction Definition of function  Function as functional relation.
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