I have posted a revised version of the abstractmath.org article on mathematical definitions
here. I previously attempted to post it on this blog but the addition of the necessary css file had various side effects elsewhere in my Word Press setup, so I deleted the post.
Send to KindleThe 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 SolvEq.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.
This post explains some basic distinctions that need to be made about the process of writing and explaining math. Everyone who teaches math knows subconsciously what is happening here; I am trying to raise your consciousness. For simplicity, I have chosen a technique used in elementary algebra, but much of what I suggest also applies to more abstract college level math.
Solve the equation "$ax=b$" ($a\neq0$).
Understanding the statement of this problem requires a lot of Secret Knowledge (the language of ninth grade algebra) that most people don't have.
The expression "$ax$" means that $a$ and $x$ are numbers and $ax$ is their product. It is not the word "ax". You have to know that writing two symbols next to each other means multiply them, except when it doesn't mean multiply them as in "$\sin\,x$".
The whole expression "$ax=b$" ostensibly says that the number $ax$ is the same number as $b$. In fact, it means more than that. The phrase "solve the equation" tells you that in fact you are supposed to find the value of $x$ that makes $ax$ the same number as $b$.
How do you know that "solve the equation" doesn't mean find the value of $a$ that makes $ax$ the same number as $b$? Answer: The word "solve" triggers a convention that $x$, $y$ and $z$ are numbers you are trying to find and $a$, $b$, $c$ stand for numbers that you are allowed to plug in to the equation.
The conventions of symbolic math require that you give a solution for any nonzero value of $a$ and any value of $b$. You specifically are not allowed to pick $a=1$ and $b=33$ and find the value just for those numbers. (Some college calculus students do this with problems involving literal coefficients.)
The little thingy "$(a\neq0)$" must be read as a constraint on $a$. It does not mean that $a\neq0$ is a fact that you ought to know. ( I've seen college math students make this mistake, admittedly in more complex situations). Nor does it mean that you can't solve the problem if $a=0$ (you can if $b$ is also zero!).
So understanding what this problem asks, as given, requires (fairly sophisticated in some cases) pattern recognition both to understand the symbolic language it uses, and also to understand the special conventions of the mathematical English that it uses.
This problem could be reworded so that it gives an explicit description of the problem, not requiring pattern recognition. (Warning: "Not requiring pattern recognition" is a fuzzy concept.) Something like this:
You have two numbers $a$ and $b$. Find a number $c$ for which if you multiply $a$ by $c$ you get $b$.
This version is not completely explicit. It still requires understanding the idea of referring to a number by a letter, and it still requires pattern recognition to catch on that the two occurrences of each letter means that their meanings have to match. Also, I know from experience that some American first year college students have trouble with the syntax of the sentence ("for which…", "if…").
The following version is more explicit, but it cheats by creating an ad hoc way to distinguish the numbers.
Alice and Bob each give you a number. How do you find a number with the property that Alice's number times your number is equal to Bob's number?
If the problem had a couple more variables it would be so difficult to understand in an explicit form that most people would have to draw a picture of the relationships between them. That is why algebraic notation was invented.
Algebra is a difficult foreign language. Showing the problem visually makes it easier to understand for most people. Our brain's visual processing unit is the most powerful tool the brain has to understand things. There are various ways to do this.
Visualization can help someone understand algebraic notation better.
You can state the problem by producing examples such as
where the reader has to know the multiplication symbol and, one hopes, will recognize "$\boxed{\text{??}}$" as "What's the value?". But the reader does not have to understand what it means to use letters for numbers, or that "$x$ means you are suppose to discover what it is". This way of writing an algebra problem is used in some software aimed at K-12 students. Some of them use a blank box instead of "$\boxed{\text{??}}$".
Such software often shows the algorithm for solving the problem visually, using algebraic notation like this:
I have put in some buttons to show numbers as well as $a$ and $b$. If you have access to Mathematica instead of just to CDF player, you can load SolvEq.nb and put in any numbers you want, but CDF's don't allow input data.
You can also illustrate the algorithm using the tree notation for algebra I used in Monads for high school I (and other posts). The demo below shows how to depict the value-preserving transformation given by the algorithm. (In this case the value is the truth since the root operation is equals.)
This demo is not as visually satisfactory as the one illustrating the use of the associative law in Monads for high school I. For one thing, I had to cheat by reversing the placement of $a$ and $x$. Note that I put labels for the numerator and denominator legs, a practice I have been using in demos for a while for noncommutative operations. I await a new inspiration for a better presentation of this and other equation-solving algorithms.
Another advantage of using pictures is that you can often avoid having to code things as letters which then has to be remembered. In Monads for high school I, I used drawings of the four functions from a two-element set to itself instead of assigning them letters. Even mnemonic letters such as $s$ for "switch" and $\text{id}$ for the identity element carry a burden that the picture dispenses with.
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Many technical words and phrases in math are ordinary English words ("commonwords") that are assigned a different and precisely defined mathematical meaning.
Beginning students come with the (generally subconscious) expectation that they will pick up clues about the meanings of words from connotations they are already familiar with, plus things the teacher says using those words. They think in terms of refining an understanding they already have. This is more or less what happens in most non-math classes. They need to be taught what definition means to a mathematician.
Other technical names in math don't cause the problems that commonwords cause.
Named after somebody The phrase "Hausdorff space" leads a math student to understand that it has a technical meaning. They may not even know it is named after a person, but it screams "geek word" and "you don't know what it means". That is a signal that you can find out what it means. You don't assume you know its meaning.
New made-up words Words such as "affine", "gerbe" and "logarithm" are made up of words from other languages and don't have an ordinary English meaning. Acronyms such as "QED", "RSA" and "FOIL" don't occur often. I don't know of any math objects other than "RSA algorithm" that have an acronymic name. (No doubt I will think of one the minute I click the Publish button.) Whole-cloth words such as "googol" are also rare. All these sorts of words would be good to name new things since they do not fool the readers into thinking they know what the words mean.
Both types of words avoid fooling the student into thinking they know what the words mean, but some students are intimidated by the use of words they haven't seen before. They seem to come to class ready to be snowed. A minority of my students over my 35 years of teaching were like that, but that attitude was a real problem for them.
You can write for several different audiences.
Math fans (non-mathematicians who are interested in math and read books about it occasionally) In my posts Explaining higher math to beginners and in Renaming technical concepts, I wrote about several books aimed at explaining some fairly deep math to interested people who are not mathematicians. They renamed some things. For example, Mark Ronan in Symmetry and the Monster used the phrase "atom" for "simple group" presumably to get around the cognitive dissonance. There are other examples in my posts.
Math newbies (math majors and other students who want to understand some aspect of mathematics). These are the people abstractmath.org is aimed at. For such an audience you generally don't want to rename mathematical objects. In fact, you need to give them a glossary to explain the words and phrases used by people in the subject area.
Postsecondary math students These people, especially the math majors, have many tasks:
It is appropriate for books for math fans and math newbies to try to give an understanding of concepts without necessary proving theorems. That is the aim of much of my work, which has more an emphasis on newbies than on fans. But math majors need as well the traditional emphasis on theorem and proof and clear correct explanations.
Lately, books such as Visual Group Theory have addressed beginning math majors, trying for much more effective ways to help the students develop good intuition, as well as getting into proofs and rigor. Visual Group Theory uses standard terminology. You can contrast it with Symmetry and the Monster and The Mystery of the Prime Numbers (read the excellent reviews on Amazon) which are clearly aimed at math fans and use nonstandard terminology.
I have been thinking about the section of Abstracting Algebra on binary operations. Notice this terminology:
I came up with the names in the right column in an attempt to make some sense out of them. The design is somewhat like the names of some chemical compounds. This would be appropriate for a text aimed at math fans, but for them you probably wouldn't want to get into such an exhaustive list.
I wrote various pieces meant to be part of Abstracting Algebra using the terminology on the right, but thought better of it. I realized that I have been vacillating between thinking of AbAl as for math fans and thinking of it as for newbies. I guess I am plunking for newbies.
I will call groups groups, but for the other structures I will use the phrases in the middle column. Since the book is for newbies I will include a table like the one above. I also expect to use tree notation as I did in Visual Algebra II, and other graphical devices and interactive diagrams.
In the sixties magmas were called groupoids or monoids, both of which now mean something else. I was really irritated when the word "magma" started showing up all over Wikipedia. It was the name given by Bourbaki, but it is a bad name because it means something else that is irrelevant. A magma is just any binary operation. Why not just call it that?
Well, I will tell you why, based on my experience in Ancient Times (the sixties and seventies) in math. (I started as an assistant professor at Western Reserve University in 1965). In those days people made a distinction between a binary operation and a "set with a binary operation on it". Nowadays, the concept of function carries with it an implied domain and codomain. So a binary operation is a function $m:S\times S\to S$. Thinking of a binary operation this way was just beginning to appear in the common mathematical culture in the late 60's, and at least one person remarked to me: "I really like this new idea of thinking of 'plus' and 'times' as functions." I was startled and thought (but did not say), "Well of course it is a function". But then, in the late sixties I was being indoctrinated/perverted into category theory by the likes of John Isbell and Peter Hilton, both of whom were briefly at Case Western Reserve University. (Also Paul Dedecker, who gave me a glimpse of Grothendieck's ideas).
Now, the idea that a binary operation is a function comes with the fact that it has a domain and a codomain, and specifically that the domain is the Cartesian square of the codomain. People who didn't think that a binary operation was a function had to introduce the idea of the universe (universal algebraists) or the underlying set (category theorists): you had to specify it separately and introduce terminology such as $(S,\times)$ to denote the structure. Wikipedia still does it mostly this way, and I am not about to start a revolution to get it to change its ways.
In the olden days, people thought of groups in this way:
This is a better way to describe a group:
This definition makes it clear that a group is a structure consisting of a set and three operations whose axioms are all equations. It was formulated by people in universal algebra but you still see the older form in texts.
The old form is not wrong, it is merely inelegant. With the old form, you have to prove the unity and inverses are unique before you can introduce notation, and more important, by making it clear that groups satisfy equational logic you get a lot of theorems for free: you construct products on the cartesian power of the underlying set, quotients by congruence relations, and other things. (Of course, in AbAl those theorem will be stated later than when groups are defined because the book is for newbies and you want lots of examples before theorems.)
Send to KindleThe 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 Monad.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.
This is the second part of a series of posts describing how I will lead up to introducing monads in my proposed e-book Abstracting Algebra (AbAl). It follows Monads for high school I. Comments in red are meta and mostly will not be included in the book.
A list is a specific kind of mathematical object. This is a reasonable specification for lists:
A list of length $n$ determines and is determined by what its first, second, $\ldots$, $n$th entries are.
In this post, lists will always be finite in length.
For doing rigorous proofs you need a precise definition of a list, such as a function from $\{1,2,…,n\}$ to a set, or a recursive definition. This book is not about proofs.
The most common way in the symbolic language of math to represent a finite list is to use a comma-delimited expression in parentheses. For example, \[(4,4,2,8)\] is the list of length 4 whose first and second entries are both $4$, third entry $2$ and fourth entry $8$.
If $S$ is any set, finite or infinite, $\textrm{Lists}(S)$ denotes the set of all lists of finite length whose entries come from $S$. Thus the set $\textrm{Lists}(\{1,\ 2,\ 3\})$ contains:
$\textrm{Lists}$ is a function from sets to sets. Its input is any set and its output is the set of all finitely-long lists whose entries are from the input set. We will also use the similar function $\textrm{Lists}^+$ which takes a set to the set of nonempty lists with entries from the set.
(Review from Monads for high school I.) If a binary operation is associative, then the operation is defined on any (finite) list of inputs in its underlying set. For example, the sum of the list $\boxed{4\ 4\ 2\ 8}$ is 18. It follows from associativity that you can add it up as $(4+4)+(2+8)$, $4+(4+(2+8))$, $4+((4+2)+8)$, $(4+(4+2))+8$ or as $((4+4)+2)+8$. They all give the same answer. In other words, Plus is in fact an operation on lists of numbers. It is customary to extend associative binary operations to lists of length $0$ and $1$ by setting the value at the empty list to be the identity element of the operation, and the value at a one element list to be its only entry. Thus Plus($\boxed{4\ 4\ 2\ 8}$)$=18$, Plus($\boxed{\ \vphantom{0} }$)$=0$, Times($\boxed{\ \vphantom{0} }$)$=1$ and Plus($\boxed{3 }$)$=3$.
The order matters. If you join $\boxed{5\ 7}$ to $\boxed{2\ 12\ 7 }$ in that order you get $\boxed{5\ 7\ 2\ 12\ 7}$.
Join is in fact an associative binary operation on lists. Example:
This means we can define an operation on lists of lists that joins all the lists inside together to make one list.
Notice the blue rectangle disappears when you do the operation. What I have defined here is a function that has a list of lists as input and a list of numbers as output.
The operation of joining lists to get a single list has a property shown by the drawing below (which will be interactive when I work on it some more). Start on the left with a list of lists of lists. The border colors distinguish the innermost lists, bordered in black, from the second level lists, in blue, and the outside list, bordered in green.
The two lists on the right are the same. That always happens, whatever lists you start with. (Try it with others, and include some singleton and empty lists while you are at it.)
You might not have thought of this property, and now that you see it, it may look like some sort of second-rate phenom to take note of. Or not. But in fact, it turns out that it means that our modest function $\textrm{Lists}^+$, that takes a set to the nonempty set of lists of its elements is a monad. (So is $\textrm{Lists}$.) In order to say this we must define some other concepts: functor and natural transformation, and we have to verify a number of other properties of the $\textrm{List}^+$ function: It is not just a function, it is a functor on the category of sets, the join function is a natural transformation, and some other technicalities.
Once we do that, we can define what the algebras of the join monad are, and it turns out that they are exactly all the associative binary operations.
In other words:
The binary operation of join on nonempty lists is the mother of all associative binary operations.
But that will have to wait for the next post.
Send to KindleI propose the phrase "shared mental object" to name the sort of thing that includes mathematical objects, abstract objects, fictional objects and other concepts with the following properties:
It is the name "shared mental object" that is a new idea; the concept has been around in philosophy and math ed for awhile and has been called various things, especially "abstract object", which is the name I have used in abstractmath.
I will go into detail concerning some examples in order to make the concept clear. If you examine this concept deeply you discover many fine points, nested ideas and circles of examples that go back on themselves. I will not get very far into these fine points here, but I have written about some of them posts and in abmath (see references). I am working on a post about some of the fine points and will publish it if I can control its tendency to expand into infinite proliferation and recursion.
There is a story about the early days of telegraphy: A man comes into the newly-opened telegraph station and asks to send a telegram to his son who is working in another city. He writes out the message and gives it to the operator with his payment. The operator puts the message on a spike and clicks the key in front of him for a while, then says, "I have sent your message. Thanks for shopping at Postal Telegraph". The man looks astonished and points at the message and says, "But it is still here!"
A message is a shared mental object.
Other examples that are similar in nature to messages are schedules and the month of September (see Math Objects in abmath, where they are called abstract objects.). In English-speaking communities, September is a cultural default: you are expected to know what it is. You can know that September is a month and that right this minute it is not September (unless it is September). You may think that September has 31 days and most people would say you are wrong, but they would agree that you and they are talking about the same month.
The general concept of the month of September and facts concerning it have been in shared existence in English-speaking cultural groups for (maybe) a thousand years. In contrast, a message is usually shared by only two or three people and it has a short life; a few years from now, it may be that none of the people involved with the message remember what it said or even that it existed.
A symbol, such as the letter "a" or the integral sign "$\int$", is a shared mental object. Like the month of September, but unlike messages, letters are shared by large cultural entities, every language community that uses the Latin alphabet (and more) in the case of "a", and math and tech people in the case of "$\int$".
The letter "a" is represented physically on paper, a blackboard or a screen, among other things. If you are literate in English and recognize an occurrence as representing the letter, you probably do this using a process in the brain that is automatic and that operates outside your awareness.
Literate readers of English also generally agree that a string of letters either does or does not represent the word "default" but there are borderline cases (as in those little boxes where you have to prove you are not a robot) where they may disagree or admit that they don't know. Even so, the letter "a" and the word "default" are shared in the minds of many people and there is general (but not absolutely universal) agreement on when you are seeing representations of them.
Fictional objects such as Sherlock Holmes and unicorns are shared mental objects. I wrote briefly about them in Mathematical objects and will not go into them here.
The integer $111$, the integral $\int_0^1 x^2\,dx$ and the set of all real numbers are all mathematical objects. They are all shared mental objects. In most of the world, people with a little education will know that $111$ is a number and what it means to have $111$ beans in a jar (for example). They know that it is one more that $110$ and a lot more than $42$.
Mathematicians, scientists and STEM students will know something about what $\int_0^1 x^2\,dx$ means and they will probably know how to calculate it. Most of them may be able to do it in their head. I have taught calculus so many times that I know it "by heart", which means that it is associated in my brain with the number $1/3$ in such a way that when I see the integral the number automatically and without effort pops us (in the same way that I know September has 30 days).
Beginning calculus students may have a confused and incorrect understanding of the set of all real numbers in several ways, but practicing mathematicians (and many others) know that it is an uncountably infinite dense set and they think of it as an object. A student very likely does not think of it as an object, but as a sprawling unimaginable space that you cannot possibly regard as a thing. Students may picture a real number as having another real number sitting right beside it — the next biggest one. Most practicing mathematicians think of the set of real numbers as a completed infinity — every real number is already there — and they know that between any two of them there is another one.
As a consequence, when students and professors talk about real numbers the student finds that some times the professor says things that sound completely wrong and the professor hears the student say things that are bizarre and confused. They firmly believe they are talking about the same thing, the real numbers, but the student is seen by the professor as wrong and the professor is seen by the students as talking meaningless nonsense. Even so, they believe they are talking about the same thing.
I tried various other names before I came to "shared mental objects".
The major advantage of "shared mental object" is that it describes the important properties of the concept: It is a mental object and it is shared by people. It has no philosophical implications concerning platonism, either. Mathematical objects do have special properties of verifiability that general shared mental objects do not, but my terminology does not suggest any existence of absolute truth or of an Ideal existing in another world. I don't believe in such things, but some people do and I want to point out that "shared mental object" does not rule such things out — it merely gives a direct evidence-based description of a phenomenon that actually exists in the real world.
Abstract objects in the Stanford Encyclopedia of Philosophy
Abstract object in Wikipedia
Mathematical objects in abstractmath
Mathematical objects in Wikipedia
What is Mathematics, Really? R. Hersh,
Previous posts
Representations of mathematical objects
Representations III: Rigor and Rigor Mortis
This post uses MathJax. If you see mathematical expressions with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.
Send to KindleI want to call your attention to the following items about ways to avoid algebra and why you should and shouldn't.
Don't kill math, by Evan Miller
Inventing on principle, video by Bret Victor
Send to KindleThe 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 Representing sets.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.
Sets are represented in the math literature in several different ways, some mentioned here. Also mentioned are some other possibilities. Introducing a variety of representations of any type of math object is desirable because students tend to assume that the representation is the object.
The standard representation for a finite set is of the form "$\{1,3,5,6\}$". This particular example represents the unique set containing the integers $1$, $3$, $5$ and $6$ and nothing else. This means precisely that the statement "$n$ is an element of $S$" is true if $n=1$, $n=3$, $n=5$ or $n=6$, and it is false if $n$ represents any other mathematical object.
In the way the notation is usually used, "$\{1,3,5,6\}$", "$\{3,1,5,6\}$", "$\{1,5,3,6\}$", "$\{1,6,3,5,1\}$" and $\{ 6,6,3,5,1,5\}$ all represent the same set. Textbooks sometimes say "order and repetition don't matter". But that is a statement about this particular representation style for sets. It is not a statement about sets.
It would be nice to come up with a representation for sets that doesn't involve an ordering. Traditional algebraic notation is essentially one-dimensional and so automatically imposes an ordering (see Algebra is a difficult foreign language).
In Visible Algebra II, I experimented with the idea of putting the elements at random inside a circle and letting them visibly move around like goldfish in a bowl. (That experiment was actually for multisets but it applies to sets, too.) This is certainly a representation that does not impose an ordering, but it is also distracting. Our visual system is attracted to movement (but not as much as a cat's visual system).
One possibility would be to extend the machinery in a visible algebra system that allows you to make a box you could drag elements into.
This box would order the elements in some canonical order (numerical order for numbers, alphabetical order for strings of letters or words) with the property that if you inserted an element in the wrong place it would rearrange itself, and if you tried to insert an element more than once the representation would not change. What you would then have is a unique representation of the set.
An example is the device below. (If you have Mathematica, not just CDF player, you can type in numbers as you wish instead of having to use the buttons.)
This does not allow a representation of a heterogenous set such as $\{3,\mathbb{R},\emptyset,\left(\begin{array}{cc}1&2\\0&1\\ \end{array}\right)\}$. So what? You can't represent every function by a graph, either.
The tree notation used in my visual algebra posts could be used for sets as well, as illustrated below. The system allows you to drag the elements listed into different positions, including all around the set node. If you had a node for lists, that would not be possible.
This representation has the pedagogical advantage of shows that a set is not its elements.
Infinite sets are sometimes represented using the curly bracket notation using a pattern that defines the set. For example, the set of even integers could be represented by $\{0,2,4,6,\ldots\}$. Such a representation is necessarily a convention, since any beginning pattern can in fact represent an infinite number of different infinite sets. Personally, I would write, "Consider the even integers $\{0,2,4,6,\ldots\}$", but I would not write, "Consider the set $\{0,2,4,6,\ldots\}$".
By the way, if you are writing for newbies, you should say,"Consider the set of even integers $\{0,2,4,6,\ldots\}$". The sentence "Consider the even integers $\{0,2,4,6,\ldots\}$" is unambiguous because by convention a list of numbers in curly brackets defines a set. But newbies need lots of redundancy.
Setbuilder notation is exemplified by $\{x|x>0\}$, which denotes the positive reals, given a convention or explicit statement that $x$ represents a real number. This allows the representation of some infinite sets without depending on a possibly ambiguous pattern.
A Visible Algebra system needs to allow this, too. That could be (necessarily incompletely) done in this way:
The Can't Tell answer is a necessary requirement because the general question of whether an element is in a set defined by a first order sentence is undecidable. Perhaps the system could add some choices:
Even so, I'll bet a system using Mathematica could answer many questions like this for sentences referring to a specific polynomial, using the Solve or NSolve command. For example, the answer to the question, "Is $3\in\{n|n\lt0 \text{ and } n^2=9\}$?" (where $n$ ranges over the integers) would be "No", and the answer to "Is $\{n|n\lt0 \text{ and } n^2=9\}$ empty?" would also be "No". [Corrected 2012.10.24]
Send to KindleThe interactive example 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 algebra2.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.
I am in the process of writing an explanation of monads for people with not much math background. In that article, I began to explain my ideas about exposition for readers at that level and after I had written several paragraphs decided I needed a separate article about exposition. This is that article. It is mostly about language.
There are many recent books explaining some aspect of math for people who are not happy with high school algebra; some of them are listed in the references. They must be smart readers who know how to concentrate, but for whom algebra and logic and definition-theorem-proof do not communicate. They could be called interested laypeople, but that is a lousy name and I would appreciate suggestions for a better name.
My post on monads is aimed at people who have some math, and who are interested in "understanding" some aspect of "higher math"; not understanding in the sense of being able to prove things about monads, but merely how to think about them. I will call them math newbies. Of course, I am including math majors, but I want to make it available to other people who are willing to tackle mathematical explanations and who are interested in knowing more about advanced stuff.
These "other people" may include people (students and practitioners) in other science & technology areas as well as liberal-artsy people. There are such people, I have met them. I recall one theologian who asked me about what was the big deal about ruler-and-compass construction and who seemed to feel enlightened when I told him that those constructions preserve exactly the ideal nature of geometric objects. (I later found out he was a famous theologian I had never heard of, just like Ngô Bảo Châu is a famous mathematician nonmathematicians have never heard of.)
If you are aiming at interested laypeople you absolutely must avoid algebra. It is a foreign language that simply does not communicate to most of the educated people in the world. Learning a foreign language is difficult.
So how do you avoid algebra? Well, you have to be clever and insightful. The book by Matthew Watkins (below) has absolutely wonderful tricks for doing that, and I think anyone interested in math exposition ought to read it. He uses metaphors, pictures and saying the same thing in different words. When you finish reading his book, you won't know how to prove statements related to the prime number theorem (unless you already knew how) but you have a good chance of understanding the statement of some theorem in that subject. See my review of the book for more details.
If your article is for math newbies, you don't have to avoid algebra completely. But remember they are newbies and not as fluent as you are — they do things analogous to "Throw Mama from the train a kiss" and "I can haz cheeseburger?". But if you are trying to give them some way of thinking about a concept, you need many other things (metaphors, illustrative applications, diagrams…) You don't need the definition-theorem-proof style too common in "exposition". (You do need that for math majors who want to become professional mathematicians.)
In writing expositions for interested laypeople or math newbies, you should not introduce an unfamiliar notation system (which is like a minilanguage). I expect to write the monad article without commutative diagrams. Now, commutative diagrams are a wonderful invention, the best way of writing about categories, and they are widely used by other than category theorists. But to explain monads to a newbie by introducing and then using commutative diagrams is like incorporating a short grammar of Spanish which you will then use in an explanation of Sancho Panza's relationship with Don Quixote.
The abstractmath article on and, or and not does not use any of the several symbolic notations for logic that are in use. The explanations simply use "and", "or" and "not". I did introduce the notation, but didn't use it in the explanations. When I rewrite the article I expect to put the notation at the end of the article instead of in the middle. I expect to rewrite the other articles on mathematical reasoning to follow that practice, too.
This is about the technical terminology used in math. Technical terminology belongs to the math dialect (or register) of English, which is not a foreign language in the same sense as algebra. One big problem is changing the meaning of ordinary English words to a technical meaning. This requires a definition, and definitions are not something most people take seriously until they have been thoroughly brainwashed into using mathematical methodology. Math majors have to be brainwashed in this way, but if you are writing for laypeople or newbies you cannot use the technology of formal definition.
"You say the Monster Group is SIMPLE??? You must be a GENIUS!" So Mark Ronan in his book (below) referred to simple groups as atoms. Marcus du Sautoy calls them building blocks. The mathematical meaning of "simple group" is not a transparent consequence of the meanings of "simple" and "group". Du Sautoy usually writes "group of symmetries" instead of just "group", which gives you an image of what he is talking about without having to go into the abstract definition of group. So in that usage, "group" just means "collection", which is what some students continue to think well after you give the definition.
A better, but ugly, name for "group" might be "symmetroid". It sounds technical, but that might be an advantage, not a disadvantage. "Group" obviously means any collection, as I've known since childhood. "Symmetroid" I've never heard of so maybe I'd better find out what it means.
In beginning abstract math courses my students fervently (but subconsciously) believe that they can figure out what a word means by what it means already, never mind the "definition" which causes their eyes to glaze over. You have to be really persuasive to change their minds.
Matthew Watkins referred to the prime factorization of an integer as a cluster. I am not sure why Watkins doesn't like "prime factorization", which usually refers to an expression such as $p^{n_1}_1p^{n_2}_2\ldots p^{n_k}_k$. This (as he says) has a spurious ordering that makes you have to worry about what the uniqueness of factorization means. The prime factorization is really a multiset of primes, where the order does not matter.
Watkins illustrates a cluster of primes as a bunch of pingpong balls stuck together with glue, so the prime factorization of 90 would be four smushed together balls marked 2, 3, 3 and 5. Below is another way of illustrating the prime factorization of 90. Yes, the random movement programming could be improved, but Mathematica seduces you into infinite playing around and I want to finish this post. (Actually, I am beginning to think I like smushed pingpong balls better. Even better would be a smushed pingpong picture that I could click on and look at it from different angles.)
Any exposition of math should use metaphors, pictures and graphs, especially manipulable pictures (like the one above) and graphs. This applies to expositions for math majors as well as laypeople and newbies. Calculus and other texts nowadays have begun doing this, more with pictures than with metaphors.
I was turned on to these ideas as far back as 1967 (date not certain) when I found an early version of David Mumford's "Red Book", which I think evolved into the book The Red Book of Varieties and Schemes. The early version was a revelation to me both about schemes and about exposition. I have lost the early book and only looked at the published version briefly when it appeared (1999). I remember (not necessarily correctly) that he illustrated the spectrum as a graph whose coordinates were primes, and generic points were smudges. Writing this post has motivated me to go to the University of Minnesota math library and look at the published version again.
Send to KindleThe 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 Wolfram website. The code for the demos is in the Mathematica notebook algebra2.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.
I have written about visible algebra in previous posts (see References). My ideas about the interface are constantly changing. Some new ideas are described here.
In the first place I want to make it clear that what I am showing in these posts is a simulation of a possible visual algebra system. I have not constructed any part of the system; these posts only show something about what the interface will look like. My practice in the last few years is to throw out ideas, not construct completed documents or programs. (I am not saying how long I will continue to do this.) All these posts, Mathematica programs and abstractmath.org are available to reuse under a Creative Commons license.
Times and Plus are commutative and associative operations. They are usually defined as binary operations. A binary operation $*$ is said to be commutative if for all $x$ and $y$ in the underlying set of the operation, $x*y=y*x$, and it is associative if for all $x$,$y$ and $z$ in the underlying set of the operation, $(x*y)*z=x*(y*z)$.
It is far better to define a commutative and associative operation $*$ on some underlying set $S$ as an operation on any multiset of elements of $S$. A multiset is like a set, in particular elements can be rearranged in any way, but it is not like a set in that elements can be repeated and a different number of repetitions of an element makes a different multiset. So for any particular multiset, the number of repetitions of each element is fixed. Thus $\{a,a,b,b,c\} = \{c,b,a,b,a\}$ but $\{a,a,b,b,c\}\neq\{c,b,a,b,c\}$. This means that the function (operation) Plus, for example, is defined on any multiset of numbers, and \[\mathbf{Plus}\{a,a,b,b,c\}=\mathbf{Plus} \{c,b,a,b,a\}\] but $\mathbf{Plus}\{a,a,b,b,c\}$ might not be equal to $\mathbf{Plus} \{c,b,a,b,c\}$.
This way of defining (any) associative and commutative operation comes from the theory of monads. An operation defined on all the multisets drawn from a particular set is necessarily commutative and associative if it satisfies some basic monad identities, the main one being it commutes with union of multisets (which is defined in the way you would expect, and if this irritates you, read the Wikipedia article on multisets.). You don't have to impose any conditions specifically referring to commutativity or associativity. I expect to write further about monads in a later post.
The input process for a visible algebra system should allow the full strength of this fact. You can attach as many inputs as you want to Times or Plus and you can move them around. For example, you can click on any input and move it to a different place in the following demo.
Other input notations might be suitable for different purposes. The example below shows how the inputs can be placed randomly in two dimensions (but preserving multiplicity). I experimented with making it show the variables slowly moving around inside the circle the way the fish do in that screensaver (which mesmerizes small children, by the way — never mind what it does to me), but I haven't yet made it work.
A visible algebra system might well allow directly input tables to be added up (or multiplied), like the one below. Spreadsheets have such an operation In particular, the spreadsheet operation does not insist that you apply it only as a binary operation to columns with two entries. By far the most natural way to define addition of numbers is as an operation on multisets of numbers.
Operations that are associative but not commutative, such as matrix multiplication, can be defined the monad way as operations on finite lists (or tuples or vectors) of numbers. The operation is automatically associative if you require it to preserve concatenation of lists and some other monad requirements.
Some binary operations are neither commutative nor associative. Two such operations on numbers are Subtract and Power. Such operations are truly binary operations; there is no obvious way to apply them to other structures. They are only binary because the two inputs have different roles. This suggests that the inputs be given names, as in the examples below.
Later, I will write more about simplifying trees, solving the max area problem for rectangles surmounted by semicircles, and other things concerning this system of doing algebra.
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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 algebra1.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.
In my previous post Visible algebra I constructed a computation tree for calculating the area of a window consisting of a rectangle surmounted by a semicircle. The visual algebra system described there constructs a computation by selecting operations and attaching them to a tree, which can then be used to calculate the area of the window.
I promised to produce a live computation tree later; it is below.
Press the buttons from left to right to simulate the computation that would take place in a genuine algebra system. Note that if you skip button 2 you get the effect of parallel computation (the only place in the calculation that can be parallelized).
In Visual Algebra I the tree was put together step by step by reasoning out how you would calculate the area of the window: (1) the area is the sum of the areas of the semidisk and the rectangle, (2) the rectangle is width times height, (3) the semidisk has half the area of a disk of radius half the width of the rectangle, and so on. So the resulting tree is a transparent construction that lets you see the reasoning that created it.
The resulting tree could obviously be simplified. But if you were designing a few such windows, why should you simplify it? You certainly don't need to simplify it to speed up the computation. On the other hand, if you are going on to solve the problem of finding the maximum area you can get if the perimeter is fixed, you will have to do some algebraic manipulation and so you do want a simplified expression.
Later, I will write more about simplifying trees, solving the max area problem, and other things concerning this system of doing algebra.
What I am showing in these posts is a simulation of a possible visible algebra system. I have not constructed any part of the system; these posts only show something about what the interface will look like. My practice in the last few years is to throw out ideas, not construct completed documents or programs. (I am not saying how long I will continue to do this.) All these posts, Mathematica programs and abstractmath.org are available to reuse under a Creative Commons license.
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