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exposition, language, math, understanding math

Mathematical Information I

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Introduction

The January, 2016 meeting of the American Mathematical Society in Seattle included a special session on Mathe­matical Information in the Digital Age of Science. Here is a link to the list of talks in that session (you have to scroll down a ways to get to the list).

Several talks at that session were about communi­cating math, to other mathe­maticians and to the general public. Well, that’s what I have been about for the last 20 years. Mostly.

Overview

These posts discuss the ways we communi­cate math and (mostly in later posts) the revolution in math communication that the internet has caused. Parts of this discussion were inspired by the special session talks. When they are relevant, I include footnotes referring to the talks. Be warned that what I say about these ideas may not be the same as what the speakers had to say, but I feel I ought to give them credit for getting me to think about those concepts.

Some caveats

  • The distinctions between different kinds of math communi­cation are inevitably fuzzy.
  • Not all kinds of communication are mentioned.
  • Several types of communication normally occur in the same document.

Articles published in journals

Until recently, math journals were always published on paper. Now many journals exist only on the internet. What follows is a survey of the types of articles published in journals.

Refereed papers containing new results

These communications typically containing proofs of (usually new) theorems. Such papers are the main way that academic mathematicians get credit for their researchG for the purpose of getting tenure (at least in the USA), although some other types of credit are noted below.

Proofs published in refereed journals in the past were generally restricted to formal proofs, without very many comments intended to aid the reader’s under­standing. This restricted text was often enforced by the journal. In the olden days this would have been prompted by the expense of publishing on paper. I am not sure how much this restriction has relaxed in electronic journals.

I have been writing articles for abstractmath.org and Gyre&Gimble for many years, and it has taken me a very long time to get over unnecessarily restricting the space I use in what I write. If I introduce a diagram in an article and then want to refer to it later, I don’t have to link to it — I can copy it into the current location. If it makes sense for an informative paragraph to occur in two different articles, I can put it into both articles. And so on. Nowadays, that sort of thing doesn’t cost anything.

Survey articles and invited addresses

You may also get credit for an invited address to a prestigious organi­zation, or for a survey of your field, in for example the Bulletin of the AMS. Invited addresses and surveys may contain considerably more explanatory asides. This was quite noticeable in the invited talks at the AMS Seattle meeting.

Books

There is a whole spectrum of math books. The following list mentions some Fraunhofer lines on the spectrum, but the gamut really is as continuous as a large finite list of books could be. This list needs more examples. (This is a blog post, so it has the status of an alpha release.)

Research books that are concise and without much explanation.

The Bourbaki books that I have dipped into (mostly the algebra book and mostly in the 1970’s) are definitely concise and seem to strictly avoid explanation, diagrams, pictures, etc). I have heard people say they are unreadable, but I have not found them so.

Contain helpful explanations that will make sense to people in the field but probably would be formidable to someone in a substantially different area.

Toposes, triples and theories, by Michael Barr and Charles Wells. I am placing our book here in the spectrum because several non-category-theorists (some of them computer scientists) have remarked that it is “formidable” or other words like that.

Intended to introduce professional mathematicians to a particular field.

Categories for the working mathematician, by Saunders Mac Lane. I learned from this (the 1971 edition) in my early days as a category theorist, six years after getting my Ph.D. In fact, I think that this book belongs to the grad student level instead of here, but I have not heard any comments one way or another.

Intended to introduce math graduate students to a particular field.

There are lots of examples of good books in this area. Years ago (but well after I got my Ph.D.), I found Serge Lang’s Algebra quite useful and studied parts of it in detail.

But for grad students? It is still used for grad students, but perhaps Nathan Jacobson’s Basic Algebra would be a better choice for a first course in algebra for first-year grad students.

The post My early life as a mathematician discusses algebra texts in the olden days, among other things.

Intended to explain a part of math to a general audience.

Love and math: the heart of hidden reality. by Edward Frenkel, 2014. This is a wonderful book. After reading it, I felt that at last I had some clue as to what was going on with the Langlands Program. He assumes that the reader knows very little about math and gives hand-waving pictorial expla­nations for some of the ideas. Many of the concepts in the book were already familiar to me (not at an expert level). I doubt that someone who had had no college math courses that included some abstract math would get much out of it.

Symmetry: A Journey into the Patterns of Nature, by Marcus du Sautoy, 2009. He also produced a video on symmetry.

My post Explaining “higher” math to beginners, describes du Sautoy’s use of terminology (among others).

Secrets of creation: the mystery of the prime numbers (Volume 1) by Matthew Watkins (author) and Matt Tweed (Illustrator), 2015. This is the first book of a trilogy that explains the connection between the Riemann $\zeta$ function and the primes. He uses pictures and verbal descriptions, very little terminology or symbolic notation. This is the best attempt I know of at explaining deep math that might really work for non-mathe­maticians.

My post The mystery of the prime numbers: a review describes the first book.

Piper Harron’s Thesis

The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields: an Artist’s Rendering, Ph.D. thesis by Piper Harron.

This is a remarkable departure from the usual dry, condensed, no-useful-asides Ph.D. thesis in math. Each chapter has three main parts, Layscape (explanations for nonspecialists — not (in my opinion) for nonmathe­maticians), Mathscape (most like what goes into the usual math paper but with much more explanation) and Weedscape (irrelevant stuff which she found helpful and perhaps the reader will too). The names of these three sections vary from chapter to chapter. This seems like a great idea, and the parts I have read are well-done.

These blog posts have useful comments about her thesis:

Types of explanations

Any explanation of math in any of the categories above will be of several different types. Some of them are considered here, and more will appear in Mathematical Information II.

The paper Varieties of Mathematical Prose, by Atish Bagchi and me, provides a more fine-grained description of certain types of math communication that includes some types of explanations and also other types of communication.

Images and metaphors

In abstractmath.org

I have written about images and metaphors in abstractmath.org:

Abstractmath.org is aimed at helping students who are beginning their study of abstract math, and so the examples are mostly simple and not at a high level of abstraction. In the general literature, the images and metaphors that are written about may be much more sophisticated.

The User’s GuideW

Luke Wolcott edits a new journal called Enchiridion: Mathematics User’s Guides (this link allows you to download the articles in the first issue). Each article in this journal is written by a mathematician who has published a research paper in a refereed journal. The author’s article in Enchiridion provides information intended to help the reader to understand the research paper. Enchiridion and its rationale is described in more detail in the paper The User’s Guide Project: Giving Experential Context to Research Papers.

The guidelines for writing a User’s Guide suggest writing them in four parts, and one of the parts is to introduce useful images and metaphors that helped the author. You can see how the authors’ user’s guides carry this out in the first issue of Enchiridion.

Piper Harron’s thesis

Piper Harron’s explanation of integrals in her thesis is a description of integrals and measures using creative metaphors that I think may raise some mathematicians’ consciousness and others’ hackles, but I doubt it would be informative to a non-mathematician. I love “funky-summing” (p. 116ff): it communicates how integration is related to real adding up a finite bunch of numbers in a liberal-artsy way, in other words via the connotations of the word “funky”, in contrast to rigorous math which depends on every word have an accumulation-of-properties definition.

The point about “funky-summing” (in my opinion, not necessarily Harron’s) is that when you take the limit of all the Riemann sums as all meshes go to zero, you get a number which

  • Is really and truly not a sum of numbers in any way
  • Smells like a sum of numbers

Connotations communicate metaphors. Metaphors are a major cause of grief for students beginning abstract math, but they are necessary for understanding math. Working around this paradox is probably the most important problem for math teachers.

Informal summaries of a proofW

The User’s Guide requires a “colloquial summary” of a paper as one of the four parts of the guide for that paper.

  • Wolcott’s colloquial summary of his paper keeps the level aimed at non-mathematicians, starting with a hand-waving explanation of what a ring is. He uses many metaphors in the process of explaining what his paper does.
  • The colloquial summary of another User’s Guide, by Cary Malkiewich, stays strictly at the general-public level. He uses a few metaphors. I liked his explanation of how mathematicians work first with examples, then finding patterns among the examples.
  • The colloquial summary of David White’s paper stays at the general-public level but uses some neat metaphors. He also has a perceptive paragraph discussing the role of category theory in math.

The summaries I just mentioned are interesting to read. But I wonder if informal summaries aimed at math majors or early grad students might be more useful.

Insights

The first of the four parts of the explanatory papers in Enchiridion is supposed to present the key insights and organizing principles that were useful in coming up with the proofs. Some of them do a good job with this. They are mostly very special to the work in question, but some are more general.

This suggests that when teaching a course in some math subject you make a point of explaining the basic techniques that have turned out very useful in the subject.

For example, a fundamental insight in group theory is:

Study the linear representations of a group.

That is an excellent example of a fundamental insight that applies everywhere in math:

Find a functor that maps the math objects you are studying to objects in a different branch of math.

The organizing principles listed in David White’s article has (naturally more specialized) insights like that.

Proof stories

“Proof stories” tell in sequence (more or less) how the author came up with a proof. This means describing the false starts, insights and how they came about. Piper Harron’s thesis does that all through her work.

Some authors do more than that: their proof stories intertwine the mathe­matical events of their progress with a recount of life events, which sometimes make a mathe­matical difference and sometimes just produces a pause to let the proof stew in their brain. Luke Wolcott wrote a User’s Guide for one of his own papers, and his proof story for that paper involves personal experiences. (I recommend his User’s Guide as a model to learn from.)

Reports of personal experiences in doing math seem to add to my grasp of the math, but I am not sure I understand why.

References

The talks in Seattle

  • List of all the talks.

  • W. Timothy Gowers, How should mathe­matical knowledge be organized? Talk at the AMS Special Session on Mathe­matical Information in the Digital Age of Science, 6 January 2016.
  • Colloquium notes. Gowers gave a series of invited addresses for which these are the notes. They have many instances of describing what sorts of problems obstruct a desirable step in the proof and what can be done about it.

  • Luke Wolcott, The User’s Guide. Talk at the AMS Special Session on Mathe­matical Information in the Digital Age of Science, 6 January 2016.

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

Explaining “higher” math to beginners

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The 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.

Notes on viewing

Explaining math

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.

Who is it written for?

Interested laypeople

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. 

Math newbies

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.)

Algebra and other foreign languages

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.) 

Unfamiliar notation

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.

Technical terminology

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.

Groups, simple groups

"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.

Prime factorization

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.)

Metaphors, pictures, graphs, animation

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.

References

Expositions for educated non-mathematicians

Previous posts in G&G

Relevant abmath articles

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