The role of social interaction in mathematics

Charlotte Simmons started her career as a pure mathematician, later moving into the history of mathematics. She is currently an administrator at her institution, but still finds time to do research and write wonderful pieces. Perhaps a “pure” mathematician can do research and also write about the history of the subject. I only included the statement starting as a “pure mathematician,” since it was in her own bio.

Today I want to talk about an article that she wrote entitled “Augustus De Morgan behind the Scenes.”

The article is in the 2012 yearly collection of the best mathematical writing of each year, which is regularly edited by Mircea Pitici. As this is the start of the new year, it is time to start writing great pieces on mathematics. Perhaps yours will be included in the future.

What struck me about Simmons’ article were two things. First, and her main theme, is the role that De Morgan played as a mentor to two other extremely famous mathematicians. I did not know that he was so instrumental in helping Sir William Hamilton and George Boole in their great careers. Between them, two more different personality types, approaches to research, and areas of interest could hardly be found. Yet De Morgan could and did unselfishly help both in many ways.

## Social Net Work

The second point is not emphasized by her—at least not explicitly. But her article struck me as making a wonderful case for how mathematics is done. In particular, how people interact via the means that are available to them.

De Morgan worked in the mid 1800’s, yet I believe that how he operated and mentored is not much different from how many of us do the same today. Yes now we often write papers via the Internet. Sometimes we use e-mail, sometimes we chat, sometimes we just share paper drafts, but it is less personal than we might like. I thought the impersonal nature of working was a new phenomenon, but Simmons’ paper makes it clear that it is not. So I thought it would be interesting to look back when almost all communication was via mail, snail mail, and see how it relates to today’s world. I believe that while the world has vastly changed, distances are less important, and communication is almost instant, these changes have affected how we work less than I thought.

Let’s start by looking at De Morgan’s own work, and then moving on to how he mentored.

## De Morgan’s Work

Computer scientists know De Morgan for many things, but perhaps the most famous are his “laws”—which are trivial but extremely valuable:

$\displaystyle \begin{array}{rcl} \neg ( x \vee y) &=& (\neg x) \wedge (\neg y) \\ \neg ( x \wedge y) &=& (\neg x) \vee (\neg y), \end{array}$

where ${x}$ and ${y}$ are boolean values. Thus the negation of an “OR” can be converted into the “AND” of the negated terms, and dually.

These laws are named after De Morgan. It is pretty neat having “laws” named for you, but he was not the first to discover them. As I had stated before, credit often goes to the one who last discoveres something, rather who first does. A good case can be made that the Greeks, in particular Aristotle, knew these laws; much later, in the fourteen century, William of Ockham wrote the laws down in words not symbols. The laws are easy to prove, easy to use; they are simple. But De Morgan gets the credit for them. The laws are routinely used in complexity theory, especially in work on boolean circuits. The power of these simple “laws” is that negations can be pushed down to the inputs of circuits. It is often a nice simplification to only have negations at the inputs.

De Morgan also named one concept that is central to modern mathematics: mathematical induction. His fame is rightly recognized in a number of other ways. The headquarters of the London Mathematical Society is called De Morgan House, and the student society of the Mathematics Department of University College London is called the August De Morgan Society. There even is a crater on the moon named after him—cleverly called “the De Morgan crater.”

## De Morgan’s Mentoring

Hamilton was already well established when he began a long correspondence with De Morgan. This lasted over twenty-five years, and on Hamilton’s death De Morgan said:

And yet we did not know each other’s faces. I met him about 1830 at [Charles] Babbage’s breakfast table, and there for the only time in our lives we conversed ${\dots}$

One of the most helpful things that De Morgan did for Hamilton was to agree to review his text, Lectures on Quaternions. Recall that Hamilton did many things, but was perhaps personally most excited about his creation of the four-dimensional algebraic system that generalized complex numbers. The task was more work than probably either envisioned at the beginning, since the writing took five years and the final text was over 700 pages. The text was later described as one that would “take any man a twelvemonth to read, and near a lifetime to digest.”

Boole, in contrast to Hamilton, was not well known when he began corresponding with De Morgan. He sent a letter to De Morgan in 1842 that was extremely modest. Boole had no formal degree, and said that he had “no claim to the notice of those mathematicians who did.” De Morgan encouraged Boole to submit his work to the Royal Society, where it soon became an award-winning paper. This paper helped launch Boole’s career.

It is important to note, as Simmons points out, that De Morgan was working on logic of exactly the same kind as Boole when his letter arrived. Boole’s masterpiece, An investigation of the Laws of Thought appeared on the very same day—how is this possible?—as De Morgan’s own publication. Boole’s soon became the publication. Ken spent many a day in the math-science room of the library at Merton College, Oxford, where an 1854 first edition of Boole’s book nestled on a prominent shelf below a window alongside modern texts.

De Morgan then began to help Boole in many ways: he got him a pass for the British Museum, a position at Queen’s College, and more. They corresponded for years, and De Morgan helped Boole especially with his insecurities. Simmons notes that De Morgan knew exactly how to treat Boole and Hamilton. With Boole he was also supportive and gentle, while he was not afraid to say this to the strong willed Hamilton:

Ink must be cheap in Ireland if you can afford to waste it on such a supposition as that.

## Social Aspects

All this work in common and co-operation were done by mail. Now in the 1800’s the mail system was better than it is today in many respects. Letters were often delivered more than once per day. This was true also of Oxford’s internal “pigeon post” when Ken was there, though he recalls plans to cut it to once a day to save costs. There were also telegrams—several of the Sherlock Holmes stories use that form of communication with almost the immediacy of telephones today.

In another famous correspondence between Richard Dedekind and Georg Cantor, Dedekind answers a letter on June 22, 1877 in reply to letter of June 20 from Cantor. This is about the set theory result that Cantor could not believe—that the unit square and the unit line had the same cardinality. Well, that is another story.

What we speculate is that a kind of conservation law is in effect. We each have room for yea-much of another person, be it a lot or a little. When we have many e-mails, that much is divided into many little pieces. But having one letter confers the whole in one go, with more time to reflect in one’s drawing room than the time between glances at one’s computer screen and beeps of a smartphone today.

Ken has been feeling this during iterations of the open letter and report about chess cheating which he has been preparing in consultation with two of his previous co-authors. The touchiness of the subject is reflected in adverse reader reactions posted here to others’ early scattershot renditions of the kind of statistical evidence he has researched comprehensively. This may be the first major case where no physical evidence is known, at present. There have been queries and adjustments and changes. This has been great—being able to revise in multiple communications per day across an ocean and other time zones—but Ken also wishes that one’s mind could be settled and conveyed in one communication.

Such as in a letter. Ken even recalls writing love letters—physical letters—do our young couples still do that anymore? Some mathematicians’ letters have been collected into interesting volumes—will anyone do that for our colleagues’ e-mails? But we are most interested in the nature and coherence or dishevelment of personal contect that goes into research.

## Open Problems

How does networking affect your research? Do you have some big personal influences, or a lot of little ones?

1. January 9, 2013 10:26 pm

Re: “How does networking affect your research?”

Reflecting on this draws me back to questions that I’ve mused on and worried over for many a year —

What is communication, really?

How does information about an object system get from a subsystem that has it to a subsystem that needs it?

What is information that a sign may bear it?

January 9, 2013 11:31 pm

Jon,

“What is information that a sign may bear it?”

Could you elaborate?

• January 9, 2013 11:48 pm

Jim,

I was trying to link to an old essay I wrote on “Semiotic Information” — where I made a first stab at relating C.S. Peirce’s theory of information from 1865–1866 to the brand of information theory we know from Shannon — but the target site is having some problems right now. I alerted the management and maybe it will be fixed tomorrow. If not, I’ll make a fresh attempt at explanation.

• January 10, 2013 10:42 am

It looks like the problem is fixed now, but, just in case, I added a link to an archive version of the essay.

I am thinking I should write another blog post to develop this theme.

January 10, 2013 8:07 am

Bill Thurston’s Foreword to Mircea Pitici’s The Best Writing on Mathematics 2010 is among the finest meditations (as it seems to me) upon the role of social interaction in mathematics:

Mathematics is commonly thought to be the pursuit of universal truths, of patterns that are not anchored to any single fixed concept. But on a deeper level the goal of mathematics is to develop enhanced ways for humans to see and think about the world. Mathematics is a transforming journey, and progress in it can better be measured by changes in how we think than by the external truths we discover. […] As I read, I stop and ask: What’s the author trying to say? What is the author really thinking? Studying mathematics transforms our minds.

Thurston’s essay explains why (as it seems to me) the great question-askers of mathematics are comparably worthy of our respect to the great question-answerers.

• January 11, 2013 11:00 am

Why am I asking this question?

How will I answer this question?

• January 24, 2013 3:04 pm

See “Prospects for Inquiry Driven Systems § 1.1” on my Bibliography page for a better copy of the above essay.

3. February 8, 2013 6:56 pm

interesting history! another great study in great mathematical mentoring/collaborations from history is the Hardy-Ramanujan relationship. theres a great chapter in World of Mathematics by Newman & info in various other places