Mathematical thinking and software

Further to my post citing Keith Devlin on the difficulties of doing mathematics online, I have heard from one prominent mathematician that he does all his mathematics now using LaTeX, not using paper or whiteboard, and thus disagrees with Devlin’s (and my) views.   Thinking about why this may be, and about my own experiences using LaTeX, it occurred to me that one’s experiences with thinking-support software, such as word-processing packages such as MS-WORD or  mark-up programming languages such as LaTeX, will very much depend on the TYPE of thinking one is doing.
If one is thinking with words and text, or text-like symbols such as algebra, the right-handed folk among us are likely to be using the left hemispheres of our brains.  If one is thinking in diagrams, as in geometry or graph theory or much of engineering including computing, the right-handed among us are more likely to be using the right hemispheres of our brains.  Yet MS-WORD and LaTeX are entirely text-based, and their use requires the heavy involvement of our left hemispheres (for the northpaws among us).  One doesn’t draw an arrow in LaTeX, for example, but instead types a command such as \rightarrow or \uparrow.   If one is already using one’s left hemisphere to do the mathematical thinking, as most algebraists would be, then the cognitive load in using the software will be a lot less then if one is using one’s right hemisphere for the mathematical thinking.  Activities which require both hemispheres are typically very challenging to most of us, since co-ordination between the two hemispheres adds further cognitive overhead.
I find LaTeX immeasurably better than any other word-processor for writing text:  it and I work at the same speed (which is not true of MS-WORD for me, for example), and I am able to do my verbal thinking in it.  In this case, writing is a form of thinking, not merely the subsequent expression of thoughts I’ve already had.     However, I cannot do my mathematical or formal thinking in LaTeX, and the software is at best a tool for subsequent expression of thoughts already done elsewhere – mentally, on paper, or on a whiteboard.    My formal thinking is usually about structure and relationship, and not as often algebraic symbol manipulation.
Bill Thurston, the geometer I recently quoted, said:

I was interested in geometric areas of mathematics, where it is often pretty hard to have a document that reflects well the way people actually think.  In more algebraic or symbolic fields, this is not necessarily so, and I have the impression that in some areas documents are much closer to carrying the life of the field.”  [Thurston 1994, p. 169]

It is interesting that many non-mathematical writers also do their thinking about structure not in the document itself or as they write, but outside it and beforehand, and often using tools such as post-it notes on boards; see the recent  article by John McPhee in The New Yorker for examples from his long writing life.
References:
John McPhee [2013]: Structure:  Beyond the picnic-table crisisThe New Yorker, 14 January 2013, pages 46-55.
William F. Thurston [1994]:  On proof and progress in mathematicsAmerican Mathematical Society, 30 (2):  161-177.

Vale: Dave Brubeck

The BBC Radio 3 program Jazz Record Requests had a special edition yesterday in memory of Dave Brubeck.  It is available to listen for another 6 days, here.   
I heard Brubeck and his quartet play a concert in Liverpool about 10 years ago. He was old enough to have to shuffle slowly onto stage, but once at the piano, his playing was alive and energetic. My only disappointment was that he performed a concert in Liverpool and not once made any reference to the music of the city’s most famous musical sons. We could have been in Outer Woop Woop, for all the difference it had on his choice of repertoire. Not even an allusion in an improvisation was just churlish.
Brubeck’s reknown was remarkable.   I once requested a busking middle-aged violinist in a Kiev cafe in the mid 1990s to play Take Five, and saw his face light up with delight.  As it happened, he also knew The Hot Canary.

Thurston on mathematical proof

The year 2012 saw the death of Bill Thurston, leading geometer and Fields Medalist.   Learning of his death led me to re-read his famous 1994 AMS paper on the social nature of mathematical proof.   In my opinion, Thurston demolished the views of those who thought mathematics is anything other than socially-constructed.  This post is just to present a couple of long quotes from the paper.
Continue reading ‘Thurston on mathematical proof’

Mathematical hands

With MOOCs fast becoming teaching trend-du-jour in western universities, it is easy to imagine that all disciplines and all ways of thinking are equally amenable to information technology.   This is simply not true, and mathematical thinking  in particular requires hand-written drawing and symbolic manipulation.   Nobody ever acquired skill in a mathematical discipline without doing exercises and problems him or herself, writing on paper or a board with his or her own hands.   The physical manipulation by the hand holding the pen or pencil is necessary to gain facility in the mental manipulation of the mathematical concepts and their relationships.
Keith Devlin recounts his recent experience teaching a MOOC course on mathematics, and the deleterious use by students of the word-processing package latex for doing assignments:

We have, it seems, become so accustomed to working on a keyboard, and generating nicely laid out pages, we are rapidly losing, if indeed we have not already lost, the habit—and love—of scribbling with paper and pencil. Our presentation technologies encourage form over substance. But if (free-form) scribbling goes away, then I think mathematics goes with it. You simply cannot do original mathematics at a keyboard. The cognitive load is too great.

Why is this?  A key reason is that current mathematics-producing software is clunky, cumbersome, finicky, and not WYSIWYG (What You See Is What You Get).   The most widely used such software is Latex (and its relatives), which is a mark-up and command language; when compiled, these commands generate mathematical symbols.   Using Latex does not involve direct manipulation of the symbols, but only their indirect manipulation.   One has first to imagine (or indeed, draw by hand!) the desired symbols or mathematical notation for which one then creates using the appropriate generative Latex commands.   Only when these commands are compiled can the user see the effects they intended to produce.   Facility with pen-and-paper, by contrast, enables direct manipulation of symbols, with (eventually), the pen-in-hand being experienced as an extension of the user’s physical body and mind, and not as something other.   Expert musicians, archers, surgeons, jewellers, and craftsmen often have the same experience with their particular instruments, feeling them to be extensions of their own body and not external tools.
Experienced writers too can feel this way about their use of a keyboard, but language processing software is generally WYSIWYG (or close enough not to matter).  Mathematics-making software  is a long way from allowing the user to feel that they are directly manipulating the symbols in their head, as a pen-in-hand mathematician feels.  Without direct manipulation, hand and mind are not doing the same thing at the same time, and thus – a fortiori – keyboard-in-hand is certainly not simultaneously manipulating concept-in-mind, and nor is keyboard-in-hand simultaneously expressing or evoking concept-in-mind.
I am sure that a major source of the problem here is that too many people – and especially most of the chattering classes – mistakenly believe the only form of thinking is verbal manipulation.  Even worse, some philosophers believe that one can only think by means of words.     Related posts on drawing-as-a-form-of-thinking here, and on music-as-a-form-of-thinking here.
[HT:  Normblog]

Time, gentlemen, please

Much discussion again over at Language Log over a claim of the form “Language L has no word for concept C”.  This time, it was the claim by Wade Davis (whose strange use of past tense indicates he has forgotten or is unaware that many Australian Aboriginal languages are still in use) that:

In not one of the hundreds of Aboriginal dialects and languages was there a word for time.”

The rebuttal of this claim by Mark Liberman was incisive and decisive.   Davis was using this claim to support a more general argument:  that traditional Australian Aboriginal cultures had different notions of and metaphors for time to those we mostly have in the modern Western world.
We in the contemporary educated West typically use a spatial metaphor for time, where the past is in one abstract place, the present in another non-overlapping abstract place, and the future in yet a third non-overlapping abstract place.    In this construal of time, causal influence travels in one direction only:  from the past to the present, and from the present to the future.   Nothing in either the present  or the future may influence the past, which is fixed and unchangeable.   Events in the future may perhaps be considered to influence the present, depending on how much fluidity we allow the present to have.  However, most of us would argue that it is not events in the future that influence events in the present, but our present perceptions of possible future events that influence events and actions in the present.
Modern Western Europeans typically think of the place that represents the past as being behind them, and the future ahead.   People raised in Asian cultures often think of the abstract place that is the past as being below them (or above them), and the future above (or below).   But all consider these abstract places to be non-overlapping, and even non-contiguous.
Traditional Australian Aboriginal cultures, as Davis argues, construe time very differently, and influences may flow in all directions.   A better spatial metaphor for Aboriginal notions of time would be to consider a modern city, where there are many different types of transport and communications, each viewable as a network:  rivers, canals, roads, bus-only road corridors, railways, underground rail tunnels, underground sewage or water drains, cycleways, footpaths, air-transport corridors, electricity networks, fixed-link telecommunications networks, wireless telecommunications networks, etc.    A map of each of these networks could be created (and usually are) for specific audiences.  A map of the city itself could then be formed from combining these separate maps, overlaid upon one another as layers in a stack.   Each layer describes a separate aspect of reality, but the reality of the actual entire city is complex and more than merely the sum of these parts.  Events or perceptions in one layer may influence events or perceptions in other layers, without any limitations on the directions of causality between layers.
Traditional Aboriginal notions of time are similar, with pasts, the present and futures all being construed as separate layers stacked over the same geographic space – in this case actual geographic country, not an abstract spatial representation of time.  Each generation of people who have lived, or who will live, in the specific region (“country” in modern Aboriginal English) will have created a layer in the stack.   Influence travels between the different layers in any and all directions, so events in the distant past or the distant future may influence events in the present, and events in the present may influence events in the past and the future.
Many religions – for example, Roman Catholicism, Hinduism, and African cosmologies – allow for such multi-directional causal influences via a non-material realm of saints or spirits, usually the souls of the dead, who may have power to guide the actions of the living in the light of the spirits’ better knowledge of the future.   Causal influence can thus travel, via such spirit influences, from future to present.  Similarly, the view of Quantum Mechanics of space-time as a single 4-dimensional manifold allows for influences across the dimension of time as well as those of space.
I am reminded of an experience I once witnessed where the only sensible explanation of a colleague’s passionate enthusiasm for a particular future course of action was his foreknowledge of the specific details of the outcome of that course of action.  But these details he did not know and could not have known at the time of his enthusiasm,  prior to the course of action being executed.  In other words, only a causal influence from future to present provided a sensible explanation for this enthusiasm, and this explanation only became evident as the future turned into the present, and the details of the outcome emerged.  Until that point, he could not justify or explain his passionate enthusiasm, which seemed to be a form of madness, even to him.    Contemporary Western cosmology does not provide such time-reversing explanations, but many other cultures do; and current theories of quantum entanglement also seem to.
Contemporary westerners, particularly those trained in western science, have a hard time understanding such alternative cosmologies, in my experience.  I have posted before about the difficulties most westerners have, for instance,  in understanding Taoist/Zen notions of synchronicity of events, which westerners typically mis-construe as random chance.

Snow jobs

Advice from Geoffrey Pullum, when faced with people who tell you that Eskimos have multiple words for “snow”:

Stand up and tell the speaker this:  C.W. Schultz-Lorentzen’s Dictionary of the West Greenlandic Eskimo Language (1927) gives just two possibly relevant roots: qanik, meaning ‘snow in the air’ or ‘snowflake’, and aput, meaning ‘snow on the ground’. Then add that you would be interested to know if the speaker can cite any more.

References:
G. K. Pullum [1989]: The great Eskimo vocabulary hoax.  Natural Language and Linguistic Theory, 7: 275-281.  Available from here.
C. W. Schultz-Lorentzen [1927]:  Dictionary of the West Greenlandic Eskimo Language. Meddelelser om Grønland, 69, Reitzels, Copenhagen, Denmark.
 

Public appearances

I have argued before that it is an abuse of power for major newspapers to run obituaries of obscure back-office staff, simply because they can.   This is an abuse since a newspaper is a public organization, playing a very public role in public life,  not some private family newsletter shared, like samizdat, between close relatives over kitchen coffee.
The Guardian now runs, as the cover story in its Saturday Review section, extracts from the piano-learning diary of its Editor, Alan Rusbridger.    Perhaps there is nothing actually unethical about a major newspaper running long cover stories about its Editor’s private hobbies, and promoting his new book.  But one has to ask:   Is The Guardian now the private family newsletter of its Editor?

PhD Vivas

Awhile back, I posted some advice from my own experiences on doing a PhD.  Since then, several people have asked me for advice about the viva voce (or oral) examination, which most PhD programs require at the end of the degree.    Here are some notes I wrote for a candidate recently.
It is helpful to think about the goals of the examiners.   In my opinion, they are trying to achieve the following goals:
1. First, they simply want to understand what your dissertation says.   This means they will usually ask you to clarify or explain things which are not clear to them.
2.  Then, they want to understand the context of the work.  This refers to the previous academic literature on the subject or on related subjects, so they will generally ask about that literature.  They may consider some topic to be related to your work which you did not cover; in that case, you would normally be asked to add some text on that topic.
3.  They want to assess if the work makes a contribution to the related literature.    So they will ask what is new or original in your dissertation, and why it is different from the past work of others.  They will also want to be able to separate what is original from what came before (which is sometimes hard to do in some dissertations, due to the writing style of the candidate or the structure of the document).   To the extent that Computer Science is an engineering discipline, and thus involves design, originality is usually not a problem:  few other people will be working in the same area as you, and none of them would have made precisely the same sequences of design choices in the same order for the same reasons as you did.
4.  They will usually want to assess if the new parts in the dissertation are significant or important.  They will ask you about the strengths and weaknesses of your research, relative to the past work of others.   They will usually ask about potential future work, the new questions that arise from your work, or the research that your work or your techniques make possible.  Research or research techniques which open up new research vistas or new application domains are usually looked upon favourably.
5.  Goals #3 and #4 will help the examiners decide if the written dissertation is worth receiving a PhD award, since most university regulations require PhD dissertations to present an original and significant contribution to knowledge.
6.  The examiners will also want to assess if YOU yourself wrote the document.  They will therefore ask you about the document, what your definitions are, where things are, why you have done certain things and not others, why you have made certain design choices and not others,  etc.     Some examiners will even give the impression that they have not read your dissertation, precisely to find out if you have!
7.  Every dissertation makes some claims (your “theses”).  The examiners will generally approach these claims with great scepticism, questioning and challenging you, contesting your responses and arguments, and generally trying to argue you down.   They want to see if you can argue in favour of your claims, to see if you are able to justify and support your claims, and how you handle criticism.   After all, if you can’t support your claims, no one else will, since you are the one proposing them.
The viva is not a test of memory, so you can take a copy of your thesis with you and refer to it as you wish.  Likewise, you can take any notes you want.    The viva is also not a test of speed-thinking, so you can take your time to answer questions or to respond to comments.    You can ask the examiners to explain any question or any comment which you don’t understand.   It is OK to argue with the examiners (in some sense, it is expected), but not to get personal in argument or to lose your temper.
The viva is one of the few occasions in a research career when you can have an extended discussion about your research with people interested in the topic who have actually read your work.   Look forward to it, and enjoy it!

Your local neighbourhood top-secret Global Military Command Centre

In WW II, the British military paid friendly nationals in neutral Sweden, Switzerland, and elsewhere to subscribe to provincial German newspapers in order to garner intelligence about life in Germany.  Among other things, printed death notices were used to estimate casualty numbers in German military units, since particular units tended to recruit from particular regions; casualty rates were a means to assess the degree of success of Nazi military campaigns those units were involved in.
Let us hope now that Britain’s enemies are not reading provincial newspapers such as The Wiltshire Times (14 June 2011):

It is hard to believe that the central communications hub for the entire British Army sits unassumingly on the outskirts of the quiet Wiltshire market town of Corsham.
. . .
The centre, at Westwells Road, in Neston, is home to GOSCC – the Global Operations Security Control Centre – a top secret centre which houses up to 600 specialists working behind the scenes to make huge military operations such as those in Libya, Afghanistan and Iraq a reality.”

I guess not revealing the street number keeps the location of the top-secret centre safe.