Archive for the 'Human intelligence' Category

Courage and luck

A dialog from Nadine Gordimer’s novel, A Guest of Honour (page 222):

‘Why does playing safe always seem to turn out to be so dangerous?’

‘It’s unlucky . . .

. . . because you’re too scared to take a chance.’

‘It’s unlucky to lack courage?’

‘That’s it.  You have to go ahead into what’s coming, trust to luck.  Because if you play safe you don’t have any, anyway.’

‘It’s forfeited?’

‘Yes.’




The Way

A recurring theme here has been the complexity of most important real-world decision-making, contrary to the models used in much of economics and computer science.   Some relevant posts are here, here, and here.    On this topic, I recently came across a wonderful cartoon by Michael Leunig, entitled “The Way” (click on the image to enlarge it):

Michael-Leunig-The-Way-2012

I am very grateful to Michael Leunig for permission to reproduce this cartoon here.




The Yirrkala Bark Petition

Australia has just commemorated the 50th anniversary of the presentation of the Yirrkala bark panels as a petition to the Australian Commonwealth Parliament in August 1963.   The panels are an example of visual art as argument, as I noted here.

Related posts on visual arguments here and here.  

 




Green intelligence

Are plants intelligent?   Here are 10 reasons for thinking so.    I suspect the reason we don’t naturally consider the activities of plants to be evidence of intelligent behaviour is primarily because the timescales over which these activities are undertaken is typically longer than for animal behaviours.    We humans have trouble seeing outside our own normal frames of reference.   (HT: JV)




What do mathematicians do?

Over at the AMS Graduate Student Blog, Jean Joseph wonders what it is that mathematicians do, asking if what they do is to solve problems:

After I heard someone ask about what a mathematician does, I myself wonder what it means to do mathematics if all what one can answer is that mathematicians do mathematics. Solving problems have been considered by some as the main activity of a mathematician, which might then be the answer to the question. But, could reading and writing about mathematics or crafting a new theory be considered as serious mathematical activities or mere extracurricular activities?”

Not all mathematics is problem-solving, as we’ve discussed here before, and I think it would be a great shame if the idea were to take hold that all that mathematicians did was to solve problems.  As Joseph says, this view does not account for lots of activities that we know mathematicians engage in which are not anywhere near to problem-solving, such as creating theories, defining concepts, writing expositions, teaching, etc.

I view mathematics (and the related disciplines in the pure mathematical universe) as the rigorous study of structure and relationship.   What mathematicians do, then, is to rigorously study structure and relationship.  They do this by creating, sharing and jointly manipulating abstract mental models, seeking always to understand the properties and inter-relations of these models.

Some of these models may arise from, or be applied to, particular domains or particular problems, but mathematicians (at least, pure mathematicians) are typically chiefly interested in the abstract models themselves and their formal properties, rather than the applications.  In some parts of mathematics (eg, algebra) written documents such as research papers and textbooks provide accurate descriptions of these mental models.  In other parts (eg, geometry), the written documents can only approximate the mental models.     As mathematician William Thurston once said:

There were published theorems that were generally known to be false, or where the proofs were generally known to be incomplete. Mathematical knowledge and understanding were embedded in the minds and in the social fabric of the community of people thinking about a particular topic. This knowledge was supported by written documents, but the written documents were not really primary.

I think this pattern varies quite a bit from field to field. 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. But in any field, there is a strong social standard of validity and truth.

. . .

When people are doing mathematics, the flow of ideas and the social standard of validity is much more reliable than formal documents. People are usually not very good in checking formal correctness of proofs, but they are quite good at detecting potential weaknesses or flaws in proofs.”




Listening to music by jointly reading the score

Another quote from Bill Thurston, this with an arresting image of mathematical communication:

We have an inexorable instinct to convey through speech content that is not easily spoken.  Because of this tendency, mathematics takes a highly symbolic, algebraic, and technical form.  Few people listening to a technical discourse are hearing a story. Most readers of mathematics (if they happen not to be totally baffled) register only technical details – which are essentially different from the original thoughts we put into mathematical discourse.  The meaning, the poetry, the music, and the beauty of mathematics are generally lost.  It’s as if an audience were to attend a concert where the musicians, unable to perform in a way the audience could appreciate, just handed out copies of the score.  In mathematics, it happens frequently that both the performers and the audience are oblivious to what went wrong, even though the failure of communication is obvious to all.” (Thurston 2011, page xi)  

Reference:

William P. Thurston [2011]:   Foreword.   The Best Writing on Mathematics: 2010.  Edited by Mircea Pitici.  Princeton, NJ, USA:  Princeton University Press.




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 occured 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.  Acivities 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 2103, pages 46-55.

William F. Thurston [1994]:  On proof and progress in mathematicsAmerican Mathematical Society, 30 (2):  161-177.




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]




The value of an education

In a letter to Rupert Hart-Davies on 29 November 1956 George Lyttelton included this statement from William Johnson Cory (1823-1892, Master of Eton 1845-1872) on education:

At school you are engaged not so much in acquiring knowledge as making mental efforts under criticism. A certain amount of knowledge, you can indeed with average faculties acquire so as to retain; nor need you regret the hours you spent on much that is forgotten, for the shadow of lost knowledge at least protects you from many illusions.  But you go to a great school not so much for knowledge as for arts and habits; for the habit of attention, for the art of expression, for the art of assuming at a moment’s notice a new intellectual position, for the art of entering quickly into another person’s thoughts, for the habit of submitting to censure and refutation, for the art of indicating assent or dissent in graduated terms, for the habit of regarding minute points of accuracy, for the art of working out what is possible in a given time, for taste, for discrimination, for mental courage, and for mental soberness.”

Reference:

Rupert Hart-Davis (Editor) [1978-79]: The Lyttelton Hart-Davis Letters:  Correspondence of George Lyttelton and Rupert Hart-Davis, 1955-1962. London: John Murray.




Suddenly, the fog lifts . . .

Andrew Wiles, prover of Wiles’ Theorem (aka Fermat’s Last Theorem), on the doing of mathematics:

Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it’s dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it’s all illuminated. You can see exactly where you were.

This describes my experience, over shorter time-frames, in studying pure mathematics as an undergraduate, with each new topic covered: epsilon-delta arguments in analysis; point-set topology; axiomatic set theory; functional analysis; measure theory; group theory; algebraic topology; category theory; statistical decision theory; integral geometry; etc.    A very similar process happens in learning a new language, whether a natural (human) language or a programming language.     Likewise, similar words describe the experience of entering a new organization (either as an employee or as a management consultant), and trying to understand how the organization works, who has the real power, what are the social relationships and dynamics within the organization, etc, something I have previously described here.

One encounters a new discipline or social organization, one studies it and thinks about it from as many angles and perspectives as one can, and eventually, if one is clever and diligent, or just lucky, a light goes on and all is illuminated.    Like visiting a new city and learning its layout by walking through it, frequently getting lost and finding one’s way again,  enlightenment requires work.  Over time, one learns not to be afraid in encountering a new subject, but rather to relish the state of inchoateness and confusion in the period between starting and enlightenment.  The pleasure and wonder of the enlightenment is so great, that it all the prior pain is forgotten.

Reference:

Andrew Wiles [1996],  speaking in Fermat’s Last Theorem, a BBC documentary by S. Singh and John Lynch: Horizon, BBC 1996,  cited in Frans Oort [2011 ]:  Did earlier thoughts inspire Grothendieck? (Hat tip).