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Multiple ways of understanding

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In his wonderful On Proof and Progress in Mathematics, Thurston begins his second section “How do people understand mathematics?” as follows:

This is a very hard question. Understanding is an individual and internal matter that is hard to be fully aware of, hard to understand and often hard to communicate. We can only touch on it lightly here.

People have very different ways of understanding particular pieces of mathematics. To illustrate this, it is best to take an example that practicing mathematicians understand in multiple ways, but that we see our students struggling with. The derivative of a function fits well. The derivative can be thought of as:

  1. Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.
  2. Symbolic: the derivative of x^n is nx^{n-1}, the derivative of \sin(x) is \cos(x), the derivative of f \circ g is f' \circ g * g', etc.
  3. Logical: f'(x) = d if and only if for every \epsilon there is a \delta such that when 0 < |\Delta x| < \delta,

    \left|\frac{f(x+\Delta x) - f(x)}{\Delta x} - d\right| < \epsilon.

  4. Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.
  5. Rate: the instantaneous speed of f(t), when t is time.
  6. Approximation: The derivative of a function is the best linear approximation to the function near a point.
  7. Microscopic: The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power.

This is a list of different ways of thinking about or conceiving of the derivative, rather than a list of different logical definitions. Unless great efforts are made to maintain the tone and flavor of the original human insights, the differences start to evaporate as soon as the mental concepts are translated into precise, formal and explicit definitions.

I can remember absorbing each of these concepts as something new and interesting, and spending a good deal of mental time and effort digesting and practicing with each, reconciling it with the others. I also remember coming back to revisit these different concepts later with added meaning and understanding.

The list continues; there is no reason for it ever to stop. A sample entry further down the list may help illustrate this. We may think we know all there is to say about a certain subject, but new insights are around the corner. Furthermore, one person’s clear mental image is another person’s intimidation:

  1. The derivative of a real-valued function f in a domain D is the Lagrangian section of the cotangent bundle T^{\ast}(D) that gives the connection form for the unique flat connection on the trivial \mathbf{R}-bundle D \times \mathbf{R} for which the graph of f is parallel.

These differences are not just a curiosity. Human thinking and understanding do not work on a single track, like a computer with a single central processing unit. Our brains and minds seem to be organized into a variety of separate, powerful facilities. These facilities work together loosely, “talking” to each other at high levels rather than at low levels of organization.

This has been extended on the MathOverflow question Different ways of thinking about the derivative where you can find even more ways of thinking about the derivative. (Two of the interesting pointers are to this discussion on the n-Category Café, and to the book Calculus Unlimited by Marsden and Weinstein, which does calculus using a “method of exhaustion” that does not involve limits. (Its definition of the derivative is also mentioned at the earlier link, as that notion of the derivative closest to [the idea of Eudoxus and Archimedes] of “the tangent line touches the curve, and in the space between the line and the curve, no other straight line can be interposed”, or “the line which touches the curve only once” — this counts as another important way of thinking about the derivative.)

It has also been best extended by Terence Tao, who in an October 2009 blog post on Grothendieck’s definition of a group gave several ways of thinking about a group:

In his wonderful article “On proof and progress in mathematics“, Bill Thurston describes (among many other topics) how one’s understanding of given concept in mathematics (such as that of the derivative) can be vastly enriched by viewing it simultaneously from many subtly different perspectives; in the case of the derivative, he gives seven standard such perspectives (infinitesimal, symbolic, logical, geometric, rate, approximation, microscopic) and then mentions a much later perspective in the sequence (as describing a flat connection for a graph).

One can of course do something similar for many other fundamental notions in mathematics. For instance, the notion of a group {G} can be thought of in a number of (closely related) ways, such as the following:

  1. Motivating examples: A group is an abstraction of the operations of addition/subtraction or multiplication/division in arithmetic or linear algebra, or of composition/inversion of transformations.
  2. Universal algebraic: A group is a set {G} with an identity element {e}, a unary inverse operation {\cdot^{-1}: G \rightarrow G}, and a binary multiplication operation {\cdot: G \times G \rightarrow G} obeying the relations (or axioms) {e \cdot x = x \cdot e = x}, {x \cdot x^{-1} = x^{-1} \cdot x = e}, {(x \cdot y) \cdot z = x \cdot (y \cdot z)} for all {x,y,z \in G}.
  3. Symmetric: A group is all the ways in which one can transform a space {V} to itself while preserving some object or structure {O} on this space.
  4. Representation theoretic: A group is identifiable with a collection of transformations on a space {V} which is closed under composition and inverse, and contains the identity transformation.
  5. Presentation theoretic: A group can be generated by a collection of generators subject to some number of relations.
  6. Topological: A group is the fundamental group {\pi_1(X)} of a connected topological space {X}.
  7. Dynamic: A group represents the passage of time (or of some other variable(s) of motion or action) on a (reversible) dynamical system.
  8. Category theoretic: A group is a category with one object, in which all morphisms have inverses.
  9. Quantum: A group is the classical limit {q \rightarrow 0} of a quantum group.

One can view a large part of group theory (and related subjects, such as representation theory) as exploring the interconnections between various of these perspectives. As one’s understanding of the subject matures, many of these formerly distinct perspectives slowly merge into a single unified perspective.

From a recent talk by Ezra Getzler, I learned a more sophisticated perspective on a group, somewhat analogous to Thurston’s example of a sophisticated perspective on a derivative (and coincidentally, flat connections play a central role in both):

  1. Sheaf theoretic: A group is identifiable with a (set-valued) sheaf on the category of simplicial complexes such that the morphisms associated to collapses of {d}-simplices are bijective for {d < 1} (and merely surjective for {d \leq 1}).

The rest of the post elaborates on this understanding.

Again in a Google Buzz post on Jun 9, 2010, Tao posted the following:

Bill Thurston’s “On proof and progress in mathematics” has many nice observations about the nature and practice of modern mathematics. One of them is that for any fundamental concept in mathematics, there is usually no “best” way to define or think about that concept, but instead there is often a family of interrelated and overlapping, but distinct, perspectives on that concept, each of which conveying its own useful intuition and generalisations; often, the combination of all of these perspectives is far greater than the sum of the parts. Thurston illustrates this with the concept of differentiation, to which he lists seven basic perspectives and one more advanced perspective, and hints at dozens more.

But even the most basic of mathematical concepts admit this multiplicity of interpretation and perspective. Consider for instance the operation of addition, that takes two numbers x and y and forms their sum x+y. There are many such ways to interpret this operation:

1. (Disjoint union) x+y is the “size” of the disjoint union X u Y of an object X of size x, and an object Y of size y. (Size is, of course, another concept with many different interpretations: cardinality, volume, mass, length, measure, etc.)

2. (Concatenation) x+y is the size of the object formed by concatenating an object X of size x with an object Y of size y (or by appending Y to X).

3. (Iteration) x+y is formed from x by incrementing it y times.

4. (Superposition) x+y is the “strength” of the superposition of a force (or field, intensity, etc.) of strength x with a force of strength y.

5. (Translation action) x+y is the translation of x by y.

5a. (Translation representation) x+y is the amount of translation or displacement incurred by composing a translation by x with a translation by y.

6. (Algebraic) + is a binary operation on numbers that give it the structure of an additive group (or monoid), with 0 being the additive identity and 1 being the generator of the natural numbers or integers.

7. (Logical) +, when combined with the other basic arithmetic operations, are a family of structures on numbers that obey a set of axioms such as the Peano axioms.

8. (Algorithmic) x+y is the output of the long addition algorithm that takes x and y as input.

9. etc.

These perspectives are all closely related to each other; this is why we are willing to give them all the common name of “addition”, and the common symbol of “+”. Nevertheless there are some slight differences between each perspective. For instance, addition of cardinals is based on perspective 1, while addition of ordinals is based on perspective 2. This distinction becomes apparent once one considers infinite cardinals or ordinals: for instance, in cardinal arithmetic, aleph_0 = 1+ aleph_0 = aleph_0 + 1 = aleph_0 + aleph_0, whereas in ordinal arithmetic, omega = 1+omega < omega+1 < omega + omega.

Transitioning from one perspective to another is often a necessary first conceptual step when the time comes to generalise the concept. As a child, addition of natural numbers is usually taught initially by using perspective 1 or 3, but to generalise to addition of integers, one must first switch to a perspective such as 4, 5, or 5a; similar conceptual shifts are needed when one then turns to addition of rationals, real numbers, complex numbers, residue classes, functions, matrices, elements of abstract additive groups, nonstandard number systems, etc. Eventually, one internalises all of the perspectives (and their inter-relationships) simultaneously, and then becomes comfortable with the addition concept in a very broad set of contexts; but it can be more of a struggle to do so when one has grasped only a subset of the possible ways of thinking about addition.

In many situations, the various perspectives of a concept are either completely equivalent to each other, or close enough to equivalent that one can safely “abuse notation” by identifying them together. But occasionally, one of the equivalences breaks down, and then it becomes useful to maintain a careful distinction between two perspectives that are almost, but not quite, compatible. Consider for instance the following ways of interpreting the operation of exponentiation x^y of two numbers x, y:

1. (Combinatorial) x^y is the number of ways to make y independent choices, each of which chooses from x alternatives.

2. (Set theoretic) x^y is the size of the space of functions from a set Y of size y to a set X of size x.

3. (Geometric) x^y is the volume (or measure) of a y-dimensional cube (or hypercube) whose sidelength is x.

4. (Iteration) x^y is the operation of starting at 1 and multiplying by x y times.

5. (Homomorphism) y → x^y is the continuous homomorphism from the domain of y (with the additive group structure) to the range of x^y (with the multiplicative structure) that maps 1 to x.

6. (Algebraic) ^ is the operation that obeys the laws of exponentiation in algebra.

7. (Log-exponential) x^y is exp( y log x ). (This raises the question of how to interpret exp and log, and again there are multiple perspectives for each…)

8. (Complex-analytic) Complex exponentiation is the analytic continuation of real exponentiation.

9. (Computational) x^y is whatever my calculator or computer outputs when it is asked to evaluate x^y.

10. etc.

Again, these interpretations are usually compatible with each other, but there are some key exceptions. For instance, the quantity 0^0 would be equal to zero [ed: I think this should be one —S] using some of these interpretations, but would be undefined in others. The quantity 4^{1/2} would be equal to 2 in some interpretations, be undefined in others, and be equal to the multivalued expression +-2 (or to depend on a choice of branch) in yet further interpretations. And quantities such as i^i are sufficiently problematic that it is usually best to try to avoid exponentiation of one arbitrary complex number by another arbitrary complex number unless one knows exactly what one is doing. In such situations, it is best not to think about a single, one-size-fits-all notion of a concept such as exponentiation, but instead be aware of the context one is in (e.g. is one raising a complex number to an integer power? A positive real to a complex power? A complex number to a fractional power? etc.) and to know which interpretations are most natural for that context, as this will help protect against making errors when manipulating expressions involving exponentiation.

It is also quite instructive to build one’s own list of interpretations for various basic concepts, analogously to those above (or Thurston’s example). Some good examples of concepts to try this on include “multiplication”, “integration”, “function”, “measure”, “solution”, “space”, “size”, “distance”, “curvature”, “number”, “convergence”, “probability” or “smoothness”. See also my blog post below in which the concept of a “group” is considered.

I plan to collect more such “different ways of thinking about the same (mathematical) thing” in this post, as I encounter them.

Written by S

Sat, 2016-03-26 at 10:05:09

Posted in mathematics, quotes

The rest is commentary

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Famous verses appear in many variants. Thanks to Google, it is easy to find many of them. For “paropakāraḥ puṇyāya, pāpāya parapīḍanam”, Google throws up a lot of variants for the first half.

The Vikramacarita has:

śrūyatāṃ dharmasarvasvaṃ, yad uktaṃ śāstrakoṭibhiḥ /
paropakāraḥ puṇyāya, pāpāya parapīḍanam

Other variants are:

saṅkṣepāt kathyate dharmo janāḥ kiṃ vistareṇa vaḥ |
paropakāraḥ puṇyāya pāpāya para-pīḍanam ||Panc_3.103||


aṣṭādaśapurāṇeṣu vyāsasya vacanadvayam /
paropakāraḥ puṇyāya pāpāya parapīḍanam //


ślokārdhena pravakṣyāmi yaduktaṃ grantha-koṭibhiḥ
paropakāraḥ puṇyāya pāpāya parapīḍanam

Going by the first line gives other verses:

śrūyatāṃ dharmasarvasvaṃ śrutvā caivāvadhāryatām | (or caiva vicāryatām ।)
ātmanaḥ pratikūlāni pareṣāṃ na samācaret ||

[Cāṇakya-nīti, Pañcatantra, Subhāṣitāvalī etc.]

See also:

prāṇā yathātmano ‘bhīṣṭā bhūtānām api te tathā |
ātmaupamyena gantavyaṃ buddhimadbhir mahātmabhiḥ

तस्माद्धर्मप्रधानेन भवितव्यं यतात्मना ।
तथा च सर्वभूतेषु वर्तितव्यं यथात्मनि ॥ Mahābhārata Shānti-Parva 167:9

05,039.057a na tatparasya saṃdadhyāt pratikūlaṃ yadātmanaḥ
05,039.057b*0238_01 ātmanaḥ pratikūlāni vijānan na samācaret
05,039.057c saṃgraheṇaiṣa dharmaḥ syāt kāmād anyaḥ pravartate

As Hillel says, the rest is commentary.

For (some) commentary, go here.

Written by S

Fri, 2014-06-06 at 23:38:30

Posted in quotes, sanskrit, unfinished

Big O() notation: a couple of sources

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This post contains, just for future reference, a couple of primary sources relevant to the O (“Big O”) notation:

  1. Some introductory words from Asymptotic Methods in Analysis by de Bruijn
  2. An letter from Donald Knuth on an approach to teaching calculus using this notation.

Read the rest of this entry »

Written by S

Thu, 2014-03-13 at 16:33:20

Stephen Fry’s “The Ode Less Travelled”: Foreword

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I reproduce below the line below Stephen Fry’s entire foreword to his book The Ode Less Travelled, because I find myself frequently referring to it and would like to be able to direct friends to some place to read it‌ — this is now such a place.

I HAVE A DARK AND DREADFUL SECRET. I write poetry. This is an embarrassing confession for an adult to make. In their idle hours Winston Churchill and Noël Coward‎ painted. For fun and relaxation Albert Einstein played the violin. Hemingway hunted, Agatha Christie gardened, James Joyce sang arias and Nabokov chased butterflies. But poetry?

I have a friend who drums in the attic, another who has been building a boat for years. An actor I know is prouder of the reproduction eighteenth-century duelling pistols he makes in a small workshop than he is of his knighthood. Britain is a nation of hobbyists—eccentric amateurs, talented part-timers, Pooterish potterers and dedicated autodidacts in every field of human endeavour. But poetry?

An adolescent girl may write poetry, so long as it is securely locked up in her pink leatherette five-year diary. Suburban professionals are permitted to enter jolly pastiche competitions in the Spectator and New Statesman. At a pinch, a young man may be allowed to write a verse or two of dirty doggerel and leave it on a post-it note stuck to the fridge when he has forgotten to buy a Valentine card. But that’s it. Any more forays into the world of Poesy and you release the beast that lurks within every British breast—and the name of the beast is Embarrassment.

And yet…

I believe poetry is a primal impulse within us all. I believe we are all capable of it and furthermore that a small, often ignored corner of us positively yearns to try it. I believe our poetic impulse is blocked by the false belief that poetry might on the one hand be academic and technical and on the other formless and random. It seems to many that while there is a clear road to learning music, gardening or watercolours, poetry lies in inaccessible marshland: no pathways, no signposts, just the skeletons of long-dead poets poking through the bog and the unedifying sight of living ones floundering about in apparent confusion and mutual enmity. Behind it all, the dread memory of classrooms swollen into resentful silence while the English teacher invites us to ‘respond’ to a poem.

For me the private act of writing poetry is songwriting, confessional, diary-keeping, speculation, problem-solving, storytelling, therapy, anger management, craftsmanship, relaxation, concentration and spiritual adventure all in one inexpensive package.

Suppose I want to paint but seem to have no obvious talent. Never mind: there are artist supply shops selling paints, papers, pastels, charcoals and crayons. There are ‘How To’ books everywhere. Simple lessons in the rules of proportion and guides to composition and colourmixing can make up for my lack of natural ability and provide painless technical grounding. I am helped by grids and outlines, pantographs and tracing paper; precise instructions guide me in how to prepare a canvas, prime it with paint and wash it into an instant watercolour sky. There are instructional videos available; I can even find channels on cable and satellite television showing gentle hippies painting lakes, carving pine trees with palette knives and dotting them with impasto snow. Mahlsticks, sable, hogs-hair, turpentine and linseed. Viridian, umber, ochre and carmine. Perspective, chiaroscuro, sfumato, grisaille, tondo and morbidezza. Reserved modes and materials. The tools of the trade. A new jargon to learn. A whole initiation into technique, form and style.

Suppose I want to play music but seem to have no obvious talent. Never mind: there are music shops selling instruments, tuning forks, metronomes and ‘How To’ books by the score. And scores by the score. Instructional videos abound. I can buy digital keyboards linked to programmes that plug into my computer and guide me through the rudiments, monitoring my progress and accuracy. I start with scales and move on to chords and arpeggios. There are horsehair, rosin and catgut, reeds, plectrums and mouthpieces. There are diminished sevenths, augmented fifths, relative minors, trills and accidentals. There are riffs and figures, licks and vamps. Sonata, adagio, crescendo, scherzo and twelve-bar blues. Reserved modes and materials. The tools of the trade. A new jargon to learn. A whole initiation into technique, form and style.

To help us further there are evening classes, clubs and groups. Pack up your easel and palette and go into the countryside with a party of like-minded enthusiasts. Sit down with a friend and learn a new chord on the guitar. Join a band. Turn your watercolour view of Lake Windermere into a tablemat or T-shirt. Burn your version of ‘Stairway to Heaven’ onto a CD and alarm your friends.

None of these adventures into technique and proficiency will necessarily turn you into a genius or even a proficient craftsman. Your view of Snow on York Minster, whether languishing in the loft or forming the basis of this year’s Christmas card doesn’t make you Turner, Constable or Monet. Your version of ‘Fur Elise’ on electric piano might not threaten Alfred Brendel, your trumpet blast of ‘Basin Street Blues’ could be so far from Satchmo that it hurts and your take on ‘Lela’ may well stand as an eternal reproach to all those with ears to hear. You may not sell a single picture, be invited even once to deputise for the church organist when she goes down with shingles or have any luck at all when you try out for the local Bay City Rollers tribute band. You are neither Great Artist, sessions professional, illustrator or admired amateur.

So what? You are someone who paints a bit, scratches around on the keyboard for fun, gets a kick out of learning a tune or discovering a new way of rendering the face of your beloved in charcoal. You have another life, you have family, work and friends but this is a hobby, a pastime, FUN. Do you give up the Sunday kick-around because you’ll never be Thierry Henry? Of course not. That would be pathologically vain. We don’t stop talking about how the world might be better just because we have no chance of making it to Prime Minister. We are all politicians. We are all artists. In an open society everything the mind and hands can achieve is our birthright. It is up to us to claim it.

And you know, you might be the real thing, or someone with the potential to give as much pleasure to others as you derive yourself. But how you will ever know if you don’t try?

As the above is true of painting and music, so it is true of cookery and photography and gardening and interior decoration and chess and poker and skiing and sailing and carpentry and bridge and wine and knitting and brass-rubbing and line-dancing and the hundreds of other activities that enrich and enliven the daily toil of getting and spending, mortgages and shopping, school and office. There are rules, conventions, techniques, reserved objects, equipment and paraphernalia, time-honoured modes, forms, jargon and tradition. The average practitioner doesn’t expect to win prizes, earn a fortune, become famous or acquire absolute mastery in their art, craft, sport-or as we would say now, their chosen leisure pursuit. It really is enough to have fun.

The point remains: it isn’t a burden to learn the difference between acid and alkaline soil or understand how f-stops and exposure times affect your photograph. There’s no drudgery or humiliation in discovering how to knit, purl and cast off, snowplough your skis, deglaze a pan, carve a dovetail or tot up your bridge hand according to Acol. Only an embarrassed adolescent or deranged coward thinks jargon and reserved languages are pretentious and that detail and structure are boring. Sensible people are above simpering at references to colour in music, structure in wine or rhythm in architecture. When you learn to sail you are literally shown the ropes and taught that they are called sheets or painters and that knots are hitches and forward is aft and right is starboard. That is not pseudery or exclusivity, it is precision, it is part of initiating the newcomer into the guild. Learning the lingo is the beginning of our rite of passage.

In music, tempo is not the same as rhythm, which is not the same as pulse. There are metronomic indications and time signatures. At some point along the road between picking out a tune with one finger and really playing we need to know these distinctions. For some it comes naturally and seems inborn, for most of us the music is buried deep inside but needs a little coaxing and tuition to be got out. So someone shows us, or we progress by video, evening class or book. Talent is inborn but technique is learned.

Talent without technique is like an engine without a steering wheel, gears or brakes. It doesn’t matter how thoroughbred and powerful the V12 under the bonnet if it can’t be steered and kept under control. Talented people who do nothing with their gifts often crash and burn. A great truth, so obvious that it is almost a secret, is that most people are embarrassed to the point of shame by their talents. Ashamed of their gifts but proud to bursting of their achievements. Do athletes boast of their hand-eye coordination, grace and natural sense of balance? No, they talk of how hard they trained, the sacrifices they made, the effort they put in.

Ah, but a man’s reach should exceed his grasp
Or what’s a heaven for?

Robert Browning’s cry brings us back, at last, to poetry. While it is perfectly possible that you did not learn music at school, or drawing and painting, it is almost certain that you did learn poetry. Not how to do it, almost never how to write your own, but how, God help us, to appreciate it.

We have all of us, all of us, sat with brows furrowed feeling incredibly dense and dumb as the teacher asks us to respond to an image or line of verse.

What do you think Wordsworth was referring to here?
What does Wilfred Owen achieve by choosing this metaphor?
How does Keats respond to the nightingale?
Why do you think Shakespeare uses the word ‘gentle’ as a verb?
What is Larkin’s attitude to the hotel room?

It brings it all back, doesn’t it? All the red-faced, blood-pounding humiliation and embarrassment of being singled out for comment.

The way poetry was taught at school reminded W. H. Auden of a Punch cartoon composed, legend has it, by the poet A. E. Housman. Two English teachers are walking in the woods in springtime. The first, on hearing birdsong, is moved to quote William Wordsworth:

TEACHER 1: Oh cuckoo, shall I call thee bird
Or but a wandering voice?
TEACHER 2: State the alternative preferred
With reasons for your choice.

Even if some secret part of you might have been privately moved and engaged, you probably went through a stage of loathing those bores Shakespeare, Keats, Owen, Eliot, Larkin and all who came before and after them. You may love them now, you may still hate them or perhaps you feel entirely indifferent to the whole pack of them. But however well or badly we were taught English literature, how many of us have ever been shown how to write our own poems?

Don’t worry, it doesn’t have to rhyme. Don’t bother with metre and verses. Just express yourself. Pour out your feelings.

Suppose you had never played the piano in your life.

Don’t worry, just lift the lid and express yourself. Pour out your feelings.

We have all heard children do just that and we have all wanted to treat them with great violence as a result. Yet this is the only instruction we are ever likely to get in the art of writing poetry: Anything goes.

But that’s how modern poetry works, isn’t it? Free verse, don’t they call it? Vers libre?

Ye-e-es…And in avant-garde music, John Cage famously wrote a piece of silence called ‘4 Minutes 33 Seconds’ and created other works requiring ball-bearings and chains to be dropped on to prepared pianos. Do music teachers suggest that to children? Do we encourage them to ignore all harmony and rhythm and just make noise? It is important to realise that Cage’s first pieces were written in the Western compositional tradition, in movements with conventional Italian names like lento, vivace and fugato. Picasso’s early paintings are flawless models of figurative accuracy. Listening to music may inspire an extraordinary emotional response, but extraordinary emotions are not enough to make music.

Unlike musical notation, paint or clay, language is inside every one of us. For free. We are all proficient at it. We already have the palette, the paints and the instruments. We don’t have to go and buy any reserved materials. Poetry is made of the same stuff you are reading now, the same stuff you use to order pizza over the phone, the same stuff you yell at your parents and children, whisper in your lover’s ear and shove into an e-mail, text or birthday card. It is common to us all. Is that why we resent being told that there is a technique to its highest expression, poetry? I cannot ski, so I would like to be shown how to. I cannot paint, so I would value some lessons. But I can speak and write, so do not waste my time telling me that I need lessons in poetry, which is, after all, no more than emotional writing, with or without the odd rhyme. Isn’t it?

Jan Schreiber in a review of Timothy Steele’s Missing Measures, says this of modern verse:

The writing of poetry has been made laughably easy. There are no technical constraints. Knowledge of the tradition is not necessary, nor is a desire to communicate, this having been supplanted in many practitioners by the more urgent desire to express themselves. Even sophistication in the manipulation of syntax is not sought. Poetry, it seems, need no longer be at least as well written as prose.

Personally, I find writing without form, metre or rhyme not ‘laughably easy’ but fantastically difficult. If you can do it, good luck to you and farewell, this book is not for you: but a word of warning from W.H. Auden before you go.

The poet who writes ‘free’ verse is like Robinson Crusoe on his desert island: he must do all his cooking, laundry and darning for himself. In a few exceptional cases, this manly independence produces something original and impressive, but more often the result is squalor—dirty sheets on the unmade bed and empty bottles on the unswept floor.

I cannot teach you how to be a great poet or even a good one. Dammit, I can’t teach myself that. But I can show you how to have fun with the modes and forms of poetry as they have developed over the years. By the time you have read this book you will be able to write a Petrarchan sonnet, a Sapphic Ode, a ballade, a villanelle and a Spenserian stanza, among many other weird and delightful forms; you will be confident with metre, rhyme and much else besides. Whether you choose to write on the stupidity of advertising, the curve of your true love’s buttocks, the folly of war or the irritation of not being able to open a pickle jar is unimportant. I will give you the tools, you can finish the job. And once you have got the hang of the forms, you can devise your own. The Robertsonian Sonnet. The Jonesian Ode. The Millerian Stanza.

This is not an academic book. It is unlikely to become part of the core curriculum. It may help you with your English exams because it will certainly allow you to be a smart-arse in Practical Criticism papers (if such things still exist) and demonstrate that you know a trochee from a dactyl, a terza from an ottava rima and assonance from enjambment, in which case I am happy to be of service. It is over a quarter of a century since I did any teaching and I have no idea if such knowledge is considered good or useless these days, for all I know it will count against you.

I have written this book because over the past thirty-five years I have derived enormous private pleasure from writing poetry and like anyone with a passion I am keen to share it. You will be relieved to hear that I will not be burdening you with any of my actual poems (except sample verse specifically designed to help clarify form and metre): I do not write poetry for publication, I write it for the same reason that, according to Wilde, one should write a diary, to have something sensational to read on the train. And as a way of speaking to myself. But most importantly of all for pleasure.

This is not the only work on prosody (the art of versification) ever published in English, but it is the one that I should like to have been available to me many years ago. It is technical, yes, inasmuch as it investigates technique, but I hope that does not make it dry, obscure or difficult-after all, ‘technique’ is just the Greek for ‘art’. I have tried to make everything approachable without being loopily matey or absurdly simplistic.

I certainly do not attempt in this book to pick up where those poor teachers left off and instruct you in poetry appreciation. I suspect, however, that once you have started writing a poem of any real shape you will find yourself admiring and appreciating other poets’ work a great deal more. If you have never picked up a golf club you will never really know just how remarkable Ernie Els is (substitute tennis racket for Roger Federer, frying pan for Gordon Ramsay, piano for Jools Holland and so on).

But maybe you are too old a dog to learn new tricks? Maybe you have missed the bus? That’s hooey. Thomas Hardy (a finer poet than he was a novelist in my view) did not start publishing verse till he was nearly sixty.

Every child is musical. Unfortunately this natural gift is squelched before it has time to develop. From all my life experience I remember being laughed at because my voice and the words I sang didn’t please someone. My second grade teacher, Miss Stone would not let me sing with the rest of the class because she judged my voice as not musical and she said I threw the class off key. I believed her which led to the blockage of my appreciation of music and blocked my ability to write poetry. Fortunately at the age of 57 I had a significant emotional event which unblocked my ability to compose poetry which many people believe has lyrical qualities.

So writes one Sidney Madwed. Mr Madwed may not be Thomas Campion or Cole Porter, but he believes that an understanding of prosody has set him free and now clearly has a whale of a time writing his lyrics and verses. I hope reading this book will take the place for you of a ‘significant emotional event’ and awaken the poet that has always lain dormant within.

It is never too late. We are all opsimaths.

Opsimath, noun: one who learns late in life.

Let us go forward together now, both opsimathically and optimistically. Nothing can hold us back. The ode beckons.

Written by S

Sun, 2013-08-04 at 18:27:35

Posted in language, literature, quotes

Wellerisms &c.

with 2 comments

[Originally posted to paronomasia/pun-ctilious.]

Charles Dickens at 24 was writing his first novel The Pickwick Papers, which was being published serially like all novels of the era. Sales were chugging along decently for the first three months, until the character Sam Weller was introduced. The career of Dickens would never be the same. The novel became a publishing phenomenon and from that moment on he was a star, and new instalments of Dickens’s novels were often more eagerly awaited than any Harry Potter book has been.

Among the characteristics that made Sam Weller so popular with the masses were his linguistic charms, one of them a form of quotation known as a Wellerism. This survives in American popular culture as the rather lame and narrow-in-scope “…that’s what she said” (or the British “…as the actress said to the bishop”), but turning to samples from Dickens himself:

“out vith it, as the father said to his child, when he swallowed a farden.”

“How are you, ma’am?” said Mr. Weller. “Wery glad to see you, indeed, and hope our acquaintance may be a long ‘un, as the gen’l’m’n said to the fi’ pun’ note.”

“All good feelin’, sir—the wery best intentions, as the gen’l’m’n said ven he run away from his wife ‘cos she seemed unhappy with him,” replied Mr. Weller.

“There; now we look compact and comfortable, as the father said ven he cut his little boy’s head off, to cure him o’ squintin’.”

“Yes, but that ain’t all,” said Sam, […] “vich I call addin’ insult to injury, as the parrot said ven they not only took him from his native land, but made him talk the English langwidge arterwards.”

“Sorry to do anythin’ as may cause an interruption to such wery pleasant proceedin’s, as the king said wen he dissolved the parliament,” interposed Mr. Weller, who had been peeping through the glass door;…

More examples not from Dickens, from Wikipedia and elsewhere:

“We’ll have to rehearse that,” as the undertaker said when the coffin fell out of the car.

“Simply remarkable,” said the teacher when asked her opinion about the new dry-erase board.

“Don’t move, I’ve got you covered”, as the wallpaper said to the wall.

‘It all comes back to me now’, said the Captain as he spat into the wind.

‘Eureka!’ said Archimedes to the skunk.

“Each moment makes thee dearer,” as the parsimonious tradesman said to his extravagant wife.

“Capital punishment,” as the boy said when the teacher seated him with the girls.

“I’ve been to see an old flame,” remarked the young man returning from Vesuvius.

“I hope I made myself clear,” as the water said when it passed through the filter.

“I’m at my wit’s end,” said the king as he trod on the jester’s toe.

“These are grave charges,” murmured the hopeless one, as he perused the bill for the burial of his mother-in-law.

“Notice the foot-note at the bottom of the page,” laughed the court fool, as the royal attendant’s shoes emitted a squeak.

“That’s my mission in life,” said the monk, as he pointed to his monastery.

“Oh, how blue I am,” mourned the poet, as his fountain pen spattered upon him.

“That’s an old gag,” said the cashier, as the bandit stopped up his mouth.

“My business is looking good,” said the model.

See also this post by Krish Ashok, which has a stream of examples culminating in

“Looks like we still have gaps”, he pointed out, like Aamer Sohail to Venkatesh Prasad.

A subgenre is the “Tom Swifty”, with a pun on the adverb:

“The doctor had to remove my left ventricle,” said Tom half-heartedly.

“The situation is grave,” Tom said cryptically.

“I’ve joined the navy,” Tom said fleetingly.

“I have a split personality,” said Tom, being frank.

“This is the real male goose,” said Tom producing the propaganda.

“I won’t finish in fifth place,” Tom held forth.

[See the paronomasia archives for more Tom Swifties from its members, like

“Let’s put them in to bat now and bowl them out,” Tom declared.

and of course everywhere on the internet.]

Written by S

Sun, 2011-08-14 at 06:16:21

Posted in funny, language, quotes

The invitation

with 5 comments

Translated from the शार्ङ्गधर-पद्धति by Octavio Paz:

The invitation

Traveler, hurry your steps, be on your way:
the woods are full of wild animals,
snakes, elephants, tigers, and boars,
the sun’s going down and you’re so young to be going alone.
I can’t let you stay,
for I’m a young girl and no one’s home.

Translated from the गाहा-सत्तसई (= गाथा-सप्तशती) by Andrew Schelling:

sleeps over there
so does the
rest of the household but
    this is my bed
    don’t trip over
    it in the dark

Written by S

Tue, 2011-06-21 at 18:51:21


with one comment


“Is my team ploughing,
That I was used to drive
And hear the harness jingle
When I was man alive?”

Ay, the horses trample,
The harness jingles now;
No change though you lie under
The land you used to plough.

“Is football playing
Along the river shore,
With lads to chase the leather,
Now I stand up no more?”

Ay, the ball is flying,
The lads play heart and soul;
The goal stands up, the keeper
Stands up to keep the goal.

“Is my girl happy,
That I thought hard to leave,
And has she tired of weeping
As she lies down at eve?”

Ay, she lies down lightly,
She lies not down to weep:
Your girl is well contented.
Be still, my lad, and sleep.

“Is my friend hearty,
Now I am thin and pine,
And has he found to sleep in
A better bed than mine?”

Yes, lad, I lie easy,
I lie as lads would choose;
I cheer a dead man’s sweetheart,
Never ask me whose.

Read the rest of this entry »

Written by S

Fri, 2011-05-27 at 16:16:04

Posted in literature, quotes