Unambiguity is not a state of grace but a matter of economics -- weighing case by case the cost of over-specification for some readers) against the penalty for under-comprehension by others; always with the proviso that at every point the estimates have large and uncertain margins of error.

Within these constraints we do the best we can, and in the tone appropriate to those who are only doing the best they can. This last duty many find it easy to forget, who offer liminary definitions in a tone which suggests that the number of possible misunderstandings is quite small, that they have all been anticipated, and that they will never arise if only we read their definitions carefully enough. Worst of all, their tone suggests that for the sufficiently attentive reader the meaning that may accrete gradually as the defined terms are used should already be half-apparent in the Definition itself, and more than half-apparent after one or two examples of the defined term in use. In fact the only appropriate tone is one of extreme and almost apologetic courteousness, as of one requesting co-operation in what may be a long haul.

But it is one thing to understand the reason for these caveats, and quite another to live up to them. It was clear enough to Bentham (for example) that "Of a set of Definitions read by themselves in a suite, no use appears" ; and he is fully conscious that the range of possible misunderstandings is very wide. He `knows' how long it would take to cover even the main ones fairly; unfortunately, he lacks the necessary time, or patience, or perhaps the help of a sub-editor. As a result, as Hazlitt notes, he "writes as if he was allowed but a single sentence to express his whole view of a subject in, and as if, should he omit a single circumstance or step of the argument, it would be lost to the world for ever, like an estate by a flaw in the title-deeds. This is over-rating the importance of our own discoveries, and mistaking the nature and object of language altogether."


`Got that?' The manner conveys a longer unspoken message: `A lot rests on this, so do say if you're not quite happy, even if everyone else seems to be, and even though we're so pressed that two monosyllables must do for all this part of the message, and even though there's a penalty clause if we don't finish the building on time, and the next slab is already dangling from the crane, awaiting only the smallest of gestures from you -- yes, even that non-committal one.'

What matters to Inquisitors is the acquiescence, not the amount of credence attached to admissions extracted on the rack.

It is minimum prudence never to take either party's word for it that understanding has occurred. With an achievement so gratifying, how to resist the temptation to count its occurrences too generously ?

We are proud of the understanding we have attained, and at the same time dimly conscious of the degree of charity which posterity will need in looking back at the pathetic limitations of that understanding -- few of which were imposed by the backward state of our technology, i. e. not our fault.

By assertion and reassertion (in the right words and the right tone of voice) I can arrange that what you mean by `understanding' is more like what I mean by it. What greater victory could there be for socialization, or indoctrination ?

We set out to judge whether our interlocutors or students share enough of the intricate set of our own tenuous understandings. But correspondence with the tenuous must be even more tenuous. It would be irrational to expect much rationality here.

I explain p with admirable clarity, but had better not take it for granted I have been understood, even by those who claim to have done so. Rather, I can take it for granted I have not been understood, especially by those whose main concern is avoid indicating whether they have understood or not. After all, the number of legitimate misapprehensions is as large as anyone could hope. Legitimate misapprehensions exculpate those who have not understood p, but not, I fear, those who have not understood my test of having understood p.

The key assumption is that exposition and understanding proceed in miraculously isochronous step, one point at a time. If true, this would be extremely consoling to teachers. Similarly: `It is certain that if for education inducation were possible, and one man could actually impart his talent instead of its performances, a mountain of guineas would be readily paid for tuition fees.' (Emerson)

There is an analogy between understanding a sentence and having a theory (about whatever). But to suppose that your theory is the only possible one is `a sign that you have neither understood the theory nor the problem which it was intended to solve.' (Popper)

In educational circles there are vested interests which need to believe that the term `understanding' can properly be applied to something first considered five minutes ago.

Some find stimulus to theoretical reflection in our full and instant understanding of sentences invented for the purpose; others, in the less than full and much less than instant understanding of sentences laboriously collected from the annals of the species. With this goes a concomitant disagreement on how much preliminary simplification is too high a price to pay, and on how much return for that initial sacrifice is reward enough. All this comes very close to incommensurability; at least, there are two dialects, two speakers using the same words in very different senses and not equally conscious that they are. The situation demands extra courtesy from both sides; but the second of the above groups will be more ready to accept that necessity, which is grist to its theory.

To make demands on one's readers is to accept responsibility for training those readers to use a wider band of attentiveness than usual. The task is the more perilous in that, if it fails, the broad-band message has an odd effect on narrow-band readers, inducing a kind of hypnotic op-art dazzlement. Such readers are very likely to retaliate by adoration. This won't happen to you or me; but if it did you can see how easy it would be to resign ourselves to that deplorable state of affairs.

A curriculum-item or unit-of-teaching is not a summary of results, not the end-point or even the provisional end-point of a process of puzzlement, but its history (`Nobody learns significantly from conclusions' [Carl Rogers]). It aspires to be a contribution to the understanding of non-understanding, to the identification of what it was in the way of set or presumption which had previously blocked comprehension. The same might be said of the ideal response to a unit-of-teaching.

I. A. Richards: "I got enthralled by G. E. Moore. I don't think I ever understood anything. But it was complete subjugation. I got really interested in language because I felt something must be done to stop the leakage of information that was going on there all the time. I knew I didn't understand Moore or what he was at. I always thought if I went back to him for a whole course of lectures once again an inkling would come. But... no."

Joseph Agassi: "I was a devoted student of Popper's and worked closely with him for over seven years, yet I really understood him only a few years later. I liked Popper's ideas better when I understood them less."

Coleridge: "Methinks I see him sitting, the heroic Student, in his Chamber in the Warteburg, with his midnight Lamp before him... Before it lies the Hebrew Bible open, on which he gazes, his brow pressing on his palm, brooding over some obscure Text, which he desires to make plain to the simple Boor and to the humble Artizan, and to transfer its whole force into their own natural and living Tongue. And he himself does not understand it!"

Geprge Santayana: "When I quoted some spiritual saying of St. Augustine's, or of some other master, you looked up; there was light in your eye; you understood. Not that any of us is likely to reproduce a thought just as it once arose in another man's heart. How should we do that? Or, if we did, how should we know we had done so? But we may swim in the same sea; we may understand the prophets as we do the poets, each time differently, each time plunging in some direction a little farther into the divine harmonies of things, and welding their sacred bonds."

There would be no point in remarking that `Nothing has changed' unless there were reasonable expectation that something important might well have, something whose non-happening was therefore important too. That much would have to be the presumption of an overhearer who had arrived too late to get any of the co-text. For such an overhearer, only some conjectural candidate for that something would count as a possible Understanding.

We are all overhearers who have arrived too late to get all the necessary co-text.

A worries when he doesn't understand a text, but keeps a poker face, for everybody else seems to understand it, as is evidenced by their poker faces.

B, on the other hand, understands that there are many texts whose merit is that they release their meanings slowly. There are also all too many texts which he doesn't understand but which don't do enough to suggest that the effort to decode them will be worth while. Not understanding such texts leaves B with no feelings of guilt or shortcoming: it's as much their fault as his.

For B all the interest lies in the process (and the recollection of the process); arriving is the end of interest. Whence the dogged hunt for evidence that the fullness of his understanding was illusory, -- for this would give the process a new lease of life. This may be part of what Ernest Gellner is getting at when he remarks on the "widespread tendency of all who feel intensely about some object, to raise the standards of what is conclusive about it."

Otherwise, "Understanding makes the mind lazy" (Penelope Fitzgerald)

Understanding mathematics

K. Pople: "In spite of the latest efforts at an attempt to inject a system of measurement into mathematical attainment and understanding, one is still left with the feeling that understanding must be in many ways akin to happiness: that any attempt to achieve it directly is almost invariably doomed to failure; it is essentially a by-product of work and activity."


The University's function is to stimulate the social digestion of explainings -- a sort of monastery bringing together in ascetic isolation those elders supposed to be specially fertile in new or new-dressed explainings and those acolytes supposed to be specially receptive to them. It's a place for reducing those perplexities which are reducible and restoring to their full dauntingness those perplexities which turned out to be less reducible than the salesman claimed.

The university of London, it is agreed, ought to be allowed the chief merit of this general dissemination of learning and knowledge.

The World, No. 152, 28 Nov. 1755

Oliver Wendell Holmes sr: "In looking at our planet equipped and provisioned for a long voyage in space, -- its almost boundless stores of coal and other inflammable materials, its untired renewal of the forms of life, its compensations which keep the atmosphere capable of supporting life, the ever growing control over the powers of Nature which its inhabitants are acquiring, -- all these things point to its fitness for a duration transcending all our ordinary measures of time. These conditions render possible the only theory which can `justify the ways of God to man,' namely, that this colony of the universe is an educational institution so far as the human race is concerned. On this theory I base my hope for myself and my fellow-creatures."


It takes a real effort to empathize with my younger self who didn't know what I now know, and confidently knew what I have learnt to unknow. How could reliance have been placed on, satisfaction found in, such superficialities? Salutary wryness helps of course, but its returns are beginning to diminish. Still the older-&-wiser undertakes to teach those who have the same limitations as his younger self -- in that they lack, for example, a sense of what answers are just too slick to stand a chance of holding up.

C. A. Mace: "An author re-reading his philosophical juvenilia is apt to say `I wonder what I meant when I wrote that.' But another curious thing may happen. He finds that he agrees with his younger self, but now means by the words he uses something very different from what he meant before. This latter is what has happened in the case of my paper on `Representation and Expression.' I would even now try to defend most of what I wrote in that paper, but what I should be defending would differ from what I think I thought I was defending in 1934. In 1934, I was just going into `Behaviourism' (of a kind). In 1954 I am coming out on the other side. I was taking into Behaviourism much that I had learnt from Brentano (or from Brentano as he had been ingested by Stout). And I am now taking it all out again. But, oh, how different it all feels!"

E. R. Dodds: "I must confess that I know very little about early Orphism, and the more I read about it the more my knowledge diminishes. Twenty years ago, I could have said quite a lot about it (we all could at the time). Since then, I have lost a great deal of knowledge; for this loss I am indebted to Wilamovitz, Festugière, Thomas, and Linforth. Let me illustrate my present ignorance by listing a few of the things I once knew.


Devise a curriculum of Objectives To Be Avoided, with specified disachievement targets, and tests to confirm that X has been successfully unlearned -- a challenge indeed!


Nietzsche: "Philosophers all pose as though their real opinions had been discovered and attained through the self-evolving of a cold, pure, divinely indifferent dialectic (in contrast with mystics, who, fairer and foolisher, talk of `inspiration'); whereas, in fact, a prejudiced proposition, idea, or `suggestion', which is generally their heart's desire abstracted and refined, is defended by them with arguments sought out after the event. They are all advocates who do not wish to be regarded as such, generally astute defenders, also, of their prejudices, which they dub `truths' -- and very far from having the conscience which bravely admits this to itself; very far from having the good taste or the courage which goes so far as to let this be understood, perhaps to warn friend or foe, or in cheerful confidence and self-ridicule. The hocus-pocus of mathematical form, by means of which Spinoza has as it were clad his philosophy in mail and mask in order thereby to strike terror into the heart of the assailant who should dare to cast a glance on that invincible maiden, that Pallas Athene: -- how much of personal timidity and vulnerability does this masquerade of a sickly recluse show!'"

Georges Gusdorf: "Toute philosophie est l'expression, et peut-être le déguisement, d'un homme. l'exégèse valable serait celle qui arrache le masque, rétablissant par delà le sens direct et apparent l'affirmation indirecte, seule valable. Sans se laisser arrêter par la magnificence des superstructures, le malade Nietzsche dépiste le malade Spinoza.

Il y a tout de même un inconvénient à cette réduction de l'histoire de la philosophie à une sorte de roman policier. Le danger est celui de toute psychanalyse: le philosophe ne devant plus être cru sur parole, le passage du contenu manifeste au contenu latent permet d'étonnantes divergences. finalement, la profession de foi de l'exégète se substitue à celle du penseur lui-même, simplement pris à témoin de révélations qu'il est appelé à patronner, sans d'ailleurs pouvoir se défendre, puisqu'il est mort, -- et surtout puisque, en principe, il ne sait pas ce qu'il dit."


That we understand a new sentence is taken to be a fact at once unexpected and immensely significant. Isn't it just as much in need of explanation, and even more unexpected, when the (n+1)th understanding turns out to be the same as the nth? Room must in any case be left for late understanding and re-understanding and the renunciation of earlier understanding. This last was the phenomenon for which we once had the useful verb `to unteach.'

Sir Thomas Browne: "We do but learn to day, what our better advanced judgements will unteach to morrow: and Aristotle doth but instruct us, as Plato did him; that is, to confute himself."


"Children can only be taught what is already known to them." This is the sort of slogan that one side in the educational debate quotes derisively, as representative of the other side. That reaction I take to be a misreading. The slogan is evidently intended to sound paradoxical (always a risky tactic, though necessary if all else fails). It grabs the attention just because it can so obviously be taken at first reading to be absurd. Only a myopic reader could suppose that the sloganeer was blind to this. A more charitable reader might think: Since that first interpretation can't possibly be the one intended, I am being challenged to find a better reason for offering the slogan; and if I find the challenge intriguing enough, I'll accept it.

The background is a perplexity: the never-obvious relation between what is added-value and what was there already. The extreme cases are clear enough: no value can be added if there's nothing there already; and some learners seem entirely self-propelled. In between is a greyish area where no-one can be quite sure, in a given case, what the proportions were which led to success or failure, as the case may be.

The dilemma is perennial, with a history oscillating between the two horns. It is hard for campaigners against the excesses of the current fashion (whichever it is) to resist overstating their case; that is, to suppose that the merits of what is after all only a programme are as self-evident as the defects of the other lot, and does not call for any different degree of asseveration.

The idea we now mock once had the same justification as any current demolition of its crudified descendant. Once it was the protest against the formulaic vestiges of a still older idea. All we can do is to try a little harder to break out of this cycle, to stop the same thing happening to us, to so phrase our necessary simplifications as to make it harder for those unwelcome allies to turn them into over-simplifications.

It may be that teachers have to orient themselves by reference to an ideal line from a well-defined prerequisite to a well-defined end-product. Sensible teachers don't need to be told that this is an extreme idealization, and are therefore open to the constant signals that assumptions about either end, or the route(s) between them, have become comfortable habits and are due for rethink.

Others of more fundamentalist temperament choose some part of that idealization to defend at any cost. So some are happy to wait until a postulated prerequisite is attained naturally. Others just know there can be no other route from A to B than the one that seemed, in rosy retrospect, to work for them. A third lot suppose that their sole duty is to reach the official target, minimally interpreted. Between such groups there is pseudo-debate, which distracts from the only campaign worth waging, against what they have in common.


Hume: "The term for a slave, born and bred in the family, was verna. Note. As servus was the name of the genus, and verna of the species, without any correlative, this forms a strong presumption, that the latter were by far the least numerous. It is an universal observation which we may form upon language, that where two related parts of a whole bear any proportion to each other, in numbers, rank or consideration, there are always correlative terms invented, which answer to both the parts, and express their mutual relation. If they bear no relation to each other, the term is only invented for the less, and marks its distinction from the whole. Thus man and woman, master and servant, father and son, prince and subject, stranger and citizen, are correlative terms. But the words seaman, carpenter, smith, tailor, &c. have no correspondent terms, which express those who are no seamen, no carpenters, &c.

Languages differ very much with regard to the particular words where this distinction obtains; and may thence afford very strong inferences, concerning the manners and customs of different nations. The military government of the Roman emperors had exalted the soldiery so high, that they balanced all the orders of the state: Hence miles and paganus became relative terms; a thing, till then, unknown to ancient, and still to modern languages. Modern superstition exalted the clergy so high, that they overbalanced the whole state: Hence clergy and laity are terms opposed in modern languages; and in these alone. And from these same principles I infer, that if the numbers of slaves bought by the Romans from foreign countries, had not extremely exceeded those which were bred at home, verna would have had a correlative, which would have expressed the former species of slaves."

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