Further issues about the nature of knowledge may be found in the multiple resonances that Plato establishes in his writing. Plato starts by presenting a conversation between two philosophers, Euclid of Megara (i.e., not the mathematician) and Terpsion. This is said to take place on the day in 369BC that Theatetus is dying from dysentery and wounds incurred in the defence of Corinth. Euclid recalls that many years previously Socrates had described a discussion that he had had with the then young Theatetus and his teacher, Theodorus, Euclid recounts that he himself had taken notes during Socrates' spoken account and had then written up a version in direct speech, which he claims he had afterwards carefully checked over with Socrates. Terpsion suggests this might be a good time to go through the written version. The two philosophers then settle down to hear a reading of the piece by a boy servant, and Plato provides the script for us. It is not until the very end that Terpsion-and the reader of Plato's book-realizes that the discussion about knowledge must be supposed to have taken place in 399BC on the same day that Socrates was charged with corrupting Athenian youth. Thirty years later, the philosophers are reminded of the death of Socrates at the very time that they are aware of the dying Theatetus.
Plato presents some arguments about the nature of knowledge in the form of an account (his) of an account (Euclid's) of an account (Socrates') of a conversation between three people. And the present reader is now reminded that what is being read is my account of someone's translation of this sequence of accounts. Which
of these most truly reflects what is supposed to have happened? Can there be knowledge without some sort of representation, without some sort of account?
Or, to put the problem in another context, suppose someone gives you an account of a transcript of a film of some classroom dialogue: how confident are you in your understanding of the original event? Consider, for example, the account given by David Pimm (see Chapter 9 of this book) of a transcript of a sequence from an edited videotape of a classroom episode: what do you feel you know about the original event? Would you have felt more confident if you had been there yourself when the episode was being filmed? Some knowledge may be immediate; but much of what we know comes to us mediated through representation, through preconception, through theory. Socratic ignorance may be preferable where the only other choice seems to lie-in the words of Seth Bernadete, a recent translator of the Theatetus-'between an immediacy that is not available and a mediacy that is uneliminable.' (Bernadete, 1986a, p. 88).
But another alternative might be to embrace mediacy and make it serve one's ends. Socrates seeks to persuade Theatetus to discard an understanding of knowledge as 'true opinion'. He is critical of the view-ascribed to the philosopher Protagoras and advanced here by the two mathematicians-that knowledge is based on perception and that man is the measure of all things. For then knowledge becomes relative, and this was unacceptable, at any rate for Plato. But it is more commonly accepted nowadays that truth is a problematic notion and that what we can demand of opinion is that it be fruitful. And according to Goethe, was fruchtbar ist, allein ist wahr (only that which is fruitful is true). This is certainly the approach taken by psychoanalysts, who are of course very centrally concerned with interpretation in their work. Analysts invoke a distinction between narrative truth and historical truth: interpretation is understood to be a creative construction rather than a supposedly accurate historical re-construction.
So, in considering classroom accounts, it may be helpful to consider various interpretations, judging them not for some supposed veracity but in terms of their fruitfulness for the matter in hand, which may be supposed ultimately to be the improvement of our understanding and practice of the teaching and learning of mathematics. It may also be helpful to borrow some of the notions invoked by psychoanalysts when considering our own interpretations of classroom events, whether mediated or not. Thus, I hope to show that it could be fruitful to invoke the unconscious processes of displacement and condensation. These have also been characterized by linguists as associated respectively with the grammatical notions of metonymy and metaphor, and I have described elsewhere how interpretations of classroom practice, and of mathematical history, might be made in these terms (see Tahta, 1991, p. 229). Here, I want to try to illustrate the working of displacement and condensation in our understanding, or interpretation, of a mathematical theorem, and then in some further comments on Pimm's account of a classroom episode.
A Classical Theorem
It seems appropriate to take as an example the irrationality of the square root of 2, the classical result which Plato describes Theodorus and Theatetus as having generalized. It is not known for certain how it might originally have been proved that the diagonal of a square was not commensurable with its side: there are some plausible, geometric reconstructions. The well-known, dramatically elegant, arithmetic proof-ascribed by later Greek writers to Pythagoras-is a typical (and often claimed to be the first) mathematical example of proof by so-called reductio ad absurdum, a logical argument used over and over again by Socrates in Plato's dialogues. The proof is by analysis (as opposed to synthesis) in the sense that it starts with what is required, only in this case with its negative: you pretend (liar!) that you know the square root of 2 to be rational. In modern algebraic terms, the pretence is that √2 is a fraction a/b where a and b may be supposed to have no common factor. A sequence of transformations yields in the first place the result that a must be even-in which case b, having no common factor with a, much be odd; but further transformation then yields that b must be even. It seems that b must be both odd and even. Not accepting this, you backtrack to find the mistake in your argument, to find (surprise, surprise!) that it can only have been with your initial pretence.
The sequence of steps from one algebraic equation to the next can be associated with the psychoanalytic notion of displacement, a process in which links are established through a chain of connections some of which may not always be conscious. A typical example would be the slip of the tongue, in which a word gets substituted by another word. An example-from The Merchant of Venice-is Portia's hint to Bassanio that she is wholly his: 'One half of me is yours, the other half is yours-mine own, I would say.' A sequence of such shifts often occurs in dreams or in the activity of free association of words. Algebraic transformation of the equation √2 = a/b also involves a chain of substitutions. Seymour Papert has suggested that many students faced with equations of this sort engage-'almost as if they have read Freud'-in a process of mathematical free association (Papert, 1980, p. 198). And just as the psychoanalyst occasionally interrupts the flow of associations, the algebraic transformations are punctuated by an occasional conscious interpretation-for instance, when d = 2b2 is understood to mean that a², and so also a, mustbe even.
In the course of the proof, the square root sign is got rid of, but the 2 reappears, not in its original form but as a sign of evenness. Papert describes this feature as a sort of mathematical pun and associates this with the psychoanalytic notion of condensation, a process in which a single idea or symbol is at an intersection of several associative chains and coagulates a cluster of meanings. Dreams, for example, are shorter and more compressed than their verbal descriptions and eventual interpretations; a play on words-a pun-operates in the same way. Mathematics is itself a powerful condensation of experience; its 'abstractions' are derived from a number of different 'concrete' examples.
Moreover, as mathematical theorems are transmitted across generations, they condense the accumulated experience of past interpretations. This yields another sequence of accounts of accounts; but, as in the game of 'Chinese whispers', the final account may be quite unlike the initial one. Irrationality is not quite the same notion as incommensurability: in my end is not my beginning. John Mason has
suggested that many named theorems would not now be recognized by their originators:
Cayley's theorem, a fundamental theorem in group theory, doesn't exist in his writings. Unless you dig and dig and dig and recognise-because of your current awareness of what is a significant result-a sort of little kernel, a little something or other which you want to hold on to and say, 'that is a theorem'. It seems to me that in mathematics we transform everytime we present a theorem. (Mason, 1991, p. 16)
There is a further sense in which the proof of irrationality condenses a number of very dramatic, external associations that do not illuminate the mathematics as such, but have an expressive, mythical quality of their own. Such are the stories-and they may well be fictions-about the supposed consternation of the Pythagoreans at the discovery of irrationality. Some historians of mathematics would now deny that these are historical truths, and they would not wish to multiply narrative truths. Others-and I declare my own sympathy with these-are prepared to weave as rich a tapestry of associations as possible. For example, Michel Serres reads the supposed crisis at the dawn of Greek mathematics in terms of three deaths: not only the legendary death of the shipwrecked Hippasus who was held to have divulged the discovery of irrationality, and the historical death of Theatetus who developed a classification of irrational numbers, but also the turning away from some of the teachings of the revered Parmenides, which Plato somewhat startlingly called a form of parricide (Serres, 1982, p. 130).
The third death-the parricide-arises in the following way. Parmenides had formulated the law of contradiction invoked in proofs by reductio ad absurdum. But he was emphatic that Being (that which is) is One; its opposite, non-Being (that which is not) cannot be. Whatever exists can be thought of, and conversely, everything that can be thought of exists. But, he claimed, you cannot think of what is not, you could not know it, you could not even say it. However, in another Platonic dialogue, the Sophist, Theodorus brings along a philosopher, referred to as a Stranger, who tells Theatetus,
Don't take me to be, as it were, a kind of parricide…It will be necessary…to put the speech of our father Parmenides to the torture and force it to say that 'that which is not' is in some respects, and again, in turn, 'that which is' is not. (Bernadete, 1986b, p. 33)
The Stranger is prepared-despite Parmenides-to take opposites into account; the supposed parricide may be seen as a return to the dualism of the early Pythagoreans. (Theatetus, the mathematician for whom everything is countable, is certainly already prepared to take opposites into a count!) 'That which is not' can be conceived: for 'thinking makes it so', as Hamlet claimed, illustrating this with a pair of opposites, namely good and bad. How would the early Pythagoreans have conceived of that which is not commensurable? John Fauvel has pointed out that incommensurability is not in itself plausible without some sort of proof: its first proof must have constituted its discovery (Fauvel, 1987, p. 18). Proving irrationality made it so!
The proof establishes the contradiction that a number can be both even and
odd. By a series of intriguing displacements, Serres suggests that because 'even means equal, united, flat, same while odd means bizarre, unmatched, extra, left over, unequal, in short, other, the contradictory result may be described as asserting that 'Same is Other'. This cannot be so; but because that which is not rational has been proved to exist as well as what is rational, then numbers may just as well be irrational as rational. This dualism-masterfully explored by Theatetus-means that irrationality, or what Parmenides would have called non-Being, is in a sense on the same footing as rationality, or Being. So, according to Serres, Same is indeed Other 'after a fashion'; and this is the parricide of Parmenides.
'Legend, myth, history, philosophy, and pure science have common borders over which a unitary schema builds bridges', writes Serres, stretching the condensations even further to include a play on the name, Metapontum, of the birthplace of he who was shipwrecked. The 'unitary schema' reasserts the oneness of Being with a vengeance; but the crossing of boundaries brings disparate and sometimes contradictory things together. Following Parmenides, we usually assume that contradictions cannot-must not-be simultaneously entertained: either something is or it is not. But condensations demand otherwise. David Pimm has suggested that this is why metaphor can be so disquieting in mathematics:
for the very essence of metaphor…is to be able to claim at one and the same time that 'it is and it is not'! I assert 'a function is a machine' (and yet I also know it is not one)-the strength of the metaphoric assertion comes through the use of the verb to be-yet it carries with it implicitly its own negation. (Pimm, 1995, Chapter 10)
Narrative Truths
The ambivalence of simultaneous assertion and negation is familiar in psychoanalysis. It is perhaps this accommodation of the contradiction inherent in metaphor that makes some psychoanalytic narratives seem, at first sight, implausible. David Pimm quotes some startling examples from the work of Melanie Klein, including that of a 17-year-old patient, known as Lisa, who had understood addition when the things being added were the same, but not when they were different. Klein suggests, in effect, that addition was a metaphor that condensed various meanings for Lisa, including her difficulty in entertaining the idea of parental coitus, where different genitals are brought together. These remarks occur in a very comprehensive survey-based on some of her cases-of the role of the school in the libidinal development of children. Lisa is mentioned quite often in this survey; for example, it is reported that she recalled always finding it difficult to divide a large number by a smaller one. She associated this difficulty with a dream involving mutilation of a horse, and her going shopping for an orange and a candle. This-and its inevitable (Kleinian) interpretation in terms of castration-may seem far-fetched to some people, but it is precisely the way of the unconscious to be far-fetched. Another analyst might have made a less severe interpretation, but the issue must always be how fruitful this particular interpretation had been in the course of Lisa's analysis, and this, of course, we do not know.
For Klein, 'the tendency to overcome [the fear of castration] seems in general to form one of the roots from which counting and arithmetic have evolved' (Klein, 1950, p. 80). That arithmetical operations might symbolize such matters is supported by evidence from many other analysts. Klein assembles a lot of further detail about Lisa's-and other children's-unconscious thinking in her account. For instance, Lisa disliked the number 3 'because a third person is of course always superfluous'. Similar oedipal conflicts have been reported recently by Lusiane Weyl-Keiley, a therapist and teacher, who has shown how some typical problems encountered in remedial work in mathematics may be interpreted psychoanalytically. For example, 3 may be associated with the family triple-mother, father, child-and Weyl-Keiley describes a depressed adolescent for whom 5-2 was always 2. Asked to display this by folding fingers of one hand, he was unable to sustain the display of three fingers and had to fold down another. 'You see it makes two', he announced, keeping himself, it is suggested, out of the family conflict (Weyl-Keiley, 1985, p. 38).
Another example of the possible psychic significance-for some individuals-of a mathematical topic may be found in Klein's report that Lisa recalled never understanding an equation with more than one unknown. The exercise of interpreting that in psychoanalytic terms is, as they say, left to the reader. But another sort of condensed meaning may be found in the reminder that in the early development of algebra there must also have been something particularly difftcult about the notion of two unknowns. For Diophantos, who tackled various problems with apparently two or more unknowns, seemed unable-or unwilling-to symbolize more than one. A second unknown was always arbitrarily given a particular numerical value, the problem then being expressed in terms of one variable. When this caused some inconsistency or infelicity, the value of the second unknown was modified in order to satisfy all the conditions of the problem.
For example, Diophantos sought a Pythagorean triple such that the hypot-enuse less each side is a cube (see van der Waerden, 1988, p. 288). His method is here more conveniently described in modern notation. Pythagorean triples can be expressed in terms of two parameters; calling one of the associated parameters 5, Diophantos arbitrarily took the other to be 3. The base, height and hypoteneuse are then s² - 9, 6s and s² + 9, respectively. Hypotenuse less base is 18, which is not a cube. To make it one, you need to have assigned the second parameter a number whose square, doubled, is a cube. So try 2! The sides are then s²-4, 4s, s²+4. Hypotenuse less height is now the square of s - 2, which has to be a cube. So set 5 = 10! The sides are now as required. This was also the approach to such problems taken later-and independently-by the Arab mathematician, Al-Khwarizmi; it re-appears once again in the mediaeval 'rule of false position'.
We admire the crucial step that Diophantos made-the awareness that he could name the as-yet-unknown by a special symbol (it looked like an s and may have been an abbreviation of the Greek word for number, arithmos). We have no idea why he was unable, or unwilling, to invoke a second symbol at the same time. But it was clearly a difficult issue. The point here is, of course, not to interpret Diophantos as having some kind of unconscious block, but to establish that entertaining two unknowns may be problematic in itself, and to suggest that in Lisa's case, where other relevant evidence is also available, this might be seen as having come also to symbolize unresolved issues about her oedipal conflicts. It is not, of course, that two unknowns will always represent two parents for everyone, but rather that they may attract such unconscious condensations for some individuals at some time of their lives.
Listening to, and trying to make sense of, other people's accounts of their experience can be difficult. Interpretation is a fragile instrument which is often made to bear too much ontological weight. Socrates explains to Theodorus that he is afraid that they will never be able to understand the thought of Parmenides, and that in their discussion of the nature of knowledge they can only multiply interpretations which will never allow them to reach a satisfactory conclusion. All that Socrates, whose mother was a midwife, can do is to use his 'maieutic art'-or intellectual obstetrics-to help Theatetus deliver his own understanding.
So I'm afraid that we'll fail as much to understand what he [Parmenides] was saying as we'll fall far short of what he thought when he spoke, and-this is the greatest thing-that for whose sake the speech has started out, about knowledge, whatever it is, that that will prove to be unexamined under the press of the speeches that are bursting in like revellers, if anyone will obey them. And this is all the more the case now, since the speech we now awaken makes it impossible to handle by its immensity, regardless of what one will do. For if one will examine it incidentally, it would undergo what it does not deserve, and if one will do it adequately, it will by its lengthening wipe out the issue of knowledge. We must do neither, but we must try by means of the maieutic art to deliver Theaetetus from whatever he's pregnant with in regard to knowledge. (Bernadete, 1986a, p. 51)
Finding the Truth in the Mistake
Introducing his own discussion of a transcript of a videotape of a classroom dialogue, Pimm emphasizes how difficult it is to present convincing descriptions of supposed unconscious processes. His interpretations of the quoted extract in terms of the linguistic dimensions of metonymy and metaphor seem to be both apt and convincing. His account could also be seen-more or less equivalently-in terms of the associated psychoanalytical processes of displacement and condensation. In the first place, when the teacher asks Lorna to recall the word 'infinity', her answer 'fidelity' may be interpreted in terms of displacement: the associated, but possibly disturbing, word 'infidelity' is swiftly disavowed and replaced by 'fidelity'. On the other hand, taking infinity to be a powerful condensation of meanings, her answer may be interpreted as directly relating to an associated meaning, that of 'lasting for ever'; unable to recall the actual word required she responds with the word 'fidelity' that has a similar, and for adolescents a particularly potent, meaning. Both interpretations can be taken into account at the same time, for the unconscious accommodates contradictions and converses.
The example underlines how irrelevant the quest for historical truth may be in such cases. But why, someone may ask, would anyone be interested in some narrative truth, let alone a proliferation of such truths? One possible answer is clear from a further reading of the transcript, which like so many classroom reports, is a record of someone's public success (here, significantly, that of the two boys, David and Gary) at the expense of someone's humiliating public failure.
The teacher is described as repeating Lorna's word 'fidelity' with 'rising tones of surprise and disbelief'. The rest of the class can be left in no doubt that Lorna has got it wrng. This is common enough in the contemporary classroom. But, to paraphrase an oft-quoted sentence of Jules Henry's, To a Zuni, Hopi or Dakota Indian, David and Gary's performance would seem cruel beyond belief, for competition, the wringing of success from somebody's failure, is a form of torture foreign to these non-competitive cultures.'
The point about the possible interpretations of Lorna's mistake is that they suggest ways in which she may, in fact, have got it right. They support a classroom attitude which encourages everyone-teacher and fellow students-to find the truth in the mistake. Whatever our view of what psychoanalysts say, we know, at least, that they listen, and this is something that we should all do more of in our classrooms. We can also strive to contain our premature interpretations and let our students deliver their own.
Andrew, aged 5, is enormously energetic and egocentric. The world revolves round him. He has an exuberant preoccupation with words-talks incessantly, rhyming when he can and following up word-associa-tions with unconcealed glee and self-delight. He pushes people and things, jumps up and down, shouts, tears and throws paper, stamps his feet. His aggressive masculinity is at once charming and intolerable. Andrew and friends are measuring. lan solemnly records that he is '3 recorders and 1 pen tall'. Andrew has been measuring the length of the table using a block of wood. He asks me to write out an appropriate sentence for him. The table is…pieces of wood long.' Gripping the pencil like a dagger, he inserts a 7. He then stabs the paper a few times and produces what looks like a row of 7's under the sentence. I say nothing, but I feel puzzled. Is Andrew being exuberant and generous with his numerals in the way he is with spoken words? After a brief, shared silence, Andrew explains to me: 'you write it seven times because there were seven of them.' (Tahta, 1975, p. 10)
In a moving postscript to his chapter, David Pimm relates what he heard when he listened to 3-year-old Katie. She counts, pointing a finger in turn at him, her mother and herself: 'One, two, two'. She says she knows it should be one, two, three, but adds, 'I am pretending'. The previous evening she had told him, 'I thought you were daddy' and earlier that morning she had announced, 'daddy dead'. He offers some sensitive possible interpretations of her apparent avoidance of' three'. It seems overwhelmingly obvious that feelings about the missing father are being processed in all this. This confirms the authenticity of Klein's account of Lisa's version of the (not always homely!) maxim, 'two's company, three's a crowd', and of the other examples of number avoidance mentioned by Lusiane Weyl-Keiley among others. I cannot resist adding a further possible interpretation that links with Andrew's explanation of his seven sevens: in counting the two two times because there were two of them in her family, perhaps Katie was also unobtrusively telling David that she knew that he was not her father and that was all right. The unconscious can be a creative as well as a destructive force, and young children often tell it like it is. Moreover, alternative interpretations can be
simultaneously entertained, even where they may seem to be contradictory. For that is the way of the unconscious: same is other, after a fashion.
Note
| 1. | I am particularly grateful to John Mason, David Pimm and David Wheeler for their perceptive comments on earlier drafts of this chapter. |
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References
| BERNADETE, S. (1986a) Plato's Theatetus, University of Chicago Press. |
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| BERNADETE, S. (1986b) Plato's Sophist, University of Chicago Press. |
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| FAUVEL, J. (1987) Mathematics in the Greek world, MA290-unit 2, Milton Keynes, Open University. |
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| KLEIN, M. (1950) Contributions to Psychoanalysis, Hogarth Press. |
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| MASON, J. (1991) in TAHTA, D. (Ed) The history of mathematics in terms of awareness, transcript of a seminar with Caleb Gattegno, Reading, Educational Solutions. |
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| PAPERT, S. (1980) Mindstorms, Basic Books. |
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| PIMM, D. (1995) Symbols and Meanings in School Mathematics, London, Routledge. |
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| SERRES, M. (1982) Hermes, John Hopkins University Press. |
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| SPENCE, D. (1982) Narrative Truth and Historical Truth, Norton. |
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| TAHTA, D. (1975) 'Seven times', Recognitions, I, Derby, Association of Teachers of Mathematics. |
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| TAHTA, D. (1991) 'Understanding and desire', in PIMM, D. and LOVE, E. (Eds) Teaching and Learning School Mathematics, Hodder and Stoughton. |
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| VAN DER WAERDEN, B. (1988) Science Awakening, Amsterdam, Kluwer. |
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| WEYL-KEILEY, L. (1985) Victoires sur les maths, Paris, Robert Laffont. |
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