Geometry Restored

The new book Language, Discourse, and Purpose: The Specialisation and Dissemination of Knowledge, to be posted here (I hope) in a few months, contains a quite different account of geometry and its LSP(s), with focus on the quest for a science of universal space, and on the successive “decompressions” of Euclidian geometry by reference to vision, physical experience, real-world analogies, and so on. This paper is more of an exploratory nature.

Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung 6, 1991, 771-827; and Journal of the International Institute for Terminology Research 3/2, 1992, 29-125

Geometry is at the same time a science and an art, mathematics and philosophy.

Mathematics is often a lonely, impersonal experience of manipulating symbols in accordance with rules learned by rote.

1.1 The familiar aphorism about the trees obscuring the forest seems nowhere more apt than in the domain of public education. Both theory and practice are intensely preoccupied with the specific content and materials of the respective instructional domains. Educators readily take it for granted that schooling should dwell on the time-honoured offerings, such as native language, foreign language, history, chemistry, biology, algebra, and geometry; the main question is how these ‘subjects’ can be ‘taught’ and ‘learned’ most efficiently, not whether or why. If these subjects are the ‘trees’, then the ‘forest’ is the higher-level cognitive development of the child, the overall complex of processes and consequences of knowledge acquisition during education.

1.2 Fragmentation of perspective also pervades the standard approaches to these individual ‘subjects’. Each one is not only isolated from the rest, but is broken down into incidental ‘lessons’, ‘facts’, and ‘quizzes’. The ensuing mosaic of specific question/answer or problem/solution episodes creates a rather specious format of organisation. A more genuine format would reflect a comprehensive assessment of the contribution of any given episode to the learner’s development. Such an assessment could establish reliable, integrative criteria for designing a curriculum, and for deciding what should be taught in which grade and how.

1.3 Seymour Papert, a pupil of Jean Piaget, has diagnosed a symptomatic dissonance between the intuitive experience of a child and the daily practices of schooling:

The occupational activities of children are learning, thinking, playing and the like. Yet we tell them nothing about those things. Instead, we tell them about numbers, grammar, and the French revolution; somehow hoping that from this disorder, the really important things will emerge all by themselves. (Papert 1971, cit. McCorduck 1979: 290)

Perhaps an apt metaphor for these commonplace instructional practices might be the now-fashionable eating disorder known as ‘bulimia’, a compulsive cycle of gorging and purging the body. ‘Bulimic education’ force-feeds the learner with a feast of ‘facts’ which are to be memorised and used for certain narrowly defined tasks, each leading to a single ‘right answer’ already decided by teacher or textbook. After this use, the facts are ‘purged’ to make room for the next feeding. ‘Bulimic education’ thus enforces an intensely local or short-range focus, irrespective of any long-range benefits that might arise from the succession of feed-purge cycles.

1.4 The activities for utilising a set of ‘facts’ could be broadly classed into two categories. In reproductive activities, the facts are simply recalled and recorded in response to specific probes; in productive activities, the facts are deployed for solving new sets of problems or explaining new sets of phenomena. In history, for example, a reproductive approach might be a chronological enumeration of names, dates, and places to be held ready for who-when-where questions, whereas a productive approach might be a non-chronological analysis of historical situations and processes we might observe in typical cycles over the centuries, such as the symptomatic stages of colonialism.

1.5 The balance between these two classes of activities has been skewed for so long because the reproductive approach seems more congenial for short-range instructional practices embodied in brief, closed-ended episodic tasks generating ‘right’ or ‘wrong answers’. The productive approach, in contrast, would be more congenial for gradual, open-ended conceptual tasks whose results should be judged as more or less helpful appropriations of new insights through analogical or creative reasoning. The complexities involved in the latter type of judgement have doubtless encouraged educators to favour a reproductive approach even if they are quite ready grant the greater merits of productive activities for the development of human intelligence.

1.6 The productive use of knowledge can be further divided along two parameters. Power would be gained by applying one’s knowledge to a wide range of problems and phenomena (cf. 2.53). Creativity would be exercised by generating qualitatively new constructs or explanations from modifications of the given ones. To encourage these two parameters, we would treat a domain by asking not ‘what are facts?’ but ‘how can the acquisition and use of knowledge here broaden the learner’s general ability to acquire and use knowledge?’ This question could provide a heuristic for assessing the function of a subject in the curriculum and the means for reflecting that function in practical methods.

1.7 However, commonsense views about the proper ways to acquire and use knowledge have long been dominated by a wishful thinking that sees intuitive reasoning as distinctly inferior to rigorous, logical reasoning. To quote Seymour Papert again (cit. McCorduck 1979: 292):

We all have these horrible moments of confusion when [....] nothing looks clear [and] we work our way out using all sorts of odd little rules of thumb, by going down blind alleys and coming back again, but since everyone else seems to be thinking logically, or at least they claim they do, then we figure we must be the only ones in the world with such murky thought processes. We disclaim them and make believe we think in logical orderly ways, all the time knowing very well that we don’t. And the worst offenders here are teachers, who present crisp, clean batches of knowledge to the students, [who] groan inwardly, feeling so hopelessly dumb.

Papert’s rumination points toward the many real-life situations in which logical reasoning cannot be applied, because the exact nature of the problems and solutions is not at all clear. Even if the learner fully masters the strategies of formal logic, they cannot simply be substituted for the intuitive strategies of ordinary heuristic reasoning that predominate in everyday activities, including those of learning mathematics or geometry (cf. Papert 1980 for his alternative approach to learning).

1.8 This wishful thinking fosters a unbalanced view of the interaction of knowledge with the mind by implying that education should be designed to make learners abandon intuitive reasoning and embrace logical reasoning as swiftly and completely as possible. The tendency is then to estimate the relevance or merit of a given domain or school subject in terms of its potential contribution to this conversion.

1.9 A predictable upshot is a hierarchy of rank, widely accepted by folk-wisdom and academia alike, exalting formal over ordinary reasoning:

It is common to accord more prestige to mathematics classes than to native or foreign language classes, or more to chemistry than to music appreciation, and so on. The irony is rich, and no one seems to be aware that learning a foreign language or responding to a symphony is a vastly more complex and creative activity than learning long division or running a hydrolysis experiment.

1.10 This hierarchical vision, by exalting formal reasoning, also isolates it from other kinds. Learners are obliged to attempt difficult leaps into an utterly unfamiliar domain for which they believe their ordinary reasoning has poorly prepared them (3.7ff). This dilemma makes the acquisition of subjects like algebra and geometry unnecessarily harder, thereby impeding the overall process of education and narrowing one’s chances for future success.

2.1 Geometry stands out as a ‘school subject’ that has survived the centuries rather well. If Plato really did inscribe over the entrance to the Greek Academy ‘let no one unversed in geometry enter my doors’, many schools since then seem to have ordained: ‘let no one unversed in geometry exit my doors’. Despite the introduction of some alternatives, such as ‘practical mathematics’ in U.S. high schools, concerted attempts to remove geometry from the curriculum have been relatively rare. This persistence begins to seem remarkable when other ancient ‘essentials’, such as the venerable ‘trivium’ of ‘logic’, ‘rhetoric’, and (most recently) ‘grammar’, have been progressively phased out of mainstream schooling and are retained today largely as specialisation’s at post-secondary levels, e.g., in programs of philosophy, communications, and linguistics.

2.2 The considerations raised in section 1 may help to account for the privileged status of geometry: it appears to require and encourage the formal-logical modes of thought traditionally prized as ideals. Moreover, geometry ostensibly offers the impressive advantage of abstracting out the ‘thick’ or ‘grainy’ qualities of ‘true facts’ (with their wealth of individual circumstances) or the ‘noise’ of real objects (due to temperature, pressure, roughness, decay, etc.) without at all blurring the dichotomy between ‘right’ versus ‘wrong answers’; indeed, ‘rightness’ reaches a peak of purely mechanical decidability found elsewhere in the curriculum only in kindred domains of mathematics such as arithmetic, trigonometry, and calculus. Such considerations could explain why geometry has not merely survived, but why its content has changed so little that a single ancient work, Euclid’s Elements, remained the almost universally accepted textbook up into the nineteenth century (2.13, 93).

2.3 Yet a systematic uncertainty persists about the contribution of geometry to the learners’ epistemological development and its relation to their intuitive or ordinary experience of the world. Seven possible theses might be formulated:

(a) Geometry is directly related to everyday experience of shapes, dimensions, volumes, and so on, in intuitively obvious ways.

(b) Geometry is approximatively related to such everyday experience via rough but useful resemblances.

(d) Geometry is related not to everyday experience, but to higher-order powers of reasoning and problem solving.

(g) Geometry is a special-purpose language for representing shapes, dimensions, volumes, and so on.

At the one end stands greater proximity to everyday experience and the other end greater abstractness. Specific groups might prefer a thesis which suits their own positions and tasks. Groups with a practical orientation, such as teachers and textbook authors, might favour theses like (a) and (b), whereas (d) might appeal to educational theorists, (e) to philosophers, (f) to logicians, and (g) to linguists. However, the available evidence, some of which we shall examine below, indicates an informal mixture, compromise, or oscillation among these theses. In effect, the responsibility of determining the status is left up to the learners, who are the least qualified and the most heavily burdened with short-range tasks.

2.4 The thesis of a direct relation between geometry and everyday experience might be called ‘geometric realism’, having an ‘internal’ and an ‘external’ version. Internal realism reifies geometric objects by treating them as if they were real, visible objects which happen to have particular properties. Since they do not seem to occur spontaneously in the environment, the question arises of how the shapes came to possess these properties.

2.5 One answer has been to refer the shapes to the procedures for constructing them. This tactic is ancient:

The use of actual construction as a method of proving the existence of figures having certain properties is one of the characteristics of the Elements. (HE 234)1 (cf. 3.21)

In the Elements Euclid supposes that the reader can use the ruler and compass and no other instruments are allowed. [...] In the geometry originally formulated by Euclid the figures and constructions considered are restricted to those possible with the use of only the straight line and the circle [...] straight lines are drawn on (or in) a plane by means of a ‘straight edge’ or ruler, and circles or parts of circles by means of the compass. (TOM 10, 23 i.a. [= italics added])

This tactic implies that straightness and circularity are natural results of using certain tools to guide the inscribing of lines on a surface. When not in use, the straight edge reassuringly persists in embodying straightness, whereas the compass embodies an angle whose size may be varied from one act of inscription to another yet must not vary during any single act. Hence, the straight edge itself intuitively manifests the constancy of real objects, whereas the compass must be endowed with constancy through a prescribed method of use.

2.6 The traditional straight edge was not marked off into units for indicating distances, a restriction which intuitively matches the textbook notion that ‘a straight line’ ‘extends indefinitely’ (MR 2) (cf. 2.33, 36). The lack has been offset by using the compass as a pair of ‘dividers’ and thereby strategically deploying its capacity to assume a range of constancies, viz.:

By setting the points of the dividers at any chosen distance, this distance or length may be stepped off on a line by swinging each point in turn around the other as a centre, thus dividing the line into equal segments. (TOM 27, i.a.)

The actual distance (e.g. 3 cm) between the points of the dividers was not so important as long as it remained constant and thus generated equalities (cf. 3.29).

2.7 However, the Euclidean approach seems to have sensed the perils of undue reliance on tools, which do not have or produce the accuracy and perfection of Euclidean objects. According to Heath, Euclid’s first postulate about ‘drawing a straight line from any point to any point’ provides

an answer to a possible objector who should say that you cannot, with the imperfect instruments at your disposal, draw a mathematical straight line at all, and consequently (in the words of Aristotle, Analytica posteriora, I 10, 76 b 41) that a geometer uses false hypotheses, since he calls a line a foot long when it is not or straight when it is not straight. It would seem (if Gherard [of Cremona]’s translation is right) that [the Arabic commentator] an-Nairizi [died ca. 922] saw that one purpose of the Postulate was to refute this criticism: ‘the utility of the first three postulates is (to ensure) that the weakness of our equipment shall not prevent scientific demonstration’ (ed. Maximilian Curtze [1899], p. 30). (HE 195)

In practice, though, students and textbooks are extremely fastidious about drawing geometric objects to look exact. This visual exactitude probably functions to enhance the conviction or authority of the proof, especially in the face of difficulties (cf. 2.63). However hard they are to define (as seen in 2.33-45), the point, the line and the plane are easily accepted if properly drawn on a reasonably (though not geometrically) regular surface.

2.8 A less obvious but more serious danger of internal realism is to regard the tools as an independent source of proof. This would encourage a misunderstanding of the popular school practice of drawing the object to scale on paper, cutting it out, and handling or measuring it, e.g.:

The five regular solids may be formed of cardboard or stiff paper by drawing and cutting out the figures shown in solid lines, folding these on the dotted lines, and sticking the free edges together. (TOM 314)

If the figure below is traced on a piece of paper, cut out, and folded along the broken lines, it will look like a cube [...] One way to find the surface area of a solid is to add the area of each face. For certain types of solids, another way is to use formulas. (MR 346)

Geometry instruction must resist the notion that treating geometric objects as real constructs of paper or cardboard could serve as a substitute for understanding the rules and doing the computations. Such visual aids are not themselves proofs, although they can make a proofs intuitively plausible and comprehensible (3.33-43). The Elements devoted much effort to the formal proof that the five ‘regular solids’ with equal equilateral faces (the triangular pyramid, the cube, the octahedeon, the dodecahedron, and the icosahedron) are the only possible ones (3.29), which would have been a hopeless task to achieve by drawing, cutting, and pasting. The second quote is more circumspect in saying that the paper construction will look like a cube rather than be a cube, but insouciantly implies that brute arithmetic (‘adding’) is a valid alternative to higher-powered computation (‘using formulas’) — a fallacy we will examine later (3.26).

2.9 In sum, internal geometric realism can be helpful only if we are careful not to fixate the constructed objects and the physical properties enforced by the use of tools. Instead, we must keep in mind that the tangible qualities of constructed objects only supply an intuition or premonition of the relations we still have to demonstrate with formal procedures (cf. 3.32f). Archimedes (ed. Heiberg [1880-81]: II, 428.18-430) himself utilised this recourse:

Some of what became ‘mechanically’ real to me was later proven in geometric means, because the ‘mechanic’ point of view lacks strict powers of proof [Yet] it is easier to achieve the proof, if one has gained an anticipatory ‘mechanic’ notion of the matter.

Similarly, a modern textbook resolves to ‘facilitate learning’ with ‘many appealing photographs illustrations, charts, graphs, and tables’, which should ‘make it easy for the students to visualise the ideas presented’ (MR vi).

2.10 External geometric realism encourages students to identify geometric objects with ostensible instances or correlates in the real world, e.g.:

The soup can, the basketball, and the pyramid are examples of geometric solids. (MR 221).

Designs in Native American weavings often contain polygon shapes. The rugs in this photograph were designed by the Navajos. (MR 193)

Many real-life phenomena may be described using parallel lines. Fences are stretched on parallel posts [...] The rails on a railroad track never meet. What would happen if they did meet? (MR 159f)

Each floor of an A-frame house is the base of a triangle with parts of the roof as sides. [...] Because the floors are parallel, the triangles formed are similar. [photo: front view of three-story house] (MR 249)

The last example deploys the real-world analogy to derive a geometric statement about the ‘similarity of triangles’. The same textbook hopes to ‘build a solid foundation’ in this manner:

Geometric concepts are introduced intuitively, drawing upon students’ past experience with geometry in real life. (vi)

2.11 External realism is customarily accredited in accounts of the origins of geometry, e.g.:

The first geometrical knowledge was acquired accidentally and without design by way of practical experience and in connection with the most varied employments [during] the history of primitive civilisation at large, where technical geometrical appliances are known to have existed at so early and barbaric a day as to exclude absolutely the assumption of scientific effort. All savage tribes practised the art of weaving, [...] drawing, painting, and woodcutting, [and some] constructed mosaics and pavements (MA 54f)

Most of the ancient records show that definite methods and knowledge of measurement arose in connection with land measurement, building, and astrology [...] The measurement of simple figures formed of straight lines and circles required a knowledge of their properties, and it was the study of these properties which led to the development of the science of geometry. (TOM 3)

From the outset it is fundamental to note that all geometry before the Greeks was not a ‘pure’ or ‘free’ science, but an ‘applied’ one, namely, the ‘art’ of calculating and measuring spatial dimensions, surfaces, volumes etc. This was not considered essentially different from the measurement and calculation of weights or monetary amounts, although here certain material constants were included. [...] The Babylonians made no distinction between mathematical ‘a priori’ constants and physical or technical ‘empirical’ constants. (BK 18)

2.12 The force of such origins can be seen in traditions which preserve apparent anomalies. One of them comes from the ancient tradition which calls to mind the derivation of the ‘geometry’ from Greek meaning ‘earth measure’:

The Babylonians supposed that the heavens revolved around earth and that the year consisted of 360 days; this led them to divide the circle into 360 parts and thus probably originated the present degree system of angle measure (TOM 3).

We are thus the heirs of ‘the sexagesimal system of the Babylonians that has survived in our own divisions of angles into degrees, minutes and seconds, and time’ (BK 12). If Euclid or his proximate successors had incorporated an algebra into geometry, as is commonplace today (2.74), they might have seen fit to institute a decimal-based measure to simplify calculations, e.g. a circle with 100 degrees.

2.13 On the other hand, it is also customary to salute the accomplishment of geometry in transcending its early practical and experiential groundings, e.g.:

In the overall history of the foundations of mathematics the achievement of the Greeks had a great, indeed decisive significance. They founded mathematics as a science; they clearly grasped the basic concepts of infinity and continuity; they discovered irrationality as a fundamental problem [...] There can be no doubt that here already, at the same time as philosophy, exact science began and with it the ‘discovery of the mind [German Geist]’ in Europe. (BK 22f)

Some geometers take pride in this vision of geometry, e.g., as embodied in Euclid’s Elements:

all internal evidence shows, and in particular the essentially theoretical character of the work [...] its aloofness from anything of the nature of ‘practical’ geometry (HE vi)

Being himself a translator of Euclid, Sir Thomas Heath expediently opined that ‘a new textbook on the more “practical” lines’ would lead to ‘a loss of due sense of proportion’ (ibid.) (cf. 2.93).

2.14 Modern geometry textbooks are nonetheless noticeably disposed toward practical orientations. The Merrill Geometry professes to ‘use relevant, real-life applications’ to ‘provide motivation by showing how concepts have practical value and how geometry can help prepare for the future’ (MR vi). The ‘special features’ of the textbook include: ‘Using Geometry, illustrating how geometry can be and is used in everyday life’; ‘Careers, depicting a variety of people’ ‘using mathematics’ in ‘careers that students may pursue’; and ‘Using Money, providing insights into how mathematics can be used in making consumer decisions’ (ibid.).

2.15 Such practical appeals tend either to spill outside of geometry proper or to entail some tidying-up of the reality to make it more geometric (cf. 2.17, 23, 28, 34). Only a few real-world problems in the textbook have an indisputably geometrical nature, e.g.:

Julie Newton is a carpenter. The floor plan below shows a room she will panel on her next job. To determine the amount of molding for ceiling trim [...] (MR 218)

Hexagonal metal bars are made by cutting the sides from cylindrical bars. The radii of the bars is 2.5 cm Find the volume of waste metal from making a bar 60 cm long. Use 3.14 for p and round to the nearest hundredth. (MR 356)

These cases are supposed to help student grasp the formulas ‘P = l + w + l + w’ and ‘V = pr² ´ h’, respectively.

2.16 But in other real-world problems, geometry is only tangentially implicated. The commonsense question, ‘what is the minimum number of legs needed to support’ ‘a piece of stiff cardboard’ ‘in a fixed position?’ (MR 4) would not be sensible in Euclidean geometry, which constructs figures that do not have to be supported or affixed in place by physical means. The student will doubtless answer by recalling experience with stools or tables and may go astray by including ones supported by a single broad leg. Or, the statement that ‘to keep a player from scoring, a hockey goalie reduces the angle from which the player shoots’ (MR 63) is a circumscription of the ratio between the narrowness of an angle and the size of the opposite side, but an odd description of the task of the goalie, whose main job is to block the shots.

Imagine that a triangle is formed by connecting Atlanta, Cleveland, and New York [...] It is 600 miles by air from Cleveland to New York. How far is it from New York to Atlanta? (MR 240)

The accompanying drawing is a map — suggesting a flat earth — upon which is superposed an apparent right triangle, with the leg going from Cleveland to New York labelled ‘2 cm’ and the hypotenuse going from New York to Atlanta labelled ‘4 cm’. For the solution to the problem, the triangle is either superfluous if the student sees and uses the ratio between 2 and 4; or else misleading if the student imagines that the Pythagorean theorem is involved in the solution, and calculates the length of the other leg, the sum of the squares, and so on (cf. 3.37-43). Or, in a case where the Pythagorean theorem is needed:

Consecutive bases of a square-shaped baseball diamond are 90 feet apart. Find the distance from first base to third base. (MR 278).

it gets applied by tidying up the ‘diamond’ into a ‘square’, and the solution is an irrelevant trajectory for the game. Since the batter must run all bases, the relevant distance would be 180 feet, and not the 127.27922 feet the student would compute from the theorem.

2.18 Connecting the real world to geometry may require the stated problem to be abruptly converted into a more abstract and less realistic one, e.g.:

Jack Stanley worked for Monticello Builders on the construction of the curved wall shown at the right. He had to order the correct number of bricks. To complete this task, he determined the length of each curve. (MR 311)

The student is shown how to compute the length of the wall by breaking it up into arcs, each of them being a segment of an imagined circle with two radii of known length and a known angle between them. No attempt is shown to compute the bricks, an operation which could be geometrically exact only if they were curved — much less to justify the idea that the bricklayer must order the exact number before starting.

2.19 The practical use may get entirely pushed aside in order to bring in the geometry:

Jennings Lee is a chemist at Jupiter Lane Chemical. He tests products to be sure they are safe for consumers. Even some of the smallest particles of products, called atoms, are examined. The atoms are often bonded together. Chemists sometimes must estimate the distance between bonded atoms. (MR 57)

The student is then shown pair of atoms as Euclidean circles and is to asked to ‘estimate the distance’ between their centres, which (as Euclidean points) would have zero dimensions (2.35f, 47; 3.39). The textbook neither apologises for the physically inaccurate view of atoms nor suggests how such a computation could conceivably be helpful for ‘testing the safety of products’.

2.20 Sometimes, the student is told how to tidy up real-world problems. For a ‘roof truss’ whose ‘height’ and ‘span’ (shown to make the legs of right triangles) are given, the problem is to ‘find the lengths of both the long rafters and the short rafters [shown as hypotenuses], not counting the overhang’ and ‘rounding the answer to the nearest foot.’ (MR 274, i.a.). The rounding was probably proposed to avert the tiresome calculation from decimals into inches.

2.21 Other times, the untidiness is just suppressed or ignored. As a analogy to ‘similar figures’, the student is given the geometrically absurdity that ‘the clothes in the photograph’ (pants) ‘all have same shape, but differ in size’ (MR 238). Or, the student is asked to ‘tell which unit’ ‘you would use’ to state the ‘area’ of ‘a thumbnail’, ‘a cow pasture’, ‘a national park’, or ‘North Carolina’, and ‘the volume’ of ‘a seam of coal’ or ‘a walnut shell’ (MR 322, 358) — none of which are geometric objects.

2.22 Real world applications may also be annexed by extending the domain of geometry beyond the Euclidean type. A favoured extension is set theory (cf. 2.22, 74, 87f), e.g.:

Geometries can be used to describe a variety of situations. Telephone trunk lines, electrical circuits, and utility lines [...] are some examples. Another example is a commission of people that has several committees. [...] [In one problem] any two committees have at least one person in common; any two people are exactly on one committee together [etc.] Replacing the names of the terms in a system does not change the system itself. [...] Replace ‘person’ with ‘satellite’ and ‘committee’ with ‘network’ to see another situation. (MR 21)

The plural ‘geometries’ is the only signal that the frame of reference has been shifted from the Euclidean type. The example with ‘commissions’ and ‘committees’ suggest a bizarre institution, while the example with electrical circuits does not lead to Boolean logic, which would be a genuinely useful application.

2.23 Predictably, set theoretical applications may also require some tidying up, e.g. by setting tasks under strange conditions:

The three boxes [...] are labelled ‘apples’, ‘oranges’, and ‘apples and oranges’. Each label is incorrect. Suppose you may pull out one piece of fruit from one box. You are not allowed to feel around in the box or peek. Tell from which box you would choose [and] how to label the boxes correctly. (MR 25)

2.24 Another popular domain for annexing applications is conventional arithmetic. The students may be told to compute the ‘fixed, variable costs’ of ‘keeping a car’, ‘the cost of a new air conditioner’, the ‘deductions from a paycheck’, or the ‘time a computer takes to typeset a newspaper article’ (MR 41, 188, 234, 151). Or, they may be given the sound but ungeometric advice to ‘lower the cost of a prescription’ by ‘asking your doctor if there is a generic equivalent’ (MR 376). Not a word is said about how geometry might be involved.

2.25 Algebra is yet another domain for annexations. Here too, problems may be unrelated to geometry, e.g.:

A racing car uses alcohol for fuel. The alcohol weighs 4.8 pounds per gallon and is burned with a fuel to air ratio of 1 to 15. Use a proportion to find how many pounds of air are used in burning 1 gallon of alcohol. (MR 231)

Neither ‘fuel’ nor ‘air’ can be a geometric object, nor is ‘pound’ a geometric measure — quite apart from the counter-intuitive notion of ‘a pound of air’.

2.26 Such carefree annexations may lead textbook authors to overlook genuine opportunities for applying geometry. To compute a ‘mover’s bill of lading’ (MR 103), the student is given only a list of ‘charges’ to be handled with simple arithmetic and does not compute, say, the total volume of the goods to be shipped. For ‘renting apartments’, the student only figures initial or ‘monthly payments’, but does not, say, compute the area of the floor space nor the price per square foot; and the questions include ‘how to check the correctness of payments’ for ‘gas’, ‘water’, and ‘electricity’ (MR 188).

2.27 In some cases, the suggested application to reality might even be disastrous, e.g.:

April Barlow is a paralegal aide for the Tucson City prosecuting attorney’s office. She is gathering information regarding a shoplifting case. [...] Suppose the definition of ‘shoplifting’ is ‘the removal of merchandise by an individual(s) without proof of payment’. (MR 15)

This ‘definition’ is then applied to the task of ‘telling which statements can be used as evidence’: the list includes ‘statements’ that a culprit named ‘E. Val Deeds was in the store at the time’ and then ‘was in the parking lot with the merchandise’ and ‘has no receipt’. The ‘definition’ would convict not merely this culprit but all shoppers who lose or discard their receipts, because it refers not to the act of payment, but to the state of possessing ‘proof of payment’. A reason for this oddity might be that the definitions and proofs of geometry systematically refer to object, quantities, or features, rather than to event, agents, or intentions. The event here gets telescoped in between two states of the agent, ‘in the store;’ and ‘in parking lot’, thus omitting her/his intervening actions as well as those of other agents such as salespersons, floorwalkers, or clerks. (The dummy statements to be rejected as ‘evidence’ stipulate that ‘E. Val Deeds is 16 years old’ and ‘attends high school’, which diverts the ‘career’ profile into a cautionary tale against ‘evil deeds’ by the textbook user.)

2.28 In sum, external geometric realism, though more appealing than internal realism, turns out to be more unmanageable. The purported applications tend to need tidying up, or to be impractical or episodic, or to annex domains outside geometry proper. Some of these difficulties could be controlled by a more careful selection and formulation of the problems, but it seems significant that textbook authors apparently sense no special pressure to do so. In section 3, I shall suggest a revised approach using some tactics resembling geometric realism, but mitigating the unruly consequences (cf. 3.13f, 17, 21).

2.29 Perhaps due to the perplexities of geometric realism, it has often been claimed or implied that geometry stands in an approximative rather than a direct relation to everyday experience. We can thus retain the systematic appeal to intuition while cautioning that the real objects involved show only rough but useful resemblances to the corresponding geometric objects.

2.30 An interesting attempt was made by the eminent scientist Ernst Mach to consider ‘space and geometry in the light of physiological, psychological, and physical inquiry’ (English version 1906). Like the realists but on a higher plane, Mach argued that ‘the source of our geometric concepts’ lies in ‘experience’ (MA 142):

Initially, we have the general experience that movable bodies exist, to which, in spite of their mobility, a certain spatial constancy, [...] a permanently identical property, must be attributed — a property which constitutes the foundation of all notions of measurement. But in addition [...] in the pursuit of the trades and the arts, a considerable variety of special experiences have contributed their share to the development of geometry. In part appearing in unexpected form, in part harmonising with one another, and sometimes [...] becoming involved in apparently paradoxical contradictions, these experiences disturb the course of thought and incite it to the pursuit of orderly logical connections. (MA 53f)

This ‘pursuit’ would focus on ‘relations’ and their perceptible ‘constancy’:

geometry has sprung from the interest centring on the spatial relations of physical bodies. It bears the distinctest marks of this origin [as does] the course of its development. [The] assumption of spatial constancy, [which] our physiological and psychological organisation is independently predisposed to emphasise [when] the space-sensations [from] distinct acts of sense-perception remain unaltered, finds its direct expression in geometry. [...] The fundamental assumption of geometry thus reposes on an experience, although one of an idealised kind. (MA 41ff)

The search for precision encouraged these relations to be clarified in ways that led to geometry proper and eventually to physics:

If it is a question of the exact spatial relationship of bodies to one another, we must provide characteristics that depend as little as possible on the physiological conditions which are so difficult to control. This is accomplished by comparing bodies with bodies. Whether a body A coincides with another body B, whether it can be made to occupy exactly the shape filled by the other [...] can be estimated with great precision. We regard such bodies as spatially or geometrically equal in every respect — as congruent. The character of the sensations is here no longer authoritative; it is now solely a question of their equality or inequality. [...] The most convenient bodies of comparison [...] whose invariance during transportation we always have before our eyes, are our hands and feet, our arms and legs, [as reflected in] the names of the oldest measures [...] Nothing but a period of greater exactitude in measurement began with the introduction of conventional and carefully preserved physical standards; the principle remains the same. (MA 44f)

2.31 Following the mechanist outlook of the times, Mach suggested a ‘physiological’ motive for the tie between geometry and experience:

Since physiological space, as a system of sensations, is much nearer at hand than the geometric concepts that are based thereon, the properties of physiological space will be found to assert themselves quite frequently in our dealings with geometric space [...] the straight line and the plane are especially marked out by their physiological properties [as are] symmetry [and] the division of space into right angles [...] also, similitude was investigated previously to other geometric affinities because of physiological factors. (MA 35f)

The Cartesian geometry of co-ordinates in some way finally liberated geometry from physiological influences [even though] positive and negative co-ordinates [...] are reckoned to the right or to the left, upward or downward. [...] A fourth co-ordinate plane, or the determination of a point by its distances from four fundamental points not lying in the same plane, exempts geometric space from the necessity of constantly recurring to physiological space [But] the historical influences of physiological space on the development of the concepts of geometric space are, of course, not to be eliminated. (MA 36)

Expressly acknowledging the distinctions involved, Mach appealed to ‘perception’ to explain why ‘the sensible space of our immediate perception, which we find ready at hand on awakening to full consciousness, is considerably different from geometrical space’ (MA 5). On the one hand:

geometric space is not cognisant of any relation to our body, but only of relations of the points to one another [...] The space of Euclidean geometry is everywhere and in all directions alike; it is unbounded and infinite in extent. (MA 35, 5)

By the comparison of space with other manifolds, more general concepts have been reached, of which the geometric represents a special case. Geometric thought has thus been freed of conventional limitations, heretofore imagined insuperable. (MA 142f) (cf. 2.97f)

[In] the space of sight, or ‘visual space’, [...] entirely different feelings are associated with ‘upness’ and ‘downness’ as well as ‘nearness’ and ‘farness’, [...] and objects cannot be moved about without suffering expansion or contraction [...] An endless series of sensational qualities or intensities is psychologically inconceivable [...] our visual space is of unequal extent even in different directions [...] Visual space in its origin is in no wise metrical. The localities, distances, etc. of visual space differ only in quality, not in quantity [So] physiological and particularly visual space appears as a distortion of geometric space when derived from the metrical data of geometric space (MA 5ff, 11, 35). (cf. 2.38)

Haptic space, or the space of touch [later also called ‘tactual space’] has as little in common with metric space as has the space of vision [...] it also is anisotropic and non-homogeneous. (MA 10, 18f)

Citing William James, Mach offered a ‘teleological explanation’ in the terms typical of late 19th-century science, namely that the ‘properties of visual space are adapted to biological needs’ (MA 11f):

The perfect biological adaptation of large groups of connected elementary organs among one another is very distinctly expressed in the perception of space. (MA 13)

In contrast, ‘geometric space embraces only the relations of physical bodies to one another, and leaves the animal body’ ‘out of account’ (MA 32). This claim is interesting in view of the common assumption stated by Prigogine and Stengers (1984: 171):

modern science was born when Aristotelian space, for which one source of inspiration was the organisation and solidarity of biological functions, was replaced by the homogeneous and isotropic space of Euclid.

In contrast, their own ‘theory of dissipative structures moves us closer to Aristotle’s conception’ (ibid.), which would have pleased Mach.

2.32 Mach was distressed that the instruction of geometry often disregarded the experiential substrate:

A surface is commonly defined as the boundary of a space. Thus the surface of a metal sphere is the boundary between the metal and the air; it is not part of either, [and] two dimensions only are ascribed to it, [but] this concept suffers from the drawback that it does not exhibit, but on the contrary artificially conceals, the natural and actual way in which the abstractions have been reached (MA 48)

He also lamented that ‘our refined geometrical methods have become entirely estranged’ from ‘the measurement of spaces, surfaces and lines by means of solids’ (MA 50) (cf. 2.71):

A more homogeneous conception is reached if every measurement be regarded as a counting of space by means of immediately adjacent, spatially identical or at least hypothetically identical, bodies [...] (MA 49)

Measurement is experience involving a physical reaction, an experiment of superposition [...] the possibility of such a procedure must be actually experienced with material objects accounted as unalterable. (MA 62)

For Mach, ‘the frank and natural alliance of geometry with the physical sciences was not restored until’ Karl Friedrich Gauss (MA 50) (who did not formulate his ideas in a treatise). Yet he issued a warning as well:

The fact that only a minimum of inconspicuous and unobtrusive experiences is requisite [...] should not lure us into the error [...] of believing that visualisation and reasoning are alone sufficient for the construction of geometry. Geometry is concerned with the ideal objects produced by the schematisation of experiential objects (MA 67f) (cf. 3.11-14)

2.33 Still, we do find many textbooks encouraging a strategic transition from experience to geometry by drawing examples from reality while indicating that they are only approximative. This tactic is quite ancient, having been used for instance in the school of Apollonius (approximation signals italicised):

we have the notion of a line when we ask for the length of a road or a wall measured merely as length [...] Further we can obtain sensible perception of a line when we look at the division between the light and the dark when a shadow is thrown on the earth by the moon (cit. HE 159, after Proclus)

Modern presentations follow suit, e.g. by relating a real object to some model or idea of it:

In geometry, terms are described in a mathematical way. These terms, however, are used as models for real-life phenomena. For example, a point is a model for the position of a source of light. (MR 1, i.a.)

A location or a pinhole suggests the idea of a point. Points have no size. [...] A flight path or a taut wire suggests the idea of a line. Lines extend indefinitely and have no thickness or width [...] A flat map or a pane of glass in a window suggests the idea of a plane. A plane extends indefinitely in two directions and has no thickness. (MR 2, i.a.)

The ordinary ideas conveyed by the words ‘straight’, ‘curved’, ‘flat’, ‘square,’ ‘round’, ‘long’, ‘wide’, ‘thick’, ‘space’, and the like are familiar to everyone [...] If a beautiful building or a complicated machine is carefully observed however, and an attempt is made to describe it in detail [...] it will be realised that special significance attaches to this body of ideas and knowledge and that it is of the highest importance in our life and thought and in the industrial world. (TOM 21, i.a.)

A stretched string, e.g. a plummet, a ray of light entering by a small hole into a dark room are ‘rectilineal’ objects. The image of them gives us the abstract idea of the limited line which is called a ‘rectilineal segment’. (Veronese 1904: 10, i.a.)

Thompson’s textbook for the Practical Man goes on to suggest that a ‘scientific’ stance is achieved by seizing these ‘ideas’ and handling them in a disciplined way:

When each idea [...] is clearly stated and the relations between them are analysed, and when their logical consequences are followed out according to definite stated rules, there results a complete and clearly defined system or body of knowledge. This branch of knowledge is called geometry. Since the knowledge is systematically classified and all results obtained in it are subjected to logical processes, it is called a science. (TOM 21).

For Mach, on the other hand, the approximations should be offset by ‘physical experience’:

A stretched thread furnishes the distinguishing visualisation of the straight line [...] characterised by its visual simplicity. All its parts involve the same sensation of direction. [But] the geometer can accomplish little with this physiological characterisation. To be geometrically available the visual image must be enriched by physical experience concerning corporeal objects. (MA 61) (cf. 3.33)

His proposed demonstration has a somewhat Piagetian quality appropriate for young children:

Let a string be fastened at one extremity at A and let its other extremity be passed through a ring fastened at B. If we pull on the extremity at B. we shall see parts of the string, which before lay between A and B, pass out at B, while at the same time the string will approach the form of a straight line. [So] a smaller number of like parts of the string, identical bodies, suffices to compose the straight line [than] a curved line. (MA 61f )

2.34 As these quotes indicate, approximations are especially vital for conveying the simplest and most austere entities of geometry and thereby mediating the abstractness and tidiness of geometric space:

We learn early in our life to distinguish between an object which has ‘body’ and occupies space, such as a block or box, and the flat smooth surface of an object, which does not occupy space but does have size and extent. If [an object] occupies a portion of space, [it is] called a ‘solid’. If a flat surface such as the smooth top of a table can be thought of as existing apart from the table, so that it has no thickness, and if there is no bend or irregularity in it, then it becomes the ‘plane’ of geometry. (TOM 22, i.a.)

This account, however, blurs the very ‘distinction’ it invokes by implying that ‘life’ confronts us with ‘surfaces which do not occupy space’.

2.35 To grasp the most basic entities, students are encouraged to accept the convenient simplification that dimensions which appear very small can be treated as having the — physically impossible — value of zero. Thus, small dimensions are disregarded, while large, obvious ones are focused:

consider a surface as a body of very minute but unvarying thickness, which for that reason is uninfluential [...] A straight line is primarily a unique concrete image characterised by physiological properties — an image which we have obtained from a physical body of a definite specific character, which in the form of a string or wire of indefinitely small but constant thickness interposes a minimum of volume between the positions of its extremities. (MA 73, 75f, i.a.)

Surfaces may be regarded as corporeal sheets [with a] constant thickness [made] vanishingly small; lines are strings or threads of constant, vanishingly small thickness. A point then becomes a small corporeal space from the extension of which we purposely abstract. [...] Nothing prevents us from idealising [...] by simply leaving out of account the thickness of the sheets and the threads (MA 49, i.a.)

‘Exploratory exercises’ in one textbook ask the students to state whether items like these ‘suggest a point, a line or a plane’: ‘corner of a box’, ‘straw’, ‘telephone wire,’ ‘parking lot’, ‘grain of salt’, ‘clothesline’, ‘star in the sky’, ‘city on a map’ (MR 4, i.a.). Again, the actual but small dimensions such objects must have (e.g. the thickness of a pane of glass) are ironed out, along with physically necessary deviations (e.g. of a telephone wire from absolute straightness). The apparent dimensions are so much more decisive here rather than the actual dimensions that a ‘star’, which is probably many times greater than the earth, is suggested to ‘have no size’.

2.36 A complimentary approximation increases the admitted dimensions by remarking that the geometric object ‘extends indefinitely’ (2.33). This simplification implies that dimensions which appear relatively large are indeed unlimited. Such a vision might be harder to accept than the zero-value of the very small, except that in practice every geometrical object constructed during the course of study appears in a stable, limited representation. This practice in turn favours the reification of the geometric objects that we saw to be typical of internal realism (2.14-27).

2.37 The essential fountainhead of approximations involving indefinitely small and large dimensions must have been the venerable project of deriving all geometric objects from the point and the line. One ancient method was invoked by Proclus (410-485 A.D.), an assiduous commentator on Euclid, under the term ‘definition by genetic cause’. Here, the line is defined as the ‘“flux of a point” (“gßF4H F0:g4@L”), i.e. the path of a point when moved’ (HE 159) (cf. 2.44; 3.42). We find this view also cited by Aristotle (in De anima I.4., 409 a4): ‘they say that a line by its motion produces a surface, and a point by its motion produces a line’. Such a maxim is less simple than it looks. To create a line, the moving point must continually reproduce itself at uninterrupted intervals, i.e., at successive instants between which no division at all occurs (Proclus tried to escape this difficulty by suggesting that only ‘the immaterial line’ ‘is produced by a point’). To create a plane, a moving line engenders full traces only at its initial and the final positions, while its two endpoints continually reproduce themselves like the moving point. These assumptions are surely nourished by the standard practices in drawing figures — points and lines are marked whereas the insides of figures like planes are left blank (cf. 3.10, 14).

2.38 A different account was implicated in some attempts to define the straight line. Euclid’s fourth definition at the start of Book I, namely that ‘a straight line is a line which lies evenly with the points on itself’, has received ‘any number’ of ‘interpretations’, ‘but none that can be described as quite satisfactory; some authorities’ ‘have confessed that they could make nothing of it’ (HE 153, 166). Heath read the definition as indicating ‘a line which presents the same shape at and relatively to all points on it without any irregular or asymmetrical feature distinguishing one part or side from another’ (HE 167). He attributed the ‘obscure language’ to Euclid’s attempt at tidying up the previously accepted definition by filtering out the visual factor (stressed by Mach, 2.31):

The only definition of a straight line authenticated as pre-Euclidean is that of Plato, who defined it as ‘that of which the middle covers the ends’ (relatively, that is, to the eye placed at either end and looking along the straight line). It appears in the Parmenides (137 E): ‘straight is whatever has its middle in front of (i.e. so placed as to obstruct the view of) both its ends’ [...] This definition is ingenious, but implicitly appeals to the sense of sight and involves the postulate that the line of sight is straight. [...] Euclid was a Platonist, and what was more natural than that he should have adopted Plato’s definition in substance while regarding it as essential to change the form of words in order to make it independent of vision, which, as a physical fact, could not properly find a place in a purely geometrical definition? (HE 165f, 168)

Heath failed to acknowledge that Euclidean geometry consistently does entail a visual position: in front of the object and exactly perpendicular to it. The most prominent Euclidean equality, namely, the equality of all radii in a circle, would vanish if the circle were rotated in any plane away from the perpendicular and thereby converted to an ellipse (cf. 2.31). The same holds for the equalities of squares, equilateral triangles, and so on. Problems seldom arise because, as experiments have proven, people mentally correct for shapes when our line of sight is not actually perpendicular to the plane (cf. Weiner 1975).

2.39 What may have disturbed Heath was Plato’s decision to adopt a different visual position from the customary implied one. The first of Heath's two interpolations (both shown in parentheses) moved the eye or point of vision from the ‘middle’, where Plato imagined it, to ‘either end’, while the second interpolation retained Plato’s viewpoint. This inconsistency may suggest Heath's unconscious intent to rescue some of the customary viewpoint by adopting a position that is still at least outside the geometric object rather than (like Plato’s) inside it.

2.40 A less resourceful but popular solution has been to accept the concept as self-explanatory:

‘Straight’ is a simple notion and hence all definitions of it must fail [...] But if the proper idea of a straight line has been grasped, it will be recognised in all the various definitions usually given of it, [which] must therefore be regarded as explanations (Unger 1833, cit. HE 169)

It seems as though the notion of the straight line, owing to its simplicity, cannot be explained by any regular definition which does not introduce words already containing, by implication, the notion to be defined (e.g. direction, equality. uniformity) [...] and as though it were impossible, if a person does not already know what the term ‘straight’ means, to teach it to him unless by putting before him a picture or drawing of it (Pfleiderer 1826-27, cit. HE 168)

The problem again involves the divergence between visual space, wherein we can easily register complex relations as gestaltlike properties, and geometric space, wherein we strive to attend to only the simplest single property and to derive the rest from it (cf. 2.63; 3.10).

2.41 This problem assumes its most acute form in traditional attempts to define the ‘point’. Euclid ’s very first ‘definition’ in the Elements was: ‘A point is that which has no part’. His definition understandably gave rise to disputes:

Martianus Capella (5th century A.D.) [...] translated differently, ‘Punctum est cuius pars nihil est’, ‘a point is that a part of which is nothing’. [...] I cannot think that it gives any sense. If a part of a point is nothing, Euclid might as well have said that ‘a point is itself nothing’, which of course he does not do. (HE 155)

It appears a bit abstruse to build a system upon a fundamental entity which has zero parts without being nothing. Neither could the point be asserted to consist of exactly one part if we take literally the one of Euclid ’s axioms (in his parlance ‘Common Notions’) stipulating that ‘the whole is greater than the part’ (HE 155).

2.42 Still, some such assertion had been made well before Euclid . The Pythagoreans had defined the point as ‘a monad having position’ or ‘with position added’ (cit. HE 155). Similarly, Aristotle had said (Metaphysics 1016 b 24) ‘that which is indivisible in every way in respect of magnitude’ ‘but has not position is a “monad,” while that which is similarly indivisible and has position is a “point”’ (ibid.) He also felt it strategic to stipulate ‘that a point is not a body’ and ‘has no weight’ (De caelo II.13, 296 a 17, III.1. 299, a 30), presumably to ward off geometric realism of a substantialist turn (cf. 3.12).

2.43 Euclid ’s second definition went on to say that ‘a line is breadthless length’ (HE 155), which, in Heath’s view, ‘may safely be attributed to the Platonic school’ (HE 155, 158). Aristotle, who objected that this definition ‘divides the genus by negation’ (Topics VI.’6, 143 b 11), offered his own: ‘a magnitude “divisible in one way only”’ and ‘“continuous in one way or in one direction”’ (Metaphysics 1016 b 25, 1020 a 11)’ (ibid.). Whereas Euclid ’s Platonic definition refers only to an entity lacking a property (‘breadth’), Aristotle's offers two complementary properties, one which the holds the entity together (continuity) and one which dismantles it (divisibility).

2.44 To relate the line to the point, Aristotle encountered the philosophical problems inherent in

the transition from the indivisible or infinitely small to the finite or divisible magnitude. A point being indivisible, no accumulation of points, however far it may be carried, can give us anything divisible. [...] Hence he held that points cannot make up anything continuous like a line. (HE 156)

Yet he rescued the ‘definition by genetic cause’, i.e. that ‘it is only by motion that point can generate a line’ (De anima I.4, 409 a 4) (cf. 2.37), with an ingenious analogy between space and time:

A point, he says, is like the ‘now’ in time; ‘now’ is indivisible and is not a part of time, it is only a beginning or end, or a division, of time. (cit. HE 156; cf. De Caelo III.1 300 a 14)

This analogy opens up the prospect of a temporal or dynamic interpretation of geometry, which has received inadequate attention so far (cf. 3.33, 35f, 42). Usually, the dynamic aspects of spatial objects have been fudged by such devices as the ambiguity in terms like ‘generate’ with a technical sense of ‘trace out mathematically by a moving point, line, or surface’ alongside the ordinary sense of ‘produce’ or ‘originate’ (Webster’s Dictionary, p. 348).

2.45 Before Euclid , it seems to have been customary to start more conveniently with the solid three-dimensional object and work back. Aristotle (Topics VI.4, 141 b 20) ‘speaks of “the definitions”’ as ‘all defining the prior by means of the posterior, a point as the extremity of a line, a line of a surface, and a surface of a solid’ (HE 155, i.a.). The commentator Simplicius (flourished 500 A.D.), in whose definition ‘a point is the beginning of magnitudes and that from which they grow’, conjectured that

Euclid defined a point negatively because it was arrived at by detaching surface from body, line from surface, and finally point from line. ‘Since the body has three dimensions it follows that a point (arrived at after successively eliminating all three dimensions) has none of the dimensions, and has no part.’ (HE 157, his interpolation)

Heath’s portrayal of ‘modern views’ (citing Pasch [1882], Veronese [1891], Enriques & Amaldi [1903)], and Hilbert [1903]) recalls the tactics of approximation we saw above:

In the new geometry, [...] points come before lines, but the vain effort to define them a priori is not made; instead of this, the nearest material things in nature are mentioned as illustrations, with the remark that it is from them that we get the abstract idea. (HE 157)

Heath’s illustration of ‘the notion of a point in Weber and Wellstein’ (1905: 9) is reminiscent of Simplicius, albeit more guarded against epistemological difficulties:

This notion is evolved from the notion of the real or supposed material point by the process of limitation, i.e. by an act of the mind which subjects a term to a series of presentations that is itself unlimited. Suppose a grain of sand or a mote in a sunbeam continually becomes smaller [...] the possibility of determining still smaller atoms in the grain of sand also diminishes [up to] a point incapable of division [But] it is unthinkable that this procedure comes to an end; we can only believe or postulate a term beyond which it cannot go but which we never. It is a pure act of will, not of understanding. (cit. HE 157)

2.46 The approximations invoked in these definitions and their attendant problems are indifferent to problems of inexactitude arising from mechanical rather than epistemological sources. Those problems are freely conceded, e.g.:

all measurements are approximate, no matter how small the unit of measure (MR 50)

all these theorems and therefore idealised and schematised experiences; for real measurements will always show slight deviations from them [and] experimenting [leads to] inevitable errors (MA 59)

Until the formulation of Heisenberg’s ‘uncertainty principle’, it was complacently believed that continued refinements of our instruments would eventually dispose of mechanical inexactitude (cf. Heisenberg 1971). But this inexactitude still has no special bearing on geometry.

2.47 In sum, the prospect of an approximative relation between ordinary experience and geometry, while alleviating some of the most intractable implications of geometric realism, turns out to have its own historical and epistemological array of perplexities. Their roots may well lie in the ambitions of geometers to create a world free of approximation, and to do so by deriving everything in that world, step by step, from premises which are both maximally simple and maximally precise. As we have observed, steady radical reductions of simplicity eventually lead to objects with zero dimensions and zero parts ( 2.35f ), and these make highly inconvenient building-blocks. The exclusion of dimensions is too central to Euclidean geometry to be dispensed with; but the uses of approximation to manage the exclusion, as found from ancient treatises down to modern textbooks, must be carefully handled to avoid blurring the exactitude which motivates the whole enterprise of geometry.

2.48 Another alternative thesis would maintain that geometry stands in an ultimate rather than an approximative or direct relation to everyday experience. Here, we advance no claims about the intuitive groundings and thereby avoid stirring up conflicts. Instead, we merely assert that the nature of the relation could eventually be demonstrated, once the field is in place and has been mastered by the student. Taken literally, an ‘ultimate’ demonstration would be the very final step.

2.49 But it could easily be pointed out that the concerns of advanced geometry for some foundational account (e.g. Saccheri 1733; Bolyai 1832; Riemann 1867; Lobachevsky 1887) have on occasion led away into non-Euclidean geometries which, far from regrounding Euclid’s basic postulates, revise them, especially his parallel postulate (the Fifth Postulate), ‘which he found necessary to the validity of his whole system of geometry’ but which has remained ‘indemonstrable’ despite ‘the countless successive attempts made through more than twenty centuries’, ‘many of them by geometers of ability’ (HE 292). This revision could emerge even against the express intent of the geometer. Saccheri’s (1733) determined attempt to ‘vindicate Euclid of every fault’ by proving the parallel postulate only led to what is now regarded as ‘the first work on non-Euclidean geometry’ (TOM 18). ‘Lobachevski [1887] also undertook his investigations in the hope of becoming involved in contradictions by the rejection of the Euclidean axiom; but after he found himself mistaken in this expectation, he had the intellectual courage to draw all the consequences of this fact’ (MA 126).

2.50 The key point here is that the non-Euclidean geometries, revising the parallel postulate by imagining lines in a space with a curvature, proceeded by adducing abstract mathematical arguments, not by finalising the relation to ordinary experience. They did not, for example, rely on the intuitive case that the two parallel lines would eventually meet because a human cannot create an infinitely long and perfectly flat surface to inscribe them on. Nor did they invoke the modern and more theoretical case that a meeting could be caused if the lines were bent by powerful gravitational fields on a journey through outer space, which would entail the non-Euclidean provision that the leading points of the lines have mass.

2.51 The quote from Weber and Wellstein in 2.45 hints at another possible means for an ultimate step: ‘a pure act of will, not of understanding’. But the implications seem a bit theological, and understanding is after all the chief goal. What is needed, I shall argue in section 3, is a different kind of understanding from the one encouraged by appeals to intuition or to ordinary experience with real-world objects. This understanding could liberate geometry from its own version of the time-worn and slippery maxim that ‘seeing is believing’, where the ambivalence of the term ‘see’ can serve to short-circuit visual perception onto comprehension (cf. 2.31; 3.10).

2.52 In any event, hosts of geometers have proceeded tranquilly for centuries without having considered it urgent to expound the ‘ultimate’ link to reality. In fact, the famous disputes (aired in 2b) over Euclid’s most basic definitions indicate that an ‘ultimate’ statement of the relation between geometry and real-world experience cannot be achieved by rigorous deduction from inside the bounds of conventional geometry (cf. 3.1). Instead, we will need to introduce a special ‘meta-geometrical’ epistemology expressly designed to unify the theoretical with the real. There, any given geometric entity could be comprehended as a virtual entity, and Euclidean space as an indefinite expanse filled with simultaneous virtual entities even when it appears totally empty (3.13). This approach would make it easier to assign a sense to such geometric statements as Max Simon’s (1901), which troubled Heath: ‘content of space vanishes, relative position remains’ (cit. HE 158).

2.53 A wholly different thesis would be to relate geometry not to experience, but to the higher-order powers of reasoning it purportedly helps to develop. The ‘learning of geometry, it would be claimed, fosters a mode of reasoning at so high a power (in the sense of 1.6) that learners become not merely able to use individual theorems and proofs to solve specific problems, but able to reorganise their mental processes for general problem-solving and to follow the traditional ideals of ‘logical thinking’ described in section 1. Such a thesis might be advanced by educators to justify retaining mathematical or computational subjects in schools. And textbooks often juxtapose ‘goals’ like ‘developing proficiency with geometric skills’ and ‘expanding the understanding of geometric concepts’ alongside ‘goals’ like ‘learning to organise ideas’ and ‘improving logical reasoning’ (MR vi).

2.54 An argument along these lines has long been invoked by philosophers. The commentator Proclus (ed. Friedlein 1873: 27f) proceeded in a Platonist mode:

Mathematic science must be considered desirable in itself, though not with reference to the needs of daily life [...] the benefit arising from it [is due to the] intellectual knowledge to which it leads and is a propaedeutic, clearing the eye of the soul and taking away the impediments which the senses place in way of the knowledge of universals.

(Ironically, Proclus [ibid., 31] also quoted Plato’s opinion that ‘mathematics, as making use of hypotheses, falls short of the non-hypothetical and perfect science’ — as if it were not Platonic enough!) Modern praises of the intellectual benefits of geometric study are also common, though more sober, e.g.:

The Euclidean method is frequently used in school textbooks because of its conciseness and its convenience for reference, and on account of the fact that a school course in geometry is intended to be a training in mental discipline as much as in the useful properties of geometrical figures. (TOM 31)

2.55 In the idealist tradition, geometry has been adduced as a prime exemplar of a priori knowledge that is anterior to, or presupposed for, human experience. Bertrand Russell (1956 [1897]: 1) observed:

Geometry, throughout the 17th and 18th centuries, remained, in the war against empiricism, an impregnable fortress of the idealists. Those who held — as was generally held on the continent — that certain knowledge, independent of experience, was possible about the real world, had only to point to Geometry; none but a madman, they said, would throw doubt on its validity and none but a fool would deny its objective reference. The English empiricists [...] were driven into the apparently paradoxical assertion that Geometry, at bottom, had no certainty of a different kind from that of Mechanics — only the perpetual presence of spatial impressions, they said, made our experience of the truth of the axioms so wide as to seem absolute certainty.

The idealist argument resurfaced in the rationale for modern ‘mentalist linguistics’ claiming descent from 17th- and 18th-century ‘rationalism’. Chomsky (1965: 50) called on Leibniz (1873[1702-03]) to testify that ‘the senses’ are ‘necessary’ but ‘not sufficient’ for ‘actual knowledge’, furnishing only ‘examples, i.e. particular’ ‘truths’, whereas ‘the truths of numbers’, i.e. ‘all arithmetic and geometry, are in us virtually’ to ‘set in order what we already have in the mind’. The conclusion was drawn that ‘necessary truths must have principles whose proof does not depend on examples nor consequently on testimony of the senses’, and which ‘form the soul’ of ‘our thoughts’, ‘as necessary thereto as the muscles’ for ‘walking’. This account resembles Plato’s vision of ‘mathematical science’ ‘taking away the impediments which the senses place in way of the knowledge of universals’.

2.56 Yet this venerable argument entails two serious problems. One problem is that, if ‘all arithmetic and geometry are in us virtually’, then they should not have to be taught so laboriously, but should be acquired as naturally as, say, speaking or ‘walking’;2 and their ‘truths’ should be as perspicuous as those established by sensory observation. The notion that geometry is ‘in us’ would seem to make a handy alibi when geometers are hard-pressed to legitimise basic notions, e.g.:

Here we have not defined ‘space’. It is one of the ideas or notions which are inherent in us and is undefinable. (TOM 21f, i.a.)

The notion of a ‘straight line’ is fundamental and inherent in us and is difficult to define in simpler terms. (TOM 23, i.a.)

The alibi is ironic in that the objects of geometry are experienced outside us — usually on a blackboard or a paper and seen from a perpendicular line of vision (cf. 2.38). Moreover, Euclidean objects are hardly found in the construction of the human body; the ‘notions’ of ‘space’ or ‘line’ that really are ‘inherent in us’ are more likely the kinds described by Mach as ‘visual’ and ‘haptic’ (2.31) than geometric.

2.57 The second problem is that the argument foments an unhelpful Platonist tension between abstract ‘truth’ versus the sensory experiences of reality by limiting the latter to a mere triggering or instigative function. This tension contradicts the wide acceptance of perceived or perceptible ‘facts’ as the essential foundation of knowledge and science. The traditional ranking in both folk-wisdom and academia, depicted in 1.9, would be redrawn by placing the formal sciences above the empirical sciences, while these move closer to the humanities, e.g., to the study of literature, a domain of communication whose creations are widely admitted to be (fictional) examples of (not necessary but interesting) principles (Beaugrande 1986, 1988):

2.58 Evidently, the appeal to ‘higher-order powers of reasoning’ does not so much settle the relation between geometry and ordinary experience as resituate it in the mind. We then face the new question of how these ‘powers’ may interact with the presumably ‘lower-order’ activities of ordinary reasoning and problem-solving which students have already developed and which they will mainly rely on in their future lives. Plato’s vision of ‘clearing the eye of the soul’ seems to preclude any major interaction. The perplexities aired in 2a and 2b suggest that additional powers of reasoning beyond those cultivated by geometry would be needed if the latter are to be extended to objects in reality

2.59 An intermediary resolution would be to assert the usefulness of the ‘higher-order’ powers only for limited specialised domains, e.g. ‘architecture, machine work, surveying, and engineering’ (TOM xiii). The task of interfacing the powers with ordinary reasoning could then be transferred to these domains, e.g., for demonstrating how engineers invest their knowledge of geometric principles in the construction of machines. Textbooks might incorporate systematic references to possible investments, e.g.

Because it is a rigid form, a triangle is used in many constructions. Notice the triangles in the electric towers. [shown in photo] (MR 96)

The next step would be to replace the traditional uniform, general geometry with a range of specialised tracks leading toward fields of potential applications. This development would in turn presuppose a comprehensive assessment of the cognitive organisation of those fields — a task which is only beginning to be seriously pursued, especially for purposes of designing computer-assisted data base management. So far, we have mainly more theoretical deliberations, e.g. comparing the use of ‘the axiomatic method’ in geometry and physics (Henkin, Suppes, & Tarski eds. 1959).

2.60 In sum, the appeal to ‘higher powers of reasoning’ doesn’t so much provide an answer about the status of geometry as raise new questions about the nature and use of such powers. Nor do we have good reason to suppose that such powers are necessarily promoted by every engagement with geometry, but only when conditions are suitable; and the task of describing those conditions is still before us.

2.61 According to Russell’s portrayal (2.55), the rationalists and idealists claimed a priori certainty for geometry, while the empiricists argued that ‘only the perpetual presence of spatial impressions’ ‘makes our experience of the truth of the axioms so wide as to seem absolute certainty’. Thesis 2e might attenuate this confrontation by conceiving the notion of spatial certainty to be a property of a system rather than of an object (or a set of objects). The demonstrations and proofs within this system would be construed as propagations of certainty, starting from a premise that is certain because it is either (1) axiomatic, (2) given, or (3) precisely constructed with valid procedures and admissible tools. As each step in the proof is presented, it joins the certainty of its predecessors, until the final conclusion is certified.

2.62 The specific certainty of geometry arises from the network of relations among respective sizes and positions. In this sense, every polygon is a relational system regulating the certainty among its properties; for example, ‘of the six metrical magnitudes discoverable in a triangle (three sides and three angles), three, including at least one side, suffice to determine the triangle’ (MA 71). Prominently featured entities like the ‘right angle’ will be those which provide reliable and readily accessible certainty about other relations within the same figure. As we saw, some of these entities, notably the ‘straight line’, must be taken as certain a priori; and some of the relations they can enter, notably parallelism, cannot be made completely certain by formal proof (cf. 2.38ff, 49). Also, purely geometric certainty is immune to physically or visually inaccurate drawings due to faulty tools or careless handling (cf. 2.7).

2.63 The spatial aspect of the geometric system of certainty has considerable sensory and visual advantages. The figure shown in a drawing makes it feasible to grasp whole configurations of formal proportions as a unified gestalt, even though the proportions can. be conclusively established only through the steps in a proof (cf. 2.40, 3.10). Thus, the learner can ‘see’ the equality between the two sides of a square or the base angles of an isosceles triangle much more easily than the equality between the two sides of any moderately complex algebraic equation (cf. 3.26f). In effect, the spatial representation provides a dividend of holistic surplus certainty to counterbalance the stepwise parsimonious certainty delivered by the proofs.

2.64 Many textbooks, however, are not very clear about the epistemological role of these two modes of certainty. One textbook projects a conflict between visual certainty versus scientific certainty by showing a photo and asking ‘Are you convinced? Does this photograph prove there are flying saucers?’ (MR 16). The textbook goes on to make the puzzling remark tha

Not everyone has the same idea of what is convincing. Some arguments are more convincing than others. Geometry uses a system of reasoning that many people find convincing. (MR 17)

On the one hand ‘convincing’ is said to depend on people’s ‘ideas’ or what they ‘find’ so; on the other, some arguments are said to be more ‘convincing’ in and of themselves, the latter being illustrated with the first proof in the book. The ‘main parts’ of a proof are said to include ‘a diagram’ which ‘may be provided’ and which ‘should picture the information in the hypothesis of the theorem’ and be ‘labelled with names from the given statement’ (ibid.). So the diagram would an optional main part, to be omitted if it suits the convenience of the authors (or the thriftiness of the publishers) (cf. 2.93).

2.65 The propagation of certainty in traditional geometry is conspicuously close-meshed (cf. 2.84, 107; 3.7f). Aristotle stipulated that ‘in any syllogism one of the propositions must be affirmative and universal’ (cit. HE 252), i.e., must embody a certainty stated by properties (rather than by the lack of properties, 2.43) and must hold for all instances. In ancient times, geometers and philosophers devoted much of their efforts to filling out the range of proven instances guaranteed by this certainty, while treating all unproven ones with artificial (or at least official) scepticism.

2.66 The striving for close-meshed certainty naturally encouraged the Euclidean project of ‘attempting to define every term’ (TRY 7). Although the project failed and is no longer undertaken in modern geometry (cf. 2.33-45), students may be perplexed by the close-meshed use of terms and definitions to prove statements nobody would think of doubting. In response to the task, ‘prove that the congruence of line segments is reflexive’, the proposed solution runs as follows (MR 53, 528):

We take a straight line AB, assert its equality to itself by ‘postulate 2.4’ that states ‘the reflexive property of equality’ (‘for any number a, a = a’) (MR 46), and then assert its congruence to itself by the ‘definition of congruence’. The textbook had not in fact given any definition of ‘congruence’, but only of ‘congruent line segments’ — those ‘having exactly the same length’ (MR 51). The proof has thus introduced a seemingly gratuitous doubt and then removed it by citing a ‘postulate’ and a ‘definition’. The net gain in certainty is to have established that if every number is equal to itself, then every segment has the same length as itself. The task seems difficult, I suspect, because this gain is so minuscule compared to the gain we expect in ordinary reasoning, which allows us to say quite sensibly, ‘I haven’t been myself lately’, implying a complex ‘core self’ from which one may deviate under special circumstances. The self-identity of geometric objects is both a prerequisite and a consequence of their extreme simplicity and their immutability over time — although a temporal perspective can be deployed for strategic insights (cf. 2.44; 3.42). In proofs like the above, the steps are so close together that the intuitive problem is to see any difference between what is already proven and what is yet to be proven.

2.67 The need to mistrust the ‘obvious’ seems to have been a topos in early geometry. Some authorities were not overly concerned, viz.:

The three kinds of angles are among the things which, according to the Platonic Socrates (Republic VI. 510 C), the geometer assumes and argues from, declining to give any account because they are obvious. (HE 181)

When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is ‘right’ [...] an ‘obtuse angle’ is an angle greater than a right angle. An ‘acute angle’ is an angle less than a right angle. (HE 153).

Euclid accepts a priori two absolute certainties, ‘straightness’ and ‘equality’, and constructs two right angles, each determining the other. He then deploys the relative and complementary certainties of the relations ‘greater than’ and ‘less than’ for stipulating the obtuse and the acute, this time without stating how to construct them — a task we could execute in Euclidean terms like this:

When a straight line set up on a straight line makes the two adjacent angles inequal to one another, the greater angle is obtuse and the lesser is acute.

In stipulating the construction for the right angle and omitting it for the others, Euclid seems to have rated absolute over relative certainty; also, one only one shape of diagram can be drawn of equality, whereas many can be drawn for inequality. The question of how to rank angles made Aristotle confront the familiar tension between concrete and abstract, represented here by ‘matter’ and, ‘form’. He was

discussing the priority of the right angle in comparison to the acute (Metaphysics 1084 b 7): [...] the right angle is prior, i.e. in being defined, [yet] the acute angle is prior in being a part [of] a right angle [...] the acute angle is prior as matter, the right angle in respect of form (HE 181)

Aristotle juxtaposed the Euclidean definitional viewpoint with a componential viewpoint that provides no certainty of how to construct the acute angle.

2.68 Obviousness was also a point of contention for the notion expressed in Euclid ’s Proposition 20: ‘In any triangle the two sides taken together in any manner are greater than the remaining side’ (HE 286). Once more, visual reasoning was subordinated to formal reasoning:

It was the habit of the Epicureans, says Proclus ([ed. Friedlein], 322), to ridicule this theorem as being evident even to an ass and requiring no proof [...] if fodder is placed at one angular point and the ass at another, he does not, in order to get his food, traverse the two sides of the triangle but only the one separating them [...] Proclus replies truly that a mere perception of the truth of the theorem is a different thing from a scientific proof of it and a knowledge of the reason why it is true. Moreover, as Simson [1756] says, the number of axioms should not be increased without necessity. (HE 287)

The citation from the Euclid translator Robert Simson suggests that a reliance on visual perception might encourage a proliferation of unnecessary axioms.

2.69 When Euclid did happen to accept the obvious as certain without proof, his reverent commentators are eager to exonerate him. A famous case was his very first ‘Proposition’, ‘on a given finite straight line to construct an equilateral triangle’ (HE 241). He craftily proposed a superposition of circles making the given line be their common radius (Fig. 1).

He thereby ‘recycled’ the certainty he had built into the definition of a circle (a figure on whose perimeter all points are equidistant from the centre) to propagate equalities by ensuring that all three sides of his triangle are also radii of equal circles meeting at the top in ‘the point in which the circles cut one another’ (ibid.) (cf. 3.29). Yet this cautious procedure was still incomplete by Euclid’s own standards. So the commentators have defended him for assuming the obvious. Simson said:

Some authors blame Euclid because he does not demonstrate that the two circles made use of in the construction of the problem must cut one another, but this is very plain from the determination he has given [...] for who is so dull, though only beginning to learn the ‘Elements’, as not to perceive that the circle [etc.] (cit. HE 293, my emphasis)

Other commentators have supplied the missing demonstration from elsewhere in Euclid’s system, e.g.:

It is a commonplace that Euclid has no right to assume, without premising some postulate, that the two circles will meet in a point C. [...] Euclid seems to assume it as obvious, though it is not so. [...] The deficiency can only be made good by the principle of continuity [...] Suppose a line belongs entirely to a figure which is divided into two parts; then, if the line has at least one point common with each part, it must also meet the boundary between the parts. (HE 242, 235) (cf. Killing 1893-98: II/43)

In Euclid’s first proposition, however, we do not have a ‘figure divided into two parts’, but a superposition of figures, and superposition needs to be conceived or defined with reference to ‘points in common’.

2.70 The demonstration of certainties by superposition was itself a major strategy since earliest times. According to Heath, Euclid’s ‘Common Notion’ that ‘things which coincide with one another are equal to one another’ ‘was intended to assert that superposition is a legitimate way of proving the equality of two figures which have the necessary parts respectively equal, or, in other words, to serve as an axiom of congruence’ (HE 224f). But this method did not suit Euclid very well:

The phraseology of the propositions [...] in which Euclid employs the method indicated, leaves no room for doubt that he regarded one figure as actually moved and placed upon the other [yet] it is clear that Euclid disliked the method and avoided it wherever he could. [...] as though he found the method handed down by tradition [...] and followed it in the few cases [where] he had not been able to see his way to a satisfactory substitute. But seeing how much of the Elements depends on [this ‘common notion’], directly or indirectly, the method [...] is fundamental. Nor, as a matter of fact, do we find in the ancient geometers any expression of doubt as to the legitimacy (HE 225)

2.71 Again there seems to have been a conflict between the logical and the physical (cf. 2.32). Superposition presupposes a specific motion making one figure enter the space of another, but motion is not usually considered a logical process, viz. Aristotle’s assertion that ‘mathematical objects are among the things which exist apart from motion, except such as relate to astronomy’ (Metaphysics 989 b 32). Even in their most abstract formulations, the conditions upon motion must refer to such entities as mass, force, rigidity, and friction — none of them properly at home in geometry. The arguments of geometers and commentators on this problem are correspondingly abstruse and tortuous, e.g. (cit. HE 226f):

coinciding [sich decken] is either mere tautology, or something entirely empirical, which belongs not to pure contemplation [Anschauung], but to external sensuous experience. It presupposes in fact the mobility of figures; but only matter is movable in space. Thus the appeal to coincidence means departing from pure space, the sole element of geometry (Schopenhauer 1819: II/130)

Helmholtz [1850] maintained that geometry requires us to assume the actual existence of rigid bodies and their free mobility in space, whence he inferred that geometry is dependent on mechanics. Veronese [1891] exposed the fallacy, his argument being as follows [...] the notion of the equality of spaces is truly prior to that of rigid bodies or motion without deformation. Helmholtz supported his view by reference to the process of measurement in which the measure must be, at least approximately, a rigid body, but the existence of a rigid body as a standard to measure by, and the question of how we discover two equal spaces to be equal, are matters of no concern to the geometer. The method of superposition, depending on motion without deformation, is of use only as a practical test; it has nothing to do with the theory of geometry. (HE 226f ).

In Heath’s judgement, Schopenhauer’s ‘acute observation’ was ‘a criticism in advance of Helmholtz’ theory’ (HE 227), but in my judgement anticipates him by declaring that any non-empirical definition of ‘coincidence’ would be ‘a mere tautology’. The crucial difference lay in Schopenhauer’s assertion that the empirical definition by experience belongs outside geometry, whereas Helmholtz envisioned it inside, with geometry being a subdivision of ‘mechanics’.

2.72 A similarly abstruse argument advanced by Bertrand Russell (Encyclopaedia Britannica, Supplement, Vol. 4, 1902) proposed to transpose the motion entirely into the observer’s mind:

the apparent use of motion here is deceptive; what in geometry is called a motion is merely the transference of our attention from one figure to [...] a new one defined by the position of some of its elements and by certain properties it shares with the original figure [...] actual superposition, nominally employed by Euclid, is not required (cit. HE 227)

This argument would guarantee equality by replacing Helmholtz’ notion of a rigid movable body with the notion of a rigid movable frame of selective perception that ‘attends’ only to ‘certain shared properties’. As a means of demonstration, this recourse is less useful than the appeal to sensory experience which Russell wanted to disavow, since we would now be vulnerable to optical illusions and visual incompatibilities of the type portrayed by the artist M.C. Escher.3

2.73 Heath conceded that ‘if the method of superposition is given up as a means of defining theoretically the equality of two figures, some other definition is necessary’ (HE 227). The candidate he favoured (after Ingrami 1904) proceeded by breaking the figure down:

any two figures whatever will be called ‘equal’ when to the points of one the points of the other can be made to correspond univocally in such a way that the segments which join the points, two and two, in one figure are respectively equal to the segments which join, two and two, the corresponding points in the other. (cit. HE 228, i.a.)

Ingrami used the term ‘equal’ in the definition, merely propagating it from ‘segments’ to whole ‘figures’, yet proposing to see the figures as ‘arrays of points’. The ‘newer systems’ cited by Heath (228ff) from an early twentieth-century ‘review’ also take as given the notion of ‘congruence’ among equal segments (e.g. Veronese 1891), and some also that among angles (Hilbert 1903) or complete figures (Pasch 1882).

2.74 Tryon’s recent textbook, on the other hand, which also sees a ‘defect of Euclid’ in his use of ‘superposition’, argues that ‘a theoretical rather than a physical means of making figures correspond’ can be derived from ‘a concept of numerically relating the sizes of parts and ascertaining the order in which the parts are arranged’ (TRY 12). Here, congruence is attained by propagating the certainty of numbers over to the points to which they correspond and for which they act as ‘co-ordinates’. Euclid’s ‘lack of precise notation by which to locate points, lines, and geometric figures relative to each other’ would be offset by means of a ‘scale with which to give the comparison numerical values’, thus ‘taking advantage of algebraic concepts to strengthen the traditional geometry’ (TRY 13):

Introductory geometry until recently stood apart from other major areas of mathematics, in that points in a Euclidean plane or space were not associated directly with numbers and hence not with algebraic methods of study. Modern mathematics has changed this: geometry uses algebra and the ideas of set theory to extend and improve its system of logic. (TRY v)

2.75 On the face of it, the recourse to numerical relations should provide a means to break out of the circularity in the traditional conception of equality, which derived equality of the whole from the equality of the parts and vice versa, but could not adduce an abstract theoretical definition apart from any figure or line segment. Far messier problems confront definitions of partial equality, such as similarity, or of non-equality, which must theoretically derived across sets of cases, whereas equality can be established from the single case (cf. 2.67).

2.76 The recourse to numerical relations thus introduces a non-experiential and non-visual means for determining equality or non-equality; we need only assume a priori that the numbers always occur in the same order, and that each number represents a unique amount of mutually uniform units of increase and decrease. This certainty can be transferred to any line if we grant the ‘postulate’ that ‘there is a one-to-one correspondence between the set of real numbers and the set of points of a line such that to each point the corresponds exactly one co-ordinate’ and vice versa (TRY 14). This recourse frees us from implicating the concepts of superposition and motion, because it in effect makes the line segment itself act out the function Helmholtz envisioned for the rigid, movable measuring object (2.71).

2.77 The traditional problem of objects with less than three dimensions (cf. 2.19, 35f , 47) is also attenuated. A line on a measuring stick or on a tape is routinely accepted to have no thickness we need to account for when we make a measurement. For example, if we measure two consecutive segments, each extending to the line marked ‘10’ for 10 centimetres , we do not figure the total as 10 + 10 plus the thickness of the line marker. Yet ignoring the thickness creates no philosophical unease of the kind Aristotle felt (2.44) .

2.78 The key question is then how far this ‘co-ordinate geometry’ is still a system of spatial certainty in a sense comparable to the Euclidean method. The whole purpose of not introducing numerical values and scales for the size or position of geometric objects must have been to assist Euclidean geometry in making universal statements. For a proof like the one given in 2.66 above, i.e. ‘that the congruence of line segments is reflexive’, it would not be a legitimate Euclidean proof to measure the line AB drawn on the page of the textbook, find it to be 8 millimetres long, then measure it again with the same result, and to conclude with the statement ‘8 = 8’ . Nor would such a proof become more acceptable if we made a hundred measurements, or spaced the measurements out at intervals of one day or one week, even though such tactics are widely accredited as empirical means of determining equality. But if we incorporated the scale of units into the very construction of the line, any point must have its co-ordinate stated in terms of equal units. Drawing the line in a visually or mechanically inaccurate way would not endanger the equality of the unit determined mathematically, not empirically (2.7, 62).

2.79 Hence, the ‘spatiality’ of geometry assumes a different character upon the introduction of co-ordinates. Absolute space is displaced by relational space, such that the congruence of two triangles might now be shown with reference to where they are positioned, a factor that was largely irrelevant for Euclidean geometry unless the triangles touched, overlapped, or coincided. Relational space in turn provides the guarantee of equality needed for Russell’s ‘transference of our attention from one figure to another’ (2.72).

2.80 Describing geometry as a spatial system of certainty seems attractive. We come closer to the spirit of Euclid and his major successors when we recognise the source of certainty in the system rather than in the individual geometric object or in some experiential real-world correlate of it. However, the inner organisation of this system may be perplexing to students, especially the sceptical obligation to prove the obvious. Also, the notion of ‘spatiality’ itself is less straightforward than students are likely to assume, witness the problems entailed in superposition. Consequently, this conception of geometry also still needs to be carefully worked out.

2.81 If thesis 2e is accepted, then 2f , stating that geometry is a type of logic within or alongside algebra, calculus, and so on, ought to hold automatically, since every ‘logic’ in the strict sense must be a system of certainty. Indeed, Thompson claims that ‘geometry’ ‘supplies us the only perfect system of logic’ (TOM xiiif). But we must exercise caution, because the term ‘logic’ has been used in many disparate senses ranging from the commonplace to the technical. In ordinary usage, ‘logic’ can refer to any orderly mode for reasoning about temporal or causal sequences or for being consistent in one’s theses and conclusions, whence Webster’s definition of ‘logical’: ‘in accordance with inferences reasonably drawn from events or circumstances’ (p. 497). In formal usage, however, a ‘logic’ is a closed system of procedures for using established validity to determine the validity or falsity of further hypotheses. The wishful thinking described in section 1 is fomented by a symptomatic confusion between these two senses, so that ‘logical‘ reasoning becomes a slippery evaluative term for procedures connected, in indefinite or fluctuating ways, to the formal ideal at the technical end.

2.82 The danger impends of trading the problem of how geometric objects are related to experiential objects for the problem of how formal logic is related to everyday logic. Uncertainties about the latter relation abound, viz.:

The great strength of geometry as a important discipline through the centuries lies in its careful use of logic. The whole spirit of the Elements was to use fully this technique developed so thoroughly in Greek culture. Logic is a system of intellectual honesty, the same mental attitude which has made the development of modern science possible. But logic is grossly misused in our modern culture. Reasoning in heated arguments or in advertising is sadly illogical. Advertising claims by inference that every new development is effected only with the best interests of the customer at heart. Scientific research on the product may indicate that this is not true. In our scientific age it is part of our training to test every new idea with the evidence of facts, but people continue to believe all sorts of ideas not supported by fact or good judgement and to defend them with emotional fervour. (TRY 33)

This portrayal confuses the use of logic with ethical mandates (‘intellectual honesty’), taste (‘good judgement’), and truthfulness (‘evidence of facts’). These issues fall under ordinary logic, but not under geometric logic, where — in contrast to advertising — no one gains material advantages from making false statements, and — in contrast to heated arguments — emotional motivations are unlikely to arise.

2.83 The portrayal is rectified, however, by differentiating types of ‘truth’:

One of the real problems in logical thinking comes from obscurity of the meaning of the word ‘truth’ [...] In logical reasoning it is a relationship rather than a fact which is true. An argument is composed of statements, any one of which may, or may not, conform to the facts of real experience. A statement which could not be true to scientific fact can still be part of a valid argument. The requirement is that each statement conform to the others. One does not refute any other. (TRY 33f

This passage portrays logic as a system of certainty being propagated from statement to statement, independently of the ‘facts of real experience’. Traditionally, though, uses of logic have eschewed pursuing this independence to the point of intuitive conflict. Like the relation the between geometric objects and real objects, the relation between logical truth and ordinary facts has widely been fudged. The popularity of such classical ‘syllogisms’ as this:

is surely due to the factual plausibility of the premises. The validity of the conclusion, moreover, rests on a neutral and restrictive reading of the premises; it would not be a valid challenge to say that Socrates was not mortal because his life and works have lived on, nor could we read the minor premise as an elaborate, qualitative statement like that made by Antony about Brutus: ‘This was a man’ because ‘the elements were so mixed in him’ (Julius Caesar, V, v, 72-75). Indeed, we don't need to know at all who Socrates was or what is meant by ‘man’ and ‘mortal’. If the wording got changed to ‘all men are moral’ or ‘Xanthippe is a man’, we could only dispute the facts, not the conclusion.

2.84 The syllogism resembles the geometric proof in the close-meshed quality of the series of statements (cf. 2.65f ). In effect, this syllogism is using language to restate the set-theoretical truism that a property holding for all members of a set must also hold for each individual member. The words and phrases are scarcely more than counters or placeholders in a categorical truth that could be stated in infinitely many terms. And if we try to interpret them as more, we soon leave behind the logical system and the certainty it provides.

2.85 In some respects, geometric proofs are even more closed-meshed than their general logical counterparts. Such is the case for the relations between a proposition (‘If X, then Y’), its converse (‘If Y, then X’), its inverse (‘If not X, then not Y’), and its contrapositive (‘If not Y, then not X’):

The theory of logic shows that the contrapositive is always true or false, the same as the proposition is true or false, and the converse and inverse are true or false together. With all our geometric theorems which are not just statements of algebraic expressions, it is required to prove the converse of each theorem before we can accept it. (TRY 38

A purely logical use of the relations would simplify geometric procedures considerably by allowing us to propagate certainty not by proofs, but by the format of the statements themselves. We could thus use the process of conversion to move from Euclid ’s fifth postulate in Book I to his sixth (HE 251, 255):

In isosceles triangles the angles at the base are equal to one another. [...] If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to another

Instead, Euclid provides a fresh proof for the sixth by using ‘the method of reductio ad absurdum’ ‘for the first time in the Elements’ (HE 256). According to Proclus ([ed. Friedlein], 255) this method ‘assumes what conflicts with the desired result, then, using that as a basis, proceeds until it arrives at an admitted absurdity’ (HE 136). Euclid relies on a forced choice: if two lines are not ‘equal’, one is ‘greater’. He then creates a fictional segment out of the difference (Fig. 2).

Let ABC be a triangle having the angle equal to the angle ACB; I say that the side AD is also equal to the side AC. For if AB is unequal to AC, one of them is greater. Let AB be greater, and from AB the greater let DB be cut off equal to AC the less; let DC be joined.

Euclid goes on to derive the equality of the two triangles DBC and ABC and declares it ‘absurd’; ‘therefore AB is not unequal to AC; it is therefore equal to it. Q.E.D.’ (HE 256). Yet from a strict standpoint, even this meticulous proof is not complete:

Euclid assumes that because D is between A and B, the triangle DBC is less than the triangle ABC. Some postulate is necessary to justify this tacit assumption. (HE 256)

On the other hand, the proof seems gratuitously long and dissonant, since the joining of D to C created an angle BCD which as a part of BCA is necessarily smaller than the whole (2.41), thereby destroying the equality of angles stated at the outset. Euclid may have taken the longer way because he felt that the absurdity should properly apply to the entire figure, yet did not supply all the postulates needed for this result.

2.86 Heath remarks that the ‘geometrical conversion’ between the fifth and sixth postulates ‘must of course be distinguished from the logical conversion of a proposition’ (HE 256). ‘From the proposition that “all isosceles triangles have the angles opposite to the sides equal”, logical conversion would only enable us to conclude that some triangles with two angles equal are isosceles’. To ensure a ‘geometrical conversion’, wherein ‘the hypothesis and the conclusion of a theorem change places exactly, the conclusion of the theorem being the hypothesis of the converse theorem’ (HE 257) (cf. 2.104), we can use sets. We take the set of triangles with two equal angles and the set with two equal sides, and declare the two sets equal. Each postulate is then a true statement with respect to the other through the postulate of symmetry (if ‘all X is Y’, ‘all Y is X’).

2.87 Today, it is commonplace that ‘geometry uses’ ‘the ideas of set theory to extend and improve its system of logic’ (TRY v). One popular form of set notation has a spatiality of its own, demonstrating concepts like ‘intersection’ and ‘union’ with the aid of circles or ovals that are superpositioned (Fig. 3):

We are accustomed to the overlapping of two plane geometric figures, a part of one being also a part of the other. The same can be true of sets [...] The intersection of two figures can contain all their points, a few definite points, or no points at all. In the last case the figures do not intersect, but they have an intersection— the empty set. [...] The union is all the points made by combining set E and set F, with no duplication of points. (TRY 19)

But the ‘intersection’ of sets is not ‘the same’ as the ‘overlapping of two plane geometric figures’. The spatiality of set diagrams is not to be interpreted proportionally. A figure like the above in no way implies that the two sets have the same number of points merely because they appear as two circles with the same size. Nor could we argue that if the intersection contains three points and its area is calculated to be one-fifth as large as the area of the whole circle, then the full set contains 15 points.

2.88 Hence, the spatiality of set-theoretical diagrams is non-Euclidean and non-intuitive, allowing virtually no conclusions from visual impressions or physical measurements. Spatial position is relevant and informative only to a very limited extent. Defining a line as its endpoints and the set of all points between them is far less determinate than grasping these points in terms of the co-ordinates they correspond to (2.74). The question of ‘how many points a line contains’ can be answered in set theory only by enumerating the points taken as given, but not by some calculation of so many points on a scale. By using numericals, however, the answer can easily be calculated from the length for the set of points whose co-ordinates are, say, rational numbers, or real numbers to two decimal places.

2.89 The contributions of mathematics must be weighed in view of its own long-standing disputes over the correlation between spatial or visual entities versus formal entities (cf. Volkert, in press). Ambitions for pure formality are occasionally expressed, e.g.:

The condition for the formulation of a general arithmetic is [...] purely intellectual mathematics, which is completely separated from all kinds of intuition, a pure theory of forms, in which not quantities and their images, the numbers, are combined, but purely intellectual objects, entities of thought to which actual objects or relations may, but need not, correspond. (Hankel 1867: 10, cit. Volkert, his trans.)

the recent philosophy of mathematics is characterised by the tendency to mediate between formalism and intuitionism, between physical marks without meaning and mental constructions without signs. (Volkert 1990: 131)

Due to such disputes, appropriations from mathematics into geometry require further interpretations, among which the foregoing sketches of co-ordinates and set theory are merely provisional.

2.90 In sum, the inclusion of geometry into logic within or alongside algebra, calculus, and so on, may offer some advantages in respect to traditional difficulties such as zero-valued dimensions, but requires extensive qualification. The prospects are fairly strong that such an inclusion would entail significant alterations in the spatial groundings of geometry, making them either more concrete through the use of co-ordinates that specify position and dimension or more abstract by treating figures as sets of points or lines irrespective of relative size and position. In addition, the use of logic might create pedagogical obstacles by ‘drawing a sharp line between the instinctive, the technical, and the scientific acquisition of geometric notions’ as Mach suggested; to his mind, ‘it is wrong in elementary geometric instruction to cultivate predominantly the logical side of the subject’ (MA 68f ).

2.91 The thesis of geometry being a special-purpose language might well seem surprising. Most sciences and specialised domains favour a hierarchy ranking discourse below knowledge (1.9). And geometry appears to possess a repertory of figures, proofs, equations, and so on, that provide a well-developed mode of representing things independently of language. Indeed, the spatial quality of the figures to which geometric procedures ostensibly ‘refer’ should make geometry better equipped even than other mathematical domains to dispense with language and yet retain clarity.

2.92 From a historical standpoint, it might be hard to establish how far geometry was in fact developed independently from language (3.3). But it can hardly be denied that geometry was systematised, preserved and transmitted with the aid of language by the philosophical Greeks and continues to be so today by their ‘scientific’ descendants. Nor is it easy to imagine how this aid could have been or could be abandoned.

2.93 The leading historical discourse was naturally Euclid’s austere treatise. The reverence shown for the Elements is significantly reminiscent of that shown for authoritative scripture in a credo or a religion:

Probably no book except the Bible has contributed more to the intellectual life of the world. (TRY 4)

one of the very greatest classics of all time [...] one of the supreme models in all history of rigorous reasoning (cover blurb for HE)

it is one of the noblest monuments of antiquity [and] the greatest elementary textbook in mathematics that the world is privileged to possess (HE vii, ix)

To be sure, an editor and translator of Euclid has good reason to acknowledge the language-dependence of geometry with unusual candour:4

no mathematician worthy of the name can afford not to know Euclid, the real Euclid as distinct from any revised or rewritten versions which will serve for schoolboys or engineers. And, to know Euclid it is necessary to know his language (HE vii, i.a.)

The same reason doubtless moved another editor, Dionysius Lardner (1828), to the gloomy prediction:

Euclid once superseded [...] all that rigour and exactitude which have so long excited the admiration of men of science would be at an end. These very words would lose all definite meaning [...] until at length Geometry, in the ancient sense of word, would be altogether frittered away or be only considered as a practical application of Arithmetic and Algebra.

2.94 Thompson’s ‘practical’ textbook, however, portrays Euclid’s formal mode as a parallel option to the discursive mode (even though, as we see from the ‘reductio ad absurdum’ sample in 2.85, Euclid himself wrote discursively):

[In] the method used by Euclid [...] the statements are usually very careful and precise and as short as possible, and the whole is very formal and exact. Nothing necessary is omitted and nothing unnecessary is included. [...] Another form of proof is one which begins with a general discussion of a previously constructed figure, and in an informal manner, as in conversation or description, brings out successive facts or properties, using additional construction wherever necessary, until an important conclusion is reached. The full theorem is then stated for the first time, containing the conclusion just reached. This method — also very old — is easier to follow than that of Euclid but is not so convenient for reference. (TOM 31

The Merrill textbook, despite its sporadic attempts to be practical (cf. 2.14-27), steers the student away from discursive resources. Not merely the proofs, but even the student’s own notes should assume a rigorous format:

a formal proof really is a summary. It does not show the thinking and planning done to make a good logical argument (MR 22)

Taking notes as you move from step to step in a mathematical passage helps your reading comprehension. These notes are not summaries, but aids enabling you to continue reading. Such notes are usually thrown out after the passage is completed. The following pictures show how James Walker took notes [...] (MR 122)

In the ‘pictures’, the notes consist entirely of diagrams and equations. The textbook goes on to suggest that a wise use of notes is to ‘do the computation’ when a ‘mathematical passage’ does not ‘show the steps involved in reaching a conclusion’ (ibid.). This suggestion again serves the convenience of the authors and publishers rather than of the students (cf. 2.64)

2.95 Apparently, geometers are uneasy about having to rely on language, which they consider an unduly wide and open medium. They have accordingly enlisted a variety of resources for specialising their language. As in the discourse of all mathematics, the most conspicuous resources are the special symbols, notations, and representations. These have been hailed as a major key to progress:

Nothing in the history of mathematics is to me so surprising and impressive as the power it has gained by its notation of language. No one could have imagined that such ‘trumpery tricks of abbreviation’ [...] could have led to the creation of a language so powerful that it has actually itself become an instrument of research which can point the way to future progress [...] The ability of a mathematician [...] is displayed almost as much in finding the true means of representing his results as in the discovery of the results themselves. (Glaisher 1915: 75)

Glaisher’s estimation raises the prospect that a sufficiently ‘powerful language’ might invert the usual hierarchy by occupying a position superior to knowledge (‘results’) (cf. 2.90).

2.96 The ‘abbreviation’ enabled by these resources was applauded as a means to make information more dense and perspicuous than ordinary language can:

The quantity of meaning compressed into a small space by algebraic signs [...] facilitates the reasoning we are accustomed to carry on by their aid. The assumption of lines and figures to represent quantity and magnitude, was the method employed by the ancient geometers to present to the eye some picture by which the course of their reasonings might be traced; it was however necessary to fill up this outline by a tedious description, which in some instances even of no particular difficulty became nearly unintelligible, simply from its extreme length: the invention of algebra almost entirely removed this inconvenience. (Babbage 1827)

It may seem paradoxical that Babbage characterised discursive ‘descriptions’ as ‘tedious’, ‘inconvenient’, and ‘nearly unintelligible’, when formal notations seem so to many people, viz.: ‘mathematics is often considered a difficult and mysterious science, because of the numerous symbols it employs’ (Whitehead 1911: 59). Obviously, two different standards of ‘intelligibility’ are at stake, and higher mathematics may remain inaccessible to society at large until they can be reconciled (cf. section 3).

2.97 Whitehead (ibid.) gave a curious psychological account for the advantages of formal notation:

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race [...] by the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye, which would otherwise call into play the higher faculties of the brain. It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. Civilisation advances by extending the number of important operations we can perform without thinking about them.

Ironically, the rapid and efficient operations enabled by a ‘notation’ are attributed here not to higher-order powers of reasoning (in the sense of 2d), but to decidedly lower-order ones.

2.98 Mach suggested that symbols make it possible to reason beyond the bounds of our understanding:

Symbols which initially appear to have no meaning whatever, acquire gradually, after subjection to what might be called intellectual experimenting, a lucid and precise significance. [...] Think only of the so-called imaginary quantities with which mathematicians long operated, and from which they obtained important results before they were in a position to assign to them a perfectly determinate and visualisable meaning. (MA 103f )

He envisioned an ‘idea germinating’ during the ‘struggle for adapting concept and symbol to each other’ until it ‘finally finds expression’, as happened with ‘exponentials having imaginary exponents’ (MA 104).

2.99 The advantages lauded by such experts hinge on the crucial condition that we are indeed ‘fluent’ in the symbolic notation being used; ‘nothing is more incomprehensible than a symbolism which we do not understand’ (Whitehead 1911: 59). Moreover, the user must genuinely intend to achieve clarity rather to disguise a lack of it behind the notations. Admonitions on this point are easy to find:

But symbolic representation likewise has the disadvantage that the object represented is very easily lost sight of, and that operations are continued with symbols to which frequently no object whatever corresponds. As a young student I was always irritated with [...] the tendency to mysticism which is so easily fostered and bred by the unthinking employment of these methods (MA 104)

Algebra [...] may have been employed to cover, under a complication of symbols, abstruse doctrines that could not bear the light so well in a plain geometrical form; but, without doubt, obscurity may be avoided in this art as well as in geometry, by defining clearly the import and use of the symbols, and proceeding with care afterwards. (Maclaurin 1742: 575f )

The ‘plainness’ Maclaurin envisioned in the ‘geometrical form’ was doubtless enhanced by the spatial groundings (2.63).

2.100 Since fluency presupposes a systematic, consistent, and limited notation, mathematicians have long advocated establishing one and complained about the drawbacks of failing to do so. Alexandre Savérien’s (1753) article on ‘Characters’ in the Dictionnaire universel de mathematique et de physique avowed that ‘nothing is more pernicious than a diversity in expressions’; ‘the fewer characters one uses, the more one learns of mathematics’. His argument resembled Whitehead’s (2.97): ‘the memory is less burdened and consequently mathematical propositions are more easily mastered’. Similarly, Augustus de Morgan’s (1842) article on ‘Symbols’ in the Penny Cyclopedia declared that ‘the progress of mathematics’ ‘now depends more than at any previous time on the protection of established notation’. He optimistically opined that ‘the language of the exact sciences is in a continual state of wholesome fermentation which throws up and rejects all that is incongruous, obstructive and even useless’. But his estimation of the actual situation was pessimistic:

Mathematical notation, like language, has grown up without much looking to, at the dictates of convenience and without the sanction of the majority. Resemblance, real or fancied, has been the first guide, and analogy has succeeded. [...] Until some mathematician shall turn printer, or some printer mathematician, it is hardly to be hoped that this subject will be properly treated.

De Morgan’s complaints hinted at a competition between formal and ordinary languages: ‘among the worst of barbarisms is that of introducing symbols which are quite new in mathematical language, but perfectly understood in common language’.

2.101 In our own century, Florian Cajori (1923/II: 350) laments that ‘our mathematical sign language is still heterogeneous and sometimes contradictory’. Rather than ‘speculating as to how much more might have been achieved with greater symbolic uniformity’, he urges us to ‘contemplate the increasingly brilliant progress that may become possible when mathematicians readdress themselves to the task of breaking the infatuation of extreme individualism on a matter intrinsically communistic’. He advocated a ‘strong and representative international committee whose duty it shall be to pass on the general adoption of new symbols and the rejection of outgrown symbols’. He hopefully remarked that ‘the adoption’ of ‘a universal sign system’ for ‘mathematicians’ would be far easier than of a ‘universal spoken or written language by all intellectual workers’ (1923: II/348).

2.102 Several of these quotations from authorities imply a disparity between geometric versus algebraic notations. Viewed as special-purpose languages, geometry and algebra constitute distinctive ‘dialects’, each appearing a bit obtuse to the users of the other. For Babbage, the traditional (‘ancient’) reasoning in geometry seemed ‘inconvenient’, ‘tedious’, or ‘nearly unintelligible’ whereas algebraic reasoning did not. For Maclaurin, in contrast, ‘geometrical form’ seemed ‘plain’, whereas ‘algebra’ seemed ‘complicated’ and ‘abstruse’, a judgement which might fit de Morgan’s dictum (op. cit.) that ‘pictorial or descriptive notation is preferable to any other when it can be obtained by simple symbols’. Perhaps to obviate the whole disparity, Henry Frederick Baker (1922: 70) stipulated: ‘the object of a geometrical theory is to reach such a comprehensive scheme of conceptions, that every geometrical result may be obvious on geometrical grounds alone’, without the ‘use of algebraic symbols.’ The actual trend, to be sure, was just the opposite, as we saw in 2.74ff; contemporary geometry has made extensive use of algebraic ‘symbols’, which are ‘subject to the laws of computation’ (Cajori 1923: II/333).

2.103 The key role of discourse in geometry is also indicated when textbooks assiduously recommend careful treatment of constructions which consist mainly of language. A typical piece of advice is that ‘the student’ ‘learn the meaning of words in the vocabulary before attempting to write proofs’ (TRY v). Another proposal goes much further:

We need to make certain that every word in a scientific discussion is used according to a dictionary meaning or is defined in the text before it is put to use. (TRY 7)

This naive appeal to ‘dictionary meaning’ is either trivial because every word will normally be found to have such a definition if we look it up, or else overly vague because the dictionary lists many meanings that are quite irrelevant to any one ‘scientific discussion’. Far from being obligatory, as the textbook implies, such attempts to ‘make certain’ are extremely rare and would be in some cases quite impracticable.

2.104 The same source prescribes ‘definitions’ with equally exaggerated rigour:

A definition is an agreed use of words or symbols, each expression having a single limited meaning which remains unvaried [...] The term being defined can always be substituted for its definition (TRY 7f)

Unlike Tryon’s, the Merrill textbook acknowledges that ‘to define a word, you use other words that are left undefined, the simplest undefined terms of geometry being point, line, and plane’ (MR 5). But a margin note misleadingly states that ‘undefined terms are words whose meaning is readily understood’ — not at all the case with ‘point’ or ‘line’, as we saw in 2.33-45.

2.105 What is or is not to be ‘defined’ is predictably an issue of much concern:

First, certain information is taken for granted: in geometry, undefined terms, definitions and postulates. [...] ‘Undefined terms’ are basic terms from which other terms are defined; this information is not proved. ‘Definitions’ are explanations of how words are to be used. ‘Postulates’ are statements that describe the fundamental properties of the basic terms. The information from undefined terms, definitions, and postulates is used to find new information, written as a ‘theorem’. Theorems are statements that must be proved. (MR 17)

Since ‘good definitions must be written very carefully’, ‘guidelines’ are recommended :

1. Name the term being defined. 2. Use only undefined terms or already defined terms. 3. Identify the set to which the term belongs. 4. State the properties which distinguish the term from others in its set. 5. Make it reversible [by] using ‘if and only if’. (MR 5f)

The illustration ‘satisfying all the guidelines’ is: ‘points are collinear if and only if they lie on the same line’ (ibid.). Construed as an ordinary discourse transaction, every step in these guidelines is a bit curious. For 1, the idea that one might give a definition without ‘naming the term being defined’ is gratuitous. For 2, the set of ‘undefined terms’ plus the set of ‘defined terms’ should logically add up to all possible terms. For 3, the need to ‘identify the set’ e.g., that ‘collinear points are points’ raises the odd implication that they might somehow not be points. For 4, the properties ought to both sufficient and necessary to ‘distinguish the term’. And for 5, ‘reversibility’ could be adopted as a general condition in geometric definitions (cf. 2.86 on conversion) and not redundantly spelled out in every one with ‘if and only if’

2.106 In return, it is not stipulated that a ‘good definition’ be clear, comprehensible, sensible, useful, or memorable. When students are asked to state whether sample statements qualify ‘as good definitions’ (‘assuming the terms in the statement are previously defined’!), several distracting counter-examples would be good enough in ordinary discourse, such as ‘a person who lives in Detroit lives in Michigan’, or ‘a bedroom is a room where people sleep’ (MR 7). More bothersome is the task of ‘writing your own definition of “pencil”’ to ‘satisfy the qualities of a good definition’ (ibid.). Webster’s Dictionary by no means meets the ‘guidelines’: ‘an instrument for writing, drawing, or marking, consisting of or containing a slender cylinder or strip of solid marking substance’. Perhaps the students are to treat the pencil as a simple line or a cylinder, but that would not adequately ‘state the properties which distinguish’ it.5

2.107 The meticulous handling of ‘statements’ is required above all for the propagation of certainty:

We can make hundreds of statements about geometric objects and their relations. Geometry requires that we have adequate bases for accepting each one, for verifying that it conforms to all the rest of the system. The proof of each is based solely on previously adopted statements. We are thus pushed back eventually to [...] foundational statements mutually agreed upon without proof. The primary ones are postulates and properties. (TRY 8f)

Every discussion, every study of any particular subject, must have some starting point. There must be some truth or fact which is known or some statement which is assumed, admitted, or taken for granted. When these statements are once allowed they are not thereafter to be changed or denied, and every result or conclusion drawn from them by a correct logical process must be admitted as also true, or at least as true as the original assumption of statements. [...] In geometry the selection of these fundamental statements has been guided by four considerations: (1) They must be such as will be accepted by every person and offer no contradiction to our common knowledge and experience; (2) they must be mutually consistent; (3) all results and conclusions obtained from them must be consistent with one another and with human experience; (4) they should be as few as possible. (TOM 32f)

These ‘four considerations’ pull in different directions, because ‘common knowledge and experience’ are not organised to maximise consistency or economy. Conversely, the close-meshed quality of the system (2.65f, 84) makes basic statements in geometry appear unduly elementary:

Most of the postulates seem like reasonable statements of fact, some of them so obvious that we wonder why they were included. They seem too simple to be given a place of importance. This may be why Euclid did not specifically express them. But postulates do not need to be self-evident truths, conforming to our observation of our objective world. These declarations about geometric objects are creations of man’s mind, adopted because they fit into a total pattern of reasoning. (TRY 9)

‘The basic fundamental statements in geometry’ are called ‘the foundation stones of the structure; they must obviously be laid upon “bedrock”, so deep down that it is never necessary or desirable to go lower’ (TOM 33).

The axioms and postulates must be stated in terms of words already defined or generally understood, they must be put in as simple form as possible, and it should be unnecessary to analyse or explain them further. (TOM 33)

As the proof proceeds, each subsequent statement is to be certified by a ‘reason’:

The logic upon which a proof depends [is a] flow of concepts though a necessary series of steps. The direction taken by each step is controlled by the reason selected to validate the step; that reason must be based soundly on preceding steps. (TRY v)

Tryon proposes to enhance comprehension by relying more on discourse than many other textbooks (e.g. Merrill’s, 2.66, 94):

In proofs each reason is given as a condensed sentence rather than being referred to by a title, such as ‘Definition of a bisector’. This helps the student understand not only the true meaning of each proposition, but why one is correct above all others as the reason for each step (TRY vf )

2.108 In light of so much concern, it is surprising to notice the offhanded or inconsistent handling of geometric discourse, e.g., the hyperbole that ‘plane and solid geometry’ ‘include consideration of every conceivable form of figure and object’ (TOM 22). In another book, ‘conditional statements’ are described by form: ‘the part following “if”’ ‘is called the hypothesis,’ and ‘the part following “then”’ ‘is called the conclusion’; but they are distinguished from other types of statements by content: they ‘may be true or false’, whereas ‘questions and exclamations are ‘usually’ ‘neither true nor false’ (MR 9). Gratuitous philosophical disputes over truth impend when the textbook gives samples for students to ‘identify the hypothesis and the conclusion’, including formal ones like ‘if n is even then n² is even’ alongside commonsensical discursive ones like ‘if it rains then the grass gets wet’; if you live in Texas then you are an American’, and ‘if you ride a bicycle, then you have strong legs’ (MR 10) — credible but not true, and certainly not conditional in a geometric sense.

2.109 The offhand treatment of discourse issues even foments inconsistencies in the basic definitions. Whereas Euclid ’s own definition of ‘a circle’ started with ‘a plane figure contained by a line’ (HE 153), Thompson offered an ostensibly ‘stricter’ interpretation:

The familiar ‘round’ plane figure’ [...] is called a circle. Strictly speaking, the curved line forming the figure is itself the circle. This line is sometimes called the circumference of the circle though, strictly speaking, the word ‘circumference’ should be reserved to designate the measure of the length of this curved line. (TOM 25)

He evidently took seriously his own argument that ‘the lines not only form the figure, they are the figure, and the plane figure is its lines’ (TOM 222). Perhaps he was combating the tendency to reify or substantialise geometric objects, but at the expense of fudging the distinction between ‘plane’ and ‘line’, which is intuitively crucial, though hard to state rigorously (cf. 2.37).

2.110 Similarly, an ‘angle’ may be defined as ‘a pair of rays not on the same line’ or as ‘the union of two noncollinear rays’, but then ‘a straight angle’ may be introduced as ‘a pair of opposite rays’ (MR 65, 169). This inconsistency, which other textbooks avoid with a more circumspect definition of ‘angle’ (e.g. ‘the union of two rays with a common endpoint’ TRY 26), was probably encouraged by the commonsense notion that an angle has to be bent, i.e. ‘angular’. Also, a ‘straight angle’ is visually unappealing and indistinct (but see 3.34).

2.111 In sum, the notion of geometry as a special-purpose language has some merits, but substantial work would be needed to make it fruitful for traditional and current practices. Most geometers want to proceed as formally as possible, and may be uncomfortable because their reasoning cannot be detached from the discourse conveying it. Hence, insufficient attention has been devoted to determining how extensive, perspicuous, and controlled that discourse should be. Instead, we find a mix of stern theoretical advice and offhand practices which may confuse students or force them to work intuitively through their own discourse and to reconstruct and what had been rarefied or omitted for the sake of formality. The benefits of formal language praised by authorities (2.95-98) are precariously prone to turn into obstacles.

3.1 In section 2, we examined a set of possible theses about the status of geometry, ranging from a mode of ‘reality’ over to a mode of language. Like the hoary controversy between realism versus nominalism, this status has resisted fixation since ancient times. Indeed, the documentation assembled in section 2 indicates that the status of geometry might not be resolvable from within, either for the Euclidean method or for the more recent algebraic ones (cf. 2.52). Instead, we might need to attain a ‘meta-perspective’ for contemplating the overall potential of geometry as a multiplex modality for reasoning: spatial, visual, constructive, argumentative, logical, and linguistic. This perspective might not only help to clear up theoretical uncertainties about the nature of geometry and its objects, but might also suggest ways for alleviating the practical difficulties encountered by students.

3.2 With these goals in view, the question is not essentialist (‘what is geometry?’) but teleological (‘how can geometry be construed, presented, and deployed to best advantage?’). Our topmost goal would be to improve the freedom of access to knowledge (Beaugrande 1997). The commonplace impression that mathematical and geometric studies are inordinately difficult has serious human consequences, both in schooling and in career opportunities later on. Thus, after recording and describing the prevailing practices, we should go on to provide a critique of the obstacles and imbalances impeding access, and to propose more strategic alternatives.

3.3 The documentation above points to the key role of discourse but does not determine it in any conclusive manner. Whether or not knowledge can exist independently of discourse, it can — aside from very concrete and simple cases — hardly be elaborated, transmitted, or appropriated independently of discourse (cf. 2.92). By underestimating or disregarding this factor, educational practices imply doubtful theses like these:

(a) The ease or difficulty of acquiring a given body of knowledge is constant, however it might be expressed.

(b) Discourse about knowledge presents no special problems beyond those inherent in the knowledge itself.

I suspect that many of the fluctuations in educational and professional performance, which are customarily attributed to learner-inherent constructs like ‘aptitude’ and ‘intelligence’, are in fact due to communicative problems arising from inadequate attention to the discourse about knowledge, including mathematics (cf. Cohen & Stover 1981; Riley, Greeno, & Heller 1982; van Dijk & Kintsch 1983).

3.4 The learning of geometry might be rendered easier and more reliable if we could handle it as a special case of language acquisition, making systematic and strategic use of ordinary discourse rather than treating it as an inelegant or cumbersome necessity and hastening to recode mathematical and geometric reasoning into formal language. A major goal of this discourse would be to make geometry available to the modes of ordinary reasoning that are already developed among learners.

3.5 For reasons sketched in the foregoing sections, the communicability of geometry has tended not to be recognised as an authentic problem. Attempts to improve communicability may even be misunderstood as a compromise the subject-matter. Thompson’s textbook for ‘those who would study or review without a teacher’ opens with an apology for having ‘made the treatment as informal as the nature of the subject would allow’:

On account of the severely formal nature of the conventional treatment of the subject of geometry, the precise and logical form of its historical presentation, and the generally prescribed classification and arrangement of if its content, a considerable courage would be required of one who should present a school textbook on geometry of so informal a nature as the present treatment. [...] The introduction to the main facts of geometry has been put into intuitive form (TOM vf)

The Merrill textbook hopes to ‘facilitate learning’ by ‘carefully controlling the reading level to ease the special problems that occur with geometry’ (MR vi). However, this goal cannot be met without a much more circumspect and comprehensive use of discourse than the textbook achieved.

3.6 The key factor in the acquisition of knowledge can be termed ‘information load’: not the gross quantity of content, but the degree of relatedness to what the learner already knows (Beaugrande 1980, 1982, 1984). A fluent acquisition process requires careful regulation. If the load is too low, interest and motivation flag; if the load is too high, comprehension is impeded. The most acute danger is ‘information overload’: when the known or familiar is intensely overbalanced by the unknown or unfamiliar, comprehension and performance undergo a major degradation. In extreme cases, this phenomenon may be experienced as a ‘gestalt switch’ where clarity is abruptly inverted into opacity and access is entirely blocked. The result is widespread bewilderment and frustration among the learners, who come to regard the whole domain as inaccessible because they cannot isolate and overcome the chief factors of resistance.

3.7 Anecdotal evidence I have gleaned from schools and colleges (e.g. interviews with students) indicates that mathematical subjects are widely regarded as leading causes of information overload in education. Evidently, the standard presentations make information load precariously high. The issues raised in section 2 imply that the difficulty of geometry lies not just in the use of symbols and equations, but also in the special qualities of its procedures of reasoning. Paradoxically, the domain can appear terribly complex and opaque due to the same factors that were designed to make it relentlessly simple and transparent by taking the least possible as given and deriving everything else by minute close-meshed steps of ‘complication’ (cf. 2.65f, 84, 107). Hence, a learner may encounter overload when attempting to conceptualise a proof not because a large amount of information is involved, but because the internal interdependence of respective bits of information must be meticulously established and maintained. Intuitively, the mind wants to hurry over the ‘obvious’, or to grasp holistically a complete proof and ‘see’ its validity directly rather than to generate or discover it by tiny steps. Even Euclid did not escape objections for having omitted some of these steps and taken things for granted (2.69).

3.8 The close-meshed system of geometry tends to seem closed, as if you need to be firmly inside it before you can navigate properly. The practised insider, i.e. the geometer or the teacher, does not appreciate the difficulties of the outsider, who seems to be obtusely resisting lines of reasoning that are the epitome of plainness. Information overload may be aroused by the learner’s necessity to switch between outside and inside in makeshift ways. This switching is probably unavoidable in early stages, but could be managed and controlled more effectively and efficiently than it usually is.

3.9 The adjunct resources of geometry, such as symbols, formulas, and drawings, should be evaluated in terms of how they affect information load. Because they increase the density of information, these resources can enhance perspicuity and conciseness only up to a certain threshold and then rapidly steer toward overload (cf. 2.96). A formula is meaningful only insofar as the learner is able to reason through it as a statement about a set of relations, some of them being stated in the formula itself and others being inherent in the definition of the symbols. Without special training, most learners cannot stack and store the meanings of very many symbols at a time; yet fetching them on-line can be distracting and time-consuming. Also, the complexity of the relations stated by the formula among the symbols and quantities can easily explode in a ‘reverberating loop’, such that missing information at several points mutually blocks the flow at any one point. The learners are frustrated because they cannot manage to assign each symbol an appropriate interpretation while at the same time keeping track of the entire statement or proof.

3.10 Drawing figures provides the learner the opportunity to refer the statements and symbols to visible points, lines, or planes, rather than to purely hypothetical or logical entities. More importantly, the drawing seems to foreshadow or promise a visual and gestaltlike understanding of the whole proof, an unmediated perception of the equalities or inequalities that ‘were to be demonstrated’ (the ‘Q.E.D.s’) (cf. 2.40, 63). This affordance is enabled not merely by the plainness of the geometric figure, but by its sensory impoverishment, deprived of specific texture, colour-shading, perspective, and so on. The austerity of black lines and white spaces matches the anonymity of letter-names in a representation designed to present a maximum of information relevant to the proof and a minimum of non-relevant information. The insider gauges relevance by the power of the information to apply to a large set of possible objects (1.6), whereas the outsider tends to fixate the exemplar drawn for inspection (cf. 3.13f).

3.11 This divergence between insider and outsider might be mitigated by developing a revised discourse of geometry to fully instate the precept — Euclidean at least in spirit — that all geometric objects are virtual objects. This ‘virtuality’ lends geometric statements a special criterion of truth, but not the ‘necessary truth’ claimed by the rationalists (2.55). A ‘geometrically true statement’ is always a consequence of the system and an abstraction across a set of virtual objects for whom (at least) the stated properties and relations hold. The ‘more advanced’ statements represent further possible interpretations of the system, but not genuine extensions or modifications, because they are all simultaneously inherent, along with the simpler statements. The ‘trick’ is to register the inherently guaranteed status of the ‘Q.E.D.’ by chaining it, link by link, to those statements already perceived to be true within the system. Problems are solved not by injecting ‘missing information’, but by uncovering information deductively provided in principle through the system (cf. 3.26).

3.12 The subject matter of geometry would therefore be not the size and shape of actual objects, but the modes of connectedness among virtual objects. It is therefore perfectly natural that the contours of geometric objects themselves have no physical properties, no thickness, weight; nor are they dimensions, but only product of our acknowledgement of dimensions. The contours can be understood as non-substantive ‘connectednesses’ to which we have ‘loaned’ substance only as an aid to our reasoning. What happens in a successful proof is that at least one implicit connectedness becomes accessible and certified.

3.13 Perhaps the clearest demonstration of virtuality in geometry is the common and very ancient method of treating a given figure by inscribing on it one or more non-given figures, e.g. Euclid’s overlapping circles added to aid the construction of an equilateral triangle (2.69). But virtuality is obscured if we fixate the given object and do not attend to potential superpositions or adjacencies which could make obvious the proposition we are trying to prove (3.10). It would be productive here to perceive the given object not isolated at its position within any realistic space (of the kind suggested by comparisons with table tops, 2.34) but correlated with other positions within a computational space where manifested properties such as length and angles are embodiments of certain classes of computations and have — in the name of convenience but not of essence — been made perceptible and distinct to visual judgement. This space contains an unlimited quantity of other virtual objects which are not currently visible but could be made so for relevant purposes. The given object is thus contained in or superposed upon these virtual objects, and can be reproduced in virtual copies or moved around into new positions (3.38, 40, 42f). Moreover, the objects of geometry could be visualised not as arrays of lines created and assembled in advance, but as ongoing productions and transformations according to relations, such as length and angle, that are now conceived as operations for doing this.

3.14 Realism would demur that we must constrain what we assume to be ‘there’ in a space, lest we erase all differences and it no longer matter if we assume anything, everything, or nothing. But the demurral would have no force, because geometry itself is a system that precisely specifies the consequences of generating and manipulating any object we require. On that basis, geometry is a dimensional codification of productional and transformational procedures for passing from one version of a virtual object to another, e.g., from one position to another (3.42f). So geometry can afford to limit its realism through the relative simplicity and straightforwardness of these procedures, which are not called upon to generate many shapes found in nature, such as the coastlines at the seashore modelled in ‘chaos theory’ (cf. Gleick 1987:94ff). Literal realism leads toward a substantialist reification that shatters the unity and power of the procedures by focusing our attention on specific products like an isolated black-lined triangle drawn in one place on a white page.

3.15 In computational space, a dimension would be conceived not as a fixator of a specific size and shape like an imaginary measuring stick, but also as a mode of connectedness. (Beaugrande 1989). To state a dimensional quantity would be to state the mode in which the endpoints or terminal values are connected. A statement like ‘the circumference of the earth at the equator is very nearly exactly 24,912 land miles’ (TOM 70) tells us that a revolution around the equator would connect us to our original point by the stated distance in the mode of ‘land miles’ but not, say, in the mode of ‘nautical miles’ that ‘correspond to the minutes of angle on the equatorial circle’ (ibid.). This interpretation is easier to accept for the astronomical dimension of ‘light years’, where potential motion was the basis for defining the unit in the first place.

3.16 Euclidean geometry reassuringly stays within the familiar three dimensions of length, width, and depth, although recent developments in particle physics and astronomy suggest that other dimensions are theoretically necessary to account both for the basic construction of matter and for the origin of the universe. In particular, the forces that hold matter together have been expounded in terms of the ‘interchange of virtual particles’, which suggests a special mode of connectedness across infinitesimal but critical distances (cf. Beaugrande 1989 for references).6

3.17 We might make the correlation between the visual and the computational more systematic by regarding every geometric object as a computational field whose apparent internal order is the expression of interacting — mutually describing or determining —relationships such as length and angularity . A given object would be registered not as a realistic entity having just this size and shape, but as a dialectical entity to be understood in terms of certain values it does not have but might assume. Special importance would go to the boundary values where it ceases to be an object of that class and becomes an object of another class.

3.18 In this interpretation, statements about regular polygons would be specifications of a concept of the angle as an efficient gradient for the regulation of distances. The obtuse, acute, and right angles would be described in terms of how they serve in this function, and would thus be reunified with their seeming opposite, the straight line as a virtual transitional angle (cf. 2.110; 3.34f, 41).

3.19 Until now, the precept of ‘virtuality’ has influenced the discourse of geometry, though not in conspicuous or consistent ways. Geometric entities are often presented in conditional or counterfactual statements. For example, Euclid’s first ‘proposition’ refers us to ‘a given finite straight line’, irrespective of any actual line we might draw to assist the computation. His imperative ‘let AB be the given finite straight line’ could be read as an injunction to allow a given line to represent a virtual line but without restricting the scope of the demonstration to the given instance.

3.20 Some of the epistemological nicety of Euclid’s injunction may be obscured by the English translation. According to Heath, ‘the Greek usage differs from ours in that the definite article is employed in such a phrase as this where we have the indefinite, “on the given finite straight line”, i.e. the finite straight line we choose to take’ (HE 242). Heath also notes ‘the elegant and practically universal use of the perfect passive imperative in constructions’, e.g. ‘let it have been described’ or ‘suppose it described’ (ibid.), which could be read as an announcement of the geometer’s engagement with the virtual.

3.21 Neglecting virtuality can foster evasive divisions between essence versus existence:

According to Aristotle, the geometer must in general assume what a thing is, or its definition, but must prove that it is, i.e. the existence of the thing corresponding to the definition; only in the case of the two most primary things, points and lines, does he assume without proof both the definition and the existence. (HE 195)

The use of actual construction as a method of proving the existence of figures having certain properties is one of the characteristics of the Elements. (HE 234) (cf. 2.5)

Virtuality tends to be marginalised when we conceive the given object to be the tangible result of a constructive performance, rather than an exemplary possibility allowed within the overall system of geometry (cf. 2.52; 3.13). Construction is prone to be registered as a full-fledged entry into the domain of the actual, even where the procedures omit all reference to scalar size. The trivial fact that any drawing of a line or figure has one size and not an unlimited range of virtual sizes can encourage the premature geometric realism we have examined above (cf. 2.14-27; 3.13f, 32, 45).

3.22 The uses of mathematics in geometry (cf. 2.74ff, 87ff) may create the impression of higher determinacy, because numbers specify the dimensions, and operations like multiplication are precisely described. But the virtuality of mathematics is if anything even more pronounced than that of geometry, witness Einstein's memorable dictum (cited in Rosen 1978: 195): ‘in so far as the propositions of mathematics are certain they do not refer to reality, and in so far as they refer to reality they are not certain’. Our corollary might be: ‘In so far as objects fulfil the axioms of geometry, they do not occur in reality, and in so far as objects occur in reality, they do not fulfil the axioms of geometry’. Apparently, ‘islands’ of high certainty can be constructed if we accept an ‘outer margin’ of uncertainty about their relation to reality (cf. Beaugrande 1989, 1991a).

3.23 Therefore, the introduction of scalar co-ordinates, hailed as a major advance of modern geometry over ancient (2.74ff), does not so much resolve the virtuality of the geometric objects as transpose it from the objects themselves over to their ‘demarcability’. In place of ‘the finite straight line’ standing for all such lines, we would have the finite demarcable co-ordinate scale, where the actual size (e.g. 1 cm) of the distance between co-ordinates drawn on paper in any single case is irrelevant as long as that distance is theoretically scaled and uniform. Or, if the increments are not uniform but nonetheless vary according to a consistent differential, then differential functions can be regarded as further modes of connectedness, computationally rather than qualitatively distinct from the familiar three dimensions.

3.24 Mathematical symbols also entail a characteristic virtuality, although it may not be registered as such. As Benchara Branford (1908: 370) remarked, ‘progress in mathematical science’ depends on the seemingly ‘paradoxical fact’ that

mathematical symbols are to be temporarily regarded as rigid and fixed in meaning, but in reality are continually changing and actually fluid. But this change is so infinitely gradual and so wholly subconscious that we are not sensibly inconvenienced in our operations. [...] An excellent instance is the gradual evolution of algebra from arithmetic

Stated in my terms, mathematical symbols stand for virtual meanings which are ‘fixed’ only for purposes of demonstrating relations among quantities that can vary to any degree as long as these relations hold. If we are ‘not sensibly inconvenienced’, it is because ‘our operations’ have this virtual meaning whether or not we are ‘conscious’ of it.

3.25 Many learners, however, are ‘inconvenienced’ because they feel self-conscious about the symbols and insist on seeking ‘rigid and fixed meanings’. The symbols for unstated or unknown quantities are then regarded as ominous informational vacuums to be filled as quickly as possible, rather than as placeholders whose non-specification lends real generality to the formula or equation. The headlong rush to reckon up the symbols into numbers naturally favours information overload and reverberating loops, especially when the learner has no coherent plan for deciding which computation to do in which order.

3.26 Problems are accordingly frequent in the standard move of school algebra, a move calling to mind the origin of the term in Arabic ‘al-jabr’, ‘the reduction’: to attain a ‘solution’ by progressively reducing more complex equations to simpler ones. Non-numerical symbols are typically given the intimidating name of ‘unknowns’, which, for ordinary reasoning, might convey an aura of mystery and helplessness; the term ‘uncertains’, though clumsy, might sound better. The discovery of a number for the ‘unknown’ may be pursued with such effort and anxiety as to overshadow the insight that the number is merely a recodification of information already latent in the equation, i.e., an ‘explicitation’ of quantities implicitly predetermined within the set of stated relations (cf. 3.11). The real power algebra obtains by performing operations of conversion (e.g., multiplying on both sides without altering the equality relation) may not be properly appreciated as the genuinely reliable means for steadily reducing and isolating the uncertainty until it becomes trivial and ultimately resolved.

3.27 The outcome is a lingering resistance against the ‘evolution from arithmetic to algebra’ that Branford believed so convenient (3.24). Whereas the insider is able to ‘see’ the equality of a fairly complex equation, the outsider may be so anxious to reduce it that mechanical computation displaces relational insight (cf. 3.7f). Hence, when the algebraic means are not immediately evident, the learner may short-circuit’ the task by guessing at the solution. Within the system of algebra, an example like:

is both an assertion of equality and an assertion that the value of x is appropriately predetermined. The uncertainty is perceptual rather than absolute or structural, a transitory information deficit whose resolution has already been guaranteed and requires only a few algebraic steps, e.g., multiply out the parentheses, reduce 4x - 3x to x, move the 6 to the right side, and subtract (4x - 3x - 6 = 41, x + 6 = 41, x = 41 - 6, x = 35). But a learner who is obsessed with escaping from ‘unknownness’ might instead focus on the purely numerical expression ‘41’, which is perceptually the simplest and most familiar, and reason from there by trial and error. If a number close to 41 is left over after taking away three x from four x, then x might be a number about 2 higher than 41. If 43 is tried out as a candidate to work the equation, the answer is wrong : 4 x 43 = 172, 43 - 2 = 41, 3 ´ 41 = 123, but 172 - 123 = 49, not 41. We could then make x steadily smaller by 1 until we get through all the numbers down to 35 and the equation finally works; or we could notice that decreasing the candidate by 1 has the same effect on the result and could skip over some intermediary candidates. But the whole procedure would be wrong, irrespective of the rightness of the final answer, because we would have substituted bulky brute arithmetic (if we try all the integers and no decimals, itself an unwarranted assumption, we do 4 ´ 9 computations instead of 4) for the power of algebra to use operational conversions for attaining relational insight.

3.28 Accordingly, the use of equations in geometry should be handled with close attention to their perceptual consequences lest they frighten or disorient the learner by obscuring the very relations among the properties of one or more figures that they are intended to symbolise. The Euclidean method is naturally hospitable to equations because it has been strongly devoted to establishing the relations of equality and separating them by exact boundaries from inequality (cf. 2.30, 63, 67, 75f ). For the same reason, figures with inherent visual equalities, such as the square, the circle, and the equilateral triangle, have been preferred for the basic demonstrations.

3.29 Geometry derived its highest power for propagating certainty from the insight that relations of inequality could nonetheless possess regularity. Perhaps the prime inspirational case was the mutual determinacy of interior angles, which was early recognised as an authentic and universal scalar value. The scale of 360 degrees — a solitary fossil of the sexagesimal system of the Babylonians (2.12) — is numerical and virtual at the same time, irrespective of the actual distance (e.g. 2 cm ) traversed at some point by widening or narrowing an angle by so many degrees. Yet the ratio between the size of angles and the size of the sides in Euclidean polygons, being clearest in regular figures, encouraged a shift of the numerical toward the actual by enshrining a gallery of familiar ‘necessary’ shapes. Euclid’s own proof of the five regular solids — a crowning achievement of his Elements (2.8) — might have been the product of a drive to see how far the mutual determinacy of angles and sides could serve to circumscribe the full extent of complete regularity within the universe of solids. The apparently surprising result that only five are possible has a modern correlate in the so-called ‘Waltz effect’ in computer vision, whereby the explosive numbers of possible interpretations of intersections and vertices ‘seen’ by a computer is dramatically reduced by the mutual constraints within any single object (Waltz 1975).

3.30 The geometric counterpart of the algebraic solution by reduction would be the proof which deploys ratios of angles and lines to show that perceived or constructed inequalities can be precisely accounted for in terms of axiomatic equalities. The counterpart of the direct visual perception of the equality of the algebraic equation would be the perception that the asserted equality of the geometric equation is guaranteed by necessary relations within or among the figures introduced during the proof. We could, for example, informally construct an equilateral triangle for a line by stipulating that we thrice draw the line from a single length marked off on a ruler; but if, as Euclid did, we superpose circles such that every side is also a radius, the equality of the sides is guaranteed by the universal definition of ‘circle’ (2.69). To ‘see’ the equality, we need merely attend to each circle alternately and to the shared radius formed by the original line.

3.31 The next question is how visual reasoning might assist or hinder a ‘short-circuit’ effect like that described for algebra in 3.27. Some textbooks seem to worry that it will assist, e.g. when they warn students ‘not to assume’, ‘unless a figure is marked’, any ‘equality’ or ‘inequality of measure, or any other specific measures’ (MR 137). But the same books may also permit visual short-cuts, e.g.: ‘unless stated otherwise, betweenness and collinearity of points may be assumed if they are given in a diagram’ (MR 42).

3.32 I would argue that reasoning from the perceived proportions of presented drawings does not obstruct the power of geometric logic, provided we can succeed in accrediting the virtuality of geometric objects as a dynamic or ‘transformational’ means to reason about their properties. We then no longer face the forced choice between either appropriating intuitive experience directly through a misguided and untidy realism or else rejecting it out of hand in favour of total abstraction. Instead, we can encourage strategic experiences of and with virtual objects by handling them freely in ways which bring out whatever properties we wish to establish. In this manner, geometry could be allowed to hover between rational and empirical, or abstract and concrete, or deductive and inductive, or logical and sensorial, without leading to confusion or overload.

3.33 If the learner wants to observe the ratio dynamically between the size of a angle and the length of the opposite side, we can make a triangle ABC by running a string around three pins in a board. If the pin at the right end C of the ‘bottom’ side AC is pulled out and moved with a stick, keeping the string taut at all times, the learner will see that every increase in the length of the bottom side also increases the size of the opposite angle, and vice versa for every decrease (Fig. 4).

Mach suggested that such a tactic might be performed ‘in thought’, but warned that ‘the mental experiment is never anything more than a copy of the physical experiment’ (MA 71). My proposal would be to turn this around: the ‘physical experiment’ with the actual should be construed as a model of the ‘mental experiment’ with the virtual, and should be so designed to maintain harmony between the two

3.34 Suppose a learner is trying to grasp why the angles of a triangle always add up to 180°. If, as Mach maintains, ‘the knowledge that the angle-sum of a triangle angles of the plane triangle is equal to a determinate quantity’ was ‘reached by experience’ (MA 58), it could be helpful to use the same approach with learners today. We can start from a dynamic principle we could term the ‘conservation of space’: when a line or figure is moved, the space before and after remains constant. Thus, if we take a line AB with a perpendicular forming two right angles (as in Euclid’s definition, 2.67) and move CD to reduce one angle (CDB) by 45°, then the other angle (CDA) grows by the same amount to 135° (Fig. 5).

S If we swing CD by 90° instead, the coincides with a simple straight line, though its production makes it a superposition of two angles, one of 0° and one of 180°, respectively (cf. 2.110). We could therefore regard a straight line as a ‘collapsed’ angle we can unpack to make a figure such as a triangle (Fig. 6).

First, we make the left hand side by ‘swinging up’ the leg a at the pivot point P; then we make the right hand side by ‘swinging up’ the leg l at the pivot point Q until this meets the other leg at the top, point O. According to the principle of conservation, the size of every angle so created is subtracted from 180°, and our two bottom angles have the sizes 180° minus ‘ANGLE 1’ and 180° minus ‘ANGLE 2’ . We now imagine a line intersecting point O and parallel to the bottom leg d, so that ‘ANGLE 3’ equals 180° - (X + Y). We also extend the left and right legs a and l downward, giving us the two new angles V and W. Since a straight line intersecting another straight line yields corresponding angles, and since a straight line intersecting two parallel lines produces identical angles, angle V must be equal both to ‘ANGLE 1’ and to angle X, while angle W. must be equal both to ‘ANGLE 2’ andto angle Y. So the size of ANGLE 3 can be rewritten from 180° - (X + Y) over to 180° - (ANGLE 1 + ANGLE 2). Thus, ANGLE 1, ANGLE 2, and ANGLE 3 together total up to 180°, regardless of how many degrees each one might measure. Such a demonstration can lend direct experiential substance to Mach’s remark:

the theorem of parallels and the theorem of the angle-sum of a triangle are inseparably connected and represent merely different aspects of the same experience (MA 60)

The certainty of straightness and parallelism has been dynamically ‘propagated’ over to the sum of angles by ‘unpacking’ a simple line and ‘swinging’ its implicit parts to make the sides of a figure.

3.35 Another dynamic experience could come from the movement of a ‘side-like object’ during the construction of a triangle (Fig. ) (cf. MA 57).

We use a ruler with the markings facing up to draw one line, then swing the ruler such that it sweeps across the distance of the first angle (1), then repeat this step of drawing and sweeping for each of the next two angles (2 and 3); the ruler comes to rest at the bottom in the opposite position, with the markings facing down instead of up. This result indicates that sweeping across the three angles has caused an overall passage of exactly half of a full revolution of 360°, so the total must be 180°. This test could be done with a variety of triangles until the learner feels convinced that the half-revolution is always completed regardless of the size of the angles or sides.

3.36 Yet another dynamic experience could from manipulating a triangle to juxtapose all three angles. We could draw a triangle ABC on paper, cut it out and set it to rest on one side AC. We would draw the altitude DB perpendicular to the ‘bottom’ side AC, bisect the altitude at E, and draw a parallel FG at the bisection point (Fig. 8). We would then draw two more perpendiculars FH and GI from the ‘bottom’ side AC up to the points F and G where this parallel intersects the two ‘upper sides’ AB and BC. If we now fold the triangle along the parallel FG and along the two perpendiculars FH and GI (broken lines), all three angles must converge exactly, and visibly total up to the straight line (the ‘180° angle’) along the bottom side AC. After doing this on a range of triangles, the learner will again see that the size of the angles or sides doesn’t affect the total. 7

3.37 The celebrated Pythagorean theorem, which is sometimes announced in textbooks as if it were some arbitrary truth or a miraculous discovery, also seems to have stemmed from intuitive, experiential, insights. Following Cantor (1892), Mach suggests that the theorem both came from and was applied to one particular numerical ratio:

The mode in which the sides and angles depend on one another is, naturally, first recognised in special instances. In computing the areas of rectangles and of the triangles formed by their diagonals, the fact must have been noticed that a rectangle having sides 3 and 4 units in length gives a right-angled triangle having sides 3, 4, and 5 in length [...] The knowledge of this truth was employed to stake off right angles by means of three connected ropes respectively 3, 4, and 5 units in length. (MA 72)

As Cantor remarks, Vitruvius’ treatise ‘On Architecture’ (ca. 14 B.C.) claims that Pythagoras himself used just such a triangle and taught his pupils ‘how to make a right angle by means of three lengths measured by the numbers 3, 4, 5’ (HE 352). Or, inspiration may have come from the special case of a right triangle with two equal legs; if their length is a rational number, the length of the hypotenuse is irrational, and vice versa

The simplest case (geometrically) to investigate was that of the isosceles right-angled triangle; and the truth of the theorem in this particular case would easily appear from the mere construction of a figure [...] the investigation of the same fact from the arithmetical point of view would ultimately lead to the other momentous discovery of the irrationality of the length of a diagonal of a square expressed in terms of its side. (HE 352f)

Otherwise, early mathematicians might not have suspected the existence of ‘irrational’ non-terminating, non-repeating decimals, which fit the concept of ‘endlessly improving approximations’ entailed also in attempts to square the circle by inscribing a polygon inside the circle and steadily doubling the number of sides.

3.38 These two simplifications, the isosceles case and the 3-4-5 ratio, can be deployed for a visual spatial appropriation of the Pythagorean theorem, thus enforcing the serendipitous association between the ‘square’ figure and the ‘squaring’ operation whereby mathematics raises a number to its next highest exponent. If we take an isosceles right angled triangle MNO (heavy lines) with sides a, b, and c and make three more ‘virtual copies’, we can arrange the resulting four touching at their right angles (N) into a ‘main square’ (ALSO IN heavy lines) entirely filled by them and having the hypotenuse as its side and thus the total area of c² (Fig. 9).

We then make four more ‘virtual copies’ and place them outside with hypotenuses touching the first four, such that each triangle makes up exactly half of one ‘hypothetical square’ such as MNOP whose other half (broken lines) lies ‘outside’ our previous ‘main square’. Since a and b are equal in an isosceles triangle, the area of a hypothetical square MNOP could be designated either as ‘a²’ or as ‘b²’. If we choose each designation twice, we get two a² plus two b² for the total area of the figure inside the broken lines, whereas the area excluding the broken lines must be half as much, and so totals up to one a² plus one b². Since we started the original square with the side c, we now find that a² + b^2
=c². The square figure and the operation of squaring have visually converged to give a transparent experience of the theorem.

3.39 If the right triangle is not isosceles, the same method will give us a figure with a small square in the middle which is not inside any triangle. The hypothetical figures (extended with broken lines) halfway inside the ‘main square’ are now rectangles rather than squares, and each has the area ab. (Fig. 10).

Our main inner square (in heavy lines) adds up to four halves of these rectangles, plus the small leftover square whose side was obviously generated from the difference between the longer side a and the shorter side b of the triangle MNO and so must have the area of (a - b)². The main square thus has the more complicated total area:

For the 3-4-5 ratio (not shown to scale), ab = 12, 12 ÷ 2 = 6 ,nad 4 x 6 = 24; the side of the left-over square is 4 - 3 = 1 so its area is also 1, giving us the 25 needed to match the square of 5. Every right triangle with unequal legs will yield a figure with the same ratios, so the proof must be general.8

3.40 Using such tactics, the problem of how to prove something about a ‘given’ triangle is solved not by fixating or measuring it, but by creating ‘virtual copies’ and positioning them to build another figure (e.g., a square). This figure is more complex and yet, by virtue of its identical constituents, entirely transparent. The transparency allows the inexactness resulting from non-equalities to be ‘visually tamed’. If the right triangle is not isosceles, the ‘taming’ takes on the gestalt of a leftover square whose area is visibly decided by the difference between the longer leg and the shorter leg in the original triangle. Varying the length or using lengths expressed in irrational numbers would no longer be a messy or disturbing task, but a reassuring one that generates a familiar regular shape.

3.41 Learning by these means should reduce effort by offsetting the abstractness of formal mathematical properties like ‘exponentiality’ and operations like ‘squaring’ by anchoring them to perceptible results of concrete transformations. The same tactic could offset the static, isolated quality of the individual figure with its fixed size and shape by handling the figure as ‘raw material’ generated from or invested in transformations under geometrically specified conditions. A basic entity of geometry that resists axiomatic definition — e.g. the straight line — could be conceived as a procedural ‘transition marker’ whereat the transformation flips over into the converse transformation — e.g., the straight line being the marker where the sweep of an increasingly obtuse angle switches over to a reflex angle or, reversing perspective, to an increasingly acute angle. Or, we could envision two points A and B in Euclidean space, connected by a straight line as the shortest distance a runner might adopt; if our runner mistakenly began running toward point D instead, he could correct his error either early on (e.g. at point C) with a left turn at an obtuse angle, or else very late (e.g. at point D) where an acute angle is demanded (Fig. 11).

The right angle can be intuitively anchored as the transition marker between the set of distance-saving turns and the set of distance-wasting turns. The three classes of angles in Euclidean geometry are then grasped as options for regulating projected motions between two points (cf. 3.18).

3.42 he celebrated Pythagorean theorem, though sometimes announced in textbooks as if it were some arbitrary truth or miraculous discovery, also seems to have originated in intuitive, experiential, insights. Following Cantor (1892), Mach suggests that the theorem both came from and was preferentially applied to one particular numerical ratio:

[64] The mode in which the sides and angles depend on one another is, naturally, first recognised in special instances. In computing the areas of rectangles and of the triangles formed by their diagonals, the fact must have been noticed that a rectangle having sides 3 and 4 units in length gives a right-angled triangle having sides 3, 4, and 5 in length. [...] The knowledge of this truth was employed to stake off right angles by means of three connected ropes respectively 3, 4, and 5 units in length. (MA 72

As Cantor remarks, the treatise On Architecture ( De Architectura ) (ca. 14 B.C.) of Marcus Vitruvius Pollo claims that Pythagoras himself used just such a triangle and taught his pupils “how to make a right angle by means of three lengths measured by the numbers 3, 4, 5” (HE 352). By contrast, if the length of the sides of an isosceles a right triangle is a rational number, the length of the hypotenuse is irrational, and vice versa. Otherwise, early mathematicians might not have suspected the existence of “irrational” non-terminating, non-repeating decimals, which also emerge among the “endlessly improving approximations” in attempts to square the circle by inscribing a polygon inside the circle and steadily doubling the number of sides.

3.43 Dynamic geometry does not strictly require numbers at all. It can rather offer a visual dynamic experience of the Pythagorean theorem, thus exploiting the serendipitous association in LSP between the “square” figure and the “square of the hypotenuse”. We start with a right triangle PKD “sitting on” its hypotenuse DK (heavy line) (Fig. 12).

We perform SLIDE 1 rightwards and downwards along the axis of side b by a distance equal to side a (as measured with a compass) so that it now rests in the position shown with dotted lines as JLQ. We then perform SLIDE 2 leftwards and downwards along the axis of side a by a distance equal to side b (again as measured with a compass), so that the triangle rests in the position shown in dotted lines as RMN. Since all we did was to slide the same figure, all three triangles are trivially congruent because identical.

3.44 SLIDES 1 and 2 have thus relayed the a copy of the original hypotenuse (heavy line) down to the bottom position NM, whilst also leaving the original at the top position DK. Now if we connect them with perpendicular lines of the same length, DN and KM ; connect L to M and D to Q; and reinterpret QN as a side; we can see two more copies of our original triangle rotated by 90 degrees, DQN and KLM. We also see a square DKMN, all of whose sides are the hypotenuse we started out with – in fact, the “square of the hypotenuse”. And we see, at a slant, one square PJQD with side a and another JMLR with side b – the “squares of the other two sides” -- both partially protruding outside DKMN.

3.45 SLIDE 4 and SLIDE 5 move the triangles PKD and KLM from outside the square to the inside and superpose them onto the respective mutual congruent (identical) triangles DQN and NRM. If we then trim off whatever is left over outside, the “square of the hypotenuse” stands clearly revealed as equal to the sum of those two slanted squares of the other two sides (Fig.13).

3.45 The key factor in this dynamic proof is that all the steps remain visually and spatially under control through a series of straightforward operations. One need merely use a compass to ensure that the lengths of the sides and the distances of the SLIDES do not wilfully vary, and the proof will succeed.

3.44. Euclid’s own proof of the Pythagorean theorem (Proposition I.47), in contrast, was based on an ‘extraordinarily ingenious construction’ which Schopenhauer (1873-74: § 15) called ‘stilted’,even perfidious’ (‘stelzbeinig, ja hinterlistig’) (HE 354). Instead of copying the triangle or moving it about, Euclid left it in place, and drew not merely a square on each of the three sides, but lines producing six additional triangles to use in the proof (plus a dozen or so that are not used), none of them congruent to the original triangle (Fig. 14).

I shall not rehearse his tortuous reasoning (51 lines of print in HE) but I would contemplate with uneasiness the vast amount of gratuitous labour this proof must have occasioned for generations of students who had to use the Elements as their textbook (cf. 2.2, 93).

3.45 A dynamic approach to basic entities of geometry, such as is illustrated in 3.33-43, has the advantage of emphasising the experiential aspects without recourse to a misguided geometric realism of static objects in the world, such as electric towers (2.59). The relations involved are construed as criteria for performing certain kinds of constructions, transformations, and calculations within a simplified virtual world where any object can be made to ‘appear’, or copied or moved, if it will support useful insights, e.g., to ‘see’ why the Pythagorean theorem must be true. From among the large number of proofs accrued in the various geometric traditions and the perhaps infinite number of proofs that could be devised, we would prefer those which provide a transparent visual and spatial grasp, e.g. the small residual square for absorbing deviations from the isosceles right triangle in the Pythagorean demonstration (3.39f), yet which do not sacrifice geometric accuracy or skip, over the proof procedures.

3.46 Many dilemmas in the traditional learning of geometry may have arisen from not utilising the power of virtuality to provide experiential demonstrations without reifying the ‘given object’. This power makes it feasible to ‘see’ the object in other modes, e.g. (1) within a superpositioning of objects, (2) as a piece or constituent of a larger object, (3) as a momentary stage in an operation of shape transformation, or (4) as a step in a elaboration which can pursued through a series of intuitively appealing ‘connectednesses’ until its relations collapse into transparencies or identities.

3.47 Demonstrations like the above are ‘user-friendly’ also by virtue of their discursive quality. They can be conversationally presented to keep information load manageable yet without incurring the explosively lengthy or tedious quality invoked by Babbage (2.96). As the proofs grow move advanced, this quality will need to be restricted, but will also not be so crucial as it is in the stage where the basic entities are being mastered with visual and experiential support.

3.48 Since I have only illustrated the dynamic approach in the briefest way, I cannot predict how much designing would be needed to create a whole functioning curricular module corresponding to a traditional ‘geometry course’. No doubt such an approach would be most strategically developed for the critical early stages of instruction, when the information load is heaviest and the danger of overload most imminent. But it would be a rewarding task to see how far the approach can be extended to more advanced stages and onward into trigonometry.

3.49 More than ever before, technological education is rapidly expanding the demand for mathematical facilities on a higher level than the computational skills required from students in the past. The predominantly reproductive activities must be yield to productive ones, and the content of instruction must be organised to encourage the development of power and creativity (1.4, 6). In domains like physics and engineering, ‘higher-level reasoning’ must deal with steadily more complex abstractions that resist ordinary experiential appropriations through familiar ‘realism’. However, alternative means for visual and/or spatial appropriations can in principle be found, witness the metaphorics of ‘quarks’ as coloured objects (‘green’, ‘red’, ‘blue’) or directional ones (‘up’, ‘down’) in recent physics.

3.50 The role of geometry in this changing scene is far from settled. The epistemological uncertainties and peculiarities of geometry still represent a challenge to human comprehension and imagination, and merit continual fresh engagements. In this section, I have suggested some potential interpretations of the geometric domain as a computational space of virtual objects whose dimensions constitute modes of connectedness. We can derive from this space any perceptual affordance that supports our reasoning; yet for the purposes of training and instruction we should prefer and design those which are accessible without overloading either the visual or the computational field. Creating such a design so on a consistent substantive basis encourages us to rethink our traditional ambitions and conceptions, including those about what it means to be ‘scientific’’ and ‘logical’, and about how formal reasoning might be productively integrated with ordinary reasoning.

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1 To conserve space, the main sources will be cited by abbreviations plus page numbers. The key is as follows: BK: Becker (1954); HE: Heath’s (1956) edition and translation of the Elements; MA: Mach (1906); MR: the Merrill Geometry (Foster, Cummings, & Yunker 1981); TOM: Thompson (1934); and TRY: Tryon (1967). Quotes from German texts available to me (e.g. BK) are in my own translation; some quotes from other people’s translations (e.g. of MA) I have improved here and there.

2 Some such consequence was indeed drawn by mentalist linguistics for language, following Humboldt (1836): ‘the rationalist view’ implies that ‘one cannot really teach language, but can only present the conditions under which it will develop spontaneously in the mind in its own way’ (Chomsky 1965: 51; cf. Beaugrande 1991b)]

3 Hofstaedter (1979) suggests that the paradoxical gestalts of Escher’s drawings can be related to the incompleteness of formal systems of certainty proven by Gödel (1934).

4 In 3.19f, I suggest in fact that some of linguistic fine points Heath comments upon, such as the use of definite articles or of the perfect passive imperative, might lead toward an interpretation of Euclidean geometry quite unlike the traditional one in the schools.

5 Elsewhere, the book talk of ‘definitions’ in mystified, commonsensical ways. One passage says that ‘the definition of “angle” includes “rays”, “noncollinear”, “endpoint”, “sides”, and “vertex”, but soon afterward that ‘the word “angle” includes all the other ideas’ (MR 169, i.a.). In one ‘example’, the student is to ‘find the term from the following group which includes all others: “snow”, “winter”, “cold”, “Christmas” (ibid.), although such terms would never suffice for a ‘good definition’ of ‘winter’.

6 The notion in the new physics of the vacuum as a space filled with ‘virtual particles’ offers a congenial analogue to the virtual space proposed in 3.13.

7 The English translator of Mach’s volume cites a 1901 textbook by Sundara Row on Geometric Exercises in Paper-Folding. Also, some of my students have recently reported similar exercises appearing in their textbooks.

8 It appears under the picturesque but puzzling names of ‘bride’ and ‘bride’s chair’, e.g., in an old Indian source by Bhaskara from the twelfth century (cf. HE 417f ).

9 To enhance clarity I have modified, and provided with a more suitable and complete drawing, a demonstration by Mach (MA 74f ), who remarks that the demonstration can also be done ‘with an oblique-angled triangle’ to give the more general proposition, c² = a² + b² - 2cosg ’.