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
Knowledge and discourse in geometry:
Intuition, experience, logic
Robert de Beaugrande
Mathematics is more an activity than a doctrine.
— Luitzen Egbert Brouwer
Geometry is at the same time a science and an art, mathematics and philosophy.
— James Edgar Thompson
Mathematics is often a lonely, impersonal experience of manipulating symbols in accordance with rules learned by rote.
— Pamela McCorduck
1. The trees and the forest
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:
formal reasoning:
hard sciences natural sciences mathematical languages. . .
ordinary reasoning :
soft sciences human sciences natural languages. . .
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. Seven theses on the status of geometry
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.
(c) Geometry is ultimately related to everyday experience, but in mediated or abstract ways.
(d) Geometry is related not to everyday experience, but to higher-order powers of reasoning and problem solving.
(e) Geometry is a spatial system of certainty, or a model for such a system.
(f) Geometry is one type of logic alongside algebra, calculus, and so on.
(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.
2a. A direct relation
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)
The upshot was an emphasis upon the tangible tools to be employed:
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.
2.17 Tangential appeals to geometry may only make the problem harder, e.g.:
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).
2b. An approximative relation
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)
This tie persists even through it is no longer crucial:
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)
This is contrasted with how humans perceive space by sight or by touch:
[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
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’.
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,
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
2.42
Still, some such assertion had been made well before
2.43
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 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.
2c. An ultimate relation
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).
2d. Higher-order powers of reasoning
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)
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 entail