ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science

The Epistemology of Geometry I: The Problem of Exactness

A G Newstead (a.newstead@unsw.edu.au)
School of Mathematics and Statistics, Red Centre 6102 UNSW
Sydney, NSW 2052 Australia

James Franklin (j.franklin@unsw.edu.au)

School of Mathematics and Statistics, Red Centre 6109 UNSW
Sydney NSW 2052 Australia

Abstract
We show how an epistemology informed by cognitive science
As is evident from Table 1, abstract objects are typically
promises to shed light on an ancient problem in the defined negatively by contrast with concrete objects. There
philosophy of mathematics: the problem of exactness. The is no consensus as to whether abstract objects must be
problem of exactness arises because geometrical knowledge is singular objects (particulars) or universals. Frege, for
thought to concern perfect geometrical forms, whereas the example, treats numbers as both particular objects and as
embodiment of such forms in the natural world may be abstract objects. By contrast, mathematical structuralists
imperfect. There thus arises an apparent mismatch between treat mathematical patterns as universals rather than
mathematical concepts and physical reality. We propose that
the problem can be solved by emphasizing the ways in which particulars.
the brain can transform and organize its perceptual intake. It In contemporary metaphysics and epistemology, abstract
is not necessary for a geometrical form to be perfectly objects are usually thought to be causally inactive on the
instantiated in order for perception of such a form to be the grounds that efficient causal action requires location in the
basis of a geometrical concept. spatiotemporal order. If mathematical objects are abstract--
Keywords: geometrical knowledge; philosophy of in the traditional sense captured in Table 1-- then it follows
mathematics; perception and mathematics; visualization that they are causally inert. Benacerraf (1973) in his classic
paper, ‘What is Mathematical Truth?’ points out that it is
Mathematical Knowledge extremely mysterious how knowledge of abstract objects is
The dominant problem in the epistemology of mathematics possible. Our best naturalistic theory of knowledge appears
for many decades has been to give a naturalistic account of to be—at least at a base level—to involve a causal condition
mathematical knowledge. Such naturalistic account will on knowing. That is, generally we think that if a subject S
inevitably draw on cognitive science and what it shows knows that p (for some proposition p), then there must be a
about how the brain does mathematics. The major obstacle causal chain that connects S suitably with the fact that
to giving such an account has been the assumption that makes p true. This fact—the truthmaker for p—must be
mathematical objects are abstract. Philosophers do not realized in the natural, spatiotemporal world somewhere. If
completely agree on the notion of what it is for an object to mathematical objects are abstract, then the truthmakers for
be abstract. The truth may well be that ‘abstract object’ is a mathematical truths will not lie in the natural realm. This
cluster concept that is largely defined by opposition to would be a clear violation of naturalism, which D.M.
features associated with ‘concrete object’. In the western Armstrong helpfully characterizes as the view that ‘space-
philosophical tradition starting with ancient Greek time is all there is’ (Armstrong 1997:5).
philosophy, the notion of an abstract object does have a Benacerraf’s problem is posed as a problem for realists
paradigm: the Platonic form. Platonic forms are ideals that about abstract objects. Such realists believe abstract objects
exist in an intelligible realm, outside of the concrete, entities exist independently of the human mind and that
material spatiotemporal order. statements about them have a determinate truth-value even
These considerations suggest that the contrast between if that truth-value is yet to be discovered. To be sure, it is
abstract and concrete objects is captured by the following: only a problem for realists about abstracta who feel the pull
of naturalism. Thus, although Benacerraf’s objection is
Table 1: The contrast between abstract and concrete posed as a problem for ‘Platonism’ in the generic sense of
‘realism about abstract objects’, it need not apply to Plato’s
Features Abstract Objects Concrete Objects
realism.
Spatial-temporal No Yes
Plato would not have granted the assumption that abstract
location?
objects are causally inert. Plato repeatedly speaks of
Particular? Some Yes (usually)
concrete objects as ‘partaking’ or ‘participating’ in the
Causally active? No (?) Yes Forms, which suggests at least a kind of one-way interaction
Material No Yes between objects and Forms. Furthermore, Plato accepts the

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principle that power to affect and be affected is the mark of between the perfect mathematical form and what is found
reality, and holds that the Forms—despite being immaterial- (perceived, constructed) in the physical world. This problem
- are real.1 This provides Plato with an argument for the is known as ‘the problem of exactness’ and it constitutes a
reality and power of Platonic forms. We may not accept the major objection to Aristotelian realism.3 Of course one
argument, but it is highly plausible that Plato did view his possible solution to the problem is to give up the philosophy
Forms as having some kind of causal power (Fine 2003).2 that generates it. However, it may be possible to retain the
The problem, then, is not that Plato’s forms lack causal spirit of Aristotelian realism and naturalism while solving
power, but that the notion of efficient causality which is our the problem. That is what we will suggest. We distinguish
contemporary scientific notion does not apply to the forms. between three ways to solve the problem:
Plato’s realism is incompatible with scientific naturalism. (a) inexactness theory: that geometry is about the real
If we are to be naturalists, how then do we solve the shapes of things: a cartwheel doesn't have an exact circle
Benacerraf problem for mathematics? Hartry Field shape, but it does have an exact near-circular shape, and
(1989:25) points out that the Benacerraf problem survives one can explain why studying circles is relevant to
even an objection to the causal theory of knowledge. The studying near-circles;
Benacerraf problem can be re-instated using the reigning
(b) the sub-perception theory, that there are perceivable
epistemological theory, such as reliabilism. According to
shapes that are perceptually indistinguishable from perfect
one version of reliabilism, if S knows that p, then there must
circles;
be a reliable connection between the subject S and the fact
that makes p true. Once again, the problem arises that there (c) "rectification" theory: where the mind actively extracts
is no explanation for a subject’s reliable connection to facts the perfect from the visibly imperfect shapes of things.
about abstract objects. These three possible solutions to the problem can be jointly
Nominalism is the view that there are no abstract objects. maintained and are mutually compatible. In what follows
However, nominalism is not an attractive solution to the we will especially emphasize solutions (b) and (c), although
Benacerraf problem. Putnam (1971) argued persuasively we should be understood to accept (a).
that nominalism lacks the resources—the notion of a
linguistic type, which is after all a universal that transcends The problem of exactness in ancient philosophy:
its concrete instantiations--- to articulate its very doctrine. Plato and Aristotle
Aristotelian realism recommends itself as a variety of Plato himself was reluctant to locate mathematical forms in
realism suitable for naturalism. In contemporary the physical, sensible world because the perfection and
metaphysics, the position of Aristotelian realism is precision of mathematical forms seems unparalleled aby
represented by D.M. Armstrong’s theory of universals, many of the real-world exemplars of mathematical forms.
which holds that they are immanent in the world. Applied In the Phaedo, Plato notes that sensible, concrete objects in
to mathematics, Aristotelian realists hold that many basic the material world often fail to instantiate the perfect
mathematical entities (patterns, properties, facts, and mathematical forms found in the intelligible world of the
objects) are instantiated in the natural world. As a forms.4 For example, at Phaedo 74a-c, Plato says that two
consequence of this metaphysics, Aristotelian realism sticks will not instantiate equality as perfectly as the form of
promises to give a naturalistic account of mathematical the Equal itself. Presumably we cannot be sure that two
knowledge. On this account mathematical knowledge is sticks that look equal are actually equal, because our sense-
grounded in perception of patterns (universals) in the world. perception may not be able to discriminate between small
A major obstacle to locating mathematical patterns in the differences in size. Plato regards judgements about
world is that, at least in some cases, the patterns do not mathematical form made on the basis of perception as
appear to be there exactly. There seems to be a ‘mismatch’ inherently less precise and prone to inaccuracy.
It is no accident that Plato’s examples are geometrical, not
1
Plato entertains this line of thought in the Sophist at 247E. arithmetical. The problem of exactness seems to have more
See F. Cornford (trans.), (1957). Plato’s Theory of Knowledge: of a bite in geometry than arithmetic. It is plausible to think
The Theateatus and Sophist of Plato, New York: Macmillan, 234. that arithmetical forms are precisely instantiated. For
The principle has been called ‘the Eleatic Principle’ in example, a certain flower has an exact number of petals, say
contemporary literature on the grounds that Plato has the Eleatic five petals. A certain book has an exact number of pages,
stranger enunciate it. However, there is no reason to think that
such as two-hundred and twenty-nine pages. Once a sortal
Plato did not accept it.
In contemporary metaphysics one is more likely to find the concept is supplied, we can count out a precise number of
Eleatic principle—that to be real is to have causal power—used as units of an item that falls under the appropriate sortal
an argument against the existence of abstract objects (as in
3
Armstrong 1997: 41). The related problem of ‘the perfect circle’ is discussed (not
2
Rosen (2009) suggests that the distinction between abstract and using the label ‘the problem of exactness’) in F. Copleston, (2003).
concrete does not go back to Plato’s philosophy, but probably History of Western Philosophy I, Ancient Greece and Rome, New
originates with Locke’s transposition of the grammatical York: Continuum Press, pp.297ff.
4
distinction between abstract terms (like ‘whiteness’) and concrete Plato, (c.380BC), Phaedo. In J. Cooper (ed.), 1997, Plato’s
terms (‘white’) onto the realm of ideas. Complete Works, Indianapolis: Hackett.

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concept. In the case of geometrical forms, however, we deal neo-Aristotelian view has a lot to offer (Franklin 2009). In
with continuous variation of curves in (mathematical) space what follows, we will focus on how cognitive science
rather than with discrete differences in quantity. There supports solutions (b) and (c) to the problem of exactness.
seems to be plenty of scope for some patterns to fall short of
ideal mathematical patterns. A table-top may not be Cognitive science to the rescue?
perfectly square, as its edges may be bumpy.5 An artist’s Perhaps cognitive science can help us grapple with the
drawing of a house, or a geometry teacher’s drawing of a problem of exactness. Giaquinto (2007) develops an account
triangle on the blackboard, may be imperfect but sufficient of the epistemology of mathematics that goes some way to
to convey the appropriate ideas to their audience. Perhaps solving the problem of exactness; moreover, he does so in a
some geometric forms will be perfectly instantiated, but for way that draws on the psychology of perception. In what
those that are not the problem remains to give an account of follows here we endorse his solution to the problem of
how the brain recognizes in the imperfect illustration the exactness and point out its limitations.
perfect geometrical form. Consider the perfect square. How do we get the geometric
Aristotle is well aware of the objection from exactness. concept of a perfect square? Not by mere perception. It may
He notes himself that a hoop in reality will not touch a well be that we only come into contact with imperfect
straight edge normal to it just at a point as it is supposed to squares. For example, perhaps the squares in Susan’s
do in geometry: homemade brownies (biscuits) are not really square. The
For neither are perceptible lines such lines as the edges are not perfectly straight, or the symmetry isn’t quite
geometer speaks of --for no perceptible thing is straight or right. Nonetheless, an encounter with merely imperfect
curved in this way; for a hoop touches a straight edge not squares may suffice for us to acquire the concept of a
at a point, but as Protagoras said it did, in his refutation of perfect (geometrical) square. This geometrical concept may
the geometers…’ (Metaphysics B2, 997b34-998a6).6 in turn structure our perceptual experience so that we take
The philosophical problem, then, is that mathematical truth ourselves to be experiencing a perfect square. As Giaquinto
seems to be about exact mathematical objects with exact explains,
properties. Some idealization and approximation is involved It can also be part of experience that a square is perfect.
in moving from the hoop of our everyday experience to the Since there is a finite limit to the acuity of experience,
perfect circle of geometry. there are lower limits on perceptible asymmetry and
The problem with idealization is that it is not always perceptual deviation from (complete) straightness.’
truth-preserving. If Don Quixote is in reality an old man, (Giaquinto 2007: 28).
then his idealized conception of himself as a young knight is Asymmetry or other imperfections that fall beneath our
actually false. Similarly, if the earth is actually a lumpy threshold of perceptual discrimination will not be perceived.
oblate spheroid and not a perfect sphere, then it is actually Call this view ‘the sub-perception theory’. The idea is that
false that the earth has the properties properly attributed to a we can perceptually experience a perfect mathematical form
sphere (such as every point on its surface being equidistant even if objectively the form is not perfectly instantiated in
from its centre). If we engage in mathematical deduction nature. Curiously enough our perceptual limitations enable
concerning perfect objects (perfect spheres and the like), us to experience, as it were, perfect geometrical forms.
then there is no guarantee that the result of the deduction There is thus no need to be committed absolutely to the
will perfectly apply to anything in the real world! Yet the existence of perfect forms in nature: it is enough if the
beauty of the Aristotelian view is that it supposed to offer a forms in nature approximate mathematical forms.
straightforward explanation of how mathematics applies to To be sure, there are cases of imperfection that do not fall
the real world, and how mathematical knowledge is beneath the threshold of perfection: they are noticeably
obtained by learning about features of the world. imperfect. We can speculate that such cases—such an oval
Can Aristotelian realism survive? Some philosophers (rather than a perfect circle) or a shape that fails to be an
would say ‘No’. Stewart Shapiro regards the problem of enclosed triangle—are something that we can either learn to
exactness as a very serious problem for Aristotelianism and recognize as approximating a perfect shape but failing in
Platonism (Shapiro 2000: 70) However, we still think the some respect, or else we can learn to recognize them as
perfect exemplars of some new kind (ovals rather than
5
The neo-Platonists blamed matter for failing to receive form circles, for example). Reflection on such cases might make
perfectly in some cases. See Glenn Morrow (ed. and trans.) (1970). some philosophers conclude that the debate between
Proclus: A commentary on Euclid’s First Book of Elements, Platonists and Aristotelians focuses too much on the notion
Princeton University Press, 1970, reprinted in 1992 with an of pre-existing ‘geometrical forms’ that are ready to be
introduction by I. Mueller. imposed on the world. The aim of proper Aristotelianism,
6 Similar passages: Meta VII.10, 1035a25-b. ed. J. Barnes, (1984)
though, is to discover those forms already instantiated in
“Metaphysics” (based on a translation by WD Ross), The Complete
Works of Aristotle: the Revised Oxford Translation, vol. II,
the world.
Princeton University Press. For discussion of the problem of Some allowance has to be made for imperfect
exactness in Aristotle, see R. Pettigrew, (2009), ‘Aristotle and the instantiations of geometrical forms. We have suggested that
subject matter of geometry’, Phronesis vol. 54. an appeal to perceptual limitations can help the Aristotelian
Article DOI: 10.5096/ASCS200939 256
ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science

to explain how acquisition and knowledge of geometrical A Dangerous Concession? In his seminal book,
concepts is possible. Mathematical Knowledge (1984), Philip Kitcher discusses
Another psychological phenomenon that helps in solving the problem of exactness in relation to Kant’s constructivist
the problem of exactness is the ability to the visual system philosophy of geometry. Kant’s view of geometry is that it
in the brain to organize perceptual scenes using gestalt is a rule-governed construction that makes it possible for the
principles. Consider the famous Kanizsa triangle: geometer to ‘discern the universal in the particular’
(A714/B742).8 That is to say, the geometer draws figures
that represent universal geometrical patterns, and considers
the features of those figures that would be common to any
instantiation of the pattern. For example, if a teacher draws
a triangle with white chalk, the whiteness of the chalk is
irrelevant to the figure of the triangle. Only the purely
geometrical properties are relevant: these are the angles and
their total sum, the lengths of the three sides, and the
arrangement of the sides in an enclosed figure. If the figure
is used properly, these geometrical properties will be present
[Figure 1] in any figure of the triangle. Kant’s philosophy of geometry,
as with any such philosophy that gives perception (intuition)
When we look at these patterns, we seem to see an a central role has to explain how the geometrical figure that
inverted and very bright white triangle covering a less bright is the object of perception possesses exactly the properties
upright triangle. The inverted triangle has illusory of the ideal geometrical concept. As Kitcher puts it, the
contours—its edges are not really there. Yet somehow the problem is that “we cannot assume that mental perception
brain is tempted to see the bright inverted triangle as having will give us exact knowledge even of the particular figures
edges of its own. Perhaps the black lines of the upright we construct” (Kitcher 1984: 51).9
triangle ‘spread’ or ‘smear’ a bit making the upright triangle How might a Kantian respond to the challenge? Kitcher
appear to be a grayish white and therefore darker than the suggests that the Kantian can appeal to the fallible and
bright white inverted triangle on top. The ‘top-down’ limited nature of our powers of perception. In particular,
imposition of form of the bright white triangle depends also “We should concede that we might not be able to
on lower-level perceptual phenomena, such as the visual distinguish a straight line from a curved one” (Kitcher 1984:
system registering the dramatic shifts in colour across 51). This response is essentially the same as the ‘sub-
boundaries.7 The Kanizsa triangle is a good example for the perception’ view we found in Giaquinto (2007) and which
philosopher of geometry to consider because it is a case we have endorsed as one solution to the problem of
where the brain interprets the visual display in a way that exactness. Kitcher goes on to suggest that this response is “a
adds something geometrically to what is strictly speaking dangerous concession”, because it would destroy the a priori
present in the given visual form. warrant (justification) for the geometrical belief. That is, if
The implications of the preceding observations are that we acknowledge the limits and fallibility of our perception,
geometrical knowledge cannot be entirely a matter of direct we may no longer be inclined to trust perception as a
perception of perfectly instantiated geometrical forms. That reliable source for forming geometrical beliefs.
kind of naïve realism is too naïve. Geometrical knowledge We are not concerned here to defend the a priori nature of
does involve an element of abstraction which allows the the justification of geometrical beliefs. However, we are
perceiver to move away from some of the imperfections or concerned to defend the idea that geometrical knowledge
limitations inherent in a perceptual scene. Perceptual can be obtained by having perceptual experiences. Kitcher
experience does not have to be completely veridical in order attacks both claims in a serious assault on the notion of
to trigger or give rise to geometrical concepts and, in due mathematical knowledge.
course, geometrical knowledge.
If this account is right, then perceptual experience is not
completely veridical, but it is not a hallucination either. Our 8
We are indebted to Quassim Cassam for drawing attention to
perceptual concepts are formed in response to real objects this quotation from Kant. The quotation is from I. Kant, ‘The
with real properties which we perceive but whose flaws Discipline of Pure Reason’, in N. Kemp Smith, ed. (1911),
(deviations from ideal forms) we may or may not perceive. Immanuel Kant’s Critique of Pure Reason, London: Macmillan. A
Our geometrical concepts are in turn closely linked to our crisp and clear discussion of Kant’s view of geometry is found in
perceptual concepts, abstracting away from the Q. Cassam (2007), The Possibility of Knowledge, Oxford:
imperfections of some of the exemplars. Clarendon Press.
9
We wish to acknowledge Giaquinto (2007) and Norman (2006)
for putting us onto the problem of exactness, which is discussed by
Norman (2006) with special reference to Kitcher (1984) in
7
For more on perceptual completion, see L. Pessoa, E. particular at p.121 as well. Our innovation is to illustrate the
Thompson, and A. Noe, Behavioral and Brain Sciences (1998) 21, problem in ancient philosophy and to draw on psychology for its
723–802. solution.

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Kitcher’s objection is misguided. Perception can still be a found a significant correlation between ability to recognise
reliable source of geometrical beliefs even if there is not an and draw geometrical patterns and subsequent early school
absolutely perfect fit between the approximate instantiation mathematical achievement.11
of a form in reality and a perfect ideal geometrical form. One of the interesting aspects of Mulligan’s work is the
What matters is that the process that takes us from way in which her longitudinal studies have exhibited the
perception of geometrical figures in reality to geometrical growth and development of a grasp of pattern
beliefs should be a reliable one. Reliability does not require (‘mathematical form’) in young children. For example,
complete identical replication of an image. Rather what Mulligan’s data set includes the following set of four
reliability requires is this: for a given perceptual input X illustrations of a child’s progress in completing a simple
there is a reliable transform f(X) that takes us to the output geometrical drawing task:
Y, which is the geometrical concept, every time. If
background conditions are fixed, then perception can (i) (ii)
provide us with a reliable means of transforming perceptual
intake into geometrical concepts and representations.

Platonism by the back door? Someone might object to our

view that it looks like Platonism by the back door. After all, (iii) (iv)
we have admitted that in many cases geometrical forms are
not perfectly instantiated in the world. We have also hung
onto the idea that geometrical forms are perfect. There
seems to be nowhere to locate these perfect forms save in a
Platonic realm. This problem is just an aspect of the general [Figure 3] Taken from Mulligan’s ‘Understanding Young
problem of where to locate perceptual experience, with its Children’s Difficulties in Mathematics Learning’,
Janus-faced nature, both pertaining to the conscious subject Macquarie University presentation, slide 24.
and pertaining to the objective world. Our account is quite
opposed to Platonism in seeking to ground basic Mulligan et. al. rightly interpret the illustrations as
geometrical knowledge in perception rather than in showing the emergence of an ability to grasp structure and
communion with Platonic forms. attend to the geometrically relevant aspects.
It is worth noting that our account is neutral, though, on The relevance of this work to the philosophical problem
whether geometrical concepts are innate or acquired. There of exactness is clear. This data demonstrates that children
is some evidence that geometrical competence is shared learn how to focus on the geometrically relevant and salient
across cultures, which may be taken as an argument for its properties of a figure and to disregard—‘abstract away
innateness by some scientists (see Dehaene et. al. 2006). from’—other aspects of the figure that may be interesting
However, all that is required by our account is that but not relevant for geometry. For example, only in (iv) is
perceptual experience—of some modality, not necessarily there the right number of squares with the right number of
visual, but usually visual--- is a necessary trigger to the edges and approximately the right symmetry.
development of geometrical concepts. Some indirect Drawing figures requires motor skills but also requires
evidence for the necessity of perceptual experience in visual and spatial cognition. The development we see in the
acquiring geometrical knowledge comes from studies of young children’s drawings reflects a refinement of
congenitally blind children. The general consensus is that geometrical concepts: from a sloppy experimental working
while congenitally blind children can acquire spatial and concept to the final pure geometrical concept. The
geometrical concepts through touch, they are delayed in geometrical concept is an ideal and somewhat abstract one
proficiency of skills involving those concepts relative to that appears to distance itself from the messy world of
their non-visually impaired peers (Millar 1994: 133). perception. The idea of ‘abstraction’ figures heavily in
Presumably the explanation for this delay is that vision is Aristotelian philosophy of mathematics and has equally
one powerful but though not absolutely necessary means of been heavily criticized by Frege. Frege’s criticisms were
acquiring geometrical knowledge.10 immediately directed against John Stuart Mill’s crude
empiricism in which a number was identified with a
Relevant Empirical Work in Mathematics collection of ‘pure’ units (Shapiro 2000: 67). However,
Education Mill’s empiricism is a descendant of Aristotle’s empiricism,
Lastly we wish to draw the readers’ attention to some
relevant empirical work in the field of mathematics 11
Mulligan, J., Prescott, A., & Mitchelmore, M. C. (2004).
education. Mulligan, Prescott, and Mitchelmore (2004) Children's development of structure in early mathematics. In M. J.
Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th annual
conference of the International Group for the Psychology of
10
For a summary of the research on blind children’s shape Mathematics Education (Vol. 3, pp. 393-400). Bergen, Norway:
concepts, see S. Millar, Understanding and Representing Space, Program Committee.
Oxford: Clarendon Press 1994/2002.

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ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science

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Citation details for this article:

Newstead, A., Franklin, J. (2010). The Epistemology of
Geometry I: The Problem of Exactness. In W. Christensen,
E. Schier, and J. Sutton (Eds.), ASCS09: Proceedings of the
9th Conference of the Australasian Society for Cognitive
Science (pp. 254-260). Sydney: Macquarie Centre for
Cognitive Science.
DOI: 10.5096/ASCS200939
URL:
http://www.maccs.mq.edu.au/news/conferences/2009/ASCS
2009/html/newstead.html

Article DOI: 10.5096/ASCS200939 260