Commentary on Theories of Mathematics Education

606 N. Sinclair
was responsible for presenting to him the most fruitful and beautiful combination of
ideas it could.
Note that for Poincaré, as for Nimier, the unconscious is not exactly the same
thing as that theorized in more recent cognitive science such as Lakoff and Núñez
(2000), who argue that most thought is unconscious. In Freud’s terms, these authors
are describing the subconscious, which refers to the thinking below the level of conscious
awareness, and which is more accessible and not actively repressed. Lakoff
and Núñez access the subconscious by looking at human speech and gesture; the
psychoanalyst, in contrast, must engage in different methodologies—for Freud, this
involved analyzing dreams, but it can also involve analyzing slips of tongue, memories,
and, as shall be seen later, mathematical errors. Evidence for Poincaré’s use
of the Freudian unconscious can be seen in his own description. For example, he
frequently uses the term “subliminal ego” to describe the unconscious mind, and
contrasts “le moi inconscient” with “le moi conscient.” The subliminal can evoke,
but never cross the threshold of the conscious. Poincaré even presages the views
of Jacques Lacan about the unconscious: “il est capable de discernement, il a du
tact, de la delicatesse; il sait choisir, il sait deviner” (it is capable of discernment,
it has tact, sensitivity; it knows how to choose, it knows how to guess) (p. 126, my
translation).
Poincaré goes on to assert that the new combinations that come to the surface are
the ones that most profoundly affect the mathematician’s emotional sensibility. The
mathematician is sensitive to the “elements that are harmoniously disposed in such
as way that the mind without effort can embrace their totality while at the same time
seeing through to the details” (p. 25). The notion of embracing a totality, or making
whole, described by Poincaré surfaces time and time again in the admittedly small
corpus of mathematicians writing about mathematics. Indeed, the mathematician
Philip Maher (1994) offers an explanation for the ontogenesis of the characteristically
mathematical activity of making whole by drawing on Lacan’s notion of the
mirror stage, which describes the transition state during which an infant comes to
recognise her reflection in the mirror as being her own. At this point, the “infant sees
a totality which he or she can control through his or her own movement” (p. 138).
Maher goes on to articulate the significance of the mastery of the body, by anticipating
“the infant’s later real biological mastery—and intellectual mastery, too, if
e.g., a future mathematician. The mirror image, then, in representing to the infant a
total and unified whole (caused and controlled by the infant) prefigures the mind’s
desire to make whole” (pp. 138–9). Maher’s conclusion resonates strongly with the
work of Poincaré (1908) who speaks of an aesthetic rather than psychological need:
“this harmony is at once a satisfaction of our aesthetic needs and an aid to the mind,
sustaining and guiding” (p. 25).
Maher’s account proposes two other explanations of characteristically mathematical
activity. One involves the use of psychoanalyst Donald Winnicott’s theory
of transitional objects to explain the way in which mathematicians are attracted to
the Platonic idea of mathematical reality. Nicolas Bouleau (2002) alsodrawson
Winnicott to interpret the unconscious experience of the mathematician—pleasure,
desire and sublimation—as being analogous to the experience of the child playing at