BERNARD GROSSMAN
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Piaget's work can once again be used as a focal point. His theory of
cognitive development is based upon internal mental structures
which accommodate the observed world while assimilating external
reality to the existing mental structures. While the theory's level of
abstraction approaches that of theoretical physics, simplification
again occurs as a result of the high level of symmetry . Symmetry
considerations play an important role in the child's understanding of
concepts like volume, number, logic, space or time. In a famous ex–
ample , understanding the concept of volume means being able to
compare the amount of water in a short fat cup with the same
amount of water in a tall thin one. According to Piaget , a mature
concept of volume is invariant under the distortions imposed by
changes in physical shape. Within this new way of thinking about
the problem , one should replace the notion of a concept or of an ob–
ject by the symmetry associated with it. This conceptual framework
also sheds light on the concept of a geometry without points . The
usual notion of a space and time made of points is replaced by the in–
variants (like rotation) of the loops or strings in space and time . The
search for all the invariants of string theory has been one direction
for new formulations of string theory.
Equally important in theoretical physics is the concept of sym–
metry breaking . As an example, consider the places one sets for a
circular dining table .
If
the set table is going to be symmetric, one
can place all the knives on the left of the plates or on the right.
However, once a choice is made for one setting, that determines the
side for all the other settings . Neglecting consideration of etiquette,
both choices are equal. One might say that the laws of table setting
(neglecting etiquette) have a left-right symmetry (what's called a
mirror or parity symmetry), yet the actual physical setting requires a
choice which irrevocably breaks this symmetry for the table .
The equations of physics also have certain symmetries. Some of
these reflect symmetries that are actually observed in the universe,
such as light. Light is associated with the electromagnetic interaction
characterized by a fundamental symmetry of the universe, that
resulting from the conservation of charge . Therefore, the conserva–
tion of charge carried by a current in a battery is a consequence of
the existence of an electromagnetic symmetry. However, some of
the symmetries of the equations of physics are, in fact, not sym–
metries of the observed universe, much as the mirror symmetry of
the table was not a symmetry of the setting. One example of this is
mirror symmetry in nature. It is observed that the results of certain
