EMS Newsletter June 2010 - European Mathematical Society ...

Then there is a higher kind of model, which Manin
refers to as belonging to their aristocracy. These are referred
to as ‘theories’ and correspond to the second level.
The purpose of a theory is ultimately to explain. Simplicity
and beauty are valued aspects of a theory and often
indicative of it being ‘true’. The notion of truth does not
apply to a plebeian model; it is but a convenient representation,
something the Catholic Inquisition understood
well when it accepted the helio-centric view as merely a
convenient mathematical model simplifying computations.
With theories we think otherwise; somehow they
are supposed to reveal the inner workings of reality. Of
course beauty and simplicity are not the only criteria on
which the truth of a theory is judged; it also has to be congruent
with the so-called facts. However, if the theory is
beautiful enough, discrepancies with facts can be thought
of as artefacts of reality and not of the theory itself. Dirac
had this attitude and he was always ultimately vindicated.
Theories are definitely higher in the hierarchy; the
very conception of a mathematical model requires some
underlying theory. I would personally propose that the
Stokes equation is a mathematical model while the Maxwell
Equations constitute a theory. Much more comes
out of the latter than is put into it. They have a value
beyond their predictions. As an example, the invariance
group foreshadowed special relativity. To model-builders,
mathematics is a convenience, a kind of language, only
useful as far as it is applicable. To theory-builders, on the
other hand, mathematics does have an independent reality
that underlies manifest reality –a Platonic view; it is
deeply satisfying to mathematicians, regardless of their
interests in physical applications. It is noteworthy that
mathematical theories only exist in physics, which may
explain that of all sciences, only physics has applications
to mathematics, something we will return to below.
Finally, there is an even more rarefied way mathematics
can exert its influence and that is through metaphor,
hence the very title of this collection. A metaphor should
be a stimulation to thought, not a substitute for it. A metaphor
by its very nature should never be taken literally,
as it then collapses and becomes merely silly. In particular,
metaphors are naturally evocative and cannot be manipulated
– hence the meaning of the initial comments
on philosophy. As an archetypical example of a mathematical
metaphor Manin suggests the idea of the brain
as a computer. Metaphors are not theories but if they are
potent they may inspire the formation of theories. Theories
are scientific; metaphors are philosophical.
Physics is another magnet of compelling interest and
Manin claims that no mathematical development of the
20th century has matched the revolutionary change of
paradigm characterizing the fate of classical physics over
the same period. This is a fact that should come as no
surprise, he remarks, except possibly to a few ‘autistic’
mathematicians. Mathematics and physics used to be intimately
connected, from the time of Newton until the
end of the 19th century, but then there was a split and
a divergence, after which neither camp became particularly
interested in what the other was preoccupied with,
a cultural rift that survives to this day. Paraphrasing the
Book review
words of Manin, the mathematicians became obsessed
with our relation to thought, while the physicists with our
relation to reality. The former led to a kind of neurotic
introspection and a subsequent hygiene of precision and
formal reduction, while the latter led to flights of fancy, at
least temporarily unfettered by precision and rigour.
In addition, the intuitions of physicists have recently
proved more fertile in solving mathematical problems
than the techniques of mathematicians, something the
latter, to their credit, have freely acknowledged. In fact,
as indicated initially, some parts of physics, e.g. string
theory, have had more significant applications to pure
mathematics than they have had to physics and its ostensible
subject – physical reality. Why this should be
is still something of a mystery. Some parts of modern
physics, such as relativity theory, have been successfully
mathematized and can be, but for the rigid boundaries
of academic disciplines, acceptably classified under the
mathematical banner. In contrast, quantum theory, which
is far more influential and ‘useful’ than general relativity,
is still intellectually marred by internal contradictions
and an incompatibility with the latter. This does not seem
to really bother the general physicist. The closer we view
the material world, the less solid and commonsensical it
appears. It is a world that can only be approached and
described mathematically, i.e. in the sense of using mathematics
as both a language and as a method of manipulation.
The reason for this growing effervescence, according
to Manin, is that in quantum physics the observer can
no longer be separated from what he observes, unlike
the classical paradigm in which one could (at least as a
thought experiment) isolate pieces of the world and view
them as closed systems, in which one was able in principle
to control every variable (i.e. its position in an idealized
phase-space), as well as its value, without affecting it.
It is clear that when it comes to pure adventure, modern
physics has provided a far more thrilling ride than modern
mathematics.
Language is another one of Manin’s preoccupations.
Are humans defined by language? In fact, are we simply
social creatures to whom individuality is just a consequence?
In other words individual consciousness would
be impossible without a shared language? Karl Popper
claims that sociology is not applied psychology, that it
predates psychology and, in fact, is a prerequisite for it.
There is a natural evolutionary explanation for this, as
various manifestations of social cohesion exist among
other mammals. Clearly modern humans evolved from
humanoid creatures in which a strong social cohesion was
already present. Organisms are not just genes; they are
also part of cultural traditions and contingencies. It takes
more than genes to make up an organism. The extinction
of a species also involves the rupture of a continuous tradition
with roots going back to the beginning of life (and
beyond), a tradition that cannot be encoded and which
‘just happened to be’. In particular it is impossible to resurrect
extinct animals simply by reviving their DNA. It is
also this sense of language being something autonomous,
subjective maybe from the point of view of humanity but
objective as far as the individual, which makes sense of
EMS Newsletter June 2010 59