Modern Age - Summer 2014 - 60

MODERN AGE

has the character of a fiction, a mere symbol,
for the real nature [is] unknown in itself"
(15). This is radically false: the structure is
"unimaginable" because it is foreign to our
ordinary experience, not because it is ontologically deficient in itself. With time many
of the structures hypothesized by science
have been actually seen, and even those that
cannot be seen because they are too small or
too distant become familiar through measurement, and thus their ontological content
becomes manifest. Dark matter, for example,
is almost certainly real, but we can never see
it; nonetheless, we can investigate it and try
to determine its characteristics.
Similar strictures apply to the author's
claim that "physico-mathematical science
is not formally physical science, although
it is physical as regards the matter in which
it verifies its judgments, and although it is
oriented toward physical reality and physical
causes as the terminus of its investigations.
It does not, however, aim to grasp the inner
ontological nature of its subject matter" (12).
To anyone in the physical sciences, this is
simply absurd. Physical science and scientists
are in fact interested in the nature of their
subject matter, in all respects; it's just that
that nature is accessible to us-insofar as it is
accessible at all-only through the language
of mathematical physics. Through that and
experimentation, we have to develop whatever knowledge and intuition we can possess
of the nature of the objects and processes
we study. And indeed, we have developed
an intuition of atoms that did not exist two
hundred years ago, and perhaps two hundred years from now our intuition of forces
of nature and their unification will be better.
But this will not come from contemplation
but further research and study by scientists.
As already noted, what science cannot do by
itself is to formulate a comprehensive picture
of reality; its role there is restricted to provid60

SUMMER 2014

ing us with the basic material needed to form
an intuitive understanding of reality. It does
not appear that we shall ever attain anything
like a "grasp of the inner ontological nature"
of most of the objects we study, nor is that
really necessary.

ougherty's discussion of mathematics seems rather superficial, and I can
only conclude that his acquaintance with the
subject-especially modern mathematics-
is marginal at best. He attempts to place
Euclidean geometry-supposedly more
intuitable-front-and-center for philosophy,
as is typical for Aristotelian thinkers, with
the justification that it is how we intuit the
world. But this won't do; mathematics has
long since moved beyond Euclidean geometry, however important it was for creating
the enterprise of mathematics as we know it.
And even with respect to Euclidean space,
anything beyond three dimensions ceases to
be intuitable, though it can be handled quite
well with the tools of mathematics.
This is not just an academic exercise-
large parts of the modern world function on
the basis of higher-dimensional Euclidean
spaces. For example, all types of regression
analysis utilize higher-dimensional Euclidean space. The key is to have a structure for
the space and a metric. Everyone is familiar with the Euclidean metric x + y in two
dimensions, which measures distance. But
metrics exist in other spaces, spaces that may
be composed of functions, for example, and
that may be infinite in dimension. These
spaces are no more or less real than Euclidean space, and often serve well to characterize aspects of the world. So when the author
says, "Whereas Euclidean space is directly
grasped by intuition, others of necessity are
referable to the Euclidean notion of space for
their intelligibility," this is wide of the mark.
These spaces share some notions with Euclid2

Modern Age - Summer 2014

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