Modern Age - Summer 2014 - 61

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ean geometry, but their "intelligibility" is
strictly a function of their mathematical
structure. Very little of modern mathematics
can be immediately grasped, but with time
and a lot of concentrated thought, it can
be understood. We intuitively understand
Euclidean concepts because they happen to
fit our day-to-day world well and we have
used them since childhood; if we spent that
much time in a non-Euclidean world, we
would understand it quite well too.
The author believes it is necessary "to philosophically study the first principles of mathematical science, which alone can provide a
rational account of the true nature of mathematical abstraction and the mental objects
it considers" (12). But what does this mean?
"Such a study would embrace the mutual
relationships of the continuous and discontinuous, the real meaning of surds [irrational
numbers] and transfinite numbers, the infinitesimal, non-Euclidean space, and the validity
of mathematical transcripts of physical reality
such as quantum mechanics and the theory
of relativity" (12-13). This sentence does not
explain anything: the mathematical definition of continuity clarifies what it is, and thus
the discontinuous. Irrational numbers do not
have any "real" meaning: they just are, and
there are lots of them.
There are questions to investigate about
transfinite numbers, such as the truth of the
continuum hypothesis, but not their real
meaning-they too just are. And quantum
mechanics is not a "mathematical transcript"
of reality but a theory about reality, which
requires the language of mathematics to
be understood. Nonetheless, this points
in the right direction: we can ask metamathematical questions, and since Gödel,
we know that the task of mathematicians is
to investigate the properties of the real structures they create. The language of mathematics clarifies this: "Let A be a Hilbert

space," and then investigate it by proposing
and proving theorems.
The issue of final causality is addressed in
lecture 5, entitled "The Principle of Final
Causality." This chapter jumps around
quite a bit, with other types of causality,
especially efficient causality, discussed at
length. Dougherty correctly points out
that empiricist tradition, including science,
has restricted itself to material and efficient
causality (56), though some have argued that
scientific laws themselves represent a type
of formal causality. Be that as it may, the
important question is the nature of causality and the extent to which science actually
probes causal relations. Causality is an especially difficult problem for the Aristotelian
tradition because of its commitment to a
particular deterministic version.
Though Dougherty obliquely criticizes
Averroës's view of causality, he doesn't address
the relevant aspects of it that have been
defended even in our own time by philosophers of science such as Stanley Jaki, namely
determinism and contiguity, both of which
are severely challenged by the indeterminism and nonlocality of quantum mechanics.
Perhaps the question can be put this way:
do we really have knowledge of something
if we do not know how it acts causally on
other things through its own essence? This
question comes up partly in the context of
final causality. Is final causality necessary to
understand science? Or is it something with
which science needs to be concerned? In fact,
the real question isn't whether final causality
is needed for science (it isn't), but whether
scientific explanations are or can be sufficient
to explain all of reality.
The question of final causality is not of
great interest except in biological contexts,
where it seems that organisms and their
component systems have been designed
to achieve some end. On the other hand,
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Modern Age - Summer 2014

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