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The mechanical calculator was
one of Leibniz’s many inventions. Its
creation is a testament to his interest
in mathematics and logic—fields in
which he was a great innovator.RENAISSANCE AND THE AGE OF REASON
methods to discover the answer,
but had my rational faculties been
better I could also have discovered it
through rational reflection. Whether
it is a truth of reasoning or a truth
of fact, therefore, seems to depend
on how I arrive at the answer—but
is this what Leibniz is claiming?
Necessary truths
The trouble for Leibniz is that he
holds that truths of reasoning are
“necessary”, meaning that it is
impossible to contradict them,
while truths of fact are “contingent”;
they can be denied without logical
contradiction. A mathematical
truth is a necessary truth, because
denying its conclusions contradicts
the meanings of its own terms.
But the proposition “it is raining
in Spain” is contingent, because
denying it does not involve a
contradiction in terms—although
it may still be factually incorrect.
Leibniz’s distinction between
truths of reasoning and truths of
fact is not simply an epistemological
one (about the limits of knowledge),
but also a metaphysical one (about
the nature of the world), and it is
not clear that his arguments
support his metaphysical claim.
Leibniz’s theory of monads seems
to suggest that all truths are truths
of reasoning, which we would have
access to if we could finish our
rational analysis. But as a truth of
reasoning is a necessary truth, in
what way is it impossible for the
temperature on Betelgeuse to be
2,401 Kelvin rather than 2,400
Kelvin? Certainly not impossible
in the sense that the proposition
2 + 2 = 5 is impossible, for the latter
is simply a logical contradiction.
Likewise, if we follow Leibniz
and separate neccesary and
contingent truths, we end up with
the following problem: I can
discover Pythagoras’s theorem
simply by reflecting on the idea of
triangles, so Pythagoras’s theorem
must be a truth of reasoning. But
Betelgeuse’s temperature and
Pythagoras’s theorem are both just
as true, and just as much part of
the monad that is my mind—so
why should one be considered
contingent and the other necessary?
Moreover, Leibniz tells us that
whereas no-one can reach the end of
an infinite analysis, God can grasp
the whole universe at once, and so
for him all truths are neccessary
truths. The difference between a
truth of reasoning and a truth of fact,
therefore, does seem to be a matter
of how one comes to know it—and
in that case it is difficult to see why
the former should always be seen
to be necessarily true, while the
latter may or may not be true.An uncertain future
In setting out a scheme in which an
omnipotent, omniscient God creates
the universe, Leibniz inevitably
faces the problem of accounting for
the notion of freedom of will. How
can I choose to act in a certain way
if God already knows how I am
going to act? But the problem runs
deeper—there seems to be no room
for genuine contingency at all.
Leibniz’s theory only allows for adistinction between truths whose
necessity we can discover, and
truths whose necessity only God
can see. We know (if we accept
Leibniz’s theory) that the future of
the world is set by an omniscient
and benevolent god, who therefore
has created the best of all possible
worlds. But we call the future
contingent, or undetermined,
because as limited human beings
we cannot see its content.Leibniz’s legacy
In spite of the difficulties inherent
in Leibniz’s theory, his ideas went
on to shape the work of numerous
philosophers, including David Hume
and Immanuel Kant. Kant refined
Leibniz’s truths of reasoning and
truths of fact into the distinction
between “analytic” and “synthetic”
statements—a division that has
remained central to European
philosophy ever since.
Liebniz’s theory of monads
fared less well, and was criticized
for its metaphysical extravagance.
In the 20th century, however, the
idea was rediscovered by scientists
who were intrigued by Leibniz’s
description of space and time as
a system of relationships, rather
than the absolutes of traditional
Newtonian physics. ■God understands
everything through eternal
truth, since he does not
need experience.
Gottfried Wilhelm
Leibniz