2 The Meaning of Understanding
27
for the material we wish to cover later. In the West, Aristotle, whose work
inspired Euclid, Archimedes, Apollonius and many others, pursued a bottomup approach to rational thought. They began from simple concepts that could
be accepted readily (axioms), and deduced more complicated consequences
by following strict rules of reasoning. The geometry of Euclid that we all
learned in high school is probably the best-known example of this approach.
A phrase such as the “strict rules of reasoning” might give the impression
that these rules are universally understood and applied in all daily human
intercourse. This is certainly not the case. As an example of logic, consider
the two statements: “no desire is voluntary” and “some beliefs are desires”.
The logical consequence of these two statements is the conclusion: “some
beliefs are not voluntary”. The underlying logic of this thought process can
be extracted and formalised by writing it in the form:
“If no Q is R and some Ps are Qs, then some Ps are not R.”
Here Q is shorthand for “desire”, R for “voluntary”, and P for “belief ”. The
same logic is applicable to countless other examples. For instance, if Q =
“flyer”, R = “pig”, and P = “bird”, the above logical statement leads to: “If
no flyers are pigs, and some birds are flyers, then some birds are not pigs.” It
might be tempting to cry out: “but surely no birds are pigs.” 2 While undoubtedly true, this stronger conclusion cannot be deduced from our two premises.
Such is the uncompromising nature of formal logic.
So confident was Aristotle in the reliability of this method that, when he
applied it to the physical world, he seldom bothered to check whether his
inferences were in accord with actual reality. For instance, he asserted that a
moving body in a vacuum came to rest immediately after the force that had
induced the motion was removed. In air, he believed the motion persisted
because the surrounding air sustained the force for a while, until it ultimately dissipated. This “fact” was accepted by scholars until finally refuted
by Galileo, almost two millennia later. It is the combination of logic and
observation, beginning with the work of Galileo, that is the characteristic of
modern science.
Before leaving this topic, there are several questions that are worth posing,
even if there is little hope of agreeing on the answers:
(1) Why does our mathematical logical approach explain the observed
physical world so well?
2 Strictly speaking, we should define what we mean by birds, pigs and flyers before introducing them
into such a logical statement.
27
for the material we wish to cover later. In the West, Aristotle, whose work
inspired Euclid, Archimedes, Apollonius and many others, pursued a bottomup approach to rational thought. They began from simple concepts that could
be accepted readily (axioms), and deduced more complicated consequences
by following strict rules of reasoning. The geometry of Euclid that we all
learned in high school is probably the best-known example of this approach.
A phrase such as the “strict rules of reasoning” might give the impression
that these rules are universally understood and applied in all daily human
intercourse. This is certainly not the case. As an example of logic, consider
the two statements: “no desire is voluntary” and “some beliefs are desires”.
The logical consequence of these two statements is the conclusion: “some
beliefs are not voluntary”. The underlying logic of this thought process can
be extracted and formalised by writing it in the form:
“If no Q is R and some Ps are Qs, then some Ps are not R.”
Here Q is shorthand for “desire”, R for “voluntary”, and P for “belief ”. The
same logic is applicable to countless other examples. For instance, if Q =
“flyer”, R = “pig”, and P = “bird”, the above logical statement leads to: “If
no flyers are pigs, and some birds are flyers, then some birds are not pigs.” It
might be tempting to cry out: “but surely no birds are pigs.” 2 While undoubtedly true, this stronger conclusion cannot be deduced from our two premises.
Such is the uncompromising nature of formal logic.
So confident was Aristotle in the reliability of this method that, when he
applied it to the physical world, he seldom bothered to check whether his
inferences were in accord with actual reality. For instance, he asserted that a
moving body in a vacuum came to rest immediately after the force that had
induced the motion was removed. In air, he believed the motion persisted
because the surrounding air sustained the force for a while, until it ultimately dissipated. This “fact” was accepted by scholars until finally refuted
by Galileo, almost two millennia later. It is the combination of logic and
observation, beginning with the work of Galileo, that is the characteristic of
modern science.
Before leaving this topic, there are several questions that are worth posing,
even if there is little hope of agreeing on the answers:
(1) Why does our mathematical logical approach explain the observed
physical world so well?
2 Strictly speaking, we should define what we mean by birds, pigs and flyers before introducing them
into such a logical statement.