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If induction cannot be brought within the framework of the syllogism, this
does not mean that it operates without any rules of its own. Mill sets out five
rules, or canons, of experimental inquiry to guide the inductive discovery of
causes and effects. We may consider, as illustrations, the first two, which Mill calls
respectively the method of agreement and disagreement.
The first states that if a phenomenon F appears in the conjunction of the
circumstances A, B, and C, and also in the conjunction of the circumstances C,
D, and E, then we are to conclude that C, the only common feature, is causally
related to F. The second states that if F occurs in the presence of A, B, and C, but
not in the presence of A, B, and D, then we are to conclude that C, the only
feature differentiating the two cases, is causally related to F. Mill gives as an
illustration of this second canon: ‘When a man is shot through the heart, it is by
this method we know that it was the gunshot which killed him: for he was in the
fulness of life immediately before, all circumstances being the same, except the
wound.’
Like all inductive procedures, Mill’s methods seem to assume the constancy of
general laws. As Mill explicitly says, ‘The proposition that the course of Nature is
uniform, is the fundamental principle, or general axiom, of Induction.’ But what
is the status of this principle? Mill sometimes seems to treat it as if it was an
empirical generalization. He says, for instance, that it would be rash to assume
that the law of causation applies on distant stars. But if this very general principle
is the basis of induction, surely it cannot itself be established by induction.
It is not only the law of causation which presents difficulties for Mill’s system.
So too do the truths of mathematics. Mill did not think – as some other empiri-
cists have done – that mathematical propositions were merely verbal propositions
which spelt out the consequences of definitions. The fundamental axioms of
arithmetic, and Euclid’s axioms of geometry, he maintains, state matters of fact.
Accordingly, he had in consistency to conclude that arithmetic and geometry, no
less than physics, consist of empirical hypotheses. The hypotheses of mathematics
are of very great generality, and have been most handsomely confirmed in our
experience; none the less, they remain hypotheses, corrigible in the light of later
experience.
Mill’s assertion that mathematical truths were empirical generalizations was
inspired by his overriding aim in The System of Logic, which was to refute the
notion which he regarded as ‘the great intellectual support of false doctrines and
bad institutions’, namely, the thesis that truths external to the mind may be
known by intuition independent of experience. His view of mathematics was very
soon to be shown as untenable by the German philosopher Gottlob Frege, and
after Frege’s work even those who had great sympathy with Mill’s empiricism –
including his godson Bertrand Russell – abandoned his philosophy of arithmetic.
After MillUs death at Avignon in 1873 an engaging Autobiography was pub-
lished posthumously, and some essays on religious topics. In his essay Theism,
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