Routledge History of Philosophy Volume IV

SPINOZA: METAPHYSICS AND KNOWLEDGE 277
suggest an interpretation which relates this kind of knowledge both to
Descartes’s views about knowledge and to Spinoza’s moral philosophy.
Four features can safely be ascribed to intuitive knowledge.
1 Like reason, it is necessarily true; 84 that is, the truths known by such
knowledge are necessary truths.
2 Like reason, intuitive knowledge conceives and understands things ‘under a
species of eternity’. 85
3 Unlike reason, intuitive knowledge is (as has just been pointed out)
knowledge of particular things. 86
4 Unlike reason, again, intuitive knowledge is, as its name suggests,
‘intuitive’.
It is this last feature of intuitive knowledge which causes most difficulty. The
difficulty arises out of a passage in the second Scholium to Proposition 40 of
Part II of the Ethics, in which Spinoza illustrates all three kinds of knowledge by
a single mathematical example. 87 He takes the problem of finding a fourth
proportional. One is given three numbers, and one is required to find a fourth
number which is to the third as the first is to the second. Spinoza points out that,
in this case, there is a well-known rule: multiply the second number by the third
and divide the product by the first. The use of this rule, however, raises two
questions. First, on what grounds is the rule accepted? And second, is the use of
such a rule always requisite if the problem is to be solved? With regard to the
first question, Spinoza points out that some people may have found the rule to
work for small numbers, and generalize it to cover all numbers. This would be a
case of inductive reasoning, and belongs to the first kind of knowledge,
imagination. Others may accept the rule because they know the proof given in
Euclid VII, 19. This is based on what Spinoza calls a ‘common property’ of
proportionals, and belongs to the second kind of knowledge, reason. But when the
numbers involved are small numbers—say, 1, 2 and 3—there is no need to make
use of a rule. In such a case, Spinoza says, everyone sees that the fourth
proportional is 6; this is because ‘we infer the fourth number from the very ratio
which, with one intuition, we see the first bears to the second’. Or, as Spinoza
says in the Tractatus de Intellectus Emendatione (G ii, 12), we see the ‘adequate
proportionality’ of the numbers ‘intuitively, without performing any operation’.
To see things in this way is to make use of the third kind of knowledge, ‘intuitive
knowledge’.
The problem is that what Spinoza says about the third kind of knowledge
appears to be self-contradictory. On the one hand, he seems to hold the view that
intuitive knowledge is immediate, in the sense that no process of inference, no
application of a general rule to a particular instance, is involved. Yet when he
describes intuitive knowledge in the Ethics he speaks of inferring a fourth
number. However, it may be that one can get some help here from Descartes.
Although Spinoza does not refer to Descartes in the context of the third kind of