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Horst Steinke
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correct to speak of the “axiomatic method”
as it is not at all
uniquely followed only in “geometry”. And even the term “axi-
omatic” has reference only to a moment of the total process.
The more comprehensive and accurate terms would be “deduc-
tive”, “deduction”, or even more descriptively, “deductive logic”.
Now, the most fundamental property or characteristic of deduc-
tive logic is that it is “truth-preserving”. When the rules of logical
inference are followed, they guarantee that the conclusions
drawn from certain premises are correct
83
. So the only other key
requirement is the provision of acceptable premises; they may be
provided by explicit definition for the subject matter at hand, or
ideas that are taken for granted in the circumstances, or, often, a
mixture of both. The truth-preserving quality of deductive logic
had a powerful hold on early modern thinkers
84
. Spinoza was no
exception when he wrote that «a doctrine [that God’s judgments
far transcend human understanding] might well have sufficed to
conceal the truth from the human race for all eternity, if mathe-
matics had not furnished another standard of verity»
85
. In Spino-
za’s philosophy, this can be said to have been true in two ways:
first, in the sense of providing “cognitive certitude”, and second-
ly, mirroring the essence and deepest structure of reality
86
. It is of
course true that modern students of Spinoza, including those
who find his philosophy persuasive and congenial, have taken
exception to the validity of his “logico-geometrical” reasoning in
Ethics
. As one scholar stated: «It is generally acknowledged that it
is impossible in the
Ethics
to deduce geometrically any of the par-
ticular beings of the natural world»
87
. It is often apparent that
important steps in the chain of deduction are absent; in Spino-
za’s defense, however, on this level, it could be pointed out that
it is not uncommon in mathematics either to leave out many in-
termediate steps in proofs
88
, and it is possible often to add to the
logical coherence of Spinoza’s line of reasoning through auxiliary
constructions
89
.