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Horst Steinke


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


. 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


. 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»


. 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


. 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


. As one scholar stated: «It is generally acknowledged that it

is impossible in the


to deduce geometrically any of the par-

ticular beings of the natural world»


. 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


, and it is possible often to add to the

logical coherence of Spinoza’s line of reasoning through auxiliary