Spinoza: consistency, completeness, and independence

Spinoza wrote his Ethics in the form of a mathematical theory and treatise. He starts with definitions and axioms and proceeds to prove all manner of propositions from them. One question is whether he actually proved his propositions. However, since it is written in the form of a mathematical theory we might also ask some basic questions regarding mathematical logic. Is it consistent, complete, and are the axioms independent of each other.

Should some of his proofs be flawed, that does not necessarily show that his theory is inconsistent. A valid proof of his flawed propositions may yet be forthcoming by some ingenious soul. However, another ingenious soul might show that he has proved a proposition and its negation making the theory inconsistent and thereby being able to prove any proposition from it.

The completeness of the theory seems problematic also. Can he prove every true proposition from his axiom system? That gets into sticky questions about models of the universe that satisfy his axiom system.

Are his axioms independent of each other? Can one prove an axiom from the other axioms? Let’s say his axioms are independent: you cannot prove an axiom from the other axioms. Suppose I negate one of them such as one might negate the Euclidean parallel postulate to arrive at consistent non-Euclidean geometries. What strange and counterintuitive propositions might arise from the theory?

When one does philosophy in the form of a mathematical theory, one damn well better be able to answer some basic mathematical logic questions about what one has done. If one can’t do that, one has a real problem with one’s credibility.

Of course, in Spinoza’s case, there is so much more than the logic contained in it. There are all those things in Ethics regarding the feeling of what happens.

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Published in: on March 20, 2008 at 11:32 am  Leave a Comment  

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