1. Do not leave undefined any terms at all obscure or ambiguous.
2. Use in definitions only terms perfectly well known or already explained.
3. Ask only that evident things be granted as axioms.
4. Prove all propositions, using for their proof only axioms that are perfectly self-evident or propositions already demonstrated or granted.
5. Never get caught in the ambiguity of terms by failing to substitute in thought the definitions which restrict or explain them.
Blaise Pascal (1623-1662)Pascal's thinking remains instructive to this day for all forms of analysis.
Source: Pascal, B (1952), On Geometrical Demonstration (On the Geometrical Mind) (R Scofield, Trans). In R M Hutchins & M Adler (Eds), Great Books of the Western World (Vol 33, pp. 430-446), Chicago, IL: Encyclopedia Britannica.
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3 comments:
I am a fan of Pascal, but "Ask only that evident things be granted as axioms" is VERY dangerous. What is 'evident' may well be wrong. For example, one may have a theory of economics which is perfectly valid from a mathematical perspective and in which the axioms are 'evidence' to economists. The theory may be dangerously wrong yet anyone reading Pascal might be forgiven for supposing that the mathematical validity somehow gives credence to its conclusions. It doesn't. It only shows that the conclusions follow from the axioms.
For 100s of years Geometry was based on flawed set of 'evident' axioms. It was also evident that atoms were atomic (indivisible) and that light was a wave in 'the ether' etc etc. Let these be our lessons!
Hi David, the question becomes, "what is self-evident?" Of course, that changes over time as art and science increase our understanding of the universe. I tend to agree pragmatically with Popper that all theories remain theories until they are falsified and/or replaced by more valid and reliable theories. Thanks for your astute comments...
Godel's incompleteness theorem essentially states that, at some point when building a body of knowledge, you must resort to axioms. However, as Dave argues, relying on self-evidence is a slippery slope (and I have to believe that Pascal was probably as aware of that as we are). Self-evidence is often a place to start in defining axioms. However, ultimately, we are forced to demonstrate that our systems of knowledge agree with all of our observations and not only the apparently self-evident types of observations. If our axioms or consequent theories do not coincide with our observations, then something must be adjusted. In some notable cases, it is the axioms which are adjusted.
http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems
http://en.wikipedia.org/wiki/Self-evidence
http://en.wikipedia.org/wiki/Axiom
I like to think of our ultimate reliance on observation for epistemology in the same way that we ultimately rely on observation in language. With language, we can define a dog to be a mammal with particular bone/tooth structure, but then we need to define what a mammal is and what bones and teeth are. Ultimately, some words must be associated in our brains to our observations. Without that, language would be useless and the same can be said for epistemology.
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