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Faith and Axioms?
Posted on: January 23, 2007 - 12:44am
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Faith and Axioms?
I've been wondering this: Do axioms need faith to be accepted? Or is faith the wrong word here?
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I would say that axioms are not accepted on faith but chosen in order to produce a mathematical or logical system which has practical value. For example, we accept that -(-x)=x for all reals not because we have faith that it is true but because we want to define "-x" in a non-contradictory manner which makes addition possible.
No. Depending upon what you are are referring to, axioms of metaphysics are atomic statements presumed within a system to make that system work. Axioms are defend through retortion.
Anyone who holds that 'axioms are held by faith' is basically committing two errors:
1) He's conflating theistic faith (non contingent belief) with an 'assumption' required for a system.
1a) Assumptions required for a system do not suppose any more than is required for the specified system
2) He or she is expressing a fundamental ignorance of what axioms are, or are intended be...
However, to proceed, I would need you to specify what you mean, specifically, by axioms.... I'd also suggest that anyone who brings up 'faith' in such a discussion ought to be avoided like a schizophrenic with a switchblade...
"Hitler burned people like Anne Frank, for that we call him evil.
"God" burns Anne Frank eternally. For that, theists call him 'good.'
There are basically three meanings and axiom can take: An axiom as used by logicians, an axiom used by metaphysicians, and an axiom used by the general public.
An axiom, as used by logicians, are simply propositions which do not get derived from something else, but for which other propositions are derived from. So, what justifys a good axiom? Its highly intuitive plausibility. The Law of Non-Contradiction, is highly intuititve. Except for a few logicians, everyone accepts this. Can a proof be given for it, in a non-circular way? nope. The goal of a good axiom, is to state some proposition SO obvious that it is hard to doubt. For example, in number theory, there is a mathematical axiom which states that 0 is not the successor or any other number.
Metaphysical axioms are those which lie at the very bedrock of metaphysics. Take Descartes favorate "cogito." This is a metaphysical axiom.
an axiom, as used by people uneducated in philosophy or mathematics, is simply what one takes as a given. Theists, often times, take the existence of God as their "axiom."
In the first two cases, these axioms are not taken on faith. in the latter, it is.
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
Just a pedantic nit-pick. -(-x)=x for real numbers is not taken to be an axiom. The real numbers form a field, and from the field axioms, -(-x)=x is proven.
Mathematical axioms are much different from axioms in other fields, as was pointed out. There are some axioms that are not very intuitive (like the axiom of choice) but are used mainly because it makes proving theorems so much easier.