Aye, causation is still a strangely vexed issue in the philosophy of science (in particular Hume argued for a skeptical position with regards to causation). Though I should ask, what do you mean by 'validity'?Sastrawan said:Just a couple of thoughts, and these haven't been fully thought through yet, so feel free to criticise them any way you want.
1) Science and reason has no greater validity to describe the world than any other form of belief system. It just so happens that science is very adept at describing and predicting the behaviour of physical things in controlled conditions. This is science's primary and only aim. Newton himself said "I make no hypotheses." He did not claim to be able to explain why things happened, only the ways in which they would happen.
I think you may potentially undermine your claims in (2) and (3) with your ensuing argument. Think about the claim that a sentence S (e.g. "I am not proveable in axiomatic system A") is unproveable in a system A but is nonetheless true, or that 'there is an infinite set of unproveable but true sentences'. In what sense can such claims be said to be true? Arguably we need some method of judging mathematical truth independent of the axiomatic system in order to make such a pronouncement. But once you start falling back on mathematical intuition and judgement (on the part of the mathematician, say) you start to invoke notions of platonic categories or objectively, obviously true logical axioms - i.e. the very things you start to deny later in your argument. Thus your argument suggests that it may not even be fully meaningful to say that a sentence is not proveable but is nonetheless 'true'.Sastrawan said:2) Logic itself is founded on axioms which are asserted, much like religion. If Christianity begins with the premise "God exists and He is good", logic is based on things like "If all As are Bs, and all Bs are Cs, then all As are Cs" (not quite, but my technical knowledge of logical axioms is shaky). In fact, 20th century logicians have proven, among other things, that from the axioms of set theory and logic, it is possible to come to a conclusion that is independent of those axioms. That is, the axiom system is incomplete.
3) Another proven theorem is that "there are an infinite number of statements that are true but unprovable" - Kurt Gödel.
[By the way, none of this is my material; it comes from my Maths teacher Dr. Bill Pender, whom you may know from his 3U Cambridge Maths textbook]
What (3) means is that mathematical proof is an insufficient method for reaching mathematical truth.
What I tried to get at by (2) and (3) is to show that logic is not a perfect and complete system in and of itself; let alone one to describe reality.
Another issue worth considering relating to intuitive approaches to verifying mathematical statements (basic logical claims in particular) is that our intuitions can lead us to contradiction (some have argued that we should tolerate some contradictions, notably Australian philosopher Graham Priest). A beautiful example of this is Naive Set Theory which falls prey (much to Frege's dismay) to Russell's Paradox whereby one can construct the set of all sets which are not members of themselves... contradiction ensues. The paradox is not too disimilar to that of the liar, i.e. "this sentence is false". In any case, Naive Set Theory is based on axioms which all have that objective logical ring to them but which nonetheless, taken collectively, allow for the emergence of contradiction. 'So much for intuitive mathematical judgement', one might say.
What if there are objectively true facts but we just can't prove it?Sastrawan said:True. But notice that he has shifted the criterion of value from "objective truth" to "usefulness in specific situations". This, I believe, is a better criterion to use for systems of belief. Objective truth, in my opinion, does not exist. Science is useful for engineering and sending rockets to the moon and making a model of how sub-atomic particles interact and healing people.