The Axioms Of Scientific Investigation

From this source there are three axioms that must be accepted to pursue any 'truths'/knowledge.

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• The First Fact: The fact of our existence. "I exist."
• The First Principle: The principle of contradiction. "A thing can not be and not-be at the same time in the same respect."
• The First Condition: The essential capability of the mind to know truth. "My intellect can reason and discover truth."
  

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The first Condition seems to cover your first axiom, but your axiom might be worth noting as well. After that - well, things can get a little more difficult to pin down since each science tends to go down its own branching paths of assumptions.Further assumptions might beIf an idea has perfect correspondence with something in the external/objective reality world, then the idea is true for that reality.A bit clumsy perhaps, but a short way of suggesting that if I have the idea that it is raining, and all the things that must be happening in the external world that corresponds to the idea that it is raining (eg there is x amount liquid water falling from nearby clouds) then the idea 'it is raining' is true.Of course, we can't always know what all the things that must be happening out there are - and if we did we can't always check them, and if we could we couldn't do it with perfect precision or accuracy. But that's what leads us to fallibilism.How's that, for a start?

You may be stretching 'axiom' to breaking point, but I think I see where you are going. Certain 'principles' of science are 'unproven' (and instead they rely on strong arguments in favour of). Obviously between Science! and what I've outlined there needs a lot of filling in, but I was trying to keep as close to axioms and avoid as much as possible straying into 'principles' as I could.The philosophy side of 'knowing', or epistemology is simply about trying to figure out when something can be reasonably called 'knowledge'. Then there needs to be acceptance of certain real-world limitations, followed by figuring out what those limitations are, what compromises need to be made (such as the problem of induction) and finally some kind of methodology can be constructed.The methodology (or rather, methodologies) we have ended up with is called 'science', and there is a lot of metaphysical/epistemological groundwork that goes behind it all - not all of it (any of it!?) 'provable'.So yeah - science shouldn't be considered an epistemology per se, it is a practical pursuit rather than an idealised one. In an ideal world we'd look at the evidence and be able to deduce reliable conclusions from it, just as Rob implied. However - the real world is not ideal and so various compromises and gaffer tape patches are needed.The philosophy of science is an interesting subject (even if certain science based posters around here would flat out deny that philosophy and science have anything to do with one another), let's hope someone that disagrees with us pops along and that might spur a bit of creative posting.So science does lean on some other ideas possibly includingPragmatism
Coherentism
Parsimony
Fallibilism
CorrespondenceAnd so on and so forth. These kinds of things cannot be proven so are they the kinds of things you are looking for? My favourite is the argument against verificationism which says that since verificationism itself cannot be verified, verificationism by its own standards is incoherent or meaningless. Such wonderful circles one can run in.Does that help at all?