For example, in Zermelo–Fraenkel set theory (ZF), the addition of the axiom of determinacy(AD) is inconsistent with the addition of the axiom of choice(AC). Is it legitimate to adopt ZFC (ZF+AC) as the background mathematical theory when working on problem X and ZFD (ZF+AD) when working on problem Y?
If AC and AD are scientific hypotheses, I don't see a problem, but I don't know if the mathematical axioms are the same as the scientific hypotheses.
I'm venturing an answer to this even though I don't know much about abstract mathematics in hope that it will be of some use. Please apply elevated skepticism to the below.
Scientists generally follow the lead of mathematicians on esoteric matters of mathematics, on the rare occasions where esoteric matters of mathematics concern them at all, unless it happens to be something that physicists got to first. Since ZFC is widely accepted by mathematicians, it is safe to assume that it is widely accepted by scientists (at least those few who care enough about abstract mathematics to know about it).
Science is concerned with predicting reality. This necessitates relevant mathematical axioms which define some aspect of reality mathematically, in addition to foundational mathematical axioms such as are needed by mathematics (and largely taken for granted or ignored by scientists) to define relationships between logical objects.
These more physically salient axioms may indeed be different for different predictions of different phenomena. In fact, multiple incompatible sets of axioms may underlie a single model, which still makes predictions that comport with measurements.
1: Defining axioms: An axiom is a mathematically meaningful definition of a quantity or relationship, including physical quantities or relationships. Axioms inform hypotheses, but they are not themselves hypotheses: they can be useless if hypotheses based on them predict falsely, but they don't make predictions themselves, so they can't be true or false.
2: The role of axioms in science: A set of axioms and the logical (mathematical) extrapolation therefrom constitute the mathematical formulation of a model.
3: The role of models in science: Models map past measurements to predictions of future measurements. Such a mapping between measurements, especially if a mechanism is identified, is a hypothesis. Models are discarded if their predictions of future measurements are false, kept if they are true within measurement error, and may be adjusted or retained as approximations if they are moderately close most of the time.
4: Domains, measurements, and reality: A domain is the range of all possible input measurements that a model is supposed to translate to predictions which comport with reality within a certain acceptable threshold of error. The domain may be knowable a priori or it may be discovered by measurement.
5: Axioms and domains: Axioms may define domains if the axioms themselves are based on approximations or necessarily imply a special case. For an obvious example: the axioms that underlie inviscid flow theory define a domain of inviscid flows.
6: Exceeding your domain: ...except there aren't any inviscid flows in reality. But if you apply the model to situations that are (mathematically) close to the domain, you might get useful predictions anyway. And since this is science, not mathematics, we have recourse to experiment: we can ask reality whether and how well the model works outside of its logically implied domain.
7: Effective theories: Pushing the boundary of your domain can only take you so far. Eventually your predictions and measurements will diverge. For this reason we come up with effective theories. Effective theories either leverage models known to work in other contexts, rebuilt for a new context in which the physically real mechanisms that they originally described are absent; or they mathematically mash something together out of multiple models known to work in other contexts, even if they have incompatible axiomatic foundations. In either case, unlike a theory, an effective theory doesn't purport to describe a physically real mechanism, just to be a model that successfully maps past measurements to predictions of future measurements.
8: If it works, it works: So now we're operating in a domain which is prohibited by our axioms and/or using a model based on other, contradictory axioms, which also prohibit the domain in which we're operating. But sometimes when you do this, you can reliably predict reality. Experiment, not mathematical neatness, is the ultimate arbiter of scientific truth.
Scientific knowledge is just some type of knowledge (it is generated by means of the scientific method, etc.).
Your problem is not about science, but about logic.
Following the principle of explosion, if you accept two axiomatic systems that contradict, any conclusion (theorem, in the formal systems jargon) is subject to a contradiction, and even more: any axiom is true and false at the same time.
Now, if you use such flawed logic system as a foundation to make science, you will produce flawed knowledge. In fact, strictly, the scientific method cannot be compatible with such kind of logic. Take the definition of the method by Wikipedia for example:
The overall process involves making conjectures (hypotheses), deriving predictions from them as logical consequences, and then carrying out experiments based on those predictions...
As you can see, the foundation is a consistent logic, which evidently excludes any possibility of contradictions.
Every factual science (physics, chemistry...) has a common background of formal sciences: first-order-predicate logic and mathematics. Formal sciences are independent of factual sciences and the latter only may serve for inspiration or motivation for new developments. The rationality of factual science demands for the coherence of its discourse, so no contradiction is admissible in factual science (of course this is true for formal science too!). Hypothesis in factual sciences are different from axioms in factual sciences in that they must be proved to be false o true in the factual world, normally, by means of experiments. But, for the purposes of deductive reasoning inside the respective theories, axioms and hypothesis are the same. For a concise but very complete reference on these matters, consult the book by the philosopher Mario Bunge: Philosophical Dictionary (Prometeus Book).
Scientists want to do science. They do not want to worry about foundational concerns until they have to.
It will not be decided by the scientific establishment if it is legitimate or not to use contradictory axioms, until those contradictory axioms prove themselves any use at all for doing science.
For example we might make predictions using AC and they seem reliable. Then we make other predictions using AD and they also seem reliable. But I am unaware of any such predictions.
Please inform me if they exist.
