In math, numbers and addition are logically defined by Zermelo Set Theory, a small group of axioms upon which everything else can be built. Could it be possible to have a working theory, (in any field not just math), without any preexisting axioms?
This entire question is a case of classical equivocation.
Theories in sciences don't have axioms. In fact theory, in the sense of Set Theory, or The Theory of Complex Variables has nothing in common with the notion of a scientific theory whatsoever.
A scientific theory is a proposed explanation for a set of phenomena, and a theory in the mathematical sense does not explain anything, it is a subject in itself.
In math, a theory is a major part of a field of study that is somewhat separable from related parts of the same field. It is often defined as the body that proceeds from the understanding of a given model.
The modern way of looking at a model is as the embodiment of a set of axioms. But that is not the only approach to take. A lot of people do not think of Euclidean geometry as what proceeds from Euclid's axioms, but what proceeds from a given shared understanding that humans develop naturally by living in macroscopic three-dimensional space. So by that notion, things like geometry, real analysis and number theory do not really involve axioms. The axioms are created to solidify our understanding, but they are in some sense superfluous, and a concession to a level of excessive modern formalism in logic, and to the success of abstract algebra. These domains that focus intently on a single model really have that model in a more informal, but shared set of ideas.
To think of addition as defined by Set Theory ignores millennia of history during which people really did math.
These two things don't really overlap.
Even in math, axioms are only an aspect of a given approach, although an extremely common one.
Could it be possible to have a working theory, (in any field not just math), without any preexisting axioms?
I doubt it. Without some set of axioms, the observer will almost always draw a conclusion that is random in relation to the event observed; further, in the absence of guidance from fundamental principles, the observer can never understand when a conclusion is accurate, even when it is so.
“An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.” (Wikipedia, Axiom) I think of Euclid’s Elements as stating axioms of mathematics, particularly of geometry. Without them, a mathematician could not conclude, one way or the other, that the three angles of a triangle total 180 degrees.
I think of the Laws of Thought as axioms. Without them, there would be no assumptions about identity, non-contradiction, or the excluded middle. Such an absence of guidance would permit any number of random theories, none of whic could be attached to reality, nor to any larger theory about the world.
Is the answer to this question, to ask another question? Can a question exist in nothing?
ie. it takes an intelligence to ask a question and an intelligence exists on certain principles that pre-exist that intelligence, upon which it relies.
The foundation is an axiom. All we know is this very experience. One argument for evolution is it came out of nothing. But equally within the theory there are assumed rules and points of stability that exist for no known reason, and therefore are axioms. There is a faith or belief that if something exists, it exists in isolation and cannot have been created, even though its origin is unknown and what it is is also unknown, just its apparent effect that can be measured.
I suspect we like absolutes, and conclusions rather than processes and a process of becoming and discovering which never ends, though this is our experience of life.
Existence precedes essence. Axioms are great, but I put it to you that in practice they come after finding an interesting theory, in the process of generalising and universalising it.
If the laws of physics existed before the universe, where did they exist? Theynare the universe, and the universes behaviour is the laws. Theybare true in detqil and axiomatically at the same time, without hierarchy. Things just exist, and how they exist are the laws. We look for patterns, and what is invariant we call an axiom. But we begin with the pattern, generalise it a little, learn more, get deeper axioms, etc etc. The world continues to work without us knowing any axiom, and the world is the theory of all theories.