It's my understanding that the scientific method builds upon certain axiomatic assumptions, such as uniformitarianism and the principle of induction. Is faith required to believe these axiomatic assumptions?
First, let's define faith. What makes faith different from mere trust ? An effective definition is "belief in something without, or even in spite of, the evidence".
Boarding a plane trusting it will not crash is not faith, because thousands of planes take of everyday and don't crash. Sitting on a chair trusting it won't break is not faith, because we sat on hundreds of chairs that did not break. The decision to board or seat is based on evidence.
Also, if the chair emits a crack sound as we seat, we can revise our judgement and decide not to seat. Presented with contrary evidence, we revise our opinion.
Contrast with the members of the sect described in when prophecy fails who had no evidence for their prophecy yet believed it enough to sell all their stuff in preparation for the end of the world, and presented with the evidence that it was false when it didn't happen chose to continue following their prophet.
I am not trying to be derogatory here: listen to religious believers' use of language, you will see that they use the word faith in the same way: someone who is said to have a strong faith is someone who does not require evidence to believe and won't doubt when faced with arguments.
Now, how about scientific axioms?
It is true that they are considered true a priori, without evidence. Sometimes they are considered trivially obvious ("let be a line and a point external to the line, there can be only one parallel to the line passing through this point"), or necessary to a rational enquiry (rejecting uniformitarianism means astronomy is pointless: if we can't assume the physics on Jupiter is the same as on Earth, we can't deduce any knowledge from what we see of it).
But that does not mean they escape scrutiny a posteriori:
first of all, the validity of the system of propositions they generate can be tested. If a set of axioms leads to a contradiction (2 propositions soundly deducted from the set contradict each other), the axioms are revised. Practical applications can also be an evidence of their soundness, like in the way Euclidean geometry helps building houses that don't collapse, or planning for the right amount of material.
on the other hand, some axioms seem necessary to have any discussion at all. For example it has been shown that, if the principle of non-contradiction (not A and A can't be both true) is false, any proposition however ludicrous and its opposite can be demonstrated, which makes any logical inquiry pointless.
they can be challenged. People have taken axioms from Euclidean geometry and see what happens when they are considered false. For example, what happens if by one point can pass more than one parallel to a line? This gave birth to non-Euclidean geometry, which also has practical applications.
people try to eliminate them, by proving one axiom of a system from the others, reducing the amount of propositions to be believed without evidence.
they are tossed away when proved to be false. For example, general relativity and quantum physics broke many principles that were hold trivially obvious in our everyday life. Even uniformitarianism has been tossed away temporarily, since both theories are still incompatible, yet give valid results each in its domain of applications. Sure, hope to find a unifying theory has not been given up, but nobody holds the position that "uniformitarianism is sacred, therefore one of either relativity or quantum mechanics must be false".
Axiomatic assumptions is oxymoronic. There is no such thing as an axiom that is an assumption, if we agree to the defintion: it is self-evident and inarguable. Axioms of Euclid in geometry are axiomtic. David Berlinsky, retired philosophy professor, argues that only mathematics qualifies as a science because it is axiomatic. Everything that can be questioned is on a level below axiomatic (a priori ... self-evident) knowledge, and therefore yes, it does take a certain amount of faith to adopt non-axiomatic understanding; hence, all the endless and fruitless debates regarding evolution--which is not an example of axiomtic knowledge ... it's not a priori, it's a posteriori--grounded on experience, and as Immanuel Kant has it: "But experience teaches us what exists and how it exists, but never that it must necessarily exist so and not otherwise. Experience therefore can never teach us the nature of things in themselves." Prolegomena, under the section "How is Pure Science of Nature Possible", 2nd paragraph.
There's no a priori certainty that can be derived from experience, there are assumptions, deductions, judgments, etc., but if you doubt this and say that for instance, Darwin's theory is self-evidently true ... then you're attributing to it the a priori certainty is does not have--nor does Darwin lend any such suggestion. Much of science is built on presuppositions, and many of those are taken on faith, not by reason of overwhelming evidence in their favour. You don't need overwhelming evidence to understand that 3 plus 3 equals 6.
First, the axiomatic assumptions must go beyond the Doctrine of Uniformity or the Problem of Induction. Those are the special cases of a certain General Principle, the Foundational Belief of a fully rational worldview: the belief in the objective1 reality.
Second, while any assumption is itself irrational,2 the reason for going with that particular one is not. These days it is known as the "street light effect",3 but make no mistake -- it has been relied upon long before humans.
Or, if you want, while it cannot be defended with pure reason alone, for practical reasons believing in the objective reality (and, hence, in science) is just as necessary.
So, answering your question, no. No faith is required. Rather than being faith-based, our belief in science is itself a leap of faith.4,5
1 or, specifically:
- We exist as parts of the One and Only, the Ultimate Reality that we all share.
- As it transforms before our eyes, the change is constant, but it is never random.
Everything is an effect by a certain cause in its past, and we can always (at least in theory) trace the former to the latter. We can always figure out how and why everything that is (or ever has been) has come to existence. And how it would have played otherwise.
2 "rational" means expalinable, and assumptions, by definition, aren't
3 “Science is a bit like the joke about the drunk who is looking under a lamppost for a key that he has lost on the other side of the street, because that’s where the light is. It has no other choice.” -- Noam Chomsky 
4 "One can understand this only if one understands that it can not be understood, and one is not able to understand this if one thinks it can be understood" -- quoting from Kena-Upanishad to emphasize that while it might look like I'm playing with words for the sake of it, that was not my intention. The subject is sure tricky though...
5 The difference, I guess, is that "requiring faith" sounds open-ended, as if we are giving up on staying rational... while "making a leap" underscores our fundamental commitment to rationality.
I think I can answer in the negative in two different ways - my personal preferred position is that an axiom isn't even the sort of thing that you need to believe, but a scientist that wants to hold that you should probably believe the content of your axioms also has a fairly natural response to suggest that their axioms are sufficiently supported by the body of evidence provided for their overall theory.
I am a mathematical Formalist. I think that numbers (also mathematical objects in general), as hypothetical spatiotemporally independent abstract objects, do not have existential import beyond their use in a language game that happens to be incredibly useful in describing, inventing and applying models of reality. That is, while it may be useful in our understanding of how things work to talk about numbers, they needn’t “really” exist, in the way that trees, atoms, quanta of energy etc. do.
However, without a doubt, the most useful way of understanding how mathematical language works is through interpreting it as a first order logical theory, describing a domain of objects that are related in certain logically useful ways. Similarly, our current “foundational” understanding of the general hierarchy of Scientific abstract model building being founded on paraphrases of the underlying domain of mathematics, it seems like much of what we propose to exist is, in effect, just a way of talking about particular abstract mathematical objects of interest to physics, chemistry, biology, psychology, sociology etc.
There is an argument that says that since mathematical domains are so pivotal to the functioning of effective science, we ought to accept that this commits us to their existence. We may not, it is supposed, have independent reason to believe that the number 2 exists - however, since the workings of Arithmetic, Calculus and Analysis (in particular) are so deeply embedded in our ways of talking about things that we do have good reason to believe to exist, this constitutes not only sufficient evidence to affirm that numbers are real, but in fact that it constitutes a comprehensive argument that one ought to believe that they are. The evidence for one’s overall scientific understanding confers epistemic obligation towards the component parts of the prima-facie ontological commitments of mathematics.
This obligation is invoked as an “Indispensability Argument”, and part of the foundation of this way of understanding evidential warrant is that 1) the language of first order logic carries the cleanest interpretation of what it means to be existentially committal, and 2) our totality of beliefs does not face the test of evidential confirmation on an individualistic piecemeal basis, but rather hang on and are applied – whether confirmed or rejected – against reality as a whole.
This is an influential position in analytic philosophy of science, and the response that this would give you for your specific question about the epistemic status of one’s axioms would be to say that “no, one does not need to have faith in one’s foundational axioms, because the system of scientific method and modelling as a whole provides sufficient evidence, via testing against reality, to ground those central axioms.” I think this is a perfectly sound way to do things, and if one is prepared to conclude on the back of this that numbers really exist, then I would have to say that’s probably a reasonable way to understand the world.
But I find something quite unsatisfying about this. Talk about “indispensability” suggests a kind of blasé approach to what it means to be practically committed to an object’s existence. If a scientist using this ontological commitment is later shown a reduction proof that reduces mathematical objects to logical relations, we ought strictly speaking to say that this is an instance of Theory Change, and that something has substantially changed in one’s understanding of the world. And yet it seems hard to affirm that we are in fact being realists about mathematical objects while simply holding that the primary reason we believe in them is that we haven’t yet shown that we can do science without them. If this is a kind of epistemic commitment, then it is a very weak kind of commitment – you don’t need to be a card-carrying subscriber to Platonism in order to say “numbers exist because science needs them”. Really, this is much more like a kind of Instrumentalism, rather than an assertion that the objects of one’s theories exist in an independent reality.
My challenge to this perspective is to push back against thesis 1) – an axiom scheme in first order logical form does not necessarily constitute any adjoint existence claims to an underlying language framework. This has its roots in the debate between David Hilbert and Gottlob Frege about the nature of mathematical axioms – while Frege thought that numbers and sets of numbers had to be objects in their own right, for Hilbert all that mattered was that there was something in the domain of discourse that realized the relational structure described by the axioms. Objects related in a successor relationship as described in models of number theory would be sufficient to realize the axioms of number theory. Taking such a paraphrase seriously would have some odd consequences (e.g. an instantiator of the number 2 being possibly identical to Julius Caesar), but this is not a result in number theory, but rather something about a particular way of talking about the world.
The road this takes us down then is to start with the idea of being an Axiom according to a kind of formal language protocol. The first part of investigating any axiom scheme is not to ask “whether it is true”, but rather to look at its formal consequences. Our axioms taken together describe abstract structures – the rules of the game that we propose that the world may or may not work by. It’s only in the process of our testing of reality that we come to determine whether reality does in fact play by some specific set of rules, which might be described according to some particular protocol or another. And in the context of formal theorising, a plurality of axiom systems can be deserving of investigation. IE: Axioms themselves belong to the world of Pure mathematics, in contrast with scientific theorising as an attempt to use that same body of structures in Applied settings.
After quite a bit of careful theorising of different axiom systems into more intricate concepts of canonical mathematical theory, it turns out that we’re actually getting quite good at understanding which mathematical structures tend to be quite good for model-building – this is not so much strong evidence for those theories describing independently existing mathematical domains, but rather strong evidence that those theories schematically provide true descriptions of the real world. They are good mathematical theories in that sense, but that is to say that the world follows those structures, not to say that the axioms themselves are any more or less true about the structures they describe. That, we must insist, is automatically true in as much as we make sense of axiom systems in pure mathematics, and one can, in the same breath, talk about radically incompatible axiom systems in mathematics without thereby developing a paradoxical commitment to the impossibility of reality.
To summarise, our axioms do not necessarily constitute independent theses about reality, requiring independent belief for their functioning. Instead it seems more productive to think of them as part of the mechanism by which good scientific models of reality are built; some axiom schemes are useful, others might not be, but nothing about their application really impacts their validity as axioms, which can be studied independently in the realm of formal languages and protocol. You don’t even need to necessarily think of an axiom as literally true in order for it to have a good contributing role to play in building an effective understanding of what our evidence points towards – what’s more important is that it’s useful in constructing a true model. We may then, as it were, kick the ladder away behind us, as it will have done its job.
Not everyone will agree with me on this, and to them I offer the following alternative - even if we suppose that we ought to include the content of our axioms in our commitments independently of the remainder of our theory, then we can still appeal to Indispensability-style arguments to provide evidential support for the content of those axioms. I'm not sure that this is necessarily very strong evidence for that stronger thesis - indeed, formalist reductions of mathematical commitments should be seen as reasonable alternative hypotheses - but nonetheless the empirical efficacy of one's wider theory, and the importance of our axioms to that theory, should be seen as transferring a reasonable degree of warrant to those axioms anyway.