I occasionally am in a finitistic mood; while I have no reason to believe that the following is applicable to all, or even most, finitists, it might be worth writing down.
Note that in everything below, I'm perfectly comfortable with classical logic. Personally I actually have much less foundational interest in nonclassical logic than in finitistic frameworks, and the two are for me largely unrelated. Of course, $(i)$ this is not to say that I have no mathematical interest in nonclassical logic, and $(ii)$ this is just my opinion.
Let's start at the far opposite end from infinitism: ultrafinitism. There are various reasons one might find ultrafinitism attractive, but I think the most common is the belief that numbers must have physical meaning to be meaningful. Let's call this the "physicalist" view. Of course, pinning down what "physical meaning" means precisely is hard, but there are some things I think we can say with reasonable certainty. For example:
Based on our current understanding of physics, Graham's "number" does not in fact have any phyiscal meaning.
However, this picture has certain features which at least to me make it unsatisfactory for capturing mathematical meaning. At an immediate level, there's no reason to believe that "number" today means the same thing as "number" tomorrow: the universe does seem to be expanding, after all, and arguably this means that more "numbers" are becoming actual numbers since tomorrow the state of the universe will take more bits to describe naively than it does today. Please ignore my complete lack of actual understanding of physics here; I'm just trying to make the broader point that the fact that the universe changes over time poses a problem for the idea that "number" is a well-defined concept in the physicalist view.
A more serious problem to my mind is that this definition ironically means that we have no actual direct access to numbers! This is because I can never be absolutely certain that my understanding of the physical world is accurate, and I'm not just a brain in a vat. Maybe the true physical world is much much bigger than, and behaves quite differently from, my guess at reality. Conversely, am I really certain that $1000000$ is meaningful? The universe might be very very small . . .
The version of physicalism that has some appeal to me is a sort of "relative physicalism" (my reverse math bias might be showing here):
Except in a small handful of cases, it's outright impossible for me to say with certainty that a mathematical concept has physical meaning (even if I'm confident what "physical meaning" is!). However, there are meaningful dividing lines between mathematical concepts based on what would have to be true of the universe in order for those concepts to have physical meaning.
For example, it's completely plausible to me that Graham's number has physical meaning. All that would take would be for our understanding of physics to be wildly off-base, or for me to be the victim of the most boring conspiracy in history - and relatively speaking, these are totally plausible. By contrast, something would have to be meaningfully weird in order for $\aleph_0$ to have direct physical meaning (personally, my interpretation of "physical meaning" is strict enough that an "infinite, locally finite" universe wouldn't necessarily cut it). Intuitively, if we start from a position of severe skepticism about the universe (which I wouldn't take in day-to-day life, but I think is reasonable in foundations of mathematics), then some dividing lines are "a priori significant" while others aren't (I don't see any particularly compelling reason why Graham's number should be substantially more implausible than one million, but I definitely do see a fundamental qualitative difference between "finite" and "infinite").
At this point I might allow a drop of dogma into my skepticism, and declare that some things are just too implausible to have physical meaning at all. E.g. I might claim:
I wouldn't be too surprised if Graham's number had physical meaning, but I truly can't imagine a universe in which $\aleph_0$ has physical meaning.
(Incidentally, for me a similar-feeling claim is: "I wouldn't be too surprised if ZFC were inconsistent, but I truly can't imagine an inconsistency in PA.")
I personally would never go that far, but I can definitely get behind:
I would be fundamentally more surprised by $\aleph_0$ having physical meaning than I would by Graham's number having such physical meaning.
I actually find this "degree-of-surprise/shock/distress" line of thought quite compelling. For me, finitism often captures "Mathematics which could be physically meaningful without spiking my blood pressure." I asked my doctor, and after rolling his eyes he told me I shouldn't worry about this, so I almost never do; but then I can't be completely certain that he's not part of The Conspiracy, so I do worry from time to time.
One feature of this perspective is that finitism isn't particularly privileged: "countable" is less surprising to me than "uncountable," so "countabilism" is a meaningful-to-me stance; ditto "not-supercompact-ism," and so on. On the other hand, finitism is the most restrictive position of this type, so it is distinguished in that sense.