I don't know why you're being downvoted, as your response is typically true of any A v. B discussion in philosophy, there's always grey lines. Rationalists can still claim categories of knowledge coming primarily, or even only, from sense experience, while also claiming other categories of knowledge that come from rationality alone.
Fwiw I subscribe to the anthropological view that human consciousness/ culture as we know it "turned on" about 70,000 years ago (this was introduced in my first anthropology textbook in college and is also the topic of the very popular "Sapiens"), and we've had a unique, if not structuralist, system of reflection and developing knowledge ever since; if I had to pick one, without doubt on the rationalist side.
On what grounds do trust your reason, senses, or understanding?
Whether I axiomatically start with Rationalism, Empiricism, or some combination of the two I always end with, at the very best, Solipsism (via the Simulation argument), and more so these days just pure philosophical Skepticism. "I think therefore I am" is only true if "I think" is true, and to say that "I think" is to take as a given that I exist, which is begging the question. We might be able to say that "thought exists" (and therefore a thinker must exist, etc) but this is premised on "truth exists", since "thought exists" is meaningless without veracity, and to say "truth exists" is to continues to beg the question in two possible ways: First truth requires a mind in order to be known which is what we're trying to get at in the first place. Or, if that set of reasoning is too shaky for you then: "truth exists" requires truth in order to be true.
I do not think it is useful to stay there, so when I approach the world I have a sort of base understanding that I'm taking certain things which may not be true (my own reason) as true for expediency because the alternative is to cease thinking. But just as arithmetic (the basis of nearly all of mathematics) isn't logically proved unless you assume the Zermelo-Frankel continuum I do the same with my understanding of existence.
Well, I don't assume the Cartesian cogito or the rational, centered world of Logos. I just assume that consciousness is this thing that flipped on about 70,000 years ago, which brings its own rules or structures that impinges on any sense of reality, and we've basically been immersed in it since, with the fortunate even accidental side effect of being able to understand things in a way we might call 'rational.' I'm much more on the side of "It thinks, then I register it" than "I think". So I guess, similarly, I too approach the world with a sort of distance towards things, while taking certain things sincerely for the sake of utility and happiness, but while also seeing that our "self" (in the form it is) and consciousness are rooted in Real capital-R structures produced by nature, so it's not a post-modernist, everything is endlessly decentered/ no substantial basis, etc... there really is something there governing in specific ways... enough to build a principled world with its own, I guess, absurd form of 'centeredness'... it just requires a weird sort of dynamic stance towards things, chalk it up to the fundamental condition of being a homo sapiens in-consciousness.
I'm not sure why you're excluding the possibility of both. Math is largely based on reason, and I think most empiricists believe that mathematicians have knowledge not based on sense experience.
>most empiricists believe that mathematicians have knowledge not based on sense experience.
Even the knowledge that 1+1=2 comes from sensing objects in the world. If you'd never seen more than 1 thing ever you would have no sense experience to base your conceptual framework of addition on.
All knowledge -mathematical or otherwise- can be traced back to sense experience. Your ability to reason mathematically comes from your experience of the world and from your experience of your own thoughts. Intuition is fundamentally a sensory experience.
Similarly, either it is possible to know certain things from observation of sensory input alone without reference to a ration framework of consideration, or it isn't.
And seem both cases seem unlikely, it seems that empiricism and rationalism are more complimentary (AND) than XOR.
Fwiw I subscribe to the anthropological view that human consciousness/ culture as we know it "turned on" about 70,000 years ago (this was introduced in my first anthropology textbook in college and is also the topic of the very popular "Sapiens"), and we've had a unique, if not structuralist, system of reflection and developing knowledge ever since; if I had to pick one, without doubt on the rationalist side.