Oh man, bro. Long-time readers know how I feel about this particular attitude. You are definitely over-committing yourself if you mean that everything we can know is mediated through the senses - I mean, mathematics? Logical tautologies? I don't need my eyes to know that 'all bros are bros' or that 1+1 = 2. Perhaps when you were writing your e-mail you felt smug; you didn't need your senses to know that you were feeling that way. You have what we call privileged access to your own mind. "Bro, you were wrong when you said you felt sad." That doesn't make sense at all.
So here's the broad rationalist position: we have knowledge, either in the form of intuitions or innate ideas, from which we can deduce true things without observation. We're pretty much all rationalists with respect to math and logic - no one insists that you gather the proper measurements to verify that addition works or that first order logic is complete. Sorry bro, but you can't just smash two numbers together in an accelerator and measure whether they add properly. By now we're mostly empiricists with respect to science - we certainly can't reason our way to gravity without watching some shit tumble. But what about all the middle ground?
So, assuming you didn't mean to endorse something ridiculous like 'absolutely all knowledge must be mediated through the senses', perhaps you meant something more like logical positivism, a philosophical movement influential in the 20's and 30's. As in most philosophical movements, not all positivists agreed with each other, but they generally agreed that if you couldn't verify something, it was meaningless. So, yeah, maybe you can deduce mathematics and logic from reason, and linguistic tautologies like "All bachelors are unmarried" and "All bros are geniuses" have to be true, but other than that, everything has to be empirically verified. They dismissed ethics, metaphysics, and theology as unverifiable shenanigans, completely meaningless. "Properties exist as abstract universals." The fuck? How would you go about determining if that was true?
Maybe logical positivism sounds appealing to you. Except here's the thing: how do you verify the proposition 'Any unverifiable proposition is meaningless'? If you're going to tell me how to evaluate everything, you sure as shit better meet your own test, and the verifiability criterion fails miserably. So even if you grant the obviously rational stuff like math, logic, and certain language-statements, you still can't establish that empiricism is the only valid method of obtaining other knowledge.
Ethics is a particularly rich minefield here. Hume famously formulated the is-ought distinction, which says that nothing about the way the world is tells us anything at all about how the world ought to be. Yeah, an earthquake fucking tore up Japan. So? What about that allows you to say anything about what relief we owe them, or how the world should respond, or anything else? 'But, people are dying!' Again, so what? And if you're appalled by that callousness, you probably have some moral intuition that resembles 'It ought not happen that people are starving in the wake of a natural disaster.' And that is itself a statement of how things ought to be. In the wake of Hume, it seems pretty clear that any ethical statement must rely on either intuitions or innate ideas - the conscience usually functions as one or the other.
W.D. Ross, an ethicist who describes himself as an intuitionist, probably articulates this idea best. He has a list of duties we owe each other - for example, to be grateful, to not be dicks to each other, to keep promises - and when these duties come into conflict, we weigh them against each other in any particular act to determine our duty at that moment. How does he justify these duties? Well, he doesn't, really. He says they're self-evident and prima facie - look, if it's not apparent to you that 'we ought not be dicks to each other' then what the fuck can I say otherwise to convince you? How is it not fucking obvious that we shouldn't be dicks to each other? It doesn't matter why not - maybe some God said so, maybe we evolved that way, whatever bro - just, you know, don't be a fucking dick.
Of course, there are people who deny that there's such a thing as objective morality, but some acts seem harder to hand-wave away than others. Not to go all Godwin's Law here, but some people thought the Holocaust was the right thing to do. If you would deny that we can know anything about ethics, you have to be willing to deny that we can be sure those people were wrong, which seems like a tall fucking order to me, bro. I'm not saying it can't be done, I'm just saying that it's a tough trigger to pull, and I'm not ready to call people stupid for being unwilling to pull it.
On the other end of the spectrum, our intuitions about some negative moral judgments seem tough to question - it is not (intrinsically) morally wrong to love waffles. Yeah, that seems trivial, but only because it's such a universal intuition. Who is mad someone loves waffles?
If it seems like rationalists are on shaky ground there, consider your own situation. How the fuck do you know empiricism works at all? Science depends on the assumption that 'the future will work the same way as the past' to make predictions, and Hume showed that there's no way at all to prove that's true - we just assume it based on beliefs we form. So that idea is based on observation, but the future conformance of observation is exactly what it questions. Karl Popper thought we never verify hypotheses, we only falsify them - we know Newtonian mechanics isn't true because we've observed some shit incompatible with Newton's Laws, but we can't be sure that the Relativities are conclusive. After all, Newton's laws lasted a long fucking time, too. In fact, even the idea that observations contrary to a thesis falsify that thesis depends on the rule that two contradictory propositions can't be true, and you absolutely can't prove that. Graham Priest does an incredibly convincing job of undermining the Law of Non-contradiction - it's an intuition, bro. U jelly?
Consider heliocentricity - are you sure the Earth goes around the Sun and not the other way around? Because here's the thing - there's a reference frame in which the Earth is at the center of the solar system, the Sun goes around the Earth, and the other planets go around the sun, and it's exactly as predictive as Kepler's Laws, because it's completely equivalent. So why do scientists insist that the Earth actually goes around the Sun and not the other way around, or no way at all? Well, it's more elegant for one; the math is simpler. But why should we prefer elegance? That's not verifiable, bro. Newton required an idea of absolute space to ground his idea of relative motion - not verifiable. Einstein uses a convention where the speed of light in any direction is always the same - not verifiable, since light has to go one way and then the results have to come back the other way - it seems logical to assume the one-way speed of light is always the same, but it's impossible to test that shit.
So, here's the deal - empiricism isn't as airtight as Stephen Hawking would have you believe. It has some problems with abstracting from a bunch of tests to universal laws - yeah, the gravity example seems ridiculous, except that it turns out that the way we thought gravity worked for hundreds of years turned out to be wrong. Kepler, Newton, Einstein - they all depended on some assumption or other that isn't testable. Of course, none of that supports rationalism, it just undermines empiricism. Fair enough. But as a scientist, you have to make some assumptions that you can't test; you have to rely on your intuitions about elegance and symmetry to make sense of your worldview. It's not possible to demand that all knowledge come from the senses.
So, dogmatic empiricism? Cool story bro, but you're left with two options: believe that we know precious little indeed, a view some people hold, or believe that some shit we know, we know based on some form of reason or other. Empiricism is pretty limited. 'Rationalists' believe that we can indeed know some things based on reason - generally, more things than Empiricists would like. But it's mostly a matter of where you draw the line on intuitions.
Ross' The Right and the Good is one of the most important ethical texts of the last 100 years, hands down.