No reason justifies disbelief.

[The Grand Nudger]

Well, I would argue that an intuitionst has not provided any reason for me to think that intuition is in any way ontologically distinct from empirical knowledge. In fact, to the contrary, we have a large body of evidence demonstrating that our brains do exactly that: subconsciously process large quantities of data. And, that this data-processing directly influences our choices and decisions. Calling X empirical because there exists evidence that X is empirical, would be a far more promising argument than calling X non-empirical without any reason to do so, and without being capable of positively describing or identifying what X actually is, lol. They're building a second assumption into the argument without any reason (that I can tell) to do so. "These two things are different, and also, the second thing is its own special, magical thing."

We do have some evidence of that, yeah...and it's definitely something for the intuitionist to grapple with, though the claim empiricism makes provides for a severe handicap.  As mentioned before, if 99.99% of all alleged intuition were actually some subconscious process of empirical data manipulation, the remaining .01 is still enough to make empiricism false as stated.  As it turns out, there's more than .01% in evidence, as we proceed from intuitivism here to nativism or innate knowledge below.  

Or, perhaps these phenomena are necessarily concomitant.

That's sort of the question we're asking ourselves...but if they were, then empiricism as envisioned would, again, be false.  All knowledge must proceed from sensory experience for the claim as made to be true. There can;t be even one other example of any other option....not even as concomitant partners.

I mean, an axiom is not absolute knowledge. I can't think of any knowledge that I have that could qualify as distinctively innate, and completely divorced from sensory input.  Can you?

Yeah, actually.  Human beings aren't born a blank slate.  I mentioned smiling in another post, not specifically directed at you..but that;s just one example.  There's a great deal of common behavior, and it would be difficult to argue that all of this is accounted for by us somehow all having the same empirical observations, at least not as the empiricists who envisioned both the tabula rasa and empiricism had thought.  We used to think that neurons didn't even form connections until they had some experience to build connections from, and the contents of those neurons, so far as we can tell, amount to all of our knowledge no matter what it's ultimate source may be.  We no longer believe that to be the case..because we've seen that it's not.  

We've always had this bouncing around in the back of our heads (and sometimes the front)..we were aware even when we lacked clarity on what it was we were aware of.  We used to call this sort of thing "instinct", our "our nature" (and hey, we still employ the terms).  Some intuition may be accounted for by it, but all of it qualifies as innate or nativist candidate. Anything we are born possessing the knowledge required to accomplish is unlikely to be a product of any empirical observation..or at least any empirical observation we can understand the means or timing of, or possess ourselves as the source. As with intuition above, this is something empiricists have to grapple with. Even if we were to rehabilitate empiricism in light of this, and far be it from me to claim that we can't, it will still have been the case that empiricism was wrong as conceived.

If we know to/how to smile before we could see a smile, then it isn't on account of the sensory experience of sight that we know this or know to do it, or know how to do it. If we know to run when the grass rustles behind us despite having never been chased by a lion...then it isn't on account of the experience of being chased by a lion that we know this, know to do this, or know how to do this.

Or, if you prefer..if it -is- based on those experiences, it's not based on our experience of either.

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As to an axiom not being absolut knowledge, well, I think that you'll find a compelling argument to the contrary.  The sorts assumptions on which the slimmest means of inference are built are taken to be "absolutely true", necessarily true, in fact..which is a kind of truth that might elude any truth based on empirical observation, whether it's one or one-thousand...and knowledge of things like that was exactly how we launched this subset of innate knowledge, things we know as a consequence of our rational nature.