Much of philosophy is concerned with providing a rigorous foundation to truths which are otherwise intuitive and uncontroversial; think of philosophy of math, for instance. Do philosophers believe that, absent an appreciation of such foundational principles, laymen don't actually "know" such truths, e.g., that 1+1=2; and if laymen do know such truths, how do they know them?

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Actually, the presumption here is wrong. It isn't the case that "much of philosophy is concerned with providinga rigorous foundation to truths which are otherwise intuitive anduncontroversial". In particular, that isn't the case in the philosophy of mathematics.

Of course, famously, Frege tried to show that the basic laws of arithmetic (and hence the proposition 1+1 = 2) can be derived from the laws of logic plus definitions. But he did this in order to defend the claim that arithmetical truths are analytic, true in virtue of logic alone, and so explain why those truths are necessarily true and why they necessarily apply to everything. He didn't claim that, prior to his attempted derivation of 1+1 = 2 from pure logic, no one knew it to be true. Rather we weren't in a good position to see clearly the sort of truth that it is, analytic according to Frege.

Unfortunately, one of Frege's putative laws of logic turned out to lead to contradiction, and his foundational edifice crumbled (though neo-Fregeans think that much can be rescued). In part as a response, Russell and Whitehead also famously tried to show that the basic laws of arithmetic can be derived from a small number of more basic laws. But again, it wasn't that they thought that, prior to deriving 1+1 = 2, there is a serious question mark over its truth. In fact, for them, it is the other way about: the rationale for accepting some of the laws of their basic "type theory" is regressive: that is to say, we are to accept their "foundational" laws here rather as we accept laws in physics -- i.e. they generate consequences which we already know to be true.

These days, students are taught how a vast amount of mathematics can be regimented in ZFC (Zermelo Fraenkel set theory with the Axiom of Choice), and in particular are told how to derive proxies for the Peano Axioms for arithmetic, and hence for 1+1 = 2, inside ZFC. One point of this regimenting project is that it gives us a way of calibrating the infinitary commitments of different areas of mathematics by measuring different bits of mathematics against a common yardstick. But again, it isn't as if set theorists think that the basic principles of ZFC are more secure than those of simple arithmetic. On the contrary.

So in fact Frege, Russell and Whitehead, and modern set theorists all knew/know that 1+1=2 in the same way the layman do. But what that involves is, indeed, another question.

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