(1) The paradigmatic type of synthetic a priori knowledge that was Euclidean geometry, when asserted as a universally true theory of space, has been shown to be severely contradicted by empirical evidence – and this is not what we would expect to find if this theory really was necessarily true and an irrefutable theory of reality.Ultimately, we can reject synthetic a priori knowledge by inference to the best explanation and Ockham’s razor.
(2) We can eliminate the problem of proposed synthetic a priori knowledge by carefully separating pure maths/pure geometry (which is analytic a priori and necessarily true, but not describing reality) from applied maths/applied geometry (which is asserted as true of reality but is synthetic a posteriori and contingent).
For example, most of mathematics can be clearly explained as an analytic a priori system, as derived from pure logic and set theory (Schwartz 2012: 19), as shown by the work of Frege, Russell, and Whitehead.
(3) From (1) and (2), we can satisfactorily explain proposed synthetic a priori knowledge either as (i) analytic a priori or (ii) synthetic a posteriori, eliminating a complex and unnecessary category.
Schwartz, Stephen P. 2012. A Brief History of Analytic Philosophy: From Russell to Rawls. Wiley-Blackwell, Chichester, UK.