just in case you didn't get kant's synthesis (p. 561 of our text)

let's see if you agree with kant's synthesis:

he thinks we know a priori that there is cause and effect (in other words, we're wired to it).

before, let's reexamine the meaning of

synthetic: according to david hume, 

all crows are black , or 
water boils at 100 celsius 

are synthetic. i.e., they are not analytic (turned into logical truths by substituting syn for syn) and they are known a posteriori, i.e., by experience.

so, kant agrees with hume that statements about causes are synthetic. but he thinks that some synthetic statements are a priori.

let's consider, for example, our knowledge that 5+7=12 or "the interior angles of any triangle add up to a 180 degrees, or a straight line."

kant believes that these (and similar) truths of mathematics are synthetic judgments,

for example: euclid's elements!

true, the sum of the interior angles of a triangle is not contained in the concept of a triangle. yet, clearly, such truths are known a priori, since they apply with strict and universal necessity to all of the objects of our experience, without having been derived from that experience itself.

how is that possible?

5 + 7 = 12 tells us something new about the world. it's self-evident, and undeniably a priori, but at the same time it is synthetic.

thus kant proved that a proposition can be synthetic and known a priori.