Kant on natural science (from in class discussions)

As pointed out in earlier posts Kant believes the basis of all knowledge is synthetic a priori. People may think to themselves how in the world can arithmetic and geometry to synthetic a priori. Aren’t we taught in school how to add, subtract and what different shapes are. Kant would argue that teachers are more like guides. They help you realize the knowledge you already have. If you think about a little kid, he intuitively knows the patterns of mathematics; he just doesn’t know the system (i.e the names of the numbers, and which number is higher or lower) by which it is applied on this earth. However, the patterns are there and they are a priori. Just as a child can sense time and space he is able to add and subtract once guided in the right direction. Kant also argues that natural sciences (mostly physics) are synthetic a priori. However, Hume uses an example that contradicts this statement with the pool ball example. According to Hume, we do not know a priori which direction the pool ball will go until we observe it. Thus, this knowledge is synthetic a posteriori. It is easier to dispute that the natural sciences are not synthetic a priori than it is arithmetic and geometry due to its relation to space and time.