Nietzsche on the Synthetic A Priori

¨How are synthetic judgments a priori POSSIBLE?" Kant asks himself--and what is really his answer? "BY MEANS OF A MEANS (faculty)"--but unfortunately not in five words, but so circumstantially, imposingly, and with such display of German profundity and verbal flourishes, that one altogether loses sight of the comical niaiserie allemande involved in such an answer."

"But such replies belong to the realm of comedy, and it is high time to replace the Kantian question, "How are synthetic judgments a PRIORI possible?" by another question, "Why is belief in such judgments necessary?"--in effect, it is high time that we should understand that such judgments must be believed to be true, for the sake of the preservation of creatures like ourselves; though they still might naturally be false judgments! Or, more plainly spoken, and roughly and readily--synthetic judgments a priori should not "be possible" at all; we have no right to them; in our mouths they are nothing but false judgments. Only, of course, the belief in their truth is necessary, as plausible belief and ocular evidence belonging to the perspective view of life."

Friedrich Nietzsche- Beyond Good and Evil

Friedrich Nietzsche takes on Kant's attempt to derive universal laws of thought from which we can derive new knowledge about the world, in response to Hume's skepticism.

A priori knowledge is before experience, a posteriori is after experience. 

Analytic truths are those which are true by definition and synthetic truths are true by their relation to the world. 

Hume had only analytic a priori and synthetic a posteriori truths: only knowledge from experience tells us about different things in the world. 

Kant's response was that mind has universal conditions which make knowledge possible and so enter into all judgements, making a priori synthetic knowledge possible because we, "by a means", always use them with regards to experience so there's no absolute wall between reason and experience. And so there are universal necessary truths from which we derive knowledge about the world, even if we have to restrict them to what humans can perceive of the world in itself. 

Nietzsche's response is that such necessary conditions of thought are only necessary for our purposes as living beings and it doesn't matter if they relate to the real world. It is our activity which makes the a priori synthetic, as we are ultimately sensual beings in the world. The a priori itself is only our belief in our own judgements. We need synthetic a priori knowledge to serve our own purposes and it doesn't matter if it is universal either to our thoughts or to the world. It is relevant to our specific needs. 

We need not consider this a form of truth though, and so Hume is right.