How can the synthetical a priori be beneficial to the world's operations?

Answer 1

These terms were first used together in this way by the philosopher Immanuel Kant, so we'll go with his definitions for them:

An 'a posteriori' statement is one that requires experience to determine whether it is true or not. An 'a priori' one is the opposite of that - it is true or not, but there is no way to determine it by collecting evidence.

Likewise, an 'analytic' statement is one that is true or false by definition. A 'synthetic' statement requires some outside reference to determine its validity.

An example of an analytic statement is, "A triangle has three sides." This is simply what a triangle is. Kant obeserved that since all analytic statements were true or false within themselves, there was no such thing as an 'a posteriori analytic'. Personally, I tend to equate analytics with definitions themselves - a dictionary should, in theory be a book of analytic statements.

An example of an 'a posteriori synthetic' statement would be, "My bed is shaped like a triangle." This is a synthetic because it is not true by definition, and it is a posteriori because the only way to find out if it's true or not is to go and take a look at my bed (or some other evidence of that nature). Pretty much all of science falls into this category - if you need to measure it or see it in any way, it's here.

'A priori synthetic' statements were of particular interest to Kant. These are statements that require some outside reference to determine their truth, but for which no amount of experience can provide that reference. Into this category falls all of metaphysics ("God has a triangular bed."), and also all systems which are purely arbitrary. Kant put mathematics into this category, though later philosophers disputed this classification.

So what good are synthetic a priori systems? Well, all legal systems, being largely arbitrary, are synthetic a priori. There's no intrinsic reason why the president stays in office for four years, nor can you measure what the punishment for speeding is (though you can measure one instance, you can't measure 'punishment' as a whole). These things are just simply decided upon by people, and so have a truth outside of their definitions.

Hope that helps!

Answer 2

As you say Synthetic(al) a priori must be like starting with a rough approximation to arrive at a better approximation or even the exact solution in Numerical methods.In that case a synthetic a priori can not be of any use in daily operations of intelligence. The best it can help is to arrive at a contradiction or a paradox that can not be resolved unless the a priory hypothesis is rejected and hence rejected.