There is a distinct contradiction, and yet, agreement, in the following two propositions:
P1. Mathematics cannot be shown to be complete
P2. We cannot but conceive of Mathematics, properly construed, as ideally composed of a set of axioms such that all and any system of mathematics can be reduced to a common simple system, or set of axioms such that shows a common genus to all mathematics.
This view, I maintain, is a Kantian view of mathematics. Kant’s constraints upon the proper conduct of science is that there ultimately originates a primary concept, but, that this concept is knowable or discoverable, or even actual, is not relevant, nor should we be too concerned if we never find it.
For science to be proper, Kant says, it must fit an ideal of knowledge, but such an ideal is projected (this entails the ideality of natural kinds) and not real. Such an ideal also seems to suggest that we use a bit of elipsis in our explanations and descriptions of science. A Kantian view of science also would set as a desideratum that there were a formalisability/mathematicisation constraint on anything if it is to be proper science at all.
The ideal is a projection, and is an “as if it were real” constraint (that is the ellipsis to which I speak of). Because it is a projection, our kinds and entities and laws within the scientific frame work not only can be subject to change, but desirably so, are they changeable, for scientific theories could always change, and are not rigidly set.
Rigidity is still present in the Kantian conception of science, however, in the desideratum of the constructability of formal langauges upon which we describe our phenomena. Consider the difference between ‘Water’ (h20) and water (that stuff we drink). Most, if not all the water we come across is not ‘water’, perhaps in some ways, ‘water’ does not exist, HOWEVER. Water necessarily presupposes ‘water’, in virtue of its ideality. For what makes water1 the same as water2 other than h20? Nothing.
H20 is criterial of water, but in a way, its pure form is never to be found in water, only ‘water’, which projects onto all thigns called water, makes sense of our empirical concept in such a way to be science. But, because ‘water’ is a priori regulatively ideal, it is also subject to change. The contradiction is, then, how is water necessarily h20, yet only indexical to our scientific understanding?
The answer to this lies in the conception of necessity. Necessity here, is defined as a criterial relation. Therefore, to say that “2 is a number” is necessarily true is to state a criteria. Necessity is criteria. But then, is not necessity similar to possibility? For criteria presupposes the conditions, and conditions is construed in the Kantian system as possibility. It would seem then that necessity can only take place as a concept where possibility is first defined, such that in a sense, necessity is only possible if, possibility allows, and this is necessarily so.
Destre (and Michael)