Noneism is a very rigourous and original philosophical doctrine, by and large superior to the classical mathematical philosophies. But there are some problems concerning the different ways of characterizing a universe of objects. It is very easy to understand the way a writer characterizes the protagonists of the novels he writes. But what about the characterization of the universe of natural numbers? Since in most kinds of civilizations the natural numbers are characterized the same way, we have the impression that the subject does not intervene in the forging of the characteristics of natural numbers. These numbers appear to be what they are, with total independence of the creative activity of the cognitive subject. There is, of course, the creation of theorems, but the potentially infinite sequence of natural numbers resists any effort to subjectivize its characteristics. It cannot be changed. A noneist might reply that natural numbers are non-existent, that they have no being, and that, in this respect, they are identical with mythological Objects. Moreover, the formal system of natural numbers can be interpreted in many ways: for instance, with respect to a universe of Skolem numbers. This is correct, but it does not explain why the properties of some universes are independent from subjective creation. It is an undeniable fact that there are two kinds of objectual characteristics. On the one hand, we have the characteristics created by subjective imagination or speculative thought; on the other hand, we find some characteristics that are not created by anybody; their corresponding Objects are, in most cases, non-existent but, at the same time, they are not invented. They are just found. The origin of the former characteristics is very easy to understand; the origin of the last ones is, a mystery.
Now, the subject-independence of a universe, suggests that it belongs to a Platonic realm. And as far as transafinite set theory is concerned, the subject-independence of its characteristics is much less evident than the characteristic subject-independence of the natural numbers. In the realm of the finite, both characteristics are subject-independent and can be reduced to combinatorics. The only difference between both is that, according to the classical Platonistic interpretation of mathematics, there can only be a single mathematical universe and that, to deductively study its properties, one can only employ classical logic. But this position is not at all unobjectionable. Once the subject-independence of the natural numbers system’s characteristics is posited, it becomes easy to overstep the classical phobia concerning the possibility of characterizing non-classical objective worlds. Euclidean geometry is incompatible with elliptical and hyperbolic geometries and, nevertheless, the validity of the first one does not invalidate the other ones. And vice versa, the fact that hyperbolic and other kinds of geometry are consistently characterized, does not invalidate the good old Euclidean geometry. And the fact that we have now several kinds of non-Cantorian set theories, does not invalidate the classical Cantorian set theory.
Of course, an universally non-Platonic point of view that includes classical set theory can also be assumed. But concerning natural numbers it would be quite artificial. It is very difficult not to surrender to the famous Kronecker’s dictum: God created natural numbers, men created all the rest. Anyhow, it is not at all absurd to adopt a whole platonistic conception of mathematics. And it is quite licit to adopt a noneist position. But if we do this, the origin of the natural numbers’ characteristics becomes misty. However, forgetting this cloudiness, the leap from noneist universes to the platonistic ones, and vice versa, becomes like a flip-flop connecting objectological with ontological (ideal) universes, like a kind of rabbit-duck Gestalt or a Sherrington staircase. So, the fundamental question with respect to the subject-dependent or subject-independent mathematical theories, is: are they created, or are they found? Regarding some theories, subject-dependency is far more understandable; and concerning other ones, subject-independency is very difficult, if not impossible, to negate.
From an epistemological point of view, the fact of non-subject dependent characteristic traits of a universe would mean that there is something like intellectual intuition. The properties of natural numbers, the finite properties of sets (or combinatorics), some geometric axioms, for instance, in Euclidean geometry, the axioms of betweenness, etc., would be apprehended in a manner, that pretty well coincides with the (nowadays rather discredited) concept of synthetical a priori knowledge. This aspect of mathematical knowledge shows that the old problem concerning the analytic and the a priori synthetical knowledge, in spite of the prevailing Quinean pragmatic conception, must be radically reset.
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