It can be shown that Wittgenstein’s Tractatus Logico-Philosophicus is not logically neutral, but already contains a hidden metaphysics.
The very first sentence — “1.1. The world is the totality of facts, not of things” — already shows that all of Wittgenstein’s further reasoning is not neutral, but is built upon a hidden metaphysics.
In Plato, metaphysics is built on the idea of eide — ideal, perfect images of things, existing objectively. In the everyday, manifest world, we see only incomplete, distorted projections of objectively existing knowledge about objects, together with all their really existing properties. Plato’s metaphysics, and after him in many respects the whole of European philosophy, is object-oriented.
In a certain sense, this corresponds to the metaphysical interpretation of early mathematics, where the unit of thought is numbers and their relations. Wittgenstein, however, places not the object but the fact, that is, the logical relation of things, as the main metaphysical principle.
The very concept of the fact already secretly shifts ontology from the thing to action or the potential for action. For example: the apple is ripe — this means it can be used as ripe. The stone lies still — an opposition to the stone moves. A fact always reports a certain action or potential for action as a system of relations.
On the other hand, Boole showed that formal logic can easily be reduced to Boolean algebra, that is, to an order of transformations. In turn, Boolean algebra is subject to the laws of abstract algebra and group theory, like any algebra. In group theory, what matters is not the objects themselves, but the possible transformations — the symmetry groups of transformations under which the object remains unchanged. If the symmetry groups coincide, then the object is isomorphic — that is, equal within the framework of group theory. The assumption of the primacy of facts, therefore, is not merely a logical but a metaphysical principle: the world is thought as a field of accomplished connections and transformations, and not as a collection of self-sufficient objects: the world is a fact means that, speaking in the language of abstract algebra, the world is built according to the principle of isomorphism: transformations over speculative, abstract objects are analogous to the group of transformations over the objects of the material, manifest world. The very act of description, the disclosure of the fact as such, says that the relations between real and logical objects are isomorphic — they coincide structurally. Or, in the language of continental philosophy, the assumption that the world is a fact already contains within itself a hidden metaphysics, which was developed in the continental philosophy of structuralism. It seems that there is, after all, a deep genetic connection between analytic and continental philosophy: from Wittgenstein’s hidden structuralist premise, he arrives at the conclusion that the world is not exhausted by structural relations and comes to the necessity of silence, which is close to the nothing of poststructuralist ideas.
In my school years, it seemed strange to me, almost mystical, that different mathematical paths lead to one and the same result. Why can a problem expressed by a cumbersome algebraic formula sometimes be solved much more simply — geometrically — while another method requires a long chain of algebraic transformations? These solutions look dissimilar, as though they belong to different languages, and yet they converge at one point. Behind this astonishment stands a more general question: what exactly is preserved in the transition from one mathematical form to another? Abstract algebra allows us to see that the identity of the solution is secured not by the external notation, but by the structure of transformations and invariant relations. Therefore, geometry and algebra here differ only in their mode of expression, but not in their deep organization.
The assertion that the world is a fact effectively transfers the principle of isomorphism from abstract objects to physical ones, thereby affirming that thought works not with objects as such, but with their system of relations.