It is important to figure out whether the contradiction is in the world, or whether it is imposed onto the world by problems of our understanding.
§ 41
These views on the relation of subject and object to each other express the determinations which constitute the nature of our ordinary, phenomenal consciousness; but when these prejudices are carried out into the sphere of reason as if the same relation obtained there, as if this relation were something true in its own self, then they are errors — the refutation of which throughout every part of the spiritual and natural universe is philosophy, or rather, as they bar the entrance to philosophy, must be discarded at its portals.”
Whether concept before object
The relation between opinion and fact, or more appropriately, the relationship between concept and object, is a difficult one to categorize, mainly because the dominate opinion about thought and object is a confused one. Normally, scientific inquiry asks: what comes first, the concept or the object?
Metaphysics presents the challenge of order for the physical sciences; whether the concept comes before the object or the object before the concept? The implication in the word “concept” suggests that there is a basis of the object, so that there is a beginning for it coming into being and that is related and carried out though the entire duration the object can be said to be its end conception. For the sensation like vision for instance, obviously the object comes first, and then a concept is based on it, e.g., you see something that think about it. However the concept that is supposedly based on the object presents a multidimensional analysis of it.
What the understanding receives from the object says that the so called “object” does not imply what the word does, it is not a single and definite kind of thing, but it also has a broad meaning like all words do, that it is the same substrate that presents many different views and angles of itself depending on the point from where it is observed. This does not mean that the object is relative, in the sense that there are different objects all uncommon to each other, this is the crude conclusion made from seeing the world purely from a visual landscape. the fundamental concept that is prior to any object, and the object that we take to be prior, which is what maintains it as a definite and specific kind of thing and not anything other, is something else, also a concept, of what it is.
Instead of seeing the need for an order as part of a logical relation, the empirical scientist abstracts from a particular relation a certain kind of order and claims that order to be the organization of the world; empiricist say that the object is obviously first before the concept because this is how the world appears to the senses before ending up in a theory — in other words they say theory is based on what we see.
Metaphysics reposes its claims on a logical nature and not on a mere observational basis because even observation is guided by some purely abstract and theoretical end. Metaphysics is distinguished from what is done in mathematics not in the substance they deal with because both deal with purely abstract notions like numbers for math, and thoughts for metaphysics.
There difference is present in the way the abstract is seen as related to the concrete. In mathematics numbers are held to be abstractions from objects, while in metaphysics, objects are abstractions of more fundamental non-physical operations, which is as much present, if not even more real, than the objects.
Hegel says about the ontological problems with math;
Ontological problem with mathematics
“45. Mathematics may plume itself on its self-evidence, but this self- evidence rests on the poverty of its aim and the defectivenes~ of its material, in which philosophy should be ashamed to follow it. Mathe- matics only seeks to establish quantitative relations which belong ex- clusively to the surface ofthings . I ts rna terials are space and the unit, an empty, lifeless, repetitive element, set forth in fixed, dead propositions, linked together only by equational identities, and never progressing through opposition to some qualitatively different outcome. The in- commensurability of the different dimensions, which for mathematics constitutes a problem, is a luminous necessity to the philosopher.” (Phen of spirit)
Mathematics is good for operating in the world, we use it to measure nature in units, we use these units to build and construct nature, however we cannot relay on those so ever practical means of measure to understand the world. Mathematics cannot provide an understanding of substance in the world even though the substance of the world is in part mathematical. The most abstract fields of science like mathematics abstracts from what the senses perceive as physical objects it’s fundamental movements or relations, excludes the object but keeps the movement, and then concludes that the movement is universal to all objects. For example, take a man walking left, throw away the man but keep the determination of going left, and say that left motion can be potentially made by any object, any object can go left.
The problems of mathematics are the subject matters of philosophy. Philosophy and mathematics are similar because they both deal with the most abstract of principles which means the most fundamental relations as they apply as predicate to any particular thing taken to be concrete. The difference between math and philosophy also constitutes a sub-part difference found in philosophy as a science. Formal logic is a mathematical ontology dominant in metaphysics and specifically epistemology.
Pierce explains that the best translation of “episteme” is “comprehension”, which is “the ability to define a thing in such a manner that all its properties shall be corollaries from its definition” (CP 1. 232). The definition of something should include all the necessary ingredients to make sense of it. And this is ultimately the principle of logic, that all the routes of possibilities are included in each single instance of itself.