Philosophy Now Forum

• What is an axiom?
  
  [WIKI] An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
  https://en.wikipedia.org/wiki/Axiom
  
  [Kant]"2. Axioms. These, in so far as they are immediately certain, are Synthetic a priori Principles.
  Now one Concept cannot be combined with another synthetically and also at the same time immediately, since, to be able to pass beyond either Concept, a third something is required to mediate our Knowledge.
  Accordingly, since Philosophy is simply what Reason knows by means of Concepts, no Principle deserving the name of an Axiom is to be found in it.
  Mathematics, on the other hand, can have Axioms, since by means of the Construction of Concepts in the Intuition of the Object it can combine the Predicates of the Object both a priori and immediately, as, for instance, in the Proposition that three points always lie in a plane." [CPR B670]

Does the above imply that 'axioms' can never be things-in-themselves existing absolutely of the human conditions [intuition]?

ChatGpt Wrote:

Yes, the passage from Kant implies that axioms are not things-in-themselves and cannot exist absolutely independent of human conditions, specifically our forms of intuition (space and time) and conceptual frameworks. Here’s why:

1. Synthetic A Priori Nature of Axioms in Mathematics
Kant characterizes axioms as synthetic a priori principles—statements that are both universal and necessary, yet not derived from mere analysis of concepts but from the construction of concepts in intuition. This construction occurs in the realm of human sensibility (space and time), which forms the basis for mathematics. For instance, the axiom that "three points lie in a plane" presupposes the spatial intuition that belongs to the human cognitive framework.

Thus, axioms are dependent on the a priori conditions of human intuition and understanding. They are not reflections of mind-independent realities, but rather truths valid within the phenomenal realm structured by our faculties.

2. Axioms Require Intuition for Immediate Synthesis
Kant argues that axioms combine concepts synthetically and immediately, which requires intuition as a mediating "third something." This contrasts with philosophy, where reasoning proceeds discursively through abstract concepts alone. Since intuition is tied to the human conditions of sensibility, axioms cannot refer to things-in-themselves, which lie beyond the bounds of intuition and conceptualization.

3. No Axioms in Philosophy
Kant asserts that philosophy lacks axioms because it operates purely through conceptual reasoning without the aid of intuition. This further reinforces that axioms are tied to human cognitive structures (specifically intuition) rather than to an independent, absolute reality.

Implications for Things-in-Themselves
Since things-in-themselves are inaccessible to human intuition and lie beyond the structures of space, time, and causality that condition our knowledge, axioms cannot pertain to them. Axioms are truths valid within the framework of the phenomenal world, not the noumenal world.

In summary, Kant's view suggests that axioms are epistemological constructs bound to human cognition and intuition, and therefore cannot exist as absolute, independent realities beyond human conditions.