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“Things in themselves” and noumena are similar in Kantian metaphysics (Critique of Pure Reason, mostly) and interchangeable much of the time. The phenomena/noumena divide is integral to Kantian metaphysics, so it is natural to wonder where “things in themselves” fall.

The Stanford Encyclopedia of Philosophy has a useful section discussing this (Things in themselves, noumena, and the transcendental object) where noumena are divided into “noumena in a positive sense” (positive noumena) and “noumena in a negative sense” (negative noumena). Then it is argued “things in themselves” are a subclass of negative noumena.

However, this provokes the question “What is the difference between positive noumena and negative noumena?”. The entry is difficult to understand on this matter.

Other natural questions are “Are there negative noumena that are not things in themselves? If so, what are they? What are some examples?”.

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  • Negative (this is not-p) and positive (this is q) determination of a thing play a role here: negative noumena are defined by "this thing, but not as we conceive it" - negatively, while positive noumena are defined by "conceived by intellectual intuition". One may object that intellectual intuition is only negatively defined as "not representing sensual objects", but that's not true: It is an intuition that realises (makes real) the things it conceives by conceiving it (§75 CoPJ, IIRC).
    – Philip Klöcking
    Sep 2 at 15:36
  • Pace SEP, noumena, negative ones especially, are not things in themselves, at least not in intension. They are abstract posits of the intellect, while the thing in itself is intended to be something ontological they aim at. There is a detailed study of fine distinctions b/w noumena, transcendental objects and things in themselves (and phenomena and appearances) that flow from the interplay of empirical and transcendental perspectives in Palmquist, Two Perspectives on the Object of Knowledge (from his book Kant’s System of Perspectives).
    – Conifold
    Sep 2 at 20:07
  • I guess mathematical things would be in the positive noumena class, as defined by the intuition we have of them?
    – CriglCragl
    Sep 2 at 23:18
  • @CriglCragl While Kant was sympathetic towards mathematics as a pure, a priori science, things generally are objects of experience, and that is what he means here. An intuition means specifically that which is the touching point between intellect and the world, as it were.
    – Philip Klöcking
    Sep 3 at 19:27
  • @Conifold I would be careful with the "ontological" label here since arguably, this kind of metaphysics is what Kant turns against, but I agree that negative noumena are rather a model of things-in-themselves (and thus to be distinguished from them).
    – Philip Klöcking
    Sep 3 at 19:31
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Kant must define noumena as a thing that is thought but not sensed. He cannot give a positive definition of noumena without contradicting the position that we cannot know anything about noumena through speculative reason.

Noumena is therefore an epistemological designation, not an ontological designation like thing-in-itself. Mathematical objects are things that are negative noumena (not sensed but thought), but aren't things-in-themselves bc they are merely formal entities (their objective reality consists of their formal reality - or the thing is merely its definition).

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  • Mathematical objects are synthetic a priori, that is, pure objects that don't come from experience, and are produced subjectively. That's out of the scope of sensible objects, which are perceived empirically by the senses, where the domain of the noumenon and phenomena correspond. Mathematical objects are NOT noumena.
    – RodolfoAP
    Oct 3 at 9:43
  • here is the problem with that. how do we do physics and engineering if mathematical objects dont relate to sense at all? they are not absolutely pure in the sense nothing related to sense but actually derived from according to kant from the sensible form of intuition of space just like the categories of substance, cause, and community are from the sensible intuition of time. My definition of negative noumena is what is not sensed but thought, positive noumena would be freedom and the moral law. You can think that noumena are physical objects, they dont even carry with them the idea of unity. Oct 4 at 22:50
  • @ Vishnu: that position fits better to empiricism, where mathematical objects are not pure, or Aristotelian Form and Matter, where forms do physically exist in nature hidden to our senses, but both are contrary to what Kant proposes.
    – RodolfoAP
    Oct 5 at 5:57
  • I get ur 2nd idea; math objects aren't "out there" in the physical reality since they are only formal. However, i dont understand why Kant can't empiricist at all? he did want to combine the two thoughts: rationalism and empiricism. I wouldnt say empiricism is "contrary" to Kants project. Oct 6 at 15:06

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