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Kant's ding-an-sich or noumena were roundly criticized by Fichte, Hegel, and other near contemporaries as incomprehensible, meaningless, or at least very unsatisfactory. How can we "know" or talk about the existence of an absolute "unknown" or "unknowable" things. Yet they do seem to have a necessary and positive value in Kant's system, an internal limit to certainties.

Have any philosophers since compared Kantian noumena to the functions of the empty set? Especially in the context of ordered sets. Perhaps this is only a very superficial notion. I have little grasp of set theory, so it is hard for me to carry the idea any further. From what I do know, both concepts seem to be regulative principles that remain ontologically controversial in similar ways.Is there anything more to it?

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    "Ding-an-sich" hasn't really been criticized by e.g. Hegel, but the epistemological access has, i.e. the strict disjunction between noumena and phenomena. Kant in his very late years called a noumenon an Unding, something that isn't even a thing. Sets and the laws and definitions on the other side are not any different if it happens to be an empty set. They are of the same sort, one could say, just different concretions of the same category. Noumena are neither ontologically, nor epistemologically of the same sort as phenomena are. They make a world of their own, the intelligible world.
    – Philip Klöcking ♦
    Jan 10, 2016 at 19:14
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    The identification of ding-an-sich and noumena is controversial, consensus seems to be that at best they are two different aspects of the "same", but even this "sameness" is not assertible in Kant's system. It even filtered down to Wikipedia en.wikipedia.org/wiki/Noumenon#Noumenon_and_the_thing-in-itself Palmquist directly addresses the issue in your first paragraph staffweb.hkbu.edu.hk/ppp/ksp1/KSP5A.html, essentially "ding-an-sich is unknowable" is a statement Kant makes on behalf of an outsider looking into his system, not on his own behalf, to draw the contrast.
    – Conifold
    Jan 10, 2016 at 22:42

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I do not see any similarity between Kant's noumenon, taken as the thing-in-itself, and the empty set from mathematical set theory.

Kant and science in accordance with him hypothesize the existence of objects in the real world. According to a constructivist epistomology we do not have direct access to these objects (thing-in-itself). But that hypothesis is the easiest hypothesis explaining why we make sensual experiences. And why the latter serve as input to our cognitive information processing.

In the field of mathematics you are free to create arbitrary ideas as long as they are free from inconsistencies. Mathematical concepts do not necessarily relate to the real world. The empty set is a clever concept from set theory. E.g., it is useful to define the intersection of two sets even when they are disjunct. Moreover the empty set is the basis of von Neumann's construction of natural numbers. Different from the things-in-themselves there exists only one empty set.

I would not name neither the concept of the thing-in-itself nor the empty set a principle. Possibly you can explain a bit why you consider both similar regulative principles.

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  • Perhaps "principle" is wrong. I was simply thinking of the functions they assume within the systems or constructed relations. Perhaps I am looking for some broader term that applies to the sort of centralizing "placeholder" functions like "zero" or "nothing," things that I picture as being both "inside" and "outside" any attempt at totalizing a system. A kind of amphibious ontology. I was mainly wondering if this was a comparison ever suggested in analytical philosophy.
    – Nelson Alexander
    Jan 10, 2016 at 19:21
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From a "conceptual" point of view, the empty set is quite simple .

Consider this analogy: numbers are used to count "things". Thus, what is the number 0 ? What does it mean to count "no things" ?

But from the "machinery" of arithmetic (addition, multiplication, etc.) the "limiting case" of the number that "count no things at all" is uncontroversial (and very uselful indeed).

In the same way, having assumed the concept of set (a collection of things), we can easily imagine the "limiting case" of a set without elements; with the basic axioms for sets (specifically : the Extensionality axiom) we can prove that all empty sets are equals, and thus speak of the empty set.

Having established its existence and basic property, the emptyset has a central role in the theory (see e.g. Von Neumann construction of the natural numbers).

Following your suggestion, we may say that "things" are known through their interactions with us; these interactions are the phenomena.

The "things" posited as the source of those interactions is the noumenon: it is completely unknowable to humans. In Kantian philosophy, the unknowable noumenon is often linked to the unknowable "thing-in-itself" (Ding an sich).

We can say that noumenon is the "limiting case" of a "thing" devoided of all "properties" (what we ascribe to it following our phenomenal experience).

A possible reference to check (but I've not read it) is Alain Badiou's ontology.

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  • Thanks. Makes sense. Yes, I was thinking also of Badiou, but am not familiar enough to know how he deals with Kant or what exactly his set theory ontology means. But note the difference between your use of "interaction" and Jo Wehler's "construction." Various interpretations of the noumenon with slightly different emphasis.
    – Nelson Alexander
    Jan 10, 2016 at 19:32
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Kant's things in themselves are more like Non-measurable sets, as opposed to the empty set. The existence of non-measurable sets can be proved, but they can never be "displayed".

Mathematicians are typically very careful in making sure that they are working with measurable sets (and more specifically Borel sets) when defining predicate calculuses or probability measures.

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