Wednesday, December 22, 2010

I said yesterday:

If we can construe a concept with an object (via a schema), then we know that it is at least the concept of an ens imaginarium (not, like the concept diangle, that of a nihil negativum [A291/B348]; and not, of course, of an ens rationis or nihil privativum, either).

But I now think I was misunderstanding the relationship between ens imaginarium and nihil negativum. A nihil negativum is indeed, in a sense, less (more nothing) than is an ens imaginarium, but only because the concept of it is more fully objective. That is: an ens imaginarium cannot properly be called either objectively, that is, transcendentally possible or impossible. (It must be subjectively, that is, formally possible; even an ens rationis is that much.) An ens imaginarium is “without substance” (A291/B347), i.e. without relation, and so the issue of possibility, i.e. modality, does not arise.

This must then affect the suggested interpretation of konstruieren. Rather than saying that the Konstruktion promotes a concept to the level where it becomes the concept of an ens imaginarium, but that then some further step is needed to promote it to the concept of a possible being, I should have said that the concept of an ens imaginarium is a concept which lacks a Konstruktion. Such a concept is not fully objective in that it fails to posit its object as heterogeneous to a ground of possibility. In particular, according to the thesis of transcendental idealism: such a concept fails to posit its object as heterogeneous to consciousness (pure space and time, entia imaginaria, are “in us”). A “construed concept” (konstruierte Begriff, see A716/B744), on the other hand, is a concept taken as a (purported) determination of the general (transcendental) relation between consciousness and its external object. As such it will be the concept of a possible object if the general relation admits of being determined in that way and, if not, not.

Our intellect is discursive. That is: there is for us a pure form of intuition, which puts specific conditions, not derivable from the unity of apperception itself, on the manifold which is to be unified in our having an object. (We have a particular species, Art, of intuition.) It is thanks to such conditions that the general relation in question, in our case — i.e. the “formal conditions of experience” (again: A218/B265) — admit of some determinations (e.g., determination to perception of a triangle) and not others (e.g., determination to perception of a diangle).1 And so, finally, what Kant actually says about the diangle (that is: about a non-Euclidean figure) in the proof of the First Postulate is that its impossibility

rests not on the concept [of the diangle] itself as such [an sich selbst], but rather on [the concept’s]2 Konstruktion in space, i.e. on the conditions of space and the determination of it [desselben, = of space]; these, in turn, have their objective reality, i.e., they pertain to possible things, because they include in themselves the form of experience in general. (KdrV A221/B268)

Which I take to mean: the geometrical Konstruktion of the concept diangle is a construal in the above sense, that is a construal upon which the possibility or, in this case, impossibility of the concept’s object can rest, because the conditions of space are (among) the specific conditions of our experience in general.

Then this is the same as what he says a little later about the concept triangle (the passage I already quote in my previous post): that its object would remain “always only a product of the imagination” (i.e., an ens imaginarium) if it were not that

The very same [eben dieselbe] imaging synthesis by which we construe a triangle [with its concept] in the imagination is wholly the same as [mit … einerlei] that which we exercise in the apprehension of an appearance, in order to make from it an empirical concept. (A224/B271)

That is: the procedure of the imagination in schematizing the concept makes it more than just a product of the imagination (posits it as heterogeneous to the imagination itself) only because the imagination in that procedure is subject to the same conditions to which it would be subject in construing an empirical concept with an object: that is, in experiencing such an object.3 The latter conditions — the conditions on experiencing the object of an empirical concept — are the formal conditions of experience in general.

This apparatus is what allows Kant to say that the Euclideanness of space is a special case, while at the same time denying that we know the possibility of any other case. We know the formal possibility of the concepts (such as diangle) which might function is such other cases, but we don’t know the possibility of such alternate geometries as geometries. That is: we don’t know the possibility of alternate specific formal conditions of experience in general. Because pure space and time are themselves mere entia imaginaria, we don’t even know the possibility of our own geometry except insofar as it is actual.

I’m interested in all of this, in part, because of a question I asked in a previous post, namely: what are the objects of mathematics? (That is: what should we say they are, taking that Kant must be at least partly wrong about that.) I want to suggest something like: that an object of mathematics is a kind of deconstrual of a concept, or of a type of concept. So, for example, one object of mathematics would be: the deconstrual of geometrical concepts which displays Euclidean geometry as a special case. Here “deconstrual” must be a form of determinate negation. What was the construal becomes absorbed into the formal content of the concept itself, and so abstraction from it becomes possible. (But, I’m suggesting, it’s the deconstrual itself, rather than the abstraction it subsequently allows, which is an object of mathematics.)


1In an intuitive intellect, so to speak, every representation which is not itself formally deficient is the representation of something possible. There is no other form with which it must be compared. But so to speak already displays how little we really understand the possibility of an intuitive intellect. I mean: that we do not know what are the formal conditions on such representations (their conditions qua determinations of the consciousness which is their subject). The law of noncontradiction is a formal restriction on discursive intellectual representations, i.e. universal concepts.

2This is the meaning according to the text of A and B. The 5th edition, apparently, reads derselben instead of desselben, so according to that we would have to understand “[the figure’s] Konstruktion.” As I pointed out in my previous post,
Kant speaks in both ways.

3Kant often, but I think not always, restricts Konstruktion to (what I am calling) the construal of a concept in the case where this can be done a priori (including, apparently, in the passage I’ve been discussing from A223–4/B271). I take it it is not too great a stretch to extend the term to the case of empirical concepts, simply by removing the qualification “a priori” in the definition “ihm gänzlich a priori einen Gegenstand geben” (A223/B271) or “die ihm korrespondierende Anschauung a priori darstellen” (A713/B741).

Tuesday, December 21, 2010

I’m in the midst of (yet again) trying to understand what Kant means by (to use the modern spelling) Konstruktion. I first had the thought, which no doubt I’ve had before, that it means “construal,” in the sense of “interpretation.”

That would fit a context such as KdrV A224/B271, where Kant equates “[den Begriff eines Triangels] konstruieren” (to construct/construe the concept of a triangle) with “to give [the concept] … an object,” i.e. to give it what he elsewhere calls “a reference [Beziehung] to objects, and hence a significance [Bedeutung]” (A145–6/B185), or, taking Bedeutung more broadly: to give it an “imagined significance [eingebildete Bedeutung]” (A84/B117), as opposed to a mere “logical significance” (A147/B186). To construe, that is, interpret, is to give significance.

Unfortunately, this appears not to be an accepted sense of the German verb (and moreover, according to the OED, is only a late and derivative sense of the English one). Moreover, we have to take into account another use of konstruieren in the exact same passage (from the proof of the First Postulate). Kant also talks there about the synthesis “wodurch wir in der Einbildungskraft einen Triangle konstruieren” (A224/B271). Whatever that means, it doesn’t mean “by which we construe (= interpret) a triangle in the imagination.”

Many would take it, instead, to mean something like: by which we construct (= build up) a triangle out of imaginary pieces, using an imaginary compass and an imaginary ruler, on a piece of imaginary paper. But that in itself makes little sense. Moreover, to the extent that it does make sense, it leads straight back to Descartes’s argument that we can’t do any such imaginary drawing in the case of a chiliagon (a point of which Kant is well aware: see A140/B179–80). Finally, such an understanding of konstruieren wouldn’t work at all in the first example, where Kant talks of constructing the concept of a triangle. A concept presumably can’t be drawn on any paper, no matter how imaginary.

So my new thought is this: that both of these ways of talking are short for: “einen Begriff mit einem Gegenstand konstruieren,” to construe a concept with an object (on the model of: to construe a verb with an object).

If we can construe a concept with an object (via a schema), then we know that it is at least the concept of an ens imaginarium (not, like the concept diangle, that of a nihil negativum [A291/B348]; and not, of course, of an ens rationis or nihil privativum, either). What he’s saying in the proof of the First Postulate (i.e., the proof that “what agrees with the formal conditions of experience (according to intuitions and concepts) is possible” [A218/B265]) is that to promote a concept from the concept of an ens imaginarium (which, although not impossible, is still a kind of nothing) to the concept of a possible being, we need to connect the procedure of the imagination in the schema of that concept to the synthesis of the imagination in perception.

So the object of the concept triangle would remain “always only a product of the imagination” if it were not that

The very same [eben dieselbe] imaging synthesis by which we construe a triangle [with its concept] in the imagination is wholly the same as [mit … einerlei] that which we exercise in the apprehension of an appearance, in order to make from it an empirical concept. (again, A224/B271)

The concept triangle passes this test, but the concepts of pure time and pure space as such, “die zwar Etwas sind, als Formen anzuschauen, aber selbst keine Gegenstände sind, die angeschaut werden” [A291/B347], do not. The synthesis by which the pure imagination produces time and space as pure images of the categories is not the same as any synthesis of the empirical imagination in perception (but is rather a condition of possibility of all such empirical synthesis). Hence the concepts of time and space are concepts of entia imaginaria.1

However, I’m not sure yet if this will work in other places or whether Kant anywhere actually uses the locution “to construe a concept with an object.”


1 The ursprünglich pure intuitions of time and space, I take it, do not have objects: intuitions without concepts are blind.

Tuesday, December 7, 2010

Here are two ways to think about mathematics (among possibly others). (1) As a discipline: an institution with some degree of political organization, a history, etc. (2) As an intellectual virtue: a habit of the cognitive faculty which consists in the enhanced potential to carry out certain cognitive acts. (I say “enhanced potential” because a faculty in itself is already a potential.)

It is in sense (2) that mathematics can be said to have objects. A cognitive act in general is a representation, and a representation, if at all successful (if not, e.g., self-contradictory) has some at least possible being as its object. If there is such a thing as mathematics at all in sense (2), then there must be something which characterizes the objects of just those acts for which it is the enhanced potential.

(Note I’m using “object” in the correct, relative sense: an object is the object of something. According to idealists of various stripes, every being, or at least every contingent being, must be the potential object of some cognitive faculty. But for just that reason it’s a bad idea, especially if, like me, you’re an idealist of some stripe or other, to use “object” as a synonym for “being.”)

It would be wrong to define the intellectual virtue (2) solely in terms of the institution (1). We should want to say that every normal human, past some fairly young age, possesses (knows) at least some mathematics. On the other hand, the people who play the leading role in institution (1) — that is, professional mathematicians — owe their position, on the whole, to being very good at something. There may be some disciplines where success depends, not on any cognitive ability, but rather, for example, on political or social skills. But if you know any even moderately good mathematicians you will know for sure that their skills don’t mostly lie in that direction. We should want the intellectual virtue (2), which we all share to some extent, to be the one successful mathematicians must have to an unusual extent.

But this puts constraints on the objects of mathematics. For example: although some may think that mathematicians are particularly good at counting to very high numbers, that is not actually the case. Hence, the cognitive act of representing a given (high) finite cardinality is not as such an act of the virtue (2). It follows that a given natural number simply as such (unless perhaps it is very small) is not an object of mathematics.