Wednesday, August 24, 2011

The following is partially in response to Jody Azzouni’s recent post, via Eric Schliesser, on “New APPS”. However, points 1–5 and 8–11 below stand on their own; I refer to Azzouni only in points 6 and 7.

In point 7, especially, I read a lot into Azzouni’s words. I may have misunderstood his motivations. Even if my response does not apply to him, however, I suspect it applies to plenty of others.

1. Realism about the object of some representation R is the doctrine that the entity object to R is per se a res: that it has its realitas qua object of R , or, in other words, that R represents it with respect to its essence. Nominalism, strictly speaking, is the denial of realism in this sense. So far this has nothing to do with abstraction.

2. If the object of R is not per se a res, then R must be a mediate representation. It represents an entity as viewable under a certain aspect, so to speak, but it does not determine which entity that is. So it is parasitic on some at least possible immediate representation R0 of the same entity. Prior to Kant it was assumed that R0 itself must be a representation whose object is per se a res. Under that assumption (which Kant calls “transcendental realism”), the representation R about which we are nominalists must always be the non-essential representation of an entity which is essentially represented by R0 .

It is here that abstracta come in. If the nominalist can maintain that R is a mediate representation of this kind, the realist must say in opposition that R represents a mediate entity. For example: where the nominalist about position maintains that Socrates sitting is just a mediate representation of the primary substance, Socrates, and hence dependent on the immediate representation Socrates, the realist about position will account for the same facts by claiming that Socrates’ sitting is an entity which can exist only by inhering in a primary substance — that is, an accident. So the alleged entity whose existence is denied by the nominalist (as a multiplication of entities without necessity) is always an abstractum, in the strict sense that it is not self-subsistent.

3. But what does this have to do with what is currently called nominalism? I fear that the answer involves a confusion. The term “abstractum” has come to be used in a new sense, to mean, roughly speaking, an incorporeal primary substance. This is a bad way to use the term, since it is not only historically and etymologically baseless, but also mutually exclusive with the old, proper sense: an abstractum in the old sense is never self-subsistent and hence never a primary substance of any kind. As of now I can’t say exactly how this new, bad sense arose, but I suspect the story revolves around Goodman and Quine, so that the terms “abstract” and “nominalist” were always on the table at the same time. In any case, the result seems to be that people know nominalism is about denying the existence of abstracta, but no longer have the appropriate understanding of what abstracta (allegedly) are.

4. Nominalism thus appears as the attempt to explain away what looks like successful representation of incorporeal substances under the assumption that all substances are corporeal. There are various different approaches to doing that (fictionalism, etc.), which needn’t detain us. The main problem, common to all such approaches, is that “substance” and “incorporeal” are terms of Aristotelian metaphysics, whose application has now grown difficult, if not wholly illegitimate. The motivation for such a nominalism must, then, be a metaphysical thesis which is difficult, if not wholly impossible, to reconcile with modern empirical science: difficult, not because science seems to indicate that the thesis is false, but rather because science seems to indicate that the thesis is put in empirically illegitimate terms.

So there are two points where I worry about Azzouni’s post.

6. First, I can’t agree that “the nominalist’s scruples at the end of the day are empiricist ones.” Nominalism, in either the old or the new sense, is in no obvious way connected to empiricism. As far as I can tell, the feeling to the contrary comes about because nominalists don’t believe in spooky entities, and scientists don’t believe in spooky entities, and science is empirical. But there is no consistent way to define “spooky” which makes both of the first two clauses true, and the sense of “spooky” required by the first clause is empirically unsound.

7. Second, although there is indeed a difference between Field et al., and Azzouni’s “genuine” nominalist, it is not clear to me that the difference is to the genuine nominalist’s advantage. Field and company are guilty of letting the original definition of spookiness drift away into something vaguer; but, since the original definition was the problem, that is not in itself a vice. The “genuine” nominalist, I fear, may be precisely one who doesn’t allow such drift.

In Azzouni’s post the question comes to a head in the sentence: “Scientific facts about such particles and their properties has been established not merely on the basis of the simplicity of the scientific theories about such, but on the basis of actual physical probings of nuclei.” What is “actual physical probing,” and why should a nominalist/empiricist favor it over “mere” considerations of simplicity? I’m afraid the answer is that both nominalism (in the new, improper sense) and something one might think of as empiricism are being derived from metaphysical corpuscularism. We should be nominalists because everything other than big or little balls of stuff (cf. Azzouni’s previous example of a handball) is spooky; and we should be “empiricists” because big balls of stuff can know about other balls of stuff only because there are little balls which interact with the latter and then return to bounce off the sense organs of the former. So “empiricism” entails rejecting any theory, whatever its virtues, unless it portrays the world and our relationship to its constituents in this way. In particular, Azzouni points out that, given the axiom of choice, regions of topologically continuous spaces can’t be identified with little and big balls; so, he concludes, a nominalist/empiricist can’t be happy with any theory which describes the world as containing such regions.

I won’t say that this is a simplistic picture of physical theory, but I do think it’s false. In fact, to my mind the above conclusion is a reductio ad absurdum.

8. The business of the science of physics is not simply to describe the world, using terms prescribed by metaphysics. Rather, a fundamental physical theory is above all a thesis about the appropriate terms in which the world is to be described. If “actual physical probing” is to mean: the type of probing which actually serves the science of physics, it must be a probing in which the question: what is a physical description? is raised and potentially given a new answer. Physicists are empiricists because and to the extent to which they do not allow traditional concepts such as body and motion to constrain the possible answers to that question.

9. Since observation and measurement, if there are such things at all, must in some way be components of the world, a change in physical theory in principle brings about a change in the physical description of observation and measurement, or in other words in the empirical significance of the term “empirical.” Or it would produce such a change, rather, if there were ever such a thing as a physical description of observation and measurement. An observations qua observation or a measurement qua measurements is not of interest to physicists: i.e., these are not, as such, objects of the science of physics. (I even think this could serve as the definition of the science.) So in fact there never is a physical description of observation and measurement as such, and “empirical” never has a well-understood empirical significance.

10. A philosophical understanding of empiricism must, therefore, focus on the transcendental, rather than the empirical, significance of the term. And this brings us back to the topic of nominalism, properly speaking. I see no call to turn back from Kant. In the pre-Kantian version, to repeat, the claim that the object of some representation R is not per se real is connected to the claim that the same entity which is object to R is also represented qua res by some other representation, R0 . R0 represents the object immediately, whereas R represents it via the mediation of R0 . But Kant rejects this view precisely on the basis of a principle of empiricism, in a sense which abstracts from all empirical content of the term “empirical”: the principle, namely, that, for finite beings like ourselves, the immediate representation of an object is not intellectual. Since the representation of an object as possessing determinate realitas — that is, as limited by possession of an essence — is necessarily intellectual (for us: conceptual), this means that the object of cognition for us is always real only qua object of mediate representation. The object is not a res per se (Ding an sich), then, because intellectual representation is, in itself, the representation of something real, but is only mediately the representation of an object. Empirical realism is transcendental idealism.

11. This view faces severe difficulties in accounting for progress in physical science. Without going into all the details here, it probably should not surprise anyone if I say that they revolve around what Kant calls the pure form of intuition. Specifically, the challenge is to show how we can learn empirically that what appeared to be a feature of this pure form in general was actually specific to a degenerate case (as the absoluteness simultaneity, for example, follows from the degenerate metric of Galilean spacetime). But I think this problem is the right problem: that is, I think this is what we in fact don’t understand about progress in physics.

Saturday, May 28, 2011

What is a pure concept, according to Kant?

In a sense, there aren’t any. In a sense, that is, all our actual concepts are empirical concepts.

A concept is a kind of representation. Specifically, it is an intellectual representation of a discursive intellect. But what Kant calls a “representation” is what Descartes calls an “idea”: an act of a psychic faculty which not only, like every being in general, has formal or actual reality, but which also has objective reality — that is, reference (Beziehung) to a possible object. Hence for an act of our soul to be, strictly speaking, a concept, it must fulfill both formal conditions — that is, conditions which arise from the fact that it is supposed to be a mode of the thinking subject — and material conditions — that is, conditions which arise from the fact that it is supposed to have an object, and in that sense a “matter” (a materia circa quam: see the explanations in Thomas, Wolff, and Baumgarten). Because an intellect which represents via concepts (conceptus communes) is discursive, i.e. relies on a non-intellectual faculty (sensible intuition) for immediate relation to an object, these material conditions concern the fitness of the concept to be applied, via the imagination, to intuitions — that is, its schematizability.

General logic is formal logic. In general logic, that is, we abstract from all the material conditions and consider intellectual representations insofar as they fall under the formal conditions only. In transcendental logic, on the other hand, we abstract from some, but not all, of the material conditions. The dividing line is this: of the conditions which allow an object to be given via an image of the concept, some are conditions on the object given, whereas others must already be in place for there to be an object given. The latter are, for us, conditions on objects qua beings which are prior to being in order of predication: that is, transcendental conditions. In transcendental logic, we abstract from the former, but not from the latter.

However, to abstract from a condition is not to make it go away.1 There are always also a posteriori conditions for the objective reality of a discursive intellectual representation. In fact, the last transcendental condition, expressed in the category of modality, is just this: that any concept must represent its object as possible (due to some satisfiable empirical conditions), hence as actual (under those conditions), and as necessary (relative to the fulfillment of those conditions).2 The first and most general such empirical condition is that the object must be a real movable in space which exerts a force on our sensorium. This is therefore the definition of “matter” (again: materia circa quam) for the purposes of pure natural science, as opposed to the definition of matter in transcendental philosophy, as the object of our outer intuition ├╝berhaupt.3

It follows that a “pure concept” is always either an empirical concept considered in abstraction from its empirical content, or, if the phrase is supposed to name a fully concrete act of our intellectual faculty, then it names one which is not, strictly speaking, a representation: it fulfills the formal conditions and some, but not all, of the material ones. In particular, to speak of the pure understanding thinking the pure manifold of sense through pure concepts (as schematized by the pure imagination), is only a metaphorical way to describe the process of actual, empirical thought in abstraction from its a posteriori content.


1Cf. KdrV A281/B387-8: “so wird, durch eine sonderbare Übereilung, das, wovon abstrahiert wird, dafür genommen, daß es überall nicht anzutreffen sei.”

2Modality is last before being in order of predication. It is first in order of definition, however (A289/B346).

3“Wenn ich den Begriff der Materie nicht durch ein Prädicat, was ihr selbst als Object zukommt, sondern nur durch das Verhältniß zum Erkenntnißvermögen, in welchem mir die Vorstellung allererst gegeben werden kann, erklären soll, so ist Materie ein jeder Gegenstand äußerer Sinne, und dieses wäre die blos metaphysische Erklärung derselben” (MAdN, Ak. 5:481).

Thursday, January 13, 2011

Just a brief note: I feel like I’ve been thinking the wrong way all along about the relationship between Newtonian celestial mechanics and Aristotelian physics. Thanks to Kepler, I keep trying to see Newton as having shown that Aristotelian motion-around-the-center is a special case of motion along a conic section. That’s very confusing, in part because there doesn’t seem to be any limit of Newton’s theory in which Aristotle’s is recovered. But it only just now occurred to me (perhaps I’m slow): it’s actually Aristotelian motion-toward-the-center which is shown to be a special case (i.e., Aristotelian fall is shown to be a special casus = Fall). The tendency to move toward the center is what Aristotelian’s call “gravitation,” so you might call Aristotelian physics — Aristotelian sublunar physics, that is — the theory of particular gravitation (assuming you could get Aristotelians to admit that all levitation is due to buoyancy).

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.

Saturday, November 20, 2010

A story about recent trends in philosophy of science:

A bunch of people are sitting in a room discussing the question, whether you can step into the same river twice. Some deny that you can, either because the river is never the same or because motion is impossible. Some say that you can, but only because the river has a defective mode of being. Or that you can, because the form of the river remains while the matter changes; or because being is nothing; or because temporality is ecstasis; or because a river is a set-theoretic construct or a mereological sum of time slices or both; or because that’s the way we use the words “river” and “step,” and if you use them differently you no longer speak for us; etc.

Suddenly someone runs in and says: “Hey guys! Believe me (and you should, because I once spent five years trying and failing to become a fisherman): you’ve all been working with a really simplistic and impoverished version of what a river is. A real river is shallow on the edge and deep in the middle. Sometimes the water is green, sometimes brown, sometimes blue. And there are fish! Not once in all of your discussion do you so much as mention fish.”

Not news, and not helpful.