Foreshadowing the Concept

This will conclude my walk-through of the Preface to Hegel’s Phenomenology. Here he gives an explicit anticipation of what he calls “the concept”, which will be a key term in the Phenomenology‘s successor work, the Science of Logic. Along with “idea”, “concept” for Hegel represents something that is already beyond the naive opposition of subject and object.

If we imagine the naive view of subject and object as a pair of distinct points, a preliminary analogy for Hegelian concept and idea would be a line between the subject and object points. This can be understood as including all the meaningful content of experience, and can be taken as close as you like to either or both endpoints, but does not include either endpoint. Then the inversion of point of view that Hegel speaks of between ordinary consciousness and the standpoint of his logic would correspond to the relation between seeing experience in terms of the static duality of subject and object, and seeing it in terms of “living” concept and idea.

He begins this part with some remarks about mathematics that are overall very negative-sounding. This is setting up for a contrast between mathematics as the paradigm of static “formal” reasoning, and the meaning-oriented, becoming-oriented “dialectical” reasoning he wants to promote. To put the negative remarks in proper perspective, it is worth knowing that he will devote nearly 200 pages in the early part of the Science of Logic to a serious and sympathetic discussion of mathematics that shows good familiarity with the leading mathematical writers of his day.

“As for mathematical truths, one would hardly count as a geometer if one only knew Euclid’s theorems by heart without knowing the proofs” (p. 25).

Here he repeats the important point that knowledge does not consist in bare conclusions or propositions.

However, I think he goes astray when adds that really, “The movement of mathematical proof does not belong to the object but is a doing that is external to the item at hand” (ibid). I would say almost the opposite: the appearance of externality between theorem and proof — the idea that theorems have a status of simple truth independent of their proofs — reflects the very same kind of error that he pointed out before in the separation of results from the development that produced them.

It is true that a mathematical proof viewed as an object does not consist in the kind of becoming of knowing that Hegel attributes to good philosophical thinking. To mix terminology from computer science and Aristotle, mathematical proofs are in principle “statically” evaluable; this means they do not depend on any runtime accidents. In Platonic terms, mathematical objects are “eternal”, and proof is a kind of strict unfolding of their essence that we can imagine after the fact to have been predetermined, even though we don’t see the full predetermination in advance.

Earlier in the Preface, Hegel has argued that in the genuine becoming of knowing, “accidents” play an essential role, just as I would say they do in any actual working out of Aristotelian teleology. The means is not irrelevant to the end to the extent that we care about the end’s actualization. Like Aristotle, Hegel treats the process of actualization as primary.

Thus he is right that the becoming of knowing that philosophy ought to aim at does not — and ought not to — follow the canons of mathematical proof. In philosophy, we learn as much from our mistakes as from our successes, but errors in mathematics do not present the same kind of opportunities for improving our wisdom. Mathematics is not philosophy but something else. It is not “conceptual” in Hegel’s sense that involves a kind of “life” and “self-movement” of the concept.

However, he goes on to say that “In mathematical cognition, insight is an external doing vis-à-vis the item at issue” (p. 26, emphasis added). I don’t find this to be true today, and think it was, if anything, further from true in Hegel’s day.

Surely the maximal externalization of human insight from proof would be today’s computer-based proofs. While it is now possible to produce purely symbolic proofs whose validity depends only on the syntactic rules of a functional programming language, and sometimes even to produce proofs in a fully automated way, the really big successes of computer-based mathematical proof in recent decades have involved automated proof checkers that eschew fully automated proof development in favor of “dialogue” with an insightful human. At least in the current and foreseeable state of the art, human insight is not at all external to the development of mathematical proofs, even though the checking of completed or partial proofs for errors can be fully automated.

I say that mathematics is not philosophy, but its practice is far from being the mindlessly formal “defective cognition” he makes it out to be here in the Preface. Mathematical objects including completed proofs are static, but I say that the doing of mathematics essentially involves the activity of human intelligence.

“[W]hat is formal in mathematical convincingness consists in this — that knowing advances along the line of equality. Precisely because it does not move itself, what is lifeless does not make it all the way to the differences of essence…. For it is magnitude alone, the inessential difference, that mathematics deals with” (p. 27).

Mathematics only deals with things that are in principle strictly univocal. Strictly univocal things lack “life” for Hegel, and are therefore inessential.

“In contrast, philosophy does not study inessential determinations but only those that are essential. The abstract or the non-actual is not its element and content; rather, its element and content is the actual, what is self-positing, what is alive within itself, or existence in its concept. It is the process which creates its own moments and passes through them all; it is the whole movement that constitutes the positive and its truth. This movement just as much includes within itself the negative ” (p. 28).

Philosophy for Hegel is especially concerned with actuality, and as with Aristotle, what is actual is not simply to be identified with what is factual.

“Appearance is both an emergence and a passing away which does not itself emerge and pass away… which constitutes the actuality and the living moment of truth…. Judged in the court of that movement, the individual shapes of spirit do not stably exist any more than do determinate thoughts, but they are also equally positive, necessary moments just as much as they are negative, disappearing moments” (pp. 28-29).

Here he is using “appearance” in a very different way from what Plato called mere appearance. It seems to be something like the concrete manifestation that is necessarily implicit in actuality.

“In the whole of the movement… what distinguishes itself in it and what gives itself existence is preserved as the kind that remembers, as that whose existence is its knowing of itself” (p. 29).

Previously, he said that the true is the whole. In this movement of self-knowing, which is quite different from being an object for oneself, the subject and object that are quite distinct for ordinary consciousness become interwoven.

“It might seem necessary to state at the outset the principal points concerning the method of this movement…. However, its concept lies in what has already been said, and its genuine exposition belongs to logic, or is instead even logic itself, for the method is nothing but the structure of the whole in its pure essentiality” (ibid).

The entry point for what Hegel calls “logic” is what I have glossed as being at home in otherness. For Hegel, logic is not about formal manipulations. It is a very non-ordinary way of looking at things that leaves distinctions of subject and object behind. The Phenomenology is supposed to provide a way into this perspective, starting out from what Aristotle would call the way things (ordinarily) are “for us” (see Otherness; At Home in Otherness).

“In everyday life, consciousness has for its content little bits of knowledge, experiences, sensuous concretions, as well as thoughts, principles, and, in general, it it has its content in whatever is present, or in what counts as a fixed, stable entity or essence…. [I]t conducts itself as if it were an external determining and manipulation of that content” (p. 30).

Ordinary consciousness regards things in the world as fixed, pre-known, and manipulable. It regards itself as somehow standing off to the side from the order of the world, and implicitly as able to act in complete independence from that order. It is “Cartesian”. The weakness of this point of view is progressively exhibited in the Phenomenology.

“Science may organize itself only through the proper life of the concept…. [D]eterminateness… is in science the self-moving soul of the content which has been brought to fulfillment. On the one hand, the movement of ‘what is’ consists in becoming an other to itself and thus in coming to be its own immanent content; on the other hand, it takes this unfolding back into itself, or it takes its existence back into itself, which is to say, it makes itself into a moment, and it simplifies itself into determinateness” (p. 33).

Hegelian rational “science”, sustained in otherness, examines a movement of “logical” unfolding and return that (unlike the unfolding and return in neoplatonism) occurs not in eternity but in worldly coming-to-be. The fact that the return occurs in becoming and in time gives it the form not of a simple circle but of an open-ended spiral that never literally returns to its origin.

“[S]cientific cognition requires… that it give itself over to the life of the object” (ibid, emphasis added).

In the main body of the Phenomenology, the Consciousness chapter shows the limitations of the ordinary view that we are wholly separate from the object, and the Self-Consciousness chapter develops a sharp critique of the attitude of the master who attempts to claim unilateral control over both objects and other people.

“[T]he stable being of existence… is itself its own inequality with itself and its own dissolution — its own inwardness and withdrawal into itself — its coming-to-be. — Since this is the nature of what exists, and to the extent that what exists has this nature for knowing, this knowing is not an activity which treats the content as alien. It is not a reflective turn into itself out of the content… [W]hile knowing sees the content return into its own inwardness, its activity is instead sunken into that content, for the activity is the immanent self of the content as having at the same time returned into itself, since this activity is pure self-equality in otherness” (p. 34).

Here we have a direct statement about what overcoming alienation ought to look like.

“Its determinateness at first seems to be only through its relating itself to an other, and its movement seems imposed on it by an alien power. However, … it has its otherness in itself…, for this is the self-moving and self-distinguishing thought, the thought which is its own inwardness, which is the pure concept. In that way, the intelligibility of the understanding is a coming-to-be, and as this coming-to-be, it is rationality” (p. 35).

Overcoming alienation is anything but the suppression of what is other. Neither is it a return to an original perfection. Rather, it consists in a non-ordinary sense of self that is not opposed to the other or to the field of otherness.

“Logical necessity in general consists in the nature of what it is [for something] to be its concept in its being. This alone is the rational, the rhythm of the organic whole, and it is just as much the knowing of the content as that content itself is the concept and the essence…. The concrete shape which sets itself into movement… is only this movement, and [its concrete existence] is immediately logical existence. It is therefore unnecessary to apply externally a formalism to the concrete content. That content is in its own self a transition into this formalism, but it ceases to be the latter external formalism because the form is the indigenous coming-to-be of the concrete content itself” (ibid).

In emphasizing the contentfulness of the concept rather than formal syntax as the true driver of logical necessity, he seems to be talking about something like what Sellars and Brandom call material inference.

“Although what is stated here expresses the concept, it cannot count as more than an anticipatory affirmation. Its truth does not lie in this narrative exposition” (p. 36, emphasis added).

Truth, once again, must lie in an extensive development that is never truly finished by us humans. This remark could reasonably apply to the whole Preface, but I am struck by the reference to the concept and by the place in which it occurs, just after an explicit reference to logic. Here he is looking forward not only to the main body of the Phenomenology, but even more so to what will become the Science of Logic.

He goes on to criticize “clever argumentative thinking” at length, and to contrast it with “comprehending thinking”.

“[C]lever argumentation amounts to freedom from content and to the vanity that stands above all content” (p. 36).

By Hegel’s high standards, any argument that assumes meanings are determined in advance at least tends toward the vanity and irresponsibility of what Plato and Aristotle denounced as sophistry.

Hegel wants to recommend instead that “This vanity is expected to give up this freedom, and, instead of being the arbitrary principle moving the content, it is supposed to let this freedom descend into the content and move itself by its own nature…. This refusal both to insert one’s own views into the immanent rhythm of the concept and to interfere arbitrarily with that rhythm by means of wisdom acquired elsewhere, or this abstinence, are all themselves an essential moment of attentiveness to the concept” (pp. 36-37).

Moreover, what plays the role of the subject of thought is not at all the same for comprehending thinking as it is for clever argumentation.

“[C]lever argumentative thinking is itself the self into which the content returns, and so too, the self in its positive cognition is a represented subject to which the content is related as accident or predicate. This subject constitutes the basis in which the content is bound and on the basis of which the movement runs back and forth” (p. 37).

He continues, “Comprehending thinking conducts itself in quite a different way. While the concept is the object’s own self, or the self which exhibits itself as the object’s coming-to-be, it is not a motionless subject tranquilly supporting the accidents; rather, it is the self-moving concept which takes its determinations back into itself. In this movement, the motionless subject itself breaks down; it enters into the differences and the content and constitutes the determinateness, which is to say, the distinguished content as well as the content’s movement, instead of continuing simply to confront that movement” (pp. 37-38).

Comprehending thinking “enters into the differences and the content”.

“[T]here is an obstacle based in the habit of grasping the speculative predicate according to the form of a proposition instead of grasping it as concept and essence” (p. 41).”

The form of a proposition is simply to be true or false. He may also have in mind the form of predication. Grasping something as concept and essence is treating it as articulable meaning to be interpreted, rather than as a mere thing to be pointed at.

“True thoughts and scientific insight can be won only by the labor of the concept. Concepts alone can produce the universality of knowing” (p. 44).

Hegel’s Preface

In Nature, Ends, Normativity I raised the question of what happens to Aristotelian teleology when we look at it through a Kantian critical lens, then made a preliminary gesture toward its resolution by invoking Hegel’s challenge and admonition to us in the Preface to the Phenomenology of Spirit to make ourselves at home in otherness. Just how making ourselves at home in otherness helps with the question about Aristotle and Kant may not be at all clear yet. For now it’s just a thought to keep in mind.

First I want to try to explore Hegel’s larger point in the Preface that I risked reducing to the phrase “make ourselves at home in otherness”, and let that lead where it may. This post won’t get to the point where the phrase is introduced. It’s even possible that I’m remembering it from a paraphrase in H. S. Harris’ outstanding commentary. I’ll walk through the Preface over the course of several posts, using Terry Pinkard’s translation published in 2018.

Hegel’s Preface is an extremely dense text that seems to very deliberately follow a non-linear order. It does have a development, but it doesn’t just proceed from beginning to middle to end. Rather, it seems to repeatedly circle around several key insights, adding a little more each time. Famously, he begins by rejecting the very idea that philosophical truth is the kind of thing that could be “introduced” or made easily digestible by a conventional preface.

He goes on to repeatedly criticize two chief ways in which philosophy is made digestible and shallow — one that treats truth as something merely formal, and one that claims to leap into absolute knowledge by means of intellectual intuition. Especially in some of the later parts, Hegel gives a number of valuable hints at what he thinks serious philosophy ought to look like.

“[C]onventional opinion holds that the opposition between the true and the false is itself fixed and set…. It does not comprehend the diversity of philosophical systems as the progressive development of truth as much as it sees only contradiction in that diversity…. However, at the same time their fluid nature makes them into moments of an organic unity in which they are not only not in conflict with each other, but rather, one is equally as necessary as the other” (p. 4).

Hegel is not at all advocating some trite relativism or erasure of distinctions here. He is objecting to the artificially “fixed and set” way in which what he calls formalism sees the truth of propositions taken in isolation. More positively, he seems to be suggesting that we view the great philosophers as participants in a common, mutually enriching dialogue rather than as competitors.

“[T]he subject matter [of philosophy] is not exhausted in its aims; rather, it is exhaustively treated when it is worked out. Nor is the result which is reached the actual whole itself; rather, the whole is the result together with the way the result comes to be…. [T]he unadorned result is just the corpse that has left the tendency behind…. The easiest thing of all is to pass judgment on what is substantial and meaningful. It is much more difficult to get a real grip on it, and what is the most difficult of all is both to grasp what unites each of them and to give a full exposition of what that is” (p. 5).

Here Hegel makes a very Aristotelian point about the essential role of actualization. What he is directly applying it to is philosophical accounts of things. We should be interested not just in philosophy’s ostensible conclusions, but in how they were arrived at. (But an analogous point could be made about the actual working out of Aristotelian teleology in the world. What is relevant to this is not just pure ends by themselves, but the whole process by which ends are actualized by means of concrete tendencies.)

“In positing that the true shape of truth lies in its scientific rigor — or, what is the same thing, in asserting that truth has the element of its existence solely in concepts — I do know that this seems to contradict an idea (along with all that follows from it), whose pretentiousness is matched only by its pervasiveness in the convictions of the present age. It thus does not seem completely gratuitous to offer an explanation of this contradiction even though at this stage such an explanation can amount to little more than the same kind of dogmatic assurance which it opposes” (p. 6).

By “scientific” he basically means rational. Hegel here aligns himself with Kant’s emphasis on the conceptual and discursive character of rationality, and with Kant’s closely related rejection of claims to immediate knowledge by intellectual intuition. He is particularly alluding to claims of intellectual intuition in the philosophy of nature by followers of Schelling, as well as to the religiosity of immediate feeling promoted by followers of the German literary figure F. H. Jacobi, from whom Kierkegaard borrowed the image of the leap of faith.

The “true shape of truth” Hegel contrasts these with lies in conceptual elaboration — interpretation and explanation, not just asserted conclusions. The measure of truth is the insight and understanding it gives us. He also notes a difficulty that I often feel in attempting to summarize the results of a substantial development: summaries always run the risk of shallowness and dogmatism.

Hegel continues ironically that for his contemporary opponents, “The absolute is not supposed to be conceptually grasped but rather to be felt and intuited” (ibid).

“There was a time when people had a heaven adorned with a comprehensive wealth of thoughts and images. The meaning of all existence lay in the thread of light by which it was bound to heaven and instead of lingering in this present, people’s view followed that thread upwards towards the divine essence; their view directed itself, if one may put it this way, to an other-worldly present. It was only under duress that spirit’s eyes had to be turned back to what is earthly and kept fixed there, and a long time was needed to introduce clarity into the dullness and confusion lying in the meaning of things in this world, a kind of clarity which only heavenly things used to have; a long time was needed both to draw attention to the present as such, an attention that was called experience, and to make it interesting and to make it matter. — Now it seems that there is the need for the opposite, that our sense of things is so deeply rooted in the earthly that an equal power is required to elevate it above all that. Spirit has shown itself to be so impoverished that it seems to yearn for its refreshment only in the meager feeling of divinity, very much like the wanderer in the desert who longs for a simple drink of water. That it now takes so little to satisfy spirit’s needs is the full measure of the magnitude of its loss” (pp. 7-8).

Hegel was critical of traditional Augustinian other-worldliness, but saved his special disdain for followers of Schelling and Jacobi.

“The force of spirit is only as great as its expression, and its depth goes only as deep as it trusts itself to disperse itself and to lose itself in its explication of itself. — At the same time, if this substantial knowing, itself so totally devoid of the concept, pretends to have immersed the very ownness of the self in the essence and to philosophize in all holiness and truth, then what it is really doing is just concealing from itself the fact that instead of devoting itself to God, it has, by spurning all moderation and determinateness, instead simply given itself free rein within itself to the contingency of that content and then, within that content, given free rein to its own arbitrariness” (ibid).

It is not enough just to have a concept like the absolute Idea.

“However, just as little of a building is finished when the foundation is laid, so too reaching the concept of the whole is equally as little the whole itself. When we wish to see an oak tree with its powerful trunk, its spreading branches, and its mass of foliage, we are not satisfied if instead we are shown an acorn. In the same way, science, the crowning glory of a spiritual world, is not completed in its initial stages. The beginning of a new spirit is the outcome of a widespread revolution in the diversity of forms of cultural formation; it is both the prize at the end of a winding path as it is the prize won through much struggle and effort” (p. 9).

He implicitly recalls Aristotle’s argument that the oak tree is logically prior to the acorn, and cautions against assuming perfection in beginnings.

“Only what is completely determinate is at the same time exoteric, comprehensible, and capable of being learned and possessed by everybody. The intelligible form of science is the path offered to everyone and equally available to all” (p. 10).

When the Idea is kept vague, it becomes the province of claims of esoteric knowledge and special genius. Here he links the idea of rational “science” to a democratic tendency. But we should also beware of premature claims.

“At its debut, where science has been wrought neither to completeness of detail nor to perfection of form, it is open to reproach” (ibid).

He goes on at length about the formalism of the Schellingians’ insistence that all is one. The rhetoric is strong, but he is standing up for the importance of difference and distinction, which I completely support.

To condense a good deal, “when what is demanded is for the shapes to originate their richness and determine their differences from out of themselves, this other view instead consists in only a monochrome formalism which only arrives at the differences in its material because the material itself has already been prepared for it and is something well known…. [N]owadays we see the universal Idea in this form of non-actuality get all value attributed to it, and we see that what counts as the speculative way of considering things turns out to be the dissolution of the distinct and determinate, or, instead turns out to be simply the casting of what is distinct and determinate into the abyss of the void…. To oppose this one bit of knowledge, namely, that in the absolute everything is the same, to the knowing which makes distinctions… that is, to pass off its absolute as the night in which all cows are black — is an utterly vacuous naivete in cognition” (pp. 11-12).

(Remaining parts of this walk-through of the Preface are, in order, Substance and Subject; At Home in Otherness; Otherness; and Foreshadowing the Concept.)

Form Revisited

My original skeletal note on form dates back to the first months of my writing here. This is intended to be the beginning of a better treatment.

When I speak of form, I have in mind first of all the various uses of the term in Aristotle, but secondly a family of ways of looking at the world largely in terms of what we call form, as one might broadly say that both Plato and Aristotle did. Then there is a very different but also interesting family of uses in Kant. There are also important 20th century notions of “structure”.

Form in its Platonic and Aristotelian senses is closely related to what we might call essence, provided we recognize that essence is not something obvious or pre-given. At the most superficial level it may refer to a kind of shape, but it may involve much more.

Plato was classically understood to assert the existence of self-subsistent intelligible “forms” that do not depend on any mind or body. I prefer to emphasize that he put a notion of form first in the order of explanation — ahead of any notion of something standing under something else, ahead of notions of force or action, ahead of particular instances of things. Related to this, he put the contents of thought before the thinker, and used the figure of Socrates to argue that a thing is not good because God wills it to be so, but rather that God wills a thing because it is good.

Aristotle identified form with the “what it is” of a thing. He put form and things like it generally first in the order of explanation, but explicitly argued that form is not self-subsistent. At the same time, he made the notion of form much more lively. While Plato had already suggested that form has an active character and that the soul is a kind of form, most of his examples of form were static, like the form of a triangle or the form of a chair. Aristotle on the other hand was very interested in the forms of the apparent motions of the stars; the marvelous variety of the forms of animals, considering not only their anatomy but patterns of activity and ways of life; and the diverse forms of human communities, their ways of life and institutionalized concepts of good. Form figures prominently in the development of the notion of ousia (“what it was to have been” a thing) into potentiality, actualization, and prior actuality in Aristotle’s Metaphysics. Aristotelian form is interdependent with logical “matter” in such a way that I think the distinction is only relative. It is also inseparable from a consideration of ends. (See also Form as Value; Form, Substance.)

At first glance, Kant’s notion of form seems like the “mere form” of formalism, contrasted with something substantive called “content”. A certain notion of formalism is so strongly identified with Kant that in some contexts it has become a name for whatever was Kant’s position. I think some of Hegel’s criticisms of Kantian formalism are legitimate, and some overstated. In any case, the categorical imperative and its consequences of respect for others and the value of seeking to universalize ethical precepts — perhaps the first really original constellation of ethical ideas since Aristotle — are deeply tied to Kant’s so-called ethical formalism. Kant seeks a formalist path to the highest good, and argues that only a formalist path can truly reach it. The fact that it is a path to the highest good has deep implications for the meaning of this kind of “formalism”, and sets it apart from what is referred to as formalism in mathematics, logic, or law. This could also be related to Kant’s idea that ethical reason comes before tool-like reason in the order of explanation.

The 20th century notion of “structure” — to hazard a simplifying generalization — is about understanding each thing in terms of its relations to other things — principally how things are distinguished from one another, and how one thing entails another. Structure is form interpreted in a relational way that transcends fixed objects and properties. Objects and properties can be defined by relations of distinction and entailment.

Formalist Existentialism?

The English translation of Alain Badiou’s Being and Event III: The Immance of Truths has just been published. There is not much in this book that I would recognize as philosophy; neither other philosophers nor questions of interpretation are discussed at any length. Badiou primarily wants to assert that actual infinity is established by classical set theory as an “absolute ontological referent”.

Badiou’s deepest influences are Sartrean existentialism and what at first appears to be a kind of extreme formalist view of mathematics. For Sartre, what distinguishes the human is an ability to make utterly arbitrary choices. Such views have historically been justified by appeals to human likeness to an omnipotent God that, while commonly raised by religious sectarians, actually diverge from more broadly accepted views of orthodoxy in religion, which temper appeals to raw infinite power by emphasizing that God is good and more reasonable than we are, and therefore does not act arbitrarily. Sartre and Badiou, however, are both militant atheists who aim to ground the argument for human arbitrariness in some other, nonreligious way.

I think what we need for ethics is to recognize that we are beings who partake of an active character. We do things, and along the way we make choices between alternatives, but real-world decisions — the only kind there are — are never made in a vacuum. I think activity necessarily involves purposefulness (seeking some good, i.e., something judged by someone to be good in some way, even if we would completely reject the judgment). Any kind of purpose at all is incompatible with complete arbitrariness. (See also Beings.) But Badiou would disqualify this whole line of thought, because he doesn’t believe in ethics or in purposes that are independent of arbitrary decision.

I call Badiou’s appeals to formalism in mathematics extreme because — utterly contrary to the spirit of the early 20th century program of David Hilbert, which is usually taken as the paradigm of mathematical formalism — Badiou claims that his formalist arguments directly apply to the real world. Even so-called mathematical Platonism only asserts the independence of mathematical objects, and nothing like the immediate relevance to politics claimed by Badiou. The whole point of Hilbert’s formalism is that it doesn’t care about the real world at all. For Hilbert, mathematics consisted in purely hypothetical elaboration of the consequences of arbitrary axioms and definitions. He likened this to a kind of game.

Badiou’s use of purely formal elaboration from arbitrary starting points is decidedly not hypothetical; it is combined with an extreme realism. According to Badiou, Paul Cohen’s theorems about generic subsets, for instance, are supposed to directly lead to political consequences that are supposed to be liberating. We are supposed to get some enlightenment from considering, e.g., immigrant workers as a generic subset, and this is supposed to represent a kind of unconditional or “absolute” truth that is nonetheless immanent to our concrete experience. But the treatment of arbitrary hypotheses as unconditional truths is utterly contrary to what Hegel meant by “absolute” knowledge, which I would argue is really supposed to involve the exact opposite of arbitrariness. Hegel’s “absolute” is about as far from Badiou’s “absolute ontological referent” as could be. (See also Hegelian Finitude.)

I am only a moderately well-informed mathematical layman and claim no deep understanding of Cohen’s results, but the basic idea of a generic set or subset seems to be that it is an arbitrary selection of elements from some pre-existing set. Being arbitrary, it has no definition or characteristic function (other than by sheer enumeration of its elements). But in classical set theories, new sets and subsets can be formed from an arbitrary set. Badiou relates this to Georg Cantor’s proof that any set has more subsets than elements. In itself, I find the latter unobjectionable. But Badiou likes classical set theory because it gives a putative mathematical respectability both to arbitrary beginnings and to actual infinity. (See also Categorical “Evil”; Infinity, Finitude.)

According to Badiou, belief in actual infinity is revolutionary and good, whereas disbelief in actual infinity is conservative and bad. Infinity is supposed to be revolutionary precisely because it is unbounded. This just means that it can be used as a putative license for arbitrariness. I want to insist on the contrary that there is nothing socially progressive about arbitrariness! Badiou’s recommended political models are the chaotic Maoist cultural revolution of the 1960s and the ephemeral May 1968 Paris uprising. I don’t see that the oppressed of the world gained any benefit from either.

Badiou explicitly endorses arguments of the notorious Nazi apologist Carl Schmitt that were used to justify a permanent “state of exception” in which absolute political power is asserted. This intellectual red-brown coalition is unfortunately being taken seriously by some academic leftists. The unifying theme is the claim that metaphysical support for arbitrariness is the key to achieving social justice. There are much better ways…

Commitment to Commitment

A commitment to the practices associated with commitment is more fundamental than any particular commitment we may have. To say it another way, taking our committedness seriously is more important than the exact content of our particular commitments as to what is good and true, or to what we will do.

A high level of seriousness about commitments does not mean sticking to our guns at all cost. If we truly take our commitments seriously, that ought to mean that we also want to improve them when we have the opportunity, and to fix them when they are broken.

The American transcendentalist Ralph Waldo Emerson (1803-1882) famously made the remark that “a foolish consistency is the hobgoblin of little minds…. With consistency a great soul has simply nothing to do”. One might hope that he really meant to distinguish between a foolish consistency and a wise one — between a kind of rigid adherence to mere formalisms, and what I might call consistency in substance or essence or deep meaning. The latter would be more akin to personal integrity.

Emerson himself was a bit intemperate in the passage that followed (“Speak what you think now in hard words, and tomorrow speak what tomorrow thinks in hard words again, though it contradict every thing you said today”). He further confuses the matter by connecting this message with the theme that “great souls are always misunderstood”. This is all in his essay “Self-Reliance”. The rhetoric is quite memorable and there is a sense in which each of these sayings has validity, but they are both what Hegel would call “one-sided” formulations that are highly vulnerable to misuse. Their combination suggests the dangerous implication that it must be our fault if we don’t understand the one who says contradictory things. This clearly goes too far.

In the course of arguing that it is actually possible for a human to have a kind of general knowledge of being, Aristotle in Metaphysics book IV chapter 3 famously defends a principle of noncontradiction that is not merely formal.

He says in part, “For that which is necessary for one who understands any of the beings whatever to have is not a hypothesis” (Sachs translation, p. 58).

“[W]hat it is, after this prelude, let us state. It is not possible for the same thing at the same time both to belong and not to belong to the same thing in the same respect (and as many other things as we ought to specify in addition for the sake of logical difficulties, let them have been specified in addition). And this is the most certain of principles” (p. 59).

“[T]he starting point… is not the demand that one say something either to be or not to be (for perhaps one might suppose that this would require from the outset the things to be shown), but that what he says must mean something to both himself and someone else; for this is necessary, if he is going to say anything” (p. 60).

Robert Brandom argues that all the most important and valuable parts of Kant’s thought can be reconstructed in terms of the process of synthesizing a unity of apperception. This process is not a sequence of events that happen in the world; it is an ethical task for which we are responsible.

No truths follow from the principle of noncontradiction alone. In particular, it is not a deductive source of metaphysical conclusions.

On the other hand, it is what in Kantian language might be called a moral imperative. To be committed to commitment, I would argue, is to embrace that imperative. Stubborn persistence in self-contradiction destroys the possibility of shareable meaning and dialogue. In real life, self-contradiction happens to good people, but that should be an occasion for learning and humility, never something to proudly affirm.

As soon as we acknowledge piecemeal responsibility for the integrity of our commitments, we implicitly have responsibility for the integrity of the whole constituted by all our commitments. Commitment to commitment is an implicit condition of all our particular commitments, and it involves a responsibility for safeguarding and improving the integrity of the whole of our commitments. However fallible it may be, by its very nature it involves at least the germ of the crucial ability to learn, to improve itself and to correct itself.

This also has important consequences for what Kantian respect for others and the related notion of Hegelian mutual recognition look like in practice. First and foremost, respect for others takes the form of recognition of their implicit commitment to commitment, even when we do not endorse all the others’ particular commitments. (See also Brandomian Forgiveness.)

Logic for People

Leading programming language theorist Robert Harper refers to so-called constructive or intuitionistic logic as “logic as if people mattered”. There is a fascinating convergence of ideas here. In the early 20th century, Dutch mathematician L. E. J. Brouwer developed a philosophy of mathematics called intuitionism. He emphasized that mathematics is a human activity, and held that every proof step should involve actual evidence discernible to a human. By contrast, mathematical Platonists hold that mathematical objects exist independent of any thought; formalists hold that mathematics is a meaningless game based on following rules; and logicists argue that mathematics is reducible to formal logic.

For Brouwer, a mathematical theorem is true if and only if we have a proof of it that we can exhibit, and each step of that proof can also be exhibited. In the later 19th century, many new results about infinity — and infinities of infinities — had been proved by what came to be called “classical” means, using proof by contradiction and the law of excluded middle. But from the time of Euclid, mathematicians have always regarded reproducible constructions as a better kind of proof. The law of excluded middle is a provable theorem in any finite context. When the law of excluded middle applies, you can conclude that if something is not false it must be true, and vice versa. But it is not possible to construct any infinite object.

The only infinity we actually experience is what Aristotle called “potential” infinity. We can, say, count a star and another and another, and continue as long as you like, but no actually infinite number or magnitude or thing is ever available for inspection. Aristotle famously defended the law of excluded middle, but in practice only applied it to finite cases.

In mathematics there are conjectures that are not known to be true or false. Brouwer would say, they are neither true nor false, until they are proved or disproved in a humanly verifiable way.

The fascinating convergence is that Brouwer’s humanly verifiable proofs turn out also to exactly characterize the part of mathematics that is computable, in the sense in which computer scientists use that term. Notwithstanding lingering 20th century prejudices, intuitionistic math actually turns out to be a perfect fit for computer science. I use this in my day job.

I am especially intrigued by what is called intuitionistic type theory, developed by Swedish mathematician-philosopher Per Martin-Löf. This is offered simultaneously as a foundation for mathematics, a higher-order intuitionistic logic, and a programming language. One might say it is concerned with explaining ultimate bases for abstraction and generalization, without any presuppositions. One of its distinctive features is that it uses no axioms, only inference rules. Truth is something emergent, rather than something presupposed. Type theory has deep connections with category theory, another truly marvelous area of abstract mathematics, concerned with how different kinds of things map to one another.

What especially fascinates me about this work are its implications for what logic actually is. On the one hand, it puts math before mathematical logic– rather than after it, as in the classic early 20th century program of Russell and Whitehead — and on the other, it provides opportunities to reconnect with logic in the different and broader, less formal senses of Aristotle and Kant, as still having something to say to us today.

Homotopy type theory (HoTT) is a leading-edge development that combines intuitionistic type theory with homotopy theory, which explores higher-order paths through topological spaces. Here my ignorance is vast, but it seems tantalizingly close to a grand unification of constructive principles with Cantor’s infinities of infinities. My interest is especially in what it says about the notion of identity, basically vindicating Leibniz’ thesis that what is identical is equivalent to what is practically indistinguishable. This is reflected in mathematician Vladimir Voevodsky’s emblematic axiom of univalence, “equivalence is equivalent to equality”, which legitimizes much actual mathematical practice.

So anyway, Robert Harper is working on a variant of this that actually works computationally, and uses some kind of more specific mapping through n-dimensional cubes to make univalence into a provable theorem. At the cost of some mathematical elegance, this avoids the need for the univalence axiom, saving Martin-Löf’s goal to avoid depending on any axioms. But again — finally getting to the point of this post — in a 2018 lecture, Harper says his current interest is in a type theory that is in the first instance computational rather than formal, and semantic rather than syntactic. Most people treat intuitionistic type theory as a theory that is both formal and syntactic. Harper recommends that we avoid strictly equating constructible types with formal propositions, arguing that types are more primitive than propositions, and semantics is more primitive than syntax.

Harper disavows any deep philosophy, but I find this idea of starting from a type theory and then treating it as first of all informal and semantic rather than formal and syntactic to be highly provocative. In real life, we experience types as accessibly evidenced semantic distinctions before they become posited syntactic ones. Types are first of all implicit specifications of real behavior, in terms of distinctions and entailments between things that are more primitive than identities of things.

Indistinct Cows, Pistol Shot

Hegel in the Phenomenology wants to teach us to be at home in what he calls “otherness”.

Plato was traditionally read as treating “the Others” as inferior to “the One” in the Parmenides, but in the Sophist he explicitly suggested that notions of Other, Same, and Being are equally fundamental.

Hegel goes further, in affirming the essential role of mediation (dependence of things on other things) — as well as the kind of differences in form that “make a difference” practically — in any kind of intelligibility. In the Preface, he sharply criticizes unnamed contemporaries for effectively denying the importance of otherness, either through excessive preoccupation with formal identity or through emphasis on a kind of immediate intuition of God or the Absolute.

Schelling never forgave Hegel for the quip that to insist that all is one in the Absolute makes of the Absolute a “night in which all cows are black”, which has often been read as directed at him. Harris in his commentary argues that the main target of this particular remark was actually the purely formal notion of truth propounded by K. L. Reinhold, who helped popularize Kant.

A bit later, Hegel goes on to denounce “the sort of ecstatic enthusiasm which starts straight off with absolute knowledge, as if shot out of a pistol, and makes short work of other points of view simply by explaining that it is to take no notice of them” (Baillie translation, pp. 88-89). In this case Harris finds it most plausible that the reference really is to Schelling’s Presentation of My Own System (1801), but adds in a note that a good case has also been made that the reference is to J. K. Fries, who apparently talked a lot about the feeling of the infinite.

Hegel shared many of the perspectives of the German Romantics, including a concern for spiritual renewal, awareness of the limits of formal reasoning, and inspiration from Greek antiquity. But by the time of the Phenomenology and for the rest of his life, he supported Kant against Schelling in denying the legitimacy of appeals to direct intuition of metaphysical truth, and had distanced himself from Romantic notions of individual immediate interiority.

For Hegel, Reason finds its home in otherness. This is closely related to the noncontrolling attitude he associates with what he calls “Science” (see The Ladder Metaphor). Hegel’s verbal emphasis on “system” and “Science” needs to be understood in the context of his defense of other-sensitive, value-oriented interpretive Reason against both its reduction to formalism and its effective rejection by Romantics and other proponents of metaphysical intuition.

Next in this series: Adverbial Otherness

Form as Value

Plato’s most famous discussions of form involved things like the form of virtue, of justice, or of the Good. These are questions that perplex the wise and the sincere inquirer. They therefore could not be the objects of any simple dogma.

In Aristotle there is a deep connection between form and ends. For both Aristotle and Plato, “essence” is never merely factual but always has what analytic philosophers call a normative dimension. It is not the kind of thing that could be simply given (see Form, Substance).

Brandom says that for Kant and Hegel, concepts always have a normative dimension, and intentionality is to be explained in terms of normativity rather than vice versa.

The necessity in formal logic and mathematics also has a normative character, but it is different from the previous examples in that it is univocal and definitely knowable. Things that are “formal” in this modern sense are quite different from form for Plato or Aristotle, which is closer to what Brandom would call conceptual content (see Mutation of Meaning). Well-founded certainty is only possible in domains that are purely formal in the modern sense.

Anything involving the “real world” involves interpretation, which is never finished. In life we work, act, and love on the basis of partial interpretations of the forms of things.