Pure Reason?

Hegel’s “logic” takes what Kant calls pure reason as its subject matter. Hegel regards Kantian pure reason as a world-changing revolution, because in contrast to early modern views, it seeks not to imitate the formal character of mathematical reasoning, but rather to achieve the discipline of a kind of self-sufficiency that does not appeal to anything external to it. Kant and Hegel differ on the scope of this self-sufficiency, but that is a different matter.

Early modern views of the world generally rely on many substantive assumptions. There is strong motivation for them to do so, because in order to yield any substantive conclusions, reasoning of a broadly formal kind requires substantive assumptions. The assumptions are typically of a sort analogous to those that Aquinas regards as grounded in the natural light of reason, which is not itself reason, but a kind of originating intuition of truth given to us by God. Descartes, for example, explicitly appeals to a variant of the Thomistic doctrine of natural light.

(The strong Thomistic notion of the natural light of reason and of reason’s relative autonomy from the simple dictates of authority is itself a development of almost inestimable importance, compared to completely authority-bound views of religion such as present-day fundamentalism. Indeed, something like the natural light of reason was never completely absent from the earlier medieval tradition either.)

But for Kant, reason is purely discursive, and cannot appeal to any intuitive source of truth like a natural light. Pure reason is nonetheless supposed to be able to stand on its own. In Kant’s language, it is “autonomous” (see also Kant’s Groundwork; Self-Legislation?). Kant’s critique of dogmatism especially targets assumptions that are naively realistic in the sense of claiming direct knowledge of external or inner objects, but it is broader than that.

Kant in the Critique of Pure Reason is most directly responding to empiricist views broadly associated with Locke, which were dominant in England and France, and popular in Germany in his day; but even more so to the rationalist system of Christian Wolff (1679-1754), which then dominated German academic teaching. (Wolff was an accomplished mathematician who had corresponded with Leibniz, and greatly contributed to popularizing the part of Leibniz’s philosophy that Leibniz had published in his own lifetime. Like Leibniz, he is associated with moderate Enlightenment, while at the same time showing a degree of sympathy for scholastic philosophy.)

Kantian pure reason effectively aims to be free of unnecessary assumptions, especially those of the Wolffian system, but also those of the empiricists. Kant also criticizes Wolff’s and Spinoza’s idea that philosophical reasoning should as much as possible resemble mathematical reasoning. What makes it possible for Kant to avoid assumptions beyond the famous “God, freedom, and immortality” (and for Hegel to avoid any assumptions at all) is a move away from the early modern ideal of reason as formal.

Without ever explicitly saying so, Kant in fact takes up and works with a notion of reason that is close to aspects of Plato and Aristotle that were generally neglected in the intervening tradition. Reason in Plato, Aristotle, Kant, and Hegel is not limited to formal reasoning. It includes what in more recent times Sellars and Brandom have elaborated under the name of material inference.

Formal reasoning is called formal because it is supposed to apply to all things, independent of any analysis of meaning. But this makes it dependent on assumptions in order to yield conclusions. Material inference — which was also present as a minor theme in scholastic logic — is on the contrary grounded in the interpretation of meaning. It is this reflective grounding that can enable reason to be autonomous and “pure”, with no reliance on anything external to it.

Sellars illustrates material inference with examples like “there are dark rain clouds in the sky, so I should take my umbrella when I go out”. Brandom elaborates with an account of how such judgments may be successively refined based on additional information. In general, if I strike a match correctly, it will light. But under certain conditions, it will not light. But under yet more specific additional conditions, it will in fact still light.

Both Sellars and Brandom — working within the tradition of contemporary analytic philosophy — tend to reach for examples that involve empirical facts, and relations of cause and effect in the broad modern sense. But material inference is more general than that. It is grounded in meaning as we encounter it in real life. Its scope is not limited to any particular kind of meaning, nor does it assume any particular theory of meaning.

Pure reason, then — far from excluding meaning, as formal logic does — is concerned with the progressive self-clarification of meaning — or Kantian “taking as”, or judgment — in a reflective context.

For Hegel, “logic is to be understood as the system of pure reason, as the realm of pure thought” (Science of Logic, di Giovanni trans., introduction, p. 29). This is what he calls the “concept of science”, and also “absolute knowledge” (p. 28). As I’ve pointed out before, in Hegel these terms have specialized meanings that are far from their ordinary connotations in English. Science need not be empirical, and “absolute” in this context just means the same thing as “pure” or “autonomous” — that reflective judgment need presuppose nothing outside itself.

For Hegel, the standpoint of pure reason (or “science”, or “absolute” knowing) is that of reflective judgment. The whole effort of the Phenomenology of Spirit is required to reach this point, which he then uses as a starting point in the Logic.

“Pure science thus presupposes the liberation from the opposition of consciousness [between itself and its object]…. As science, truth is pure self-consciousness as it develops itself and has the shape of a self, so that that which exists in and for itself is the conscious concept and the concept as such is that which exists in and for itself” (p. 29, emphasis in original).

The reflective concept has the shape of a “self” — a reflexivity — that is not to be identified with our empirical self, but rather is related to the reflective character of self-consciousness, which overcomes the simple opposition between consciousness and its object.

“This objective thinking is thus the content of pure science. Consequently, far from being formal, far from lacking the matter for an actual and true cognition, it is the content which alone has absolute truth” (ibid).

He calls reflective judgment objective thinking, precisely because it begins only after the separation of consciousness from its object ends. Reflective judgment and self-consciousness will not be separated from “the concept” in which they are embodied. Rather, we have here a case of the Aristotelian identity of pure thinking with what it thinks.

“Logic has nothing to do with a thought about something which stands outside by itself as the base of thought; nor does it have to do with forms meant to provide mere markings of the truth; rather, the necessary forms of thinking, and its specific determinations, are the content and the ultimate forms of truth itself.”

“To get at least some inkling of this, one must put aside the notion that truth must be something tangible. Such tangibility, for example, is carried over even into the ideas of Plato which are in God’s thought, as if they were, so to speak, things that exist but in another world or region, and a world of actuality were to be found outside them which has a substantiality distinct from those ideas and is real only because of this distinctness” (pp. 29-30).

Truths are not objects, and they are not given to us in the way that ordinary consciousness takes objects to be. For Hegel, moreover, spiritual values do not have to do with turning away from this world in favor of another one. They are intended to guide us in life.

“There will always be the possibility that someone else will adduce a case, an instance, in which something more and different must be understood by some term or other” (p. 28).

Reflection and interpretation are inherently open-ended.

“How could I possibly pretend that the method that I follow in this system of logic, or rather the method that the system itself follows within, would not be capable of greater perfection, of greater elaboration of detail? Yet I know that it is the one true method. This is made obvious by the fact that this method is not something distinct from its subject matter and content — for it is the content in itself, the dialectic which it possesses within itself, which moves the subject matter forward. It is clear that no expositions can be accepted as scientifically valid that do not follow the progression of this method and are not in tune with its simple rhythm, for it is the course of the fact [Sache] itself” (p. 33).

Translator di Giovanni comments in his glossary, “In non-technical contexts, [Sache] can and should be translated in a variety of ways, such as ‘substance’, or even ‘thing’. As category, however, ‘fact’ seems to be the best rendering. Sache, like ‘fact’, denotes a thing or a situation which we understand to implicitly contain all the factors required for an explanation of its existence. Its presence therefore cannot be doubted even when those factors have yet to be made explicit. The related word, Tatsache, was first coined… in order to translate the English term ‘matter of fact'” (pp. lxxi-lxxii).

To me, these sound like reasons for calling Hegel’s Sache something other than “fact”. Especially in a work of “logic” that invokes “science”, the English word “fact” would most commonly be taken taken to mean an unambiguous empirical truth. Both what I think Hegel means and the explanation di Giovanni gives of it seem better suited by the more open connotations of an English phrase like “the concrete case” or “the matter at hand”. The Sache is something objective, but it is objective in the indefinite sense of a Gegenstand [“object” in the sense of something standing over and against us, but whose nature has yet to be determined].

I used to think that reason that would be applicable to life (or to anything like Hegel’s Sache) could not possibly be pure. I now think that with the inclusive character of reflective judgment and material inference, it can be pure.

Mathematical Things and Forms

We’ve reached the end of Aristotle’s Metaphysics, though there are in fact two more books, Mu (XIII) and Nu (XIV).

Aristotle’s main point of contention with his former colleagues of the Platonic Academy is whether or not mathematical objects and forms understood as universals are independent things in their own right. Both books Mu and Nu (XIV) are concerned with this, and have a somewhat polemical character. I think Aristotle’s own distinctive views on form are better expressed in what has been said already, so I will mostly focus on the other remarks he makes in book Mu, and will skip book Nu entirely.

Book Mu does not clearly refer to the preceding book Lambda (XII), but does refer to previous discussion of “the thinghood that has being as being-at-work” (Sachs tr., ch. 1, p. 253) as well as to discussion on aporias.

“Now it is necessary, if mathematical things are, that they be either in the perceptible things, as some people say, or separate from the perceptible things (and some people also speak of them that way); or if they are not present in either way, then they do not have being or they have it in some other manner. So for us the dispute will not be about whether they have being, but about the manner of their being” (pp. 253-254).

I take especial note of the last sentence above. This could also serve as a comment on what is at stake in the Metaphysics in general — questions not really about being as if it were one thing, but about what things are, and the ways they are.

He goes on to argue that mathematical things are neither “in” perceptible things, nor are they separate things in their own right. “In” for Aristotle suggests a material constituent.

“It has been said sufficiently, then, that mathematical things are not independent things more than bodies are, nor are they prior in being to perceptible things, but only in articulation, nor are they capable of being somewhere as separate; but since they are not capable of being in perceptible things either, it is clear that either they have no being at all, or that they have being in a certain manner and for this reason do not have being simply, for we speak of being in a number of ways” (ch. 2, p. 257).

For those who insist that the whole Metaphysics is a single linear development and not just generally coherent with itself, and also that being finally acquires an unequivocal sense, it seems inconvenient that now, after book Lambda, he continues to emphasize that being is said in many ways.

I think the Metaphysics is very much coherent with itself, but is not a single linear development pointing toward Being, and that he never wavers on the emphasis that being is said in many ways, although he does sometimes use the word equivocally himself. If the whole thing points toward something, that something is the good and the beautiful, and not Being.

He goes on to make some positive remarks about mathematics.

“Now just as the things that are universal within mathematics are not about things that are separate from magnitudes and numbers, but are about these, but not insofar as they are of such a sort as to have magnitude or to be discrete, it is clear that it is also possible for there to be both articulations and demonstrations about perceptible magnitudes, not insofar as they are perceptible but insofar as they are of certain sorts” (ch. 3, p. 257).

He is saying that insofar as there is mathematical knowledge, it is not about magnitude or number as such, but about more specific things such as right triangles or even numbers. Similarly, the meaning of articulations and demonstrations about perceptible magnitudes does not depend on their perceptibility as such.

“[S]ince it is true to say simply that there are not only separate things but also things that are not separate…, it is also true to say simply that there are mathematical things and that they are of such a sort as people say….If it is about things which incidentally are perceptible, but is not concerned with them insofar as they are perceptible, mathematical knowledge will not be about perceptible things; however, it will not be about other separate beings besides these either” (p. 258).

Mathematical things are bona fide things in the broad sense, but not all things are separate or independent. Some are attributed to others.

“[I]f someone examines anything concerning these attributes, insofar as they are such, positing them to be separate, he will not on this account cause anything to be false, any more than when one draws a line on the ground that is not a foot long, and says it is a foot long, for the false assumption is not in the proposition…. [F]or this reason the geometers speak rightly” (pp. 258-259).

Mistaken belief about the independence of mathematical things is incidental to the doing of mathematics. It is irrelevant to the results of constructions or calculations.

“And since the good and the beautiful are different (for the former is always involved in action but the beautiful is also present in motionless things), those who claim that the mathematical kinds of knowledge say nothing about what is beautiful and good are wrong…. The greatest forms of the beautiful are order and symmetry and determinateness, which the mathematical kinds of knowledge most of all display. And since these make their appearance as causes of many things…, it is clear that these kinds of knowledge would also speak about what has responsibility in the manner of the beautiful as a cause in some manner” (p. 259).

Here he not only recognizes mathematical beauty, but relates it to the beauty associated with that-for-the-sake-of-which as a cause.

“The opinion about the forms came to those who spoke about them as a result of being persuaded by the Heraclitean writings that it is true that all perceptible things are always in flux, so that, if knowledge and thought are to be about anything, there must be, besides the perceptible things, some other enduring natures, since there can be no knowledge of things in flux. And then Socrates made it his business to be concerned with the moral virtues, and on account of them first sought to define things in a universal way. For among those who studied nature, only to a small extent did Democritus attain to this… and before that the Pythagoreans did about some few things…. But it is reasonable that Socrates sought after what something is…. But Socrates did not make the universals or the definitions separate, while those who came next did, and called beings of this sort forms” (ch. 4, p. 260).

Aristotle rejects the Heraclitean doctrine of radical flux that influenced Plato. He says Plato was driven to assert separate forms because he wanted to assert that there is knowledge, in spite of his Heracliteanism about perceptible things. Aristotle says that driven by a concern for ethics, Socrates — and not any of those we know as the pre-Socratics — was the first to seriously inquire about what things are. Aristotle has been inquiring about the what-it-is of things and its causes and sources, and we have seen in abundance his concern for the good and the beautiful. Aristotle is claiming a Socratic heritage, and claiming to be truer to it than the Platonists: “Socrates did not make the universals or the definitions separate”.

There follows a long argument against Platonic views about the forms, at the end of which he observes:

“[K]nowledge, like knowing, has two senses, the one as in potency, the other as at-work. The potency, being, like material, universal and indeterminate, is of what is universal and indeterminate, but the being-at-work is determinate and of something determinate; being a this it is of a this, but incidentally sight sees a universal color because this color that it sees is a color, and this A that the grammarian contemplates is an A” (ch. 10, p. 279).

If I am reading this right, he is saying here that being as universal and indeterminate is to being-at-work as potentiality is to being-at-work. If that is so, then the priority of actuality over potentiality would also seem to be a priority of actuality over being. Once again, it just doesn’t seem that being is the principal term.

In any case, he returns to the ultimately ethical theme of the priority of actuality or being-at-work or fulfillment over potentiality, and of particular concrete things over universals in the ordinary logical sense. This still has to be carefully balanced with his other view that there is no knowledge of particulars; knowledge is of universals only.

Positive concern for the priority of actuality is in my opinion the primary thing that underlies his sharp critique of the Platonists. The second — evidenced in the part I skipped over — was the popularity within the Academy of a kind of Pythagorean mystique of numbers that also identified the forms with numbers, in sometimes baffling ways. Plato himself was apparently not immune to this.

Many think Aristotle claims to have knowledge of non-perceptible particular independent everlasting things. I think this interpretation relies on ambiguous use of Aristotle’s saying of “knowledge” in different ways in different contexts. Sometimes he means it very strictly, other times much more loosely. Some translations add confusion by using the same English “knowledge” for other Greek words like gnosis, which I think for Aristotle means personal acquaintance with things nearer to us, whereas episteme is supposed to be about things in their own right.

I do not think that Aristotle means to claim knowledge in the strong sense about ultimate things, but rather that his attitude was in a way closer to that of Kant, who held them to have the highest importance but not to be knowable in the strict sense. This means we do not have to equivocate about what knowledge is.

The wisdom that is called sophia in book capital Alpha initially seems to be concerned with universals in the ordinary sense, as true episteme or knowledge genuinely is. It turns out in book Lambda that sophia‘s primary concern is not with universals in the ordinary sense at all, but with analogous relations that a uniquely positioned particular or particulars has or have to all other things.

In any case, my own view is that the wisdom or sophia concerning these highest things ought to be understood as aligned not so much with knowledge or episteme, as with the ethical or “practical” wisdom (phronesis) that is explicitly said to be a wisdom about particulars. A wisdom about particulars is not prevented from making — and indeed presumably would make — use of knowledge of any universals that genuinely apply. Nonetheless it is the wisdom about particulars that judges which universals should apply in a particular case.

System?

Fichte greatly admired the coherence of the quasi-mathematically structured “system” of Spinoza’s Ethics, even though he strenuously objected to Spinoza’s determinism. In his early work, he announces the objective of constructing a “system of freedom” that would be some sort of inverse of Spinoza’s. Rather than following Spinoza’s quasi-mathematical method of presentation, Fichte proceeded more informally. He was influenced by the early Kant interpreter K. L. Reinhold’s claim that philosophy should be derived from a single principle, and aimed to put this into practice with his notion of the universal “I” as the principle.

Schelling inherited the rhetorical emphasis on a system from Fichte (e.g., one of the works of his early period was entitled Presentation of My Own System), but in general was a less rigorous thinker.

Hegel also inherited the rhetorical emphasis on a system, but aimed to be more rigorous than Fichte. At the same time he expands upon Kant’s criticism of the quasi-mathematical presentation in Spinoza, and explicitly rejects Reinhold’s view that philosophy should be derived from a single principle. So, there is a serious question what “system” really means for Hegel.

It is clear from his explicit remarks that he put an extraordinarily high value on the coherence of philosophical thought. The advance of studies of Hegel, especially since the later 20th century, has confirmed that he largely succeeded in putting this into practice. Both his overall thought and his detailed arguments are increasingly recognized as highly coherent.

The historic negative reception of Hegel has consisted largely in caricatures of his systematic ambitions. I call them caricatures because they rely on attributing to Hegel notions of “system” that were not his.

Hegel’s rhetorical emphasis on system, I want to suggest, is a red herring. What really matters in his thought is not “system” but coherence.

The notion of systems originates in mathematics, and there it has unambiguous meaning. Systems in mathematics do have great utility, because you can’t mathematically prove anything independent of a particular presentation, but this makes mathematical systems intrinsically presentation-dependent. That is to say, the particular terms and order with which the content is developed and presented are essential to making it a system for the same reasons that they are essential to proof. Mathematicians recognize that there may be multiple equivalent formulations, presentations, and systematizations of the “same” content.

I don’t find any of the attempts to present non-mathematical “systems” very helpful or convincing as such. (The common talk about real-world “systems” in engineering and science — which does also have utility — I take to be grounded in a kind of transference from the mathematical concept of a system. It is really the mathematics that describes the things or behavior of interest that may be expressible as a system.)

On the other hand, I want to say that the notion of coherence is more universal than that of a system or systems — systems are presentation-dependent, and coherence is not. The rhetorical stance of the German idealists seems to me to have assumed that the only way to achieve coherence is through the uniform presentation of a system. Certainly it is the most straightforward way, but that does not mean it is the only way.

Coherence in Hegel, I want to suggest, is “development-dependent” but not presentation-dependent. Robert Pippin points out that none of Hegel’s works is structured in a deductive order — rather, they all follow a “developmental” order that more resembles the telling of a story or an account of a history.

Hegel’s notorious idiosyncratic and paradoxical straining of language to talk about “identities” that preserve distinctions is helpfully explainable in terms of the notion of narrative identity developed by Paul Ricoeur. Aristotle’s articulation of things “said in many ways” and his more subtle development of “substance” in the Metaphysics are relevant background for this. (See Aristotelian Identity; Univocity.)

Ricoeur is the main developer to date of a synthesis of Kant and Aristotle independent of Hegel’s. Mediation is as central to his thought as it is to Hegel’s, and he explicitly recognized the convergence. However, he strongly rejected the “system” aspect of Hegel, and his development also doesn’t explicitly include anything resembling the Hegelian absolute, even in the deflationary form in which I think Hegel really meant it.

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).

Cause

Aristotle flourished before the great flowering of Greek mathematics that gave us Euclid, Ptolemy, Apollonius, and Aristarchus. In his day, mathematics amounted to just arithmetic and simple geometry. In spite of the famous Pythagorean theorem that the square constructed from the hypotenuse of a right triangle is equal in area to the sum of the squares constructed from the other two sides, the historic reality of the Pythagorean movement had more to do with number mysticism, other superstitions, and curious injunctions like “don’t eat beans” than it did with real mathematics.

I think Aristotle was entirely right to conclude that arithmetic and simple geometry were of little use for explaining change in the natural world. I’ve characterized his physics as grounded in a kind of semantic inquiry that Aristotle pioneered. We are not used to thinking about science this way, as fundamentally involved with a very human inquiry about the meaning of experience in life, rather than predictive calculation. For Aristotle, the gap between natural science and thoughtful reflection about ordinary experience was much smaller than it is for us.

Aristotle invented the notion of cause as a semantic tool for expressing the reasons why changes occur. Aristotle’s notion is far more abstract than the metaphor of impulse or something pushing on something else that guided early modern mechanism. Even though the notion of cause was originally developed in a text included in Aristotle’s Physics, the “semantic” grounding of Aristotelian physics places it closer to logic than to modern physical inquiries.

I think the discussion of the kinds of causes could equally well have been grouped among his “logical” works. In fact, the form in which we have Aristotle’s works today is the result of the efforts of multiple ancient editors, who sometimes stitched together separate manuscripts, so there is room for a legitimate question whether the discussion of causes was originally a separate treatise. We tend to assume that there must be something inherently “physical” about the discussion of causes, but this is ultimately due to a circular argument from the fact that the more detailed version of it came down to us as part of the Physics (there is another, briefer one that came down to us as part of the Metaphysics).

Since Hume and especially since the later 19th century, many authors have debated about the role of causes in science. Bertrand Russell argued in the early 20th century that modern science does not in fact depend on what I have called the modern notion of cause.

More recently, Robert Brandom has argued that the purpose of logic is “to make explicit the inferential relations that articulate the semantic contents of the concepts expressed by the use of ordinary, nonlogical vocabulary”. I see Aristotelian causes in this light.

I want to recommend a return to a notion of causes in general as explanatory reasons rather than things that exert force. This can include all the mathematics used in modern science, as well as a broader range of reasons relevant to life. (See also Aristotelian Causes; Mechanical Metaphors; Causes: Real, Heuristic?; Effective vs “Driving”; Secondary Causes.)

Figurative Synthesis

I wanted to extract a few more key points from Beatrice Longuenesse’s landmark study Kant and the Capacity to Judge. She strongly emphasizes that judgment for Kant refers to a complex activity, not a simple reaching of conclusions. She especially stresses the role of a capacity to judge that precedes any particular judgment and is grounded in a synthesis of imagination. (See Capacity to Judge; Imagination: Aristotle, Kant; Kantian Synthesis.)

At issue here is the very capacity for discursive thought, as well as “the manner in which things are given to us” (p. 225, emphasis in original), which for Kant involves what he called intuition. (See also Beauty and Discursivity).

Through careful textual analysis, Longuenesse argues that Kant’s claim to derive logical categories from forms of judgment makes far more sense than most previous commentators had recognized. For Kant, she argues, the “forms of judgment” are not just logical abstractions but essential cognitive acts that reflect “universal rules of discursive thought” (p. 5).

She recalls Kant’s insistence that the early modern tradition was wrong to take categorical judgments (simple predications like “A is B“) as the model for judgments in general. For Kant, hypothetical and disjunctive judgments (“if A then B” and “not both A and B“, respectively) are more primitive. These correspond to the judgments of material consequence and material incompatibility that Brandom argues form the basis of real-world reasoning.

Another distinctive Kantian thesis is that space and time are neither objective realities nor discursive concepts that we apply. Rather, they are intuitions and necessary forms of all sensibility. Kantian intuitions are produced by the synthesis of imagination according to definite rules.

“[I]ntuition is a species of cognition (Erkenntnis), that is, a conscious representation related to an object. As such it is distinguished from mere sensation, which is a mere state of the subject, by itself unrelated to any object…. One might say that, in intuition, the object is represented even if it is not recognized (under a concept).” (pp. 219-220, emphasis in original).

Before we apply any concepts or judgments, “Representational receptivity, the capacity to process affections into sensations (conscious representations), must also be able to present these sensations in an intuition of space and an intuition of time. This occurs when the affection from outside is the occasion for the affection from inside — the figurative synthesis. The form of the receptive capacity is thus a merely potential form, a form that is actualized only by the figurative synthesis” (p. 221, emphasis in original).

“[A]ccording to Locke, in this receptivity to its own acts the mind mirrors itself, just as in sensation it mirrors outer objects…. Kant shares with Locke the conception of inner sense as receptivity, but he no longer considers the mind as a mirror, either in relation to itself or in relation to objects…. Just as the thing in itself that affects me from outside is forever unknowable to me, I who affect myself from within by my own representative act am forever unknowable to me” (p. 239, emphasis added).

The point that the mind is not a mirror — either of itself or of the world — is extremely important. The mirror analogy Kant is rejecting is a product of early modern representationalism. We can still have well-founded beliefs about things of which we have no knowledge in a strict sense.

“Kant’s explanation is roughly this: our receptivity is constituted in such a way that objects are intuited as outer objects only in the form of space. But the form of space is itself intuited only insofar as an act, by which the ‘manifold of a given cognition is brought to the objective unity of apperception’, affects inner sense. Thanks to this act the manifold becomes consciously perceived, and this occurs only in the form of time” (p. 240, emphasis in original).

She develops Kant’s idea that mathematics is grounded in this kind of intuition, ultimately derived from the conditions governing imaginative synthesis. In particular, for Kant our apprehensions of unities and any kind of identification of units are consequences of imaginative synthesis.

“Extension and figure belong to the ‘pure intuition’ of space, which is ‘that in which the manifold of appearances can be ordered’, that is, that by limitation of which the extension and figure of a given object are delineated. Therefore, space and time provide the form of appearances only insofar as they are themselves an intuition: a pure intuition, that is, an intuition preceding and conditioning all empirical intuition; and an undivided intuition, that is, an intuition that is presupposed by other intuitions rather than resulting from their combinations” (p. 219, emphasis in original).

“According to Locke, the idea of unity naturally accompanies every object of our senses, and the idea of number arises from repeating the idea of unity and associating a sign with each collection thus generated by addition of units…. But for Kant, the idea (the concept) of a unit is not given with each sensory object. It presupposes an act of constituting a homogeneous multiplicity…. Thus the idea of number is not the idea of a collection of given units to which we associate a sign, but the reflected representation of a rule for synthesis, that is, for the act of constituting a homogeneous multiplicity. When such an act is presented a priori in intuition, a concept of number is constructed.” (p. 260, emphasis in original).

“Mathematics has no principles in the absolute sense required by reason. Axioms are not universal propositions cognized by means of pure concepts. They may be universally and apodeictically true, but their truth is based on the pure intuition of space, not derived from pure concepts according to the principle of contradiction” (p. 287).

Incidentally, Longuenesse thinks it does not follow from Kant’s account that space is necessarily Euclidean, as many commentators have believed and Kant himself suggested.

Whitehead: Process, Events

The originally British mathematician and philosopher Alfred North Whitehead (1861–1947) was profoundly concerned with the inter-relatedness of things. His later “philosophy of organism” inspired a movement of so-called “process theology”.

Whitehead was one of the inventors of universal algebra, which extends algebraic principles to symbolic representations of things that are not numbers. He collaborated with Bertrand Russell on the famous Principia Mathematica (1910, 1912, 1913) , which sought to ground all of mathematics in the new mathematical logic, but was less attached than Russell to the goal of reducing math to logic.

He did work in electrodynamics and the theory of relativity, emphasizing a holistic approach and the nonlocal character of electromagnetic phenomena. Counter to the spirit of the time, he developed a philosophy of science that aimed to be faithful to our intuitions of the interconnectedness of nature. He characterized mathematics as the abstract study of patterns of connectedness. In Science and the Modern World (1926), rejecting the world views of Newton and Hume as understood by the logical empiricists, he developed alternatives to then-dominant atomistic causal reductionism and sensationalist empiricism. Eventually, he turned to what he and others called metaphysics.

His Process and Reality (1929) is a highly technical work that is full of interesting insights and remarks. It aims to present a logically coherent system that radicalizes the work of John Locke in particular, but also that of Descartes, Spinoza, and Leibniz. As with many systematic works, however, it doesn’t engage in depth with the work of other philosophers.

Whitehead’s radicalization involves, among other things, a systematic rejection of mind-body dualism; of representationalism; of metaphysical applications of the subject-predicate distinction; and of Locke’s distinction between “primary” (mathematical) and “secondary” (nonmathematical) qualities. Plato and Aristotle both get positive mention. Whitehead thoroughly repudiates the sensationalist direction in which Hume took Locke’s work; aims deliberately to be “pre-Kantian”; and seems to utterly ignore Hegel, though he gives positive mention to the “absolute idealist” F. H. Bradley.

He wants to promote a thoroughgoing causal realism and to avoid any subjectivism, while eventually taking subjective factors into account. He wants to reinterpret “stubborn fact” on a coherentist basis. He is impressed by the work of Bergson, and of the pragmatists William James and John Dewey.

For Whitehead, “experience” encompasses everything, but he gives this an unusual meaning. Experience need not involve consciousness, sensation, or thought. He stresses the realist side of Locke, and wants to apply some of Locke’s analysis of the combination of ideas to realities in general.

He says that the world consists fundamentally of “actual entities” or “actual occasions” or “concrescences”, which he compares to Descartes’ extended substances. However, he interprets Einstein’s theory of relativity as implying that substances mutually contain one another, a bit like the monads in Leibniz.

For Whitehead, every actual entity has a kind of self-determination, which is intended to explain both human freedom and quantum indeterminacy. On the other hand, he also says God is the source of novelty in the universe. Whitehead recognizes what he calls eternal objects, which he compares to Platonic ideas, and identifies with potentiality.

Compared to the Aristotelian notions of actuality and potentiality I have been developing here, his use of actuality and potentiality seems rather thin. Actuality is just factuality viewed in terms of the connections of things, and potentiality consisting in eternal objects amounts to a kind of abstract possibility. His notion of causality seems to be a relatively standard modern efficient causality, modified only by his emphasis on connections between things and his idea of the self-determination of actual entities. His philosophy of science aims to be value-free, although he allows a place for values in his metaphysics.

According to Whitehead, perception has two distinct modes — that of presentational immediacy, and that of causal efficacy. Humean sensationalism, as codified by early 20th century theories of “sense data”, tries to reduce everything to presentational immediacy, but it is our intuitions of causal efficacy that connect things together into the medium-sized wholes recognized by common sense. As far as it goes, I can only applaud this move away from presentational immediacy, though I have also tried to read Hume in a less reductionist way. (I also want to go further, beyond intuitions of efficient causality in the modern sense, to questions of the constitution of meaning and value that I think are more general.)

In his later works, he emphasizes a more comprehensive notion of feeling, which he sees as grounded in subjective valuations, glossed as having to do with how we take various eternal objects. Compared to the logical empiricism that dominated at the time, this is intriguing, but I want to take the more radically Aristotelian (and, I would argue, also Kantian) view that values or ends (which are themselves subjects of inquiry, not simply given) also ultimately drive the constitution of things we call objective. I also don’t see “metaphysics” as a separate domain that would support the consideration of values, over and above a “science” that would ostensibly be value-free.

Whitehead considered the scientific reductionism of his day to exemplify what he called the “fallacy of misplaced concreteness”. What I think he wanted to question by this was the idea that scientific abstractions are more real or more true than common-sense apprehensions of concrete things. I would phrase it a bit differently, but the outcome is the same. Abstractions can have great interpretive value, but they are things entirely produced by us that have value because they help us understand concrete things that are more independent of us.

Attempting to take into account the idea from quantum mechanics that reality is not only relational but also granular, he made what is to me the peculiar statement that “the ultimate metaphysical truth is atomism”. Whitehead is certainly not alone in this kind of usage; indeed, the standard modern physical notion of “atoms” allows them to have parts and internal structure. That concept is fine in itself, but “atom” is a terrible name for it, because “atom” literally means “without parts”. The word “atom” ought to denote something analogous to a point in geometry, lacking any internal features or properties whatsoever.

Be that as it may, Whitehead sees an analogy between the granularity of events in quantum mechanics and the “stream of consciousness” analyzed by William James. “Your acquaintance with reality grows literally by buds or drops of perception. Intellectually and on reflection you can divide these into components, but as immediately given, they come totally or not at all” (Process and Reality, p. 68). To me, this is an expression not of atomism but of a kind of irreducibility of medium-sized things.

Anyway, Whitehead’s “atomic” things are events. Larger events are composed of smaller events, but he wants to say there is such a thing as a minimal event, which still may have internal complexity, and to identify this with his notion of actual occasion or actual entity.

I like the identification of “entities” with occasions. For Whitehead, these are a sort of what I call “medium-sized” chunks of extension in space-time. Whitehead’s minimal events are nonpunctual.

Freed of its scholastic rigidifications, this is close to what the Aristotelian notion of “primary substance” was supposed to be. I think of the latter as a handle for a bundle of adverbial characterizations that has a kind of persistence — or better, resilience — in the face of change. Only as a bundle does it have this kind of resilience.

Although — consistent with the kind of grounding in scientific realism he is still aiming at — Whitehead emphasizes the extensional character of actual occasions, they implicitly incorporate a good deal of intensional (i.e., meaning-oriented, as distinguished from mathematical-physical) character as well. Following Brandom’s reading of Kant on the primacy of practical reason, I think it is better to explain extensional properties in terms of intensional ones, rather than vice versa. But I fully agree with Whitehead that “how an actual entity becomes constitutes what that actual entity is” (p. 23, emphasis in original), and I think Aristotle and Hegel would, too.

According to the Stanford Encyclopedia of Philosophy, Whitehead’s work was attractive to theologians especially because it offered an alternative to the traditional notion of an omnipotent God creating everything from nothing. Whitehead argued that the Christian Gospel emphasizes the “tenderness” of God, rather than dominion and power: “not… the ruling Caesar, or the ruthless moralist, or the unmoved mover. It dwells upon the tender elements in the world, which slowly and in quietness operate by love” (p. 343). “The purpose of God is the attainment of value in the world” (Whitehead, Religion in the Making, p. 100). God for Whitehead is a gentle persuader, not a ruler.

(I would not put unmoved moving in anywhere near the same bucket as ruling omnipotence. Unmoved moving in Aristotle is attraction or inspiration by a pure end, where all the motion occurs in the moved thing. It is not some kind of ruling force that drives things.)

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.

Cartesian Metaphysics

For Descartes, according to Gueroult, “metaphysics” is the universal science or the system of science, and also a kind of introduction to more concrete studies. Here we are far from Aristotle and much closer, I think, to Duns Scotus. Without knowledge of God and oneself, Descartes says, it would never be possible to discover the principles of physics. Gueroult says that Descartes insists on an “incomprehensibility” of God that is neither unknowability nor irrationality but the “formal reason of the infinite” (Descartes selon l’order des raisons, p. 17). This again has a somewhat Scotist sound to my ear.

The infinitude of God puts God absolutely first, as the first truth that founds all others. Gueroult quotes Descartes saying, “It is a ‘blasphemy’ to say that the truth of something precedes God’s knowledge of it…, because the existence of God is the first and the most eternal of all the truths that can be, and the truth from which all the others proceed” (ibid; my translation).

Descartes says that God “freely creates” eternal truths. I have no idea what creation of eternal truths could even possibly mean, though such a notion seems to be at least implicit in the teaching of Duns Scotus. To be eternal is to have no before and after. Therefore, it seems to me, all eternal things must be co-eternal. This point of view accommodates part of Descartes’ thesis, insofar as if all eternal things are co-eternal, then an eternal truth would not “precede” God’s knowledge of it. In broadly neoplatonic terms, eternal truths could plausibly be regarded as aspects of the “nature” of God. I can also grasp the idea of truths following logically from the “nature” of God, but I suspect Descartes would either follow Scotus in arguing that God’s infinite power is not a “nature”, or follow Aquinas in arguing that God is pure existence and has no other “nature”. I don’t see how anything more specific can directly follow from either infinite power or pure existence.

For Descartes, though, God’s omnipotence “excludes the possibility of error” and “alone founds the objective validity of my intellectual faculty” (ibid). Descartes aims at “a total system of certain knowledge, at the same time metaphysical and scientific, … entirely immanent to mathematical certitude enveloped in the clear and distinct intellect, … in its requirement of absolute rigor. This totality of the system is in no way that of an encyclopedia of material knowledge effectively acquired, but the fundamental unity of the first principles from which follow all possible certain knowledge” (p. 18). Descartes’ doctrine is for him “a single block of certainty” (p. 19) that would be falsified by adding or removing any detail. All this seems way too strong to me.

Gueroult points out that Descartes wants to contrast an “order of reasons” with an “order of material”, as being more principled. However, unlike geometry, the total system of metaphysical reasoning for Descartes has “psychological” as well as logical requirements. Gueroult says it is for this reason that the Meditations best represent Descartes’ paradigm of rigorous analytic demonstration.

Granted that there is a clear “psychological” aspect to the Meditations, at this point I’m unsure what it means to relate that to the claimed rigor of the system. Moreover, adding a “psychological” dimension to what was said before about mathematical reasoning affects the very meaning of the claim of rigor. I think I understand what mathematical rigor is. I do not understand what “psychological” rigor would be in this context, but I suspect it may be wrapped up with what I would call extraordinary presumptions of absolute self-transparency and immediate reflexivity.

Gueroult on Descartes

Having been greatly impressed by Martial Gueroult’s two extant volumes on Spinoza’s Ethics, I wanted to challenge myself to get some sense of the detail of his magisterial Descartes selon l’order des raisons (1968). Sometimes called a “structuralist” in the history of philosophy, Gueroult systematically developed the fine grain of argument in Spinoza’s demonstrations, and here he does the same for Descartes’ Meditations.

Beginning with a distinction between understanding and explanation, Gueroult announces his intention to subordinate the former to the latter (p. 9). Here “understanding” is a sort of intuitive or imaginative grasp of the whole, whereas “explanation” develops the details in their interrelation. I am reminded of Paul Ricoeur’s great theme of the value of the “long detour”.

Gueroult says Descartes viewed “isolated thoughts” with a sort of horror. This is already interesting. I have long puzzled over Brandom’s treatment of Descartes as a proto-inferentialist, when Descartes has seemed to me on the contrary like an arch-representationalist who plucked “truths” out of thin air. Both Gueroult and Brandom take Descartes’ “method” very seriously. Brandom’s work previously set me on a path that led me to radically change my views of Kant and Hegel. Perhaps I’ll have to revise or modulate some of my judgments of Descartes as well.

For Gueroult, it is objective structures of argument that distinguish philosophy from poetry, spiritual or mystical elevation, general scientific theory, or mere metaphysical opinions. He says that even while “excommunicating” the history of philosophy, Descartes nonetheless formulated a good principle of reading, rejecting eclectic tendencies to pull out this or that idea from a great author, in favor of a systematic approach. Descartes is quoted saying the “precious fruit” must come from “the entire body of the work” (p. 11). This is an important complement to his one-sided insistence elsewhere on beginning with what is simple. However, Descartes is also quoted insisting that all conflicts of interpretation are due to shallow eclecticism and deficiency of method, and that wherever there is such a conflict, one side must certainly be wrong (pp. 13-14).

This insistence on univocal interpretation is one of my big issues with Descartes. It works well for things like geometry, but much less well for sorting out arguments about power or potentiality, for instance. Pushing univocal interpretation as far as it can go can be a very valuable exercise, but as soon as we leave pure mathematics, it also shows its limits. I think that while mathematical necessity can be understood as something we “ought” to recognize for a multitude of reasons, sound ethical judgment must in principle reach beyond what can be expressed with certainty by formal equations. Much as I admire a good mathematical development, I therefore think ethics is more fundamental for us humans than mathematics, and philosophy is more ethical than mathematical.

According to Gueroult, the seminal idea guiding all of Descartes’ work is that human knowledge has unavoidable limits due to the limits of thought, but within those limits it is capable of perfect certainty (p. 15). For Descartes, we do not know thought by things, but we know things by thought. As a matter of principle, we should doubt everything that does not come from the certainty of thought. We are thus offered a stark division between that which is supposed to be certain beyond question, and that which is vain and useless. I think this results both in a treatment of too many things as certain, and in a premature dismissal of aspects of human reality that are uncertain, but nonetheless have real value.

I agree that mathematical reasoning is capable of (hypothetical) certainty, but I contend that we humans live mainly on middle ground that is neither certainty nor mere vanity.