Intangible Truth

Hegel wants to teach us to put aside the prejudice that a truth must be something “tangible” or discrete in itself, and thus capable of being viewed in isolation, in the way that a Platonic form is commonly supposed to be. He says that ordinary logic already gives us a clue to an alternate view of truth. Indeed, Plato’s own literary depictions of Socratic inquiry and dialogue already suggest a deeper notion of essence and truth than is promoted by standard accounts of Platonic forms.

“The Platonic idea is nothing else than the universal, or, more precisely, it is the concept of the subject matter; it is only in the concept that something has actuality, and to the extent that it is different from its concept, it ceases to be actual and is a nullity; the side of tangibility and of sensuous self-externality belongs to this null side. — But on the other side one can appeal to the representations typical of ordinary logic; for it is assumed that in definitions, for example, the determinations are not just of the knowing subject but are rather determinations of the subject matter, such that constitute its innermost essential nature. Or in an inference drawn from given determinations to others, the assumption is that the inferred is not something external to the subject matter and alien to it, but that it belongs to it instead, that to the thought there corresponds being” (Science of Logic, di Giovanni trans., introduction, p. 30).

There is a glimmer of a deeper truth even in the naive belief that ordinary logic can tell us about how the world really is (not of course how the world is, full stop, just some important things “about” how it is). What we infer by a good inference is at least as real as whatever is intuitively present to us. Neither of these is an infallible source of knowledge. Hegel’s main point, though, is that being immediately present to us is not a criterion of deeper truth.

He continues, “Everywhere presupposed by the use of the forms of the concept, of judgment, inference, definition, division, etc., is that they are not mere forms of self-conscious thinking but also of objective understanding” (ibid).

This leads to a criticism of Kant, which implies that Kant’s famous critique of dogmatism remains incomplete.

“Critical philosophy… gave to the logical determinations an essentially subjective significance out of fear of the object…. But the liberation from the opposition of consciousness that science must be able to presuppose elevates the determination of thought above this anxious, incomplete standpoint” (ibid).

The “opposition of consciousness” Hegel speaks of is its division into subject and object. For Kant, this distinction is interwoven with what Kant takes to be an uncrossable gap between knowledge on the side of the subject, and being on the side of the object. Hegel argues that we can avoid the dogmatism Kant means to criticize, without positing an uncrossable gap between knowledge and being. For him, the works of Aristotle are decisive proof of this.

Kant seeks to ensure the avoidance of dogmatism by treating logical determinations exclusively as attitudes actively taken up by a thinking being. Hegel points out that this leads inevitably to the unknowability of the Kantian thing-in-itself. In Kant, these are two sides of one coin. Thus cut off from logical determination, the thing-in-itself can only be unknowable, just as Kant says it is. According to Hegel’s analysis yet to come, meaning is grounded in judgments of determination, and so to be cut off from determination is to be devoid of meaning.

In criticizing Kant on this score, Hegel speaks of a Kantian “fear of the object”. Elsewhere he specifies that what is wrong with the Kantian thing-in-itself has nothing to do with its resemblance to a kind of essence, but rather with the putative self-containedness of that essence, and with the fact that for Kant the true essence is unknowable as a matter of principle.

Leibniz had earlier concluded that in order for the world to be intelligible in terms of self-contained essences or monads, each monad had to include within itself a microcosmic mirror of the entire universe and all the other monads, each of which also includes all the others, and so on to infinity. For Leibniz, things in the world are really only related to one another indirectly, via their individual immediate relations to God. God is ultimately the entire source of the world’s coherence.

At the very beginning of his career, Kant had argued against Leibniz that interactions and inter-relations between things are real and not just an appearance. The world therefore has a kind of objective coherence in its own right. This is a stance that Aristotle clearly would endorse.

Hegel strongly agrees with Kant on this, but thinks that Kant did not take his critique of Leibniz far enough. (I don’t mean to identify Kant’s critique of dogmatism with his earlier critique of Leibniz, only to suggest that there is a connection between the two.) Hegel in effect argues that no essence is ever really self-contained, and that once we also drop the Leibnizian notion that essences are each supposed to be self-contained in splendid Hermetic isolation, there is nothing left in Kant’s philosophy that would require them to be unknowable as a matter of principle.

Dogmatism for Hegel refers — as it also implicitly would for Plato and Aristotle — to any claim that we somehow know the things we believe to be true, when in reality the basis of our belief is potentially refutable. Dogmatism is claiming the necessity characteristic of knowledge for conclusions that Aristotle would at best call merely probable.

(For Aristotle, “necessary” is just a name for whatever always follows from certain premises; “probable” is the corresponding name for what follows most of the time. Whether or not something always follows is a disputable question. New information might require that we re-classify what previously seemed to be a necessary conclusion as a merely probable one. I would add that what therefore seemed to be knowledge — because it seemed to follow necessarily — may turn out to be only a relatively well-founded belief. Individual humans do have genuine knowledge, but no individual knower can legitimately certify herself as a knower in any specific case.)

(Beyond this, even the historic mutual recognition of any individual concrete community can also turn out to be seriously wrong on particular matters. Widespread and longstanding social acceptance does not guarantee that certain things that are believed to be known are not just shared prejudice. Just consider the history of inferences from race, sex, religion, etc., to characteristics claimed to hold for all or most individuals subject to those classifications.)

(This does not mean we should indiscriminately throw out all claims that are based on social acceptance. That would result in paralyzing skepticism. To avoid dogmatism, we just have to be open in a Socratic way to honestly, fairly examining the basis of our beliefs about what meaning follows from what other meaning, in light of new perspectives. For what it’s worth, I say that once exposed to the light, prejudice against people based on shallow classification of their “kinds” can only be perpetuated through — among other things — an implicit repudiation of fairness and intellectual honesty in these cases.)

(Hegel the man was not immune to the various social prejudices of his time and place. According to his own philosophy, we would not expect him to have been. Outside the context of his main philosophical works, he is recorded to have made a few utterly terrible prejudiced remarks, and a number of other bad ones. In cases like this, we should give heed to the philosopher’s carefully developed philosophical views, and blame the time and place for the philosopher’s spontaneous expression of other particular views that seem out of synch with these. Every empirical community’s views are subject to adjudication in light of the ethical ideal of the truly universal community of all talking animals. The core of Hegel’s philosophy provides unprecedented resources for this.)

Kant’s own response to the issue of dogmatism is to maintain that strictly speaking, certainty and necessity apply only to appearances, which he does understand in a relational manner, but not to the things-in-themselves, which — following Leibniz — he still regards as self-contained and therefore non-relational.

Kant and Hegel seem to share the view that the very nature of necessity is such that it applies to things only insofar as they are involved in relations, and is only expressible in terms of relations. Where they differ is that Hegel sees not only appearances but also reality itself fundamentally in terms of relations.

For Hegel, there is no self-contained “thing in itself”, because the world is made up of what things are “in and for themselves”. Hegel introduces the notion of what something (relationally) is “for itself”, in the context of a reflective concept, and precisely as an alternative to the still-Leibnizian self-containedness of the Kantian “in itself”. What things really are “for themselves” turns out to undo the assumption of their essences’ self-containedness.

Simple Substance?

I tremendously admire Leibniz, but have always been very puzzled by his notion of “substance”. Clearly it is different from that of Aristotle, which I still ought to develop more carefully, based on the hints in my various comments on Aristotle’s very distinctive approaches to “dialectic” and “being”. (See also Form, Substance.)

Leibniz compounds a criterion of simplicity — much emphasized in the neoplatonic and scholastic traditions — with his own very original notion of the complete concept of a thing, which is supposed to notionally encompass every possible detail of its description. He also emphasizes that every substance is “active”. Leibniz’ famous monads are identified by him with substances.

A substance is supposed to be simple. He explicitly says this means it has no parts. In part, he seems to have posited substances as a sort of spiritual atoms, with the idea that it is these that fundamentally make up the universe. The true atoms, Leibniz says, are fundamentally spiritual rather than material, though he also had great interest in science, and wanted to vindicate both mathematical and Aristotelian physics. Leibniz’ notion of spiritual atoms seems to combine traditional attributes of the scholastic “intellectual soul” (which, unlike anything in Aristotle, was explicitly said by its advocates to be a simple substance) with something like Berkeley’s thesis that what can truly be said to exist are just minds.

On the other hand, a substance is supposed to be the real correlate of a “complete” concept. The complete concept of a thing for Leibniz comprises absolutely everything that is, was, or will be true of the thing. This is related to his idea that predicates truly asserted of a grammatical subject must be somehow “contained” within the subject. Leibniz also famously claimed that all apparent interaction between substances is only an appearance. The details of apparent interaction are to be explained by the details contained within the complete concept of each thing. This is also related to his notions of pre-established harmony and possible worlds, according to which God implicitly coordinates all the details of all the complete concepts of things in a world, and makes judgments of what is good at the level of the infinite detail of entire worlds. One of Kant’s early writings was a defense of real interaction against Leibniz.

Finally, every monad is said by Leibniz to contain both a complete microcosm of the world as expressed from its distinctive point of view, and an infinite series of monads-within-monads within it. Every monad has or is a different point of view from every other, but they all reflect each other.

At least in most of his writings, Leibniz accordingly wanted to reduce all notions of relation to explanations in terms of substances. In late correspondence with the Jesuit theologian Bartholomew Des Bosses, he sketched an alternate view that accepted the reality of relations. But generally, Leibniz made the logically valid argument that it is far simpler to explain the universe in terms of each substance’s unique relation to God, rather than in terms of infinities of infinities of relations between relations. For Leibniz all those infinities of infinities are still present, but only in the mind of God, and in reflection in the interior of each monad.

Leibniz’ logically simpler account of relations seems like an extravagant theological fancy, but however we may regard that, and however much we may ultimately sympathize with Kant over Leibniz on the reality of interaction and relations, Leibniz had very advanced intuitions of logical-mathematical structure, and he is fundamentally right that from a formal point of view, extensional properties of things can all be interpreted in an “intensional” way. Intension in logic refers to internal content of a concept, and to necessary and sufficient conditions that constitute its formal definition. This is independent of whatever views we may have about minds. (See also Form as a Unique Thing.)

So, there is much of interest here, but I don’t see how these ultra-rich notional descriptions can be true of what are also supposed to be logical atoms with no parts. In general, I don’t see how having a rich description could be compatible with being logically atomic. I think the notion of logical atomicity is only arrived at through abstraction, and doesn’t apply to real things.

New Approaches to Modality

I periodically peek at the groundbreaking work on formal systems that is going on in homotopy type theory (HoTT), and in doing so just stumbled on an intriguing treatment of modal HoTT that seems much more philosophically promising to me than standard 20th century modal logic.

Types can be taken as formalizing major aspects of the Aristotelian notions of substance and form. Type theory — developed by Swedish philosopher Per Martin-Löf from early 20th century work by the British philosopher Bertrand Russell and the American mathematician Alonzo Church — is the most important thing in the theory of programming languages these days. It is both a higher-order constructive logic and an abstract functional programming language, and was originally developed as a foundation for constructive mathematics. Several variants of type theory have also been used in linguistics to analyze meaning in natural language.

Homotopy type theory combines this with category theory and the categorical logic pioneered by American mathematician William Lawvere, who was also first suggested a category-theory interpretation of Hegelian logic. HoTT interprets types as paths between topological spaces, higher-order paths between paths, and so on, in a hierarchy of levels that also subsumes classical logic and set theory. It is a leading alternative “foundation” or framework for mathematics, in the less epistemologically “foundationalist” spirit of previous proposals for categorical foundations. It is also a useful tool for higher mathematics and physics that includes an ultra-expressive logic, and has a fully computational interpretation.

There is a pretty readable new book on modal HoTT by British philosopher David Corfield, which also gives a nice introductory prose account of HoTT in general and type theory in general. (I confess I prefer pages of mostly prose — of which Corfield has a lot — to forests of symbolic notation.) Corfield offers modal HoTT as a better logic for philosophy and natural language analysis than standard 20th century first-order classical logic, because its greater expressiveness allows for much richer distinctions. He mentions Brandom several times, and says he thinks type theory can formally capture many of Brandom’s concerns, as I previously suggested. Based on admittedly elementary acquaintance with standard modal logic, I’ve had a degree of worry about Brandom’s use of modal constructs, and this may also help with that.

The worry has to do with a concept of necessity that occasionally sounds overly strong to my ear, and is related to my issues with necessity in Kant. I don’t like any universal quantification on untyped variables, let alone applied to all possible worlds, which is the signature move of standard modal logic. But it seems that adding types into the picture changes everything.

Before Corfield brought it to my attention, I was only dimly aware of the existence of modal type theory (nicely summarized in nLab). This apparently associates modality with the monads (little related to Leibnizian ones) that I use to encapsulate so-called effects in functional programming for my day job. Apparently William Lawvere already wrote about geometric modalities, in which the modal operator means something like “it is locally the case that”. This turns modality into a way of formalizing talk about context, which seems far more interesting than super-strong generalization. (See also Modality and Variation; Deontic Modality; Redding on Morals and Modality).

It also turns out Corfield is a principal contributor to the nLab page I previously reported finding, on Hegel’s logic as a modal type theory.

Independent of his discussion of modality, Corfield nicely builds on American programming language theorist Robert Harper’s notion of “computational trinitarianism”, which stresses a three-way isomorphism between constructive logic, programming languages, and mathematical category theory. The thesis is that any sound statement in any one of these fields should have a reasonable interpretation in both of the other two.

In working life, my own practical approach to software engineering puts a high value on a kind of reasoning inspired by a view of fancy type theory and category theory as extensions or enrichments of simple Aristotelian logic, which on its formal side was grounded in the composition of pairs of informally generated judgments of material consequence or material incompatibility. I find the history of these matters fascinating, and view category theory and type theory as a kind of vindication of Aristotle’s emphasis on composition (or what could be viewed as chained function application, or transitivity of implicit implication, since canonical Aristotelian propositions actually codify material inferences) as the single most important kind of formal operation in reasoning.