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.

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.

Constructive

Brandom’s inferentialist alternative to representationalism stresses material, meaning-oriented over formal, syntactic inference. Prior to the development of mathematical logic, philosophers typically used a mixture of reasoning about meanings with natural language analogues of simple formal reasoning. People in ordinary life still do this.

Where Brandom’s approach is distinctive is in its unprecedentedly thorough commitment to the reciprocal determination of meaning and inference. We don’t just do inference based on meanings grasped ready-to-hand as well as syntactic cues to argument structure, but simultaneously question and explicitate those very meanings, by bracketing what is ready-to-hand, and instead working out recursive material-inferential expansions of what would really be meant by application of the inferential proprieties in question.

For Brandom, the question of which logic to use in this explicitation does not really arise, because the astounding multiplication of logics — each with different expressive resources — is all in the formal domain. It is nonetheless important to note that formal logics vary profoundly in the degrees of support they offer for broad representationalist or inferentialist commitments.

Michael Dummet in The Logical Basis of Metaphysics argued strongly for the importance of constructive varieties of formal logic for philosophy. Constructive logics are inherently inference-centered, because construction basically just is a form of inference. (Dummet is concerned to reject varieties of realism that I would call naive, but seems to believe the taxonomy of realisms is exhausted at this point. This leads him to advocate a form of anti-realism. His book is part of a rather polarized debate in recent decades about realism and anti-realism. I see significant overlap between non-naive realisms and nonsubjective idealisms, so I would want to weaken his strong anti-realist conclusions, and I think Brandom helps us to do that.)

Without endorsing Dummet’s anti-realism in its strong form, I appreciate his argument for the philosophical preferability of constructive over classical logic. It seems to me that one cannot use modern “classical” formal logic without substantial representationalist assumptions, and a lot of assumed truth as well. If and when we do move into a formal domain, this becomes important.

As used in today’s computer science, constructive logic looks in some ways extremely different in its philosophical implications from Brouwer’s original presentation. Brouwer clouded the matter by mixing good mathematics with philosophical positions on intuition and subjectivity that were both questionable and not nearly as intrinsic to the mathematics as he seemed to believe. The formal parts of his argument now have a much wider audience and much greater interest than his philosophizing.

Constructive logic puts proof or evidence before truth, and eschews appeals to self-evidence. Expressive genealogy puts the material-inferential explicitation of meaning before truth, and eschews appeals to self-evidence. Both strongly emphasize justification, but one is concerned with proof, the other with well-founded interpretation. Each has its place, and they fit well together.