6. Sixth Lecture: “7 + 5”, “Double I” and the Inaccessible True Subjectivity

In this lecture, I explore the concept of synthetic a priori judgment using mathematics and the well-known "7 + 5" example. However, this is by no means something to simply "get out of the way." On the contrary, we connect it to our search for true subjectivity—a theme that resonates in the works of many authors following Kant and one we aim to conceptualize through the model of evolutionary truth.


The Gravity Behind “7 + 5”

Kant provides in the Critique the example of mathematics as a synthetic a priori judgment, referring to “7 + 5” adding up to “12.” Many have claimed that explaining how synthetic a priori judgments are possible is one of the main reasons Kant wrote the Critique. Furthermore, the necessity of synthetic a priori judgments may also relate to the core principles of subjectivity as far as Kant managed to develop them.

In this section, I shall not address whether mathematics in general can be based on a priori propositions in light of logicism. Instead, I will focus on Kant and subjectivity, examining how the fact that 7 + 5 equals 12 necessitates the validity of synthetic a priori judgments in mathematics. This is essential if we wish to claim self-consciousness and, in conjunction, consciousness of the world. For Kant, this would be a regressive argument and perhaps not a particularly interesting one. However, it is worth noting that, in a sense, knowing 7 + 5 equals 12 is related to the very foundations of consciousness. Refuting this would essentially mean acknowledging the impossibility of knowing whether we exist as conscious beings at all—a paradoxical situation where the refusal itself becomes a manifestation of self-consciousness.

Kant emphasizes this point in the Critique of Pure Reason, noting the impossibility of logical self-analysis due to the fact that our mind becomes affected by itself when reflecting on its own subjectivity.

“All such introspective analyses lead to an unfathomable depth, to an abyss in exploring one’s nature,” for several reasons. Firstly, the object of such analyses (the “I of empirical apperception”) is affected by the subject (the “I of pure apperception”) conducting the analysis. Kant states: “He has a representation of himself as he is affected by himself, which, according to its form, depends merely on the subjective property of his nature and should not be interpreted as belonging to the object.” Psychology, therefore, may never hope to complete this task or satisfactorily answer the question: “What is the human being?” (A31).

Kant asserts that the subject, as the object of inner sense, can only be represented as an appearance. However, it is impossible to produce this appearance as a fixed and constant set of properties. The moment we start actively reflecting on ourselves, our mind becomes affected. The appearance of subjectivity (as a manifold of appearances) is created, experienced in time as we would experience other objects. This is why we cannot intellectually analyze our true subjectivity as something stable and permanent. Each reflection produces a potentially new manifold of appearances in time, which we perceive and synthesize into cognition. While this process yields knowledge of who or what we are, none of these cognitions comes with certainty.

Synthetic a priori judgments about our true subjectivity are, therefore, impossible. Descartes simplified this issue, positing that we could know ourselves (and our existence) prior to experiencing ourselves in time.

“Double I” and the Inaccessible True Subjectivity

In Anthropology from a Pragmatic Point of View, Kant refers to the intriguing notion of the “I appearing double.” He states: “Consciousness of oneself (apperception) can be divided into that of reflection and that of apprehension. The first is consciousness of understanding, pure apperception; the second is consciousness of inner sense, empirical apperception.” This results in the “Double I” (or “doubled consciousness of this I”), where consciousness splits between the “pure apperception of reflection,” the basis of subjectivity (“S0” in the model of the Evolutionary Truth), and the “empirical apperception of apprehension,” the subjective self as an inner stream of consciousness (“S1” in the model of the Evolutionary Truth).

I as the Pure Apperception of Reflection

From a psychological perspective, the “pure apperception of reflection” is not an object of any relationship in the psyche, nor does it contain any content. It is a foundational structure and precondition for understanding, functioning to unify and structure experiences, akin to Fichte’s “Absolute I.” Although related to a single subject and not the external world (“Not-I”), it provides the transcendental ground enabling the subject to distinguish itself from the world and its thoughts.

The critical aspect of “pure apperception” as a priori transcendental self-consciousness is not just its role in perceiving the world by cognizing objects but also in creating the separation that allows intuitions to become objects “für mich” (for me). Kant writes:

“Understanding is, generally speaking, the faculty of cognitions. These consist in the determinate relation of given representations to an object. An object, however, is that in the concept of which the manifold of a given intuition is united. Now, however, all unification of representations requires unity of consciousness in the synthesis of them. … The synthetic unity of consciousness [as a function of ‘pure apperception’] is therefore an objective condition of all cognition … something under which every intuition must stand to become an object for me; otherwise, the manifold would not be united in one consciousness.”

In the simpler terms, “pure apperception” precedes and facilitates everything else, including the categories of understanding, the forms of space and time, and the synthesis of empirical apperceptions derived from sensory input. On this stage, the “I” of empirical consciousness (as inner sense) emerges as an actor.

I as the Empirical Apperception of Apprehension

“Empirical apperception of apprehension” is one’s self-awareness arising from sensory experience. It is a posteriori consciousness, always tied to internal perceptions (i.e., time). Through empirical apperception, a human represents themselves as a unified subject across various states.

If “pure apperception” is the capacity to comprehend being a person in the world (e.g., seated in a restaurant), “empirical apperception” is the experience of tasting a specific dish. The latter is impossible without the former, as the former constructs reality by enabling the perception of surroundings and distinguishing objects.

In the simplest terms, the “I of empirical apperception” refers to the stream of consciousness accompanying our experiences and perceiving ourselves as subjects in phenomenal situations.