Pathway to Japanese Literature

I

I shall begin with Kant.
However, the relationships between the concepts employed in Kant’s spatial theory are by no means clear.
I wish to organize this beforehand.
According to the “Metaphysical Exposition of the Concept of Space” in the first *Critique*, space is neither an empirical concept nor a “so-called general concept concerning the quantitative relations of things in general.”
A concept is represented as something that contains a multiplicity of representations under itself (*unter sich*), but it can never be thought to contain them within itself (*in sich*).
Yet space contains its parts *Räume* within itself (*in sich*).
Therefore, space is not a concept but must be intuition.
Moreover, since space is “an a priori necessary representation lying at the foundation of all outer intuition,” space must be an a priori intuition rather than an empirical intuition when viewed from this perspective.
This is called pure intuition.
Next, we can represent space even without objects within it, but we cannot represent the absence of space.
That is, space is “regarded as the condition for the possibility of phenomena” (which should likely be amended to “outer phenomena”).
This means that space does not belong to things themselves but rather to the subjective *Beschaffenheit* of our *Gemüt*; in this sense, it belongs to the form of intuition.
Therefore, we have here obtained two conceptual aspects regarding space.
Pure Intuition and Form of Intuition.
Even though these two conceptual aspects are indeed conceptual contents of the same entity called space, it goes without saying that they are not immediately identical as conceptual contents.
Then, apart from both being conceptual contents of the same entity called space, what direct relation do the two bear to each other?
Kant states in “The Transcendental Exposition of the Concept of Space.”
Space must first and foremost be intuition rather than a concept not only when its concept is regarded as given a priori (in the case of metaphysical exposition) but also when it is regarded as a principle enabling synthetic a priori cognition (such as geometry) (in the case of transcendental exposition).
And it must be pure intuition in that it precedes all perception.
That is, this refers to the same pure intuition as in the previous case.

However, that such pure intuition could dwell within our *Gemüt* (*dem Gemüte beiwohnen*) was said to derive from its occupying a position within the subject as the latter's formal *Beschaffenheit*—that is, as the "form of appearance in general," or in other words, as the form of intuition.
In other words, when pure intuition was considered to occupy a position within the subject, it became none other than the form of intuition.
Therefore, what was called pure intuition constituted that state during which pure intuition was not yet considered to occupy a position within the subject, while insofar as pure intuition was considered to occupy such a position, it became the form of intuition.
This demonstrated that the conceptual content of pure intuition simultaneously contained two distinct elements.
Namely, pure intuition *qua* pure intuition and pure intuition *qua* form of intuition.
[It consisted of] pure intuition that was not the form of intuition and pure intuition as the form of intuition.
To rephrase further: Regarding the former, pure intuition and the form of intuition were distinguished as in the case of metaphysical exposition, while regarding the latter, the two proved indistinguishable.
Indeed, we discovered that which corresponded to this latter case at the very outset of the "Transcendental Aesthetic."
According to this interpretation, that which corresponded to sensation within phenomena was called *Materie*, while that which brought about "the manifold of phenomena being ordered in certain determinate relations" was called the *Form* of phenomena.
Since representations containing nothing pertaining to sensation were termed pure, such a form had to be referred to as the "pure form of sensible intuition."
This constituted precisely the form of intuition.
Yet within such a pure form, "the entire manifold of phenomena were intuited in certain determinate relations."
Therefore, by virtue of intuiting in this manner, this form of intuition was itself declared to be pure intuition—so it was asserted.
That is to say, the form of intuition became pure intuition.
Thus did the apparent contradiction—that pure intuition was distinguished from the form of intuition while simultaneously being identified with it—serve solely to indicate both a distinction between two differing contents within the concept of pure intuition and their simultaneous designation as "pure intuition" through a specific relational framework (hereafter I shall refer to pure intuition distinguished from the form of intuition as the first type, and that remaining undifferentiated as the second type).

How, then, does this distinction between the first and second types of pure intuition actually manifest itself? To examine this, I considered from another perspective what space constitutes as Kant's form of intuition and as pure intuition. Even when calling space a form of intuition, Kant's meaning cannot be that of a passive framework like a mere frame or box—it must itself be something dynamically active. Moreover, it is evident that even when speaking of it as the *Beschaffenheit* of the *Subjekt* or *Gemüt*, it should not be termed an immutable structure or *Organism* of these entities. It would be most precise to say the form of intuition is that which "lies at the foundation" of intuition, enables the "possibility of phenomena," and acts as sensibility's "constraint." However, describing it as a "foundation" or "constraint" naturally does not imply temporal precedence as a cause or developmental antecedent. In this sense, we may provisionally regard it as logically prior. Yet while this characterization holds true insofar as "logically prior" means more than mere psychological precedence, interpreting it as denoting the premise in a purely logical syllogism becomes entirely meaningless here. The notion that the constraining and constrained elements exist within logic's domain is generally nonsensical in this context. Consequently, there remains no basis for claiming such an interpreted form of intuition could equate to pure intuition. What persists for us must therefore be a "foundation" or "constraint" belonging neither to temporal sequence nor so-called logical-temporal order. This would constitute a transcendental-logical anticipation in the strict sense—that without which no given entity could attain existence. Yet even this formulation does not immediately place the constraint itself within logos' domain. One might tentatively posit that while the constraint belongs to logos' realm, what is constrained only emerges through this logos. However earnestly we attempt to deduce space from logos, I maintain the ultimate result would merely yield coexistence's form. This cannot even be called Räumlichkeit. Complex numbers do not directly constitute geometric planes. While calling coexistence a constraint here may not be inherently unreasonable, it ultimately represents thought's form rather than intuition's required form. Coexistence undeniably is not space itself.

If space itself (not the concept of space) cannot be deduced from logos, then such pan-logicist interpretation becomes unnecessary in this case. In that case, how should what are termed logical foundations, grounds, or anticipations be interpreted regarding Kant's space? I now wish to consider with some depth and generality what spatial consciousness truly consists of. As implied by the term *Gesichtsraum*, at least visual perception constitutes spatial consciousness. If we divide visual objects into color and form, these two are considered to differ entirely in nature through their mutual relationship. As commonly stated, we cannot represent color without form. While one might represent color without any definite form, it remains impossible without an extension that could be called form-in-general. Conversely, we can affirm that extension may be represented even without color. Although form arises solely through boundaries between colors, since these colors themselves already presuppose extension, color merely serves to make extension assume definite forms. Extension itself does not depend on color. When we speak of seeing an object at this moment, we likely observe the surface of that object—delimited into definite form by color—but in such cases, we intuit "the act of seeing a surface" as Fichte states; we intuit *Sehen* itself. Here, what is seen becomes the object's surface while what is intuited becomes the pure form of that surface. This constitutes nothing other than space (Fichte, *The Vocation of Man*). That is to say, pure form (form/extension) consists precisely in what is intuited through *Sehen* itself. Just as visual redness cannot be called red itself, space as such pure form contains no color. The seen object's surface must be understood as pure formal intuition combined with color sensation. To state this might seem to lead to the improper conclusion that what is intuited equals both vision and space; however, intuition's essential characteristic must lie in how intuiting immediately coincides with being intuited. Even when saying vision is intuited, this does not mean the intuiting subject stands outside vision; spatial intuition must still reside within vision while simultaneously transcending mere visuality. That visual perception becomes intuited means precisely this mode of intuiting—and as its necessary counterpart, nothing less than space being intuited.

There may also be a question of how one can immediately equate [the mere fact that vision is intuited] with spatial intuition, but we are not now deducing space; we are merely examining the characteristics of spatial intuition as such. Space was anticipated from the very beginning. Spatial intuition, on such grounds, transcends sensation; it was none other than that which establishes intuited space through intuiting the *Sehensakt* itself—for to intuit is to be intuited. Therefore, this was not mere *Anschauen* but had to be called *äussere Anschauung* (Fichte, Lotze et al.) or *Hinschauen* (Fichte). Needless to say, this relationship of spatial intuition was not limited to vision. It had to be common to tactile sensation at minimum. But what did this commonality fundamentally signify? It meant that both visual space and tactile space must ultimately converge into identical space. If they did not converge into identical space, it could not be called spatial intuition. For only when one still conceived of space as sensation or perception could two spaces be considered distinct—as psychology teaches—was this possible. The convergence into an identical entity had to obtain its possibility at minimum from spatial intuition transcending sensation. However, that alone remained insufficient for its realization. For realization to occur, normativity had to be anticipated in spatial intuition. Spatial consciousness could not be exhaustively interpreted as mere immanence. It was consciousness transcending consciousness itself. It had to be consciousness of objective reality. It had to be the norm for reality cognition. I believe space’s radical characteristic lay precisely in spatial consciousness being undeducible from mere consciousness alone. Space was the norm establishing an eternally uninternalizable external world. Through this normativity alone was objective reality’s uniqueness first guaranteed.

I believed that through this normativity all spatial representations could be linked to a singular spatial intuition.
And it was precisely through this normativity alone that spatial intuition became distinguished in principle from ordinary intuition.
Even what we call *Hinschauen* could only be grounded in this [normative framework].
The a priori nature Kant ascribed to space required interpretation precisely as this normativity—as commentators had rightly observed.
I maintained that the authentic meanings of those concepts called “basis” and “constraint” necessarily resided here.
What demanded particular attention was how this normativity could only emerge through spatial intuition’s status as actual consciousness—consciousness rooted in transcending sensation through its very operation.
At least within this framework, one found it absolutely inconceivable that norms like spatiality could subsist independently of actual consciousness.
The normativity of space proved inseparable from its fundamental condition as spatial intuition.
The norm itself was spatial intuition.
It was precisely in being both representation’s norm and norm’s representation that space possessed its distinctive spatiality—that quality distinguishing it from all else.
Space was first intuition and, within those bounds, secondarily norm.

Now that we had first clarified that spatial consciousness was spatial intuition in this sense, let us examine the distinction between the first and second pure intuitions that constituted our initial problem.
As had been stated earlier, intuition must generally possess both the aspect of that which intuits and the aspect of that which is intuited.
Therefore, spatial intuition necessarily contained both an objectified dimension and a not-yet-objectified subjective dimension.
The second pure intuition—that is to say, the form of intuition—had to precede this [intuition] in a certain sense by virtue of being intuition's form and its indispensable condition.
Yet this proved none other than the characteristic belonging to spatial intuition's not-yet-objectified aspect as previously discussed.
Thus the second pure intuition became the intuiting aspect, while consequently the first pure intuition became the intuited aspect.
In practice, Kant consistently invoked geometrically objectified space when illustrating the first pure intuition.
Only through such reasoning could one affirm the necessity of both being called pure intuition simultaneously—a necessity arising from spatial intuition's dialectical character.
This growing awareness of opposition between form of intuition and pure intuition within such parameters meant nothing less than Kant's transition from his theory of sensibility into the spatial theory of his *Transcendental Logic*.

Kant stated the following in his “Deduction”:
“Space, which is the mere form of outer sensible intuition, is not yet cognition at all.
That space merely provides the manifold of a priori intuition to some possible cognition.
However, to recognize something—for example, a line—within space, I had to draw that line and observe it.
Thus, a determinate combination of the given manifold needed to be synthetically established.
The unity of this procedure was simultaneously the unity of consciousness (the concept of a certain line).
‘And it is only through this that an object (a determinate space) is recognized for the first time.’”
According to this interpretation, the space initially termed “not yet cognition at all” as a mere form corresponded to what we had earlier defined as the form of intuition. Consequently, what Kant distinguished from this—the so-called “determinate space”—could tentatively be identified with pure intuition in its primary sense.
Indeed, when Kant wrote that “space is represented not merely as the form of sensibility but as intuition itself” (*Critique of Pure Reason*, 2nd ed., p. 160), this “intuition itself” had to be understood specifically as that which is intuited. This reaffirmed that the aforementioned “determinate space” was none other than pure intuition in its primary sense.
Here, as I had earlier noted, Kant must be seen as standing at the opposition between primary pure intuition and the form of intuition.
Yet alongside this opposition, he appeared to introduce a reorientation.
For according to Kant’s own formulation, space is “represented as intuition itself that contains a manifold—that is, represented *a priori* with the determination of the unity of this manifold within this intuition” (p. 160). However, I maintained that “being represented as intuition itself” held no legitimate meaning beyond denoting what is intuited.
Kant further claimed to unify intuitive representations.
This necessitated distinguishing between intuition *as* represented (i.e., qua intuition itself) and intuition *in* itself.
The former possessed the determination of manifold unity, whereas the latter lacked such determination.
While the latter corresponded merely to what was intuited—pure intuition in its primary sense—the former had to be an intuition equipped with unity’s determination, which Kant described as “synthesizing [zusammenfassen] the manifold into a single intuitive representation.”
Thus, in this former case, it could no longer be considered identical to pure intuition in its primary sense.
Kant specifically termed this formal intuition.
Yet he seemed not to have explicitly clarified the relationship between formal intuition and pure intuition.
I resolved to examine this distinction more thoroughly.

What did it mean that formal intuition possessed the property of unity?
This was none other than formal intuition itself being constituted through such unification.
That is to say, formal intuition represented the result of unification.
However, when analyzing this unification by dividing it into two elements—that which unified and that which was unified—the element corresponding to what was being unified could not be this formal intuition itself.
For formal intuition had already been unified.
Therefore, what was sought had to correspond to intuition not yet unified.
That could only be what had earlier been termed the first pure intuition.
Yet clearly there existed neither purely unifying elements nor purely unified elements—only unified entities stood established.
That is to say, there was no such thing as mere pure intuition; what existed was solely formal intuition as unified pure intuition.
Properly speaking then, pure intuition became constituted for the first time through formal intuition’s unification.
To rephrase—only when considering first pure intuition as a component of formal intuition did spatial intuition’s essential character manifest itself.
Pure intuition must indeed be formal intuition itself.
As Kant himself stated: through formal intuition’s unification did space first become “given” as intuition (S. 161).
Having proceeded from examining pure intuition’s opposition to intuitive form—if pure intuition was thus reducible to formal intuition—what relationship then existed between such formal intuition and intuitive form?
“The form of intuition provides mere manifold; whereas formal intuition furnishes representational unity” (S. 160).
Therefore formal intuition had to be considered more than mere intuitive form.
Moreover: “When space becomes represented as an object (as geometry indeed requires), it contains more than merely intuitive form’s [content]”
(Ibid.).
From this perspective one must recognize formal intuition as already objectified content.
Then what remained unobjectified within formal intuition?

Given that it is intuition, such a thing must necessarily exist. What is this? That clearly cannot be this form of intuition. For through this, only a mere manifold is provided. However, given that the first pure intuition is subsumed under formal intuition (objectified) in the previously discussed sense, its opposing form of intuition must likewise be subsumed under formal intuition (non-objectified). The form of intuition must also be considered as ultimately subsumed under formal intuition in this sense. There exists no form of intuition that merely provides a manifold; what exists is solely the form of intuition that provides the unity of the manifold—that is, unification into an intuitive representation. Therefore, the opposition between pure intuition and the form of intuition shifts to an opposition between objectified formal intuition and formal intuition that has not yet been objectified. This transition, I believe, is none other than what signifies Kant’s shift from his “Deduction” to the theory of space in the „Analytik der Grundsätze“.

According to Kant, the unity of this representation of formal intuition—which should be regarded as objectified—was none other than “the unification that combined the manifold of given intuition in general with fundamental consciousness in accordance with the categories” (p. 161). How was this possible? According to Kant, this unity presupposed a certain synthesis that did not belong to the *Sinn* through which “all concepts of space” (*alle Begriffe*) were first made possible. That is to say, behind the unity of this representation, there had to be a certain synthesis. That which clarified this synthesis was none other than the “axioms of intuition.” For all phenomena to be incorporated into empirical consciousness (meaning actual consciousness), they had to rely on the combination of *das Gleichartige* (the homogeneous) and the consciousness of the synthetic unity of this manifold of the same kind. However, the consciousness of this manifold of the same kind was, within intuition, the concept of quantity. “Yet we cannot represent even the shortest line unless we ‘draw it within the mind’ (*in Gedankenzu ziehen*)—that is, unless we successively produce all its parts from a single point and thereby first exhibit the intuition of a line.” We called that which preceded and underlay the representation of the whole through the representation of its parts an extensive magnitude. Therefore, space was none other than such an extensive magnitude. According to the “principle” of the axioms of intuition, “all intuitions were extensive magnitudes.” The intuition unified as an extensive magnitude became possible through the successive synthesis of its parts as described above. The intuition of space unified as an extensive magnitude had to, from the perspective of containing objectified unity, be that objectified formal intuition. Then, what was this successive synthesis of parts that was considered to be based on this objectified formal intuition? According to Kant, that which successively synthesized these parts in the consciousness of space was none other than productive imagination (*produktive Einbildungskraft*). Generally speaking, it was indeed this productive imagination that brought about actual consciousness. The rule (*Regel*) by which productive imagination established spatial intuition was none other than this successive synthesis. This *Regel* was none other than the very unity of spatial representation. And it was precisely something like figures as “schemata”—the *Produkt* generated by this *Regel*—that constituted that objectified formal intuition. Therefore, it was precisely this successive synthesis of productive imagination that had to correctly correspond to that formal intuition which had not yet been objectified.

When we interpret space not merely as something objective but as concrete formal intuition, must we not think of it in the manner described above? The opposition between objectified formal intuition and formal intuition that has not yet been objectified must be directly connected through formal intuition as productive imagination. I believe that the reason space is an intuition manifests in Kant as this formal intuition. Moreover, while formal intuition has already been interpreted as productive imagination, according to Kant, productive imagination must necessarily be linked to the categories. The *Regel* of productive imagination’s successive synthesis means nothing other than that productive imagination follows the categories. In other words, formal intuition can be called categorial to that extent. However, categories are, as people also say, nothing other than the norms of cognition. Formal intuition must be normative. The a priori that Kant recognized in space is none other than this normativity. Intuitive space is a norm.

II

Assuming that Kant's space could be interpreted in this manner, I sought to examine more deeply the relationship between geometry and this intuitive space.
The geometry Kant understood was likely three-dimensional Euclidean geometry, one might reasonably surmise.
Naturally, we could not expect to find in Kant's own words any definitive statement about whether spatial intuition itself was Euclidean or still lacked such determination.
However, since Kant maintained that Euclidean geometry was based on such intuition—and given that Euclidean geometry was then presumed to be the only geometry—it seemed natural to imagine that the intuitive space Kant conceived had been Euclidean.
Must we not then conclude that so-called non-Euclidean geometry cannot be grounded in Kant's intuitive space precisely because it proves incompatible with Euclidean geometry?
In fact, refutations of Kant's spatial theory typically proceed from this very point.
When assuming Kant's intuitive space was Euclidean and therefore incapable of supporting non-Euclidean geometry, Medicus mounted a defense of Kant true to his philosophical spirit (Kants transzendentale Ästhetik und Nichteuklidische Geometrien. Kantstudien 3.).
Kant's space demanded interpretation strictly as a constraint governing phenomena or experience.
Consequently, whether such constraint-laden space should be called Euclidean or non-Euclidean became essentially meaningless.
The Euclidean property of space—its flatness—could not be deduced as a necessary experiential constraint.
This characteristic merely reflected features spatial intuition had acquired through experience; thus it possessed no necessary validity but only factual applicability.
At most it held cosmological significance.
Space constituted the principle of spatial relations that defied categorization as either planar or non-planar.
Though this principle provided foundations for both Euclidean and non-Euclidean geometries through spatial relations themselves, such general laws of spatial synthesis had to remain singular despite geometry's internal distinctions.
Yet despite this theoretical unity, people persisted in believing intuitive space was planar.
Guided by this conviction, they found themselves able to conceptualize non-Euclidean spaces.
Thus Euclidean geometry maintained normative supremacy over all other geometries, its planar space appearing as something immediately given.
But since this normative significance itself stemmed from psychologically subjective experience, it stood apart from apriority.
Even while accepting intuitive space's apparent planarity, we lacked grounds for a priori rejecting its potential non-planar nature.
Setting aside this conclusion's validity for now, its foundational premises seemed fraught with lingering difficulties.

Even if one speaks of space as a “constraint of experience” or a “principle of spatial relations,” it goes without saying that this is not merely a constraint of experience but a special “constraint of experience” called “space,” and not merely a principle but a “principle of spatial relations.” Therefore, even if one discusses definitions of general “constraints of experience” or “principles” that have stripped away what is intrinsic to space, this bears no relation to space whatsoever. We must understand why space is both a constraint of experience and a principle. Moreover, if we adopt Kant’s position—as previously stated—it is precisely because space constitutes formal intuition that we find grounds to regard it as both a “constraint of experience” and a “principle.” If one were to claim that space could be considered a “constraint of experience” without being formal intuition, then space would become indistinguishable from the categories as Kant himself conceived them. When one forgets that space belongs to intuition (even if it cannot subsist without relation to categories), space must inevitably collapse into an empty concept. As Kant himself confesses in the *Prolegomena* (§39): “In my study of the pure elements of human cognition, it was only after long months of deliberation that I could definitively distinguish and separate the pure elementary concepts of sensibility (space and time) from those of the understanding”—the constitutive quality that makes space truly *space* must reside in the inseparability of regarding it as both constraint and principle from recognizing its belonging to intuition. If space can be interpreted exhaustively as formal intuition, then whether it is a priori planar or cannot yet be considered a priori as such becomes, I believe, a provisional question worth raising. Consequently, we must tentatively acknowledge the possibility that its planarity could be deduced as a “necessary constraint.” To be sure, advancing this argument may invite criticism that one still naively conceives of space objectively; however, as I have previously stated, spatial intuition must contain a point where that which intuits and that which is intuited are immediately connected—hence where the act of intuiting (what might be called the *principle* of space) must directly bind to objective space. One might object that what is intuited is not necessarily objective, but were this so, space would become indistinguishable from inner sense. Space must retain some aspect that can be regarded as the framework of the external world—a notion said to persist even in Kant’s conception. It is only when one fails to discover an internal principle-as-force within this framework that it becomes an idea warranting critique for the first time. Furthermore, while Medicus asserts that intuitive space’s planarity constitutes merely our psychological belief—a norm derived from experience—what precisely does it mean for such beliefs and norms to “originate from experience”? Pushed to its logical conclusion, this can only mean that spatial planarity has been empirically demonstrated through experience. Yet as this very paper concludes, such empirically demonstrated planarity in fact pertains to physical space—not intuitive space. Nevertheless, we persist in believing intuitive space to be planar. Therefore, this “belief” can no longer be deemed empirical or psychological in the aforementioned sense. It must instead be recognized as nothing other than a “belief” inherent to intuitive space itself—a belief that may well belong to intuitive space’s essential nature.

When considering this matter, it became imperative to question whether space was a priori planar.
And indeed, as long as grounds existed for imagining Kantian space as planar, criticisms that Kant could not provide foundations for non-Euclidean geometry seemed destined to persist.

But was intuitive space fundamentally Euclidean from its inception?
If so, in what sense did this hold true?
Oskar Becker (*Contributions to the Phenomenological Foundation of Geometry and Its Physical Applications*, Jahrbuch VI) phenomenologically differentiated three types of spatial representation.
First: *präspatiale* or *Ausbreitungsfelder* manifested as fields of visual-tactile-auditory perception and those arising from ocular/tactile organ movements; second: *orientierter Raum* emerged through self-external world relations possessing depth.
Within this space, the self's central point functioned as singularity; through repetition of this singularity arose the third *homogener Raum*.
Our so-called intuitive space clearly resided exclusively within this third stratum.
Becker separated Euclidean properties into two aspects: spatial curvature value K=0 and its connexus being infinitely open (*einfach* in unitary terms) - where 'connexus' denotes topological fundamentals like edge possession in spatial figures or figure division through closed line rotations (degree of connectivity).
(Just as spatial curvature constitutes metric geometry's foundational concept).
The third homogeneous space first required K=0.
This necessity stemmed from three factors: First, for enabling repetitions distinguishing *orientierter Raum* from homogeneous space, rigid-body motion groups needed translation groups as subgroups (*Untergruppen*) - necessitating K=0 per Lie's formulation.
Second, homogeneous space permitted partitioning into congruent figures (e.g., sphere-filled space with uniform interstitial gaps per Einstein's *Geometry and Experience*).
Thirdly, similarity propositions held valid here without directional variance by position.
These conditions collectively demonstrated K=0 through indirect proof.
Thus homogeneous space curvature necessarily equaled zero.
As homogeneous space embodied *principium individuationis*, it lacked periodicity (*Periodizität* - interpretable as connectivity degree) restricting spatial content freedom.
"Therefore when presupposing Euclidean metrics," Becker concluded, "we must select non-periodic unified forms - neither excluding nor imposing cycles."
"This constitutes Euclidean space's unitarily infinite connexus."

Thus, homogeneous space is said to be infinitely *offen*.
Therefore, intuitive space appears to necessarily be Euclidean.
However, if principles such as relativity demand the existence of non-Euclidean space—that is, their validity within spatial intuition—we are confronted with the contradiction that spatial intuition is simultaneously Euclidean and non-Euclidean.
How can this contradiction be avoided?
Becker here distinguishes between space’s *Konstitution* and its *Struktur*, arguing that intuitive space (*Phantomraum*) arising from the former is Euclidean, while metric space derived from the latter can be non-Euclidean (ibid., p. 151).
Measurement or quantification, being phenomenologically based on that second *orientierter Raum*, does not contradict the intuitive space as the third homogeneous space being a metric space.
However, when we measure, it is not merely measuring with the eye (which is appropriate in the *orientierter Raum*); it is nothing other than determining quantities physically.
However, determining quantities physically only becomes possible through intervention in what is called homogeneous space.
Even the structure of space must ultimately be founded upon the constitution of homogeneous space.
I think that structure and constitution are not two positions but a distinction of stages of determination within a single position.
Therefore, the contradiction is not eliminated by this distinction.

It was said that intuitive space—considered Euclidean—was infinitely *offen*; however, this in fact signified periodicity—that is, independence from connexus.
Therefore, one had to conclude that it could not be considered either infinitely *offen* or finite *geschlossen*.
Could the same not be said regarding curvature?
In other words, intuitive space was neither *K* = 0 nor *K* ≠ 0.
For this was because the connexus of space itself could only become a matter of question insofar as it was considered to be based on spatial curvature—and consequently, the connexus of space itself had to be said to be metric in this sense.
I would pursue this point further.

I thought it was Lotze who most directly illuminated the inner nature of intuitive space.
In Lotze's framework, intuitive space was understood as foundational to geometry.
"Through comparing one line with others," he wrote, "we distinguish length and direction—yet even the simplest propositions about these cannot be established except through intuition."
(Metaphysik, pp. 223–4).
Synthetic judgments in geometry become possible only through spatial intuition.
What then constituted Lotze's intuitive space?
The relationship *r* between two elements intuited as lines and relationship *w* between two such *r* as angles combined to form our intuitive space—yet this hypothetical *Raumoid*, imagined when *r* and *w* formed different combinations than those in our actual space—proved fundamentally impossible (ibid., p. 241).
Thus intuitive space had to be singular.
We maintained that straight lines could be conceived as limits of curves but noted: "In their determination and measurement, constructing this series of curves proves impossible without employing some intuition of straight lines" (ibid., p. 246).
A straight line recurring as a circle with infinite diameter amounted to logical barbarism.
One could not invert the principle that parallel lines never meet by claiming they intersect at infinity.
Logical demonstration failed to resolve the problem of parallels.
No contradiction could arise within intuition itself.
Since parallel lines rested on "intuition's perfectly clear fact," observed deviations in physical triangles' angle sums required interpreting bent light from special causes rather than spatial alteration.
Spatial relations themselves had to remain absolutely invariant (ibid., pp. 246–9)—hence intuitive space remained Euclidean.
Yet according to Lotze, curves became conceivable against straight lines through intuitive space's linearity principle.
The Euclidean character of intuitive space consisted precisely in this linearity.
How then should we understand such linearity?
Though ordinary lines were said to lack curvature, I found it necessary to distinguish between "having no curvature" and "having zero curvature."
We asserted that projective geometry involved no curvature while metric geometry contained curvature.

And within the latter, we considered that only in Euclidean geometry was the curvature zero.
Therefore, this distinction was not merely a classification of terms.
When we said that the straight line was a principle, I thought this could only mean that curvature was not being considered here—in other words, that there was no curvature.
For since zero was also considered a number, the case where curvature was zero should hold an equal position to where it was not zero; consequently, the former becoming a principle that grounded the latter was something that could not arise unless we assigned a special meaning to zero.
For linearity to be a principle did not mean the special case where curvature was zero; rather, it had to mean there was no curvature.
In non-Euclidean geometry, even its so-called straight lines possessed curvature; yet did not the fact that straight lines were still unambiguously distinguished from other curves mean that linearity did not depend on curvature?
I thought that linearity as a principle was one that consistently underlay all geometries.
Poincaré argued that projective geometry presupposed straight lines and that since straight lines were based on measurement, projective geometry was also quantitative (*Dernières Pensées*, p. 58); however, originally in projective geometry, there had been no such measurement-based straight lines.
A simple line sufficed.
Moreover, as Poincaré had us state, we could not have a reason to consider this line particularly as a curve; that is, we still had to consider the line a straight line in a certain sense.
A distinction had to be made between quantitative straight lines and qualitative straight lines.
The principle of linearity was precisely manifested by this qualitative straight line.
Projective geometry was qualitative.
Therefore, if one thoroughly examined Lotze’s intuitive space, I believed it could be said that intuitive space was qualitatively Euclidean.
And since non-Euclidean nature was a quantitative specification where *K* ≠ 0, it followed that this did not contradict qualitative Euclidean nature.
When those like Jonas Cohn asserted the a priori nature of Euclidean space (J. Cohn, *Voraussetzungen und Ziele des Erkennens*, p. 249), if one argued that by distinguishing between quality and quantity, Euclidean space became a priori in a qualitative sense, this would become all the more evident.
If Kant’s space was interpreted as Euclidean, then it could be interpreted as this qualitative flatness (linearity).
And only by interpreting it in this way would we be able to grasp the qualitative characteristics of spatial intuition.
While allowing that spatial intuition was Euclidean, I thought it could still retain the possibility of grounding non-Euclidean geometry.
Then, what became of the finite or infinite nature of space that appeared connected to planarity?

In ordinary space, we distinguished between metric (*Metrik*) and connection (*Connexus*, *situs*). Of course, regarding each individual object of geometry—that is, any figure within space—this could not be doubted. However, the connection of space itself could not be considered independent from its metric. When *K* = 0 or *K* < 0, it necessarily followed that space was infinitely *offen*; when *K* > 0, it necessarily followed that space was finite and *geschlossen*. That is, the connection of space itself was in fact nothing but a consequence of its metric determination. Then, was distinguishing between connection and metric in space entirely meaningless? When we spoke of Riemannian space as finite, we considered that we had been able to represent it in intuitive space by first representing a spherical surface and then analogizing it to the next dimension. However, since the spherical surface represented as a plane in Riemannian space was no longer two-dimensional but could only be represented in three dimensions, it had to be said that the representation of three-dimensional Riemannian space required four dimensions. However, in reality, we could not possess such four-dimensional representations. We could only convince ourselves that the image of a sphere within space actually signified Riemannian space. Only through this convention of representing space itself as an image within space did the concept of spatial connection come into being, and to that extent, it could merely be considered independent of the spatial metric. The so-called positional analysis centered on the concept of connection had to be called a specialized geometry that did not deal with the determination of space itself but rather handled the determination of figures within space. Lotze criticized Riemann’s finite space for conflating space with its interior and figures (*Metaphysik*); however, representing space as a figure within space was not a direct representation of Riemannian space itself but merely a representation based on this convention. This, of course, did not constitute a criticism of Riemannian geometry. However, I thought this pointed out the fallacy of considering that spatial connection could essentially come into being even without relying on those conventions. Therefore, it was fundamentally impossible to classify space based on its connection. The reason it appeared as if this were possible was merely because the connection was in fact considered as a consequence of the metric—that is, as a connection indistinguishable from the metric. Clifford surfaces might also have been considered a spatial *Raumform*, but just as a spherical surface was a representation of Riemannian space within Euclidean space, I rather thought that they might have been considered a representation of Euclidean space within Riemannian space—one of its forms. The combination of figures within space had to not be considered as the combination of space itself. There existed no qualitative combination in the sense that space itself was pure—that is, independent of metric.

Certain things were merely quantitative combinations—in essence amounting to nothing more than measurement.
Just as Riemann had distinguished between space’s unboundedness (*Unbegrenztheit*) and its infinity (*Unendlichkeit*), designating the former as an extension relation and the latter as a metric relation (*Ueber die Hypothesen, welche der Geometrie zugrunde liegen*), the finite or infinite nature of space was in fact not a matter of space’s combination itself but rather a combination based on its metric.
The ability to designate space as *offen* or *geschlossen* was possible solely through certain conventions that represented space within its figures.
Now, since I had previously clarified that intuitive space was Euclidean even without containing measurement, such intuitive space consequently not only did not permit the concept of qualitative combination but also did not possess the necessity of including quantitative combination through measurement.
The concept of connection in intuitive space generally did not hold.
Space therefore generally could not be said to be finite *geschlossen* or infinite *offen*.
Just as Riemann stated "eine unbegrenzt ausgedehnte Mannigfaltigkeit," it was generally considered unbounded.
Such unbounded space was to be considered both finite and infinite.
The unbounded space we represented could also be considered finite as it was.
For he belonged to quality and this belonged to quantity (Kerry, *System einer Theorie der Grenzbegriffe*, p. 88).
Kant had stated in one of his antinomies that assertions about the world being finite or infinite with respect to space (*Critique of Pure Reason*, *Prolegomena*) and about space itself being finite or infinite (*Prolegomena*) both rested on improper concepts.
That was because they viewed the infinite *Prozess* of *Reihe* itself as if it were given to the understanding.
Using Lotze’s words, it was because they mistook space itself for figures within space.
The concept of spatial connection did not hold from the very beginning.
The concept of “infinite space” that Kant had pointed out did not hold as he described insofar as “infinite” was interpreted in the sense of *offen*.
If one interpreted this in Riemann’s sense of *unbegrenzt* (unbounded), space would be represented as “das unendlich Gegebene” (*Critique of Pure Reason*, p. 39) (Kant had of course used *unendlich* and *unbegrenzt* synonymously, but I distinguished between *unendlich* and *unbegrenzt* by assigning them new meanings throughout).
Now I understood why 'infinite' and 'unbounded' were easily conflated.
Just as the absence of curvature was thought to directly link to *K* = 0—which contradicted it—despite not conflicting with *K* ≠ 0, unboundedness was thought to directly link to infinity—which was incompatible with it—despite being capable of encompassing finitude.
Therefore in this sense the statement that space was qualitatively infinite could also be permitted.
And finally corresponding to Euclidean space being infinite in metric geometry that qualitative planarity and this qualitative infinity were linked even in their qualitative aspects.

The necessity could only be said to lie within spatial intuition itself.
Therefore, even if space were called Euclidean in this sense, it would not follow from anywhere that this could not provide a foundation for non-Euclidean geometry.

Having considered planarity in the above manner, how then were we to consider three-dimensionality? Kant counted the three-dimensionality of space as a necessary element, even listing it among geometry’s propositions. Did this mean Kant’s space could not provide a foundation for n-dimensional geometry? Medicus defended Kant here as well, arguing that just as with planarity, the three-dimensionality of space was not deduced as a necessary constraint of experience but remained merely factual—thus posing no difficulty for Kant on this point. Yet the three-dimensionality of intuitive space appeared far more “indubitable” compared to its planarity (the aforementioned paper). However, since Kant’s space had to be regarded as intuition—as previously established—this “indubitable” aspect now demanded consideration as a determination of intuition itself. Of course, it would not possess rational necessity. Why space must be three-dimensional was something reason could not explain at all—in that sense, it would not constitute a necessary experiential constraint. What was conceived here was not space itself but spatiality at best. Spatiality might lack rational necessity for three-dimensionality, but space had to be absolutely three-dimensional in intuition. As Kant himself had posited, one had to call it “necessary.” Only through this necessity could intuitive space establish itself—and it was precisely this intuitive space that gave rise to why space was termed an experiential constraint. The thinker who expounded on the necessity of intuitive space’s three-dimensionality was once again Lotze. Helmholtz had proposed that if intelligent beings with only two-dimensional consciousness existed—yet equal in intellect to us—their perceived space might resemble a spherical surface in Euclidean space corresponding to a Riemannian geometric plane (*Populäre wissenschaftliche Vorträge III*). Lotze refuted this, arguing that beings endowed solely with two-dimensional consciousness could not logically exist. If such a creature departed from point A on a sphere and circumnavigated it along a longitude—even returning to A—how could it recognize this as the same point? It would merely perceive an identical sensation labeled A repeating endlessly. Were it conscious of spatial relations, this would amount only to circumstance A recurring at fixed intervals along an infinitely extended straight line. Yet if the creature somehow knew it had returned to point A, it would inevitably discover a contradiction in the concept of a straight line.

For according to Lotze, a straight line was nothing other than distance; thus, in this case, the maximum distance would mean the absence of distance. Therefore, beings were logically compelled to avoid this contradiction regardless of perception. Consequently, they were forced to anticipate space's three-dimensionality. That is to say—if beings possessed spatial representation—the two-dimensionality they were assumed to possess had to necessarily expand into three-dimensionality (Metaphysik, pp. 249–252). However, a spherical surface was not Riemannian space's plane itself but rather merely its representation within Euclidean space as previously discussed. Since Lotze mistook this representation for space itself, one had to conclude this criticism missed its mark as criticism proper. Yet what Lotze expressed amounted solely to intuitive space's necessary three-dimensionality. Following Lotze's reasoning: while intelligent beings found themselves compelled—and simultaneously capable—of transitioning from two-dimensional to three-dimensional space in this manner; no equivalent claim could be made about humanity transitioning from three-dimensional to four-dimensional space. For whereas two-to-three-dimensional transition arose from inherent contradiction—no such contradiction could be discovered within three-dimensional space itself; moreover humans factually lacked capacity for four-dimensional intuition (ibid., pp. 252–257). Intuitive space must be three-dimensional. So long as one acknowledged Kantian space as intuition—even when recognizing its constitution through category-linked productive imagination—there existed no justification for considering it divorced from this three-dimensional determination intrinsic to intuition. Though connected to categories—space was not reducible to categories themselves. While conforming to categories (den Kategorien gemäß), it had to be understood as imposing spatial-specific limitations upon Category A. Even Kant's notion of in Gedanken ziehen ("drawing in thought") could not enable us to conceive drawing lines within four dimensions. Of course—though objective intuitive space was three-dimensional—one might argue that productive imagination's successive synthesis constituting it was not itself three-dimensional; yet framing it thus required recognizing this successive synthesis as fundamentally dimensionless. Productive imagination's successive synthesis itself had to lack dimensions while paradoxically generating them. Nevertheless—that which emerged could only be three-dimensionality itself. Therefore—though not three-dimensional per se—this successive synthesis had necessarily to produce three-dimensionality. Some might contend spatial intuition itself remained dimensionally indeterminate—only becoming three-dimensional through reflective processes—yet that which reflection revealed as three-dimensional must have been three-dimensional ab initio. Did this then mean Kantian space could not ground n-dimensional geometries?

Therefore, I had to examine more thoroughly the relationship between geometry and intuition as Kant conceived it.
As Kant states—"The science of extension (geometry), together with its axioms, bases itself upon this successive synthesis of productive imagination in figure production, and these axioms express the transcendental constraints of sensible intuition" (*Kritik der reinen Vernunft*, p. 204)—within his framework, geometry's object and intuitive space were immediately considered identical; however, intuition itself and intuition's axioms had to be regarded as fundamentally distinct.
Even if other propositions were derived from axioms through the law of contradiction, there remained no difference between them and axioms in that all propositions must consist of intuitive content.
Yet according to Kant, axioms were said to be "directly certain."
However, whether intuition was directly certain or not constituted a judgment about intuition rather than intuition's own standpoint.
Intuition itself had to be entirely direct.
Even the content of theorems requiring the most elaborate proofs was never merely indirect.
The "directness" of being "directly certain" belonged not to intuition itself but to judgments about intuition.
Intuition itself possessed no temporal sequence.
This sequencing existed solely within "demonstration."
Axioms were nothing but demonstration's starting point.
Therefore, one could only say that axioms first emerged through extracting specific content from intuition as particularly "direct."
It was not that intuition existed through axioms—rather, axioms existed through intuition.

What did it mean for intuition to become axioms of intuition? It had to signify nothing other than thought becoming linked to intuition. However, since this discussed intuition was formal intuition interpreted as productive imagination—and since productive imagination could not remain mere so-called intuition but had to be bound to categories—one might argue it was improper to particularly claim it "linked" to thought. Yet clearly productive imagination was not itself a category. Nor did productive imagination entirely encompass categories within itself. Productive imagination (belonging to sensibility) had to remain entirely outside categories while still being required to "follow" them. Therefore formal intuition as productive imagination was not something pre-connected to concepts from inception—it merely possessed potential connectivity capability. Thought needed fresh linkage to intuition precisely where spatial intuition revealed its distinctive feature—the capacity for true connection with thought empirical intuition could only link with empirical concepts pure thought could fully exert its *demonstrative* power exclusively within spatial intuition axioms of intuition could be called intuitions mediated by thought spatial intuition through linking with free thought axioms could define its content via thought’s own power while free thought through linking with spatial intuition axioms could transcend spatial intuition without negation while remaining grounded therein axioms of n-dimensional space could only first emerge thusly For thought itself contained no contradiction when extending three dimensions into four even if sensibly unfeasible Considering space as schema its sensibility resided in three-dimensional necessity while intellectual aspect emerged through transcending this sans contradiction Herein lay formal intuition’s raison d'être Without this distinction space would lack basis for differentiation from mere empirical intuition When interpreted thus Kant’s position contained no difficulties whatsoever.

If we do not acknowledge here this freedom that thought holds over spatial intuition—that is, if spatial intuition were bound to thought from its very inception—then not only would the concept of n-dimensional space become impossible, but we could not comprehend how diverse geometric spaces resting upon differing axioms might be conceived.
This was likely the difficulty into which Kant fell.
By unjustifiably binding spatial intuition and thought in an excessively rigid relationship—one might say immovable—Kant reduced axioms to mere tautological reflections of intuition, necessitating their fixity and immutability.
From Kant’s perspective, if intuitive space possesses qualitative flatness as I previously stated, this would entail the parallel postulate and consequently must conflict with non-Euclidean geometry.
The sole path beyond such difficulties lies in abandoning fixed axioms—that is, in recognizing thought’s freedom vis-à-vis spatial intuition.
This amounts to nothing less than rigorously distinguishing between intuition and its axiomatic formulations.
To put it plainly: “intuitive space” must be decisively distinguished from “geometric space” as geometry’s object—both in conceptual intension and extension.
We discern a fragment of this distinction in intuitive space’s three-dimensionality.

III

In reiterating what had been discussed thus far here, we arrived at two points: first, that intuitive space possessed normativity; and second, that intuitive space was distinct from the object of geometry.
Having established these points, I considered how physical space came into being.
The term “physical space” was needless to say commonly used; however, its independent existence was not necessarily acknowledged.
“There was no basis for distinguishing between geometric space and physical space.”
“However, whereas for mathematicians space was a spatial image abstracted from physical objects, for physicists it was a physical object, and the spatial image was nothing more than its form.”
“This distinction was nothing more than a matter of our subjectivity—that is, our observational perspective—whereas space itself remained an indivisible unity.”
(L・Nelson、*Kant und Nichteuklidische Geometrie*、p・26 ff・) Could such a view not be considered?
In this way, physical space would ultimately be subsumed into geometric space.
However, I believed I could answer such questions neither affirmatively nor negatively.
The necessity that two spaces must be identical in some sense was not so much a result of investigation as it had rather to be the starting point.
For if there had been no direct connection between the two, it would have become utterly meaningless for us to call them both “space” in the first place.
However, even if space had ultimately to be connected to a single thing, we had not the slightest reason to think that space must therefore be of only one kind.
As I had already pointed out the distinction between geometric space and intuitive space—even when considering how the former, while being grounded in the latter, ultimately had to be distinguished from it—regarding space as merely one kind was either dogmatism or carelessness.
We had essentially to permit the possibility of various spaces.
It was none other than the so-called “observational perspective” that signified the existence of these various spaces.
However, if one were to seek something positive from Nelson’s words, it would have been the inherently close relationship between geometry and physics.
By this inherent nature, it might have been speculated that geometric space and physical space constituted an indistinguishable single entity.
However, when considering the possibility of various spaces, this relationship should have served not to reach a conclusion denying the independence of physical space but rather to clarify how physical space came to be independent.
I believed I could find in Helmholtz’s theory of space something that—while reaching the same conclusion as those views seeking to negate the distinction between geometric space and physical space in this manner—in a certain sense constituted a diametrically opposed position.
According to Helmholtz, measurement in geometry was based on the “principle of congruence,” which in turn was founded on the possibility of perfectly free motion of rigid bodies (*Ueber die Ursprung und die Bedeutung der geometrischen Axiome*.).

And the fact that this congruence was performed irrespective of a figure’s position, direction, or path of motion constituted the *Tatsache* that made spatial measurement possible (*Ueber die Tatsache, welche der Geometrie zugrunde liegen.*).
The question of how the axioms of geometry could be applied to physics first became comprehensible when geometry was considered to be founded upon such facts.
If spatial quantities in which identical physical phenomena occurred under identical conditions and elapsed time were called "physically equivalent," then such equivalent spatial quantities were defined through the motion of rigid bodies—for example, by compasses and rulers.
This fact of physical equivalence alone constituted the completely determined, objective, and unique property of spatial quantities.
Therefore, geometry based on such facts indeed had to be called physical geometry.
The geometry applied in physics was this physical geometry; consequently, the possibility of its application left no room for doubt as something natural (*Die Tatsache in der Wahrnehmung*, Beilagen.).
However, this physical geometry—which contained the *realer Inhalt* of facts and whose axioms were determined not merely by representational forms but by relations in the *reale Welt*—had to be clearly distinguished from pure geometry according to Helmholtz, the latter consisting of space’s innate, immutable form of intuition and its axioms as conceived by Kant (ibid.).
That such pure geometry was applied to physics remained purely coincidental; it bore no relation whatsoever to physical geometry, which “expressed not merely spatial relations but solely the dynamical relations of rigid bodies’ motion.”
Now, since physical geometry took Riemann’s “n-dimensional manifold” as its object per Helmholtz’s own statement, this object had on one hand to signify geometric space.
Yet simultaneously, since it expressed only dynamical relations, it necessarily signified physical space.
Herein lay the denial of distinction between the two in accordance with Nelson.
However, if one allowed for the possibility of Kant’s so-called pure geometry, then geometric space as its object had to be declared entirely independent of both aforementioned spaces.
In this sense, geometric space and physical space were not merely distinguished but had their relationship completely severed.
Nevertheless, such mutual non-interaction would have found it profoundly difficult to gain sympathy from the many scientists and philosophers who believed in mathematics and physics’ pre-established harmony—the necessary application of mathematics to physics (see A. Kneser, *Mathematik und Physik*).
Moreover, Helmholtz’s thought on geometry and space—asserting their mutual implication in one sense while resulting in complete non-interaction in another—concealed within it a difficulty inherent to empiricism.
That was the question of how one could find *apodiktische Geltung* in geometry from a thinker like Helmholtz.
The solution to this difficulty was none other than Kant’s theory of space.
Therefore, physical space would for the first time find its legitimate foundation for formation in Kant’s theory of space.

In the Third Analogy (*dritteAnalogie*) of the *Critique of Pure Reason*, it is stated that “all substances, insofar as they are perceived as simultaneous in space, stand in thoroughgoing interaction”; yet given that Kant himself provides no particular explanation of this space, it seems natural to interpret this as what I have previously identified in Kant’s work as intuitive space.
If that is the case, then in the sense of serving as the field of interaction for such physical entities, physical space appears to lack any basis for being distinctly distinguished from intuitive space.
If, as is commonly said, we call Kant’s intuitive space absolute space, then in this case it may also be referred to as physical absolute space.

However, on the other hand, Kant denied all existence of absolute motion and consequently asserted that every physical space ("empirischer Raum") must be relative.
That which contained relative space and relative space had to itself be relative space.
Absolute space was posited solely as an idea conceived to ultimately encompass all possible relative spaces (Metaphysische Anfangsgründe der Naturwissenschaft. Phoronomie.).
If the absolute space from the initial Third Analogy and the absolute space as a later idea had been identical, no contradiction should have arisen here.
However, in that case, the question immediately emerged of how the initial absolute space—conceived as intuitive space—related to the later physical relative space.
We arrived at the problem of how intuitive space becomes physical space.
If, furthermore, the earlier absolute space and the later absolute space as an idea could not be immediately considered identical, then by asserting that physical space must be regarded as absolute while simultaneously requiring all physical spaces to be relative, one appeared to fall into a contradiction.
The only way to escape this contradiction now lay in noting two distinct meanings of "absolute" and "relative."
That is, the former physical space was absolute in being considered intuitive space, while the latter physical space had to be relative as a system of motion.
However, even in this case, we reached the problem of how what could be considered physical space might become motion-system space derived from intuitive space.
Therefore, in any case, within Kant's framework, physical space had to first receive its formative foundation through being brought into relation with intuitive space.

In Kant’s view, pure intuition was considered the limit of empirical intuition.
When a fixed quantity of sensation in empirical intuition gradually diminished to zero, it became pure intuition (Kritik der reinen Vernunft, S. 208).
Intuitive space could not lose its meaning as such a leerer Raum (empty space) in one aspect, as Kant had also conceived.
How should we further consider the relationship between intuitive space and sensation or perception?
The "pure determinations" such as form and quantity within space—because they a priori represented that which could always be given a posteriori in experience—we might not find it impossible to call them the "material of phenomena" (S. 209).
That is to say, intuitive space was considered to prefigure sensation.
Sensations entering into intuitive space had to already be prefigured within it.
Sensation was nothing more than the mere content of intuition relative to intuition’s form.
It occupied a position that was, in principle, entirely subordinate to intuitive space itself, asserting no unique position relative to intuitive space.
Viewed thus, one could only conclude that sensation in this case was entirely passive.
Yet when we considered the matter apart from Kant, we in fact encountered a perspective that placed greater weight on sensation within spatial representation.
That which viewed sensation more actively within spatial representation had to be the concept of spatial perception most commonly employed in psychology.
Since one viewed space not as intuition in Kant’s manner but as spatial perception—which might be called the sum of intuition and perception—perception accordingly became that much more active.
If psychology generally pointed out that due to its methodological constraints it inherently faced difficulties immediately relevant to our current problem—and thus argued discussions on spatial perception were unnecessary—we might replace it with phenomenology, which observed consciousness from its own unique standpoint.
I had already previously cited Becker and touched upon spatial perception’s analysis.
However, as Becker demonstrated, various stages had to be distinguished within spatial perception.
Moreover, such stages represented not merely a division but an order.
It could only mean a hierarchy ranging from the most primitive to the most developed.
That homogeneous space had been located at this hierarchy’s highest position.
That is, it had to be considered spatial perception’s developmental endpoint.

But what did such an endpoint of development fundamentally mean? Moreover, on the other hand, it had to be said that this homogeneous space possessed the property of being able to contain the spaces of prior stages as its determinations.Even an *orientierter Raum* had to ultimately find its orientation within homogeneous space.What did it fundamentally mean for it to encompass all other spatial perceptions within itself?This clearly demonstrated that homogeneous space possessed a special value.Homogeneous space held the value of being both the developmental terminus and foundational basis for all other spatial perceptions.When speaking from phenomenology's methodological standpoint，it might indeed have been impossible to proceed beyond this point.Yet when viewed through cognition's modal validity，this special value of homogeneous space had to be declared—by my initial determination—as none other than reality's cognitive norm.It was solely due to this norm that so-called homogeneous space emerged as something ultimate.If we reasoned thus，we could only conclude that even spatial perception became subsumed within normative space.Spatial perception itself lacked inherent normativity.The mere coexistence of visual and tactile spaces could never guarantee their unification into singular spatiality.That which compelled their reconciliation into unified space had necessarily been termed spatial normativity existing beyond perceptual grasp.As previously established，Kant's intuitive space had been precisely this.Hence even spatial perception could never escape its principled subordination beneath intuitive space qua norm.I could only conclude that sensation's apparent authority within spatial perception remained ultimately passive.

The reason sensation cannot become truly active is that this normativity—which might be called the uniqueness of the objective world—is always sought in intuitive space that transcends sensation. If one could find normativity within sensation or perception itself in some sense, then and only then would it become possible to depart from the standpoint of intuitive space. However, even if we say this, if what is found within oneself is indeed the norm of intuitive space itself, then ultimately it remains the normativity of intuitive space rather than that of active sensation or perception. Therefore, to state it correctly, for sensation to become independent requires finding within oneself not the norm of intuitive space itself but a normativity equivalent to it. Of course, even if such independent normativity could be discovered, this does not mean sensation thereby becomes completely independent from intuitive space. While remaining thoroughly governed by intuitive space's norm, it nevertheless creates a new domain that is independent in itself. Then what constitutes the normativity corresponding to intuitive space? I have already noted the distinction between intuitive space and geometry's object; however, we must say geometry's *apodictic validity* arises precisely where geometry originates from intuitive space. That is, it must derive from intuitive space's norm of reality cognition. Intuitive space is *a priori* in that it precedes experience as a norm, and accordingly geometry is *a priori* in maintaining validity independent of experience. Of course, geometry's *a priori* validity itself does not directly constitute this normativity. It can bear normative meaning only when applied to reality's cognition. Yet it remains evident that geometry alone constitutes something equivalent to intuitive space's norm. Therefore, seeking normativity equivalent to intuitive space's norm must occur precisely where geometry finds application. Having clarified the distinction between intuitive and geometric space, it goes without saying that the normativity thus obtained differs from intuitive space's own norm. Only when geometry applies to sensation or perception can independent normativity emerge for it. Sensation becomes truly active. Only when geometric relations form between sensation and perception does it gain meaning independent of intuitive space. However, if one discusses this application to experience in Kant's sense—where geometry becomes intuitive space's "pure determination" (as with schematic figures)—then it remains merely what was earlier termed phenomena's material, devoid of sensation's active right whatsoever. Since Kant discussed geometry's application to experience without recognizing sensation's active right, the applied result remains confined to intuitive space's content determination, utterly precluding any possibility of something independent arising therefrom.

If what Kant conceived could be called geometry's direct application, our present case might well be termed its indirect application. What then does this so-called indirect application of geometry to experience—the establishment of geometric relations between sensation and perception—signify? It signifies nothing less than the establishment of measurement, which constitutes physics' foundational principle. For physics, only that which can be measured may exist. I propose to rigorously distinguish herein between measurement and quantification. While quantification denotes a direct a priori correspondence between number and extension, measurement implies that this correspondence has been empirically verified through specific procedures; thus, as relativity theory demonstrates, spatial measurement necessarily incorporates temporal measurement, and temporal measurement only becomes possible through physical elements like light speed. Moreover, measurement inherently presupposes a measurer's existence—an origin point—yet this origin differs fundamentally from metric geometry's coordinate origin. For measurement origins must permit mutual relative motion. Nevertheless, measurement undeniably requires quantification's prior framework. To establish geometric relations between sensation and perception therefore amounts to applying metric geometry to these domains. I maintain that geometry's most thoroughgoing application to experience occurs precisely in this context. The system of measured quantities—the coordinate system emerging from such application—transcends both metric geometry's abstract object and intuitive space's mere content determination. It incorporates sensation itself. This alone constitutes what we call "physical space."

Physical space undoubtedly originated from intuitive space.
However，first and foremost，it differed from intuitive space in its material by including the activation of sensation，which had been considered passive in intuitive space.
Secondly，when it incorporated the activation of sensation，it had necessarily absorbed geometry’s quantitative determinations as measurement into its content.
Moreover，in intuitive space，such quantitative determinations were not necessarily essential.
Thirdly，since geometry’s quantitative determinations transcended intuitive space’s three-dimensionality，physical space did not necessarily retain any reason to remain confined to three dimensions.
Minkowski’s four-dimensional world constituted one example of this.
Of course，as some argued，Minkowski’s world-space might have been deemed nothing more than a construct insofar as it did not fully reside within intuitive space’s content determinations.
However，that which was a construct relative to intuitive space was not wholly fictitious in every sense.
Rather，I consider this one among those possessing legitimate existence as physical space.
The necessity of adding temporal changes in objects as an independent dimension—for sensory content to genuinely bind with space—had rather required recognition as physical space’s crucial characteristic.
I maintain that what we commonly and naively call space corresponds precisely to this rudimentary form of physical space.
Minkowski’s world-space must be acknowledged as its most refined achievement.
Physical space demanded resolute distinction from spatial intuition.
Next，physical space clearly did not coincide with geometric space.
Even when geometric space was conceived not as something purely qualitative like projective geometry’s object but rather as metric geometry’s coordinate-based object，it remained a system of metric coordinate systems—never yet a system of measurement foundations.
Physical space required absolute differentiation from geometric space too.
Consequently，physical space established itself as a spatial category endowed with independent significance.
When contemplating how profoundly physical space underpinned physics’ foundations，this fact would manifest ever more strikingly.

(1924/11/20)